manfredkaltenbacher - wordpress.com · chap. 12 demonstrates the applicability to real-life...

Manfred Kaltenbacher

Numerical Simulation of Mechatronic Sensors and Actuators

Manfred Kaltenbacher

Numerical Simulationof Mechatronic Sensorsand Actuators

With 286 Figures and 41 Tables

123

Dr. Manfred Kaltenbacher

Universität ErlangenLS SensorikPaul-Gordon-Str. 3/591052 Erlangen, [email protected]

Library of Congress Control Number: 2007924154

ISBN 978-3-540-71359-3 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the materialis concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad-casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication ofthis publication or parts thereof is permitted only under the provisions of the German Copyright Lawof September 9, 1965, in its current version, and permission for use must always be obtained fromSpringer. Violations are liable for prosecution under the German Copyright Law.

Springer is a part of Springer Science+Business Media

springer.com

© Springer-Verlag Berlin Heidelberg 2007

The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.

Typesetting: Digital data supplied by the authorProduction: LE-TEX Jelonek, Schmidt & Vöckler GbR, LeipzigCover: eStudio Calamar S.L., F. Steinen-Broo, Girona, Spain

SPIN 11801375 60/3180/YL - 5 4 3 2 1 0 Printed on acid-free paper

Preface to the second edition

The second edition of this book fully preserves the character of the first editionto combine the detailed physical modelling of mechatronic systems and theirprecise numerical simulation using the Finite Element (FE) method. Most ofthe text and general appearance of the previous edition were retained, whilethe coverage was extended and the presentation improved.

Starting with Chap. 2, which discusses the theoretical basics and computerimplementation of the FE method, we have added a section describing the FEmethod for one-dimensional cases, especially to provide a easier understandingof this important numerical method for solving partial differential equations.In addition, we provide a section about a priori error estimates. In Chap. 3,which deals with mechanical fields, we now additionally discuss locking effectsas occurring in the numerical computation of thin structures, and describe twowell established methods (method of incompatible modes and of enhanced as-sumed strain) as well as a recently newly developed scheme based on balancedreduced and selective integration. The physical discussion of acoustic soundgeneration and propagation (see Chap. 5) has been strongly improved, includ-ing now also a description of plane and spherical waves as well as a sectionabout quantitative measures of sound. The treatment of open domain prob-lems has been extended and include a recently developed Perfectly MatchedLayer (PML) technique, which allows to limit the computational domain towithin a fraction of the wavelength without any spurious reflections.

Recently developed flexible discretization techniques based on the frame-work of mortar FE methods for the numerical solution of coupled wave propa-gation problems allow for the use of different fine meshes within each computa-tional subdomain. This technique has been applied to pure wave propagationproblems (see Sec. 5.4.3) as well as coupled mechanical-acoustic field prob-lems (see Sec. 8.3.2), where the computational grids of the mechanical regionand the acoustic region can be independently generated and therefore do notmatch at the interface. Furthermore, we have investigated in the piezoelectriceffect and provide in Chap. 9 an extended discussion on the modelling andnumerical computation of nonlinear effects including hysteresis.

In the last three years, we have established a research group on computa-tional aeroacoustics to study the complex phenomenon of flow induced noise.Therewith, the totally new Chap. 10 contains a description of computational

VI Preface to the second edition

aeroacoustics with a main focus on a recently developed FE method for effi-ciently solving Lighthill’s acoustic analogy.

Within Chap. 12, which deals with industrial applications, we have rewrit-ten Sec. 12.5 to discuss latest computational results on micromachined capac-itive ultrasound transducers, and have added a section on high power ultra-sound sources as used for lithotripsy as well as a section on noise generationby turbulent flows.

Most of the formulations described in this book have been implementedin the software NACS (see www.simetris.de).

Acknowledgment

The author wishes to acknowledge the many contributions that colleaguesand collaborators have made to this second edition. First of all I would liketo express my gratitude to the members of the Department of Sensor Tech-nology and its head Prof. Reinhard Lerch for the pleasant and stimulatingworking atmosphere. Amongst many, I wish to specially thank M.Sc. Max Es-cobar, M.Sc. Andreas Hauck, M.Sc. Gerhard Link, Dipl.-Ing. Thomas Hege-wald and Dipl.-Ing. Luwig Bahr for fruitful discussions and proof reading.Much is owned by many intensive discussions with my wife Prof. BarbaraKaltenbacher with whom I work on hysteresis models and parameter identi-fication for electromagnetics and piezoelectrics. Special thank is dedicated toDr. Stefan Becker and his co-workers M.Sc. Irfan Ali and Dr. Frank Schaferfor the contribution on computational aeroacoustics and the intensive coop-eration within the current research project Fluid-Structure-Noise founded bythe Bavarian science foundation BFS. Furthermore, the author would liketo thank Dr. Bernd Flemisch and Prof. Barbara Wohlmuth for the fruitfulcooperation on nonmatching grids. A common research project on Numer-ical Simulation of Acoustic-Acoustic- and Mechanical-Acoustic-Couplings onNonmatching Grids founded by German Research Foundation DFG has juststarted. Moreover, the author wants to acknowledge the excellent workingenvironment at the Johann Radon Institute for Computational and AppliedMathematics in Linz, Austria, where the author stayed for one semester in2005/06 as an invited lecturer for coupled field problems within a specialsemester on computational mechanics. Special thank is dedicated to Prof.Ulrich Langer, who organized this event, and who did a great job in bring-ing together different researchers from all over the world. During this time, Ialso started the cooperation with Prof. Dietrich Braess on enhanced softeningtechniques to avoid locking in thin mechanical structures, to whom I wouldlike to express my gratitude for revealing new and interesting perspectives tome.

Manfred KaltenbacherFebruary 2007

Preface to the first edition

The focus of this book is concerned with the modelling and precise numericalsimulation of mechatronic sensors and actuators. These sensors, actuators, andsensor-actuator systems are based on the mutual interaction of the mechanicalfield with a magnetic, an electrostatic, or an electromagnetic field. In manycases, the transducer is immersed in an acoustic fluid and the solid–fluid cou-pling has to be taken into account. Examples are piezoelectric stack actuatorsfor common-rail injection systems, micromachined electrostatic gyro sensorsused in stabilizing systems of automobiles or ultrasonic imaging systems formedical diagnostics.

The modelling of mechatronic sensors and actuators leads to so-calledmultifield problems, which are described by a system of non-linear partialdifferential equations. Such systems cannot be solved analytically and thusa numerical calculation scheme has to be applied. The schemes discussed inthis book are based on the finite element (FE) method, which is capable ofefficiently solving the partial differential equations. The complexity of thesimulation of multifield problems consists of the simultaneous computationof the involved single fields as well as in the coupling terms, which intro-duce additional non-linearities. Examples are moving conductive (electricallycharged) body within a magnetic (an electric) field, electromagnetic and/orelectrostatic forces.

The goal of this book is to present a comprehensive survey of the mainphysical phenomena of multifield problems and, in addition, to discuss calcula-tion schemes for the efficient solution of coupled partial differential equationsapplying the FE method. We will concentrate on electromagnetic, mechanical,and acoustic fields with the following mutual interactions:

• Coupling Electric Field – Mechanical FieldThis coupling is either based on the piezoelectric effect or results from theforce on an electrically charged structure in an electric field (electrostaticforce).

• Coupling Magnetic Field – Mechanical FieldThis coupling is two-fold. First, we have the electromotive force (emf),which describes the generation of an electric field (electric voltage re-

VIII Preface to the first edition

spectively current) when a conductor is moved in a magnetic field, andsecondly, the electromagnetic force.

• Coupling Mechanical Field – Acoustic FieldVery often a transducer is surrounded by a fluid or a gaseous medium inwhich an acoustic wave is launched (actuator) or is impinging from anoutside source towards the receiving transducer.

In Chap. 2, we give an introduction to the finite element (FE) method.Starting from the strong form of a general partial differential equation, wedescribe all the steps concerning spatial as well as time discretization to arriveat an algebraic system of equations. Both nodal and edge finite elements areintroduced. Special emphasis is put on an explanation of all the importantsteps necessary for the computer implementation.

A detailed discussion on electromagnetic, mechanical, and acoustic fieldsincluding their numerical computation using the FE method can be foundin Chap. 3 – 5. Each of these chapters starts with the description of therelevant physical equations and quantities characterizing the according phys-ical field. Special care is taken with the constitutive laws and the resultantnon-linearities relevant for mechatronic sensors and actuators. In addition,the numerical computation using the FE method is studied for the linear aswell as the non-linear case. In Chap. 4, where the electromagnetic field isdiscussed, we explain the difficulties arising at interfaces of jumping materialparameters (electric conductivity and magnetic permeability), and introducetwo correct formulations adequate for the FE method. At the end of eachof these chapters, we present an example for the numerical simulation of apractical device.

In Chap. 6, we study the interaction between electrostatic and mechanicalfields and concentrate on micromechanical applications. After the derivationof a general expression for the electrostatic force, applying the principle ofvirtual work, we focus on the numerical calculation scheme. The simulationof a simple electrostatic driven bar will demonstrate the complexity of suchproblems, and will show the necessity of taking into account mechanical non-linearities.

The physical modelling and numerical solution of magnetomechanical sys-tems is presented in Chap. 7. In this chapter, we first discuss the correctphysical description of moving and/or deforming bodies in a magnetic field.Later, we derive a general expression for the electromagnetic force, again (asfor the electrostatic force) by using the principle of virtual work. The dis-cussion on numerical computation will contain a calculation scheme for theefficient solution of magnetomechanical systems and, in addition, electric cir-cuit coupling as arise for voltage-driven coils. Especially for the latter case,we give a very comprehensive description of its numerical computation.

Chapter 8 deals with coupled mechanical-acoustic systems and explainsthe physical coupling terms and the numerical computation of such systems.The simulation of the sound emission of a car engine will illustrate different

Preface to the first edition IX

approaches concerning time-discretization schemes and solvers for the alge-braic system.

A special coupling between the mechanical and electrostatic field occursin piezoelectric systems, which are studied in Chap. 9. After explaining thepiezoelectric effect and its physical modelling, we concentrate on the efficientnumerical computation of such systems. Whereas for sensor applications alinear model can be usually used, in many actuator applications non-linear ef-fects play a crucial role, which we here account for by applying an appropriatehysteresis model.

Since the efficiency of the solution (both with respect to elapsed CPU timeand computer memory resources) is of great importance, Chap. 11 deals withgeometric and algebraic multigrid solvers. These methods achieve an optimalcomplexity, that is, the computational effort as well as memory requirementgrows only linearly with the problem size. We present new especially adaptedmultigrid solvers for Maxwell’s equation in the eddy current case and demon-strate their efficiency by means of TEAM (Testing Electromagnetic AnalysisMethods) workshop problem 20 established by the Compumag Society [223].

After these rigorous derivations of methods for coupled field computation,Chap. 12 demonstrates the applicability to real-life problems arising in indus-try. This includes the following topics:

• Analysis and optimization of car loudspeakers• Acoustic emission of electrical power transformers• Simulation-based improvements of electromagnetic valves• Piezoelectric stack actuators such as used, e.g., in common-rail diesel in-

jection systems• Ultrasonic imaging system based on capacitive micromachined ultrasound

transducers

The appendices provide an introduction to vector analysis, functionalspaces, and the solutions of non-linear equations.

The structure of this book has been designed in such a way that in eachof Chaps. 3 – 9 we first discuss the physical modelling of the correspondingsingle or coupled field, then the numerical simulation, followed by a simplecomputational example. If the reader has no previous knowledge of vectoranalysis, she/he should start with the first section of the Appendix. Chapter2 can be omitted if the reader is only interested in the physical modelling ofmechatronic sensors and actuators. The three chapters concerning the singlefield problems (mechanical, electromagnetic, acoustic) are written indepen-dently, so that the reader can start with any of them. Clearly, for the coupledfield problems, the reader should have a knowledge of the involved physicalfields or should have read the corresponding preceding chapters. Chapter 11presents the latest topics on multigrid methods for electromagnetic fields andrequires some knowledge on this topic. For a basic introduction of multigridmethods we refer to classical books [36, 185, 186]. Chapter 12 demonstrates

X Preface to the first edition

the use of numerical simulation for industrial applications. For each of them,we first discuss the problem to be solved, followed by an analysis study ap-plying numerical simulation to allow a better understanding of the differentphysical effects. For most applications, we also demonstrate measurements ofthe CAE-optimized prototype.

Most in this book described formulations for solving multifield problemshave been implemented in the software CAPA (see www.wissoft.de).

Acknowledgements

The author wishes to express his gratitude to all the people who have in-spired, increased, and sustained this work. Of course we feel obliged towardsthe Department of Sensor Technology in Erlangen and the previous Instituteof Measurement Technology in Linz under the competent and generous lead-ership of Prof. Reinhard Lerch. The dynamic and stimulating atmosphere atthe institute was certainly essential for this work; the author therefore thanksall his present and former colleagues among whom especially Dr. ReinhardSimkovics, Dr. Martin Rausch, Dr. Johann Hoffelner, Dr. Manfred Hofer, andDipl.-Ing. Michael Ertl have to be mentioned. Much is owed to the long andfruitful cooperation with Dipl.-Math Hermann Landes and his company Wis-Soft. The author also thanks Dr. Stefan Reitzinger for many intensive andproductive hours of work and discussion, and my wife Dozent Dr. BarbaraKaltenbacher for her assistance on mathematical problems and the coopera-tion on precise material parameter determination applying inverse methods.Moreover, we acknowledge the constructive working environment within thespecial research programs SFB 013 Numerical and Symbolic Scientific Com-puting in Linz (Prof. Ulrich Langer, Dr. Joachim Schoberl, Dr. Michael Schin-nerl) funded by the Austrian science foundation FWF, and SFB 603 Mod-ellbasierte Analyse und Visualisierung komplexer Szenen und Sensordaten inErlangen (Dipl.-Math. Elena Zhelezina, Dr. Roberto Grosso, Dipl.-Inf. FrankReck) funded by the DFG (Deutsche Forschungsgemeinschaft) (German Re-search Foundation) as well as the BMBF project Entwurf komplexer Sensor-Aktor-Systeme (Prof. Peter Schwarz, Dipl.-Ing. Rainer Peipp). Additionally,the author thanks the industrial partners involved in this work for the oppor-tunity of doing research on real-life problems.

Finally, I would like to thank my copyeditor Dr. Peter Capper for readingthe book very carefully and pointing out many errors and misspellings.

Manfred KaltenbacherDecember 2003

Notation

Mathematical symbols

e unit vectorn unit normal vectort unit tangential vectorI unit matrixIR set of real numbersr position vector∫C

ds contour integral∮C

ds closed contour integral∫Γ

dΓ surface integral∮Γ

dΓ closed surface integral∫Ω

dΩ volume integral

∇ nabla operator

curl , ∇× curl

div , ∇· divergence

grad , ∇ gradient

∂/∂x partial derivative

∂/∂n partial derivative innormal direction

d/ dx total derivative

Finite Element Method

u, a, etc. nodal vectors of displ.,acceleration, etc.

C damping matrixK stiffness matrixM mass matrixnn number of nodesnen number of nodes per

finite elementne number of elementsneq number of equationsnd space dimensionJ Jacobi matrix|J | Jacobi determinantx, y, z global coordinatesΩ whole simulation domainΩ simulation domain

without boundaryΓ boundary of simulation

domainΓe Dirichlet boundaryΓn Neumann boundaryγP integration parameter

(parabolic PDE)βH, γH integration parameters

(hyperbolic PDE)ξ, η, ζ local coordinates

XII Notation

Acoustics

b diffusivity of soundB/A parameter of non-linearityc speed of soundIa sound-field intensityk wave numberKs adiabatic bulk modulusKT isothermal bulk modulusLp , SPL sound-pressure levelLIa sound-intensity levelLPa sound-power levelp′ acoustic pressurePa acoustic powerv′ acoustic particle velocitywa acoustic energy densityZa acoustic impedanceρ′ acoustic densityψ scalar acoustic potentialκ adiabatic exponentλ wavelengthζv bulk viscosityµv shear viscosityxs shock formation distance

Electromagnetics

A magnetic vector potentialB magnetic flux densityD electric flux densityE electric field intensityFel electric forceFmag magnetic forceH magnetic field intensityI, i electric currentJ current densityJi impressed current densityM magnetizationqe electric charge densityQe total electric chargeP electric polarizationR ohmic resistoruind induced voltageVe scalar electric potentialwel electric energy density

Wel total electric energywmag magnetic energy densityWmag total magnetic energyρe specific electric resistanceγ electrical conductivityµ magnetic permeabilityν magnetic reluctivityε electric permittivityσe electric surface chargeφ magnetic fluxψm reduced magnetic scalar

potentialΨ total magnetic fluxδ skin depth

Mechanics

a acceleration[c] tensor of mechanical moduluscL velocity of longitudinal wavecT velocity of shear waveEm elasticity modulefV volume force[Fd] deformation gradient[Hd] displacement gradientG shear modulusm massPmech mechanical powerS vector of linear strains[S] tensor of linear strainsT Piola-Kirchhoff stress vector[T] 2nd Piola-Kirchhoff stress tensoru mechanical displacementv velocityV Green-Lagrangian strain vector[V] Green-Lagrangian strain tensorαM, αK damping coefficientsρ densityνp Poisson ratioσ Cauchy stress vector[σ] Cauchy stress tensorµL, λL Lame-parameters

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 The Finite Element (FE) Method . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Finite Element Method for a 1D Problem. . . . . . . . . . . . . . . . . . . 132.3 Nodal (Lagrangian) Finite Elements . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Quadrilateral Element in IR2 . . . . . . . . . . . . . . . . . . . . . . . . 232.3.3 Triangular Element in IR2 . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.4 Tetrahedron Element in IR3 . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.5 Hexahedron Element in IR3 . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.6 Global/Local Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.7 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Finite Element Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.1 Parabolic Differential Equation . . . . . . . . . . . . . . . . . . . . . . 362.5.2 Hyperbolic Differential Equation . . . . . . . . . . . . . . . . . . . . 40

2.6 Integration over Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.7 Edge (Nedelec) Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.8 Discretization Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Mechanical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Navier’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Deformation and Displacement Gradient . . . . . . . . . . . . . . . . . . . 553.3 Mechanical Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.1 Plane Strain State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.2 Plane Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.3 Axisymmetric Stress–Strain Relations . . . . . . . . . . . . . . . . 63

3.5 Waves in Solid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.6 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

XIV Contents

3.7 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.7.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.7.2 Damping Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.7.3 Geometric Non-linear Case . . . . . . . . . . . . . . . . . . . . . . . . . 703.7.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.8 Locking and Efficient Solution Approaches . . . . . . . . . . . . . . . . . . 783.8.1 Incompatible Modes Method . . . . . . . . . . . . . . . . . . . . . . . . 813.8.2 Enhanced Assumed Strain Method . . . . . . . . . . . . . . . . . . 833.8.3 Balanced Reduced and Selective Integration . . . . . . . . . . 85

4 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 Quasistatic Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2.1 Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2.2 Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.3 Electrostatic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.1 Magnetic Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.4.3 Dielectric Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.5 Electromagnetic Interface Conditions . . . . . . . . . . . . . . . . . . . . . . 1094.5.1 Continuity Relations for Magnetic Field . . . . . . . . . . . . . . 1094.5.2 Continuity Relations for Electric Field . . . . . . . . . . . . . . . 1114.5.3 Continuity Relations for Electric Current Density . . . . . 113

4.6 Numerical Computation: Electrostatics . . . . . . . . . . . . . . . . . . . . . 1134.7 Numerical Computation: Electromagnetics . . . . . . . . . . . . . . . . . . 114

4.7.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.7.2 Discretization with Edge Elements . . . . . . . . . . . . . . . . . . . 1214.7.3 Discretization with Nodal Finite Elements . . . . . . . . . . . . 1224.7.4 Newton’s Method for the Non-linear Case . . . . . . . . . . . . 1254.7.5 Approximation of B–H Curve . . . . . . . . . . . . . . . . . . . . . . . 1294.7.6 Modelling of Current-loaded Coil . . . . . . . . . . . . . . . . . . . . 1314.7.7 Computation of Global Quantities . . . . . . . . . . . . . . . . . . . 1314.7.8 Induced Electric Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.8 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.8.1 Ferromagnetic Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.8.2 Thin Iron Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5 Acoustic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1 Wave Theory of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.1.1 Conservation of Mass (Continuity Equation) . . . . . . . . . . 1405.1.2 Conservation of Momentum (Euler Equation) . . . . . . . . . 1415.1.3 Pressure-Density Relation (State Equation) . . . . . . . . . . . 1435.1.4 Linear Acoustic Wave Equation . . . . . . . . . . . . . . . . . . . . . 1455.1.5 Acoustic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Contents XV

5.1.6 Plane and Spherical Waves . . . . . . . . . . . . . . . . . . . . . . . . . 1485.2 Quantitative Measure of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.3 Non-linear Acoustic Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 1565.4 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.4.1 Linear Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.4.2 Non-linear Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.4.3 Non-matching Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.4.4 Discretization Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.5 Treatment of Open Domain Problems . . . . . . . . . . . . . . . . . . . . . . 1735.5.1 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 1755.5.2 Perfectly Matched Layer (PML) Technique . . . . . . . . . . . 176

5.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.6.1 Transient Wave Propagation in Unbounded Domain . . . 1845.6.2 Harmonic Wave Propagation in Unbounded Domain . . . 1865.6.3 Acoustic Pulse Propagation over a Non-Matching Grid . 1885.6.4 Non-linear Wave Propagation in a Channel . . . . . . . . . . . 190

6 Coupled Electrostatic-Mechanical Systems . . . . . . . . . . . . . . . . . 1956.1 Electrostatic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1966.2 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.2.1 Calculation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2026.2.2 Voltage-driven Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7 Coupled Magnetomechanical Systems . . . . . . . . . . . . . . . . . . . . . . 2077.1 General Moving/Deforming Body . . . . . . . . . . . . . . . . . . . . . . . . . . 2077.2 Electromagnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2097.3 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.3.1 Electric Circuit Coupling: Voltage-loaded Coil . . . . . . . . . 2117.3.2 Force Computation via the Principle of Virtual Work . . 2137.3.3 Calculation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2167.3.4 Moving Current/Voltage-loaded Coil . . . . . . . . . . . . . . . . . 218

8 Coupled Mechanical-Acoustic Systems . . . . . . . . . . . . . . . . . . . . . 2298.1 Solid–Fluid Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.2 Coupled Field Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2318.3 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8.3.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 2338.3.2 Non-matching Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2348.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

9 Piezoelectric Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2439.1 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2439.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2449.3 Piezoelectric Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 2459.4 Models for Non-linear Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . 249

XVI Contents

9.5 Hysteresis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2539.6 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

9.6.1 Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2579.6.2 Non-linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

9.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2619.7.1 Computation of Impedance Curve . . . . . . . . . . . . . . . . . . . 2619.7.2 Non-linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

10 Computational Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26710.1 Requirements for Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . 26710.2 Lighthill’s Analogy and its Extension . . . . . . . . . . . . . . . . . . . . . . 27010.3 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27510.4 Validation: Co-Rotating Vortex Pair . . . . . . . . . . . . . . . . . . . . . . . 279

11 Algebraic Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28311.1 Preconditioned Conjugate Gradient (PCG) Method . . . . . . . . . . 28311.2 Multigrid (MG) Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.3 Geometric MG Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

11.3.1 Geometric MG for Edge Elements . . . . . . . . . . . . . . . . . . . 28811.3.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

11.4 Algebraic MG Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29311.4.1 Auxiliary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29411.4.2 Coarsening Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29511.4.3 Prolongation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29811.4.4 Smoother and Coarse-grid Operator . . . . . . . . . . . . . . . . . 29911.4.5 AMG for Nodal Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 30011.4.6 AMG for Edge Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 30111.4.7 AMG for Time-harmonic Case . . . . . . . . . . . . . . . . . . . . . . 30411.4.8 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

12 Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31312.1 Electrodynamic Loudspeaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

12.1.1 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31412.1.2 Verification of Computer Models . . . . . . . . . . . . . . . . . . . . 31612.1.3 Numerical Analysis of the Non-linear Loudspeaker

Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31712.1.4 Computer Optimization of the Non-linear Loudspeaker

Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32012.2 Noise Computation of Power Transformers . . . . . . . . . . . . . . . . . . 320

12.2.1 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32112.2.2 Verification of the Computer Models . . . . . . . . . . . . . . . . . 32512.2.3 Verification of the Calculated Winding and

Tank-surface Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32612.2.4 Verification of the Sound-field Calculations . . . . . . . . . . . 32812.2.5 Influence of Tap-changer Position . . . . . . . . . . . . . . . . . . . . 328

Contents XVII

12.2.6 Influence of Stiffness of Winding Supports . . . . . . . . . . . . 33012.3 Fast-switching Electromagnetic Valves . . . . . . . . . . . . . . . . . . . . . 330

12.3.1 Modelling and Solution Strategy . . . . . . . . . . . . . . . . . . . . 33112.3.2 Actuator Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 33312.3.3 Actuator Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33512.3.4 Dynamics Optimization I: Electrical Premagnetization . 33612.3.5 Dynamics Optimization II: Overexcitation . . . . . . . . . . . . 33712.3.6 Switching Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

12.4 Cofired Piezoceramic Multilayer Actuators . . . . . . . . . . . . . . . . . . 34012.4.1 Setup of Multilayer Stack Actuators . . . . . . . . . . . . . . . . . 34112.4.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34212.4.3 Measured and Simulated Results . . . . . . . . . . . . . . . . . . . . 343

12.5 Capacitive Micromachined Ultrasound Transducers . . . . . . . . . . 34512.5.1 Requirements to Numerical Simulation Scheme . . . . . . . . 34612.5.2 Single CMUT Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34812.5.3 CMUT Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35012.5.4 Controlled CMUT Array . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

12.6 High-Intensity Focused Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . 35712.6.1 Piezoelectric Transducer and Input Impedance . . . . . . . . 35712.6.2 Pressure Pulse Computation . . . . . . . . . . . . . . . . . . . . . . . . 35712.6.3 High-Power Pulse Sources for Lithotripsy . . . . . . . . . . . . . 361

12.7 Noise Generated from a Flow around a Square Cylinder . . . . . . 36612.7.1 3D Flow and 2D Acoustic Computations . . . . . . . . . . . . . 36612.7.2 3D Flow and 3D Acoustic Computations . . . . . . . . . . . . . 377

A Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381A.1 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381A.2 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

B Scalar and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383B.1 The Nabla (∇) Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386B.2 Definition of Gradient, Divergence, and Curl . . . . . . . . . . . . . . . . 386B.3 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387B.4 The Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389B.5 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390B.6 Divergence Theorem (Gauss Theorem) . . . . . . . . . . . . . . . . . . . . . 392B.7 The Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392B.8 The Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393B.9 Stoke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395B.10 Green’s Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396B.11 Application of the Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397B.12 Irrotational Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397B.13 Solenoidal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

C Appropriate Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

XVIII Contents

D Solution of Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 405D.1 Fixed-point Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407D.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

1

Introduction

Each modern industrial process environment needs sensors to detect the phys-ical quantities involved (e.g., electric current, mechanical torque, temperature,etc.), a signal-conditioning circuit, and an interface to computers, where theprocess parameters are controlled. According to the controlling signals, power

Fig. 1.1. Industrial process

electronic circuits supply the actuators, which will steer the process (see Fig.1.1). As indicated by the gray boxes in Fig. 1.1, the main topic of this bookis concerned with sensors and actuators, especially their sophisticated design.Therefore, we provide a comprehensive discussion on the precise physical mod-elling of sensors and actuators and, furthermore, on the numerical solution ofthe governing partial differential equations. To be precise, we concentrate ourinvestigation on mechatronic sensors and actuators. Mechatronic sensors, ac-tuators and sensor-actuator systems are based on the mutual interaction of

2 1 Introduction

(a) Electromagnetic valve

(b) Capacitive micromachined ultrasoundtransducers (CMUT)

Fig. 1.2. Examples of mechatronic sensors and actuators

the mechanical field and an electrostatic, an electromagnetic, or an acous-tic field. Typical examples are: electromagnetic valves for injection systemsin vehicles, capacitive micromachined ultrasound transducers (CMUTs) formedical imaging systems, electrodynamic loudspeakers, surface acoustic wave(SAW) devices for telecommunications (e.g., in a mobile phone), piezoelec-tric or magnetostrictive actuators for ultrasound cleaning, etc. (see Fig. 1.2).The main transducing mechanisms for mechatronic sensor and actuators aredisplayed in Fig. 1.3 (those in boldface will be discussed within this book).

In most cases, the fabrication of prototypes within the design process ofmodern mechatronic transducers is a lengthy and costly task. The still widelyused experimental-based design, as shown in Fig. 1.4(a) exhibits many disad-vantages. First, it is very time-consuming, since for each change in the designa new prototype has to be fabricated and the relevant parameters have to bemeasured. In particular, if a design failure is recognized late, high costs result.

1 Introduction 3

Fig. 1.3. Transducing mechanisms of mechatronic sensors and actuators

Furthermore, not all parameters of interest can be measured (e.g., mag-netic field or mechanical stresses inside a solid body) and the measurementsetup may influence the (dynamical) behavior of the prototype.

(a) Experimental-based design (b) CAE-based design

Fig. 1.4. Design process

Since for the development of modern mechatronic transducers all the dif-ferent coupling mechanisms of the involved physical fields have to be consid-ered, the design process is a very complex task. Therefore, an increasing needfor reliable and usable computer modelling tools capable of precisely sim-ulating the multifield interactions arises. Such appropriate computer-aided

4 1 Introduction

engineering (CAE) tools offer many possibilities to the design engineer. Arbi-trary modification of transducer geometry and selective variation of materialparameters are easily performed and the influence on the transducer behav-ior can be studied immediately. In addition, the simulation provides access tophysical quantities that cannot be measured, and simulations strongly supportthe insight into physical phenomena. Thus, a CAE-based design as displayedin Fig. 1.4(b) can tremendously reduce the number of necessary prototypeswithin the design process.

However, we want to emphasize that a direct physical control of the trans-ducer design is possible only with the help of experiments, whereas the com-puter simulation is always based on a model of reality. Therefore, the qualityof the results depends on the suitability of the physical model as well as thematerial parameters. Moreover, numerical effects may spoil the results unlessproper computational methods are used. For these reasons, one should alsobe aware of the risk of trusting every result the CAE environment computes.These facts make it very important that the user of such tools has both a deepphysical understanding of the ongoing processes and mathematical knowledgeof the simulation algorithms. Summarizing, one can say that an appropriateuse of CAE tools for the design of mechatronic transducers can tremendouslyreduce the number of necessary prototypes within the design process, but theuser should always critically question each result.

The use of these CAE tools in the design of mechatronic transducersstarted only some years ago. The main reason for the lack of CAE environ-ments, capable of performing multifield simulations that arise in the designof mechatronic sensors and actuators, is the complexity of such problems. Wewill demonstrate this by discussing the requirements on a CAE tool for thenumerical simulation of an electrodynamic loudspeaker. Figure 1.5 shows anelectrodynamic loudspeaker with all its components. A cylindrical, small lightvoice coil is suspended freely in a strong radial magnetic field, generated by apermanent magnet. The magnet assembly, consisting of pole, back plate andtop plate, helps to concentrate most of the magnetic flux within the magnetstructure and, therefore, within the narrow radial gap. When the coil is loadedby an electric voltage, the interaction between the magnetic field of the per-manent magnet and the current in the voice coil results in an axial Lorentzforce. The voice coil is wound onto an aluminum holder, which is attached tothe rigid, light cone diaphragm in order to couple the forces more effectivelyto the air and, hence, to permit acoustic power to be radiated from the assem-bly. The dust cap does not usually fulfill any acoustic function, but preventsthe penetration of dust into the air gap of the magnetic assembly. The mainfunction of the suspension and the surround is to allow free axial movementof the moving coil driver, while nonaxial movements are suppressed almostcompletely. To obtain a precise description of an electrodynamic loudspeakerwe have to model the electromagnetic, mechanical, and acoustic fields, and inaddition, the following coupling mechanisms:

1 Introduction 5

Fig. 1.5. Electrodynamic loudspeaker

• Mechanical Field – Acoustic FieldThe normal component of the surface velocity of the solid must meet thenormal component of the particle velocity of the fluid.

• Mechanical Field – Magnetic FieldThe interaction between these two physical fields is given by the Lorentzforce and the motional electromotive force (emf) term leads to additionaleddy currents in the coil.

• Electric Circuit – Magnetic FieldDue to the fact that the coil is voltage driven, the magnetic field equationhas to be solved together with the electric circuit equation.

For the design of electrodynamic loudspeakers, the frequency dependence ofthe axial pressure response at 1 m distance and the electrical input impedanceof the voice coil are the two most important parameters. Since the overallphysical model consists of a system of coupled, non-linear partial differentialequations a transient analysis has to be performed and then the results aretransformed to the frequency domain. To obtain a good frequency resolutionwithin the frequency range of interest (0 − 20 kHz) around 10 000 time stepshave to be computed. Since the coupled system of partial differential equationscannot be solved analytically, a numerical scheme has to be applied.

The finite element (FE) method has been established as the standardmethod for numerically solving the partial differential equations describingthe physical fields including their couplings. Thus, a static, transient andtime-harmonic analysis including non-linearities (e.g., material non-linearities,geometric non-linearities, etc.) can be performed very efficiently. This methodis currently used in most commercial computer codes. Of course it has tobe mentioned that for different physical field problems (e.g., fluid dynamics,high-frequency electromagnetics, etc.) different numerical methods (e.g., finite

6 1 Introduction

volume, finite difference, finite integration, boundary element) might be themethods of choice [22, 33, 64, 66, 213].

2

The Finite Element (FE) Method

The finite element (FE) method has become the standard numerical calcu-lation scheme for the computer simulation of physical systems [16, 104, 226].The advantages of this method can be summarized as follows:

• Numerical efficiency: The discretization of the calculation domain withfinite elements yields matrices that are in most cases sparse and symmet-ric. Therefore, the system matrix, which is obtained after spatial and timediscretization, is sparse and symmetric, too. Both the storage of the sys-tem matrix and the solution of the algebraic system of equations can beperformed in a very efficient way.

• Treatment of non-linearities: The modelling of non-linear material be-havior is well established for the FE method (e.g., non-linear curves, hys-teresis).

• Complex geometry: By the use of the FE method, any complex domaincan be discretized by triangular elements in 2D and by tetrahedra in 3D.

• Analysis possibilities: The FE method is suited for static, transient,harmonic as well as eigenfrequency analysis.

The two essential disadvantages of the FE method are given by

• Discretization: The effort for the discretization of the simulation domainis quite high, since in 2D the whole cross section and in 3D the whole vol-ume has to be subdivided into finite elements.

• Open domain problems: Models that need the treatment of an openboundary, e.g., the simulation of radiation characteristics of an ultrasoundarray, lead in the general case to errors due to the limitation of the sim-ulation domain. One of several approaches to overcome this problem is

8 2 The Finite Element (FE) Method

the use of absorbing boundary conditions, perfectly matched layer (PML)techniques or so-called infinite elements (see Sect. 5.5).

Fig. 2.1. From the strong formulation to the algebraic system of equation

The general approach of the FE method is shown in Fig. 2.1. Starting fromthe partial differential equation (PDE) with given boundary conditions, wemultiply it by appropriate test functions and integrate over the whole simu-lation domain. Performing a partial integration, we arrive at the variationalformulation, also called the weak formulation. Applying Galerkin’s approxi-mation method using finite elements (FE) results in the algebraic system ofequations.

As already mentioned, the use of the FE method requires the discretizationof the whole domain (see Fig. 2.2). For the discretization triangular as well asquadrilateral finite elements are used in 2D and tetrahedral as well as hexahe-dron finite elements in 3D. The physical quantity of interest (e.g., temperature,mechanical displacement, etc.) is approximated by so-called shape functionsand the solution of the algebraic equation yields the physical quantity in thediscretization points, the so-called finite element nodes, for Lagrangian finiteelements and along the edges for Nedelec finite elements.

Fig. 2.2. FE method: Discretization of the domain with quadrilateral finite elements

2.1 Finite Element Formulation 9

2.1 Finite Element Formulation

In the following, all steps—from the strong formulation of the partial differ-ential equation (PDE) to the algebraic equation—will be briefly described bymeans of the following simple PDE with the searched for quantity u(r, t),the known source term f(r, t) at each time t of the interval (0, T ), and thecorresponding initial and boundary conditions.

Given: f : Ω × (0, T ) → IRu0 : Ω → IR

Find: u : Ω × [0, T ] → IR

∂u

∂t= ∇ · ∇u+ f (2.1)

u = ue on Γe × (0, T )

∂u

∂n= un on Γn × (0, T )

u(r, 0) = u0 , r ∈ Ω .

In this so-called strong formulation of the initial-boundary value problem IRdenotes the set of real numbers, Ω the simulation domain, Ω the simulationdomain without the boundary Γ = Γe ∪ Γn, Γe the boundary with prescribedDirichlet boundary condition, and Γn the boundary with prescribed Neumannboundary condition. Now, let us introduce for any t ∈ [0, T ] the space Tt

Tt = u(·, t) | u(·, t) ∈ H1(Ω), u(r, t) = ue(r, t) on Γe , (2.2)

and G, the space of so-called test functions, as

G = w | w ∈ H1(Ω), w = 0 on Γe , (2.3)

with H1 the standard Sobolev space (see Appendix C). It has to be notedthat the spaces Tt, t ∈ [0, T ] vary with time, whereas the space G is time-independent.

In the first step, we multiply the partial differential equation with anarbitrary test function w and perform an integration over the whole domainΩ ∫

Ω

w

(∂u

∂t− ∇ · ∇u− f

)dΩ = 0 .

Applying Green’s first integration theorem to the above equation results in∫Ω

w∂u

∂tdΩ +

∫Ω

(∇w) · (∇u) dΩ =∫

Ω

wf dΩ +∫

Γn

w∂u

∂ndΓ . (2.4)


Thus, the weak formulation (often also called variational formulation) for theinitial-boundary problem is as follows:

Given: f : Ω × (0, T ) → IRu0 : Ω → IR

Find: u(t) ∈ Tt such that for all w ∈ G and t ∈ [0, T ]

∫Ω

w∂u

∂tdΩ +

∫Ω

(∇w) · (∇u) dΩ =∫

Ω

wf dΩ +∫

Γn

wun dΓ (2.5)

u = ue on Γe × (0, T )∫Ω

wu(0) dΩ =∫

Ω

wu0 dΩ .

Since the Neumann boundary condition is now incorporated into the equationof the weak form, it is also called natural. The Dirichlet boundary condition onu still has to be explicitely forced, and is therefore called essential. Formallyit can be proven that the two formulations according to (2.1) and (2.5) aremathematically equivalent, provided u is sufficiently smooth [99].

To discretize (2.5), which is still infinite dimensional, we now apply theapproximation according to Galerkin’s method. Let us define the finite di-mensional spaces Th

t and Gh according to

Tht ⊂ T Gh ⊂ G .

Therefore, we perform the domain discretization (see Fig. 2.2), and approxi-mate the searched for quantity u(t) as well as the test function w by

u(t) ≈ uh(t) w ≈ wh , (2.6)

with h the discretization parameter (defining the mesh size). Furthermore, wedecompose uh(t) into the searched for value vh(t) and the known Dirichletvalues uh

e (t). For vh, wh, and uhe we choose the following ansatz

vh(t) =neq∑a=1

Na(r)va(t) (2.7)

wh =neq∑a=1

Na(r)ca (2.8)

uhe (t) =

ne∑a=1

Na(r)uea(t) , (2.9)

where Na(r) denotes appropriate shape functions (often also called interpola-tion or basis functions), neq the number of unknowns, which is equal to the


number of finite element nodes with no Dirichlet boundary condition, and ne

the number of finite element nodes with Dirichlet boundary condition. Sub-stituting (2.7) – (2.9) into (2.5) results in

∫Ω

(neq∑a=1

Naca∂

∂t

neq∑b=1

Nbvb

)dΩ +

∫Ω

∇(neq∑

a=1

Naca

)· ∇

(neq∑b=1

Nbvb

)dΩ

+∫

Ω

(neq∑a=1

Naca∂

∂t

ne∑b=1

Nbueb

)dΩ +

∫Ω

∇(neq∑

a=1

Naca

)·(

∇ne∑

b=1

Nbueb

)dΩ

=∫

Ω

neq∑a=1

Nacaf(ra) dΩ +∫

Γn

neq∑a=1

Nacaun(ra) dΓ . (2.10)

Now, since we can put the sums before the integrals and having in mind that∇ just operates on the shape functions N(r) (ca as well as ub are constantswith respect to the space variables), we may write (2.10) for the 2D planecase as follows

neq∑a=1

ca

neq∑b=1

( [∫Ω

NaNb dΩ]∂vb

∂t

+[∫

Ω

(∂Na

∂x

∂Nb

∂x+∂Na

∂y

∂Nb

∂y

)dΩ

]vb

)

−∫

Ω

Naf dΩ −∫

Γn

Naun dΓ

+ne∑

b=1

(∫Ω

(∂Na

∂x

∂Nb

∂x+∂Na

∂y

∂Nb

∂y

)dΩ

)ueb

+ne∑

b=1

(∫Ω

NaNb dΩ)∂ueb

∂t

= 0 .

Since the equation has to be fulfilled for all coefficients ca, we obtain thedefining equations for the searched for finite element node values vb: for eacha (a = 1, .., neq) we have to solve an equation as follows


neq∑b=1

([∫Ω

NaNb dΩ]∂vb

∂t+[∫

Ω

(∂Na

∂x

∂Nb

∂x+∂Na

∂y

∂Nb

∂y

)dΩ

]vb

)

−∫

Ω

Naf dΩ −∫

Γn

Naun dΓ

+ne∑

b=1

[∫Ω

(∂Na

∂x

∂Nb

∂x+∂Na

∂y

∂Nb

∂y

)dΩ

]ueb

+ne∑

b=1

[∫Ω

NaNb dΩ]∂ueb

∂t= 0 . (2.11)

Thus, the semidiscrete Galerkin formulation can be written in matrix form asfollows

Mv + Kv = f , (2.12)

with v = ∂v/∂t, v the nodal unknowns and f the right-hand side vector.

• Mass matrix M:

M = [Mab]

Mab =∫

Ω

NaNb dΩ (2.13)

1 ≤ a, b ≤ neq

• Stiffness Matrix K:

K = [Kab]

Kab =∫

Ω

(∂Na

∂x

∂Nb

∂x+∂Na

∂y

∂Nb

∂y

)dΩ (2.14)

1 ≤ a, b ≤ neq

• Right-hand side f :

f = [fa]

fa =∫

Ω

Naf dΩ +∫

Γn

Naun dΓ

−ne∑

b=1

[∫Ω

(∂Na

∂x

∂Nb

∂x+∂Na

∂y

∂Nb

∂y

)dΩ

]ueb

−ne∑

b=1

[∫Ω

NaNb dΩ]∂ueb

∂t(2.15)

1 ≤ a ≤ neq

1 ≤ b ≤ ne (2.16)

2.2 Finite Element Method for a 1D Problem 13

This example was supposed to illustrate the main steps of the FE method.Note that the mass and stiffness matrix may take different forms dependingon the physical phenomena they model and on the material parameters (seeSects. 3, 4, and 5). The resulting equation (2.12) is still infinite dimensionaldue to the time dependence. Therefore, in Sect. 2.5 we will discuss time-discretization schemes, in order to arrive at the algebraic system of equations.Before, we will provide a detailed description of the FE method by means ofapplying it to a 1D problem, followed by discussing all steps necessary forcomputer implementation.

2.2 Finite Element Method for a 1D Problem

In order to illustrate the main idea of the FE method, we will consider thefollowing 1D differential equation

−∂2u

∂x2+ c u = f(x) (2.17)

u(a) = ua

u(b) = ub ,

where [a, b] defines the computational domain. As described in Sec. 2.1, thefirst step is to derive the weak form of (2.17). For this purpose, we choose anappropriate test function v, multiply (2.17) by this test function and integrateover the whole domain

b∫a

v

(−∂

2u

∂x2+ c u− f(x)

)dx . (2.18)

For the first term in (2.18) we perform an integration by parts

b∫a

v∂2u

∂x2dx = v

∂u

∂x

∣∣∣∣ba

−b∫

a

∂v

∂x

∂u

∂xdx .

Provided that the test function v(x) vanishes on the Dirichlet boundary (firstrestriction on the test function, second one will be the existence of a first-orderderivative in the weak sense, see Sec. C), we obtain for (2.18)

b∫a

(∂v

∂x

∂u

∂x+ cvu

)dx =

b∫a

vf dx . (2.19)

Therewith, the weak (variational) formulation reads as follows:


Given:f, c : [a, b] → IR

Find: u ∈ V = u ∈ H1(a, b)|u(a) = ua , u(b) = ub such that for allv ∈ W = v ∈ H1(a, b)|v(a) = v(b) = 0

a(u, v) =< f, v > (2.20)

with

a(u, v) =

b∫a

(∂v

∂x

∂u

∂x+ cvu

)dx

< f, v > =

b∫a

vf dx .

In (2.20) a(u, v) is called a bilinear form and < f, v > an inner product in thespecified functional space.

In the next step, we divide the computational domain into cells, so-calledfinite elements. Therewith, in our case, we divide the interval [a, b] into a setof smaller intervals [xi−1, xi], i = 1, ...,M such that the following propertiesare fulfilled

• Ascending order of node positions

xi−1 < xi for i = 1, ...,M

• Complete covering of the domain

[a, b] =M⋃i=1

[xi−1, xi] x0 = a, xM = b

• No intersection of intervals

[xi−1, xi] ∩ [xj−1, xj ] = 0 for i = j

For simplicity, we choose an equidistant discretization, so that we obtain (seeFig. 2.3)

xi = a+ ih h =a− b

Mi = 0, ...,M .

The unknown quantity u(x) is now approximated by a linear combination offinite functions with local support, which means that these functions are justdifferent from zero in a ’small’ interval (see Fig. 2.4). Such a choice is givene.g., by piecewise linear hat-functions, defined as follows (see Fig. 2.5)


Fig. 2.3. Subdivision of the computational domain into finite elements

Fig. 2.4. Finite element function with local support: suppN(x) = [α, β]

Ni(x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0 a ≤ x ≤ xi−1

x−xi−1h xi−1 < x ≤ xi

xi+1−xh xi < x ≤ xi+1

0 xi+1 < x ≤ b

(2.21)

Our chosen ansatz (shape) functions fulfill the requirement

Fig. 2.5. Piecewise, linear hat-functions

Ni(xj) = δij =

1 i = j

0 i = j(2.22)

and the approximation of the unknown u(x) is given by


u(x) ≈ uh(x) =M−1∑i=1

Ni(x)ui +N0(x)ua +NM (x)ub (2.23)

uh(x = xi) = ui . (2.24)

The discretized weak (variational) formulation reads as

Given:f, c : [a, b] → IR

Find: uh ∈ Vh = uh(x) =M−1∑i=1

Ni(x)ui +N0ua +NMub such that for all

vh ∈Wh = vh(x) =M−1∑i=1

Ni(x)vi

a(uh, vh) = < f, vh > (2.25)

with

a(uh, vh) =

b∫a

(∂vh

∂x

∂uh

∂x+ cvhuh

)dx

< f, vh > =

b∫a

vhf dx .

For the finite dimensional functional spaces V h, Wh we have the propertyV h ⊂ V , Wh ⊂W .

Now, we have to set up the algebraic system of equations in order toobtain the unknowns ui at all the finite element nodes within our grid. Usingthe approximation according to (2.23) for u as well as the test function vresults in

b∫a

∂

∂x

(M−1∑i=1

Ni(x)vi

)∂

∂x

⎛⎝M−1∑

j=1

Nj(x)uj +N0ua +NMub

⎞⎠ dx

+

b∫a

c

(M−1∑i=1

Na(x)va

) ⎛⎝M−1∑

j=1

Nj(x)uj +N0ua +NMub

⎞⎠ dx

−b∫

a

(M−1∑i=1

Ni(x)vi

)f dx = 0 .

Considering that we can interchange the integrals and the sums, and that allvi as well as uj are constants (no function of x), we obtain


M−1∑i=1

vi

⎛⎝ M−1∑

j=1

uj

b∫a

(∂Ni

∂x

∂Nj

∂x+ cNiNj

)dx

+

b∫a

(∂Ni

∂x

(∂N0

∂xua +

∂NM

∂xub

)+ cNi(N0ua +NMub)

)dx

−b∫

a

Ni f dx

⎞⎠ = 0 .

Letting vi with i = 1, ..,M − 1 run through all unit vectors in IRM , we obtainfor each i an equation

M−1∑j=1

Sij uj = fi i = 1, ...,M − 1

Su = f (2.26)

with

Sij =

b∫a

(∂Ni

∂x

∂Nj

∂x+ cNiNj

)dx (2.27)

fi =

b∫a

Ni f dx−b∫

a

∂Ni

∂x

(∂N0

∂xua +

∂NM

∂xub

)dx

−b∫

a

cNi(N0ua +NMub) dx . (2.28)

According to the properties of our chosen ansatz functions (see (2.22)), weget the following pattern for our system matrix S (see Fig. 2.6)

S =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∗ ∗ 0 0 · · · 0 · · · 0

∗ ∗ ∗ 0 · · · 0 · · · 0

0 ∗ ∗ ∗ · · · 0 · · · 0

......

......

. . ....

......

0 · · · · · · ∗ ∗ ∗0 · · · · · · 0 ∗ ∗

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Sij =

0 j ∈ i− 1, i, i+ 1∗ j ∈ i− 1, i, i+ 1

∗...nonzero entry

Now, let us compute the nonzero entries of S by evaluating (2.27) and using(2.21)


Fig. 2.6. Shape functions for nodes xi−1, xi and xi+1

Si,i−1 =

xi∫xi−1

(∂Ni

∂x

∂Ni−1

∂x+ cNiNi−1

)dx

=

xi∫xi−1

((1h

) (−1h

)+ c

(x− xi−1

h

)) (xi − x

h

)dx

=−1h

+ch

6

Si,i =

xi+1∫xi−1

(∂Ni

∂x

∂Ni

∂x+ cNiNi

)dx

=

xi∫xi−1

((1h

) (1h

)+ c

(x− xi−1

h

) (x− xi−1

h

))dx

+

xi+1∫xi

((− 1h

) (− 1h

)+ c

(xi+1 − x

h

) (xi+1 − x

h

))dx

=2h

+2ch3

Si−1,i = Si,i−1 =−1h

+ch

6.

For simplicity, we set our source term f(x) equal to 1 over the whole compu-tational domain and assume ua = ub = 0, which results in


fi =

xi+1∫xi−1

Ni(x) dx

=

xi∫xi−1

x− xi−1

hdx+

xi+1∫xi

xi+1 − x

hdx = h .

Let us choose L = 10 and M = 5, so that our mesh size h is 2. For theparameter c we choose once the value 0 and once 0.5, and we set the boundaryvalues ua as well as ub to zero. Substituting these values, results in

Kc=0 =

⎛⎜⎜⎜⎜⎜⎝

1 − 12 0 0

− 12 1 − 1

2 0

0 − 12 1 − 1

2

0 0 − 12 1

⎞⎟⎟⎟⎟⎟⎠ f =

⎛⎜⎜⎜⎜⎜⎝

2

2

2

2

⎞⎟⎟⎟⎟⎟⎠

Kc=0.5 =

⎛⎜⎜⎜⎜⎜⎝

53 − 1

3 0 0

− 13

53 − 1

3 0

0 − 13

53 − 1

3

0 0 − 13

53

⎞⎟⎟⎟⎟⎟⎠

Solving the two algebraic systems of equations results in the solutions dis-played in Fig. 2.7. It is worth mentioning that the FE solution corresponding

Fig. 2.7. Solutions for the cases c = 0 and c = 0.5

to c = 0, which solves the 1D Poisson equation (Laplace operator, see (2.17)),is exact at the FE nodes. However, the case c = 0.5 already shows an error atthe FE nodes. An a priori error estimate for the discretization error will bediscussed in Sec. 2.8.


2.3 Nodal (Lagrangian) Finite Elements

As previously discussed, the FE method subdivides the simulation domaininto small elements (e.g., triangles, tetrahedra, etc.) and the unknown quan-tities are approximated by interpolation functions that have local support.After spatial and time discretization, we end up with an algebraic systemof equations. Now our task is to discuss the computation of the matrices(stiffness, mass, etc.) as well as the right-hand side suitable for a computerimplementation.

The first important step is to rewrite the integration over the whole domain(see e.g., (2.14)) as a sum of integrations over the element domains, e.g., forthe stiffness matrix

Kab =∫

Ω

(∂Na

∂x

∂Nb

∂x+∂Na

∂y

∂Nb

∂y

)dΩ

=ne∑

e=1

∫Ωe

(∂Na

∂x

∂Nb

∂x+∂Na

∂y

∂Nb

∂y

)dΩ ,

with ne the number of finite elements within the mesh. Therefore, we canintroduce the so-called element stiffness (mass, etc.) matrix and obtain

K =ne∧

e=1

ke ke = [kpq]

kpq =∫

Ωe

(∂Np

∂x

∂Nq

∂x+∂Np

∂y

∂Nq

∂y

)dΩ

1 ≤ p, q ≤ nen ,

with nen the number of element nodes and∧

the assembly operator (for theassembling procedure see Sect. 2.4).

In the second step, we shall briefly discuss the computation of the elementmatrices and the right-hand side. For this task, we need to compute the inter-polation functions, their derivatives, and perform numerical integration. Theeasiest and most general strategy is to introduce transfer functions for thedifferent geometric elements (quadrilateral, tetrahedral, etc.) to their parentelements (see Sect. 2.3). For these parent elements, we have to develop appro-priate shape functions as well as numerical integration schemes. In addition,since we need the derivatives of the interpolation functions with respect to theglobal coordinates x, y, z (also called global derivatives), we have to developa procedure for performing this task with the help of the transformation aswell as local shape functions of the geometric elements (see Sect. 2.3.6). Inthe case that we choose for the transformation from the local to the globalcoordinate system

x(ξ) =nen∑i=1

Ni(ξ)xi

2.3 Nodal (Lagrangian) Finite Elements 21

the same interpolation functions Ni as for the unknown quantity

uh(ξ) =nen∑i=1

Ni(ξ)uhi

we call these finite elements isoparametric.

2.3.1 Basic Properties

Let us assume that we want to solve a partial differential equation with thescalar unknown u on the domain Ω with Dirichlet boundary Γe and Neu-mann boundary Γn as displayed in Fig. 2.8. After performing the domain

Fig. 2.8. Domain to be discretized

discretization—in this case with triangular finite elements—we obtain 12 fi-nite elements and 11 finite element nodes (see Fig. 2.9). The numbers in theparenthesis are the equation numbers, and as can be seen, for Dirichlet nodesthis number is zero (compare with the decomposition of u into the unknownv and known values ue, see Sect. 2.1). Figure 2.10 displays the interpolation

Fig. 2.9. Discretization of the domain with finite elements

function N5(r) for Eq. (5) (finite element node 9). As can be seen, the in-terpolation function has a local support, which means that it has the value 1


at the node, decreases to zero approaching the neighboring nodes and is zerooutside the neighboring elements. Therefore, the ansatz according to (2.7) –(2.9) is allowed, since, e.g., at any node b it exhibits exactly the value vb

vh(t)|r=rb=

neq∑a=1

Na(rb)va(t) = vb(t) .

In addition, it is now clear that the mass, stiffness as well as effective sys-

Fig. 2.10. Shape (basis) function for the unknown in FE node 9

tem matrices show a sparse profile, since the integrals defining their entries(see (2.13) and (2.14)) are nonzero only if the supports of the interpolationfunctions Na and Nb overlap, which only happens if the functions belong toneighboring nodes. For our simple example we obtain the following matrixstructure ⎛

⎜⎜⎜⎜⎜⎜⎝

a11 a12 0 0 0 0a21 a22 a23 0 0 a26

0 a32 a33 a34 a35 a36

0 0 a43 a44 a45 00 0 a53 a54 a55 a56

0 a62 a63 0 a65 a66

⎞⎟⎟⎟⎟⎟⎟⎠ .

The main properties to be fulfilled by any nodal finite element are:

1) Smoothness on each element interior Ωe

2) Continuity across each element boundary Γ e

3) Completeness

Let us assume that we have to evaluate finite element matrices with par-tial derivatives of order m in the weak formulation. Then, properties 1 and2 demand for any nodal finite element basis functions, which are m timesdifferentiable in Ωe and (m − 1) times differentiable over the boundary Γ e.For example, for m = 1 any finite element with linear interpolation functions


(C0 finite element) fulfills property 1 and 2. In general we call finite elementssatisfying property 1 and 2 conforming, or compatible.

Now, let us illustrate the property completeness. We assume the followingansatz for the unknown quantity u

uh(x) =nen∑a=1

Nauea ,

with nen the number of nodes for the finite element e, and the property

uh(xea) = ue

a .

The finite element is said to be complete if

uea = c0 + c1x

ea + c2y

ea + c3z

ea (2.29)

impliesuh(x) = c0 + c1x+ c2y + c3z , (2.30)

with arbitrary constants c0 ... c3. This means the element interpolation func-tions are capable of exactly representing an arbitrary linear polynomial.Therewith, it is guaranteed that by reducing the mesh size he, the approxi-mated solution converges towards the exact solution.

2.3.2 Quadrilateral Element in IR2

For the computation of the element matrices, it is convenient to transformeach finite element to its reference element, where the numerical integrationcan be performed easily (see Sect. 2.3.7). Let us investigate this transformationfor the bilinear quadrilateral element in two space dimension as displayed inFig. 2.11. The local coordinates

34

1

2

x

y

3 (-1,-1) 4 (1,-1)

1 (1,1)2 (-1,1)

x

x

Fig. 2.11. Bilinear quadrilateral element


ξ =(ξη

)(2.31)

are related to the global coordinates

x =(xy

)(2.32)

via the following transformation

x(ξ) =

⎧⎪⎪⎨⎪⎪⎩x(ξ, η) =

4∑i=1

Ni(ξ, η)xei

y(ξ, η) =4∑

i=1

Ni(ξ, η)yei

(2.33)

Now, in order to compute explicitly the basis functions Ni, we choose thefollowing bilinear ansatz

x(ξ, η) = α0 + α1ξ + α2η + α3ξη (2.34)y(ξ, η) = β0 + β1ξ + β2η + β3ξη . (2.35)

The shape functions have to be constructed in such a way that the relations

x(ξi, ηi) = xei

y(ξi, ηi) = yei

are fulfilled. Since the local coordinates take only the following values

node i ξi ηi

1 1 12 -1 13 -1 -14 1 -1

we obtain 8 equations for the eight unknowns α0 .. β3. Using the solution forαi and βi in (2.34) and (2.35) and comparing the coefficients with (2.33), wearrive at the following explicit form of the shape function for node i (see Fig.2.12)

Ni(ξ) = Ni(ξ, η) =14

(1 + ξiξ)(1 + ηiη) . (2.36)

Now, let us investigate the three mentioned properties for the quadrilateralelement.

1) Smoothness on each element interior Ωe:The shape functions Ni define smooth functions, if each interior angle ofthe quadrilateral is less then 180o.


Fig. 2.12. Shape function of node i for a quadrilateral element

2) Continuity across each element boundary Γ e:Figure 2.12 displays the shape function Ni for node i defined by (ξi, ηi),and it is easy to see that

Ni(ξi, ηi) = δij (2.37)

is fulfilled. Along the boundary, e.g., η = −1, we obtain

Ni(ξ,−1) =1 + ξiξ

2,

which is exactly the shape function for the 1D case. Since this shape func-tion is typically the same for all edges, the quadrilateral element fulfillsthe continuity condition.

3) Completeness:

uh =nen∑i=1

Ni(ξ, η)uei

=nen∑i=1

Ni(ξ, η)(c0 + c1xei + c2y

ei )

=

(nen∑i=1

Ni(ξ, η)

)c0 +

(nen∑i=1

Ni(ξ, η)xei

)︸︷︷︸

x(ξ,η)

c1 +

(nen∑i=1

Ni(ξ, η)yei

)︸︷︷︸

y(ξ,η)

c2

Summing up all four shape functions results innen∑i=1

Ni(ξ, η) =14[(1 − ξ)(1 − η) + (1 + ξ)(1 − η)

+(1 + ξ)(1 + η) + (1 − ξ)(1 + η)]= 1 ,

which proves the completeness.


2.3.3 Triangular Element in IR2

Fig. 2.13. Transformation from global to local domain

The linear triangular element is defined by its three nodes as displayed inFig. 2.13. The local coordinates are as follows

node i ξi ηi

1 0 02 1 03 0 1

Similar to the quadrilateral element (see above) we obtain the local shapefunctions given by

N1 = 1 − ξ − η

N2 = ξ

N3 = η .

2.3.4 Tetrahedron Element in IR3

The linear tetrahedron element is defined by its four coordinates as shown inFig. 2.14.

node i ξi ηi ζi

1 0 0 02 1 0 03 0 1 04 0 0 1

Let us compute the transformation that maps any arbitrary tetrahedralelement in the global (x, y, z)-domain to a parent tetrahedron in the local(ξ, η, ζ)-domain by choosing the following linear ansatz


xi = α0 + α1ξi + α2ηi + α3ζi (2.38)yi = β0 + β1ξi + β2ηi + β3ζi . (2.39)

We know that the transformation has to satisfy the following relations at thefour nodes of a tetrahedron element

x(ξ) = xea a = 1, ... , 4 .

Therefore, we obtain (see Fig. 2.14)

x1 = α0 y1 = β0 (2.40)x2 = α0 + α1 y2 = β0 + β1 (2.41)x3 = α0 + α2 y3 = β0 + β2 (2.42)x4 = α0 + α3 y4 = β0 + β3 . (2.43)

Solving the above system of equations, we arrive at a general expression forthe shape function Ni as a function of the local coordinates

N1 = 1 − ξ − η − ζ

N2 = ξ

N3 = η

N4 = ζ .

Fig. 2.14. Transformation from global to local domain

2.3.5 Hexahedron Element in IR3

In many 3D applications hexahedron elements are used for the domain dis-cretization, due to their good approximation property. Figure 2.15 displays


the hexahedron element in its global and local coordinate system.

node i ξi ηi ζi

1 1 1 -12 -1 1 -13 -1 -1 -14 1 -1 -15 1 1 16 -1 1 17 -1 -1 18 1 -1 1

For the element a trilinear mapping is applied between the global (definedby x) and the local (defined by ξ) element domain [99]

x(ξ) = α0 + α1ξ + α2η + α3ζ + α4ξη + α5ηζ + α6ξζ + α7ξηζ . (2.44)

The coefficients αi are determined by the relations (see Fig. 2.15)

Fig. 2.15. Hexahedron element: notation in the global and local domain

x(ξa) = xea a = 1, .., 8 , (2.45)

which results in

Na(ξ) =18

(1 + ξaξ)(1 + ηaη)(1 + ζaζ) . (2.46)

Using a simple degeneration technique [99], one can obtain a pyramid, a wedgeas well as a tetrahedron element from a hexahedron one.


2.3.6 Global/Local Derivatives

For the computation we need to evaluate derivatives of the shape functionswith respect to the global coordinate system (see e.g., (2.14)). Since the shapefunctions Ni depend on the local coordinates (ξ, η, ζ), we may write

Na,x = Na,ξξ,x +Na,ηη,x +Na,ζζ,x

Na,y = Na,ξξ,y +Na,ηη,y +Na,ζζ,y

Na,z = Na,ξξ,z +Na,ηη,z +Na,ζζ,z ,

with the notation, e.g., ξ,x = ∂ξ/∂x. In matrix form, we obtain⎛⎜⎜⎝Na,x

Na,y

Na,z

⎞⎟⎟⎠ =

⎡⎢⎢⎣ξ,x η,x ζ,x

ξ,y η,y η,y

ξ,z η,z ζ,z

⎤⎥⎥⎦⎛⎜⎜⎝Na,ξ

Na,η

Na,ζ

⎞⎟⎟⎠ . (2.47)

Now, we do not have the explicit form of the derivatives of the local coor-dinates with respect to the global coordinates. However, we can express thisrelation as follows ⎛

⎜⎜⎝Na,ξ

Na,η

Na,ζ

⎞⎟⎟⎠ =

⎡⎢⎢⎣x,ξ y,ξ z,ξ

x,η y,η z,η

x,ζ y,ζ z,ζ

⎤⎥⎥⎦⎛⎜⎜⎝Na,x

Na,y

Na,z

⎞⎟⎟⎠ . (2.48)

Comparing (2.47) and (2.48) we arrive at⎡⎢⎢⎣ξ,x η,x ζ,x

ξy η,y η,y

ξ,z η,z ζ,z

⎤⎥⎥⎦ =

⎡⎢⎢⎣x,ξ y,ξ z,ξ

x,η y,η z,η

x,ζ y,ζ z,ζ

⎤⎥⎥⎦−1

︸︷︷︸(J T )−1

, (2.49)

with J the Jacobi matrix. The computation of J can be performed by thetransformation between the global and local coordinate systems

x(ξ, η, ζ) =nen∑a=1

Na(ξ, η, ζ)xea

y(ξ, η, ζ) =nen∑a=1

Na(ξ, η, ζ)yea

z(ξ, η, ζ) =nen∑a=1

Na(ξ, η, ζ)zea .

Therefore, the explicit expression for the Jacobian reads as


J =

⎡⎣x,ξ x,η x,ζ

y,ξ y,η y,ζ

z,ξ z,η z,ζ

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎣

nen∑a=1

Na,ξxea

nen∑a=1

Na,ηxea

nen∑a=1

Na,ζxea

nen∑a=1

Na,ξyea

nen∑a=1

Na,ηyea

nen∑a=1

Na,ζyea

nen∑a=1

Na,ξzea

nen∑a=1

Na,ηzea

nen∑a=1

Na,ζzea

⎤⎥⎥⎥⎥⎥⎦ . (2.50)

The algorithm can be summarized as follows (nint denotes the number of in-tegration points, see next section):

for l := 1, nint

Determine: Wl, ξl, ηl, ζlfor a := 1, nen

Calculate: Na , Na,ξ , Na,η , Na,ζ at (ξl, ηl, ζl)end

Compute Jacobi matrix, determinant and its inverseCompute global derivatives Na,x, Na,y, Na,z at (ξl, ηl, ζl)

end

2.3.7 Numerical Integration

For the computation of the element matrices as well as element right-handsides we have to numerically evaluate an integral of the form∫

Ωe

f(x) dΩ . (2.51)

Since we perform a transformation of each finite element to its parent element,(2.51) changes, e.g., for a hexahedron, to

1∫−1

1∫−1

1∫−1

f(x(ξ))|J |dξ dη dζ , (2.52)

with |J | the Jacobi determinant (see Sect. 2.3.6). In the 1D case a Gaus-sian quadrature formula is optimal, since by using nint integration points, weachieve an accuracy of order 2nint (see e.g., [99])

1∫−1

g(ξ) dξ =nint∑l=1

g(ξl)Wl + E (2.53)

ξl ... zero positions of Legendre polynomial with order nint

Wl ... weighting factor for integration point lE ... error.


For our 3D case, we can write

1∫−1

1∫−1

1∫−1

f(x(ξ))|J |dξ dη dζ =n1

int∑l1=1

n2int∑

l2=1

n3int∑

l3=1

g(ξl1 , ηl2 , ζl3 )Wl1Wl2Wl3 + E

=nint∑l=1

g(ξl, ηl, ζl)Wl + E . (2.54)

In the following, we give for each discussed geometric element the integra-tion points as well as the weighting factors.

• Quadrilateral elements (Gaussian quadrature):

l ξl ηl Wl

1 −0.57735026919 −0.57735026919 1.02 0.57735026919 −0.57735026919 1.03 0.57735026919 0.57735026919 1.04 −0.57735026919 0.57735026919 1.0

• Triangular elements (Gaussian quadrature):

l ξl ηl Wl

1 0.166 666 67 0.166 666 67 0.166 666 672 0.666 666 67 0.166 666 67 0.166 666 673 0.166 666 67 0.666 666 67 0.166 666 67

• Tetrahedron elements (Gaussian quadrature):

l ξl ηl ζl Wl

1 0.585 410 0.138 196 0.138 196 0.041 666 72 0.138 196 0.585 410 0.138 196 0.041 666 73 0.138 196 0.138 196 0.585 410 0.041 666 74 0.138 196 0.138 196 0.138 196 0.041 666 7

• Hexahedron elements (Gaussian quadrature):

l ξl ηl ζl Wl

1 -0.57735026919 -0.57735026919 -0.57735026919 1.02 0.57735026919 -0.57735026919 -0.57735026919 1.03 0.57735026919 0.57735026919 -0.57735026919 1.04 -0.57735026919 0.57735026919 -0.57735026919 1.05 -0.57735026919 -0.57735026919 0.57735026919 1.06 0.57735026919 -0.57735026919 0.57735026919 1.07 0.57735026919 0.57735026919 0.57735026919 1.08 -0.57735026919 0.57735026919 0.57735026919 1.0


2.4 Finite Element Procedure

In the previous section we discussed the computation of the element matricesas well as right-hand side. The still-open question of the assembly procedurewill be addressed here.

In the first step we introduce the nodal equation array NE, which relatesthe global equation number P to the global node number A.

NE(A) =

⎧⎨⎩P : if the quantity is unknown at A0 : if the quantity is known at A

(e.g., Dirichlet boundary condition)

A ... global node numberP ... global equation number.

Fig. 2.16. Example: global node numbers and in parenthesis the local node numbers

Since the whole simulation domain is discretized with finite elements, andwe first compute the element matrices (right-hand side) and then assemble itto the global system matrix (right-hand side), we need information given bythe following information element node array IEN

IEN(a, e) = A

a ... local element node numbere ... element numberA ... global node number.

Assuming that we solve a scalar PDE, we have just one unknown per FEnode, and the local node number coincides with the local equation number.Combining the NE array with the IEN array results in the equation arrayEQ

EQ(a, e) = NE (IEN(a, e)) = P .

The EQ array connects the element node number (element equation number)a of element e with the global equation number P . The following simpleexample, displayed in Fig. 2.16, will demonstrate all the steps that have to

2.4 Finite Element Procedure 33

be performed for the assembly process. The FE mesh consists of two finiteelements with given Dirichlet boundary conditions u1, u2 at node 1 and 2.Let us write the algebraic system of equations for some discretized PDE onthis domain (e.g., the Poisson equation) in the following general form⎡

⎢⎢⎢⎢⎢⎢⎣

K11 K12 K13 K14 K15 K16

K21 K22 K23 K24 K25 K26

K31 K32 K33 K34 K35 K36

K41 K42 K43 K44 K45 K46

K51 K52 K53 K54 K55 K56

K61 K62 K63 K64 K65 K66

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

u1

u2

u3

u4

u5

u6

⎤⎥⎥⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎣

f1f2f3f4f5f6

⎤⎥⎥⎥⎥⎥⎥⎦ . (2.55)

Since we know u1 and u2 we can rewrite (2.55) as⎡⎢⎢⎣K33 K34 K35 K36

K43 K44 K45 K46

K53 K54 K55 K56

K63 K64 K65 K66

⎤⎥⎥⎦

⎡⎢⎢⎣u3

u4

u5

u6

⎤⎥⎥⎦ =

⎡⎢⎢⎣f3 −K31u1 −K32u2

f4 −K41u1 −K42u2

f5 −K51u1 −K52u2

f6 −K61u1 −K62u2

⎤⎥⎥⎦ . (2.56)

The nodal equation array NE is given by1 2 3 4 5 60∗ 0∗ 1 2 3 4

∗: at this node we already know the quantity (Dirichlet boundary condi-tion)

and using it for setting up the algebraic system, we recognize that we obtaina similar system as in (2.56) with four unknowns. Since there is no connectionbetween the nodes 1, 2 and 5, 6 (see Fig. 2.16), the values of K51, K52, K61,and K62 are zero.

The number of nodes for the linear quadrilateral element is nen = 4 andthe unknown quantity is a scalar. Therefore, we obtain the following elementmatrices as well as right-hand sides for the two finite elements

k1 =

⎡⎢⎢⎣k111 k1

12 k113 k1

14

k121 (k1

22) (k123) k1

24k1

31 (k132) (k1

33) k134

k141 k1

42 k143 k1

44

⎤⎥⎥⎦ f1 =

⎡⎢⎢⎣f11

< f12 >

< f13 >

f14

⎤⎥⎥⎦

k2 =

⎡⎢⎢⎣

(k211) (k2

12) (k213) (k2

14)(k2

21) (k222) (k2

23) (k224)

(k231) (k2

32) (k233) (k2

34)(k2

41) (k242) (k2

43) (k244)

⎤⎥⎥⎦ f2 =

⎡⎢⎢⎣< f2

1 >< f2

2 >< f2

3 >< f2

4 >

⎤⎥⎥⎦

( ) ... contributes to the global stiffness matrix K

< > ... contributes to the global right-hand side f

... contributes to the local right-hand side fe.


The entries of the right-hand side f1 for element 1 compute as

f1 =

⎡⎢⎢⎣f11

f12 − k1

21u1 − k124u2

f13 − k1

31u1 − k134u2

f14

⎤⎥⎥⎦ .

Combining the NE array with the element node array IEN , we arrive at theEQ array

IEN el.nr.1 2

local 1 1 3node 2 3 5number 3 4 6

4 2 4

EQ el.nr.1 2

local 1 0 1node 2 1 3number 3 2 4

4 0 2

Assembling:

EQ(1, 1) = 0

EQ(2, 1) = 1K11 ← K11 + k1

22

K12 ← K12 + k123

f1 ← f12

EQ(3, 1) = 2K22 ← K22 + k1

33

K21 ← K21 + k132

f2 ← f13

EQ(4, 1) = 0

EQ(1, 2) = 1

⎧⎪⎪⎨⎪⎪⎩K11 ← K11 + k2

11

K13 ← K13 + k212

K14 ← K14 + k213

K12 ← K12 + k214

f1 ← f21

EQ(2, 2) = 3

⎧⎪⎪⎨⎪⎪⎩K33 ← K33 + k2

22

K31 ← K31 + k221

K34 ← K34 + k223

K32 ← K32 + k224

f3 ← f22

EQ(3, 2) = 4

⎧⎪⎪⎨⎪⎪⎩K44 ← K44 + k2

33

K41 ← K41 + k231

K43 ← K43 + k232

K42 ← K42 + k234

f4 ← f23

EQ(4, 2) = 2

⎧⎪⎪⎨⎪⎪⎩K22 ← K22 + k2

44

K21 ← K21 + k241

K23 ← K23 + k242

K24 ← K24 + k243

f2 ← f24 .

2.5 Time Discretization 35

Thus, we obtain the following algebraic system of equations

K =

⎡⎢⎢⎢⎢⎢⎣k122 + k2

11 k123 + k2

14 k212 k

213

k132 + k2

41 k133 + k2

44 k242 k

243

k221 k2

24 k222 k

223

k231 k2

34 k232 k

233

⎤⎥⎥⎥⎥⎥⎦ f =

⎡⎢⎢⎢⎢⎢⎣f12 + f2

1

f13 + f2

4

f22

f32

⎤⎥⎥⎥⎥⎥⎦ .

Comparing the result with Fig. 2.16, we can set up the following rules

• Diagonal entries:K11 is the entry of K due to node 3. Since this node belongs to element 1as well as 2, the entry in K will be a sum of the contributions from element1 and element 2. The entry K33 is due to global node 5, and therefore,only element 2 will contribute to it.

• Off-diagonal entries:The same situation as described for the diagonal entries occurs.

Figure 2.17 summarizes the whole assembly procedure, and the followingpseudo-code demonstrates the computer implementation.

//loop over all finite elementsfor e := 1, ne

//loop over all element nodes (equations)for a := 1, nen

eq1 := EQ(a, e)if ( eq1 > 0 )

//loop over all element nodes (equations)for b := 1, nen

eq2 := EQ(b, e)if ( eq2 > 0 )

K(eq1, eq2) := K(eq1, eq2) + ke(a, b)end

endend

endend

2.5 Time Discretization

In the following we will describe single time-step discretization algorithms forparabolic (e.g., electromagnetic field in the eddy current case) as well as forhyperbolic (mechanical as well as acoustic field) partial differential equations.


Fig. 2.17. FE procedure

2.5.1 Parabolic Differential Equation

Let us consider the following semidiscrete Galerkin formulation (discrete inspace and continuous in time)

Mu(t) + Ku(t) = f(t) , (2.57)

with M the mass matrix, K the stiffness matrix, f the right-hand side, uthe vector of unknowns at the finite element nodes and u its derivative withrespect to time. In a first step, let us subdivide the time interval [0, T ] intoM subintervals, which are defined as follows

[0, T ] =M⋃

n=1

[tn−1, tn] , t0 = 0 < t1 < t2 < .... < tn < .. < tM = T .


For simplicity, we assume the time step size ∆t = tn − tn−1 = T/M tobe constant over the whole time interval of interest. In a second step, weapproximate the time derivative of the unknown u(t) by the forward timedifference scheme

u(t) ≈ u(tn+1) − u(tn)∆t

=un+1 − un

∆t. (2.58)

In the third step, we have to decide at which point in time we evaluate theelliptic part Ku(t) and the right-hand side f(t). Applying a convex linearcombination with the time integration parameter γP to these terms, will resultin the following system of equations

Mu(tn+1) − u(tn)

∆t+γPKun+1+(1−γP)Kun = γPfn+1

+(1−γP)fn. (2.59)

Table 2.1 is listing the different time discretization methods depending on thevalue of γP.

Table 2.1. Finite difference schemes

γP Method

0.0 forward difference; forward Euler

0.5 trapezoidal rule; Crank-Nicolson

1.0 backward difference; backward Euler

Rearranging the terms in (2.59), we arrive at

(M + γP∆tK) un+1 = ∆t(γPfn+1

+ (1 − γP)fn

)+ (M − (1 − γP)∆tK) un .

(2.60)We obtain the same system of algebraic equations, when we apply the

general trapezoidal difference scheme [99], which is defined as follows

un+1 = un +∆t((1 − γP)un + γPun+1

). (2.61)

From (2.61) we can compute un+1

un+1 =un+1 − un

γP∆t− 1 − γP

γPun

and substituting it into the time discrete version of (2.57) at t = tn+1 leadsto

(M + γP∆tK)un+1 = γP∆tfn+1+ M (un + (1 − γP)∆tun) . (2.62)


To see that (2.62) and (2.60) are equivalent, we use the relation

Mun + Kun = fn

and rewrite the right-hand side of (2.60) as follows

∆t(γPfn+1

+ (1 − γP)fn

)+ (M − (1 − γP)∆tK)un

= γP∆tfn+1+ Mun + (1 − γP)∆t

(f

n− Kun

)︸︷︷︸

Mun

= γP∆tfn+1+ M (un + (1 − γP)∆tun) .

Applying the general trapezoidal difference method, we can write the solu-tion process as a predictor-corrector algorithm, where we distinguish betweenan effective mass and effective stiffness formulation.

1. Effective Mass Formulation• Perform predictor step:

u = un + (1 − γP)∆t un . (2.63)

• Solve algebraic system of equations:

M∗un+1 = fn+1

− Ku (2.64)

M∗ = M + γP∆tK .

• Perform corrector step:

un+1 = u+ γP∆t un+1 . (2.65)

2. Effective Stiffness Formulation• Perform predictor step:

u = un + (1 − γP)∆t un . (2.66)


K∗un+1 = fn+1

+1

γP∆tMu (2.67)

K∗ = K +1

γP∆tM .


un+1 =un+1 − u

γP∆t. (2.68)


According to the choice of the integration parameter γP, one distinguishesbetween an explicit and implicit algorithm.

1. Explicit Algorithm: γP = 0For this case, M∗ = M and lumping of the mass matrix might be con-

sidered (see e.g., [99]). Thus, the system matrix of the algebraic systemis a diagonal matrix, and the only time-consuming part of the solutionprocess are the matrix vector multiplications on the right-hand side. Thedisadvantage of this algorithm is clearly the lack of stability, which impliesrestrictions on the time step value ∆t, depending on the material param-eters and the quality of the mesh. In addition, one has to consider thatthe mass matrix must be regular. This is, for example, not the case withinelectromagnetic computations, where the entries in the mass matrix arezero for regions with zero electrical conductivity (e.g., air). A solution tothis problem may be the application of a mixed explicit/implicit time dis-cretization [99].

2. Implicit Algorithm: 0 < γP ≤ 1In this case, M∗ computes as the sum of the mass matrix M and the

stiffness matrix K multiplied by γP∆t. Now M∗ is a sparse matrix and thealgebraic system of equations has to be solved by any direct or iterativesolver. For γP = 0.5 the time discretization is called the Crank-Nicolsonscheme (second-order accurate), and for γP ≥ 0.5 the algorithm is absolutestable (independently of ∆t).

In order to investigate in the accuracy of the time discretization scheme,we will expand u(t) at time t = tk+γP in a Taylor series

u(t) = u(tk+γP) + u(tk+γP) (t− tk+γP) + u(tk+γP)(t− tk+γP)2

2+O

((t− tk+γP)3

). (2.69)

By considering the relation tk+γP = tk + γP∆t we obtain for the evaluation of(2.69) at t = tk

u(tk) = u(tk+γP) + u(tk+γP) (−γP∆t) + u(tk+γP)(−γP∆t)2

2+O

((−γP∆t)3

)as well as at t = tk+1

u(tk+1) = u(tk+γP) + u(tk+γP) ((1 − γP)∆t) + u(tk+γP)((1 − γP)∆t)2

2+O

(((1 − γP)∆t)3

).

Therewith, we obtain for the difference of u(tk+1) and u(tk), which is used forapproximating the time derivative (see (2.58)), the following expression


u(tk+1) − u(tk) = u∆t+ u∆t2 − 2γP∆t

2

2+O(∆t3) . (2.70)

Now, (2.70) clearly shows that only for γP = 0.5 (Crank-Nicolson), we arriveat a second order time discretization scheme, and for all other choices thescheme is first-order accurate.

2.5.2 Hyperbolic Differential Equation

For the hyperbolic case, we arrive after the spatial discretization at the fol-lowing system of second-order ordinary differential equations (still continuousin time)

Mu(t) + Ku(t) = f(t) . (2.71)

Now, we will apply a finite difference scheme of second order to approximateu(t)

u(t) ≈ u(tn+1) − 2u(tn) + u(tn−1)∆t2

=un+1 − 2un + un−1

∆t2(2.72)

with ∆t the time step size. Similar to the parabolic case, we substitute theelliptic part Ku(t) and the right-hand side f(t) by a convex, linear combi-nation at t = tn−1, tn and tn+1 weighted by the integration parameter αH.Therewith, we obtain the following algebraic system of equations

Mun+1 − 2un + un−1

∆t2+ αHKun+1 + (1 − 2αH)Kun + αHKun−1

= αHfn+1+ (1 − 2αH)f

n+ αHfn−1

, (2.73)

which can be rewritten as

(M + αH∆t2K)un+1 = αH∆t

2fn+1

+ (1 − 2αH)∆t2fn

+ αH∆t2f

n−1

+ (2M − (1 − 2αH)∆t2K)un (2.74)− (M + αH∆t

2K)un−1 .

Choosing αH = 0 and transforming M to a diagonal matrix (mass lumping,see e.g., [99]), results in an explicit time discretization scheme, which does notneed any algebraic solver. However, such schemes will not be unconditionallystable. A stability analysis will exhibit the so-called CFL (Courant-Friedrich-Levi) condition [110]

∆t <2√

λmax(M−1K). (2.75)

In (2.75) λmax denotes the largest eigenvalue of the matrix (M−1K). Sincethis value is of the order O(h−2), we should choose the time step size ∆tand the mesh size h of the same order. For αH = 1/4 the above scheme isunconditional stable.


For practical applications, especially if an additional damping matrix C ispresent, the Newmark schemes are mainly used. Let us start at the semidis-crete Galerkin formulation

Mun+1 + Cun+1 + Kun+1 = fn+1

, (2.76)

with M the mass matrix, C the damping matrix (e.g., C = αMM + αKK,see Sect. 3.7.2), K the stiffness matrix, f the right-hand side, u the vector ofunknowns at the finite element nodes and u as well as u its first and secondderivative with respect to time. Thus, we have (see e.g., [99])

un+1 = un +∆t un +∆t2

2((1 − 2βH)un + 2βHun+1

)(2.77)

un+1 = un +∆t((1 − γH)un + γHun+1

). (2.78)

In (2.77) and (2.78) n denotes the time-step counter, ∆t the time-step valueand βH, γH the integration parameters. Substituting un+1 and un+1 accordingto (2.77) and (2.78) into (2.76) leads to the following algebraic system ofequations

M∗un+1 = fn+1

− C (un + (1 − γH)∆t un)

−K(un +∆t un +

∆t2

2(1 − 2βH)un

)(2.79)

M∗ = M + γH∆tC + βH∆t2 K .

According to the choice of the integration parameters βH and γH, onedistinguishes similarly to the parabolic case between an explicit and implicitalgorithm.

1. Explicit Algorithm: βH = 0 and C = αMMChoosing for γH the value 0.5 we achieve a second-order accurate scheme.Similar to the parabolic case, it makes sense to lump the system matrix,so that again the time-consuming part of the solution process is the ma-trix vector multiplications on the right-hand side. Of course, the stabilitydepends on the time step value ∆t, the material parameters and the qual-ity of the mesh. However, this kind of algorithm is used quite often inacoustic computations, especially when the discretization is performed bya mapped mesh.

2. Implicit Algorithm: βH = 0 , γH = 0In this case, M∗ computes as the sum of the mass matrix M, the dampingmatrix C and the stiffness matrix K with appropriate integration factors.The matrix M∗ is now sparse and the algebraic system of equations hasto be solved by any direct or iterative solver. For βH = 0.25 and γH = 0.5the time discretization is second-order accurate with respect to time, if C


vanishes. To keep the second-order accuracy even in the damped case, onehas to extend the Newmark scheme to the Hilbert-Hughes-Taylor scheme(see [99]).

Writing the solution process for one time step as a predictor-corrector algo-rithm we arrive at the effective mass as well as effective stiffness formulations.

1. Effective Mass Formulation• Perform predictor step:

u = un +∆t un + (1 − 2βH)∆t2

2un (2.80)

˜u = un +∆t (1 − γH)un . (2.81)


M∗un+1 = fn+1

− Ku− C˜u (2.82)

M∗ = M + γH∆tC + β∆t2 K .


un+1 = u+ βH∆t2 un+1 (2.83)

un+1 = ˜u+ γH∆t un+1 . (2.84)

2. Effective Stiffness FormulationAccording to (2.77) and (2.78) we can express un+1 and un+1 as follows

un+1 =un+1 − u

βH∆t2(2.85)

un+1 = ˜u+ γH∆t un+1 = ˜u+γH

βH∆t(un+1 − u) . (2.86)

Therefore, we obtain• Perform predictor step:

u = un +∆t un + (1 − 2βH)∆t2

2un (2.87)

˜u = un +∆t (1 − γH)un . (2.88)


K∗un+1 = fn+1

− C˜u+(

1βH∆t2

M +γH

βH∆tC)u (2.89)

K∗ = K +γH

βH∆tC +

1βH∆t2

M .


un+1 =un+1 − u

βH∆t2(2.90)

un+1 = ˜u+ γH∆tun+1 . (2.91)

2.7 Edge (Nedelec) Finite Elements 43

2.6 Integration over Surfaces

Very often one has to evaluate an integral over a 3D surface or along a contourin 2D. In the first case the integration is performed within a 1D finite elementin a 2D space and in the second case within a 2D finite element in a 3D space.

Let us assume a scalar function f(x, y, z) as the integrand of a surfaceintegral. According to [157], we can write∫

Γ

f(x, y, z) dΓ =∫ ∫

f(x(ξ, η), y(ξ, η), z(ξ, η)) |xξ(ξ, η) × xη(ξ, η)| dξ dη

(2.92)with

xξ =

⎛⎜⎝

∂x∂ξ∂y∂ξ∂z∂ξ

⎞⎟⎠ xη =

⎛⎜⎝

∂x∂η∂y∂η∂z∂η

⎞⎟⎠ .

For the second case of a contour integral over a scalar function f(x, y), weobtain ∫

C

f(x, y) ds =∫f(x(ξ), y(ξ)) |xξ(ξ)| dξ , (2.93)

with

xξ =

(∂x∂ξ∂y∂ξ

).

Therefore, with slight modifications we can evaluate such integrals by per-forming an integration in the local domain and instead of the Jacobian deter-minant we have to compute the expressions given in (2.92) and (2.93).

2.7 Edge (Nedelec) Finite Elements

Edge finite elements belong to the family of vector finite elements (shapefunctions are vectors) and assign the degrees of freedom to the edges ratherthan to the nodes of the element. These types of elements were first introducedby Whitney (see e.g., [79]). Their importance in electromagnetics were realizedby Nedelec (see e.g., [163]), who constructed edge elements on quadrilateraland tetrahedron elements. Many important studies followed for the furtherdevelopment of different electromagnetic field problems (see e.g., [14, 26, 121,160,214]).

Within edge elements, a physical vector quantity A (e.g., the magneticvector potential) is approximated by the following ansatz

A ≈ Ah =ne∑

k=1

AekEk . (2.94)


In (2.94) ne defines the number of edges in the finite element mesh, Ek theedge shape function associated with the k-th edge, and Ae

k the correspondingdegree of freedom, namely the line integral of the physical vector quantityalong the k-th edge

Aek =

∫k

A · ds . (2.95)

For edge shape functions Ek of lowest order, the following conditions have tobe fulfilled [197]:

1. The tangential component of Ek along the edge k has to be constant.2. The tangential component of Ek along edges l = k is zero.3. The divergence of Ek is zero inside the element.

Since, in this work, Nedelec finite elements are exclusively used for 3Dmagnetic field computation, we will restrict ourselves to this case using tetra-hedron elements. Let us consider the following vector ansatz function E1 (∇

Fig. 2.18. Tetrahedron element: nodes are denoted by 1, .. , 4 and edges by 1, .. , 6

defines here the derivatives with respect to the global coordinates x)

E1 = N1∇N2 −N2∇N1 (2.96)

along edge 1 defined by the nodes 1 and 2 (see Fig. 2.18) and N1 and N2 thenodal interpolation functions (see Sect. 2.3.4) in node 1 and 2, respectively.To check condition 1, we compute, e.g., the tangential component of E1 alongedge 1

E1 · t1 = N1t1 · ∇N2 −N2t1 · ∇N1 .

It has to be mentioned that we have to compute the derivatives with respectto the global coordinate system (see Sect. 2.3.6). Since t1 · ∇N1 = −1/(2l1)and t1 · ∇N2 = 1/(2l1) (l1 denotes the length of edge 1), we obtain

E1 · t1 =1l1.

2.8 Discretization Error 45

Condition 2 is also fulfilled, since N1 vanishes along the edges (4,5,6), N2

along the edges (2,3,6) and t2 · ∇N1, t3 · ∇N1, t4 · ∇N2, t5 · ∇N2 are zero.Therewith, shape function E1 has no tangential component along the edges(2,3,4,5,6).

Condition number 3 states that the divergence of E1 has to vanish insidethe element. Applying the divergence to E1 results in

∇ · (N1∇N2 −N2∇N1) = N1∆N2 + ∇N1 · ∇N2

− N2∆N1 − ∇N2 · ∇N1

= N1∆N2 −N2∆N1 .

Since the interpolation functions N1 as well as N2 are linear functions, thevalue of ∇ · E1 is zero.

To obtain for the tangential components of Ek the value 1, we have to scalethe vector function with the length of the corresponding edge length, whichresults in the following edge interpolation functions for a linear tetrahedron

E1 = (N1∇N2 −N2∇N1) l1E2 = (N1∇N3 −N3∇N1) l2E3 = (N1∇N4 −N4∇N1) l3E4 = (N2∇N3 −N3∇N2) l4E5 = (N4∇N2 −N2∇N4) l5E6 = (N3∇N4 −N4∇N3) l6 .

2.8 Discretization Error

When applying the FE method, it is of great importance to have some knowl-edge of the discretization error. Since the error analysis is a quite sophisticatedtask, we will just provide a short overview containing important results. Fora detailed discussion on this topic we refer to [110].For our purpose of error analysis let us consider the following variational form:

Given:f, c : Ω → IR

Find: u ∈ Vg such that for all v ∈ V0

a(u, v) = < f, v > (2.97)

with


a(u, v) =∫Ω

∇v · ∇u dΩ +∫Ω

c v u dΩ

< f, v > =∫Ω

vf dΩ

Vg = u ∈ H1|u = g on ΓV0 = v ∈ H1|v = 0 on Γ .

Applying the FE method to our problem, results in

Given:f, c : Ω → IR

Find: uh ∈ V hg such that for all vh ∈ V h

0

a(uh, vh) = < f, vh > (2.98)

with

a(uh, vh) =∫Ω

∇vh · ∇uh dΩ +∫Ω

c vh uh dΩ

< f, vh > =∫Ω

vhf dΩ

with V hg ⊂ Vg and V h

0 ⊂ V0.In general, we distinguish between a priori and a posteriori error estimates:

• The a priori error estimate is expressed as

||u− uh|| ≤ C1(u)hα (2.99)

with α > 0 and C1(u) being a positive constant. In (2.99) u denotes theexact solution, uh our solution obtained by the FE method, h the dis-cretization parameter (e.g., longest edge in the FE mesh) and || · || anadequate norm. The constant α depends on the smoothness of the solu-tion u and the polynomial degree of the chosen FE basis (shape) functions.In general, this constant is not known.

• The a posteriori error estimate reads as

||u− uh|| ≤ C2(uh, h) . (2.100)

As for the a priori error estimate, u denotes the exact solution, uh oursolution obtained by the FE method, h the discretization parameter and|| · || an adequate norm. The constant C2 depends on the FE solution andthe discretization parameter h. Providing an optimal error estimator, it ispossible to obtain C2 or at least sharp bounds for C2 (see e.g., [4]).


In the following, we will concentrate on a priori estimates, and we assumethat our bilinear form as defined in (2.98) is V0-elliptic and V0-bounded (for aproof see e.g., [110]). V0-ellipticity means that there exists a constant C1 > 0so that

a(v, v) ≥ C1||v||2H1

for all v ∈ V0. V0-boundedness states that there exists a constant C2 > 0, suchthat

|a(u, v)| ≤ C2||u||H1 ||v||H1

for all u, v ∈ V0.In addition to the norms defined in App. C, we introduce the energy norm

of the error e = (u− uh) associated with the bilinear form a(u, v)

||u− uh||a =√a(u− uh, u− uh) =

√a(e, e) .

Since V h0 ⊂ V0, we may rewrite (2.97) by

a(u, vh) =< f, vh >

for all vh ∈ V h0 . Now, let us subtract from this equation the discrete weak

form (see (2.98)). This operation results in

a(u− uh, vh) = 0

for all vh ∈ V h0 , which is known as the Galerkin orthogonality of the error.

This result implies that in a certain sense uh is the best approximation of uin V h

g .The most important theorems for the a priori estimation of the discretiza-

tion error in H1(Ω) are Cea’s lemma (see theorem 1 below) and the Bramble-Hilbert lemma [78]. The lemma of Cea allows us to transform the estimationof the discretization error to an estimation of the approximation error.

Theorem 1: Let the bilinear form a(·, ·) be V0-elliptic and V0-bounded. Then,the error estimation reads as

||u− uh||H1 ≤ C1

C2inf

wh∈V hg

||u− wh||H1 . (2.101)

Since the approximation error

infwh∈V h

g

||u − wh||H1

can be estimated from above by the interpolant (see Fig. 2.19)

Πh(u) =M∑i=0

Ni(x)ui ⊂ V hg ,


we may rewrite (2.101) by

||u− uh||H1 ≤ C1

C2inf

wh∈V hg

||u− wh||H1 ≤ C1

C2||u−Πh(u)||H1 .

x =a0 x =bM

u(x)

uu

(u)

h

h

Fig. 2.19. Exact solution u, interpolant Πh(u) and the FE solution uh

According to this important result and using the Bramble-Hilbert lemma,we will obtain practical a priori error estimates. For our FE discretization wewill use piecewise, continuous polynomial basis functions of order m, and canstate the following theorem [78]:

Theorem 2: Let a(·, ·) be a V0-elliptic and V0-bounded bilinear form. Fur-thermore, there exists derivatives in the weak sense of the exact solution u upto order (k+ 1) with k ≤ m, so that u ∈ Vg ∩Hk+1. Then, we estimate the apriori error by

||u− uh||H1 ≤ C hm|u|Hm+1 (2.102)

with C independent of the discretization parameter h and m the polynomialorder of the basis functions defined on the reference element.

In (2.102) |u|Hm+1 stands for the semi norm (see C.11). For our consideredbilinear form the exact solution has derivatives in the weak sense up tom+1 =2, so that the error will be of order O(h).It has to be noticed, that an increase of the polynomial order of our basisfunctions makes just sense for sufficiently smooth u. If u ∈ Hk with 2 ≤ k ≤m + 1, and we use polynomials of order m for the basis functions, then thefollowing estimate holds


||u− uh||H1 ≤ C hk−1|u|Hk .

This means that for such a case an increase of our assumed polynomial orderm of the basis functions, will not result in an decrease of the error. Therefore,a reduction of the discretization error can just be achieved by a refinement ofthe FE mesh (decreasing h).

If we now consider the case for

• c = 0 in our bilinear form (i.e. the bilinear form corresponds to the Laplaceoperator)

• u ∈ Vg ∩H2

• linear basis functions• a quasi uniform discretization of our computational domain,

then we arrive at the following practical estimates

||u − uh||H1 ≤ C h|u|H2 = O(h) (2.103)||u− uh||L2 ≤ C h2|u|H2 = O(h2) (2.104)

with C, C being constants and independent of the discretization. However,problems with discontinuous jumps in material parameters (e.g., magneticpermeability at an iron–air interface) usually do not permit solutions that aresmooth enough to be in H2. In such a case, the bounds deteriorate to

||u− uh||H1 ≤ C hα|u|H2 = O(hα) (2.105)||u− uh||L2 ≤ C h2α|u|H2 = O(h2α) (2.106)

with 0 < α < 1. The value of α depends on the smoothness (regularity) of thesolution u.

3

Mechanical Field

3.1 Navier’s Equation

Let us consider a solid body with prescribed volume force fV and support atequilibrium, which means that the sum of all forces as well as the sum of allmoments are zero. In the first step we cut out a small part of this solid bodyso that the faces of this small body are parallel to the Cartesian coordinatesystem (see Fig. 3.1a). Now, we have to apply forces at the cutting planes tostill guarantee equilibrium (see Fig. 3.1b). These forces correspond to inner

Fig. 3.1. (a) Cutting out a small body; (b) Mechanical stress σ on a surface of thesmall body

forces acting within the solid body. Since the applied forces are distributed allover the cutting planes, we describe them by mechanical stresses (force perunit area). The stress state of each face is defined by its stress vectors σx, σy

52 3 Mechanical Field

and σz, where the index denotes the associated face. Thus, we can describethe stress state as follows

σx = σxxex + σxyey + σxzez (3.1)σy = σyxex + σyyey + σyzez (3.2)σz = σzxex + σzyey + σzzez (3.3)

with (σxx, σyy, σzz) the normal stresses and (σxy, σxz, σyx, σyz, σzx, σzy) theshear stresses and ei defining the unit vector in direction i.

Now in a second case, we consider a body with an oblique cutting face andinvestigate on the mechanical stress σα on the face dΓα (see Fig. 3.2). We

x

z

yx

y

z

Fig. 3.2. Mechanical stresses on the surface of a small body with an oblique face

define the normal vector nα by

nα = nxex + nyey + nzez

with the property |nα| =√n2

x + n2y + n2

z = 1. According to Fig. 3.2, theequilibrium of all forces results in

dΓασα − dΓxσx − dΓyσy − dΓzσz = 0 . (3.4)

In (3.4) dΓα denotes the infinite small surface of the oblique cutting face, andthe surfaces of the three other faces can be expressed as

dΓx = dΓαnx dΓy = dΓαny dΓz = dΓαnz .

Therewith, we can rewrite (3.4) by

σα = nxσx + nyσy + nzσz ,

which results by using (3.1) - (3.3) in

3.1 Navier’s Equation 53

σα = (σxxnx + σyxny + σzxnz) ex + (σxynx + σyyny + σzynz) ey

+ (σxznx + σyzny + σzznz) ez .

To obtain a more compact expression, we introduce the mechanical stresstensor [σ], called the Cauchy stress tensor, as

[σ] =

⎡⎣σxx σxy σxz

σyx σyy σyz

σzx σzy σzz

⎤⎦ . (3.5)

The stress tensor now allows us to express any physical stress vector σα actingon a face defined by its normal vector nα as follows

σα = [σ]Tnα .

After having defined the stress state, we can now investigate in the equa-tion for translation ∫

Ω

fV dΩ +∮

Γ

[σ]T dΓ = 0 (3.6)

and the equation for rotation∫Ω

(r × fV) dΩ +∮

Γ

(r × [σ]) dΓ = 0 , (3.7)

which have to be fulfilled, if the body is at rest. First, let us consider theequation for translation in x-direction given by∫

Ω

fV · ex dΩ +∮

Γ

([σ]T ex

)· dΓ = 0 . (3.8)

Rewriting [σ]T ex by σx and applying the divergence theorem to the secondterm of (3.8), we obtain∫

Ω

fx dΩ +∫

Ω

∇ · σx dΩ = 0 , (3.9)

with fV = (fx fy fz)T . Since this result has to hold for each volume Ω, wemay write (3.9) in the following form

fx + ∇ · σx = 0 . (3.10)

Similar expressions are obtained for the y- and z-directions

fy + ∇ · σy = 0fz + ∇ · σz = 0 . (3.11)

The equilibrium condition for the rotation around the x-axis with the positionvector r = (x y z)T reads as

54 3 Mechanical Field∫Ω

(yfz − zfy) dΩ +∮

Γ

(yσz − zσy) · n dΓ = 0 . (3.12)

Applying the divergence theorem and omitting the volume integral results inthe following differential form

yfz − zfy + ∇ · (yσz) − ∇ · (zσy) = 0 . (3.13)

Using the vector identity

∇ · (ξu) = ξ∇ · u + u · ∇ξ (3.14)

we obtain

yfz − zfy + y∇ · σz + σz · ∇y − z∇ · σy − σy · ∇z = 0 . (3.15)

With the result of (3.11) we can simplify (3.15) to

σzy = σyz . (3.16)

The relations for the remaining two axes of rotation yield

σzx = σxz and σyx = σxy . (3.17)

Therefore, the equilibrium equation for a body at rest can be expressed asfollows

fV + ∇[σ] = 0 , (3.18)

where [σ] denotes the Cauchy stress tensor. Since the stress tensor [σ] issymmetric, it is convenient to write it as a vector of six components usingVoigt notation [18]

⎡⎣σxx σxy σxz

σyx σyy σyz

σzx σzy σzz

⎤⎦ =

⎡⎣σ11 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ33

⎤⎦ ;

⎛⎜⎜⎜⎜⎜⎜⎝

σxx

σyy

σzz

σyz

σxz

σxy

⎞⎟⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎝

σ11

σ22

σ33

σ23

σ13

σ12

⎞⎟⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎝

σ1

σ2

σ3

σ4

σ5

σ6

⎞⎟⎟⎟⎟⎟⎟⎠ . (3.19)

By introducing the differential operator B

B =

⎛⎜⎜⎝

∂∂x 0 0 0 ∂

∂z∂∂y

0 ∂∂y 0 ∂

∂z 0 ∂∂x

0 0 ∂∂z

∂∂y

∂∂x 0

⎞⎟⎟⎠

T

, (3.20)

(3.18) takes on the following form

BT σ + fV = 0 . (3.21)

In the dynamic case the sum of all forces is equal to the inertia force and wearrive at Navier’s equations describing the dynamical behavior of mechanicalsystems

fV + BT σ = ρa , (3.22)with ρ denoting the density of the medium and a the acceleration of the body.

3.2 Deformation and Displacement Gradient 55

3.2 Deformation and Displacement Gradient

Figure 3.3 displays the relation between an arbitrary point P0 in Ω0, the ini-tial configuration of a body, and the corresponding point P in the deformedconfiguration defined by Ω. Furthermore, X defines the position of any mate-rial point in Ω0 (often called Lagrangian coordinates), and x the position ofthe corresponding point in Ω (often called Eulerian coordinates). The motion

Fig. 3.3. Body with volume Ω0 in the initial configuration and with volume Ω inthe deformed configuration

of the body shall be described by the unique map Φ

x = Φ(X, t) . (3.23)

To locally compute the deformation, we introduce the deformation gradient[Fd], which maps the differential line element dX in Ω0 to the correspondingdifferential line element dx in Ω

dx = [Fd] dX

[Fd] =∂x∂X

= ∇X Φ

=

⎡⎢⎢⎣

∂x∂X

∂x∂Y

∂x∂Z

∂y∂X

∂y∂Y

∂y∂Z

∂z∂X

∂z∂Y

∂z∂Z

⎤⎥⎥⎦ . (3.24)

Since the map Φ is bijective, the Jacobi determinant |J | of [Fd] is differentfrom zero, and in order to exclude intersections, it has to be greater than zero.By introducing the displacement vector u according to

u(X, t) = x− X = Φ(X, t) − X , (3.25)


we obtain

Fd = ∇X (X + u)= I + ∇X u= I + [Hd] (3.26)

∇Xu =

⎡⎢⎢⎣

∂ux

∂X∂ux

∂Y∂ux

∂Z

∂uy

∂X∂uy

∂Y∂uy

∂Z

∂uz

∂X∂uz

∂Y∂uz

∂Z

⎤⎥⎥⎦ , (3.27)

with [Hd] the displacement gradient.The knowledge of Fd allows the definition of transformations for differ-

ential quantities. In particular, the transformation between the differentialsurface element dΓ0 in Ω0 to dΓ in Ω is given by [220]

dΓ = n dΓ = |J |[Fd]−T n0 dΓ0 = |J |[Fd]−T dΓ0 , (3.28)

and for a differential volume element by

dΩ0 = |J | dΩ . (3.29)

3.3 Mechanical Strain

In order to first provide a basic physical understanding of the mechanicalstrain, we will derive the relation between mechanical displacement and strainin the linear case for the configuration shown in Fig. 3.4. Due to a given

Fig. 3.4. Initial state and deformed state for a infinite small rectangle

load case, a general point P0(x, y) of the elastic body at the initial state will

3.3 Mechanical Strain 57

undergo a deformation according to the displacement components ux(x, y)and uy(x, y). We assume that the infinite small rectangle with side-length ∆xand ∆y will deform in a parallelogram with small angles α and β. Therewith,the side-length in x-direction computes for the deformed body as

ux(x+∆x, y) − ux(x, y)cosα

≈ ux(x+∆x, y) − ux(x, y) = ∆ux . (3.30)

Now, the elongation in x-direction computes for the limit ∆x→ 0

lim∆x→0

ux(x+∆x, y) − ux(x, y)∆x

.

Expanding the term ux(x +∆x, y) in a Taylor series

ux(x +∆x, y) = ux(x, y) +∂ux

∂x∆x+ ... higher oder terms

and neglecting the higher order terms, results in

sxx =∂ux

∂x. (3.31)

In (3.31) sxx is the unit elongation of the elastic body in x-direction and wecall it the normal strain in x-direction. The same derivation can be appliedto the unit elongation in y-direction, which will define the normal strain iny-direction

syy =∂uy

∂y.

Now, let us compute the shearing of the body, which means to obtain a relationbetween α, β and the displacements ux, uy. According to Fig.3.4, we obtain

tanα =uy(x+∆x, y) − uy(x, y)

∆x+ ux(x+∆x, y) − ux(x, y)=

uy(x+∆x,y)−uy(x,y)∆x

1 + ux(x+∆x,y)−ux(x,y)∆x

. (3.32)

Expanding the terms ux(x +∆x, y) and uy(x +∆x, y) in a Taylor series upto the linear term, substituting these expressions into (3.32) and performingthe limit ∆x→ 0, results in

tanα =∂uy

∂x

1 + ∂ux

∂x

. (3.33)

Assuming that the terms ∂uy/∂x and ∂ux/∂x are small compared to 1, theangle α will also be small, and we obtain the following approximation of (3.33)

α =∂uy

∂x.

Applying the same steps for the computation of β results in


β =∂ux

∂y.

The total shearing of our elastic body computes as the sum of α and β

α+ β =∂uy

∂x+∂ux

∂y= 2sxy ,

and we call sxy the shear strain.

For deriving the general relation between the mechanical strain and dis-placement, we consider the case as displayed in Fig. 3.5. The displacement umaps the initial configuration into the deformed one. The deformation stateof a body is defined by considering the change of a line element between twoneighboring points (P0(X,Y, Z), Q0(X,Y, Z)) in the initial configuration and(P (x, y, z), Q(x, y, z)) in the deformed configuration (see Fig. 3.5). Since the

Fig. 3.5. Strain measurement

metric of a space—the measure of the length and angle of the deformation—isdefined by the square of the line element, we obtain for the differential elementdl0 in the initial configuration

dl20 = dX2 + dY 2 + dZ2 = dXT dX (3.34)

and for dl in the deformed configuration

dl2 = dx2 + dy2 + dz2 = dxT dx . (3.35)

With the help of (3.24), we can express the difference as follows

dl2 − dl20 = dxT dx − dXT dX= dXT [Fd]T [Fd] dX− dXT dX= dXT

([Fd]T [Fd] − I

)dX

= dXT 2[V] dX , (3.36)

3.3 Mechanical Strain 59

where [V] denotes the Green–Lagrangian strain tensor. Thus, V measuresthe difference between the square length of an infinitesimal segment in thedeformed configuration and in the initial configuration. Since we can express[Fd] by (I + ∇X u) the Green–Lagrangian strain tensor [V] takes the form

[V] =12

((I + ∇X u)T (I + ∇X u) − I

)=

12(∇X u + (∇X u)T

)+

12((∇X u)T ∇X u

). (3.37)

As can be easily seen, the first part in the above equation defines the linearstrain, whereas the addition of the second part allows the description of largedeflections. The explicit form of the Green–Lagrangian strain tensor writtenin vector notation is given as follows

V =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂ux

∂X

∂uy

∂Y

∂uz

∂Z(∂uy

∂Z + ∂uz

∂Y

)(

∂uz

∂X + ∂ux

∂Z

)(

∂ux

∂Y + ∂uy

∂X

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸S

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

12

((∂ux

∂X

)2+(

∂uy

∂X

)2

+(

∂uz

∂X

)2)12

((∂ux

∂Y

)2+(

∂uy

∂Y

)2

+(

∂uz

∂Y

)2)12

(((

∂ux

∂Z

)2+(

∂uy

∂Z

)2

+(

∂uz

∂Z

)2)∂ux

∂Y∂ux

∂Z + ∂uy

∂Y∂uy

∂Z + ∂uz

∂Y∂uz

∂Z

∂ux

∂Z∂ux

∂X + ∂uy

∂Z∂uy

∂X + ∂uz

∂Z∂uz

∂X

∂ux

∂X∂ux

∂Y + ∂uy

∂X∂uy

∂Y + ∂uz

∂X∂uz

∂Y

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (3.38)

with [S] the tensor of the linear strains in vector notation. The linear straintensor [S] in vector form according to Voigt notation reads as

⎡⎣sxx sxy sxz

syx syy syz

szx szy szz

⎤⎦ =

⎡⎣ s11 s12 s13s21 s22 s23s31 s32 s33

⎤⎦ ;

⎛⎜⎜⎜⎜⎜⎜⎝

sxx

syy

szz

2syz

2sxz

2sxy

⎞⎟⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎝

s11s22s332s232s132s12

⎞⎟⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎝

s1s2s3s4s5s6

⎞⎟⎟⎟⎟⎟⎟⎠ . (3.39)

The factor of 2 on the shear strains results from the requirement that thecomputed energy in Voigt notation and index notation must be the same.

It should be noted that the Green–Lagrangian strain tensor [V] definesthe strains in the initial configuration, whereas the Cauchy stress tensor [σ]defines the stress for the deformed configuration. In the non-linear case, wewill introduce the second Piola–Kirchhoff stress tensor, which is also definedfor the initial configuration, and therefore fits to the Green–Lagrangian straintensor. However, in the linear case we do not have to distinguish between theinitial and the deformed configuration and can relate the Cauchy stress tensor[σ] to the linear strain tensor [S].


3.4 Constitutive Equations

The simplest and most frequently used relation between the stress and strainis the linear law of elasticity known as Hook’s law [168, 225]. Assuming anisotropic material, it can be expressed by knowledge of the shear modulus Gand the Poisson ratio νp only, as follows (sij denotes the components of thelinear strain tensor [S])

σxx = 2G(sxx +

νp1 − 2νp

(sxx + syy + szz))

; σxy = 2Gsxy

σyy = 2G(syy +

νp1 − 2νp

(sxx + syy + szz))

; σyz = 2Gsyz (3.40)

σzz = 2G(szz +

νp1 − 2νp

(sxx + syy + szz))

; σzx = 2Gszx .

The often-used elasticity modulus Em is computed via G and νp by

G =Em

2(1 + νp). (3.41)

By introducing the so-called Lame parameters λL and µL

λL =νpEm

(1 + νp)(1 − 2νp)(3.42)

µL =Em

2(1 + νp), (3.43)

we obtain the following explicit form of the stress–strain relation for isotropicmaterials⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

σxx

σyy

σzz

σyz

σxz

σxy

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

λL + 2µL λL λL 0 0 0

λL λL + 2µL λL 0 0 0

λL λL λL + 2µL 0 0 0

0 0 0 µL 0 0

0 0 0 0 µL 0

0 0 0 0 0 µL

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

sxx

syy

szz

2syz

2sxz

2sxy

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (3.44)

Therefore, Navier’s equation can be expressed by

µL∇ · ∇u + (λL + µL)∇ ∇ · u + fV = ρ∂2u∂t2

. (3.45)

For the general anisotropic case, we introduce the tensor of elasticity moduli[c] in the form

3.4 Constitutive Equations 61

[σ] = [c][S]σij = cijklskl , (3.46)

which has the following properties

cijkl = cijlk

cijkl = cjikl (3.47)cijkl = cklij .

Due to symmetry, we can write the stress as well as the strain tensors asvectors. Thus, we can combine the four indices of the cijkl to two indices cIK

as follows

ij / kl I / K

11 122 233 323 413 512 6

Expressing the linear strain vector S by Bu and combining (3.22) with (3.46),results in

BT [c]Bu + fV = ρ∂2u∂t2

. (3.48)

3.4.1 Plane Strain State

Fig. 3.6. Plane strain case


The plane strain case as depicted in Fig. 3.6 is a simplification of thegeneral case and can be used when the third dimension (assumed as the z-direction) is very large and within each cross section (in our case the xy-plane)the same boundary conditions as well as forces act on the body. Therefore, thedependence of the mechanical displacements ux and uy on the z-coordinatecan be neglected

szy = szx = 0 (3.49)szz = 0 . (3.50)

The stress–strain relation for the isotropic case simplifies to⎛⎜⎜⎜⎝σxx

σyy

σxy

⎞⎟⎟⎟⎠ =

⎡⎢⎢⎢⎣λL + 2µL λL 0

λL λL + 2µL 0

0 0 µL

⎤⎥⎥⎥⎦⎛⎜⎜⎜⎝sxx

syy

2sxy

⎞⎟⎟⎟⎠ . (3.51)

3.4.2 Plane Stress State

A demonstrative example for the plane stress case is shown in Fig. 3.7, wherea thin sheet is loaded by mechanical forces at the boundary and the forces actwithin the defined plane. By cutting an infinitely small piece out of the whole

Fig. 3.7. Plane stress case

thin sheet, we can assume to have stresses on its surface as displayed in Fig.3.7. Therefore, the following relations are fulfilled

σzx = σzy = 0 (3.52)σzz = 0 . (3.53)

By using (3.52) and (3.53), we immediately obtain for the isotropic case (see(3.44))

3.5 Waves in Solid Bodies 63

szx = szy = 0 (3.54)

szz = − λL

λL + 2µL(sxx + syy) , (3.55)

which leads to the following simplifications for the plane stress case assumingan isotropic material⎛

⎜⎜⎜⎝σxx

σyy

σxy

⎞⎟⎟⎟⎠ =

⎡⎢⎢⎢⎣

2λLµLλL+2µL

+ 2µL2λLµL

λL+2µL0

2λLµLλL+2µL

2λLµLλL+2µL

+ 2µL 0

0 0 µL

⎤⎥⎥⎥⎦⎛⎜⎜⎜⎝sxx

syy

2sxy

⎞⎟⎟⎟⎠ . (3.56)

3.4.3 Axisymmetric Stress–Strain Relations

In a cylindrical-coordinate system, the displacement components read as

ur displacement in radial direction (r-direction)uz displacement in axial direction (z-direction)uθ displacement in circumferential direction (θ-direction)

Since in the axisymmetric case the mechanical displacements do not de-pend on the θ-coordinate, the following equations must hold

uθ = 0 (3.57)srθ = szθ = 0 . (3.58)

Thus, the stress–strain relation for the isotropic case is given by⎛⎜⎜⎜⎜⎜⎝

σrr

σzz

σrz

σθθ

⎞⎟⎟⎟⎟⎟⎠ =

⎡⎢⎢⎢⎢⎢⎣

λL + 2µL λL 0 λL

λL λL + 2µL 0 λL

0 0 µL 0

λL λL 0 λL + 2µL

⎤⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎝srr

szz

2srz

sθθ

⎞⎟⎟⎟⎟⎟⎠ . (3.59)

3.5 Waves in Solid Bodies

In this section we want to investigate different wave types that can propa-gate within solids. In the general case, the mechanical displacement u canbe decomposed into an irrotational part (curl u = 0) and a solenoidal part(divu = 0) (Helmholtz decomposition)

u = gradϕ+ curlψ . (3.60)

By using this decomposition for (3.45) and setting fV to zero, we arrive at


µL∇ · ∇gradϕ+ (λL + µL)∇ ∇ · gradϕ+ µL∇ · ∇curlψ

+(λL + µL)∇ ∇ · curlψ = ρ∂2gradϕ

∂t2+ ρ

∂2curlψ∂t2

(3.61)

grad(ρ∂2ϕ

∂t2− (λL + 2µL)∇ · ∇ϕ

)

+curl(ρ∂2ψ

∂t2− µL∇ · ∇ψ

)= 0 . (3.62)

This relation is fulfilled if ϕ and ψ solve the two following equations

∂2ϕ

∂t2=λL + 2µL

ρ∇ · ∇ϕ (3.63)

∂2ψ

∂t2=µL

ρ∇ · ∇ψ . (3.64)

First, we will choose for (3.63) an ansatz expressed by

ϕ = f(ξ) = f(k · r− ωt) , (3.65)

which defines a wave propagation in direction of k with velocity c. By usingthis ansatz for (3.63), we obtain with

∂2ϕ

∂t2=

∂

∂t

(∂ϕ

∂ξ

∂ξ

∂t

)=

∂

∂t

(−ω∂ϕ

∂ξ

)= ω2 ∂

2ϕ

∂ξ2(3.66)

∂2ϕ

∂x2i

=∂

∂xi

(∂ϕ

∂ξ

∂ξ

∂xi

)=

∂

∂t

(ki∂ϕ

∂ξ

)= k2

i

∂2ϕ

∂ξ2(3.67)

(3.68)

the following result

ω2 ∂2ϕ

∂ξ2=(λL + 2µL

ρ

) 3∑i=1

k2i

∂2ϕ

∂ξ2

=(λL + 2µL

ρ

)∂2ϕ

∂ξ2

3∑i=1

k2i︸︷︷︸

k2

. (3.69)

Now, since the relation c2 = ω2/k2 holds, (3.63) is fulfilled, if c2 takes on thevalue (λL +2µL)/ρ. The mechanical displacement for the scalar component ϕcomputes

3.6 Material Properties 65

u = gradϕ =∂ϕ

∂xex +

∂ϕ

∂yey +

∂ϕ

∂zez

=∂ϕ

∂ξ

∂ξ

∂xex +

∂ϕ

∂ξ

∂ξ

∂yey +

∂ϕ

∂ξ

∂ξ

∂zez

= k∂ϕ

∂ξ. (3.70)

Thus, (3.70) clearly shows that the mechanical displacements are in the direc-tion of the wave propagation, which defines a longitudinal wave with velocitycL

cL =

√λL + 2µL

ρ=

√(1 − νp)Em

(1 + νp)(1 − 2νp)ρ. (3.71)

By choosing the ansatz

F = F(ξ) = F(k · r − ωt) (3.72)

we can solve (3.64). In this case the mechanical displacement is computed by

u = curlψ

= k × ∂ψ

∂ξ, (3.73)

and we obtain a mechanical displacement that is perpendicular to the directionof propagation. The type of wave is called a shear wave, which propagates withvelocity cT

cT =√µL

ρ=

√Em

2(1 + νp)ρ=

√G

ρ. (3.74)

The ratio of the two velocities

cLcT

=

√λL + 2µL

µL(3.75)

leads to the following inequality

cL >√

2cT . (3.76)

3.6 Material Properties

The mechanical material properties are defined by the density ρ and the tensorof mechanical moduli [c]. In the general case, the entries of [c] depend on themechanical stress σ. Figure 3.8 displays a typical stress–strain curve of ametallic material obtained by a tensile test. For the region defined by stressesup to σp, we find a strict proportionality between the stress and strain as


used in linear elasticity computations. For stresses larger than σp, we can finda super-proportional increase of the strain till the stress reaches σy, the so-called yield stress. By a further increase of the applied force, the stress againstrongly increases due to stiffening effects of the material until the samplebreaks at σb. For a more detailed discussion, especially on material models

Fig. 3.8. Stress–strain curve for a metallic material

(e.g., viscoelastic, viscoplastic, etc.) we refer to [18, 220].Furthermore, heating up a solid body will also result in a mechanical

deformation. The resulting thermal strain sthij can be modeled as follows

sthij = αi(T − T0) , (3.77)

with αi the so-called thermal expansion coefficient in direction i and T0 thereference temperature. For a homogenous and isotropic material, the value ofα is the same for all directions, so that the shear strains are zero. Therefore,Hook’s law can be written as

[σ] = [c]([S] − [Sth]

)(3.78)

sthij =

α(T − T0) for i = j0 for i = j

. (3.79)

In Table 3.1 the mechanical properties of some materials are summarized.

3.7 Numerical Computation

3.7.1 Linear Elasticity

The strong formulation for linear elasticity problems reads as follows:

3.7 Numerical Computation 67

Table 3.1. Mechanical properties of some materials

Material ρ Em νp α cL cT

(kg/m3) (N/m2) (1/T) (m/s) (m/s)

aluminum 2.7 × 103 7.20 × 1010 0.34 24 × 10−6 6.3 × 103 3.13 × 103

iron 7.7 × 103 21.6 × 1010 0.29 12 × 10−6 5.9 × 103 3.20 × 103

copper 8.9 × 103 12.5 × 1010 0.35 12 × 10−6 4.7 × 103 2.26 × 103

PVC 1.1 × 103 0.30 × 1010 0.48 11 × 10−6 2.2 × 103 1.10 × 103

Given:u0 : Ω → IRd

u0 : Ω → IRd

ρ, cij : Ω → IRfV : Ω → IRd .

Find: u(t) : Ω × [0, T ] → IRd

BT [c]Bu + fV = ρu . (3.80)

Boundary conditionsu = ue on Γe × (0, T )

[σ]Tn = σn on Γn × (0, T ) .

Initial conditionsu(r, 0) = u0 , r ∈ Ωu(r, 0) = u0 , r ∈ Ω .

For simplicity, we will set the boundary conditions to zero (ue = 0, σn = 0).Multiplying (3.80) by an appropriate test function u′ and performing a partialintegration will transform (3.80) to its variational formulation, which reads asfollows: Find u ∈ H1

0 such that∫Ω

ρu′ · u dΩ +∫Ω

(Bu′)T [c]Bu dΩ =∫Ω

u′ · fV dΩ (3.81)

for any u′ ∈ H10. Let us perform the spatial discretization with standard nodal

finite elements, which approximate the continuous displacement u as follows

u ≈ uh =nd∑i=1

n′n∑

a=1

Nauiaei =n′

n∑a=1

Naua ; Na =

⎛⎝Na 0 0

0 Na 00 0 Na

⎞⎠ , (3.82)

with nd the space dimension and n′n the number of finite element nodes withno Dirichlet boundary condition. Applying the same approximation to the


test function u′, we have the following semidiscrete Galerkin formulation forlinear elasticity

n′n∑

a=1

n′n∑

b=1

⎛⎝∫

Ω

ρNTa Nb dΩ ub +

∫Ω

(Bua)T [c]Bu

b dΩ ub

−∫Ω

NTa fV(ra) dΩ

⎞⎠ = 0 , (3.83)

with

Bua =

⎛⎜⎝

∂Na

∂x 0 0 0 ∂Na

∂z∂Na

∂y

0 ∂Na

∂y 0 ∂Na

∂z 0 ∂Na

∂x

0 0 ∂Na

∂z∂Na

∂y∂Na

∂x 0

⎞⎟⎠

T

. (3.84)

In matrix form, we may write (3.83) as

Muu+ Kuu = f , (3.85)

with

Mu =ne∧

e=1

meu ; me

u = [mpq] ; mpq =∫Ωe

ρNTp Nq dΩ (3.86)

Ku =ne∧

e=1

keu ; ke

u = [kpq ] ; kpq =∫Ωe

(Bup )T [c]Bu

q dΩ (3.87)

f =ne∧

e=1

fe ; fe = [fp] ; f

p=∫Ωe

NTp fV(rp) dΩ . (3.88)

The time discretization is performed by a standard Newmark method as ex-plained in Sect. 2.5.2. Thus, we arrive at the following time-stepping schemefor an effective mass formulation:

• Perform predictor step:

u = un +∆t un +∆t2

2(1 − 2βH) un (3.89)

˜u = un + (1 − γH)∆t un . (3.90)


M∗uun+1 = f

n+1− Kuu− Cu

˜u (3.91)

M∗u = Mu + γH∆tCu + βH∆t

2 Ku . (3.92)

In (3.91) we have introduced a damping matrix Cu according to a standardRayleigh model (see Sect. 3.7.2).



un+1 = u+ βH∆t2 un+1 (3.93)

un+1 = ˜u+ γH∆t un+1 . (3.94)

3.7.2 Damping Model

In general, vibrating mechanical systems will always show a damped behavior.The reason for the damping is mainly to friction within the material and itsmathematical model is usually an added velocity proportional damping term.Therefore, within the FE method a constant damping matrix Cu is introducedand the term Cuu is added to the semidiscrete Galerkin formulation given in(3.85). In many FE formulations, the Rayleigh damping model is applied, sothat Cu is computed via a combination of the mass matrix Mu and the linearstiffness matrix Ku

Cu = αMMu + αKKu . (3.95)

In (3.95) αM denotes the mass proportional and αK the stiffness proportionaldamping coefficients. As shown in [16], a mode superposition analysis includ-

Fig. 3.9. Damped sine curve

ing damping according to (3.95) leads to the following relation

αM + αKω2i = 2ωiξi , (3.96)

with ωi the i-th eigenfrequency (in rad/s) and ξi the modal damping for thei-th eigenmode. The modal damping ξi corresponds to the loss factor tan δifor ωi, so that we obtain


tan δi = 2ξi =αM + αKω

2i

ωi. (3.97)

In addition, it can be shown (see [16]) that (3.85) (with Cuu) can be de-composed in a system of non-coupled single degree of freedom differentialequations with unit mass as well as stiffness

xi(t) + 2ξiωixi(t) + ω2i xi(t) = fi(t) , (3.98)

with generalized displacements xi and forces fi. The technically relevant solu-tion of (3.98) will be an exponentially (with amplitude ξi) damped sine curveas displayed in Fig. 3.9. The relation between the logarithmic decrement Di

and the modal damping factor ξi computes as

Di = lnxn

xn+1=

2πξi√1 − ξ2i

(3.99)

ξi =

√D2

i

D2i + 4π2

. (3.100)

Therefore, if we measure e.g., the decay of a mechanical wave excited withfrequency ωi propagating within a solid body, we can compute the dampingfactor ξi and thus the loss factor tan δi. The computation of αM and αK forthis ξi can then be performed using (3.97) as follows

αM + αK(ωi +∆ω)2 = 2(ωi +∆ω) ξi (3.101)αM + αK(ωi −∆ω)2 = 2(ωi −∆ω) ξi . (3.102)

The value ∆ωi shall be kept small, so that we meet the prescribed ξi at ωi.However, if we have to model a wide frequency range by fixed αM and αK,which is, e.g., the case within a transient analysis, the actual ξ will differ fromξi according to (3.97). Let us suppose we perform a transient analysis of athickness mode piezoelectric transducer with resonance frequency 1 MHz, andwe set the damping coefficient ξ at resonance frequency to 0.005. Then, we willcompute the Rayleigh damping coefficients in order to meet this damping atresonance frequency, which will exhibit αM = 3.1×104 and αK = 7.9×10−10.For all other frequencies within the excitation signal, the damping ξ will beaccording to Fig. 3.10.

3.7.3 Geometric Non-linear Case

We have derived the partial differential equation for the mechanical field byconsidering the equilibrium equations (translation as well as rotation) for anelastic body (for simplicity we just consider the static case)

∇ [σ] + fV = 0u = ue on Γe (3.103)

[σ[Tn = σn on Γn . (3.104)


Fig. 3.10. Damping factor ξ as a function of frequency (ξi = 0.005 at 1MHz;computed αM = 3.1 × 104 and αK = 7.9 × 10−10)

In (3.103) [σ] denotes the mechanical stress tensor, fV any volume force, andu the mechanical displacement. However, (3.103) is just applicable for linearmechanics, since we are mixing up quantities defined in the deformed config-uration (e.g., Cauchy stress tensor [σ]) and quantities defined in the initialconfiguration (e.g., mechanical volume force fV). Now, if we perform any com-putation, we always start at the initial configuration and aim at calculatingthe deformation of the body due to any prescribed boundary conditions andvolume forces given for the initial configuration. Thus, we have to transformthe Cauchy stress tensor [σ] from the deformed to the initial configurationusing (3.28) ∫

Γ

[σ] dΓ =∫Γ0

|J |[σ][Fd]−T dΓ0 =∫Γ0

[τ ] dΓ0 . (3.105)

In (3.105), [τ ] stands for the 1st Piola–Kirchhoff tensor, which is a nonsym-metric stress tensor. Therefore, we introduce the 2nd Piola–Kirchhoff tensor[T], which represents no physical stresses, but is symmetric and computes as

[T] = [Fd]−1[τ ] = |J |[Fd]−1 [σ] [Fd]−T . (3.106)

Thus, we can rewrite (3.103) as

∇X ([Fd] [T]) + fV = 0 , (3.107)

including only quantities defined on the initial configuration (no mixing ofquantities defined in the deformed and initial state as in (3.103)). According


to (3.107), we define the non-linear operator F as a function of the mechanicaldisplacement u

F(u) = ∇X ([Fd] [T]) + fV = 0 . (3.108)

Now, a Newton step can be written as (see Appendix D.2)

uk+1 = uk + s with F ′(uk)[s] = −F(uk) . (3.109)

First, let us derive the weak formulation of (3.107). For an arbitrary testfunction u′ ∈ H1

0 and assuming homogeneous Neumann boundary conditionσn = 0, we obtain∫

Ω0

u′ (∇X · ([Fd] [T]) + fV) dΩ = 0 (3.110)

∫Ω0

[T] · [Fd]T ∇X u′ dΩ =∫Ω0

u′ · fV dΩ , (3.111)

with

∇X u′ =

⎛⎜⎜⎜⎝

∂u′x

∂X∂u′

x

∂Y∂u′

x

∂Z

∂u′y

∂X

∂u′y

∂Y

∂u′y

∂Z

∂u′z

∂X∂u′

z

∂Y∂u′

z

∂Z

⎞⎟⎟⎟⎠ (3.112)

being in general a nonsymmetric tensor. Since [T] is a symmetric tensor, thescalar product with [Fd] will always result in a symmetric tensor. Therefore,we rewrite the first term in (3.110) as∫

Ω0

[T] · [Fd]T ∇X u′ dΩ =∫Ω0

[T] · 12

([Fd]T ∇X u′ + ∇X

Tu′ [Fd])

dΩ .

(3.113)According to (3.109), we need the Frechet derivative F ′, which can be ap-proximated by F(uk + s) − F(uk) (see Appendix D.2). In its weak form, wehave to compute∫

Ω0

[T(uk + s)] · 12

([Fd(uk + s)]T ∇X u′ + ∇X

Tu′ [Fd(uk + s)])

dΩ

−∫Ω0

[T(uk)] · 12

([Fd(uk)]T ∇X u′ + ∇X

Tu′ [Fd(uk)])

dΩ . (3.114)

Now, let us remember the following relations

[T] = [c][V] =12[c]([Fd]T [Fd] − I

)(3.115)

[Fd] = I + ∇X u , (3.116)


with [c] the tensor of the mechanical elasticity coefficients (in our case weassume constant entries). The evaluation of the term [T(uk + s)] leads to

[T(uk + s)] = [c][V(uk + s)]

[V(uk + s)] =12([Fd(uk + s)]T [Fd(uk + s)] − I

)=

12

((I + ∇X (uk + s)

)T (I + ∇X (uk + s)

)− I

)=

12

((I + ∇X uk + ∇X s

)T (I + ∇X uk + ∇X s

)− I

). (3.117)

Neglecting all second-order terms, we arrive at

[V(uk + s)] ≈ 12

([Fd(uk)]T [Fd(uk)] − I

)

+12

((I + ∇X uk

)T ∇X s + ∇XT s

(I + ∇X uk

))

= [V(uk)] +12

([Fd(uk)]T ∇X s + ∇X

T s [Fd(uk)]). (3.118)

In addition, the term [Fd(uk + s)] can be expressed as follows

[Fd(uk + s)] = I + ∇X (uk + s)= [Fd(uk)] + ∇X s . (3.119)

With the help of these expressions for [V(uk + s)] and [Fd(uk + s)], we canrewrite the first term in (3.114) as∫

Ω0

([T(uk)] +

12[c]([Fd(uk)]T ∇X s + ∇X

T s [Fd(uk)]))

(3.120)

· 12

(([Fd(uk)]T + ∇X

T s)

∇X u′ + ∇XTu′

([Fd(uk)] + ∇X s

))dΩ .

Setting all second-order terms to zero, we arrive at∫Ω0

[T(uk)] · 12

([Fd(uk)]T ∇X u′ + ∇X

Tu′ [Fd(uk)])

dΩ

+∫Ω0

[T(uk)] · 12

(∇X

T s∇X u′ + ∇XTu′ ∇X s

)dΩ (3.121)

+∫Ω0

12[c]([Fd(uk)]T ∇X s + ∇X

T s [Fd(uk)])

·12

([Fd(uk)]T ∇X u′ + ∇X

T u′ [Fd(uk)])

dΩ .


Substituting this result into (3.114), we get∫Ω0

12[c]([Fd(uk)]T ∇X s + ∇X

T s [Fd(uk)])

·12

([Fd(uk)]T ∇X u′ + ∇X

T u′ [Fd(uk)])

dΩ

+∫Ω0

[T(uk)] · 12

(∇X


)dΩ . (3.122)

Therefore, the Newton step can be evaluated as follows∫Ω0

12[c]([Fd(uk)]T ∇X s + ∇X

T s [Fd(uk)])

·12

([Fd(uk)]T ∇X u′ + ∇X

T u′ [Fd(uk)])

dΩ

+∫Ω0

[T(uk)] · 12

(∇X


)dΩ

=∫Ω0

u′ · fV dΩ

−∫Ω0

[T(uk] · 12

([Fd(uk)]T ∇X u′ + ∇X

T u′ [Fd(uk)])

dΩ (3.123)

uk+1 = uk + s .

Before we go over to the discretized version of (3.123), let us apply some helpfultransformations. Using the relation [Fd(uk)] = [I + ∇X (uk)], we obtain

12

([Fd(uk)]T ∇X s + ∇X

T s [Fd(uk)])

=12

[I + ∇X (uk)]T ∇X s +12

∇XT s [I + ∇X (uk)]

=12

(∇X s + ∇X

T s)

︸︷︷︸B s

+12

(∇X

T uk∇X s + ∇XT s∇X uk

)︸︷︷︸

Bnl(uk) s

.

The differential operator B has already been defined (see (3.20)) and Bnl

computes as follows


Bnl =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂ukx

∂X∂

∂X

∂uky

∂X∂

∂X∂uk

z

∂X∂

∂X

∂ukx

∂Y∂

∂Y

∂uky

∂Y∂

∂Y∂uk

z

∂Y∂

∂Y

∂ukx

∂Z∂

∂Z

∂uky

∂Z∂

∂Z∂uk

z

∂Z∂

∂Z

∂ukx

∂Y∂

∂Z + ∂ukx

∂Z∂

∂Y

∂uky

∂Y∂

∂Z + ∂uky

∂Z∂

∂Y∂uk

z

∂Y∂

∂Z + ∂ukz

∂Z∂

∂Y

∂ukx

∂X∂

∂Z + ∂ukx

∂Z∂

∂X

∂uky

∂X∂

∂Z +∂uk

y

∂Z∂

∂X∂uk

z

∂X∂

∂Z + ∂ukz

∂Z∂

∂X

∂ukx

∂X∂

∂Y + ∂ukx

∂Y∂

∂X

∂uky

∂X∂

∂Y + ∂uky

∂Y∂

∂X∂uk

z

∂X∂

∂Y + ∂ukz

∂Y∂

∂X

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (3.124)

The second term in (3.123) contains a scalar product between two tensors,which in general for two tensors [A] and [B] computes as

[A] · [B] = A11B11 +A12B12 +A13B13

+ A21B21 +A22B22 +A23B23

+ A31B31 +A32B32 +A33B33 .

Using this relation, we can rewrite this term as follows

[T(uk)] · 12

(∇X


)= BT

x [T]B (3.125)

B =(∂

∂X

∂

∂Y

∂

∂Z

)T

. (3.126)

Approximating s and u′ by nodal finite elements (see (3.82)) will result inthe following discrete Galerkin formulation

KNLu (uk)s = f

a− f

i. (3.127)

In (3.127) KNLu , f

a(external applied forces) as well as f

i(internal forces due

to stresses) are calculated as follows

KNLu =

ne∧e=1

keu ; ke

u = [kpq]

kpq =∫

Ωe

(Bup )T [c]Bu

q dΩ +∫Ωe

(Bnlp )T [c]Bnl

q dΩ

+∫

Ωe

(BT

p [T]Bq

)I dΩ (3.128)

fa

=ne∧

e=1

fe ; fe = [fp] (3.129)

fp

=∫

Ωe

NTp fV(rp) dΩ (3.130)


fi=

ne∧e=1

fe ; fe

i= [f

p] (3.131)

fp

=∫

Ωe

(Bu

p + Bnlp

)T[T(uk)] dΩ , (3.132)

with Bup as given in (3.84), I the identity matrix and T the second Piola–

Kirchhoff tensor in vector notation. The operator Bnlp depends on uk and

computes as follows

Bnlp =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂ukx

∂X∂Np

∂X

∂uky

∂X∂Np

∂X∂uk

z

∂X∂Np

∂X

∂ukx

∂Y∂Np

∂Y

∂uky

∂Y∂Np

∂Y∂uk

z

∂Y∂Np

∂Y

∂ukx

∂Z∂Np

∂Z

∂uky

∂Z∂Np

∂Z∂uk

z

∂Z∂Np

∂Z

∂ukx

∂Y∂Np

∂Z + ∂ukx

∂Z∂Np

∂Y

∂uky

∂Y∂Np

∂Z +∂uk

y

∂Z∂Np

∂Y∂uk

z

∂Y∂Np

∂Z + ∂ukz

∂Z∂Np

∂Y

∂ukx

∂X∂Np

∂Z + ∂ukx

∂Z∂Np

∂X

∂uky

∂X∂Np

∂Z + ∂uky

∂Z∂Np

∂X∂uk

z

∂X∂Np

∂Z + ∂ukz

∂Z∂Np

∂X

∂ukx

∂X∂Np

∂Y + ∂ukx

∂Y∂Np

∂X

∂uky

∂X∂Np

∂Y +∂uk

y

∂Y∂Np

∂X∂uk

z

∂X∂Np

∂Y + ∂ukz

∂Y∂Np

∂X

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(3.133)with ∂uk

α/∂β computed within each integration point by

∂ukα

∂β=

ne∑a=1

∂Na

∂βuk

a α α , β ∈ X,Y, Z . (3.134)

The operator Bp is given by

Bp =(∂Np

∂X

∂Np

∂Y

∂Np

∂Z

). (3.135)

The iterative solution process is stopped if the incremental error as well asthe residual error fulfill

||uk+1 − uk||2||uk+1||2

< δa||fk+1

i− fa||2

||fa||2

< δr , (3.136)

with appropriate δa and δr. To guarantee that the Newton method convergesto the correct solution, a line search algorithm should be applied to obtain anoptimal relaxation parameter η in each Newton step (see Appendix D)

uk+1 = uk + ηs . (3.137)


3.7.4 Numerical Example

In order to show the influence of the discretization as well as the order of theshape functions used for the approximation, we will compute the deflection ofa beam due to a mechanical load. Figure 3.11 displays the setup and since thebeam is fully supported at both sides, we exploit the symmetry. The beam has

Fig. 3.11. Setup

a length of L = 1000µm and a thickness of d = 4µm, and we perform a planestrain analysis. On the left side (x = 0), where the beam is fully supported, weset ux = uy = 0 and on the right side (symmetry) we set ux = 0. The nodalforce has a value of 1 N and is supplied at x = L/2 and y = d. In the firststep we will perform a linear analysis using once linear and once quadraticshape functions for the approximation. The discretization in the y-directionis done just by one finite element and the discretization in the x-direction isvaried. As displayed in Table 3.2, the obtained tip displacement for the linearshape functions converges very slowly to the correct value when increasing thenumber of finite elements ne. The reason for this effect is that linear shapefunctions exhibit a very poor approximation of the true solution when appliedto thin walled structures. This effect is called locking (see e.g., [16, 99]), andwill be discussed in detail in the subsequent section.

Table 3.2. Tip displacement uy at x = L/2 and y = 0

Type of basis functionne Linear Quadratic

20 –0.145 µm –4.436 µm40 –0.947 µm –4.457 µm80 –2.182 µm –4.465 µm160 –3.238 µm –4.468 µm320 –3.683 µm –4.468 µm

In the second step we perform a non-linear analysis and compare the resultsto the linear one (see Fig. 3.12). For the discretization, quadrilateral elements


with quadratic shape functions have been used with one finite element inthe y-direction and 40 finite elements in the x-direction. Figure 3.12 clearlydemonstrates that due to the large deflection, which is of the order of thethickness of the beam, a non-linear analysis has to be performed for our setup.

Fig. 3.12. Displacement along the beam obtained by linear and non-linear analysis

3.8 Locking and Efficient Solution Approaches

For some problems in computational mechanics, we recognize that the straightforward application of the displacement-based finite element method (as de-scribed in Sec. 3.7.1) will provide inaccurate results and show slow convergencewhen reducing the mesh size. Being specific, the finite element solution resultsin too small displacements (see example in the previous section), and we willrefer to this effect as locking. In general, the three main locking phenomenaare as follows:

• Shear locking: The shear locking effect occurs by the application of thedisplacement-based finite element method to thin walled structures, whereespecially the relation t << h holds (t denotes the thickness and h themesh size).

• Volumetric (Poisson) locking: For nearly incompressible materials (Pois-son ration νP → 0.5, which means for the Lame parameters λL >> µL)

3.8 Locking and Efficient Solution Approaches 79

volumetric locking will occur when applying the displacement-based finiteelement method.

• Membrane locking: This type of locking just occurs in curved beam andshell elements.

We will concentrate ourself on the shear locking and start with a mathe-matical investigation (see e.g., [29]). We consider the following weak formula-tion for the unknown u in an appropriate Hilbert space X

a0(u, v) +1t2

(Bu,Bv) =< f, v > . (3.138)

In (3.138) t denotes a parameter with 0 < t ≤ 1, a0 : X×X → IR a continuous,symmetric and coercitive bilinear form, B : X → L2 a continuous mappingand f a given function in the computational domain Ω. Now, one can showthat for a given f , there exists a v0 such that < f, v0 >= 0 and Bv0 = 0.Then, the solution u is bounded from below by

||u|| ≥ C||f ||

with a constant C independent of t. However, in the FE setting, we obtain [30]

||uh|| ≤ t2C(h−2)||f || .

Therewith, for small values of the parameter t the standard finite elementmethod will result in too small displacements, and we call the phenomenalocking.

Let us demonstrate the shear locking effect by considering a thin Tim-oshenko beam. There, the assumption is made that a plane section originalperpendicular to the neutral axis remains plane, but according to the shear de-formations rotates as displayed in Fig. 3.13. The shear angle γ and curvature

Beam cross-section

Neutral axis

w

Fig. 3.13. Timoshenko beam assumptions


κ can be expressed as follows

γ =∂w

∂x− β κ =

∂β

∂x.

Therewith, the total potential energy Epot computes as (see e.g., [16])

Epot =EmI

2

L∫0

(∂β

∂x

)2

dx+GAk

2

L∫0

(∂w

∂x− β

)2

dx+

L∫0

pw dx . (3.139)

In (3.139), Em denotes the elasticity modulus, G the shear modulus, A = btthe cross sectional area with t the thickness and b the width of the beam,I the moment of inertia (I = bt3/12), k the shear correction factor and pthe external load. Therein, the first term defines the bending energy, thesecond term the shear energy and the last one the potential of the load. Bynormalizing Epot and neglecting for the further considerations the load term,we obtain

Epot =

L∫0

(∂β

∂x

)2

dx+1t2

L∫0

(∂w

∂x− β

)2

dx . (3.140)

By applying the variation of the potential energy Epot, we arrive at the weakformulation for the Timoshenko beam, which will be similar to (3.138) with

B(β,w) =∂w

∂x− β .

From (3.140) we can conclude that for the case of a very thin beam (t << 1),the term B(β,w) has to approach zero. This is actual the case, since the sheardeformation will approach zero, e.g., ∂w/∂x = β with γ = 0.

For the finite element method, we subdivide our computational domain[0, L] into finite elements of equal mesh size h. Choosing linear basis functions,e.g., c1x+ c0, it is easily shown that the following inequality holds

x0+h∫x0

(c1x+ c0)2 dx ≥ h3

3c31 =

h2

3

x0+h∫x0

c21 dx .

With this result, we obtain for the terms in (3.140) the following relation

x0+h∫x0

(∂wh

∂x− βh

)2

dx ≥ h2

3

x0+h∫x0

(∂βh

∂x

)2

dx .

This clearly shows us that with linear basis functions a strong locking willoccur, and the FE solution will result in too small displacements.


In the following, we will investigate efficient solution approaches for treat-ing the locking effect. Therewith, we will discuss two very popular methods,namely the method of incompatible modes and the method of enhanced assumedstrain. Furthermore, we will show that locking still occurs for second-orderbasis functions and we will apply a balanced reduced and selective integrationtechnique.

3.8.1 Incompatible Modes Method

The method of incompatible modes (IM) has been first introduced in [216]and was enhanced for general shaped bilinear quadrilateral elements in [207].It is one of the most popular methods in computational mechanics to avoidshear, as well as volumetric locking. The key idea is based on extending theshape functions of quadrilateral (hexahedron) elements to account for higher-order modes. The additional degrees of freedom are kept internal and nocompatibility of these displacements are taken into account. Therewith, theseincompatible displacements can be eliminated at the element level, so that theglobal number of degrees remain the same compared to standard quadrilateral(hexahedron) elements in structural mechanics.

The mechanical displacement u for the IM method is extended by incom-patible displacements α

u ≈ uh + αh =nen∑a=1

Naua +nIM∑a=1

NIMa αa . (3.141)

For a bilinear quadrilateral element, the number of additional shape functionsnIM is two. They are defined by

N IM1 = 1 − ξ2 N IM

2 = 1 − η2 ,

and for a hexahedron we extend the standard shape functions by

N IM1 = 1 − ξ2 N IM

2 = 1 − η2 N IM3 = 1 − ζ2 .

Therewith, in addition to the standard differential operator Bua (see (3.84)),

we define the differential operator BIMa containing the incompatible modes

BIMa =

⎛⎜⎜⎝

∂N IMa

∂x 0 0 0 ∂N IMa

∂z∂N IM

a

∂y

0 ∂N IMa

∂y 0 ∂N IMa

∂z 0 ∂N IMa

∂x

0 0 ∂N IMa

∂z∂N IM

a

∂y∂N IM

a

∂x 0

⎞⎟⎟⎠

T

. (3.142)

This will lead to the following system of equations on the element level(ke

uu keuα

keαu ke

αα

) (ue

αe

)=

(fe

0

)(3.143)


with

keuu =

∫Ωe

(Bu)T [c]Bu dΩ keuα =

∫Ωe

(Bu)T [c]BIM dΩ

keαα =

∫Ωe

(BIM

)T[c]BIM dΩ .

From (3.143) we can compute the incompatible displacements as

αe = − (keαα)−1 ke

αu ue (3.144)

and obtain

keue = fe (3.145)

ke = keuu − ke

uα (keαα)−1 ke

αu . (3.146)

Now, as pointed out in [207], the straightforward application of the IMmethod as described above does not pass the so-called patch test. This test is anecessary requirement for displacement-type elements in structural mechanicsto assess convergence. Shortly speaking, the patch test requires αe to be zero,whenever ue corresponds to a rigid body motion or constant strain condition(for a detailed discussion we refer to [226]). Now, let us define ue

0 the set ofdisplacements corresponding to one of the above cases, then we obtain from(3.144)

αe0 = − (ke

αα)−1 keαu u

e0 .

Since keαα is positive definite, we arrive for αe

0 to be zero at the followingrelation

keαuu0 =

∫Ωe

(BIM

)T[c]Bu u0 dΩ = 0 . (3.147)

The assumption on u0 results for the term [c]Bu u0 in some constant stress.Consequently, the requirement (3.147) reduces to∫

Ωe

BIM dΩ = 0 .

Investigating this relation for the bilinear quadrilateral element, we obtain(see Sec. 2.3.6)


∫Ωe

BIM1 dΩ = 2

1∫−1

1∫−1

⎛⎜⎜⎝

−ξy,η 0

0 ξx,η

ξx,η −ξy,η

⎞⎟⎟⎠ dξ dη (3.148)

∫Ωe

BIM2 dΩ = 2

1∫−1

1∫−1

⎛⎜⎜⎝

ηy,ξ 0

0 −ηx,ξ

−ηx,ξ ηy,ξ

⎞⎟⎟⎠ dξ dη . (3.149)

The derivatives of the global coordinates x = (x, y)T w.r.t. to the local coor-dinates ξ = (ξ, η)T compute as

x,ξ =nen∑a=1

ξa(1 + ηaη)xa (3.150)

x,η =nen∑a=1

ηa(1 + ξaξ)xa . (3.151)

Therewith, whenever the geometric shape of the quadrilateral element is arectangle or parallelogram, (3.150) and (3.151) are constant, and (3.148) willintegrate to zero. The simple but effective idea proposed in [207] to fulfillthe condition even in the general case is to evaluate (3.150) and (3.151) atξ = η = 0. A generalization of this idea can be found in [102] and [24].

3.8.2 Enhanced Assumed Strain Method

Perhaps the most popular method to overcome locking effects in structuralmechanics is the method of enhanced assumed strain introduced in [199]. Themethod is based on the Hu-Washizu principle, eliminating there the mechan-ical stress and resulting in a nonconforming method. Within the Hu-Washizuprinciple, all three mechanical fields (displacement, strain and stress) remainin the weak (variational) formulation, which reads as follows (already thediscrete weak form): Find uh, Sh and σh for given a volume force fh, suchthat ∫

Ω

(ηh)T (

[c]Sh − σh)

dΩ = 0 (material law) (3.152)

∫Ω

(Bvh

)Tσh dΩ −

∫Ω

vh · fh dΩ = 0 (force balance law) (3.153)

∫Ω

(τh)T (

Buh − Sh)

dΩ = 0 , (kinematics) (3.154)

for all test functions ηh, vh and τh (equivalent to Sect. 3.7.1, we have set pos-sible prescribed boundary stresses σn to zero). By introducing the enhancedstrain Sh

EAS as follows


Sh = Buh + ShEAS

and substituting it into (3.154), we obtain the following important relation∫Ω

(τ h)T

ShEAS dΩ = 0 .

Moreover, we setσh = [c]

(Buh + Sh

EAS

)and choose ηh from the same function space (space of enhanced gradients) asSh

EAS, so that ∫Ω

(ηh)T

σh dΩ = 0 .

Therewith, the discrete stresses σh and enhanced strains ShEAS are energy

orthogonal. This allows us to rewrite the weak formulation of the Hu-Washizuprinciple as∫

Ω

(Bvh

)T[c]Buh dΩ +

∫Ω

(Bvh

)T[c]Sh

EAS dΩ =∫Ω

vh · fh dΩ

∫Ω

(ηh)T

[c]Buh dΩ +∫Ω

(ηh)T

[c]ShEAS dΩ = 0 . (3.155)

The above discrete weak formulation of the assumed enhanced strain methodis fully equivalent to a mixed finite element formulation (see e.g., [29]).

In order to satisfy the above mentioned orthogonality condition, we choosethe following ansatz for Sh

EAS

ShEAS =

|J0||J | B

EAS α = BEAS α

with |J | the determinant of the Jacobian and |J0| = |J |ξ=η=0. In [199], wherethe assumed enhanced strain has been originally introduced, BEAS computesfor a bilinear quadrilateral as

BEAS =

⎛⎜⎜⎝ξ 0

0 η

ξ η

⎞⎟⎟⎠ .

For a detailed discussion on the choice of BEAS for quadrilateral as well ashexahedron elements we refer to [7] and [39].

Using the same ansatz for ηh, we may rewrite (3.155) on the element levelin matrix form

3.8 Locking and Efficient Solution Approaches 85(ke

uu keuα

keαu ke

αα

) (ue

αe

)=

(fe

0

)(3.156)

with

keuu =

∫Ωe

(Bu)T [c]Bu dΩ keuα =

∫Ωe

(Bu)T [c] BEAS dΩ

keαα =

∫Ωe

(BEAS

)T

[c] BEAS dΩ .

Similar to the method of incompatible modes, we can perform a static con-densation on the element level, and arrive at the modified element stiffnessmatrix

ke = keuu − ke

uα (keαα)−1 ke

αu .

3.8.3 Balanced Reduced and Selective Integration

The approach of a balanced reduced and selective integration technique ismotivated by developments for the Mindlin–Reissner plate by [10] as wellas [40]; see also [28]. We apply selective reduced integration only to a portionof the shear term (for the Timoshenko beam this shear term is the secondterm in (3.140)). The amount of the portion is derived by a scaling argument.In this way we obtain a robust finite element treatment and do not encounterzero-energy modes or checkerboard modes as in a pure reduced and selectiveintegration technique (see e.g., [99]), while carrying for the elimination oflocking.

We consider a 3D body with thickness 2t. Since t is small in applicationsof interest, we take special care for avoiding locking effects. To this end wewill compare the trial functions later with those of a well-established Mindlin–Reissner plate model. The finite element functions with quadratic polynomialsin z are written with orthogonal polynomials in z as

ui(x, y) = Ui(x, y) + zθi(x, y) + (z2 − t2

3)φi(x, y), i = 1, 2, 3. (3.157)

For convenience, we also write x1 = x, x2 = y and x3 = z when evaluatingthe strains Sik = 1

2 (∂ui/∂xk + ∂uk/∂xi). In particular,

Si3 =12

(θi +

∂U3

∂xi

)+

12z

(2φi +

∂θ3∂xi

)+

12(z2 − t2

3)(∂φ3

∂xi

), i = 1, 2.

(3.158)By integrating over the thickness, we obtain


t∫−t

S2iidz = 2t

(∂Ui

∂xi

)2

+23t3(∂θi

∂xi

)2

︸︷︷︸MR-plate

+845t5(∂φi

∂xi

)2

, i = 1, 2,

t∫−t

S212dz =

12t

(∂U1

∂x2+∂U2

∂x1

)2

+16t3(∂θ1∂x2

+∂θ2∂x1

)2


+245t5(∂φ1

∂x2+∂φ2

∂x1

)2

,

t∫−t

S233dz = 2t θ23 +

83t3 φ2

3,

t∫−t

S2i3dz =

12t

(θi +

∂U3

∂xi

)2


+16t3(

2φi +∂θ3∂xi

)2

+245t5(∂φ3

∂xi

)2

i = 1, 2.

The terms that are encountered in the Mindlin–Reissner plate theory areemphasized by the mark ’MR-plate’.

The internal stored energy of the solution is proportional to t3. We avoidlocking if the terms with a factor proportional to t do not spoil the finiteelement solution. It follows from Korn’s inequality that the gradients of U1

and U2 contribute to the energy if they are not small. Therefore, U1 and U2

must be small of order t. The same holds for θ3. This is consistent with thefact that these terms are set to zero in plate theory.

The critical terms are the shear terms with Si3, where the first term in theintegral is relaxed in finite element computations unless polynomials of highorder are used.

We first consider the case that only polynomials are chosen that are linearin z, i.e., we have

φi = 0, i = 1, 2, 3,

in (3.157). In this case the shear term is softened as follows. The full integralis incorporated with a factor

α =t2

h2 + t2, (3.159)

while selective reduced integration is applied to the rest of the shear term,i.e., to the portion

β = 1 − α =h2

h2 + t2. (3.160)

Specifically, quadrature formulas with 1 point in the x, y-plane are used. Thisprocedure is equivalent (on parallelograms) to incorporating only the integralof the squared mean-value of θi +∂U3/∂xi on each element. The idea to applya softening only to a portion of the shear term in order to save coercitivity was


first suggested by [10]. The factors α and β are fixed following [40]. The shearterm is a singular perturbation in comparison to the other terms of the internalstored energy. The importance of keeping the coercivity and not reducing thefull term was discussed in [29] in the framework of saddle point problems withpenalty terms. So our approach differs from the typical application of the EASconcept; see e.g., [176].

The correct softening is more involved if quadratic polynomials in z arepresent. Only the first term in

∫S2

i3 is to be reduced, but the second termmust not. This can be realized by different implementations.

The most direct way is to keep the three terms in∫S2

i3 separated duringthe assembling and the whole finite element computation. So it is possible toapply the selective reduced integration only to the critical term, i.e., the firstone.

An alternative is a separation of the even and the odd functions in z, andit goes with φ3 = 0. Recalling (3.158), we see that a softening of the even partof Si3 is required. Note that

+t∫−t

S2i3(x, y, z)dz =

12

+t∫−t

(Si3(x, y, z) + Si3(x, y,−z) )2 dz

+12

+t∫−t

(Si3(x, y, z) − Si3(x, y,−z) )2 dz. (3.161)

A separation of the first and the second term is performed here via a symmetryargument. Only a portion of the even part (with the weight β) is incorporatedinto the selective reduced integration.

A third possibility provides the EAS method. We replace the standardtrial functions for Si3 by

Si3 + Si3

with the enhanced strain being in each element of the form Si3 = a(x−xe)+b(y − ye), where xe and ye are the (local) coordinates of the center of theelement. The mean value of the enhanced strain in each element is zero andyields an appropriate softening. Since it does not depend on z, it acts only onthe even part of the shear term. Although one finds in literature investigationswith the EAS method applied to the complete shear term, the extension to aportion with a given factor is straightforward.

The shear components are proportional to the thickness t of the structure,whereas the bending terms scale with t3. We decompose the material tensor[cE ] with the scaling from (3.159) and (3.160) as follows


[cE ] = [cE ]α + [cE ]β (3.162)

[cE ]α =

⎛⎜⎜⎜⎜⎜⎜⎝

c11 c12 c13 0 0 0c21 c22 c23 0 0 0c31 c32 c33 0 0 00 0 0 αc44 0 00 0 0 0 αc44 00 0 0 0 0 c66

⎞⎟⎟⎟⎟⎟⎟⎠ , α =

t2

h2 + t2, (3.163)

[cE ]β =

⎛⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 βc44 0 00 0 0 0 βc44 00 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎠ , β =

h2

h2 + t2. (3.164)

In (3.163) and (3.164) h denotes the largest side length and t the smallest sidelength (for thin structures t corresponds to the thickness) of the elements.Therewith, the element stiffness matrix ke

u of the mechanical part computesas

keu =

14∑i=1

BTi [cE ]α B |Ji|wstd

i

+12

9∑i=5

(Bi+5 − Bi)T [cE ]β (Bi+5 − Bi) |Ji|wstd

i

+12

(B3 + B1)T [cE ]β (B3 + B1) |J1|wred

1

+ BT2 [cE ]β B2 |J2|wred

2 , (3.165)

where |Ji| denotes the Jacobian determinant at integration point i. As can beseen from (3.165), the first two terms are integrated by the standard scheme(see Tab. 3.3) and the last two terms by the reduced scheme (see Tab. 3.4). Inparticular, only the symmetrical part of the critical shear term is delt with byreduced integration, and the decomposition (3.161) is used for this purpose.

To test the developed method, we will compare our results with those ofthe Mindlin–Reissner plate bending element of Hughes and Tezduyar [100](referred to as MR HT) and the Mindlin–Reissner plate bending element ofSimo and Rifai (referred to as MT SR). The latter are based on the methodof enhanced assumed strains [199]. Contrary to these two formulations, weperform a full 3D-analysis of the structures with hexahedron elements of 2ndorder. Specifically, we apply our new approach with balanced reduced inte-gration (referred as present) as well as standard finite elements that do notcontain precautions to locking (referred as standard).


Table 3.3. Points and weights for the standard integration

ξi ηi ζi wstdi

0.7958 0.0 0.0 0.8864-0.7958 0.0 0.0 0.8864

0.0 0.7958 0.0 0.88640.0 -0.7958 0.0 0.88640.0 0.0 -0.7958 0.8864

0.7588 0.7588 -0.7588 0.33520.7588 -0.7588 -0.7588 0.3352

-0.7588 0.7588 -0.7588 0.3352-0.7588 -0.7588 -0.7588 0.3352

0.0 0.0 0.7958 0.88640.7588 0.7588 0.7588 0.33520.7588 -0.7588 0.7588 0.3352

-0.7588 0.7588 0.7588 0.3352-0.7588 -0.7588 0.7588 0.3352

Table 3.4. Points and weights for the reduced integration

ξi ηi ζi wredi

0 0 −1/√

3 8/30 0 0 8/3

0 0 1/√

3 8/3

The three structures of investigation are a square, a rhombic, and a circu-lar plate. The data is chosen as described in [199]. In these examples, a simplysupported boundary condition is used on the lateral boundary. This meansin our displacement formulation that we set the mechanical displacement inthickness direction for all nodes along the bottom line of the outer boundaryto zero (see e.g. Fig. 3.14). If a symmetry of the plate is exploited, the com-putation needs only be performed on a half or a quarter of the plate and thedisplacement in the direction of the normal is set to zero on the ’symmetryboundaries’.

In a first numerical example, we consider the bending of a square plate. Theplate has a length of L = 10 m, a thickness of 0.1 m and is simply supported.The modulus of elasticityEm and Poisson ratio νP are chosen to be 10.92 N/m2

and 0.3, and the plate is loaded by a uniform pressure p of 1 N/m2. We usesymmetry boundary conditions in order to just model a quadrant. Figure 3.14displays the mesh using 4 by 4 elements in the plane and one element in thethickness direction.

Table 3.5 shows the numerical results. [A value of 4.0644 · 104 is reportedin [199] for the result of a series.] Moreover the stiffness of the 3D model is 0.5−


Fig. 3.14. Mesh for rectangular plate (due to symmetry, only one fourth of thestructure is meshed)

Table 3.5. Bending of a square plate (all meshes just have one finite element inthickness direction)

center displacement × 104

mesh MR HT MT SR standard present

2 x 2 3.9712 3.9712 3.6688 4.00364 x 4 4.0439 4.0436 4.0438 4.17668 x 8 4.0593 4.0593 4.0711 4.1497

16 x 16 4.0632 4.0632 4.0831 4.117332 x 32 4.0913 4.103564 x 64 4.0947 4.1025

1% smaller than that of the plate. This is explained by the additional degreesof freedom of the 3D model. It is not an effect of the reduced integration sincethe standard finite elements show the same effect asymptotically.

The 30o skew plate, as displayed in Fig. 3.15, has a side length of 100 m,thickness of 1 m and is loaded by a uniform pressure of 1 N/m2. The modulusof elasticity Em is chosen to be 103 N/m2 and the Poisson ratio νP to 0.3.All along the boundary the plate is simple supported, which means for a

Fig. 3.15. Mesh for rhombic plate


displacement formulation setting the mechanical displacement in thicknessdirection to zero. The results of our computations as well as the one reportedin [199] are listed in Tab. 3.6.

Table 3.6. Bending of a rhombic plate (all meshes just have one finite element inthickness direction)

center displacement × 10−2


4 x 4 3.8803 3.9841 1.9628 3.02888 x 8 4.1565 4.2727 3.0536 3.8631

16 x 16 4.3883 4.4668 3.8283 4.321732 x 32 4.3270 4.525864 x 64 4.5474 4.6593

128 x 128 4.6460 4.6965

There is the well-known complication that we have not only the danger oflocking, but also less regularity at the 120o corner. Therefore, the convergenceof the 3D finite element computations is slower. [Note that the FE resultsof the plate models do not reflect the reported result of a series which is4.455 · 10−2.]

For the circular plate example we use the same material parameters as forthe square plate above. The radius is chosen to be 5 m, the thickness 0.1 m,and the plate is loaded by a uniform pressure of 1 N/m2. The plate is simply

Fig. 3.16. Mesh for circular plate

supported on its boundary. Again, we model just one quadrant of the platefor symmetry reasons. The thickness of the plate will be discretized for thecomputation by one element, and we will use isoparametric elements in theplane as shown in Fig. 3.16. Since isoparametric elements of second degreeare used, the curved boundary is approximated by quadratic polynomials. –Table 3.7 contains the numerical results.


Table 3.7. Bending of a circular plate (all meshes just have one finite element inthickness direction)

center displacement × 104


12 elements 3.9070 3.6966 3.7595 3.878848 elements 3.9649 3.9140 3.8366 3.8784

192 elements 3.9789 3.9664 3.8640 3.8741768 elements 3.9822 3.9791 3.8696 3.8726

4

Electromagnetic Field

4.1 Maxwell’s Equations

The full system of partial differential equations describing the electromagneticfield was published for the first time by James Clerk Maxwell in his work ATreatise on Electricity and Magnetism, Vol. I,II [153, 154] in the year 1862.He based his theory on the work and experiments of Ampere, Gauss, andFaraday. His great contribution lies in the unification of the different equationsto a set of partial differential equations. The introduction of the displacementcurrent, which generalizes Ampere’s law, allowed him to foresee the physicalphenomena of the propagation of electromagnetic waves.

In general, we distinguish two domains in electromagnetism, both are ofcourse included in Maxwell’s equations:

• The high-frequency domain, which includes the study of electromagneticwaves and propagation of energy through matter. In general, we will de-fine high-frequency domains, as domains, where the displacement currentscannot be neglected.

• The low-frequency domain includes the major part of electromagnetic de-vices like motors, relays or transformers. These are all applications atfrequencies below a few tens of kHz. Strictly speaking, any application inwhich displacement currents can be neglected is a low-frequency applica-tion. In these domains, corresponding to the quasistatic case, we can, ingeneral, study electric and magnetic fields as separate quantities. At highfrequencies, the electric and magnetic fields are interdependent.

According to Fig. 4.1 we can divide electromagnetism into different cases,where each case represents a particular aspect of Maxwell’s equations. Theseequations are a set of partial differential equations, linear in space and time,applied to electromagnetic quantities. When electromagnetic fields interactwith materials, the equations can assume non-linear forms. The electromag-netic quantities involved in Maxwell’s equations are:

94 4 Electromagnetic Field

Fig. 4.1. Classification of Maxwell’s equations [104]

Notation Unit Description

E (V/m) electric field intensityD (As/m2) electric flux density (electric induction)H (A/m) magnetic field intensityB (T) magnetic flux density (magnetic induction)J (A/m2) current densityqe (As/m3) charge densityM (T) magnetizationP (As/m2) electric polarization

In addition, we define the following material parameters

• Magnetic permeability µ (Vs/Am)• Magnetic reluctivity ν = 1/µ (Am/Vs)• Electric permittivity ε (As/Vm)• Electric conductivity γ (1/Ωm = S/m)

The four partial differential equations (PDEs), stated as Maxwell’s equa-tions in differential form, fully describe all phenomena in electromagnetic fields

∇ × H = J +∂D∂t

(4.1)

∇ × E = −∂B∂t

(4.2)

∇ · D = qe (4.3)∇ ·B = 0 . (4.4)

In addition, to get solvability of the four PDEs, the following constitutiveequations are introduced

4.1 Maxwell’s Equations 95

J = γ(E + v × B) (4.5)D = εE = ε0E + P (4.6)B = µH = µ0H + M . (4.7)

It has to be stated that (4.5) is a direct consequence of the force relationdefined by Lorentz

fV = qe(E + v × B) . (4.8)

Thus, the total electromagnetic force acting on the electric volume charge qeis given by a term defined by the product of the electric field E and a volumecharge qe at rest and a term defined by the magnetic induction B and a volumecharge qe at velocity v.

As already mentioned, Maxwell’s equations are based on experiments andconcluding empirical laws stated by Ampere, Faraday, and Gauss.

Law of AmpereWith the help of experiments, Ampere was able to prove in the year 1820that an electric current (dc or ac) generates a magnetic field (Fig. 4.2). This

Fig. 4.2. Experiments of Ampere

empirical law states that the line integral of the magnetic field intensity alonga closed contour is equal to the electric current I∮

C(I)

H · ds = I . (4.9)

The direction of the magnetic field intensity H is related to the direction ofthe electric current I via the right-hand rule. For general distributed currentswithin a cross section Γ with boundary curve C, the law of Ampere reads as∮

C

H · ds =∫Γ

J · dΓ , (4.10)


and by applying Stokes’ theorem (see Appendix B.9), we obtain∫Γ

(∇ × H) · dΓ =∫Γ

J · dΓ . (4.11)

Since this relation has to be independent of the surface, the law results in thedifferential form, which reads as follows

∇ × H = J . (4.12)

Maxwell found that (4.12) does not fully describe the general case andadded the so-called displacement current density ∂D/∂t in order to obtain adescription for the general case (see e.g., [205]). Since we are mainly concernedin the quasistatic case, we will not discuss this term in detail.

Law of FaradayAccording to his experiments, Faraday postulated that the relation betweenthe induced voltage uind in an open conductive loop and the magnetic flux φthrough it (Fig. 4.3), is given by

uind = − dφdt. (4.13)

The magnetic flux φ is defined by

φ =∫Γ

B · dΓ , (4.14)

with Γ the cross section passed by the magnetic induction B. A change of the

Fig. 4.3. Loop within a time-varying magnetic field

magnetic flux with respect to time can be obtained in two different cases. First,the loop is at rest and the magnetic field is time dependent, and secondly,


the loop changes its position as a function of time and the magnetic field isconstant.

Let us consider a surface Γ on which E and B are defined, then we canrewrite (4.2) as ∫

Γ

(∇ × E) · dΓ = −∫Γ

∂B∂t

· dΓ .

Through the use of Stoke’s theorem, we obtain∮C(Γ )

E · ds = −∫Γ

∂B∂t

· dΓ ,

where C is the contour that limits the surface Γ. We now look at the appli-cation of this expression in an example as depicted in Fig. 4.4. Let us assumethat the magnetic field is concentrated in the high-permeable cylinder andthe magnitude of B is constant at each cross section of the cylindrical core.Furthermore, an electrical conductive loop, electrically isolated from the core,encloses the core such that it forms a contour C(Γ ).

Fig. 4.4. Induced voltage in an open conductive loop, caused by a time-varyingmagnetic field

First, note that assuming the direction of dΓ as in Fig. 4.4b, the directionof ds must be as shown (right-hand rule). With the direction of ∂B/∂t asshown in Fig. 4.4b, we obtain

−∂B∂t

· dΓ < 0 ; E · ds < 0 ,

which makes E point in the direction opposite to ds.


The circulation of E along C leads to an electromotive force (emf), de-tectable by a voltmeter as an electric potential difference. The electromotiveforce computes as

uind =∮C

E · ds . (4.15)

It has to be noted that the electric field due to the time variation of themagnetic field is solenoidal, and thus, the measured voltage depends on theway the electric voltmeter is connected to the open loop [95].

Because of the fact that B only depends on time and not on its position,we conclude that ∂B/∂t = dB/ dt. Since the integration over the surface Γand the time derivative are independent operations, we can write

−∫Γ

∂B∂t

· dΓ = − ddt

∫Γ

B · dΓ = − dφdt,

and obtain the expression

uind = − dφdt. (4.16)

Now let us consider the case of a conductive slab positioned in a uniformand time-independent magnetic field B (see Fig. 4.5). As already mentioned

Fig. 4.5. Electrically conductive slab moving with velocity v in a time-constantmagnetic field B

in this section, the total electromagnetic force on a charge is given due to anelectrostatic field acting on a charge in rest, and a magnetic field acting ona moving charge. Therefore, an electromagnetic force Fmag will act on theelectric charges Qe within the conductive slab (see Fig. 4.5)

Fmag = Qe (v × B) , (4.17)


and the charges will distribute on the surface of the slab as depicted in Fig.4.5. Due to this separation of positive and negative charges, an electrostaticfield E arises, which will result in the restoring force

Fel = QeE . (4.18)

By assuming the equilibrium state, we obtain

Fmag + Fel = 0 (4.19)E = −v × B . (4.20)

If we now connect a resistor R to the slab in parallel by using a flexible cable(so that the resistor is at rest), then the electric field E will result in an electriccurrent i. Therefore, the term v×B is called the motional electromotive force.By neglecting the resistance of the connections, the potential difference acrossthe resistor, which we define by the voltage u, is given by

u = Ri =∮

E · ds = l · (v × B) , (4.21)

with l the length of the conductive slab (direction points into the direction ofthe current). Hence, the mechanical power Pmech we apply to the slab (sincewe move it with velocity v) is converted into electric power Pel

Pmech = Fmech · v = Ri2 = Pel . (4.22)

Simply speaking, we have the case of an electric generator with an ohmic load.

Law of GaussBy means of experiments, Gauss postulated that the amount of electric fluxdensity D crossing a closed surface Γ is equal to the total electric charges qewithin the volume Ω (Fig. 4.6)∮

Γ

D · dΓ =∫Ω

qe dΩ . (4.23)

Applying the divergence theorem (Appendix B.6), the above equation readsas ∫

Ω

(∇ ·D) dΩ =∫Ω

qe dΩ (4.24)

and, since the law has to hold for any volume Ω, we end up at the differentialform

∇ · D = qe . (4.25)

Therefore, the sources of the electric field are given by the electric charges,and according to this relation the electric field is irrotational.


Fig. 4.6. Sources of the electric field

Solenoidal magnetic fieldSince magnetic materials cannot be split into a piece only having a north poleor a south pole; no magnetic charges exist, and the magnetic field is alwayssolenoidal (closed field lines). This property of the magnetic field can be statedby ∮

Γ (Ω)

B · dΓ = 0 , (4.26)

which means that the magnetic flux is conservative. Applying the divergencetheorem ∫

Ω

(∇ ·B) dΩ = 0 ,

and generalizing it for any volume Ω, we obtain the differential form

∇ ·B = 0 . (4.27)

It is very interesting to note that this equation, which has been introducedby Maxwell, is necessary to guarantee that no magnetic charges exist. If weapply the operator ∇· to (4.2)

∇ · (∇ × E) = −∇ · ∂B∂t

(4.28)

we achieve the following relation

∂

∂t(∇ ·B) = 0 . (4.29)

The solution of this equation without considering the relation ∇·B = 0 wouldresult in ∇ · B = const. = 0, and therefore, in the existence of magneticcharges.

4.2 Quasistatic Electromagnetic Fields 101

4.2 Quasistatic Electromagnetic Fields

The most important case for electromagnetic sensors and actuators is the qua-sistatic case often referred to as the eddy current case. For quasistatic electro-magnetic fields we can neglect the displacement current density term ∂D/∂t,which transforms Maxwell’s equations including the constitutive equations tothe following subset

∇ × H = J (4.30)

∇ × Es = −∂B∂t

(4.31)

∇ · B = 0 (4.32)J = γ(Etotal + v × B) (4.33)B = µH = µ0H + M . (4.34)

In (4.33) Etotal denotes the total electric field intensity given as a sum of anyirrotational part Ei (defined by (4.3)) and of any solenoidal part Es

Etotal = Ei + Es . (4.35)

Now, we can rewrite (4.33) as

J = Ji + γ(Es + v × B) , (4.36)

with Ji an impressed current density due to a given electric potential difference(current- or voltage-loaded coil).

4.2.1 Magnetic Vector Potential

According to (4.32) the magnetic field is solenoidal and therefore can be de-scribed by the curl of a vector

B = ∇ × A . (4.37)

The vector A is called the magnetic vector potential. This ansatz results for(4.2) in the following relation

∇ × Es = − ∂

∂t(∇ × A)

∇ ×(Es +

∂A∂t

)= 0 . (4.38)

Since we are concerned with eddy current problems, the electric field is puresolenoidal. Thus, we set Es = −∂A/∂t and arrive at

∇ × ν∇ × A = Ji − γ∂A∂t

+ γ(v × ∇ × A) . (4.39)


It has to be noted that the introduction of the magnetic vector potential Aleads to non-uniqueness of the solution. This can be seen very easily by addingthe gradient of a scalar function to A, which leads to a new vector A∗

A∗ = A + ∇ψ . (4.40)

Since the curl of any grad is zero, we arrive at

B = ∇ × (A∗ − ∇ψ) = ∇ × A∗ . (4.41)

To obtain a unique solution, we have to gauge the magnetic vector potentialA, which is, e.g., achieved by a Coulomb gauge [22]

∇ · A = 0 . (4.42)

In the 2D case the magnetic vector potential is given by

A = A(x, y)ez (4.43)

and therefore (4.42) is automatically guaranteed. The same holds for the ax-isymmetric case

A = A(r, z)eϕ . (4.44)

Furthermore, it has to be noted that we are interested in the quantityB computed via ∇ × A, and therefore, no physical need exists to gauge themagnetic vector potential A.

4.2.2 Skin Effect

Supplying an electric conductor with cross section ΓC by an alternating cur-rent with frequency fC, we will find that the higher the frequency, the morethe current will be concentrated near the surface of the conductor and tendto zero towards the center of the conductor. To investigate this effect, let ussimplify the setup to an infinitely extended half-plane as depicted in Fig. 4.7.Therefore, the current density J and, respectively, the electric field intensityE only depend on the coordinate z and point towards the x-direction. Using(4.12), we obtain

−∂Hy

∂z= Jx (4.45)

∂Ex

∂z= −∂By

∂t. (4.46)

By applying the operation ∂/∂z to (4.45) and using (4.46) as well as theconstitutive laws (4.5) and (4.7), we arrive at

−∂2Hy

∂z2=∂Jx

∂z= γ

∂Ex

∂z= −γµ∂Hy

∂t. (4.47)

4.2 Quasistatic Electromagnetic Fields 103

Fig. 4.7. Infinite extended half-plane

Equation (4.47) is called the diffusion equation, because it models the diffusionof electromagnetic energy into the conductor. For the time-harmonic case withangular frequency ω, (4.47) gets the form

Hy = Hy(z) ejωt (4.48)

∂2Hy

∂z2= jωγµHy , (4.49)

which has the general solution

Hy(z) = C1 e√

jωγµz + C2 e−√

jωγµz . (4.50)

Since for physical reasons, the magnetic field decreases with distance z, theconstant C1 has to be zero. By using the relation

√j =

1 + j√2

and the boundary condition Hy(z = 0) = H0, we arrive at the solution forthe magnetic field inside the infinite half-plane (using (4.48))

Hy(z) = H0 e−z/δ ej(ωt−z/δ) (4.51)

Hy(z) = Re(Hy) = H0 e−z/δ cos(ωt− z/δ) . (4.52)

In (4.51) δ denotes the so-called skin (penetration) depth and as can be seenfrom (4.50), it is a function of frequency and material properties

δ =1√πfγµ

. (4.53)

Additionally, we obtain solutions for B, J, and E

By(z) = B0 e−z/δ cos(ωt− z/δ) (4.54)

Jx(z) = J0 e−z/δ cos(ωt− z/δ) (4.55)Ex(z) = E0 e−z/δ cos(ωt− z/δ) . (4.56)


Thus, the magnitudes of the electromagnetic quantities not only decay expo-nentially but also their phase changes with z/δ, so that at a certain depth,the field quantities are even in opposite directions.

Concluding, it has to be noted that the expression for the skin depth δ isnot restricted to this simplified example but can be considered as a generalrelation for eddy current problems (see e.g., [142, 201]).

4.3 Electrostatic Field

For the static case, the physical quantities do not depend on time, andMaxwell’s equations with constitutive laws can be separated into a magneticand an electric subsystem. For the electric subsystem, called the electrostaticcase, we obtain the following system of partial differential equations

∇ × E = 0 (4.57)∇ ·D = qe (4.58)

D = εE . (4.59)

Since the curl of the electric field intensity E is zero, we can express it by thegradient of a scalar potential Ve, which is called the electric scalar potential

E = −∇Ve . (4.60)

Thus, by combining (4.58), (4.59), and (4.60), we arrive at

−∇ · ε∇Ve = qe , (4.61)

which describes the electrostatic field within any media characterized by thematerial quantity ε.

4.4 Material Properties

4.4.1 Magnetic Permeability

The constitutive law within the magnetic field is given by (4.7), which relatesthe magnetic induction B and the magnetic field intensity H via the magneticpermeability µ. In general, the magnetic permeability is a tensor of rank 2,but for most cases we can assume a scalar value, which can be decomposedas follows

µ = µ0µr (4.62)µ0 = 4π 10−7 (Vs/Am) . (4.63)


Therein, µr is called the relative permeability and characterizes the magneticmaterial. Often, instead of the magnetic permeability µ the magnetic reluc-tivity ν given by

ν =1µ

(4.64)

is used. In general, we differentiate between two types of magnetic materials,namely

• Soft magnetic materials, which can be classified into diamagnetic, param-agnetic, and ferromagnetic materials

• Hard magnetic materials of which permanent magnets are fabricated

By placing a diamagnetic material within an external magnetic field, theinternal magnetization M will be opposite to the external field. Therefore,the total magnetic field within such a material will be smaller than the exter-nal applied field (see Fig. 4.8(a). Within paramagnetic materials the internal

Fig. 4.8. Cube within an external magnetic field (a) diamagnetic material; (b)paramagnetic material; (c) ferromagnetic material

magnetization will be directed already in the direction of the external mag-netic field. However, the effect is small, as can be seen in Fig. 4.8(b). Onlyfor ferromagnetic materials will the resulting internal magnetization be largeand, as shown in Fig. 4.8(c), the magnetic field concentrates inside the cube.By switching off the external field, the magnetization almost disappears, andwe call it a reversible process. However, if the amplitude of the external field islarge, irreversible processes will take place, and after switching off the exter-nal field a remanent magnetic field can be detected. Therefore, the followingdefinitions of permeabilities exists:

• Reversible permeability:

µrev = lim∆H→0

∆B

∆H. (4.65)

This case is given if a large constant magnetic field is superimposed on asmall alternating field.


• Differential permeability:

µdiff =dBdH

. (4.66)

This permeability is always larger than the reversible permeability, sinceit will also contain irreversible processes.

• Starting permeability

µs =dBdH

|B=0,H=0 . (4.67)

By first applying an external magnetic field to a ferromagnetic material,the magnetic induction will rise strongly until the material is saturated. If wenow decrease the amplitude of the external applied field, we will recognizethat the magnetic induction inside the material will decrease along a curvedifferent from the initial magnetization curve and at external field zero, wewill have a remanent magnetic induction Br of the material. This effect isdue to the already discussed irreversible processes inside the body. Finally,by driving the external magnetic field to a negative maximum amplitude andback to the positive amplitude, we will measure a similar hysteresis curveas displayed in Fig. 4.9. The quantity Hc is called the coercitive magnetic

Fig. 4.9. Hysteresis and initially magnetization curve for a soft magnetic material

field intensity (force), since we have to externally apply this value in order toobtain zero magnetic induction inside the material.

In addition to the magnetic field dependency of the magnetic permeability,magnetic materials completely lose magnetization, if the temperature exceedsthe so-called Curie temperature TCurie, which can be modelled as follows

B = µ(H)H (4.68)

µ = f(T ) =µ = µ(H) : T < TCurie

µ = µ0 : T > TCurie. (4.69)


In general, Br is small for soft magnetic materials, which also means thatthe magnetization follows the external field almost with no essential retar-dation effect. This physical phenomenon is quite contrary for hard magnetic

Fig. 4.10. Typical magnetization curve of hard magnetic material

materials, which exhibit a large remanent magnetic field Br. A typical B–Hcurve is given in Fig. 4.10 and Table 4.1 summarizes the data of importantpermanent magnets.

Table 4.1. Properties of permanent magnets

Material Br (T) Hc (kA/m) (BH)max (kWs/m3)

Alnico 1.25 60 50Ferrite 0.38 240 25Ne-Fe-B 1.15 800 230

Therefore, if we consider a magnetic assembly including permanent mag-nets, we can describe the magnetic field via the magnetic vector potential bythe following partial differential equation (assuming a known magnetizationM and no motional emf)

∇ × 1µ

∇ × A = Ji − γ∂A∂t

+ ∇ × M . (4.70)

If we just consider a static field problem, and decompose the magnetic fieldintensity H into HJ generated by a current density J and HM describingthe magnetic material, we can introduce the so-called reduced magnetic scalarpotential ψm according to

H = HJ + HM (4.71)HM = −∇ψm . (4.72)

Therefore, by using (4.4) we obtain the describing equation for ψm


∇ · µ(HJ + HM ) = 0 (4.73)∇ · µ∇ψm = ∇ · (µHJ) , (4.74)

and HJ is computed via Biot–Savart’s law (see e.g., [95])

H(x′, y′, z′) =14π

∫Ω

J × er

r2dΩ , (4.75)

with (x′, y′, z′) the field (observation) point, Ω the volume of a current loadedstructure and er the unit vector pointing from the volume Ω to the field point.

4.4.2 Electrical Conductivity

In general, the conductivity is a tensor of rank 2, but for many applicationsit can be assumed to be a scalar value. Table 4.2 gives the values for somematerials. The electrical conductivity γ mainly depends on the temperature T .

Table 4.2. Electrical conductivity of some materials

Iron Aluminum Copper Steel Carbon

Conductivity (S/m) 1 × 107 3.5 × 107 5.8 × 107 5 × 106 3 × 104

In most cases the dependency is available for the specific electrical resistivityρe = 1/γ and takes the relation

ρe = ρ20

(1 + α(T − T20 oC) + β(T − T20 oC)2

), (4.76)

with α the linear and β the quadratic temperature coefficient. In (4.76) ρ20

denotes the specific resistivity at T = T20 oC. Therefore, we obtain the electricconductivity (see Fig. 4.11)

γ =γ20

(1 + α(T − T20 oC) ,+β(T − T20 oC)2), (4.77)

with γ20 = 1/ρ20.

4.4.3 Dielectric Permittivity

The modelling of dielectric materials is quite similar to that of magnetic ma-terials. Instead of the magnetization M, we define the polarization P forcharacterizing dielectric materials [95]

D = ε0εrE = ε0E + P . (4.78)

4.5 Electromagnetic Interface Conditions 109

Fig. 4.11. Dependency of the electrical conductivity of steel on temperature

Table 4.3. Relative permittivity of dielectric materials

Air Insulating Glass Water PZT 5Apaper (3-direction)

εr 1 3 6 80 3000

The scalar permittivity ε can be decomposed into the permittivity of vacuumε0 = 8.854 × 10−12 (As/Vm) and the relative permittivity εr. The materialcan be described by a scalar value εr (see Table 4.3), and therefore D, E,and P have the same direction. However, materials with a large value ofrelative permittivity exhibit a strong anisotropy and are called ferroelectrics.This is the special case for piezoelectric materials. These materials also showa hysteresis as depicted in Fig. 4.12 with Dr denoting the dielectric remanentand Ec the coercitive electric field intensity.

4.5 Electromagnetic Interface Conditions

4.5.1 Continuity Relations for Magnetic Field

According to (4.4) and by application of the divergence theorem, we obtain

∇ · B = 0∫Ω

(∇ ·B) dΩ = 0∮Γ

B · dΓ = 0 . (4.79)


Fig. 4.12. Hysteresis and initial polarization curve

At the interface between two materials with different permeabilities, we haveto evaluate the surface integral according to Fig. 4.13 for b→ 0.

Fig. 4.13. Continuity of the normal component of B

limb→0

∮Γ

B · dΓ = B1 · n1 + B2 · n2 = 0

n1 · (B1 − B2) = 0B1n = B2n (4.80)

µ1H1n = µ2H2n . (4.81)

Therefore, at an interface of changing permeability, the normal component ofthe magnetic induction B is continuous.

By using (4.30) and assuming no current density at the interface (see Fig.4.14), we obtain the continuity relation for the magnetic field intensity H

4.5 Electromagnetic Interface Conditions 111

(applying Stoke’s theorem)

Fig. 4.14. Continuity of the tangential component of H

limb→0

∫Γ

(∇ × H) · dΓ = 0

limb→0

∮H · ds = 0

H1 · s− H2 · s = 0st · (H1 − H2) = 0

H1t = H2t . (4.82)

Therefore, at an interface of changing permeability, the tangential componentof the magnetic field intensity H is continuous. By using the material relationbetween B and H we get the defining equation for the tangential componentof the magnetic induction

B1t

µ1=B2t

µ2. (4.83)

4.5.2 Continuity Relations for Electric Field

Using the integral form of (4.3)∫Ω

(∇ · D) dΩ =∫Ω

qe dΩ , (4.84)

with the relation dΩ = b dΓ and b→ 0 (see Fig. 4.15) we obtain

limb→0

qe dΩ = σe dΓ (4.85)

limb→0

∫Ω

qe dΩ =∫Γ

σe dΓ , (4.86)

with σe the surface charge along the interface. By applying the divergencetheorem


Fig. 4.15. Continuity of the normal component of D

∫Ω

(∇ · D) dΩ =∮Γ

D · dΓ , (4.87)

and performing the step for b→ 0, we arrive at

limb→0

∮Γ

D · dΓ = D1 · n1 + D2 · n2 = n1(D1 − D2) . (4.88)

Therefore, for the normal components of the electric field quantities, the fol-lowing relations have to hold

D1n = D2n + σe (4.89)ε1E1n = ε2E2n + σe . (4.90)

Fig. 4.16. Continuity of the tangential component of E

For the tangential component of the field quantities at an interface betweentwo materials with different permittivities and no time-varying magnetic field(Fig. 4.16), we have to start at (4.2) and perform similar steps as for thetangential component of the magnetic field, and obtain

4.6 Numerical Computation: Electrostatics 113

E1t = E2t (4.91)D1t

ε1=D2t

ε2. (4.92)

4.5.3 Continuity Relations for Electric Current Density

Similar to (4.80), we can rewrite (4.30) after applying the divergence on bothsides as

∇ · J = 0 , (4.93)

which results in the following interface condition

J1n = J2n . (4.94)

Assuming a jump of the electrical conductivity γ at an interface, we obtainfor the eddy current case (see Sect. 4.2.1 and v = 0)

γ1∂A1n

∂t= γ2

∂A2n

∂t. (4.95)

4.6 Numerical Computation: Electrostatics

Let us consider the situation of a domain with given electric volume chargesqe, where we want to compute the generated electrostatic field. The strongformulation of this problem will be stated as follows

Given:qe : Ω → IRε : Ω → IR .

Find: Ve : Ω → IR

−∇ · ε∇Ve = qe . (4.96)

Boundary conditionsVe = 0 on Γ = Γe .

To obtain the variational formulation, we multiply (4.96) with an appropriatetest function ω ∈ H1

0 (see Appendix C) and perform a partial integration.Therefore, the weak form reads as: Find Ve ∈ H1

0 such that∫Ω

ε∇w · ∇Ve dΩ −∫Ω

wqe dΩ = 0 , (4.97)


for any w ∈ H10 . Using standard nodal finite elements, we approximate the

continuous electric scalar potential Ve as well as the test function w by

Ve ≈ V he =

neq∑a=1

NaVea (4.98)

w ≈ wh =neq∑a=1

Nawa , (4.99)

with neq the number of equations. Therefore, (4.97) is transformed into thefollowing discrete formulation

neq∑a=1

neq∑b=1

⎛⎝∫

Ω

ε (∇Na)T ∇Nb dΩ Veb −∫Ω

Naqe(ra) dΩ

⎞⎠ = 0 . (4.100)

In matrix form, (4.100) may be written as

KVeVe = fVe, (4.101)

with

KVe =ne∧

e=1

keVe

; keVe

= [kpq] ; kpq =∫Ωe

ε (∇Np)T ∇Nq dΩ (4.102)

fVe

=ne∧

e=1

fe

Ve; fe

Ve= [fp] ; fp =

∫Ωe

Npqe(rp) dΩ . (4.103)

4.7 Numerical Computation: Electromagnetics

The numerical computation of electromagnetic fields has been performed formore than 20 years. For the domain discretization nodal as well as edge finiteelements have been used successfully. Nevertheless, in recent years inaccurateresults at material parameter interfaces in the magnetostatic as well as inthe eddy current case, and spurious modes in Maxwell’s eigenvalue problemshave been reported. Therefore, we will perform a precise investigation in theformulation and further discretization of electromagnetic fields in the eddycurrent case.

4.7.1 Formulation

Let us consider a domain Ω with boundary Γ consisting of two subdomains Ω1

and Ω2 with interface Σ12 as displayed in Fig. 4.17. The material relations aregiven by H = νB and J = γE with ν, γ the magnetic reluctivity (ν = 1/µ) and

4.7 Numerical Computation: Electromagnetics 115

Fig. 4.17. Domain with given material properties

electrical conductivity of the material. On the boundary Γ = ∂Ω the normalcomponent of the magnetic induction (B·n = 0 on Γ ) shall be prescribed withn being the normal unit outward vector on Γ . In addition, we consider thecontinuity conditions that have to be fulfilled at an interface Σ12 of changingmagnetic permeability as well as conductivity

[B · n] = B1 · n − B2 · n = 0 (4.104)[H× n] = H1 × n− H2 × n = 0 (4.105)

[J · n] = J1 · n− J2 · n = 0 . (4.106)

The partial differential equation to be solved has been derived in Sect. 4.2.1,and assuming no moving bodies, reads as

γ∂A∂t

+ ∇ × ν∇ × A = Ji . (4.107)

The boundary conditions change to n×A = 0 on Γ , and the interface condi-tions take on the form

[A× n] = 0 (4.108)[ν n× ∇ × A] = 0 (4.109)[

γ n · ∂A∂t

]= 0 . (4.110)

Therefore, the strong formulation for the eddy current case reads as follows:

Given:A0 : Ω → IRd

γ, ν : Ω → IR .

Find: A(t) : Ω × [0, T ] → IRd


γ∂A∂t

+ ∇ × ν∇ × A = Ji . (4.111)

Boundary conditionsn× A = 0 on Γ × (0, T ) .

Interface conditionsA1 × n = A2 × n on Σ12 × (0, T )

ν1 n× ∇ × A1 = ν2 n× ∇ × A2 on Σ12 × (0, T )

γ1 n · ∂A1

∂t= γ2 n · ∂A2

∂ton Σ12 × (0, T ) .

Initial conditionA(r, 0) = A0 , r ∈ Ω .

Now, multiplying (4.111) by appropriate test functions A′ and applyingGreen’s first integral theorem in vector form (Appendix B.10) will trans-form the PDE into its variational formulation, which reads as follows: FindA ∈ HΣ

0 (curl) such that∫Ω

γA′ · ∂A∂t

dΩ +∫Ω

∇ × A′ · ν∇ × A dΩ =∫Ω

A′ · Ji dΩ , (4.112)

for any A′ ∈ HΣ0 (curl) with the Sobolev space

HΣ0 (curl) = u ∈ (L2(Ω))3 | ∇ × u ∈ (L2(Ω))3,

u × n|Γ = 0,[n × u]|Σ = 0 . (4.113)

If the conductivity γ is positive, the problem has a unique solution in H0(curl),whereas for γ = 0 (globally, or only in some regions), the solution is uniqueonly up to gradient fields in the case that Ω is simply connected. Since we areinterested in the quantity B computed via ∇ × A, there is no need to gaugethe vector potential A, though.

The fact that our variational formulation (4.112) automatically fulfils theinterface conditions (4.108) - (4.110) can be shown as follows. Starting at(4.112) and performing partial integration (Green’s first integral theorem invector form, see Appendix B.10), we obtain

4.7 Numerical Computation: Electromagnetics 117∫Ω

γA′ · ∂A∂t

dΩ +∫Ω1

A′∇ × ν∇ × A dΩ +∫Σ

(A′ × ν1∇ × A) · n1︸︷︷︸A′·(n×ν1∇×A1)

dΓ

+∫Ω2

A′∇ × ν∇ × A dΩ +∫Σ

(A′ × ν2∇ × A) · n2︸︷︷︸A′·(n×ν2∇×A2)

dΓ

+∫Γ

(A′ × ν∇ × A) · n dΓ

=∫Ω

A′ · Ji dΩ (4.114)

with n1 = −n2 = n. Now, let us choose a test function A′ being concentratednear the interface Σ and zero elsewhere. Furthermore, we assume A′ to pointinto normal direction, i.e., A′ = wn with w an arbitrary scalar function. Sincethe term (n× ν∇×A) results in a vector orthogonal to n, the scalar productwith A′ is zero. Moreover, due to (4.106) and the fact that the intersectionof the support of A′ with the interior of Ω1 and Ω2 is made arbitrary small,and the interface Σ is at distance from the boundary Γ , only the first termin (4.114) remains∫

Σ

γ1wn · ∂A1

∂tdΓ =

∫Σ

γ2wn · ∂A2

∂tdΓ . (4.115)

This holds for any w, so that [γ n · ∂A

∂t

]= 0 . (4.116)

On the other hand, when we choose A′ arbitrarily tangential to Σ and thesupport of A′ as concentrated to Σ, then the volume integrals in (4.114) arevanishing, while only the surface integrals over Σ can give nonzero values∫

Σ

A′ · (n × ν1∇ × A1) dΓ =∫Σ

A′ · (n × ν2∇ × A2) dΓ

[ν n × ∇A] = 0 . (4.117)

The different asymptotic behaviors of volume and surface integrals, as it wasexploited here for deriving (4.117), can more easily be made obvious in a spe-cial situation without losing generality. Let us specify Σ as a plane orthogonalto the z-axis

Σ = (x, y, 0)|(x, y) ∈ [0, 1]2 .

Then, our proof amounts to choose


A′(x, y, z) =

⎛⎜⎜⎝wx(x, y)

wy(x, y)

0

⎞⎟⎟⎠ φε(z)

with φε(z) as a hat function having support [−ε, ε] and value 1 at z = 0(see Fig. 4.18). Now, observe that φε equals 1 on Σ for all ε, while, since

Fig. 4.18. Hat function φε(z) with support [−ε, ε]

∫ ε

−ε φε(z) dz goes to zero as ε → 0, volume integrals containing A′ in their

integrand have to vanish as ε→ 0. Note that this situation, although lookingquite simple, is sufficiently significant, since any smooth interface can locallybe smoothly transformed to a piece of a plane.

The edge finite elements [162] for the spatial discretization of (4.112) areH0(curl)-conform, and solving the resulting algebraic system of equationsleads to correct results. Nevertheless, the solution of the algebraic systemrequires special care in order to obtain an optimal solver (see e.g., [9,92]). Wesuggest to add a fictive electric conductivity γ′ to regions with zero electricconductivity to obtain a variational form, which is elliptic [191]. Of course,this fictive conductivity γ′ has to be chosen small compared to the reluctivityof the material. The proof of convergence even in the case of γ′ → 0 is givenin [180].

For the application of nodal finite elements, we have to perform additionalsteps. As shown in [74], the space HΣ

0 (curl) has the decomposition for anyconvex domain Ω

HΣ0 (curl) =

(H1

0 (Ω))3 ⊕ gradH1

0 (Ω) . (4.118)

This is equivalent to splitting the magnetic vector potential A as follows

A = w + ∇φ ∇ ·w = 0 , (4.119)

with w ∈(H1

0 (Ω))3 and φ ∈ H1

0 (Ω). The same decomposition is done for thetest function A′ = v + ∇ψ. Since Ji is assumed to be divergence free, ψ doesnot enter into the right-hand side.


To guarantee ∇ · w = 0, we may add the term∫

Ων∇ · v ∇ · w dΩ to

the variational formulation (4.112), which corresponds to a penalty formula-tion (see e.g., [78]). The variational formulation changes to: Find (w, φ) ∈(H1

0 (Ω)3, H10 (Ω)) such that∫

Ω

ν∇ × v · ∇ × w dΩ +∫Ω

ν ∇ · v∇ · w dΩ

+∫Ω

γ(v + ∇ψ) · ∂∂t

(w + ∇φ) dΩ

=∫Ω

Ji · v dΩ , (4.120)

for any (v, ψ) ∈ (H10 (Ω)3, H1

0 (Ω)). Thus, the standard continuous nodal fi-nite elements can be used to approximate both fields w and φ and we arriveat a conforming FE approximation. It should be noted that due to the de-composition of A into w and ∇φ the magnetic induction B as well as eddycurrent density Jeddy compute as

B = ∇ × A = ∇ × (w + ∇φ) = ∇ × w (4.121)

Jeddy = −γ ∂A∂t

= −γ ∂w∂t

− γ∂∇φ

∂t. (4.122)

As reported in many scientific contributions, the discretization of (4.120)with nodal finite elements leads to correct values in the eddy current case,where the permeability is constant all over the domain (see e.g., [143]). Inac-curate results have been demonstrated in the case of domains with materialsof different magnetic reluctivities (e.g., iron–air interface). Therefore, we willconcentrate on this problem and in the following investigate the magnetostaticcase (∂/∂t = 0)∫

Ω

ν∇ × v · ∇ × w dΩ +∫Ω

ν ∇ · v∇ · w dΩ

=∫Ω

Ji · v dΩ . (4.123)

In the first step we will consider a domain as displayed in Fig. 4.17, where wehave a jump of the reluctivity at the interface Σ between the subdomains Ω1

and Ω2. If we perform similar steps as for the previous variational formulation(4.112), we will see that (4.123) fulfills the following interface conditions

[ν n12 × ∇ × w] = 0 (4.124)[ν n12∇ ·w] = 0 . (4.125)


Of course, from the numerical point of view, the approximation of these inter-face conditions using linear interpolation functions will be poor, but applyingquadratic interpolation functions will give acceptable results. So, the ques-tion arises, what is the reason for the poor results discretizing (4.123) withnodal finite elements? Let us consider the case of a ferromagnetic cube em-bedded in air (see Fig. 4.19). Assuming the case ν2 → 0 (permeability is very

Fig. 4.19. Ferrormagnetic cube in air

large in the iron), we arrive at a nonconvex domain. Now, according to [46],it is known that for nonconvex domains the discretization with nodal finiteelements produces wrong solutions due to the nondensity of smooth fields.In [46] the authors could prove that by introducing a special weighting func-tion inside the divergence integral, nodal finite elements can yet be used forthe approximation. Therefore, the second term in the variational formulation(4.123) has to be changed to∫

Ω

ν ∇ · v ∇ ·w dΩ →∫Ω

ν s ∇ · v∇ · w dΩ , (4.126)

withs =

∏a∈A

rαa . (4.127)

In (4.127) A denotes the set of all re-entrant corners, ra the distance to eachre-entrant corner, and α an exponent. This means that w has to be in HY

HY = w ∈ H0(curl) | ∇ · w ∈ Y . (4.128)

By a correct choice of the weighted Sobolev space Y, the space H10 is dense

in HY , and thus, nodal finite elements do lead to correct results, and we callit the weighted regularization method.

It has to be noted that this idea can be implemented in a simple way bysetting the weighting function s to zero for finite elements near re-entrantcorners [118]. This was exactly the treatment the authors in [172] employed,where a current vector potential formulation was used. Since the current vec-tor potential is just defined in electric conductive regions, in most cases theboundary will exhibit re-entrant corners.


4.7.2 Discretization with Edge Elements

Performing an edge finite element discretization of (4.112), we first define theapproximation of the vector potential A as

A ≈meq∑b=1

EbAb . (4.129)

In (4.129) meq defines the number of edges with unknown magnetic vectorpotential in the finite element mesh, Eb the edge shape function associatedwith the b-th edge, and Ab the corresponding degree of freedom, namely theline integral of the magnetic vector potential along the b-th edge

Ab =∫b

A · ds . (4.130)

Applying the same discretization to the test function A′, we arrive at (see(4.112))

meq∑a=1

meq∑b=1

⎛⎝∫

Ω

γEa ·Eb dΩ Ab +∫Ω

ν(∇ × Ea) · (∇ × Eb) dΩAb

−∫Ω

Ea · Ji dΩ

⎞⎠ = 0 . (4.131)

In matrix form, we obtain

MAA+ KAA = f (4.132)

with

MA =ne∧

e=1

me ; me = [mpq] ; mpq =∫

Ωe

γEp · Eq dΩ (4.133)

KA =ne∧

e=1

ke ; ke = [kpq ] ; kpq =∫Ωe

ν(Bcurl

p

)T Bcurlq dΩ (4.134)

f =ne∧

e=1

fe ; fe = [fep ] ; fe

p =∫Ωe

Ep · Ji dΩ . (4.135)

For the computation of Bcurlp we know, according to Sect. 2.7 that the edge

shape function Ep for a tetrahedra is given by

Ep = (Ni∇Nj −Nj∇Ni)lp , (4.136)


with p defining the edge and i, j the corresponding vertices. To find a moreexplicit form for Bcurl

p , let us rewrite Ep as follows

Ep = lpNi

⎛⎜⎜⎝

∂Nj

∂x

∂Nj

∂y

∂Nj

∂z

⎞⎟⎟⎠− lpNj

⎛⎜⎜⎝

∂Ni

∂x

∂Ni

∂y

∂Ni

∂z

⎞⎟⎟⎠ (4.137)

= lp

⎛⎜⎜⎝Ni

∂Nj

∂x −Nj∂Ni

∂x

Ni∂Nj

∂y −Nj∂Ni

∂y

Ni∂Nj

∂z −Nj∂Ni

∂z

⎞⎟⎟⎠ . (4.138)

Therefore, we obtain

Bcurlp = ∇ × Ep =

⎛⎜⎜⎝

∂Ez

∂y − ∂Ey

∂z

∂Ex

∂z − ∂Ez

∂x

∂Ey

∂x − ∂Ex

∂y

⎞⎟⎟⎠ = 2lp

⎛⎜⎜⎜⎝

∂Ni

∂y∂Nj

∂z − ∂Nj

∂y∂Ni

∂z

∂Ni

∂z∂Nj

∂x − ∂Nj

∂z∂Ni

∂x

∂Ni

∂x∂Nj

∂y − ∂Nj

∂x∂Ni

∂y

⎞⎟⎟⎟⎠ . (4.139)

The time discretization is performed by applying the trapezoidal method(see Sect. 2.5.1) using, e.g., an effective mass formulation, which results in thefollowing scheme:


A = An + (1 − γP)∆t An . (4.140)


M∗AAn+1 = f

n+1− KAA (4.141)

M∗A = MA + γP∆tKA . (4.142)


An+1 = A+ γP∆t An+1 . (4.143)

4.7.3 Discretization with Nodal Finite Elements

According to (4.120), we have to compute the unknowns of the vector w andthe scalar φ at all finite element nodes. To obtain four equations for the threeunknowns of w and the one of φ, we set once the test function ψ and oncethe test function v to zero, which results in


ν∇ × v · ∇ × w dΩ +∫Ω

ν ∇ · v ∇ · w dΩ +∫Ω

γ v ·(w + ∇φ

)dΩ

=∫Ω

Ji · v dΩ (4.144)

∫Ω

γ∇ψ ·(w + ∇φ

)dΩ = 0 , (4.145)

with w = ∂w/∂t and φ = ∂φ/∂t. In the first step we will perform a spatial dis-cretization of (4.144) and (4.145) by introducing the following approximationsof the continuous functions w and φ

w ≈ wh =nd∑i=1

n′n∑

a=1

Nawiaei =n′

n∑a=1

Nawa ; Na =

⎛⎝Na 0 0

0 Na 00 0 Na

⎞⎠ (4.146)

φ ≈ φh =n′

n∑a=1

Naφa . (4.147)

In (4.146) and (4.147) n′n denotes the number of nodes with no Dirichletboundary condition, nd the space dimension andNa the interpolation functionfor node a. Applying the same approximations for the test functions A′ andψ, we arrive at

n′n∑

a=1

n′n∑

b=1

⎛⎜⎝∫

Ω

γ NTa Nb︸︷︷︸B1

ab

dΩ wb +∫Ω

γ NTa ∇Nb︸︷︷︸B2

ab

dΩ φb

+∫Ω

ν (∇ × Na)T︸︷︷︸(Bcurl

a )T

(∇ × Nb) dΩ wb +∫Ω

ν (∇ ·Na)T︸︷︷︸(Bdiv

a )T

(∇ · Nb) dΩ wb

−∫Ω

Na · Ji dΩ

⎞⎠ = 0 (4.148)

n′n∑

a=1

n′n∑

b=1

⎛⎝∫

Ω

γ(∇Na)T Nb dΩ wb +∫Ω

γ∇Na · ∇Nb dΩ φb

⎞⎠ = 0 . (4.149)

The different operators B have the following explicit form


Bcurla =

⎛⎜⎜⎝

0 −∂Na

∂z∂Na

∂y

∂Na

∂z 0 −∂Na

∂x

−∂Na

∂y∂Na

∂x 0

⎞⎟⎟⎠ Bdiv

a =

⎛⎜⎜⎝

∂Na

∂x 0 0

0 ∂Na

∂y 0

0 0 ∂Na

∂z

⎞⎟⎟⎠ (4.150)

B1ab =

⎛⎜⎜⎝NaNb 0 0

0 NaNb 0

0 0 NaNb

⎞⎟⎟⎠ B2

ab =

⎛⎜⎜⎝Na

∂Nb

∂x

Na∂Nb

∂y

Na∂Nb

∂z

⎞⎟⎟⎠ . (4.151)

Thus, we can write the spatially discretized matrix equation as[Mww Mwφ

MTwφ Mφφ

](w

φ

)+

[Kww 0

0 0

](w

φ

)=

(f

mag

0

), (4.152)

with w, φ the unknowns at the nodes and the matrices as well as the right-hand side vector compute as

Mww =ne∧

e=1

meww ; me

ww = [mpq] ; mpq =∫Ωe

γB1pq dΩ (4.153)

Mwφ =ne∧

e=1

mewφ ; me

wφ = [mpq] ; mpq =∫Ωe

γB2pq dΩ (4.154)

Mφφ =ne∧

e=1

meφφ ; me

φφ = [mpq] ; (4.155)

mpq =∫Ωe

γ∇Na · ∇Nb dΩ

Kww =ne∧

e=1

keww ; ke

ww = [kpq]

kpq =∫Ωe

ν((Bcurl

p )TBcurlq + (Bdiv

p )TBdivq

)dΩ (4.156)

fmag

=ne∧

e=1

fe

mag; fe

mag= [f

a] ; (4.157)

fa

=∫Ωe

(NaJix , NaJiy , NaJiz)T dΩ ,

with ne the number of finite elements.


The time discretization is performed by applying the trapezoidal method(see Sect. 2.5.1) both to w, φ using, e.g., an effective mass formulation, whichresults in the following scheme:


w = wn + (1 − γP)∆t wn (4.158)φ = φ

n+ (1 − γP)∆t φ

n. (4.159)


[M∗

ww Mwφ

MTwφ Mφφ

](wn+1

φn+1

)=

(f

n+1

0

)−[Kww 0

0 0

](w

φ

)(4.160)

M∗ww = Mww + γP∆tKww . (4.161)


wn+1 = w + γP∆t wn+1 (4.162)

φn+1

= φ+ γP∆t φn+1. (4.163)

4.7.4 Newton’s Method for the Non-linear Case

The main non-linearity within magnetic field computation is due to the de-pendence of the permeability µ on the magnetic field, which can be modelledas follows

B = µ(H)H (4.164)H = ν(B)B , (4.165)

with µ denoting the magnetic permeability and ν the magnetic reluctivity.Since experimental setups for measuring the non-linear behavior of magneticmaterials exhibit a functional relation between H and B, we can derive thefollowing alternative formulation

H(B) = ν(B)B (4.166)

ν(B) =H(B)B

(4.167)

ν′(B) =H ′(B)B −H(B)

B2. (4.168)

For the magnetostatic case, we obtain from (4.112) the following weak formu-lation ∫

Ω

ν(|∇ × A|) (∇ × A) · (∇ × A′) dΩ =∫Ω

Ji ·A′ dΩ , (4.169)


with the test function A′ ∈ H0(curl). This equation can be expressed withthe non-linear operator F as

F(A) = 0 .

Newton’s method is now defined by (see Appendix D.2)

Ak+1 = Ak + S , where S solves F ′(Ak)[S] = −F(Ak) . (4.170)

Therefore, in the first step we must derive the linearized form of F

F ′(Ak)[S] = F(Ak + S) −F(Ak) +O(||S||2) . (4.171)

The term F(Ak + S) − F(Ak) for arbitrary test functions A′ ∈ H0(curl)computes in its weak form as∫

Ω

ν(|∇ × (Ak + S)|) (∇ × (Ak + S)) · (∇ × A′) dΩ

−∫Ω

ν(|∇ × Ak|) (∇ × Ak) · (∇ × A′) dΩ . (4.172)

By adding the term ν(|∇ ×Ak|)(∇× (Ak + S) · (∇×A′) to (4.172) as wellas at the same time subtracting it from (4.172) and combining similar termsleads to∫Ω

(ν(|∇ × (Ak + S)|) − ν(|∇ × Ak|)

) (∇ × (Ak + S)

)· (∇ × A′) dΩ

+∫Ω

ν(|∇ × Ak|) (∇ × S) · (∇ × A′) dΩ . (4.173)

Now we can perform the following approximations

ν(|∇ × (Ak + S)|) − ν(|∇ × Ak|) ≈ ν′(|∇ × Ak|)(|∇ × (Ak + S)| − |∇ × Ak|

)

|∇ × (Ak + S)| − |∇ × Ak| =|∇ × (Ak + S)|2 − |∇ × Ak|2|∇ × (Ak + S)| + |∇ × Ak|

=|∇ × Ak|2 + |∇ × S|2 + 2(∇ × Ak) · (∇ × S) − |∇ × Ak|2

|∇ × (Ak + S)| + |∇ × Ak|

≈ (∇ × S) · (∇ × Ak)|∇ × Ak|

. (4.174)

With the help of these approximations, (4.173) can be written as


ν′(|∇ × Ak|)(∇ × S) · (∇ × Ak)

|∇ × Ak|(∇ × (Ak + S) · (∇ × A′) dΩ

+∫Ω

ν(|∇ × Ak|) (∇ × S) · (∇ × A′) dΩ .(4.175)

Neglecting all terms of second order and retaining only terms linear in S, wearrive at∫

V

ν′(|∇ × Ak|)(∇ × S) · (∇ × Ak)

|∇ × Ak|(∇ × Ak) · (∇ × A′) dΩ

+∫Ω

ν(|∇ × Ak|) (∇ × S) · (∇ × A′) dΩ . (4.176)

Therefore, the resulting Newton step computes as (see (4.170))∫Ω

ν(Bk) (∇ × S) · (∇ × A′) dΩ

+∫Ω

ν′(Bk) (∇ × S) · eBkBk eBk

· (∇ × A′) dΩ

=∫Ω

Ji · A′ dΩ

−∫Ω

ν(Bk)(∇ × Ak) · (∇ × A′) dΩ (4.177)

Ak+1 = Ak + S , (4.178)

with Bk = ∇ × Ak, Bk = |Bk| and eBk= Bk/Bk.

Now, let us apply the FE method to (4.177) using, e.g., nodal finite ele-ments. Therefore, we have to consider instead of (4.169) a weak formulationsimilar to (4.123). Since we have to ensure ∇ ·w = 0 (see (4.119)), we applya penalty formulation by adding

∫Ω

ν ∇ · v ∇ · w dΩ to (4.177) (having the

decomposition A = w + ∇φ in mind)

128 4 Electromagnetic Field∫Ω

ν(Bk) (∇ × v′) · (∇ × S) dΩ +∫Ω

ν(Bk) ∇ · v′ ∇ · S dΩ

+∫Ω

ν′(Bk)Bk(∇ × v′) · eBk(∇ × S) · eBk

dΩ

=∫Ω

Ji · v′ dΩ

−∫Ω

ν(Bk)((∇ × wk) · (∇ × v′) + (∇ ·wk)(∇ · v′)

)dΩ . (4.179)

Performing the same procedure as in Sect. 4.7.3, we arrive at the followingalgebraic system of equations(

KL(wk) + KNL(wk))S = f

mag− KL(wk)wk = f

res. (4.180)

In (4.180) KL and fmag

are calculated as in (4.156) and (4.157). The matrixKNL, also called the tangent stiffness matrix, is computed as follows

KNL =ne∧

e=1

keNL ; ke

NL = [kNLpq ]

kNLpq =

∫Ωe

ν′(Bk)((Bcurl

p · eBk)Bk (Bcurl

q · eBk)T)

dΩ , (4.181)

with ne the number of finite elements and ν′(Bk) as well as eBkto be evalu-

ated at each integration point during the numerical integration. The iterativesolution process is performed, until the following two stopping criteria arefulfilled

||wk+1 − wk||2||wk+1||2

< δa||f

res||2

||fmag

||2< δr , (4.182)

with appropriate δa and δr. Since a standard Newton method is not globallyconvergent, it is necessary to apply a technique that controls the updating by

wk+1 = wk + ηS , (4.183)

with some relaxation parameter η. The scalar parameter η > 0 is introducedto control the convergence during early steps of the iteration process, or in thepresence of nonmonotonic material relations. A common algorithm to computeη is a line search (see Appendix D) defined by approximately minimizing

|G(η)| = |ST fres

(wk + ηS)| .


4.7.5 Approximation of B–H Curve

As can be seen from Sect. 4.7.4, not only values of the magnetic reluctivityν as a function of the magnetic induction B but also first derivatives of νare required for the solution of the non-linear equation. Figure 4.20(a) showsa typical B–H curve obtained by linear interpolation of the measured B–H values. Since, from the physical point of view, the B–H curve is always

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

H (kA)

B (

T)

(a) Typical B–H curve

0 0.5 1 1.5 20

2000

4000

6000

8000

10000

12000

B (T)

(A

m/V

s)

(b) Corresponding ν(B) curve

Fig. 4.20. Typical B–H curve and computed ν(B) curve


monotone, which is not the case for the ν(B) curve as displayed in Fig. 4.20(b),it makes sense to find a functional expression for H(B) and compute ν as wellas ν′ as given in (4.167) and (4.168).

At this point, the question arises if one should apply an interpolation oran approximation scheme to obtain the function H(B). Since the measuredB–H values will always just be accurate up to a measurement error, it is notappropriate to fix the curve to these values at the measurement points, aswould be done by interpolation. Instead, one should do an approximation tothe measurement points combined with some regularization to avoid oscilla-tions. We will use so-called smoothing splines [49] and minimize the followingregularized least squares functional

Jα[H ] =

Bend∫0

(H ′′(B))2 dB + α

N∑k=1

(H(Bk) −Hk)2 , (4.184)

with H ′′(B) the second derivative with respect to B and N the number ofmeasured B–H values. Therefore, we obtain for small values of α a very smoothapproximation and for large values of α a solution, which will be very closeto the discrete points. For determining the regularization parameter α a pos-teriori, we use a discrepancy principle with bisection, i.e., set α equal to thelargest member of the geometrically decreasing sequence 2−j such that for theminimizer Bα of Jα the residual√√√√ N∑

k=1

(H(Bk) −Hk)2 (4.185)

is of the order of magnitude of the measurement noise.Note that—unlike ν—the B–H curve itself always has to be monotone for

physical reasons, so in order to be able to monitor and preserve this propertywe prefer to apply the described smoothing spline technique directly to thecollection of (Bk, Hk)N

k=1 points. Practically, the monotonicity of the B–Hcurve approximation is gained by putting restrictions on the coefficients ofthe spline curve. These restrictions arise naturally, if we require H ′(B) > 0.Consequently, the task is to find a monotone approximation that fulfills thediscrepancy principle for a given data noise level. Note additionally that thenoise level is given in terms of (Bk, Hk)N

k=1 points. If we would deduce anestimate for the data noise in terms of the (νk, Bk)N

i=k points, this would, dueto the strong variation in scale of both function and derivative values of thecurve under consideration, lead to a locally too pessimistic noise estimate andhence to a poor curve approximation. For a detailed discussion on this topicwe refer to [181] and [169].


4.7.6 Modelling of Current-loaded Coil

In this section we will investigate the numerical computation of electromag-netic fields excited by a current-loaded coil. Figure 4.21 displays a typicalsetup of a coil with a ferrite core.

(a) Structure of the winding around the ferritecore

(b) Hollow cylinder

Fig. 4.21. Coil with a ferrite core

To simplify the geometric modelling of the winding structure as well asthe further meshing, we will substitute the complex structure by a hollowcylinder (see Fig. 4.21(b)). This simplification can be done since the changein the computed magnetic field can be neglected. The relation for the currentdensity J in the coil is now defined by

J = JeJ =I

ΓweJ =

INc

κΓceJ . (4.186)

In (4.186) eJ denotes the unit vector for the direction of J, I the current, Γw

the cross section of one winding, Nc the number of turns, Γc the cross sectionof the hollow cylinder, and κ the filling factor.

4.7.7 Computation of Global Quantities

In order to compare simulated results with measurements, we have to evaluatequantities derived from the computed field.


Magnetic Flux and Inductance

In general, the inductance L for a magnetic assembly is defined by [95]

L =Ψ

I. (4.187)

In (4.187) Ψ denotes the total magnetic flux and I the electric current loadingthe coil. The total magnetic flux Ψ for a coil with Nc turns computes as

Ψ = Ncφ ,

where φ is the flux defined by (see Fig. 4.22)

φ =∫Γ

B · dΓ . (4.188)

Expressing the magnetic induction B via the magnetic vector potential A

Fig. 4.22. Magnetic field crossing Γ

and using Stoke’s theorem (Appendix B.9), we arrive at

φ =∫Γ

(∇ × A) · dΓ

=∮

C(Γ )

A · ds . (4.189)

It has to be noted that the orientation of ds depends on the normal vector naccording to the right-hand rule.

General 3D Case

To obtain a very accurate numerical value for the magnetic inductance, wecompute the magnetic flux over the whole coil volume and then normalize itto the cross section of the coil Γc (see Fig. 4.21). Thus, we obtain


φ =1Γc

⎛⎜⎝ n∑

i=1

∫Ωe

i

(A · eJ) dxdy dz

⎞⎟⎠ , (4.190)

with n the number of finite elements within the coil region (hollow cylinder)and eJ the unit vector defining the direction of the current density J in thecoil.

2D Plane Case

Cut

Gc

l

(a) Full 3D setup

Region 1

x

Region 2

Gc Gc

y

(b) 2D model

Fig. 4.23. Example for a 2D plane case

According to the 2D plane model (see Fig. 4.23), we have

A = Az(x, y)ez

ds = dz ez .

Thus, the evaluation of (4.189) gives us

φ = −A1zl +A2

zl = l(A2z −A1

z) .

Since the magnetic vector potential can vary over the cross section, we com-pute the magnetic flux φ by the following averaging

φ =l

Γc

⎛⎜⎝ n1∑

i=1

∫Ωe

i

A2z dxdy −

n2∑i=1

∫Ωe

i

A1z dxdy

⎞⎟⎠ ,

with n1 and n2 the number of finite elements in regions 1 and 2.

Axisymmetric Case

The situation for the axisymmetric case is displayed in Fig. 4.24. In the ax-


(a) Full 3D setup (b) 2D model

Fig. 4.24. Example for a axisymmetric case

isymmetric case, just the circumferential component of the magnetic vectorpotential is different from zero, thus

A = Aϕ(r, z)eϕ

ds = dϕ eϕ .

Here we also perform an averaging, which means to compute the integral overthe volume of the coil and normalize it to the cross section Γc of the coil

Ψ = Ncφ

=2πNc

Γc

n∑i=1

∫Ωe

i

rAϕ dr dz , (4.191)

with n the number of finite elements in Γc.

4.7.8 Induced Electric Voltage

Let us consider an electrically open coil, and the task of computing the inducedelectric voltage due to a time-varying magnetic field. Then, according to Sect.4.1, we have to evaluate

uind = −Ncdφdt, (4.192)

with Nc the number of turns. Thus, we can simply use (4.190) (or its 2Dversions), replace A by A and multiply the result by −Nc to arrive at (4.192).

4.8 Numerical Examples 135

4.8 Numerical Examples

4.8.1 Ferromagnetic Cube

In a first study, we will consider the numerical computation of the magneticinduction B within a ferromagnetic cube surrounded with air (see Fig. 4.25). Aparameter jump in the magnetic reluctivity across the ferromagnetic cube/airinterface by a factor of 1000 is assumed. The boundary conditions for the

Fig. 4.25. Geometry setup and finite element mesh

magnetic vector potential A = (Ax, Ay, Az) are applied as follows:

z = 0 Az = 0z = H Az = 0x = 0 Az = 0 Ay = A0

x = L Az = 0 Ay = −A0

y = 0 Az = 0 Ax = −A0

y = L Az = 0 Ax = A0

The geometry of this cube is fixed by its side length L and heightH . The usedfinite element mesh is displayed in Fig. 4.25. In the first step we computedthe magnetic induction by discretizing (4.123) using nodal finite elements withlinear interpolation functions (linear hexahedra). As shown in Fig. 4.26(a), in-stead of a constant magnetic induction over the ferromagnetic cube, we obtainvery poor results near the corners and edges. In the next step, we have usednodal finite elements with quadratic interpolation functions. As can be seenfrom Fig. 4.26(b), the wrong solution at the corners has not changed much,but along the edges the solution has been improved considerably. By applyingthe weighted regularization method (see Sect. 4.7.1), we obtain for the caseof linear interpolation functions the results displayed in Fig. 4.27(a), and, forthe case of quadratic interpolation the results shown in Fig. 4.27(b). As can


(a) Nodal finite elements with linear in-terpolation functions

(b) Nodal finite elements withquadratic interpolation func-tions

Fig. 4.26. Magnetic induction: Standard formulation

(a) Nodal finite elements with linear in-terpolation functions

(b) Nodal finite elements withquadratic interpolation func-tions

Fig. 4.27. Magnetic induction: Weighted regularization

be seen, the magnetic induction is almost constant all over the ferromagneticcube when using the weighted regularization. Of course, also in this case, thequadratic interpolation functions yield the better results.

4.8.2 Thin Iron Plate

In a second example, we study the distribution of the magnetic inductionwithin a thin iron plate (Fig. 4.28(a)). At the outer boundary the tangentialcomponent of the magnetic vector potential A has been set to zero, whereasat the symmetry planes the normal component has to vanish. The givencurrent density generates the magnetic field, and within the iron plate themagnetic induction shall be constant in the x-direction and will decrease inthe y-direction. The finite element discretization using linear hexahedra andtetrahedra for the iron plate and coil is shown in Fig. 4.28(b). The computa-tional results by using the standard finite element formulation are displayedin Fig. 4.29(a), and show a strong variation of the magnetic induction in thex-direction. The formulation with the weighted regularization produces a mag-netic induction distribution within the plate as shown in Fig. 4.29(b). If we


(a) Geometry setup: Iron plate and coil (b) Finite element mesh without am-bient air

Fig. 4.28. Iron plate surrounded by a coil

(a) Standard regularization (b) Weighted regularization

Fig. 4.29. Magnetic induction in the iron plate

compare the evaluated inductances of the coil (global quantity)—computedwith the standard formulation (regularization term all over the domain) andwith weighted regularization—then the difference is less then 0.5 %. This re-sult indicates that the error occurring at interfaces of jumping permeabilityis a local one for the standard formulation.

5

Acoustic Field

Acoustic waves can propagate in non-viscous media just in the form of longi-tudinal waves. Thus, the particles of the fluid move forwards and backwardsin the direction (and opposite) of the propagation and produce locally a com-pression and expansion of the fluid (see Fig. 5.1). The repulsive force, which isessential for the propagation of the wave, is generated by the pressure changein the media.

Fig. 5.1. Compression and expansion of a fluid

5.1 Wave Theory of Sound

The wavelength λ is uniquely defined by the frequency f and the speed ofsound c via

λ =c

f. (5.1)

Assuming a frequency of 1 kHz, the wavelength in air takes on the value of0.343 m (c = 343 m/s) and in water 1.5 m (c = 1500 m/s). The propagation of

140 5 Acoustic Field

waves is described by the time and the spatial variation of the density ρ

ρ =mass

volume(kg/m3)

the pressure p

p =force

cross section(N/m2)

and the velocity v

v =distance

time(m/s) .

These quantities can be decomposed in their mean and alternating part ac-cording to

ρ = ρ0 + ρ′

p = p0 + p′ (5.2)v = v0 + v′ ,

where we denote by ρ′ the acoustic density, p′ the acoustic pressure and v′ theacoustic particle velocity. For linear acoustics, we have the following relationsbetween the mean and alternating quantities

ρ′ << ρ0 p′ << p0 v′ << v0 . (5.3)

The acoustic field is fully described by the equation of mass conserva-tion (continuity equation), the equation of momentum conservation (Euler’sequation, Newton’s law for fluids) and the pressure-density relation (stateequation). These basic physical relations will be discussed in the followingsection.

5.1.1 Conservation of Mass (Continuity Equation)

The propagation of an acoustic wave through a fluid (gas) causes local changesin the density. Therefore, we have to mathematically model the fact that a netflow across a closed surface causes a mass change due to a change in densityinside the volume in order to guarantee mass conservation. Let us consider avolume Ω enclosed by a surface Γ as displayed in Fig. 5.2. The mass m attime t and given density ρ(x, y, z, t) computes as

m(t) =∫Ω

ρ(x, y, z, t) dΩ .

Now, if we model a positive net flow across the surface Γ during the timeinterval dt, then the mass inside the volume Ω will decrease. Thus, we obtain∮

Γ

ρv dt · dΓ = −∫

Ω

(ρ(t+ dt) − ρ(t)) dΩ , (5.4)

5.1 Wave Theory of Sound 141

Fig. 5.2. Considered volume Ω with closed surface Γ

which states that a reduction of mass in the volume Ω with respect to timehas to be compensated by a mass flow through the surface Γ enclosing thevolume. Expanding the expression ρ(t+ dt) in a Taylor series up to the linearterm

ρ(t+ dt) ≈ ρ(t) +∂ρ

∂tdt

changes (5.4) to ∮Γ

ρv dt · dΓ = −∫

Ω

(∂ρ

∂tdt)

dΩ . (5.5)

Applying the divergence theorem (see Appendix B.6) to the term on the leftside of (5.5) and cancelling dt results in∫

Ω

∇ · (ρv) dΩ = −∫

Ω

(∂ρ

∂t

)dΩ . (5.6)

Since we integrate the same arbitrary volume, we can omit the integrationand obtain the continuity equation in differential form

∇ · (ρv) = −∂ρ∂t. (5.7)

5.1.2 Conservation of Momentum (Euler Equation)

Let us consider an infinitely small volume, which moves with the fluid andhas the mass dm = ρ dΩ. Then Newton’s law states

adm = df . (5.8)

For non-viscous fluids, the repulsive force is caused by the change in pressure,and the pressure is understood to be (−pn) with n the outward normal vectoron the surface. Thus, we obtain for the infinite small volume dΩ = (dxdy dz)(see Fig. 5.3)

− dfx = (p(x+ dx/2, y, z)− p(x− dx/2, y, z)) dy dz− dfy = (p(x, y + dy/2, z)− p(x, y,− dy/2, z)) dxdz− dfz = (p(x, y, z + dz/2)− p(x, y, z − dz/2)) dxdy .


Fig. 5.3. Infinite small volume with pressure state

Now we can expand the terms in a Taylor series up to the linear term, e.g,p(x+ dx/2, y, z) and p(x− dx/2, y, z)

p(x+ dx/2, y, z) ≈ p(x, y, z) +∂p

∂x

dx2

p(x− dx/2, y, z) ≈ p(x, y, z) − ∂p

∂x

dx2.

This results in

dfx = − ∂p∂x

dΩ (5.9)

dfy = −∂p∂y

dΩ (5.10)

dfz = −∂p∂z

dΩ , (5.11)

or in compact form we may write

df = −∇p dΩ . (5.12)

Since we use a coordinate system fixed in space, the position of the fluidparticles depends on time. Thus, the expression for the acceleration a as afunction of the acoustic velocity v reads as

a =dvdt

=∂v∂t

+∂v∂x

∂x

∂t+∂v∂y

∂y

∂t+∂v∂z

∂z

∂t

=∂v∂t

+ (v · ∇)v . (5.13)

Using (5.8), (5.12), and (5.13) yields Euler’s equation

ρ

(∂v∂t

+ (v · ∇)v)

= −∇p . (5.14)


5.1.3 Pressure-Density Relation (State Equation)

According to the quasistatic theory of thermodynamics, we can describe thelocal state of a fluid at any time by the density ρ and the specific internalenergy Eint. Other state variables like the pressure p or the temperature Tare determined from equations of state defined by ρ and Eint. In the contextof acoustic wave propagation, we shall also consider the specific entropy s,which is related to heat transfer. In general, the pressure is regarded to be afunction of the density and the specific entropy

p = p(ρ, s) . (5.15)

However, if the specific entropy can be considered as constant, which is ex-pressed by

ds

dt= 0 , (5.16)

the state is adiabatic and the pressure is just a function of ρ. Now, the ques-tion of sound propagation being an adiabatic or isothermal process can beanswered by considering the heat conduction process within a fluid. Actu-ally, the temperature rises and falls according to the density and pressurefluctuations in a sound wave. Since the relation

cp∂T

∂t= T

ds

dt(5.17)

holds (cp denotes the specific heat at constant pressure), we can rewrite thethermal diffusion equation by

Tds

dt= γT∆T . (5.18)

with γT the coefficient of thermal conductivity. If conduction dominates theprocess, we can approximate (5.18) by ∆T = 0 and the sound is consideredas isotherm, otherwise (5.18) is approximated by ds/dt = 0 and the sound isconsidered adiabatic. Both cases are approximations of reality, but for freelypropagating acoustic waves both in gases as well as liquids at frequencies ofinterest, the state is adiabatic (for a detailed discussion see [170]). Therewith,no thermal exchange of energy between the particles exists. Physically we canexpress this behavior by

pΩκ = const. , (5.19)

with κ the adiabatic exponent and Ω the considered volume. The adiabaticexponent κ is computed by the ratio of specific heat cp at constant pressureand the specific heat cΩ at constant volume

κ =cpcΩ

. (5.20)

Now, let us consider a fluid in rest with (Ω0, p0, ρ0) and a fluid, in which anacoustic wave propagates, defined by (Ω, p, ρ). According to (5.19) we obtain

144 5 Acoustic Field (Ω0

Ω

)κ

=p

p0. (5.21)

By assuming a constant mass, we derive a relation between the volumes anddensities

m0 = ρ0Ω0 = ρΩ

Ω0

Ω=

ρ

ρ0. (5.22)

This relation, in combination with (5.21), yields

p

p0=(ρ

ρ0

)κ

. (5.23)

For an ideal gas, the relationp = ρRT (5.24)

holds (R denotes the gas constant, e.g., in air R = 297 Ws/kgK), and we mayexpress the speed of sound c by [170]

c =√κRT (5.25)

to obtain its dependency on the temperature T .Since for linear acoustics we may assume ρ′ << ρ0, we can perform the

following approximation(ρ

ρ0

)κ

=(ρ0 + ρ′

ρ0

)κ

≈ 1 + κρ′

ρ0. (5.26)

This result combined with (5.23) yields the linearized state equation for acous-tic waves

p′

ρ′= κ

p0

ρ0= c2 . (5.27)

For liquids, such as water, the pressure-density relation is mainly expressedby the adiabatic bulk modulus Ks according to

∂p(ρ, s)∂ρ

=Ks

ρ(5.28)

c =

√Ks

ρ. (5.29)

The reciprocal 1/Ks is known as the adiabatic compressibility. In the case ofa liquid, a negligible error occurs, if we consider the state to be adiabatic orisotherm. The reason for this behavior is the little difference between the adia-batic bulk modulus Ks, which is quite difficult to measure, and the isothermalbulk modulus KT (for details see [170]).


5.1.4 Linear Acoustic Wave Equation

For linear acoustic wave propagation, the following relations hold

v ∇ρ ρ∇ · v ≈ ρ0∇ · v

(v · ∇)v ∂v∂t

.

Thus, Euler’s equation (5.14) as well as the continuity equation (5.7) read as

∂v∂t

= − 1ρ0

∇p (5.30)

∇ · v = − 1ρ0

∂ρ

∂t. (5.31)

Since any derivative with respect to time as well as space of the constantquantities (v0, p0, ρ0) are zero, we may substitute the total physical quantities(v, p, ρ) by their alternating ones (v′, p′, ρ′). Applying the time derivative to(5.31) and interchanging time and space derivatives yields

∇ · ∂v′

∂t= − 1

ρ0

∂2ρ′

∂t2. (5.32)

Expressing the term ∂v/∂t by (5.30) results in a relation between pressureand density

∇ ·(− 1ρ0

∇p′)

= − 1ρ0

∂2ρ′

∂t2. (5.33)

Using the relation ρ′ = p′/c2 according to the state equation (5.27), we obtainthe acoustic wave equation for p′

∆p′ =1c2∂2p′

∂t2. (5.34)

By applying the curl operator to (5.30), it is easy to show that the acousticvelocity v′ is irrotational

∇ ×(ρ0∂v′

∂t+ ∇p′

)= 0

ρ0∂

∂t(∇ × v′) + ∇ × ∇p′ = 0

∇ × v′ = 0 . (5.35)

Thus, the acoustic velocity v′ can be expressed as the gradient of a scalarpotential, the so-called acoustic velocity potential

v′ = −∇ψ . (5.36)


By introducing the acoustic velocity potential in Euler’s equation, we obtainfrom (5.30) the relation between ψ and p′

−ρ0∂

∂t(∇ψ) = −∇p′

p′ = ρ0∂ψ

∂t, (5.37)

where the integration constant can be set to zero, since (5.36) defines ψ onlyup to a constant. With (5.37) we may write the wave equation (5.34) for theacoustic velocity potential

∆ψ =1c2∂2ψ

∂t2. (5.38)

5.1.5 Acoustic Quantities

Let us first consider the linear version of Euler’s equation (see (5.30)) andtake the dot product with the acoustic velocity v′

v′ · ρ0∂v′

∂t= −v′ · ∇p′

= −∇ · (p′ v′) + p′∇ · v′ . (5.39)

Using the linear version of the mass conservation and the linear pressure-density relation ρ′ = p′/c2 results in

v′ · ρ0∂v′

∂t= −∇ · (p′v′) − p′

1ρ0

∂ρ′

∂t

∂

∂t

(12ρ0v′ · v′

)= −∇ · (p′v′) − 1

2∂

∂t

(p′2

ρ0c2

). (5.40)

Therewith, we obtain

∂wa

∂t+ ∇ · Ia = 0 (5.41)

wkina =

12ρ0v′ · v′ (5.42)

wpota =

p′2

2ρ0c2(5.43)

with wa the total acoustic energy density and Ia the acoustic energy flux, alsocalled the acoustic intensity. Furthermore, we refer to wkin

a as the acoustickinetic energy density and wpot

a the acoustic potential energy density.Now let us assume that the acoustic source is driven with constant fre-

quency f (angular frequency ω = 2πf), so that we can express the acousticfield variables by


p′(t) = p cos(ωt)v′(t) = v cos(ωt+ ϕ) .

Therewith, we obtain the following expressions for the time averaged acousticenergy density wa as well as intensity Ia (T = 1/f is the period time)

wava =

1T

t0+T∫t0

12ρ0v

2 cos2(ωt+ ϕ)︸︷︷︸12 (1+cos(2(ωt+ϕ))

dt+1T

t0+T∫t0

12

p2

ρ0c2cos2(ωt)︸︷︷︸

12 (1+cos(2ωt))

dt

=14ρ0v

2 +14p2

ρ0c2

Iava =

1T

t0+T∫t0

pv cos(ωt) cos(ωt+ ϕ)︸︷︷︸12 cos ϕ+ 1

2 cos(2ωt+ϕ)

dt

=pv2

cosϕ .

By using these results and substituting it in (5.41), we arrive at

∇ · Iava = 0 .

This means that the spatial variation of Iava has to be zero, or expressing it in

integral form ∫Ω

∇ · Iava dΩ =

∮Γ (Ω)

Iava · n dΓ = 0 .

However, if the considered volume encloses any acoustic sources, the abovederivation does not apply, and we obtain

P ava =

∮Γ (Ω)

Iava · n dΓ . (5.44)

In (5.44) we denote by P ava the average acoustic power radiated by all sources

enclosed by the surface Γ .Strictly speaking, each acoustic wave has to be considered as transient,

having a beginning and an end. However, for some long duration sound, wespeak of continuous wave (cw) propagation and we define for the acousticpressure p′ a mean square pressure (p′)2av as well as a root mean squared(rms) pressure p′rms

prms =

√√√√√ 1T

t0+T∫t0

(p− p0)2 dt . (5.45)


In (5.45) T denotes the period time of the signal or if we cannot strictly speakof a periodic signal, an interminable long time interval.

Any acoustic media is defined by its acoustic impedance Za, which com-putes as the quotient of sound pressure p′ and the magnitude of the acousticparticle velocity v′

Za =p′

v′. (5.46)

If the particles of the fluid have the same oscillation in the plane normal tothe direction of propagation, the corresponding acoustic wave is called a planewave. For plane waves, the acoustic impedance is constant and is called thespecific acoustic impedance

Z0 =p′

v′= ρ0c (Ns/m3) . (5.47)

In air the acoustic impedance Z0 has a value of 408 Ns/m3, in water a valueof 1.5 × 106 Ns/m3 and in solids values between (3 − 110)× 106 Ns/m3.For non-plane waves the sound field impedance is not constant, e.g., for aspherical wave Za computes as

Za =p′

v′=

jρ0ckr

1 + jkr= Z0

jkr1 + jkr

, (5.48)

with k = ω/c = 2π/λ the wave number and r the distance from the source.

5.1.6 Plane and Spherical Waves

In order to get some physical insight in the propagation of acoustic sound, wewill consider two special cases plane and spherical travelling waves. Let’s startwith the simpler case, the propagation of a plane wave as displayed in Fig. 5.4.Thus, we can express the acoustic pressure by p′ = p′(x, t) and the velocity by

Fig. 5.4. Propagation of a plane wave

v′ = v(x, t)ex. Using these relations together with the linear pressure-densitylaw, we arrive at the following 1D linear wave equation


∂2p′

∂x2− 1c2∂2p′

∂t2= 0 , (5.49)

which can be rewritten in factorized version as(∂

∂x− 1c

∂

∂t

) (∂

∂x+

1c

∂

∂t

)p′ = 0 . (5.50)

This version of the linearized, 1D wave equation motivates us to introduce thefollowing two functions

ξ = t− x/c

η = t+ x/c

with two main properties

∂

∂t=

∂

∂ξ+

∂

∂η

∂

∂x=

1c

(∂

∂η− ∂

∂ξ

).

Therewith, we obtain for the factorized operator

∂

∂x− 1c

∂

∂t= − 1

2c∂

∂ξ

∂

∂x+

1c

∂

∂t=

12c

∂

∂η

and the linear, 1D wave equations transfers to

− 14c2

∂

∂ξ

∂

∂ηp′ = 0 .

The general solution computes as a superposition of arbitrary functions of ξand η

p′ = f(ξ) + f(η) = f(t− x/c) + g(t+ x/c) . (5.51)

This solution describes waves moving with the speed of sound c in +x and−x direction, respectively.

Now let us derive the relation between acoustic pressure p′ and particlevelocity v′. For this, we substitute the above relations in the linearized, 1Dequations for momentum and mass conservation, and use the linear pressure-density relation

∂p′

∂t+ ρ0c

2 ∂v′

∂x= 0 ρ0

∂v′

∂t+∂p′

∂x= 0

∂p′

∂ξ+∂p′

∂η+ ρ0c

(∂v′

∂η− ∂v′

∂ξ

)= 0 ρ0

(∂v′

∂ξ+∂v′

∂η

)+

1c

(∂p′

∂η− ∂p′

∂ξ

)= 0

∂

∂ξ(p′ − ρ0cv

′) +∂

∂η(p′ + ρ0cv

′) = 0∂

∂ξ(ρ0cv

′ − p′) +∂

∂η(ρ0cv

′ + p′) = 0 .

Once adding and once subtracting these two obtained equations will result in


∂

∂η(ρ0cv

′ + p′) = 0∂

∂ξ(ρ0cv

′ − p′) = 0

so that (ρ0cv′ + p′) as well as (ρ0cv

′ − p′) become functions of ξ and η, andwe arrive at

v′ =1ρ0c

(f(t− x/c) + g(t+ x/c)) =p′

ρ0c. (5.52)

Therewith, the value of the acoustic pressure over acoustic particle velocityfor a plane wave is constant and results in the specific acoustic impedance (see5.47). Furthermore, the acoustic energy density wa and the acoustic intensityIa simplifies to

wa =ρ0v

′2

2+

p′2

2ρ0c2=

p′2

ρ0c2(5.53)

Ia =p′2

ρ0cex = cwaex . (5.54)

In (5.53) we realize that the acoustic kinetic and potential energy density areequal, and (5.54) demonstrates that the acoustic energy propagates with thespeed of sound c.

The second case of investigation will be a spherical wave, where we assumea point source located at the origin. In the first step, we rewrite the linearizedwave equation in spherical coordinates and consider that the pressure p′ willjust depend on the radius r. Therewith, the Laplace-operator reads as

∆p′(r, t) =∂2p′

∂r2+

2r

∂p′

∂r=

1r

∂2rp′

∂r2

and we obtain1r

∂2rp′

∂r2− 1c2

∂2p′

∂t2︸︷︷︸1r

∂2rp′∂t2

= 0 . (5.55)

A multiplication of (5.55) with r results in the same wave equations as ob-tained for the plane case (see (5.49)), just instead of p′ we have rp′. Therefore,the solution of (5.55) reads as

p′(r, t) =1r

(f(t− r/c) + g(t+ r/c)) , (5.56)

which means that the pressure amplitude will decrease according to the dis-tance r from the source. The assumed symmetry requires that all quantitieswill just exhibit a radial component. Therewith, we can express the timeaveraged acoustic intensity Iav in normal direction n by a scalar value justdepending on r

Iav · n = Iavr

and as a function of the time averaged acoustic power P av of our source

5.2 Quantitative Measure of Sound 151

Iavr =

P av

4πr2. (5.57)

According to (5.57), the acoustic intensity decreases with the squared distancefrom the source. This relation is known as the spherical spreading law.

In order to obtain the acoustic velocity v′ = v′(r, t)er as a function of theacoustic pressure p′, we substitute the general solution for p′ (see (5.56), inwhich we set without loss of generality g = 0) into the linear Euler equation(see (5.30))

∂v′

∂t= − 1

ρ0

∂p′

∂r= − 1

ρ0

∂

∂r

(f(t− r/c)

r

)

v′ = − 1ρ0

∂

∂r

(F (t− r/c)

r

)(5.58)

with f(t) = ∂F (t)/∂t. Using the relation

∂F (t− r/c)∂r

= −1c

∂F (t− r/c)∂t

and performing the differentiation with respect to r results in

v′(r, t) = − 1ρ0

1r

∂F (t− r/c)∂r

+F (t− r/c)

ρ0r2(5.59)

=1ρ0c

1r

∂F (t− r/c)∂t︸︷︷︸

f/r=p′

+F (t− r/c)ρ0r2

(5.60)

=p′

ρ0c+F (t− r/c)ρ0r2

. (5.61)

Therewith, spherical waves show in the limit r → ∞ the same acoustic be-havior as plane waves.

Now with this acoustic velocity-pressure relation, we may rewrite theacoustic intensity for spherical waves as

Ir =p′2

ρ0c+

p′

ρ0r2F (t− r/c) =

p′2

ρ0c+

12ρ0r3

∂F 2(t− r/c)∂t

,

which results for the time averaged quantity (assuming F (t−r/c) is a periodicfunction) in the same expression as for the plane wave

Iavr =

(p′2)avρ0c

.

5.2 Quantitative Measure of Sound

Sound is characterized by its pressure amplitude and its frequency spectradefining the tone color. For an average, young person (about 20 years) the


sensible frequency range is from 16 Hz to 16 kHz. It is of great interest thatthe human perception concerning the frequency of sound signals is a relativeone. This means that frequency differences in two sound signals are sensed asequal, when the ratio of sound is the same (strictly speaking this is true for fre-quencies above 500 Hz, see [228]). E.g., a change in the frequency from 500 Hzto 600 Hz and from 5000 Hz to 6000 Hz will be sensed as equal. Therewith,in most analysis of acoustic signals, 1/3-octave filters, and for some technicalreasons also octave filters are used. Their specifications are as follows:

• Filter specifications (see Fig. 5.5):

fl ... lower frequency limit

fm =√flfu fu ... upper frequency limit

fm ... center frequency• Octave filter:

fu = 2fl

Therewith, the center frequency computes by fm =√

2fl and the band-width of the filter is f = fu − fl = fu/2 = fm/

√2.

• 1/3-Octave filter:fu = 3

√2fl

Therewith, the center frequency calculates as fm = 6√

2fl ≈ 1.12fl and thebandwidth is f = ( 3

√2 − 1)fl ≈ 0.26fl.

Fig. 5.5. Filter function (band-pass)

The sound is often broadband and so makes sense to display the frequencyspectra in so-called frequency bands by applying the above mentioned filters.Furthermore, if we consider e.g., traffic noise, then a too resolved frequencyspectra will lead in most cases to non-reproducible results, so that adequatefrequency bands are used, instead. Figure 5.7 displays the frequency spectraof an acoustic signal, which is generated by a turbulent flow (see Fig. 5.6)


Fig. 5.6. Visualization of the flow structure near a forward-facing step

over a forward-facing step (flow-induced noise). As can be clearly seen, thelevels for the octave filtered spectra are the largest, followed by the 1/3-octavefiltered one and the original signal (fully resolved spectra).

Fig. 5.7. Frequency spectra of acoustic pressure level SPL (see (5.62): full resolution,1/3-octave and octave filtered

Similar to the perception of the tone color is the perception of the humanear to pressure amplitude changes, e.g., a pressure amplitude change fromp0 to 2p0 is sensed as equal with a pressure amplitude change from 5p0 to10p0 (p0 has to be larger than 40 dB, see [228]). Now, it has to be mentionedthat the threshold of hearing of an average human is at about 20µPa andthe threshold of pain at about 200 Pa, which corresponds to an interval of107. Thus, logarithmic scales are mainly used for acoustic quantities. Themost common one is the decibel (dB), which expresses the quantity as a ratio


relative to a reference value. The sound pressure level Lp (SPL) is defined by

Lp = 20 log10

prms

prefpref = 20µPa . (5.62)

The reference pressure pref corresponds to the sound at 1 kHz that an averageperson can just hear. The sound-intensity level LIa is then defined by

LIa = 10 log10

Iava

IrefIref = 10−12 W/m2

, (5.63)

with Iref the reference sound intensity corresponding to pref . Furthermore, thesound-power level LPa computes as

LPa = 10 log10

P ava

ParefParef = 10−12 W , (5.64)

with Paref the reference sound power corresponding to pref . In Tables 5.1 and5.2 some typical sound pressure and sound power levels are listed.

Table 5.1. Typical sound pressure levels SPL

Threshold Voice Car Pneumatic hammer Accelerating motor Jetof hearing at 5m at 20 m at 2m at 5m at 3m

0dB 60 dB 80 dB 100 dB 110 dB 140 dB

Table 5.2. Typical sound power levels and in parentheses the absolute acousticpower Pa

Voice Fan Loudspeaker Jet airliner

30 dB (25µW) 110 dB (0.05 W) 128 dB (60W) 170 dB (50 kW)

In general, an acoustic signal is not mono-frequent, but consists of ansuperposition of signals with different frequencies. For simplicity, let us analyzea pressure signal, which can be expressed as follows

p′(t) = A1 cos(ω1t) +A2 cos(w2t) .

Then, the mean square pressure computes as


p′2rms =1T

to+T∫t0

(A2

1 cos2(ω1t) + 2A1A2 cos(ω1t) cos(ω2t) +A22 cos2(ω2t)

)dt

=

to+T∫t0

A21

2(1 + cos(2ω1t)) dt+

to+T∫t0

A22

2(1 + cos(2ω2t)) dt

+

to+T∫t0

A1A2(cos((ω1 − ω2)t) + cos((ω1 + ω2)) dt .

Assuming that T = n1(2π/ω1) = n2(2π/ω2) = n3(2π/(ω1+ω2)) = n4(2π/(ω1−ω2)) with ni ∈ IN, then all integrals of the form

to+T∫t0

Ai cos(ωit)) dt

are zero, and we obtain

p′rms =

√A2

1

2+A2

2

2.

Generalizing the above case, we can rewrite (5.63) by

Lp = 10 log10

N∑k=1

p2rms,k

p2ref

= 10 log10

(N∑

k=1

10Lavi /10

)(5.65)

Lavi = 10 log10

p2rms,k

p2ref

(5.66)

In (5.65) Lp is the overall sound pressure level OSPL, and can be interpreted asthe formula for adding sound pressure levels generated by different, incoherentsound sources. Figure 5.8 displays the increase of the overall sound pressurelevel OSPL achieved by two sound signals in comparison to the sound pressurelevel SPL of the individual signals. Therewith, due to the logarithmic scaling,which corresponds to the perception of the human ear, the sum of two equalsound pressure levels just increase the overall sound pressure level OSPL by3dB, when both sound signals have the same sound pressure level SPL. For thecase that one sound signal has a SPL of about 20 dB smaller than the other, wecan neglect the contribution of this sound signal to the overall sound pressurelevel OSPL.

An interesting point we want to stress here is the fact that the aboveformula can be also applied to frequency bands, as e.g., obtained by an octave-filtering.


Fig. 5.8. Increase of the overall sound pressure level achieved by two sound signalsas a function of the difference in SPL of the individual signals

5.3 Non-linear Acoustic Wave Equation

In this section we will consider a general formulation for non-linear wavepropagation in lossy and compressible fluid media. Furthermore, we assumezero mean velocity v0, so that v = v′. Thus, we will derive Kuznetsov’sequation for non-linear acoustics [135], which is a second-order approximationfor viscous heat-conducting fluids. For this case, the equation of continuity asgiven in (5.7) still holds. However, Euler’s equation changes to a more generalequation for momentum conservation, known as Navier-Stokes equation (see[135] as well as [59])

ρ

(∂v∂t

+ (v · ∇)v)

+ ∇p = µv∆v +(µv

3+ ζv

)∇(∇ · v) . (5.67)

In (5.67) ζv denotes the bulk viscosity and µv the shear viscosity. Since thefollowing vector identities are fulfilled (domain is assumed to be convex)

∇(∇ · v′) = ∆v′ + ∇ × ∇ × v′ (5.68)

(v · ∇)v′ =12

∇(v′ · v′) − v′ × ∇ × v′ , (5.69)

we can rewrite (5.67) as

ρ∂v′

∂t+ρ

2∇(v′ · v′) + ∇p =

(4µv

3+ ζv

)∆v′ (5.70)

−(µv

3+ ζv

)∇ × ∇ × v′ + ρ0v′ × ∇ × v′ .

5.3 Non-linear Acoustic Wave Equation 157

The last two terms in (5.70) describe acoustic streaming, and according to [84]decrease exponentially with the distance to solid walls. Since in our case wecan still consider the acoustic velocity v as irrotational and set ρ = ρ′ + ρ0 aswell as p = p′ + p0 (assume ∇p0 = 0), (5.70) reads as

ρ0∂v′

∂t+ ρ′

∂v′

∂t+ρ0

2∇(v′ · v′) + ∇p′ =

(4µv

3+ ζv

)∆v′ . (5.71)

The state equation, defining the relation between the acoustic pressurep′ and density ρ′ within the fluid, can be expressed for the non-linear wavepropagation as follows [84]

ρ′ =p′

c2− 1ρ0c4

B

2Ap′2 − κ

ρ0c4

(1cΩ

− 1cp

)∂p′

∂t. (5.72)

In (5.72)B/A denotes the parameter of non-linearity, κ the adiabatic exponent(see (5.20)) and cp, cΩ the specific heat capacitance at constant pressure aswell as constant volume.

Now, the goal is to combine the three equations (5.7), (5.71), and (5.72)into a single wave equation with the acoustic scalar potential ψ as the un-known. Therefore, we will substitute any physical quantity in a second-orderterm by its linearized one, since the resulting errors will be of third order. Thismeans that we will express second-order terms in the continuity equation (5.7)and Euler equation (5.71) by the linearized ones obtained for the linear wavepropagation (see (5.30) and (5.31)). Let us start with the continuity equationin the form

∂ρ′

∂t+ ρ0∇ · v′ = −ρ′∇ · v′ − v′ · ∇ρ′ , (5.73)

where we have arranged all second-order terms on the right. By using

ρ′ ≈ p′

c2(linear state equation)

∇ · v′ ≈ − 1ρ0

∂ρ′

∂t(linear continuity equation)

and the relation

p′∂p′

∂t=

12∂p′2

∂t,

we derive the following approximation

−ρ′∇ · v′ ≈ ρ′

ρ0

∂ρ′

∂t≈ p′

ρ0c4∂p′

∂t=

12ρ0c4

∂p′2

∂t. (5.74)

By using the linear state as well as the linear Euler equations, we transformthe second term on the right-hand side of (5.73) to

−v′ · ∇ρ′ ≈ −v′ ·(

1c2

∇p′)

= −v′ ·(−ρ0

c2∂v′

∂t

)=

ρ0

2c2∂(v′ · v′)

∂t. (5.75)


Thus, our modified equation of continuity reads as

∂ρ′

∂t+ ρ0∇ · v′ =

12ρ0c4

∂p′2

∂t+ρ0

2c2∂(v′ · v′)

∂t. (5.76)

Furthermore, the use of (5.72) for ρ′ leads to

1c2∂p′

∂t− 1ρ0c4

B

2A∂p′2

∂t− κ

ρ0c4

(1cV

− 1cp

)∂2p′

∂t2+ ρ0 ∇ · v′

=1

2ρ0c4∂p′2

∂t+

ρ0

2c2∂(v′ · v′)

∂t. (5.77)

Now, let us apply a similar procedure to (5.71). For the term ρ′∂v′/∂t wefind the following approximation

ρ′∂v′

∂t≈ p′

c2∂v′

∂t≈ − p

′

c21ρ0

∇p′ = − 12ρ0c2

∇p′2 . (5.78)

In addition, we can rewrite according to [1] the term on the right of (5.71)within the second-order approximation by(

4µv

3+ ζv

)∆v′ ≈ 1

ρ0c2

(4µv

3+ ζv

)∇∂p′

∂t. (5.79)

Using (5.78) and (5.79) for (5.71), we arrive at our modified Euler equation

ρ0∂v′

∂t+∇p′ =

12ρ0c2

∇p′2− ρ0

2∇(v′ ·v′)− 1

ρ0c2

(4µv

3+ ζv

)∇∂p′

∂t. (5.80)

Similar to the linear case, we apply the divergence operator to (5.80)

ρ0

(∇ · ∂v

′

∂t

)+∆p′ =

12ρ0c2

∆p′2 − ρ0

2∆(v′ · v′) − 1

ρ0c2

(4µv

3+ ζv

)∆∂p′

∂t(5.81)

and a time derivative to (5.77)

1c2∂2p′

∂t2− 1ρ0c4

B

2A∂2p′2

∂t2− κ

ρ0c4

(1cV

− 1cp

)∂3p′

∂t3+ ρ0

∂

∂t(∇ · v′) =

12ρ0c4

∂2p′

∂t2+ρ0

2c2∂2(v′ · v′)

∂t2. (5.82)

Subtracting (5.82) from (5.81), allowing to interchange time and spatialderivative (means that the term ρ0(∂/∂t(∇ · v′)) cancels out) yields

∆p′ − 1c2∂2p′

∂t2= − 1

ρ0c4B

2A∂2p′2

∂t2− κ

ρ0c4

(1cV

− 1cp

)∂3p′

∂t3

−ρ0

2∆(v′ · v′) − 1

ρ0c2

(4µv

3+ ζv

)∆∂p′

∂t

− 12ρ0c4

∂2p′2

∂t2− ρ0

2c2∂2(v′ · v′)

∂t2+

12ρ0c2

∆p′2 . (5.83)

5.3 Non-linear Acoustic Wave Equation 159

Furthermore, we use the relations between time and spatial derivatives ac-cording to the linear wave equation for pressure and acoustic velocity

∆p′ =1c2∂2p′

∂t2(5.84)

∆v′ =1c2∂2v′

∂t2. (5.85)

Applying (5.84) and (5.85) to the right-hand side of (5.83) yields

∆p′ − 1c2∂2p′

∂t2= − b

c2∂∆p′

∂t− 1ρ0c4

B

2A∂2p′

∂t2− ρ0

c2∂2(v′ · v′)

∂t2, (5.86)

with b the diffusivity of sound

b =1ρ0

(4µv

3+ ζv

)+κ

ρ0

(1cV

− 1cp

).

Since the relations between the pressure as well as velocity to the scalar ve-locity potential ψ still hold in the non-linear case

v′ = −∇ψ

p′ = ρ0∂ψ

∂t,

we can rewrite (5.86) as follows

∆

(ρ0∂ψ

∂t

)− 1c2

∂2

∂t2

(ρ0∂ψ

∂t

)= − b

c2∂

∂t∆

(ρ0∂ψ

∂t

)

− 1ρ0c4

B

2A∂2

∂t2

(ρ0∂ψ

∂t

)2

−ρ0

c2∂2

∂t2(∇ψ · ∇ψ) . (5.87)

Thus, the non-linear wave equation, also called Kuznetsov’s equation, has theform

c2∆ψ − ∂2ψ

∂t2= − ∂

∂t

(b∆ψ +

1c2

B

2A

(∂ψ

∂t

)2

+ ∇ψ · ∇ψ

). (5.88)

Besides Kuznetsov’s equation, the following three partial differential equa-tions modelling non-linear wave propagation are widely used (see e.g., [84]):

• Burger’s equation:

∂p′

∂x− b

2c3∂2p′

∂τ2=

βa

ρ0c3p∂p′

∂τ, (5.89)


with τ = (t − x/c) the retarded time, b the diffusivity of sound andβa = 1 + B/2A the coefficient of non-linearity. The Burger’s equationallows to study the combined effect of dissipation and non-linearity onprogressive plane waves.

• Westervelt equation:

∇2p′ − 1c2∂2p′

∂t2+

b

c4∂3p′

∂t3= − βa

ρ0c4∂2p′2

∂t2, (5.90)

with b the diffusivity of sound. This partial differential equation can de-scribe the propagation of plane waves including the non-linearity effectsas well as dissipation. This means that we can use this equation, when cu-mulative non-linear effects dominate local non-linear effects. For example,once the propagation distance is greater than a wavelength, the waveformdistortion is dominated by the cumulative effect. The approximation is notappropriate for the simulation of standing waves.

• Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation:

∂2p′

∂x∂t− c

2∇2

⊥p′ − b

2c3∂3p′

∂t3=

βa

2ρ0c3∂2p′2

∂t2, (5.91)

with βa = 1 + B/2A the coefficient of non-linearity, b the diffusivityof sound and ∇⊥ the nabla operator including just partial derivativeswith respect to the transversal coordinates (in our case y and z, sincethe wave propagates in the x-direction). The simplification (compared toKuznetsov’s equation) is given by modelling directional wave propagationfulfilling

k⊥kx

1 ,

with k⊥ the wave number in the transversal direction and kx the wavenumber in the direction of propagation. This approximation is mainlyused for investigation of diffraction, non-linearity and dissipative effects indirectional sound beams, e.g., occurring in medical ultrasound, acousticmicroscopy or non-destructive testing.


Within this section we will discuss the FE formulation of the linear waveequation as well as Kuznetsov’s wave equation for the non-linear case. Fur-thermore, we will derive an FE formulation, which allows different mesh sizesfor the computational subregions.


5.4.1 Linear Acoustics

The strong formulation for linear acoustics expressed by the acoustic pressurereads as follows1

Given:f : Ω → IRc : Ω → IR .

Find: p′(t) : Ω × [0, T ] → IR

1c2p′ −∆p′ = f . (5.92)

Boundary conditionsp′ = p′e on Γe × (0, T )

∂p′

∂n= p′n on Γn × (0, T ) .

Initial conditionp′(r, 0) = p′0 , r ∈ Ωp′(r, 0) = p′0 , r ∈ Ω .

In (5.92) f denotes any excitation function for generating the acoustic wave.For simplicity we set the boundary as well as initial conditions to zero. Toobtain the variational formulation, we multiply (5.92) by an appropriate testfunction and perform a partial integration. Thus, the weak form reads as:Find p′ ∈ H1

0 such that∫Ω

1c2w p′ dΩ +

∫Ω

∇w · ∇p′ dΩ −∫Ω

wf dΩ = 0 (5.93)

for any w ∈ H10 . Using standard nodal finite elements, we approximate the

continuous acoustic pressure p′ as well as the test function w by

p′ ≈ p′h =neq∑a=1

Nap′a (5.94)

w ≈ wh =neq∑a=1

Nawa . (5.95)

Thus, (5.93) is transformed to the following semidiscrete Galerkin formulation

1 same formulation holds for the acoustic velocity potential ψ


neq∑a=1

neq∑b=1

⎛⎝ ∫

Ω

1c2NaNb dΩ p′b

+∫Ω

(∇Na) · ∇Nb dΩ p′b −∫Ω

Naf(ra) dΩ

⎞⎠ = 0 . (5.96)

In matrix form, (5.96) reads as

Mp′ + Kp′ = f , (5.97)

with

M =ne∧

e=1

me ; me = [mpq] ; mpq =∫

Ωe

1c2NpNq dΩ

K =ne∧

e=1

ke ; ke = [kpq] ; kpq =∫Ωe

(∇Np) · ∇Nq dΩ

f =ne∧

e=1

fe ; fe = [fp] ; fp =∫Ωe

Npf(rp) dΩ .

The time discretization is performed by applying a standard Newmark algo-rithm (see Sect. 2.5.2), which results in the following time-stepping scheme(effective mass matrix formulation):


p′ = p′n

+∆t p′n

+∆t2

2(1 − 2βH) p′

n(5.98)

˜p′ = p′

n+ (1 − γH)∆t p′

n. (5.99)


M∗p′n+1

= fn+1

− Kp′n

(5.100)

M∗ = M + βH∆t2 K .


p′n+1

= p′ + βH∆t2 p′

n+1(5.101)

p′n+1

=˜p′ + γH∆t p′n+1

. (5.102)

The question if one should solve for the acoustic pressure or acoustic ve-locity potential is a question of the boundary conditions.


• Acoustic pressure: p′ = p′e on Γ

Clearly, in this case we will solve for p′.

• Normal component of acoustic particle velocity: n · v′ = v′n on Γ

Due to the relation v′ = −∇ψ we obtain

n · v′ = v′n = −∂ψ∂n

,

and we will solve for ψ and compute p′ according to (5.37).

• Mixed boundary conditions: p′ = p′e on Γe; n · v′ = v′n on Γn

In an acoustic pressure formulation the boundary conditions will read ( byuse of (5.30))

p′ = p′e on Γe

∂p′

∂n= −ρ0

∂vn∂t

on Γn ,

whereas for the acoustic velocity potential ψ we obtain (using (5.37))

ρ0∂ψ

∂t= p′e on Γe

−∂ψ∂n

= vn on Γn .

For the time harmonic case, we will always apply an acoustic pressureformulation with boundary conditions defined by

p = pe on Γe

∂p

∂n= −jωρ0vn on Γn .

5.4.2 Non-linear Acoustics

The strong formulation for the non-linear wave equation is given as follows

Given:f : Ω → IRc : Ω → IRB/A, b : Ω → IR .

Find: ψ(t) : Ω × [0, T ] → IR

1c2∂2ψ

∂t2−∆ψ = f +

1c2∂

∂t

(b(∆ψ) +

B/A

2c2

(∂ψ

∂t

)2

+ (∇ψ)2). (5.103)


Boundary conditionsψ = ψe on Γe × (0, T )

∂ψ

∂n= ψn on Γn × (0, T ) .

Initial conditionψ(r, 0) = ψ0 , r ∈ Ωψ(r, 0) = ψ0 , r ∈ Ω .

For simplicity we set all boundary conditions as well as initial conditions tozero. Now, to obtain the variational formulation of (5.103), we multiply (5.103)by an appropriate test function and perform partial integration for the Laplaceterm on the left- and right-hand sides. Thus, the variational formulation readsas follows: Find ψ ∈ H1

0 such that∫Ω

1c2wψ dΩ +

∫Ω

(∇w) · (∇ψ) dΩ =∫Ω

ωf dΩ

−∫Ω

∇(b

c2w

)· (∇ψ) dΩ

+∫Ω

2c4

B

2Awψψ dΩ

+∫Ω

w∇(

2c2ψ

)· (∇ψ) dΩ (5.104)

for any w ∈ H10 . In the following, we will assume that c and b are constant

all over the computational domain. Using the same ansatz for the approxima-tions as in the linear case (see (5.94)), we arrive at the following semidiscreteGalerkin formulation

neq∑a=1

neq∑b=1

(−∫Ω

1c2NaNb dΩ ψb

−∫Ω

(∇Na) · (∇Nb) dΩ ψb

−∫Ω

b

c2(∇Na) · (∇Nb) dΩ ψb

+∫Ω

2c4

B

2ANaNb

(neq∑c=1

Ncψc

)dΩ ψb


+∫Ω

2c2Na(∇Nb) ·

(∇

neq∑c=1

Ncψc

)dΩ ψb

+∫Ω

Naf dΩ

)= 0. (5.105)

In matrix form, we may write (5.105) as

Mψψ + Kψψ + Cψψ − N1ψ(ψ)ψ − N2

ψ(ψ)ψ = f , (5.106)

with

Mψ =ne∧

e=1

meψ ; me

ψ = [mpq] ; mpq =∫Ωe

1c2NpNq dΩ (5.107)

Kψ =ne∧

e=1

keψ ; ke

ψ = [kpq] ; kpq =∫Ωe

(∇Np) · (∇Nq) dΩ (5.108)

Cψ =ne∧

e=1

ceψ ; ce

ψ = [cpq] ; cpq =∫

Ωe

b

c2(∇Np) · (∇Nq) dΩ (5.109)

N1ψ =

ne∧e=1

(n1ψ)e ; (n1

ψ)e = [n1pq] ;

n1pq =

∫Ωe

2c4

B

2ANpNq

(nen∑c=1

Ncψc

)dΩ (5.110)

N2ψ =

ne∧e=1

(n2ψ)e ; (n2

ψ)e = [n2pq] ;

n2pq =

∫Ωe

2c2Np(∇Nq) ·

(∇

nen∑c=1

Ncψc

)dΩ (5.111)

f =ne∧

e=1


Npf(rp) dΩ . (5.112)

The time discretization is performed by a standard Newmark algorithm (seeSect. 2.5.2). Since (5.106) is a non-linear equation, we have to apply an it-erative scheme. By shifting all non-linearities to the right-hand side of theequation system, we arrive at the following scheme:


1. Perform predictor step:

ψ = ψn

+∆t ψn

+∆t2

2(1 − 2βH) ψ

n= ψk

n+1(5.113)

˜ψ = ψn

+ (1 − γH)∆t ψn

= ψk

n+1. (5.114)

2. Solve algebraic system of equations:

M∗ψψ

k+1

n+1= f

n+1(5.115)

− Kψψ − Cψ˜ψ + N1

ψ(ψk

n+1)ψ

k

n+1+ N2

ψ(ψk

n+1)ψ

k

n+1

M∗ψ = Mψ + γH∆tCψ + βH∆t

2 Kψ . (5.116)

3. Perform corrector step:

ψk+1

n+1= ψ + βH∆t

2 ψk+1

n+1(5.117)

ψk+1

n+1= ˜ψh + γH∆t ψ

k+1

n+1. (5.118)

4. Test convergence:

‖ψk+1

n+1− ψ

k

n+1‖2

‖ψk+1

n+1‖2

≤ δψ

fulfilled : perform next time stepnot fulfilled : k := k + 1 goto step 2 . (5.119)

5.4.3 Non-matching Grids

Within this section, we will investigate flexible discretization techniques forthe approximative solution of acoustic wave propagation problems. In particu-lar, we will introduce non-matching grids, where we can allow totally differentgrids to exist at the interface of two subregions. This is in particular of greatinterest in computational aeroacoustics, where we need a much finer grid inthe source region, where the turbulent flow generates the sound, than in theregions, where we just compute the propagation of sound (see Sect. 10). Inorder to keep as much flexibility as possible, we intend to use independentlygenerated grids which are well suited for approximating the solution of decou-pled local subproblems in each subdomain. Therefore, we have to deal withthe situation of non-conforming grids appearing at the common interface oftwo subdomains. Special care has to be taken in order to define and implementthe appropriate discrete coupling operators.

For this type of problem, we can use the well-tested and well-studied frame-work of mortar methods (see e.g., [21, 217]). The global domain and its de-composition is displayed in Fig. 5.9. Thus, in each subdomain we have to solvethe wave equation for the acoustic pressure p′i : Ωi × (0, T ) → IR,


Fig. 5.9. Acoustic domain with two subregions Ω1 and Ω2 with different discretiza-tions

1c2p′i −∆p′i = fi, in Ωi × (0, T ), i = 1, 2. (5.120)

completed by appropriate initial conditions at time t = 0 and boundary con-ditions on the global boundary Γf.

Interface condition

For simplicity, we use the same equation and primal variable in both subdo-mains, and the interface is just artificial, i.e., no material change occurs. Werefer to [13, 136] for the treatment of more general situations. Therefore, inthe strong setting, it is natural to impose continuity in the trace and flux ofthe acoustic pressure, i.e.,

p′1 = p′2 and∂p′1∂n

=∂p′2∂n

on Γ I.

The flux coupling condition will be enforced in a strong sense by introducingthe Lagrange multiplier

λ = −∂p′1

∂n= −∂p

′2

∂n. (5.121)

However, the continuity in the trace will be understood in a weak sense∫Γ I

(p′1 − p′2)µ dΓ = 0 (5.122)

for all test functions µ out of a suitable Lagrange multiplier space. We notethat a correct functional framework is presented in [13].

Coupled formulation

We proceed as in Sec. 5.4.1 and obtain from (5.120), ignoring for the momentthe boundary condition on Γ f and the excitations fi,

168 5 Acoustic Field∫Ωi

1c2wi p′i dΩ +

∫Ωi

∇wi ·∇p′i dΩ −∫

Γ I

wi ni·∇p′i dΓ = 0

for all test functions wi, i = 1, 2. Inserting the definition of the Lagrangemultiplier (5.121) and summing up, we obtain the symmetric evolutionarysaddle point problem of finding p′1, p

′2, λ such that

2∑i=1

(∫Ωi

1c2wip′i dΩ +

∫Ωi

∇wi ·∇p′i dΩ)

+∫

Γ I

(w1 − w2)λdΓ = 0 (5.123)∫Γ I

(p′1 − p′2)µ dΓ = 0, (5.124)

for all µ and wi, i = 1, 2. In contrast to the purely primal coupled formulation(8.9),(8.10), we now face a primal-dual problem where the coupling is realizedin terms of Lagrange multipliers.

Spatial discretization

For the approximation of the acoustic pressures p′1 and p′2, we may again usestandard nodal finite elements on the triangulations T1 and T2, respectively.Special attention has to be paid to the construction of a discrete Lagrangemultiplier space Mh for approximating the interface flux λ. It is convenient todefine Mh with respect to the (d−1)-dimensional grid inherited from one ofthe subdomains Ω1, Ω2. The interface grid Tn chosen in this way is referredto as non-mortar or slave side, and the other interface grid Tm as mortar ormaster side. There exist several possibilities for choosing the basis functionsspanning Mh, most of them with equal mathematical stability and approxi-mation properties. In Fig. 5.10, four types of basis functions for a 1D interfaceusing linear or bilinear finite elements are presented: the standard ones co-inciding with the trace space Wh of the finite element space on Tn, the dual

Fig. 5.10. Basis functions: standard, discontinuous dual, continuous dual, and con-stant Lagrange multipliers

ones spanned by piecewise linear discontinuous basis functions satisfying abiorthogonality relation with the nodal basis functions of Wh [217], the dual


continuous ones where the discontinuous dual basis functions are modified bycubic polynomials [218], and the piecewise constant ones spanned by basisfunctions, which are constant from one edge midpoint to the next.

Again switching to the algebraic formulation and assuming that we havechosen the Lagrange multipliers with respect to Ω1, the discretization of(5.123), (5.124) yields⎛

⎜⎜⎝M1 0 0

0 M2 0

0 0 0

⎞⎟⎟⎠⎛⎜⎜⎝p′1

p′2

λ

⎞⎟⎟⎠+

⎛⎜⎜⎝

K1 0 D

0 K2 M

DT MT 0

⎞⎟⎟⎠⎛⎜⎜⎝p′1

p′2

λ

⎞⎟⎟⎠ =

(f

u

0

). (5.125)

The coupling matrices D,M are given by

D =nIe∑e=1

De; De = [Dpq]; Dpq =∫

Γ eI

Nmp φ

nq dΓ, (5.126)

M = [Mpq]; Mpq =∫

Γ I

Nmp φ

nq dΓ, (5.127)

where Nnp and Nm

p denote the finite element basis functions on Tn and Tm,respectively, and φn

q denotes the Lagrange multiplier basis function associatedwith the slave node q. We note that the assembly of D poses no difficultysince all basis functions involved are defined with respect to the same grid Tn.Moreover, if the dual Lagrange multipliers are used, D becomes a diagonalmatrix. However, the assembly of M is more involved, since Nm

p and φnq are

defined with respect to different grids. We will discuss the necessary steps fora straight interfaces in 2D, and refer to [67] for a detailed description in 2Dand 3D including curved interfaces.

In order to determine whether two surface elements en and em intersect,we loop over all vertices pm

k , k = 1, . . . , nmv , of em, where nm

v denotes thecorresponding number of vertices. The two elements intersect, if we find atleast one vertex pm

k which lies inside en. An easy way to justify this, is to usethe transformation of global to local coordinates, which is usually availablein any finite element code. If all local coordinates of pm

k with respect to theelement en are within the correct ranges, the elements en and em intersect.Otherwise, it may still be possible that the element en is completely coveredby the element em. Therefore, one has to repeat the procedure interchangingthe roles of en and em. If still no appropriate vertex is found, the two elementsdo not intersect.

We consider the case that the interface Γ is part of the x-axis. The cornercoordinates of en and em are denoted by xn

and xm with = 1, 2, respectively.

We assume that xn1 < xn

2 and xm1 < xm

2 . The possible four situations andresulting intersections are illustrated in Fig. 5.11.


Fig. 5.11. Four situations of intersecting elements en and em in 2D.

5.4.4 Discretization Error

In this section, we will investigate the discretization error of the FE methodapplied to the wave equation and will provide practical rules for choosing themesh size h as a function of the wave number k. In the first step we willrestrict to the time harmonic case to get rid off the error introduced by thetime discretization. The wave equation in the time harmonic case, also calledHelmholtz equation, reads as follows

∆p+ k2p = 0 (5.128)

with k = ω/c = 2πf/c the wave number. The comparison of the FE solutionto the analytic solution of this kind of PDE has already been discussed in Sect.2.2 (just substitute there the parameter c by k). There, we have observed thatthe FE solution for the parameter c = 0 is no longer exact at the FE nodes(see Fig. 2.7). In the context of the Helmholtz equation we decompose thetotal discretization error into a local error eint (interpolation error) defined by

eint = p− pI

and into the pollution error epol computed by

epol = pI − ph .

This decomposition is illustrated in Fig. 5.12. A very important result con-cerning the error analysis has been reported in [106], which reads as follows

eh ≤ C1(kh) + C2

(k3h2

)kh < 1 (5.129)

eh =|p− ph|H1

|p|H1

with | |H1 the H1 semi-norm (see Sect. C) and C1, C2 constants independentof k as well as h. In (5.129) the first term represents the local error and thesecond term the pollution error. The main effect of the pollution error is thatthe wave number kh of the FE solution is different from the wave number kof the analytic solution. We call this dispersive and therefore speak from thedispersion error. The behavior of the dispersion error has been extensivelystudied in [3]. This analysis provides a concrete guideline for choosing theorder q of the finite element basis functions and the mesh size h in order thatthe dispersion error is virtually eliminated. This relation is given by


p(x)

pp

p

x

eint

epol

^

^

^

^

h

I

Fig. 5.12. Decomposition of the discretization error into the interpolation erroreint

and pollution error epol

q +12>kh

2+ C(kh)1/3 (5.130)

with the constant C, which can in practice be set to one. In addition, we canprecisely define the dispersion error in the small wave number limit kh << 1and the high wave number limit kh >> 1

cos(khh) − cos(kh) =12

(q!

(2q)!

)2 (kh)2q+2

2q + 1+O(kh)2q+4 kh << 1

cos(khh) − cos(kh) =sin(kh)

2

(khe

2(q + 1)

)2q+1

kh >> 1

Let us consider the simple case of an acoustic wave in a channel with givenDirichlet boundary conditions on the left and right end (see Fig. 5.13). We

Fig. 5.13. Acoustic wave in a channel with given Dirichlet boundary conditions

keep the discretization fixed by 20 finite elements and change the wave numberk (see Table 5.3) as well as the order q of the finite element basis functions.


Table 5.3. Choice of wave number k and resulting number of finite elements perwavelength

k 9π/2 19π/2 29π/2

λ/h 8.9 4.2 2.7

For comparison we compute the accumulated error Ea

Ea =

√√√√√√√nn∑i=1

(ph − p)2

nn∑i=1

p2

with nn the number of finite element nodes. The analytic solution p computesby

p = cos(kx) − cos(kL)sin(kL)

sin(kx) .

Figure 5.14 clearly demonstrates that we effectively reduce the error by in-creasing the order q of the finite element basis functions. However, as shownby the case k = 29/2 π, we do not fulfill (5.130) at the mesh with q = 1 andso no reduction of the error is achieved by increasing the order of the finiteelement basis functions from q = 1 to q = 2. This clearly demonstrates, that(5.130) is a very important and practical guidline for acoustic computations.

Fig. 5.14. Accumulated error

In a second example, we consider a sine pulse travelling through a channeland record the wave signal at a distance of two times the fundamental wave-

5.5 Treatment of Open Domain Problems 173

length. Now, in addition to the correct choice of the mesh size h we have todeal with the time discretization. Figure 5.15 displays the setup. The choice of

Propagation direction

Excitation: sine pulse

p´(t)

t

Recording of wave signals

Fig. 5.15. Setup of the transient case study

the time and spatial discretization is taken as a function of the fundamentalexciting frequency f0, which results in a fundamental wavelength λ0 = c/f0.Table 5.4 lists the mesh size h as well as time step size ∆t for the three dif-ferent computations. In Fig. 5.16 it can be clearly noticed that around the

Table 5.4. Time and spatial discretization

Discretization 1 Discretization 2 Discretization 3

h 1/(20f0) 1/(40f0) 1/(60f0)∆t λ0/20 λ0/40 λ0/60

expected pulse wave parasitic waves arise due to the numerical dispersion. Infact, the generated parasitic waves occur since the discrete speed of soundch is a function of the wave number k and therefore of the frequency of thewave. Clearly, this is a pure numerical effect, since in a homogeneous mediathe continuous speed of sound c is constant. For a detailed discussion on thistopic we refer to [43].

5.5 Treatment of Open Domain Problems

Many applications within computational acoustics are open-domain problems.In order to correctly compute such problems with the FE method, we have todefine appropriate boundary conditions. Using a simple homogeneous Dirich-let or Neumann boundary condition will result in a total reflection of the


Fig. 5.16. Recorded pulse signals at a distance of two times the fundamental wave-length

outgoing waves at the boundary. Therefore, special boundary conditions haveto be developed for absorbing the waves impinging on the artificial boundaryimposed on the acoustic domain.

The three main approaches handling open-domain problems with the FEmethod can be briefly described as follows:

• Infinite finite elements:At the boundary an additional layer of finite elements, so-called infinite

finite elements try to model the effect of waves in the outer region (non-computational domain). For a basic discussion we refer to [166], and foran advanced formulation see [52].

• Absorbing boundary conditions:The general idea of applying at the boundary a special condition, so-calledabsorbing boundary conditions, is that only outgoing waves can pass theboundary.

• Perfectly matched layer (PML) technique:By this method the computational domain is surrounded by a dampinglayer, which is constructed in such a way that the acoustic impedance ofthis layer matches the impedance of the computational (wave propagation)domain and in which the waves are absorbed.


In the following, we will provide details about ideas and computer imple-mentation of local absorbing boundary conditions of first order and of theperfectly matched layer technique.

5.5.1 Absorbing Boundary Conditions

In [42,60] absorbing boundary conditions for general classes of wave equationsare discussed. Subsequently, a hierarchy of highly absorbing local boundaryconditions that approximate the theoretical non-local relation have been ob-tained. The locality of these conditions preserves the sparsity of the FE ma-trices. For an explanation of how absorbing boundary conditions work, let usconsider a boundary located at x = 0. A wave with one degree of freedom

Fig. 5.17. Wave propagating in the direction of negative x values

propagating in the direction of negative x values can be expressed by

u−(x, t) = u0 ej(ωt+kx) (5.131)

with ω the angular frequency and k the wave number. By considering therelation for the speed of sound c = ω/k, it can be easily proved that the wavefulfills the following condition(

∂

∂t− c

∂

∂x

)u = 0 . (5.132)

Thus, the wave can totally pass (no reflection) the boundary. However, a wave,which is propagating towards positive x values and that can be mathematicallymodeled by

u+(x, t) = u0 ej(ωt−kx) (5.133)

will not fulfill (5.132) and will be totally reflected. Indeed, (5.132) defines alocally absorbing boundary condition of first order (see [42,60]). An extension


of this method to waves in solids (especially piezoelectric materials) can befound in [93].

To derive the correct formulation including the absorbing boundary con-dition as given in (5.132), we start from the weak form (see (5.93)) withoutsetting the boundary integral to zero (which would correspond to homoge-neous Neumann boundary condition)∫

Ω

1c2w p′ dΩ +

∫Ω

∇w · ∇p′ dΩ −∫Γ

w∂p′

∂ndΓ −

∫Ω

wf dΩ = 0 . (5.134)

Now, in all directions we can express the 1D spatial derivative in (5.132) asthe normal derivative in (5.134). By doing so, special attention has to begiven to the direction of the normal vector relative to the positive directionof the x axis. In our case, the normal points out of the boundary towards theinfinite domain that is in the same direction of the outgoing wave, whereas xis positive in the direction opposite to the propagation of the normal wave.Finally, changing the direction of the normal vector and substituting (5.132)into (5.134) results in∫Ω

1c2w p′ dΩ+

∫Ω

∇w · ∇p′ dΩ+∫Γ

1cw∂p′

∂tdΓ −

∫Ω

wf dΩ = 0 . (5.135)

The additional surface integral, including a first time derivative of the acousticpressure p′, may be seen as a damping matrix C acting only on the surface ofthe computational domain. After performing the spatial discretization usingfinite elements, we obtain the following formula for the computation of C

C =nΓ

e∧e=1

ce ; ce = [cpq] ; cpq =∫Γ e

1cNpNq dΓ , (5.136)

with nΓe the number of surface elements (for the evaluation of the surface

integral see Sect. 2.6). The matrix C is almost empty, since only terms alongthe boundary Γ of the domain contribute to its non-zero entries.

5.5.2 Perfectly Matched Layer (PML) Technique

There has been much research work on the PML-technique since the firstintroduction of this technique [20] (see e.g., [45,85,97,200]). In this section wewill focus on the acoustic wave propagation in the frequency domain and investin basic ideas concerning impedance matching, describe the PML-techniqueand prove the perfect matching.

For linear acoustics in the 1D case the relation between the acoustic par-ticle velocity v′ = vξeξ and the acoustic pressure p′ is given by (conservationof mass and momentum, see Sect. 5.1)


∂p′

∂t= −ρ0c

2∂v′ξ∂ξ

∂v′ξ∂t

= − 1ρ0

∂p′

∂ξ,

which results in the linear wave equation for p′

1c2∂2p′

∂t2− ∂2p′

∂ξ2= 0 . (5.137)

For a plane wave the acoustic impedance Za is calculated by ρ0c, and thereflection coefficient R between two media as displayed in Fig. 5.18 computesas

R =Za2 − Za1

Za2 + Za1with Za1 = ρ0c Za2 = ρc . (5.138)

Clearly, to make R equal to zero, Za2 has to match Za1. However, we are free

Fig. 5.18. Plane wave impinging perpendicular (left) and at angle ϕ1 to the interface(right)

in choosing the two quantities ρ and c in the damping region in such a waythat just their product equals ρ0c. Therefore, by setting

ρ = ρ0(1 − jσξ) c =c

1 − jσξ(5.139)

in the damping region, we obtain for the acoustic impedance Za2 = Za1.Furthermore, the wave equation (5.137) reads in the time-harmonic case as

ω2

c2p+

∂2p

∂ξ2= 0 (5.140)

with ω the pulsation of the wave and p the acoustic pressure amplitude. With-out any restrictions, we choose σξ = σ0 (constant within the damping region)and obtain

ω2

c2(1 − jσ0)2︸︷︷︸

k2

p+∂2p

∂ξ2= 0 . (5.141)

Therewith, we arrive at a wave equation with a complex wave number k


k =ω

c(1 − jσ0) = k(1 − jσ0) . (5.142)

Since the general solution of (5.141) is

p = p0ej(ωt−kξ) = p0e

j(ωt−kξ) e−σ0ξ ,

we achieve our goal that we have impedance matching and damping of thewave at the same time.

However, if we consider a plane wave impinging at an angle ϕ to the normalvector of the interface (see Fig. 5.18), then the acoustic impedances computeby (see e.g., [170])

Za1 =ρ0c

cosϕ1Za2 =

ρc

cosϕ2

and our method will not be working. Therefore, as introduced in [20] we needa splitting of the acoustic pressure p′ into p′x, p′y and p′z and introduce artificialdamping. Therewith, the mass as well as momentum conservation equationfor linear acoustics change to

∂p′x∂t

+ σxp′x = −ρ0c

2 ∂v′x

∂x

∂v′x∂t

+ σxv′x = − 1

ρ0

∂p′

∂x(5.143)

∂p′y∂t

+ σyp′y = −ρ0c

2∂v′y∂y

∂v′y∂t

+ σyv′y = − 1

ρ0

∂p′

∂y(5.144)

∂p′z∂t

+ σzp′z = −ρ0c

2 ∂v′z

∂z

∂v′z∂t

+ σzv′z = − 1

ρ0

∂p′

∂z(5.145)

In the above equations σx, σy and σz are damping functions, which are zerowithin the acoustic propagation domain and which are different from zerowithin the PML-layer enclosing the acoustic propagation domain (see Fig.5.19). By applying a Fourier-transformation to (5.143) - (5.145) and rear-

Fig. 5.19. Computational setup: propagation region surrounded by a PML-region

ranging the involved terms, we arrive at the following equations


px = −ρ0c2 1jω + σx

∂vx

∂xvx = − 1

ρ0

1jω + σx

∂px

∂x(5.146)

py = −ρ0c2 1jω + σy

∂vy

∂xvy = − 1

ρ0

1jω + σy

∂py

∂y(5.147)

pz = −ρ0c2 1jω + σz

∂vz

∂zvz = − 1

ρ0

1jω + σz

∂pz

∂z. (5.148)

As can be seen from (5.146) - (5.148), we can directly compute vx, vy andvz as a function of px, py and pz. Therewith, we obtain for the total acousticpressure p = px + py + pz the following modified Helmholtz equation

ηyηz∂

∂x

(1ηx

∂p

∂x

)+ ηxηz

∂

∂y

(1ηy

∂p

∂y

)+ ηxηy

∂

∂z

(1ηz

∂p

∂z

)

+ ηxηyηz k2 p = 0 . (5.149)

In (5.149) k = ω/c denotes the acoustic wave number and the functions η1,η2 and η3 compute as follows

ηx = 1 − jσx

ωηy = 1 − j

σy

ωηz = 1 − j

σz

ω. (5.150)

This decomposition of the acoustic pressure p′ into its x-component, y-component and z-component is the key point of the PML-technique. For aphysical interpretation we will consider an interface between a propagationand a PML region, which is parallel to the y-axis and has a normal vectorpointing in x-direction (see Fig. 5.20). Now, the x-component p′x of the to-

Fig. 5.20. Physical interpretation of PML-technique

tal acoustic pressure p′ can be regarded as a plane wave propagating just inx-direction and which will be damped in the PML-region with the dampingcoefficient σx. Therewith, we have the 1D case and achieve our goal of a reflec-tionless interface, if simply the impedances matches. This simply considera-tion already provides us with the information, where to choose the individualdamping coefficients σx, σy and σz different from zero. Figures 5.21 and 5.22display the PML-technique for the 2D and 3D case.


Fig. 5.21. Construction of PML layer in 2D

Fig. 5.22. Construction of PML layer in 3D

At this point we want to note that there is a second general method to de-rive (5.149). This is known as a mapping of the solution of Helmholtz equationin the real coordinate space to a complex coordinate space, e.g., an analyticcontinuation of the solution (see e.g., [45, 208]).

Now, let us prove that the perfectly matched layer will have the sameacoustic impedance as the propagation medium (a similar proof is given in [20]for electromagnetic waves). Since any general solution of the homogeneouswave equation can be expressed by the superposition of plane waves, we canrestrict our investigation to plane waves impinging at an arbitrary angle ϕ onthe interface, where the perfectly matched layer starts. Therewith, we considera setup as displayed in Fig. 5.23. In this 2D consideration, the set of partialdifferential equations describing the plane wave in the layer are as follows


∂v′x∂t

+ σxv′x = − 1

ρ0

∂

∂x(p′x + p′y)

∂v′y∂t

+ σyv′y = − 1

ρ0

∂

∂y(p′x + p′y) (5.151)

∂p′x∂t

+ σxp′x = −ρ0c

2 ∂v′x

∂x

∂p′y∂t

+ σyp′y = −ρ0c

2∂v′y∂y

. (5.152)

Our goal is to compute the acoustic impedance in the perfectly matched layer.Since we assume a plane wave, we can perform the following ansatz for the

Fig. 5.23. Wave propagation in the PML region

general solutions of the components of the acoustic particle velocity and pres-sure

v′x = v0 sinϕej(ωt−kxx−kyy) v′y = v0 cosϕej(ωt−kxx−kyy) (5.153)

p′x = px0 ej(ωt−kxx−kyy) p′y = py0 e

j(ωt−kxx−kyy) . (5.154)

In the first step, we substitute these general solutions into (5.152) and obtain

px0 =ρ0c

2kxv0 sinϕjω + σx

=ρ0c

2v0 sinϕηx

kx

ω(5.155)

py0 =ρ0c

2kyv0 cosϕjω + σy

=ρ0c

2v0 cosϕηy

ky

ω. (5.156)

Using this intermediate results in (5.151) will lead us to the following tworelations

ηx sinϕ =jkxc

2

ω2

(kx sinϕηx

+ky cosϕηy

)(5.157)

ηy cosϕ =jkyc

2

ω2

(kx sinϕηx

+ky cosϕηy

), (5.158)

so that we can e.g., express ky by a function of kx

ky =ηy cosϕηx sinϕ

kx .


Substituting this result into (5.157) leads to a quadratic equation for kx

η2x sin2 ϕ =

k2xc

2

ω2

(sin2 ϕ+ cos2 ϕ

)︸︷︷︸=1

,

form which we just take the positive solution

kx =ω

cηx sinϕ = k sinϕ

(1 − j

σx

ω

). (5.159)

Performing similar steps results in the expression for ky

ky =ω

cηy cosϕ = k cosϕ

(1 − j

σy

ω

). (5.160)

Therewith, the substitutions of these terms into (5.155) and (5.156) lead to

px0 = ρcv0 sin2 ϕ py0 = ρcv0 cos2 ϕ ,

so that the total pressure computes as

px0 + py0 = ρcv0(sin2 ϕ+ cos2 ϕ

)= ρcv0 .

The acoustic impedance of the perfectly matched layer now results in the sameexpression as for a plane wave in the propagation media

Za =p

v= ρ0c .

The proper choice of the damping functions is of great importance, espe-cially in order to obtain a very robust and efficient PML-technique. For thispurpose, let us consider the case, in which a wave is propagating within thePML-layer in y-direction and having an angle of ϕ with respect to the y-axis.The total pressure p′ computes as

p′ = p′x + p′y = px0 ej(ωt−kxx−kyy) + py0 e

j(ωt−kxx−kyy) . (5.161)

Substituting the values for kx, ky according to (5.159), (5.160) and consideringthat for the chosen case σx = 0, we obtain

p′ = (px0 + py0) ej(ωt−kx sin ϕ−ky cos ϕ) e−(σy/c) cos ϕ = p0e−(σy/c) cos ϕ . (5.162)

Assuming a layer thickness of L, the damped wave will be totally reflected atthe outer boundary of the PML-region and this reflected wave at the interfacebetween propagation and PML region takes the value

p′r = p0e−(2/c) cos ϕ

LR

o

σy(y)dy= p0R . (5.163)


A reasonable choice of the reflection factor R is 10−3, since we have to takecare that a too strong damping in a too small PML-region can strongly disturbthe numerical solution. In addition, in order to get rid of the dependence of theoverall damping on the speed of sound c, we will choose all damping functionsσy direct proportional to c (see (5.163)).

In a first case, we will assume a constant damping σy = σ0. Therewith, weobtain from (5.163) the following relation for σ0

σ0 =−c lnR2L cosϕ

. (5.164)

In a second case, we consider a quadratically increasing damping function,hence we set

σy = σq0

y2

L2,

and assume that y is equal to zero at the interface and is increasing within thePML-region. Again, exploiting (5.163) we arrive at a relation for the constantfactor σq

0

σq0 =

3c lnR2L cosϕ

. (5.165)

In the last step, we will introduce a singular function, given by

σy =c

L− y, (5.166)

, which means that we increase the damping inverse with the distance.We want to emphasize that the damping functions according to (5.164)

as well as (5.166) are different from zero at the interface, and therefore willintroduce a discontinuity at the interface. However, due to the properties ofthe PML-technique, no spurious reflections will occur.

For the FE formulation of the wave equation within a PML region, westart at the strong setting, which reads as follows.

Given:f : Ω → ICc, ρ : Ω → IC .

Find: p : Ω → IC

ηyηz∂

∂x

(1ηx

∂p

∂x

)+ ηxηz

∂

∂y

(1ηy

∂p

∂y

)+ ηxηy

∂

∂z

(1ηz

∂p

∂z

)

+ ω2 ηxηyηz

c2p− f = 0 . (5.167)


Boundary conditions∂p

∂n= 0 on Γ (5.168)

In (5.167) and (5.168) f denotes any acoustic source term, IC the set of complexnumbers, Γ the outer boundary of the computational domain and ηx, ηy, ηz

the damping functions, which compute according to (5.150). In the first step,we multiply (5.167) by an appropriate test function v and integrate over thewhole domain Ω∫

Ω

v

(ηyηz

∂

∂x

(1ηx

∂p

∂x

)+ ηxηz

∂

∂y

(1ηy

∂p

∂y

)+ ηxηy

∂

∂z

(1ηz

∂p

∂z

)

+ ω2 ηxηyηz

c2p− f

)dΩ = 0 . (5.169)

Now, we apply Green’s integral theorem to the second-order spatial deriva-tives and incorporate the homogeneous Neumann boundary condition (5.168).These steps lead to the following weak formulation of (5.168): Find p′ ∈ H1

such that∫Ω

ηyηz

ηx

∂v

∂x

∂p

∂x+ηxηz

ηy

∂v

∂y

∂p

∂y+ηxηy

ηz

∂v

∂z

∂p

∂zdΩ − ω2

∫Ω

ηxηyηz

c2vp dΩ

=∫Ω

vf dΩ (5.170)

for any v ∈ H1. Using standard nodal finite elements, we arrive at the follow-ing discrete complex algebraic system of equations(

K − ω2M)p = f (5.171)

with K the stiffness matrix, M the mass matrix, p the nodal vector of complexacoustic pressure and f the complex nodal vector of the right-hand side. Fora stability investigation we refer, e.g., to [17, 31].


5.6.1 Transient Wave Propagation in Unbounded Domain

Let us consider a 2D case, where a wave is excited by a 10-kHz sine-signalfor a specific time (four periods) at the beginning of the computation (seeFig. 5.24). The configuration allows us to evaluate the performance of theproposed absorbing boundary method. Figure 5.25 shows simulation resultsfor the model described above, as a sequence of snapshots of the acoustic


Fig. 5.24. 2D plane case consisting of a plate excited with a 10-kHz sinusoidalpressure load (r = 375 mm, r/λ ≈ 10)

Fig. 5.25. Acoustic pressure distribution at different times showing the effect ofabsorbing boundary conditions

pressure pulse for different times. In Fig. 5.25, the acoustic pressure obtainedwhen using first-order absorbing boundary conditions is presented. At a timeof t = 0.44 ms the plate at the bottom is still being excited, and the sinusoidalgenerated wave can be observed. At a time of t = 0.96 ms, the results startto show spurious reflections and at t = 0.104 ms, the effect of these spuriousreflections has propagated back into the acoustic domain. However, the am-plitudes of these reflected waves are small. For a single point fixed in space(located at (x, y) = (0 mm, 115 mm)), Fig. 5.26 displays the pressure signal asa function of time.

At this point, we want to note, that with this proposed method, we cannot limit the computational domain in the acoustic near field. For such cases,the order of the absorbing boundary condition has to be increased. For latestresearch on this topic, we refer to [76].


Fig. 5.26. Acoustic pressure as a function of time using first-order absorbing bound-ary conditions for a fixed point located at (x, y) = (0mm, 115 mm)

5.6.2 Harmonic Wave Propagation in Unbounded Domain

In order to evaluate the PML method, we perform numerical simulationsof cylindrical waves and compare it with analytic results. As displayed inFig. 5.27, we prescribe the acoustic pressure on the surface of a cylinder andcompute the radiated sound field. Although the problem is 2D, we will perform

Fig. 5.27. Computational setup: propagation region surrounded by a PML-region

a full 3D computation.The dimension of the propagation domain is fixed by λ/2 and the PML-

thickness L is varied from L = λ/2 over λ/4 to λ/8 (see Fig. 5.28). Further-


more, we perform a discretization, where we use 20, 40, 60 and 80 bilinearhexahedron elements per wavelength λ. The analytical solution is given by

L = λ/8L = λ/4

L = λ/2

Fig. 5.28. Coarsest grids (h ≈ λ/20) of the three different setups concerning thethickness of the PML

p = p(R0)H

(2)0 (kr)

H(2)0 (kR0)

(5.172)

with H(2)0 the Hankel function, r = (x, y) the position vector, k the wave

number, and R0 the radius, where we define the excitation p(R0) = 1. Wecompute the relative accumulated error (sum over all finite element nodes Nin the propagation region) as follows

Etotal =

√√√√√√√√N∑

i=1

(phi − pi)2

N∑i=1

p2i

100% . (5.173)

In Fig. 5.29, we display the iso-lines for the three cases having differentPML-thickness L and which was computed on the coarsest grid (mesh sizeh ≈ λ/20).

Table 5.5, lists the computed error according to (5.173). It has to be notedthat in addition to the inexact approximation of the free radiation condition bythe PML, we measure with (5.173) the overall error (see e.g., [3,43,106,107]).However, since this is a practical case, we are interested in the overall errorof our numerical scheme. As can be seen from Table 5.5, we achieve a veryconsistent reduction of the error, by decreasing the mesh size h as well asincreasing the layer thickness. The great superiority of the inverse distancedamping function compared to the constant as well as quadratically increasingone can be clearly seen, when studying the error Etotal in Table 5.5 as wellas the iso-lines in Fig. 5.30. As a rule of thumb, we can summarize that oneshould use the same mesh size in the PML-region as used in the propagation


L = λ/2 L = λ/4 L = λ/8

Fig. 5.29. Iso-lines on the coarsest grid (h ≈ λ/20) of the three setups concerningthe thickness of the PML (inverse distance weighted damping function)

Table 5.5. Computational error Etotal for different mesh sizes h and different PML-thickness L

Damping constant Damping quadratically increasingh L = λ/8 L = λ/4 L = λ/2 L = λ/8 L = λ/4 L = λ/2

λ/20 2.9712 1.3760 0.4919 2.8872 1.1259 0.3685

λ/40 1.3537 0.5378 0.1817 1.0470 0.2376 0.1238

λ/60 0.9302 0.3329 0.1065 0.3721 0.0776 0.0880

λ/80 0.6163 0.2338 0.0982 0.2152 0.0413 0.0787

Damping inverse with distanceh L = λ/8 L = λ/4 L = λ/2

λ/20 0.5576 0.4137 0.3174

λ/40 0.3139 0.1119 0.0773

λ/60 0.1580 0.0557 0.0409

λ/80 0.1159 0.0374 0.0232

region and the mesh should have at least 10 bilinear finite elements withinthe thickness L of the PML-layer.

5.6.3 Acoustic Pulse Propagation over a Non-Matching Grid

To show the capability of non-matching grids, we will investigate the numer-ical simulation of a single acoustic spherical pulse of frequency 1000Hz andmagnitude 1, propagating over an interface with quite different spatial dis-cretizations. The computational domain in the (r, z)-plane is shown in theleft picture of Fig. 5.31. The pulse is imposed in form of an essential bound-ary condition on ΓD. We assume that the grid on Ω1 has to be substan-tially finer than that required by the acoustic wavelength. Therefore, we use


Constant damping Quadratic damping Inverse damping

Fig. 5.30. Iso-lines of the solution for a PML-thickness of λ/4 and the coarsest grid(h ≈ λ/20) using the three different damping functions

Fig. 5.31. Spherical pulse. Axisymmetric computational domain (dimensions inmeters)

2·40·40 = 3200 elements on Ω1, as depicted in the right picture of Fig. 5.31.In order to compare the conforming method with the non-conforming one, wetake 6400 elements on Ω2 in both cases, choosing nx = 80, ny = 40 for theconforming and nx = 160, ny = 20 for the non-conforming case.

In Fig. 5.32, the isolines for the velocity potential at time t = 1.6 ms are vi-sualized. Whereas the conforming method exhibits numerical noise before andbehind the pulse, the non-conforming approach is much closer to the expectedsolution. Inside Ω1, the radial symmetry of the isolines from the conformingmethod is observably disturbed. The poor quality of the conforming methodcan be easily explained by the fact that in order to obtain matching interfacegrids, the mesh on Ω2 is simply too coarse to correctly resolve the solution.

In order to examine the transient behavior more closely, we visualize theevolution of the acoustic potential at the point (0, 0.1 m) in Fig. 5.33. Inaddition to the comparison of both approaches, we employ a reference solution,obtained with a uniformly fine grid of 54500 elements. As expected from theobservations above, the conforming approach exhibits quite strong oscillations,while the behavior of the non-conforming method is quite smooth and visuallycoinciding with the reference solution. We remark that even the referencesolution is subject to small unphysical oscillations directly after the pulse.


Fig. 5.32. Isolines of the acoustic velocity potential at time t = 1.6 ms. Matching(left) and non-matching grids (right)

Fig. 5.33. Evolution of the acoustic potential at the point (0, 0.1)T.

5.6.4 Non-linear Wave Propagation in a Channel

Let us consider a plane wave in a semi-infinite channel as displayed in Fig.5.34. The excitation is performed by a rigid-body motion on the left side, so


Fig. 5.34. Simulation model

that we obtain a defined acoustic particle velocity. The prescribed mechanicalmotion follows a sine-burst signal as displayed in Fig. 5.35 with a frequencyof 100 kHz and an amplitude of 100µm. For the medium we use water with

Fig. 5.35. Signal form used for excitation (mechanical motion)

a parameter of non-linearity B/A equal to 5 and a diffusivity of sound valueb = 6 · 10−9 m2/s. Now, as a function of the distance x from the source(vibrating mechanical solid body), analytical solutions exist (see e.g., [25]).Introducing the dimensionless distance σ as

σ = x/x

with x the shock formation distance, we obtain the following two formulas

• σ < 1.0: Fubini solution

v(x, t) = v0

∞∑n=0

[2nσJn(nσ) sin[n(ωt− kx)]

], (5.174)

withv(0, t) = v0 sin(ωt) . (5.175)


In (5.174) v0 stands for the particle velocity, k = ω/c0 the wave number,c0 the speed of sound, Jn for the Bessel function of order n, and x thecoordinate for the propagation direction.

• σ > 3.0: Fay solution

v(x, t) = v0

∞∑n=0

[2/Γ

sinh[n(1 + σ)/Γ ]sin[n(ωt− kx)]

], (5.176)

with

Γ =B/A

b.

The shock formation distance x itself computes for a harmonic excitationwith velocity amplitude v0 and the fundamental wavelength λ as [94]

x =λ2

4π2v0(1 +B/(2A)). (5.177)

For the acoustic part, we specify homogeneous Neumann boundary conditionson the whole boundary. For the mechanical solid body, we prescribe the me-chanical displacement for all FE nodes with an amplitude of 100µm, whichvaries with time according to the sine-burst signal. The FE mesh consists oflinear quadrilateral elements with just one finite element for the width of thechannel and 250 finite elements per fundamental wavelength for the length ofthe channel. In order to also resolve higher harmonics, which will arise due tothe non-linearity, this fine mesh has been used. For the time discretization atime step value of 20 ns is specified, which corresponds to 500 time samples perfundamental time period. Again, as for the spatial discretization, this smalltime step is necessary to be able to resolve higher harmonics.

In Fig. 5.36 it is clearly demonstrated how the number of higher har-monics grows with the distance from the source. The comparison of the Faysolution with the numerical data is displayed in Fig. 5.37 for different dis-tances. A detailed discussion on parameter variations (B/A, b, spatial andtime discretization), especially how they influence the results, is given in [94].


σ = 0.2 σ = 0.2

σ = 0.6 σ = 0.6

σ = 1 σ = 1

Fig. 5.36. Time signal and frequency spectra of Fubini solution and numericalcomputation for different distances


σ = 3 σ = 4

σ = 5 σ = 6

σ = 7 σ = 8

Fig. 5.37. Time signal of Fay solution for distances between σ = 3 and 8

6

Coupled Electrostatic-Mechanical Systems

In electrostatic-mechanical systems, the structure is subject both to rigidmotions and elastic deformations by means of the electrostatic force. Thus,the change on the geometry in turn may strongly influence the electric fieldand thus the electrostatic force distribution. A typical application is a micro-machined pump [224], shown in Fig. 6.1. If an electric voltage is applied to

Fig. 6.1. Schematic view of an electrostatically driven micropump [224]

the electrodes, the elastic pump diaphragm is deformed by the electrostaticforce and bends towards the counterelectrode. Thereby, fluid will be sucked inthrough the inlet valve. When the supply voltage is switched off, the relaxationof the diaphragm will push the fluid through the outlet valve.

196 6 Coupled Electrostatic-Mechanical Systems

6.1 Electrostatic Force

An elegant method for deriving a general formula for the electrostatic forceFel is obtained by applying the principle of virtual work (displacement). Thus,a virtual displacement δr of a body due to a force Fel will result in a variationof the electrostatic energy δWel

δWel = Fel · δr . (6.1)

The electrostatic energy density wel is computed by the scalar product of theelectric field intensity E and the electric flux density D

wel =12D · E . (6.2)

By using the constitutive law D = εE (assuming constant ε), we obtain thefollowing expression

wel =ε

2E · E =

12ε

D ·D . (6.3)

The total electrostatic energy Wel is computed by integration of the energydensity wel over the volume Ω

Wel =∫Ω

wel dΩ . (6.4)

Now, the variation of the electrostatic energy δWel computes as

δWel = δ

⎛⎝∫

Ω

wel dΩ

⎞⎠

=∫Ω

(D · δDε

− D2 δε

2ε2

)dΩ , (6.5)

with D = |D|. By using the scalar electric potential Ve (E = −∇Ve), the firstterm in (6.5) can be rewritten as∫

Ω

D · δDε

dΩ = −∫Ω

(∇Ve · δD) dΩ . (6.6)

Furthermore, the vector identity

∇ · (Ve δD) = Ve∇ · δD + δD · ∇Ve

and ∇ · δD = δqe (variation of the free charges) lead to

−∫Ω

(∇Ve · δD) dΩ =∫Ω

(Veδqe) dΩ −∫Ω

∇ · (Ve δD) dΩ . (6.7)

6.1 Electrostatic Force 197

The use of the divergence theorem (see Appendix B.6) transforms the thirdintegral in a surface integral over Γ (Ω)∫

Ω

∇ · (Ve δD) dΩ =∫

Γ (Ω)

VeδD · dΓ . (6.8)

Now, we know that the electric potential Ve is proportional to 1/r (with r de-noting the distance), and therefore, the electric flux density D is proportionalto 1/r2 [221]. Since the surface only can grow proportional to r2, the surfaceintegral will tend to zero for increasing Ω.

The variation of the volume charge δqe is equal to the scalar product ofthe virtual displacement δr and the gradient of qe

δqe = δr · ∇qe .

In addition, we can use the relation

∇(Veqe) = qe∇Ve + Ve∇qe

in order to modify the second term in (6.7)∫Ω

Veδqe dΩ = δr ·∫Ω

(Ve∇qe) dΩ

= δr ·∫Ω

(∇(Veqe) − qe∇Ve

)dΩ

= δr ·

⎛⎜⎝ ∮

Γ (Ω)

Veqe dΓ −∫Ω

qe∇Ve dΩ

⎞⎟⎠ . (6.9)

The surface integral in (6.9) will tend to zero for increasing volume Ω forthe same reason as discussed above. Therefore, the differential energy δWel

computes by using (6.5), (6.9), and the relation δε = δr · ∇ε

δWel = δr ·∫Ω

qeE dΩ −∫Ω

12D2

ε2δε dΩ

= δr ·

⎛⎝∫

Ω

(qeE− E2

2∇ε

)dΩ

⎞⎠ . (6.10)

Comparing the above expression with (6.1), we obtain the equation for thetotal electrostatic force Fel

Fel =∫Ω

(qeE− E2

2∇ε

)dΩ , (6.11)


and for the electrostatic volume force fΩel

fΩel = qeE− E2

2∇ε . (6.12)

The first term models the electric force acting on electric volume charges qein an electric field E. The second term exhibits a force at surfaces of changingpermittivities, and therefore, we are looking for a surface force expression. Letus assume a small volume ∆Ω in form of a cylinder with height h, which islocated between the two materials with permittivities ε1 and ε2 (see Fig. 6.2).Having no volume charge qe within ∆Ω, we can compute the electrostatic

Fig. 6.2. Interface between two materials with different permittivity

force ∆F as follows

∆F =∫

∆Ω

fΩel dΩ = ∆Γ

r2∫r1

fΩel dr

= −∆Γ2

r2∫r1

(E2∇ε) dr . (6.13)

The scalar product of the force ∆F with the normal vector n and the relationdε = dr · ∇ε = n · dr ∇ε, lead to

n ·∆F = −∆Γ2

r2∫r1

E2 dεdr

dr

∆F12 = −∆Γ2

ε2∫ε1

E2dε . (6.14)

Now, we know that at an interface of changing permittivity the normal com-ponent of electric flux density and the tangential component of electric field


intensity are continuous (see Sect. 4.5.2). With the help of the normal vectorn and the tangential vector t, we can decompose E and D as follows

E = (E · t) t + (E · n)n = Et + En

D = (D · t) t + (D · n)n = Dt + Dn .

These relations allow us to write the term E2 = E · E in the form

E · E =(

Dn

ε+ Et

)·(

Dn

ε+ Et

)=D2

n

ε2+ E2

t .

The above result in combination with (6.14) leads to the term for the surfaceforce fΓel in the direction of n (∆F12 → dF12, ∆Γ → dΓ)

fΓel =dF12

dΓ= −1

2

⎛⎝D2

n

ε2∫ε1

dεε2

+ E2t

ε2∫ε1

dε

⎞⎠

=D2

n

2

(1ε2

− 1ε1

)+E2

t

2(ε1 − ε2) (6.15)

fΓel = fΓel n .

Now, within the FE computation we can evaluate (6.15) to obtain at eachFE node on the boundary the corresponding electrostatic force. However, themore accurate and general approach is to use the principle of virtual workand apply it to the FE formulation. According to (6.1), we can calculate theforce in a direction r by

Fr =δ

δr

∫Ω

12

E ·D dΩ . (6.16)

Let us consider the setup as displayed in Fig. 6.3, where a virtual displace-ment of the micromechanical cantilever will just lead to a deformation of thesurrounding finite elements in the air gap and thus in a change of the elec-trostatic energy just within these elements. Therewith, the force Fr can beexpressed by

Fr =ne∑

e=1

(δ

δr

∫Ωe

ε02

E · E dΩ)

=ne∑

n=1

(∫Ωi

ε0EδEδr

dΩ +∫

Ωe

ε02

E · E δ( dΩ)δr

). (6.17)

Here the first term takes account of the changing electric field within thedomain, whereas the second term deals with the geometric distortion of theelement.


Fig. 6.3. Micromechanical cantilever with FE discretization; nodes with a pointmay move during a virtual deformation and nodes with a cross are fixed

Now, since in the coupling to the mechanical field, we are interested in thelocal force, we may rewrite (6.17) for the total force acting on a node i locatedat the surface of the electromechanical cantilever

F ir =

nie∑

e=1

(∫Ωe

ε0EδEδr

dΩ +∫

Ωi

ε02

E · E δ( dΩ)δr

). (6.18)

In (6.18) nie is the number of finite elements to which node i belongs (see Fig.

6.3).In the next step, we will now transform the integration over the global

space of the finite elements to an integration over the reference elements.Thus, we substitute dΩ by |J| dξ dη dζ with |J| the determinant of the Jacobimatrix J

F ir =

nie∑

e=1

(∫Ωe

ε0E · δEδr

|J| dξ dη dζ +∫

Ωi

ε02

E · E δ|J|δr

dξ dη dζ). (6.19)

After the solution process of the electrostatic field, we obtain in all FE nodesthe electric potential Ve, form which we can compute the electric field intensity

E = −∇Ve = −nen∑a=1

∇Na(x)Vea (6.20)

with nen the number of element nodes per finite element and Na the shapefunction at node a. Performing the computation at the reference element, wemay rewrite (6.20) by (see Sec. 2.3.6)

E = −(J T )−1nen∑a=1

∇Na(ξ)Vea . (6.21)


According to (6.19), we have to evaluate the variation of E with respect tothe direction r, which computes by using (6.21) as

δEδr

= −δ(JT )−1

δr

nen∑a=1

∇Na(ξ)Vea . (6.22)

Applying the variation to (J T )−1 will lead to quite complex terms. In orderto arrive at a more convenient expression, we utilize the following relation (Idenotes the unit matrix)

δ(J−1 · J)δr

=δJT

δr(J−1)T + JT δ(J−1)T

δr=δIδr

= 0

δ(J−1)T

δr= −(J−1)T δ(J)T

δr(J−1)T .

Therewith, the variation of the electric field intensity E in the direction r canbe computed from the FE results as follows

δEδr

= −(J−1)T δ(J)T

δrE . (6.23)

Substituting this result into (6.19) allows us to directly compute the forceacting on a node i by

F ir =

nie∑

e=1

(∫Ωe

ε02

E ·E δ|J|δr

dξ dη dζ −∫

Ωe

ε0E (J−1)T δJT

δrE |J| dξ dη dζ

).

(6.24)For any integration point, the transposed Jacobian matrix computes as

J T =

⎛⎜⎜⎜⎝xξ xη xζ

yξ yη yζ

zξ zη zζ

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎝

nen∑a=1

Na,ξxa

nen∑a=1

Na,ξya

nen∑a=1

Na,ξza

nen∑a=1

Na,ηxa

nen∑a=1

Na,ηya

nen∑a=1

Na,ηza

nen∑a=1

Na,ζxa

nen∑a=1

Na,ζya

nen∑a=1

Na,ζza

⎞⎟⎟⎟⎟⎟⎠ ,

and its variation with respect to δr as

δJ T

δr=

⎛⎜⎜⎜⎜⎜⎝

nen∑a=1

Na,ξδxa

δr

nen∑a=1

Na,ξδya

δr

nen∑a=1

Na,ξδza

δr

nen∑a=1

Na,ηδxa

δr

nen∑a=1

Na,ηδya

δr

nen∑a=1

Na,ηδza

δr

nen∑a=1

Na,ζδxa

δr

nen∑a=1

Na,ζδya

δr

nen∑a=1

Na,ζδza

δr

⎞⎟⎟⎟⎟⎟⎠ .

The term δxa/δr (δya/δr, δza/δr) obtains the value 1, if and only if node ais equal with node i, which belongs to the surface and is virtually moved, and


the variation is in the x-direction (y-direction, z-direction). In all other cases,we apply 0 to those terms.

The last term in (6.24) to be evaluated, is the variation of the Jacobiandeterminant, which computes as follows

δ|J|δr

= δxξ

δr (yηzζ − zηyζ) + δxη

δr (yξzζ − zξyζ) + δxζ

δr (yξzη − zξyη) +

δyξ

δr (xηzζ − zηxζ) + δyη

δr (xξzζ − zξxζ) + δyζ

δr (xξzη − zξxη) +δzξ

δr (xηyζ − yηxζ) + δzη

δr (xξyζ − yξxζ) + δzζ

δr (xξyη − yξxη) .


The challenges in the simulation of electrostatic-mechanical transducers canbe summarized as follows. First, one has to deal with at least two differentphysical fields, usually the electrostatic and mechanical field. In many casesthe transducer is immersed in an acoustic fluid (e.g., in ultrasound applica-tions), and therefore, the acoustic wave propagation also has to be computed.In addition, all non-linearities of the involved physical fields have to be consid-ered (e.g., geometric non-linearity within the mechanical field due to large dis-placements and/or strains). Furthermore, the coupling mechanisms betweenthe physical fields have to be taken into account, which are highly non-linearin nature. Therefore, even in the case when all single fields are governed bylinear partial differential equations, the coupled system is non-linear and hasto be solved by an iterative method.

6.2.1 Calculation Scheme

The FE formulation of the electrostatic field and the mechanical field hasalready been derived in Sect. 4.6 as well as Sect. 3.7. Using predictor valuesfor the electric potential in order to compute the electrostatic force and pre-dictor values for the mechanical displacement to update the configuration forthe computation of the electrostatic field, we can split the coupled systemof partial differential equations into a mechanical part and an electric part.To guarantee a full coupling between the two fields, we perform an iterativesolution process. The resultant decoupled discrete matrix system of equationsreads as follows(

M∗u 0

0 KVe(un+1k )

)(un+1

k+1

Ven+1k+1

)=

(f

u(Ve

n+1k )

fVe

(un+1k )

). (6.25)

In (6.25) M∗u denotes the effective mechanical mass matrix, f

u(Ve) the nodal

vector of external mechanical and electrostatic forces, u the nodal vector ofmechanical displacement, f

Ve(u) the electric source vector and Ve the nodal


vector of scalar electric potential, n the time step, and k the iteration counterwithin each time step. Since the system matrix KVe and the nodal vectorf

Veare updated by the mechanical displacement un+1

k , we use a moving-meshtechnique to avoid large mesh deformations in the FE grid. Therefore, theentries of the system matrix KVe change as functions of un+1

k .The solution algorithm of the fully coupled system of equations is displayed

in Fig. 6.4. The outer iteration loop controls the iterative coupling process

(a) Linear mechanics (b) Non-linear mechanics

Fig. 6.4. Coupled electrostatic–mechanical simulation schemes

between the mechanical and electrostatic equation, which is performed in thesimplest case by a fixed-point method. The convergence test is based on thefollowing stopping criterion

||un+1k+1 − un+1

k ||2||un+1

k+1 ||2< δo , (6.26)

with u the nodal vector of mechanical displacements, δo an adjustable ac-curacy, || ||2 the L2-norm, k the iteration counter for the outer loop(electrostatic-mechanical iteration) and n the time step number.


Due to a movement and/or deformation of the mechanical structures inelectrostatic devices, the finite elements computing the electrostatic field willbe deformed (see Fig. 6.5 for a simple voltage-driven bar), and thus will lead tonumerical inaccuracy. These deformations have to be controlled, and, beforean intersection of finite elements occurs (see Fig. 6.5), a remeshing of thesimulation domain has to be performed. To avoid this problem, we use a

Fig. 6.5. Intersection of finite elements by using the standard method

special kind of moving-mesh technique. Therefore, the finite elements for theelectric field are modified in order to be able to handle mechanical degrees offreedom, too. Of course, the mechanical stiffness of these elements has to bechosen very small compared to the stiffness of the bar. Now, the deformationof the bar is not only acting on the first layer of finite elements around thebar, but on all finite elements between the bar and the counterelectrode (seeFig. 6.6).

Fig. 6.6. No intersection of finite elements by using moving-mesh technique

Since the deformations of the mechanical structures especially in micro-machined capacitive transducers are often large compared to the geometricdimensions, a non-linear formulation for the mechanical field has to be uti-lized. This leads to the calculation scheme as shown in Fig. 6.4(b). An inneriteration loop is introduced for the computation of the mechanical field. Thestopping criterion for this loop should be based on an error and on a residualprinciple

||un+1j+1 − un+1

j ||2||un+1

j+1 ||2< δia

||rn+1j+1 ||2

||fn+1

u||2

< δir . (6.27)


In (6.27) r denotes the nodal residual vector of the mechanical system, fVe

the nodal vector of external mechanical and electrostatic forces, δia as wellas δir adjustable accuracy values and j the iteration counter for the inner(mechanical) iteration loop.

6.2.2 Voltage-driven Bar

Fig. 6.7. Setup of the voltage-loaded bar

To investigate the behavior of electrostatic transducers, we will performsimulations of a voltage-loaded bar. Figure 6.7 displays the setup of a two-sided clamped bar (for symmetry reasons just half of the bar is shown). Thegoal is to compute the deflection of the bar as a function of the applied volt-age till snap-in occurs. To account for typical geometries of micromachinedelectrostatic transducers, we set the thickness of the bar to 1µm, the lengthto 100µm and the air-gap distance to 3µm.

For the discretization, quadrilateral elements with quadratic interpolationfunctions have been used. The thickness of the electrodes has been neglected.Therefore, all nodes belonging to the upper electrode are assigned inhomo-geneous Dirichlet boundary conditions with a value according to the appliedvoltage. Furthermore, homogeneous Dirichlet boundary conditions are appliedto all nodes belonging to the lower electrode. At the left end of the bar ux aswell uy are set to zero to account for the full support and ux is set to zero atthe right end of the bar to account for the symmetry.

In the first step, we perform the simulation of the coupled electrostatic-mechanical system with linear mechanics. The obtained results are displayedin Fig. 6.8. For this setup, it can be clearly seen that the mechanical field has tobe computed by the non-linear formulation. Due to the strong deflection of thebar, inner stresses arise, which increase the stiffness of the bar. In a secondcomputation, we have also modelled a prestressing of the bar. Due to thefabrication of micro-electromechanical systems (MEMS), inner stresses willremain within the structure after the fabrication process. We have assumed avalue of 10 MPa, which is a typical value for such structures, and performed


Fig. 6.8. Tip displacement as a function of the applied electric voltage (linearmechanics with dashed line and non-linear mechanics with solid line)

0 2 4 6 8 10 121.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Voltage (V)

Tip

dis

plac

emen

t (m

)

Fig. 6.9. Tip displacement as a function of the applied electric voltage using anon-linear mechanical formulation with (dashed line) and without (solid line) pre-stressing

a non-linear analysis for the mechanical field. As displayed in Fig. 6.9, theinfluence of the prestressing is quite strong. Now the voltage can reach valuesof about 10 V until snap-in occurs.

7

Coupled Magnetomechanical Systems

Let us consider a magnetic solenoid valve as depicted in Fig. 7.1. The valve

Fig. 7.1. Principal setup of a magnetic solenoid valve

consists of a ferromagnetic pot with a copper coil and a ferromagnetic arma-ture that is held in its position by a compression spring. If the copper coilis loaded by a current pulse, a magnetic force on the armature is generatedthat causes a motion of the armature, when its amplitude is larger than thespring force. Due to the time-varying magnetic field, eddy currents are gener-ated in the magnet pot as well as armature. As a consequence, a time delaybetween the applied current and the magnetic force will occur. Milling cutsinto the magnet pot will reduce these eddy currents, and therefore will leadto a shorter switching time.

7.1 General Moving/Deforming Body

Due to the movement of an electric conductive body in a magnetic field, theso-called motional emf term (electromotive force term)

208 7 Coupled Magnetomechanical Systems

γv × B , (7.1)

with γ the electric conductivity and v the velocity of the body has to be takeninto account. The magnetic induction B can be expressed by the magneticvector potential via B = ∇ × A resulting in the expression for the eddycurrent density Jv induced in an electrically conductive body

Jv = γv × B = γv × ∇ × A . (7.2)

The total partial differential equation in the quasistatic case including movingbodies reads as

∇ × ν∇ × A = Ji − γ∂A∂t

+ γv × ∇ × A . (7.3)

If the velocity is a priori known, the additional term remains linear, but willlead to a so-called convective term. Therefore, the numerical computationwill need some upwind technique for stability reasons (see e.g., [22, 184]). Atypical example, which can be modelled by (7.3), is the investigation of aneddy current sensor as displayed in Fig. 7.2, where the material under testcan be assumed to be infinitely wide and its velocity to be known.

Fig. 7.2. Eddy current sensor used for NDE (nondestructive evaluation)

However, if the magnetic field significantly changes due to moving and/ordeforming parts (e.g., moving parts within a magnetic valve or current/voltage-loaded coils), we have to apply an updated Lagrangian formulation of the elec-tromagnetic field equations [190]. Let us consider a general point P of themoving body at time t and t+∆t as shown in Fig. 7.3. Now, we can expressthe change of the magnetic vector potential ∆A within ∆t according to thefixed coordinate system (x, y, z)

∆A∆t

=A(x +∆x, t+∆t) − A(x, t)

∆t. (7.4)

By assuming small ∆x and ∆t, we can express the term A(x +∆x, t +∆t)by a Taylor series up to the linear term

7.2 Electromagnetic Force 209

Fig. 7.3. Moving point P at time t and t + ∆t

A(x +∆x, t+∆t) ≈ A(x, t) +∂A∂t

∆t

+∂A∂x

∂x

∂t∆t+

∂A∂y

∂y

∂t∆t+

∂A∂z

∂z

∂t∆t

= A(x, t) +∂A∂t∆t+ (v · ∇)A∆t , (7.5)

with v = (∂x/∂t, ∂y/∂t, ∂z/∂t)T . Now by letting ∆t → 0, we obtain from(7.4) using (7.5)

dAdt

=∂A∂t

+ (v · ∇)A . (7.6)

On the other hand, we can rewrite the term γv × ∇ × A as follows

γv × ∇ × A = γ∇A(v · A) − γ(v · ∇)A , (7.7)

where ∇A denotes the operation of the nabla operator just on the magneticvector potential A. By comparing (7.6) and (7.7), we can rewrite (7.3) for theupdated Lagrangian formulation

∇ × ν∇ × A = Ji − γdAdt

+ γ∇A(v · A) . (7.8)

It has to be noted that the term γ∇A(v · A) in (7.8) will always be zero inthe 2D plane as well as axisymmetric case, since the vectors v and A areorthogonal. However, in the general 3D case, one has to consider this term.

7.2 Electromagnetic Force

Applying the principle of virtual work (displacement) as used for deriving theelectrostatic force will lead to a general expression for the magnetic force Fmag.The variation of the magnetic energy δWmag relates the virtual displacementδr and magnetic force Fmag as follows


δWmag = δr · Fmag . (7.9)

Using the two magnetic quantities B and H, we can compute the electromag-netic energy density wmag by

wmag =12B · H . (7.10)

Assuming an isotropic body with magnetic permeability µ we obtain for (7.10)

wmag =B ·B2µ

= µH ·H

2.

Performing an integration over the whole volumeΩ gives us the total magneticenergy Wmag, stored within the body

Wmag =12

∫Ω

B · H dΩ =12

∫Ω

B ·Bµ

dΩ . (7.11)

Thereby, the variation of the magnetic energy leads to

δWmag =∫Ω

(B · δBµ

)dΩ − 1

2

∫Ω

(B · Bµ2

δµ

)dΩ . (7.12)

Since δµ can be expressed by

δµ = δr · ∇µ

we obtain for the second term in (7.12)

12

∫Ω

(B ·Bµ2

δµ

)dΩ =

12δr ·

∫Ω

(B · Bµ2

∇µ

)dΩ

=12δr ·

∫Ω

(H · H∇µ) dΩ . (7.13)

According to [222], we may rewrite the first term in (7.12) as

∫Ω

(B · δBµ

)dΩ = δr ·

⎛⎝∫

Ω

(J × B) dΩ

⎞⎠ .

This result, in combination with (7.13) and (7.9), leads to the total electro-magnetic force Fmag

Fmag =∫Ω

(J × B) dΩ − 12

∫Ω

(H ·H ∇µ) dΩ , (7.14)


and the expression for the magnetic volume force fmag

fmag = J × B− 12H ·H ∇µ . (7.15)

The first term describes a magnetic force that arises due to the current in aconductor (can also be a conductive body carrying eddy currents) placed ina magnetic field. The magnetic force described by the second term acts on allthose interfaces where the magnetic permeability µ changes.

Analogously to the electrostatic case, we can rewrite the second term in(7.15) in the form of a surface force fΓmag acting at the interface between twomaterials with permeabilities µ1 and µ2

fΓmag =12

(B2

n

(1µ2

− 1µ1

)+H2

t (µ1 − µ2))

n . (7.16)


The precise numerical simulation of magnetomechanical systems involves thecomputation of the magnetic field, the mechanical field and the non-linearcoupling terms. Since, in the general case, the single fields themselves aredescribed by non-linear partial differential equations (magnetization curvein the magnetic field and geometric non-linearity in the mechanical field),the solution of large-scale problems demands for a very efficient calculationscheme.

7.3.1 Electric Circuit Coupling: Voltage-loaded Coil

Let us assume the case as displayed in Fig. 7.4. The voltage source withamplitude u(t) is directly connected to the coil, which has an ohmic resistancedefined by Rs. Therefore, the circuit equation reads as

u(t) = Rsi(t) + Ψ . (7.17)

In (7.17) Ψ = Ncφ denotes the time derivative of the total magnetic flux,which is computed by the FE method using the magnetic vector potential Aas follows

Ψ = Ncφ = Nc

∫Γf

B · dΓ

= Nc

∫Γf

(∇ × A

)· dΓ = Nc

∮C(Γf )

A · ds , (7.18)

with Γf the cross section of the ferrite core. The evaluation of the contour


Fig. 7.4. Voltage-loaded coil with electric circuit

integral is usually done by averaging the magnetic vector potential A overthe whole volume Ωc of the coil (see Sect. 4.7.6)

Ψ =Nc

κΓc

∫Ωc

(A · eJ

)dΩ , (7.19)

with eJ a unit vector pointing into the direction of the current density J. TheFE formulation for computing the electromagnetic field in the eddy currentcase reads as

MAAn+1

+ KAAn+1 − f

n+1

A= 0 . (7.20)

In (7.20) A and A denote magnetic vector potential as well as its time deriva-tive at the FE nodes, MA the magnetic mass matrix, KA the magnetic stiffnessmatrix, f

n+1

Athe vector due to a current flowing in the coil and excited by a

voltage u(t) (see (7.17)) and n the time step counter. Now, if we can assumethat the skin effect in each winding can be neglected (see Sect. 4.2.2), wemodel the coil as in the current-loaded case, so that f

Acomputes as

fn+1

A= in+1 f

A(7.21)

fA

=ne∧

e=1

fe

Afe

A= [f

p]

fp

=Nc

κΓc

∫Ωe

(NpeJx NpeJyNpeJz)T dΩ . (7.22)

Therefore, we can set up the following matrix equation system using (7.17)and (7.20)(

MA −fA(

fA

)T

Rs

)(A

n+1

in+1

)+(

KA 00 0

)(An+1

in+1

)=(

0un+1

). (7.23)

For the time discretization of An+1

we will apply a general trapezoidal method(see Sect. 2.5.1)


An+1 = An +∆t((1 − γP)A

n+ γPA

n+1). (7.24)

Therefore, the following predictor/corrector algorithm results

Predictor:A = An + (1 − γP)∆t A

n. (7.25)

Algebraic system of equations:(M∗

A −fA(

fA

)T

Rs

)(A

n+1

in+1

)=(−KAAun+1

)(7.26)

M∗A = MA + γP∆tKA . (7.27)

Corrector:An+1 = A+ γP∆tA

n+1. (7.28)

Since, in the above equation, the system matrix is indefinite, it is preferableto solve the algebraic system by constructing a Schur complement. First, wecan express A

n+1using (7.26) as follows

An+1

= (M∗A)−1

fA︸︷︷︸

d0

in+1 − (M∗A)−1 KAA︸︷︷︸

d1

. (7.29)

Instead of computing the inverse (M∗A)−1, we will solve two algebraic systems,

both having the same system matrix M∗A

M∗Ad0 = f

A(7.30)

M∗Ad1 = −KAA . (7.31)

Now, since we know d0 and d1, we can compute in+1 as follows

Rsin+1 = un+1 + fT

Ad1 − fT

Ad0i

n+1

in+1 =un+1 + fT

Ad1

Rs + fT

Ad0

. (7.32)

Therefore, instead of solving (7.26), we solve (7.30) and (7.31), compute in+1

via (7.32) and then calculate An+1

from (7.29).

7.3.2 Force Computation via the Principle of Virtual Work

In this section we will demonstrate the use of the principle of virtual work forthe local force calculation within the FE method [47]. The magnetic force is


obtained by the derivative of the magnetic energy while keeping the magneticflux constant. Thus, the force in the direction of δr computes as

Fr = − δ

δr

∫Ω

⎛⎝ B∫

0

H · dB

⎞⎠ dΩ . (7.33)

The minus sign in (7.33) is explained as follows. We assume that the variationδr and the force Fr point in the same direction. The magnetic force alwaysmoves/deforms a body in such a way that the magnetic energy becomes aminimum. Therefore, the variation of the energy will be negative, and wehave to apply a minus in (7.33).

Fig. 7.5. Movable body with discretization including the layer of virtually distortedfinite elements

Now let us assume an FE mesh of the simulation domain as displayed inFig. 7.5. The total force acting on the movable/deformable body computes asa sum over all finite elements ne surrounding the body

Fr = −ne∑

e=1

δ

δr

∫Ωe

⎛⎝ B∫

0

H · dB

⎞⎠ dΩ . (7.34)

Since the magnetic permeability within the virtually distorted elements is thatof air, we may write

B∫0

H · dB dΩ =B ·B2µ0

.

The variation now leads to


Fr = −ne∑

e=1

⎛⎝∫

Ωe

Bµ0

· δBδr

dΩ +∫Ωe

B ·B2µ0

δ( dΩ)δr

⎞⎠ . (7.35)

The local force F ir , which will act at the node i located on the surface of the

body (see Fig. 7.5), computes as a sum over all elements surrounding the bodyand containing node i

F ir = −

nie∑

e=1

⎛⎝∫

Ωe

Bµ0

· δBδr

dΩ +∫Ωe

B ·B2µ0

δ( dΩ)δr

⎞⎠ . (7.36)

First, we will change the integration over the global elements to an integrationover the reference elements, which means that we have to substitute dΩ by|J| dξ dη dζ with |J| the determinant of the Jacobi matrix J . Applying thissubstitution to (7.36), we arrive at

F ir = −

nie∑

e=1

⎛⎝∫

Ωe

Bµ0

· δBδr

|J | dξ dη dζ +∫Ωe

B · B2µ0

δ|J |δr

dξ dη dζ

⎞⎠ . (7.37)

Since we will evaluate the integral numerically, we need the expression forδB/δr for each integration point. After the computation of the magnetic vec-tor potential Ah, we can compute B as follows

B =nd∑i=1

nen∑a=1

∇ × (NaAhai) =

nen∑a=1

∇Na × Aha , (7.38)

with nen the number of nodes per finite element,Na the shape function at nodea and Ah

a the calculated vector potential at node a (all three components).Performing the computation at the reference element, (7.38) changes to

B =nen∑a=1

(J T )−1∇ξNa × Aha ,

with ∇ξ = (∂/∂ξ ∂/∂η ∂/∂ζ)T , and the variation of B leads to

δBδr

=δ(J T )−1

δr

nen∑a=1

∇ξNa × Aha . (7.39)

Since the variation of the term δ((JT )−1J T )/δr is zero, we obtain

δ(J T )−1

δrJ T + (J T )−1 δJ T

δr= 0

δ(J T )−1

δr= −(J T )−1 δJ T

δr(J T )−1 .


Using this relation, we may write (7.39) as follows

δBδr

= −(J T )−1 δJ T

δrB , (7.40)

which changes (7.37) to

F ir =

nie∑

e=1

⎛⎝∫

Ωe

Bµ0

· (J T )−1 δJ T

δrB |J | dξ dη dζ −

∫Ωe

B ·B2µ0

δ|J |T

δrdξ dη dζ

⎞⎠ .

(7.41)The computation of ∂δJ T /δr and δ|J |T /δr is analogous to the electrostaticcase (see Sec. 6.1).

7.3.3 Calculation Scheme

Applying the FE method to both partial differential equations, we achieve thefollowing semidiscrete Galerkin formulation of the problem

Muun+1 + Cuu

n+1 + Ku(un+1)un+1

−fu(An+1, An+1) = 0 (7.42)

MA(un+1)An+1

+ KA(An+1, un+1)An+1

−fn+1

A(An+1, un+1, un+1) = 0 . (7.43)

In (7.42) and (7.43) Mu denotes the mechanical mass matrix, Cu the mechan-ical damping matrix, Ku the mechanical stiffness matrix, f

uthe mechanical

force vector, un+1 the nodal displacements, MA the magnetic mass matrix,KA the magnetic stiffness matrix, An+1 the nodal magnetic vector potentials,and f

Athe nodal magnetic source vector.

For the time discretization, we apply the Newmark scheme for the mechan-ical equation and the general trapezoidal scheme for the magnetic equation(see Sect. 2.5). Using predictor values for the magnetic vector potential tocalculate f

uand predictor values for the mechanical displacement to com-

pute MA, KA and fA

a decoupling into a mechanical and a magnetic matrixequation is achieved. To still ensure a strong coupling between the magneticand mechanical quantities, we have to solve these equations within each timestep iteratively (see Fig. 7.6). The outer iteration loop controls the iterativecoupling process between the magnetic and mechanical equation, which is per-formed in the simplest case by a fixed-point method. The convergence test isbased on the following displacement stopping criterion

||un+1k+1 − un+1

k ||2||un+1

k+1 ||2< δo , (7.44)


(a) Linear magnetics and mechanics (b) Non-linear magnetics and me-chanics

Fig. 7.6. Coupled magnetomechanical simulation

with u the nodal vector of mechanical displacements, δo an adjustable accu-racy, || ||2 the L2-norm, k the iteration counter for the outer loop (magne-tomechanical iteration) and n the time-step number.

When we have to consider a non-linear magnetization curve, we have an ad-ditional inner loop for the magnetic equation denoted by the iteration counteri, which has to fulfill the following convergence criteria

||An+1i+1 −An+1

i ||2||An+1

i+1 ||2< δmag

ia

||rn+1i+1 ||2

||fn+1

A||2

< δmagir . (7.45)

In (7.45) rn+1i+1 denotes the nodal residual vector of the magnetic equation,

i the iteration counter for the inner (magnetic) iteration loop, and δmagia as

well as δmagir adjustable accuracy values. The first convergence criterion in

(7.45) is based on the magnitude of the magnetic vector potential increments∆A = An+1

i+1 − An+1i , the second criterion on the magnitude of the residual


vector r (right-hand side criterion). By applying also a non-linear formulationfor the mechanical field, we need an additional inner loop for the mechanicalequation. This loop will be denoted by the iteration counter j and has to fulfillthe following stopping criteria

||un+1j+1 − un+1

j ||2||un+1

j+1 ||2< δmech

ia

||rn+1j+1 ||2

||fn+1

u||2

< δmechir . (7.46)

In (7.46) rn+1j+1 is the nodal residual vector of the mechanical equation, δmech

ia

as well as δmechir adjustable accuracy values, and j the iteration counter for

the inner (mechanical) iteration loop.

7.3.4 Moving Current/Voltage-loaded Coil

In this section we will consider the case of a current/voltage-loaded coil, andwill study different approaches to overcome the problem of moving partswithin the electromagnetic field. In particular, we will discuss the simula-tion of an electrodynamic loudspeaker as displayed in Fig. 7.7. The magnetic

Fig. 7.7. Design of an electrodynamic loudspeaker

field in the air gap is dominated by the constant radial magnetic field of thepermanent magnet. When the coil is driven by a voltage signal, the interactionbetween the resulting electric current and the magnetic field in the air gapleads to an axial magnetic force (Lorentz force), acting on the coil. Therefore,the whole structure, consisting of the coil, former, suspension, surround andcone diaphragm, starts to move and generates the acoustic sound field.

The computation of the magnetic field by applying the FE method requiresthe discretization of the whole simulation domain (permanent magnet, mag-netic assembly, coil, former, diaphragm, suspension, surround, and ambientair). Therefore, in an updated Lagrangian formulation the moving coil of theelectrodynamic loudspeaker causes mesh distortion of the surrounding finiteelements (see Fig. 7.8). In addition, if the displacements exceed the element


Fig. 7.8. Mesh deformation due to movement

size, a new mesh of the simulation domain is required. In recent years severalapproaches have been proposed to overcome this problem:

1. Coupled Finite-Element–Boundary-Element Method [91,113,134,164,167]:Applying a finite-element–boundary-element method, where the solids

are discretized by finite elements and the magnetic field in the surround-ing air with boundary elements, results in a total separation of stationaryand moving parts (see Fig. 7.9). This method requires no remeshing of thedomain, as long as no contacts between the different parts occur. Sinceonly the solid parts have to be discretized, the preprocessing, especially in3D, is easier and open-domain problems can be computed exactly. Due tothe updating of the boundary element matrices according to the mechani-cal displacements, the motional emf term is implicitly taken into account.However, the main disadvantage of this method is the large amount ofmemory and CPU time needed by the BEM.

2. Motional EMF-Term Method [44, 71, 83, 103,109]:Since only nonmagnetic parts move within the electrodynamic loud-speaker, the motional emf term

γ(v × B) (7.47)

fully describes the movement of parts with conductivity γ and velocity vin a magnetic field with flux density B. Therefore, no remeshing of thecalculation domain has to be performed. It has to be pointed out thatthe velocity of the moving parts in an electrodynamic loudspeaker is nota constant value, but changes from time step to time step resulting in anew setup of the system matrices. Furthermore, the motional emf term


Fig. 7.9. FE–BE discretization of stationary and moving part as well as of thesurrounding air

leads to a nonsymmetric system matrix and, depending on the conductiv-ity, the velocity as well as the mesh size, an upwind technique has to beutilized. In addition, the change of the overall magnetic field due to thedifferent positions of the moving coil is not taken into account.

3. Coupling via Lagrange multiplier [77, 137, 152,184]:This technique can be considered as a general method for coupling inde-pendent meshes. As displayed in Fig. 7.10 for the example of a rotatingmachine, the rotor-mesh and stator-mesh have a common interface Γc

within the air gap. The two meshes are coupled by forcing the normal

Rotor

Stator

Gc

Air gap

Fig. 7.10. Common interface in an air gap of a rotary electric machine

component of the magnetic induction [B · n] and the tangential compo-nent of the magnetic field intensity [H × n] at Γc of the rotor as well asstator to be equal. A common method for solving this problem is a La-


grange multiplier formulation. Thus, the velocity term is fully included,and no remeshing is necessary. The main disadvantage of this method isthat the condition of the system matrix becomes poor. Furthermore, thismethod is restricted to problem cases in which the common interface Γc

does not change its geometry, as is the case in rotary and translationalelectric machines.

4. Moving-Mesh Method [48, 73, 77, 108,140,188,189]:For this method, the whole volume (moving as well as stationary parts)is discretized by finite elements. If any movements occur, the coordinatesof the finite elements, discretizing the moving parts, are updated in sucha form that the distortions of the elements remain small. This can beachieved by also solving the mechanical equation within the air gap (ofcourse with a very low mechanical stiffness and no mass) together withthat of the moving parts (in our example: coil, holder, suspension, conediaphragm). Therefore, the structure of system matrices will remain andthe motional emf term is implicitly taken into account. However, the num-ber of unknowns within the mechanical equation increases, the entries ofthe magnetic system matrix change and, if the movement is too largea remeshing has to be performed. In addition, one has to be careful inchoosing the mechanical stiffness for the surrounding air region, so thatthe reaction to the moving/deforming body is negligible.

5. Moving-Material Method [19, 202]:The whole simulation domain is discretized by finite elements and will beunchanged throughout the simulation. Any movements of parts within thesimulation domain are considered by changing the material values withineach finite element according to the motion (see Fig. 7.11). Thus, thestructure of the finite element matrices will not change and no remeshingis required. However, the entries of the system matrices change, and themotional emf term is not implicitly taken into account.

z

10 f z( )|t

Region ofcoil

z

r

z

r

z

10 f z( )|t t+D

Dz

Surroundingair

Fig. 7.11. Moving-material method: Configuration at time step t and t + ∆t. Thefunction f(z) defines a spatial weighting function for the current


In the following, we will introduce two new methods only, based on the FEmethod, which perfectly fit to a moving current/voltage-loaded coil. Therefore,the system matrices will remain constant throughout the whole simulation,and no problem with mesh distortion will arise.

Modified Motional EMF-Term Method

Due to the motional emf, an additional electric current iemf will be induced inthe coil with opposite direction to coil current i excited by the applied voltageu. Thus, (7.17) changes to

u(t) = Rs(i(t) − iemf(t)) + Ψ . (7.48)

The current iemf computes according to (7.2)

iemf =Nc∑i=1

∫Γwi

γ(v × ∇ × A) · dΓ . (7.49)

Now, by modelling the complex winding structure as a hollow cylinder, (7.49)reads as

iemf = γ

∫Γc

(v × ∇ × A) · eJ dΓ . (7.50)

Since we can express the ohmic resistor Rs as

Rs =lNc

γΓw=

lN2c

γ κΓc, (7.51)

with l the length of one winding, we can rewrite (7.50) in the form of a volumeintegral over the volume Ωc of the hollow cylinder suitable for the FE method(see Sect. 4.7.6)

Rs iemf =N2

c

κΓc

∫Ωc

(v × ∇ × A) · eJ dΩ . (7.52)

Therefore, the coupled FE-network matrix equation results in (for comparisonsee (7.23)) (

MA −fA

fT

ARs

)(Ai

)+(

KA 0cTA 0

)(Ai

)=(

0u

), (7.53)

with cA including the FE discretization of (7.52). Applying the time discretiza-tion as shown in Sect. 7.3.1, we arrive at(

MA + γP∆tKA −fA

fT

ARs

)(A

n+1

in+1

)=(−KAA

un+1 − cTA A

). (7.54)


The emf term has been treated explicitly, since we have used the predictorvalue A instead of the unknown vector potential A. This method has beensuccessfully applied in the dynamic simulation of the electrodynamic loud-speaker within small-signal parameter studies (see Sect. 12.1). However, ifthe amplitude of the exciting voltage increases, as is the case for large-signalstudies, the motion of the coil will not remain within the homogeneous fieldregion. In this case, the change of the overall magnetic field has to be takeninto account, which is not possible by this calculation scheme.

Modified Moving-Material Method

The restriction of the modified motional emf term method to small movementscan be overcome by using the standard moving material method with thefollowing changes [174]

• Same handling of the motional emf term as in the modified motional emfterm method, which means, in an explicit way (using the predictor valueA instead of the unknown vector potential An+1).

• Evaluating the spatial weighting function f(z) (see Fig. 7.11) within eachnumerical integration point.

• Applying the solution strategy for a voltage-loaded coil as shown in Sect.7.3.1. Therefore, in each time step first of all the current in+1 is computedand then the magnetic vector potential An+1.

The third point allows the voltage-loaded coil to be treated as if it were acurrent-loaded coil, since we first compute the current in the coil and then themagnetic vector potential (see Sect. 7.3.1). Since the permeability of the coiland of the surrounded air is the same (both have relative permeability of 1),the movement of the coil and therefore the change of the material within thefinite elements will not change the entries of the system matrices. However,the vector f

Ahas to be recomputed according to the motion of the coil.

Verification

In the following, we will set up a simple example of a moving voltage-loadedcoil, and compare the moving-mesh, modified motional emf, and modifiedmoving-material methods. Figure 7.12 displays the test case, which is a re-duced setup of an electrodynamic loudspeaker. The coil is held by a springand a permanent magnet generates the constant magnetic field Bconst in theair gap. The values have been chosen as follows:


Bconst = 0.75T constant magnetic field in the air gapNc = 100 number of turnsγ = 5.8 × 107 S/m conductivity of the winding wireρ = 8900kg/m3 density of winding wireΓc = 0.5 mm2 cross section of the coilk = 249.6N/m spring constantc = 0.131 Ns/m damping coefficientU = 3V amplitude of applied voltagef = 131 Hz frequency of applied voltage

Fig. 7.12. Test case: voltage-loaded coil in a homogeneous static magnetic field(axisymmetric setup)

Due to the small frequency of the applied voltage, we can neglect the inductionof the coil in a semianalytical approach. Furthermore, due to the axisymmetricsetup, the magnetic induction B is orthogonal to the electric current densityJ, and we assume that within the coil the magnetic induction B and thevelocity v are also orthogonal. The movement of the coil can be described byNewton’s law as

md2x

dt2+ c

dxdt

+ k x = Fmag(t) = Fmag sin(ω t) . (7.55)

In (7.55)m denotes the mass of the coil, Fmag the magnetic force and ω = 2 π fthe angular frequency. Introducing the normalized coefficients δ and ω0, weobtain


d2x

dt2+ 2δ

dxdt

+ ω20 x =

Fmag

msin(ω t) (7.56)

δ =c

2m= 177.7

1s

(7.57)

ω20 =

k

m= (2 π 131 Hz)2, (7.58)

with ω0 the angular frequency of the free, undamped oscillation, which is equalto the angular frequency ω = 2 π f . For t 1/δ, we obtain for the amplitudeof the forced oscillation

x =Fmag

m√

(ω2 − ω20)2 + 4δ2ω2

(7.59)

ω = 2 π f = ω0. (7.60)

The amplitude of the electromagnetic force Fmag can be computed via

Fmag = B l I (7.61)l = 2 r π Nc , (7.62)

with r the average radius of the coil (r = 13.225mm) and I the amplitude ofthe coil current.

In the first step, we will neglect the motional emf term in computing theelectric current InoEMF

InoEMF =U√

R2E + (2πfLE)2

≈ U

RE= 104.70 mA (7.63)

RE =2πrN2

c

γΓc= 28.653Ω, (7.64)

with RE denoting the ohmic resistance and LE the inductance of the coil.Using (7.57) and (7.60) – (7.63), we can compute the amplitude x of the axialmotion via (7.59)

xnoEMF =B 2 r π Nc InoEMF

c ω0= 6.05 mm. (7.65)

In a second step, let us also consider motional emf. Thus, the eddy currentinduced within the coil due to the motion, computes as

IEMF = γ v B Γc /Nc = γ 2 π f xB Γc /Nc , (7.66)

with v = 2 π f x the amplitude of the velocity and x the amplitude of the axialmotion. The total current in the coil changes to

I = InoEMF − IEMF. (7.67)


In Table 7.1 we have summarized the results obtained by the semianalyticalmethod and the numerical simulation for InoEMF and xnoEMF. The differencesin the values are due to the fact that within the semianalytical computationwe have neglected the inductance of the coil (this will reduce the difference to0.08% assuming an inductance of 3.9 mH, obtained by numerical computation)and any magnetic stray field.

Table 7.1. Comparison of semianalytical and numerical computation without con-sidering motional emf

Semianalytical method Numerical simulation Relative difference

InoEMF 104.70 mA 104.10 mA −0.6%xnoEMF 6.05 mm 6.00 mm −0.8%

Table 7.2. Comparison of the different numerical calculation schemes

Motional emf term Moving material Moving mesh

x 0.534 mm 0.533 mm 0.532 mm

To see the influence of the motional emf, we have computed the motion ofthe coil with all three numerical methods and the results for the calculatedamplitude of the axial motion are given in Table 7.2. Since the moving coilalways remains within the homogeneous magnetic field, the three numericalmethods show almost no difference. It can be clearly seen that the amplitudex due to the motional emf term has strongly decreased (e.g., the case of anelectromagnetic brake). The comparison of the effective current in the coilincluding motional emf, is summarized in Table 7.3. The value of I for thesemianalytical method has been obtained by using the numerically computedamplitude of the motion x and evaluating (7.67) with the help of (7.66). The

Table 7.3. Comparison of semianalytical and numerical methods including motionalemf

Semianalytical Motional emf Moving material Moving mesh

I 9.28 mA 9.267 mA 9.265 mA 9.20 mArel. difference −0.14% −0.16% −0.9%

advantages and disadvantages of the three numerical calculation schemes aresummarized in Table 7.4.


Table 7.4. Comparison of the numerical calculation schemes

Motional emf Moving material Moving mesh

Including theemf term yes yes yesCoil positionconsidered no yes yes

Mesh deformation no no yes

Matrix entries change no no yes

Concluding, we can state that for the special application electrodynamicloudspeaker the modified moving-material method is the method of choice,since it includes all physical effects and no mesh-updating or even remeshingis necessary.

8

Coupled Mechanical-Acoustic Systems

In many technical applications, the sensor/actuator is immersed in an acousticfluid. Therefore, mechanical vibrations will generate acoustic waves, whichitself will act as a surface pressure load on the vibrating structure. In general,we distinguish between the following two situations concerning mechanical-acoustic systems:

• Strong Coupling:In this case, the mechanical and acoustic field equations including theircouplings have to be solved simultaneously. A typical example is a piezo-electric ultrasound array immersed in water (see Fig. 8.1).

Fig. 8.1. Acoustic sound field of a piezoelectric ultrasound array antenna

• Weak Coupling:If the pressure forces of the fluid on the solid are negligible, a sequentialcomputation can be performed. For example, the acoustic sound field ofan electric transformer as displayed in Fig. 8.2 can be obtained in this way.

230 8 Coupled Mechanical-Acoustic Systems

Thus, in a first simulation the mechanical surface vibrations are calculated,which are then used as the input for an acoustic field computation.

Fig. 8.2. Acoustic sound field of an electric transformer due to the Lorentz forcesacting on its winding

8.1 Solid–Fluid Interface

Fig. 8.3. Solid–fluid interface

At a solid–fluid interface, the continuity requires that the normal componentof the mechanical surface velocity of the solid must coincide with the normalcomponent of the acoustic velocity of the fluid (see Fig. 8.3). Thus, the follow-ing relation between the velocity v of the solid expressed by the mechanicaldisplacement u and the acoustic particle velocity v′ expressed by the acousticscalar potential ψ arises

8.2 Coupled Field Formulation 231

v =∂u∂t

v′ = −∇ψ

n · (v − v′) = 0

n · ∂u∂t

= −n · ∇ψ = −∂ψ∂n

. (8.1)

In addition, one has to consider the fact that the ambient fluid causes on thesurface a mechanical stress σn

σn = −np′ = −nρ0∂ψ

∂t, (8.2)

which acts like a pressure load on the solid. In (8.2) ρ0 denotes the meandensity of the fluid.

When modelling special wave phenomena, we often arrive at a partial dif-ferential equation for the acoustic pressure. Therewith, we will also derive thecoupling conditions between the mechanical displacement and acoustic pres-sure at a solid–fluid interface. For the first coupling condition, the continuityof the velocities, we have to establish the relation between the acoustic parti-cle velocity v′ and the acoustic pressure p′. According to the linearized Eulerequation (see 5.30), we can express the normal component of v′ by

n · ∂v′

∂t= − 1

ρ0

∂p′

∂n. (8.3)

Therewith, since n · v = n · v′ holds, we get the relation to the mechanicaldisplacement by

n · ∂2u∂t2

= − 1ρ0

∂p′

∂n. (8.4)

The second coupling condition as defined in (8.2) is already establishedfor an acoustic pressure formulation.

8.2 Coupled Field Formulation

Let us consider a setup of a coupled mechanical-acoustic problem as shown inFig. 8.4, where at the interface Γ I we have to consider the solid–fluid coupling.Now, within the domain Ωs the partial differential equation for the mechani-cal field (see 3.80), within the domain Ωf the partial differential equation forthe acoustic field (see 5.92) and along the interface Γ I the coupling conditionsaccording to (8.1) and (8.2) have to be fulfilled. In a first step, let us trans-form the partial differential equations to their weak form without setting theboundary integral (obtained by using integration by parts) to zero. Therefore,we obtain for the mechanical system∫

Ωs

ρu′ · u dΩ∫Ωs

(Bu′)T [c]Bu) dΩ −∫Γ I

u′ · σn dΓ =∫Ωs

u′ · fV dΩ , (8.5)


Fig. 8.4. Setup of a coupled mechanical-acoustic problem

and for the acoustic system (acoustic excitation f being zero)∫Ωf

1c2w ψ dΩ +

∫Ωf

∇w · ∇ψ dΩ +∫Γ I

w n · ∇ψ dΓ −∫Γ f

w n f · ∇ψ dΓ = 0 .

(8.6)It should be noted that the plus sign in (8.6) in front of the the boundaryintegral over Γ I is due to the choice of n (see Fig. 8.4). In a second step, wewill now incorporate the coupling conditions. With the help of (8.2), we canrewrite the boundary integral in (8.5) as follows∫

Γ I

u′ · σn dΓ = −∫Γ I

u′ · n p dΓ = −∫Γ I

u′ · n ρ0∂ψ

∂tdΓ . (8.7)

Using (8.1), we obtain for the boundary integral along the interface Γ I (atthe outer boundary Γ f we set for simplicity ∂ψ/∂n = 0) in (8.6) the followingform ∫

Γ I

w∂ψ

∂ndΓ = −

∫Γ I

w n · ∂u∂t

dΓ . (8.8)

Thus, we arrive at the following coupled system of equations∫Ωs

ρu′ · u dΩ +∫Ωs

(Bu′)T [c]Bu) dΩ +∫Γ I

u′ · n ρ0∂ψ

∂tdΓ =

∫Ωs

u′ · fV dΩ(8.9)

∫Ωf

1c2w ψ dΩ +

∫Ωf

∇w · ∇ψ dΩ −∫Γ I

w n · ∂u∂t

dΓ = 0 . (8.10)



8.3.1 Finite Element Formulation

Before we perform a domain discretization, we multiply (8.10) by −ρ0 in orderto obtain in (8.10) a boundary integral similar to the one in (8.9). Thus, thematrices, occurring from an FE discretization of these two boundary integrals,will be transposed to each other and hence symmetry of the overall systemmatrix will be obtained.

Using nodal finite elements, we approximate the mechanical displacementu as well as the scalar acoustic potential ψ as follows

u ≈ uh =nd∑i=1

n1∑a=1

Nauhiaei =

n1∑a=1

Nauha ; Na =

⎛⎝Na 0 0

0 Na 00 0 Na

⎞⎠ (8.11)

ψ ≈ ψh =n2∑

a=1

Naψha , (8.12)

with n1 the number of nodes with unknown mechanical displacement and n2

the number of nodes with unknown acoustic potential. Following the sameprocedure as described in Sect. 3.7.1 for the mechanical equation and in Sect.5.4.1 for the acoustic equation we obtain a coupled system of equations(

Mu 0

0 −Mψ

)(u

ψ

)+

(0 Cuψ

CTuψ 0

)(u

ψ

)+

(Ku 0

0 −Kψ

)(u

ψ

)

=

(f

u

0

). (8.13)

The new matrix Cuψ computes as follows

Cuψ =nIe∑e=1

Ceuψ ; Ce

uψ = [Cpq] ; Cpq =∫Γ e

ρ0

⎛⎝NpNqnx

NpNqny

NpNqnz

⎞⎠ dΓ , (8.14)

with nIe the number of finite elements along the interface. At this point, itshould be emphasized that the coupled system of equations remains symmet-ric. This is not the case if instead of the acoustic velocity potential an acousticpressure formulation is used.

The time discretization is performed by a standard Newmark algorithm(see Sect. 2.5.2), which reads for the effective mass matrix formulation as fol-lows



u = un +∆t un +∆t2

2(1 − 2βH) un (8.15)

˜u = un + (1 − γH)∆t un (8.16)

ψ = ψn

+∆t ψn

+∆t2

2(1 − 2βH) ψ

n(8.17)

˜ψ = ψn

+ (1 − γH)∆t ψn. (8.18)


(M∗

u C∗uψ

(C∗uψ)T −M∗

ψ

)(un+1

ψn+1

)=

(f

u

0

)−(

Ku 0

0 −Kψ

)(u

ψ

)

−(

0 Cuψ

CTuψ 0

)⎛⎝ ˜u

˜ψ

⎞⎠ (8.19)

M∗u = Mu + βH∆t

2 Ku (8.20)M∗

ψ = Mψ + βH∆t2 Kψ (8.21)

C∗uψ = γH∆tCuψ . (8.22)

• Perform predictor update:

un+1 = u+ βH∆t2 un+1 (8.23)

un+1 = ˜u+ γH∆t un+1 (8.24)

ψn+1

= ψ + βH∆t2 ψ

n+1(8.25)

ψn+1

= ˜ψ + γH∆t ψn+1. (8.26)

8.3.2 Non-matching Grids

In many practical cases, we wish to perform the spatial discretization withinthe elastic body independent to the discretization of the surrounding fluid. Atypical example is given in Fig. 8.5, where on the one hand we need a muchfiner grid within the mechanical structures (plates) and on the other hand wishto have a regular grid within the fluid domain. In particular, the advantages ofusing non-matching grids are demonstrated in Fig. 8.5, where the mechanicalregions have to be resolved by a substantially finer grid than the fluid region.Therewith, we will investigate in such flexible discretization techniques for theapproximative solution of coupled mechanical-acoustic problems.

For the mechanical-acoustic coupling, the problem formulation for non-matching grids remains essentially the same as for the matching situation. This


Fig. 8.5. Non-matching discretization of mechanical structure and surrounding fluid

means, in contrast to the problem setting considered in Sec. 5.4.3 (acoustic-acoustic coupling over non-matching grid), no additional Lagrange multiplierhas to be introduced.

For the spatial discretization, we use two independently generated triangu-lations Ts and Tf onΩs andΩf, respectively, and approximate the displacementu on Ts and the potential ψ on Tf by finite elements. The two triangulationsinherit two (d−1)-dimensional grids ∂Ts and ∂Tf on Γ I. Due to the flexibleconstruction of both grids, the finite element nodes on ∂Ts and ∂Tf will ingeneral not coincide. On the contrary, motivated by different spatial scales re-quired for the resolution of the local subproblems, the difference in the meshsizes can become quite large.

The discretized version of (8.9), (8.10) results in the same matrix-equationsas for the matching grid approach (see 8.13). The coupling between the twogrids is represented by the matrices Cuψ and CT

uψ which realize the boundaryintegrals in (8.9) and (8.10). Their entries are given by

Cuψ = [Cpq]; Cpq =∫

Γ I

ρfNspN

fqn dΓ ∈ IRd, (8.27)

where N sp is the scalar basis function associated with the node p on ∂Ts, and

N fq is the one for node q on ∂Tf. For a detailed discussion of the element wise

assembly of Cuψ we refer to [67].

8.3.3 Numerical Examples

Noise Radiation from an Oil Pan and a Car Engine

A typical problem that is encountered in numerical acoustics of automotiveapplications consists of the calculation of the sound radiated from machineparts. In the standard approach, first the structural response due to dynamicloads is calculated based on the FE method. In a second step, the acousticradiation is calculated, again applying the FE method. The surface grid is dis-


Fig. 8.6. Surface grid of an oil panFig. 8.7. Embedding of oil pan intofinite element grid

played in Fig. 8.6, whereas a detail of the finite element model, showing theembedding of the oil pan structure in a sphere consisting of acoustic finite ele-ments, is shown in Fig. 8.7. The total finite element model consisted of approx.500 000 3D acoustic finite elements and the resulting sound field at a drivingfrequency of 600 Hz as computed by this model is shown in Fig. 8.8. Next,

Fig. 8.8. Sound field radiated at 600 Hz

this modelling scheme was applied in the simulation of a complete car engine.Here, the surface grid consists of 8 200 2D elements. This model was embeddedin a sphere of radius 2 m and the mesh generator NETGEN [194] was usedto mesh the unfilled volume with acoustic finite elements. The embedding ofthe engine in the finite element mesh is displayed in Fig. 8.9, whereas theinnermost layer of acoustic elements (tetrahedra) is shown in Fig. 8.10. Theresults of the transient structural simulation were used as excitations for theacoustic calculations. In the simulations, two different models have been con-sidered, consisting of 210 000 and 1 340 000 finite elements, respectively. Forthe numerical computation three different approaches have been performed:

• Direct, implicit:An implicit Newmark time-integration scheme has been used and the re-


Fig. 8.9. Embedding of car engineinto finite element grid

Fig. 8.10. Innermost tetra-hedra layer around car en-gine

sulting algebraic system of equation has been solved by a direct solver.

• Iterative, implicit:Again, an implicit Newmark time-integration scheme has been used, butnow we have applied an iterative solver (GMRES with ILU as precondi-tioner, see [127]).

• Explicit:An explicit Newmark time-integration scheme with lumped mass matrixhas been used. Therewith, the system matrix is a pure diagonal and thealgebraic solver just divides the right-hand side by the diagonal entries.

In Table 8.1 and 8.2, the required computer resources are summarized.

Table 8.1. Computer resources, small model

Solver Time steps Memory

Direct, implicit 2 000 364 MBExplicit 20 000 20 MB

Iterative (GMRES) 2 000 38 MB

The direct, implicit solution was not available for the large model, since itrequired 10 GB of main memory. For the iterative solver, convergence wasachieved for an accuracy of 10−8 (relative residual norm) within six iterations


Table 8.2. Computer resources, large model

Solver Time steps Memory

Direct, implicit – –Explicit 20 000 102 MB

Iterative (GMRES) 2 000 206 MB

Fig. 8.11. Comparison of normalized acoustic pressure at evaluation point for smallmodel

per time step. Of course, this convergence behavior strongly depends on thequality of the generated grid. For the small model, no difference was foundbetween the solution for the direct and the iterative solver. However, as canbe seen from Figs. 8.11 and 8.12, due to the diagonal mass matrix formulationfor the explicit time scheme, the solution based on the explicit time steppingdiffers from both solutions obtained with implicit time stepping. This differ-ence vanishes, when an even finer time discretization is used for the explicitscheme (for our examples we applied a time step of ten times smaller as usedfor the implicit scheme).

Acoustic Wave Generation by a Multiple Plate Structure

To demonstrate the applicability of the non-matching grid technique, wepresent the generation and radiation of acoustic waves by multiple struc-tures, which admits the steering of the waves by exciting the structures ina specified chronological order. In particular, we use for the structure Ωs

25 cylindrical silicon plates with diameter 50µm and height 1µm. They areplaced as a (5× 5)-array, each plate having a distance of 50µm to its nearest


Fig. 8.12. Comparison of normalized acoustic pressure at evaluation point for largemodel

neighbors. An excitation force with frequency f = 1 MHz is applied on theirlower end. For the acoustic domain Ωf , which is assumed to be water, a cuboidof length and width 1200µm and height 420µm is chosen. Due to symmetryreasons, we use as computational domain one quarter of the original one. In

Fig. 8.13. Cylindrical plates attached to the fluid domain and isosurfaces of theacoustic potential, deformed plates

Fig. 8.13 a part of the finite element meshes is shown, for which a uniformgrid of 40×40×28 cubes is used to discretize the acoustic domain and a gridof 768 hexahedra is employed for each plate. Thus, having a mesh width ofha = 600µm /40 = 15µm, we use ten elements per wavelength for Ωf . If onehad to employ matching grids, it would be quite difficult to generate them,


and if the mesh-width could not be very small over the whole domain, theresulting element shapes would possibly result in a poor approximation of thesolution. The nonconforming approach admits to use the grid desired for eachsubdomain regardless of the grids for the other subdomains. Figure 8.14 showssnapshots, taken every ten time steps of 3.5 ns, of the evolution of the acousticvelocity potential ψ along with the deformation of the structures (magnifiedby a factor of 1000).

Fig. 8.14. Evolution of the acoustic velocity potential and of the deformed struc-tures, successive excitation: snapshots after 10, 20, . . . , 80 time steps


For the results presented in Fig. 8.14, the plates are excited successively,and as can be seen, the waves emitting from the structures add up as ex-pected to constitute the superposed global sound beam. Given a target point,it is possible to optimally steer the acoustic wave towards this point by ap-propriately adjusting the chronological order of excitation of the silicon chips.This principle is used in so-called capacitive micro-machined ultrasound trans-ducers (CMUTs) (see Sec. 12.5). There, the deformation of the structure isinduced by an electrostatic surface force acting on the loaded electrodes.

9

Piezoelectric Systems

The piezoelectric transducing mechanism is based on the interaction betweenthe electric quantities, electric field intensity E and electric induction D, withthe mechanical quantities, mechanical stress [σ] and strain [S]. By apply-ing a mechanical load (force) to a piezoelectric transducer (e.g., piezoelectricmaterial with top and bottom electrode), one can measure a change of theelectric charge (sensor effect, see Fig. 9.1). This mechanism is called the direct

Fig. 9.1. Piezoelectric force sensor with a charge amplifier

piezoelectric effect, and is due to a change in the electric polarization of thematerial. The so-called inverse piezoelectric effect is observed when loadinga piezoelectric transducer with an electric voltage, resulting in mechanicaldeformations (actuator effect). This setup can be used, e.g., in a positioningsystem (see Fig. 9.2).

9.1 Constitutive Equation

Having the numerical computation of the dynamic behavior of piezoelectrictransducers in mind, we choose D and σ as the physical quantities depending

244 9 Piezoelectric Systems

Fig. 9.2. Piezoelectric actuator (bimorph setup)

on the electric field intensity E and mechanical strain S. Therefore, afterapplying a Taylor series up to the linear term, we obtain

σij = σij(S0ij , E

0k) +

∂σij

∂Skl|E(Skl − S0

kl) +∂σij

∂Ek|S (Ek − E0

k) (9.1)

Di = Di(S0ij , E

0k) +

∂Di

∂Skl|E (Skl − S0

kl) +∂Di

∂Ek|S (Ek − E0

k) . (9.2)

Without loss of generality, we can set S0kl = 0 and E0

i = 0. As shown in [198]the partial derivatives in (9.1) yield the material tensors

cEijkl =∂σij

∂Sklekij = −∂σij

∂EkεS

ik =∂Di

∂Ek. (9.3)

Therefore, we arrive at the material law describing the piezoelectric effect

[σ] = [cE ][S] − [e]TE (9.4)D = [e][S] + [εS ]E . (9.5)

The material tensors [cE ], [εS ], and [e] appearing in (9.4) and (9.5) are thetensor of elastic modulus, of dielectric constants, and of piezoelectric moduli.The superscriptsE and S indicate that the corresponding material parametershave to be determined at constant electric field intensity E and at constantmechanical strain S, respectively.

9.2 Governing Equations

For the linear case, we will use the Cauchy stress tensor [σ] and the linearstrain tensor [S] as mechanical quantities. Starting at Navier’s equation (3.22)applying Voigt notation

fV + BT σ = ρa , (9.6)

and using (9.4) as well as (3.38) for the strain–displacement relation, we arriveat

ρu − BT([cE ]Bu− [e]T E

)= fV . (9.7)

9.3 Piezoelectric Material Properties 245

Since piezoelectric materials are insulating, i.e., do not contain free-volumecharges, the electric field is determined by

∇ ·D = 0 . (9.8)

However, considering any piezoelectric sensor/actuator, which is set up by apiezoelectric material coated at the top and bottom by a metallic layer actingas electrodes, we rewrite (9.8) as

∇ ·D = qe (9.9)

to account for free electric charges qe on the electrodes. By using the relationbetween electric field intensity E and the scalar electric potential Ve (see(4.60)), (9.7) reads as

ρu− BT([cE ]Bu + [e]T ∇Ve

)= fV . (9.10)

Now, utilizing (9.5) we can transform (9.9) to

∇ ·([e]Bu− [εS ]∇Ve

)= qe . (9.11)

Therefore, the describing partial differential equations for linear piezoelectric-ity read as

ρu − BT([cE ]Bu + [e]T ∇Ve

)= fV (9.12)

∇ ·([e]Bu− [εS ]∇Ve

)= qe . (9.13)

9.3 Piezoelectric Material Properties

Piezoelectric materials can be subdivided into the following three categories[144]

1. Single crystals, like quartz2. Piezoelectric ceramics like barium titanate (BaTiO3) or lead zirconate

titanate (PZT)3. Polymers like PVDF (polyvinylidenfluoride)

Since categories 1 and 3 typically show a weak piezoelectric effect, these mate-rials are mainly used in sensor applications (e.g., force, torque or accelerationsensor). For piezoelectric ceramics the electromechanical coupling is large,thus making them attractive for actuator applications. These materials ex-hibit a polycrystalline structure and the key physical property of these mate-rials is ferroelectricity. In order to provide some physical understanding of thepiezoelectric effect, we will perform some discussions about the microscopicstructure of piezoceramics, partly following the exposition in [122]. A piezo-electric ceramic material is subdivided into grains consisting of unit cells with


different orientation of the crystal lattice. The unit cells consist of positive andnegative charged ions, and their charge center position relative to each otheris of main importance for the electromechanical properties. We will call thematerial polarizable, if an external load, e.g., an electric field can shift thesecenters with respect to each other. Let us consider BaTiO3 or PZT, whichhave a polycrystalline structure with grains having different crystal lattice.Above the Curie temperature Tc – for BaTiO3 Tc ≈ 120 oC - 130 oC and forPZT Tc ≈ 250 oC - 350 oC, these materials have the perovskite structure. Thecube shape of a unit cell has a side length of a0 and the centers of positiveand negative charges coincide (see Fig. 9.3). However, below Tc the unit cell

a0

a0

a0

a0

a0

c0

Ba2+

Ti4+

O2-

T>Tc T<Tc

Fig. 9.3. Unit cell of BaTiO3 above and below the Curie temperature Tc

deforms to a tetragonal structure as displayed in Fig. 9.3, e.g., BaTiO3 atroom temperature changes its dimension by (c0 − a0)/a0 ≈ 1 %. In this ferro-

E>EcE=0Inital state

Fig. 9.4. Orientation of the polarization of the unit cells at initial state, due to astrong external electric field and after switching it off

electric phase, the centers of the positive and negative charges differ and theunit cell posses a spontaneous polarization. Since the single dipoles are ran-

9.3 Piezoelectric Material Properties 247

domly oriented, we call this the thermally depoled state or virgin state. Thisstate can be modified by an electric or mechanical loading with significantamplitude. In practice, a strong electric field E ≈ 2 kV/mm will switch theunit cells such that the spontaneous polarization will be more or less orientedtowards the direction of the externally applied electric field as displayed inFig. 9.4. Now, when we switch off the external electric field the ceramic willstill exhibit a non-vanishing residual polarization in the macroscopic mean(see Fig. 9.4). We call this the irreversible or remanent polarization and theprocess as poling.

Now let us consider a mechanically unclamped piezoceramic disc at virginstate and load the electrodes by an increasing electric voltage. Initially, theorientation of the polarization within the unit cells is randomly distributedas shown in Fig. 9.5 (state 1). The switching of the domains starts when the

Fig. 9.5. Polarization P as a function of the electric field intensity E

external applied electric field reaches the so-called coercitive intensity Ec1. At

this state, the increase of the polarization is much faster, until all domains areswitched (see state 2 in Fig. 9.5). A further increase of the external electricloading results in an increase of the polarization with a quite smaller slopeand the occurring micromechanical process remains reversible. Reducing theexternal applied voltage to zero will preserve the poled domain structure,and we call the resulting macroscopic polarization the remanent polarizationPr. If the previous excitation has aligned all domains, Pr corresponds to thesaturation polarization. Psat. Loading the piezoceramic disc by a negative1 It has to be noted that in literature Ec often denotes the electric field intensity

at zero polarization. According to [122] we define Ec as the electric field intensityat which domain switching occurs.


voltage of an amplitude larger than Ec will initiate the switching processagain until we arrive at a random polarization of the domains (see state 4 inFig. 9.5). A further increase will orient the domain polarization in the newdirection of the external applied electric field (see state 5 in Fig. 9.5).

Measuring the mechanical strain during such a loading cycle as describedabove for the electric polarization results in the so-called butterfly curve de-picted in Fig. 9.6. Here we also observe that an external applied electric field

Fig. 9.6. Mechanical strain S as a function of the electric field intensity E

intensity E > Ec is required in order to obtain a measurable mechanicalstrain. The observed strong increase is a superposition of two effects: First,we achieve an increase of the strain due to the domains oriented with thec-axis in the direction of the external electric field. Secondly, the orientationof the domain polarization leads to the macroscopic piezoelectric effect yield-ing the reversible part of the strain. As soon as all domains are switched (seestate 2 in Fig. 9.6), the further increase of the strain just results from themacroscopic piezoelectric effect. A separation of the switching (irreversible)and the piezoelectric (reversible) strain can be best achieved by decreasingthe external electric load to zero. Only the strain induced by the alignmentof the c-axis remains and we denote this part the saturation strain Ssat.

Alternatively, or in addition to, this electric loading, we can perform amechanical loading, which will also result in switching processes. For a detaileddiscussion on the occurring effects we refer to [122].

The linear material tensors [cE ], [εS ], and [e], which relate the mechanicaland electric quantities, show a certain sparsity pattern according to the crystalstructure and polarization of the piezoelectric material. The 6 mm crystalclass, which also represents the equivalent class for piezoelectric ceramics, hasthe following pattern:

9.4 Models for Non-linear Piezoelectricity 249

[cE ] =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

c11 c12 c13 0 0 0c12 c11 c13 0 0 0c13 c13 c33 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 (c11 − c12)/2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(9.14)

[e] =

⎛⎜⎝ 0 0 0 0 e15 0

0 0 0 e15 0 0e31 e31 e33 0 0 0

⎞⎟⎠ [εS ] =

⎛⎜⎝ ε11 0 0

0 ε11 00 0 ε33

⎞⎟⎠ . (9.15)

Properties of some widely used piezoelectric materials are summarized inTable 9.1 (PZT 5A/5H from [144], 3202 (Motorola) from [114]).

Table 9.1. Material data for some materials of class 6 mm

cE11 cE

12 cE13 cE

33 cE44 cE

66

(N/m2) (N/m2) (N/m2) (N/m2) (N/m2) (N/m2)

PZT-5A 12.1 × 1010 75.4 × 109 75.2 × 109 11.1 × 1010 21.1 × 109 22.8 × 109

PZT-5H 12.6 × 1010 79.5 × 109 84.1 × 109 11.7 × 1010 23.0 × 109 23.2 × 109

32032 14.6 × 1010 96.2 × 109 10.2 × 1010 13.8 × 1010 25.5 × 109 24.9 × 109

e15 e31 e33

(C/m2) (C/m2) (C/m2)

PZT-5A 12.3 –5.4 15.8PZT-5H 17.0 –6.5 23.33202 15.3 –11.5 20.4

ε11 ε33

(Vs/Am) (Vs/Am)

PZT-5A 919 ε0 824 ε0

PZT-5H 1730 ε0 1437 ε0

3202 1378 ε0 1290 ε0

9.4 Models for Non-linear Piezoelectricity

In most actuator applications, piezoceramic materials, e.g., PZT are used,which exhibits a strong non-linear behavior for large signal excitations. Thisnon-linear behavior is characterized by the hysteresis loop of the polarization


(see Fig. 9.5) and the butterfly curve of the mechanical strain (see Fig. 9.6). Ingeneral, we can divide the physical/mathematical models into three categories:

1. Thermodynamically consistent modelsThese models are based on a macroscopic view to describe microscopicalphenomena, see e.g., [123, 138,193].

2. Micromechanical modelsThese models are sometimes also based on thermodynamic fundamentals,however they are constructed by breaking the material down to the sizeof single grains, see e.g., [11, 69, 195].

3. Models with hysteresis operatorThese models are mostly restricted to the actuator working range andconsider either the strain or the polarization hysteresis, see e.g., [98, 133,203].

In a first simplified approach we follow the ideas developed in [198]. There-with, we just consider the polarization hysteresis and rewrite the linear con-stitutive law (9.5) by

D = [e][S] + ε0E + P . (9.16)

In (9.16) P denotes the polarization modelled by a hysteresis model (see nextsection)

P = H[E] eP

with H the hysteresis operator and eP the unit vector of the polarization Pbeing equal to the direction of the applied electric field intensity E. Therewith,the describing PDEs read as follows

ρu − BT([cE ]Bu + [e]T ∇Ve

)= fV (9.17)

∇ · ([e]Bu− ε0∇Ve + P) = qe . (9.18)

In a second approach, we follow the basic ideas discussed in [123] anddecompose the physical quantities in a reversible and an irreversible part.Therefore, we introduce the reversible part Dr and the irreversible part Di ofthe dielectric displacement according to

D = Dr + Di . (9.19)

In our case, using the general relation between dielectric displacement D,electric field strength E, and polarization P we set Di = Pi. Analogously to(9.19), the mechanical strain S is also broken up into a reversible part Sr andan irreversible part Si

S = Sr + Si . (9.20)

9.4 Models for Non-linear Piezoelectricity 251

The decomposition of the strain S is done in compliance with the theory ofelastic-plastic solids with the assumption of very small deformations [15] thatis generally true for piezoceramic materials with maximum strains not morethan 0.2 %.

The reversible parts of mechanical strain Sr and dielectric displacementDr are described by the linear piezoelectric constitutive law (see (9.4) and(9.5)).

Now, in contrast to the thermodynamically motivated approaches, we com-pute the irreversible polarization from the history of the driving electric fieldE by a hysteresis operator H (see next section)

Pi = H[E] eP (9.21)

with the unit vector of the polarization eP , set equal to the direction of theapplied electric field.

The butterfly curve for the mechanical strain can also be modelled byan enhanced hysteresis operator. Nevertheless, as it can be seen in Fig. 9.7,the mechanical strain S33 seems to be proportional to the squared dielectricpolarization P3

2 (S ∝ P 2). The relation Si = β · (H[E])2, with a modelparameter β, seems obvious. However, to keep the model more general, theset-up

Si = β1 · H[E] + β2 · (H[E])2 + ...+ βn · (H[E])n (9.22)

is chosen. Similar to [124] we define the tensor of the irreversibel strains asfollows

[Si] = 32

(β1 · H[E] + β2 · (H[E])2 + · · ·

+ βn · (H[E])n) (

eP ePT − 1

3 [I]). (9.23)

The parameters β1 . . . βn need to be fitted to measured data. In practice, afourth-degree polynomial has been sufficient.

Furthermore, the entries of the tensor of the piezoelectric modulus are nowassumed to be a function of the irreversible dielectric polarization Pi. Herethe underlying idea is that the piezoelectric properties of the material onlyappear once the material is poled. Without any polarization, the domains inthe material are not aligned and therefore coupling between the electrical sideand the mechanical side does not occur. If the polarization is increased, thecoupling also increases. Hence, the following relation is defined

[e(Pi)] =|Pi|Psat

Q [e]RT . (9.24)

Herein, Psat defines the saturation polarization, [e] the tensor of the constantpiezoelectric moduli, and [e(Pi)] the tensors of the variable piezoelectric mod-uli. The two matrices R and Q perform a rotation of [e] in the direction of the


Fig. 9.7. Measured mechanical strain S33 and squared dielectric polarization P 23 on

a piezoceramic actuator on different axis

polarization eP . Using the solid angles (α1, α2, α3) of the polarization vectorwe can compute R by

R =

⎛⎜⎜⎝

cosα3 cosα2 cosα1



⎞⎟⎟⎠

and Q according to

Q =

(R 0

0 R

).

Finally, the constitutive relations for the electromechanical coupling can beestablished

D = [e(Pi)]Sr + [εS ]E + H[E]eP (9.25)

S = Sr + Si [Si] =

(32

n∑i=0

βi (H[E])i

) (eP eT

P − 13

I)

(9.26)

σ = [cE ]Sr − [e(Pi)]T E . (9.27)

In (9.26) [Si] denotes the symmetric tensor of the irreversible strain and Si

the corresponding six-component vector using Voigt notation. For a furtherdetailed discussion on this model we refer to [89]. Combining the new consti-tutive relations (9.25) - (9.27) with the governing equations defined by (9.6)and (9.9) we arrive at the following non-linear coupled system of PDEs

9.5 Hysteresis Model 253

ρu− BT([cE ]

(Bu− Si

)+ [e(Pi)]T ∇Ve

)= fV (9.28)

∇ ·([e(Pi)]

(Bu− Si

)− [εS ]∇Ve + Pi

)= qe (9.29)

with

Pi = H[−∇Ve]eP

[Si] =

(32

n∑i=0

βi (H[−∇Ve])i

) (eP eT

P − 13

I).

9.5 Hysteresis Model

One of the most general hysteresis models used is named after F. Preisach,who developed it in 1935. Preisach’s approach was purely intuitive and wasbased on plausible hypotheses concerning magnetic material behavior [173]. Amathematical-based investigation was performed by M. Krasnoselskii in the1970s (see e.g., [132]).

In order to get some physical as well as mathematical understanding, let usinvestigate some properties of Preisach’s hysteresis model. Thus, we consideran infinite set of elementary hysteresis operators Rβ,α, where each of them canbe represented by a rectangular loop (see Fig. 9.8). Since we want to model

Fig. 9.8. Rectangular hysteresis loop

the hysteresis within dielectric materials, we choose for the input quantity thenormalized electric field intensity e and for the output quantity the normalizedpolarization p according to

e =E

Esatp =

P

Psat. (9.30)

In (9.30) Esat denotes the saturated electric field intensity and Psat the satu-rated electric polarization. In Fig. 9.8 α and β are the up and down switching


values and according to these switching values, the input will lead to an out-put value +1 or −1. Restricting the switching values to α ≥ β and |α|, |β| ≤ 1leads to the following set S (see Fig. 9.9)

(α, β) ∈ S with S = (α, β) ∈ IR2, |α|, |β| ≤ 1, β ≤ α . (9.31)

Therewith, we describe the class of hysteresis loops with closed major loop

Fig. 9.9. Set S for possible switching values α and β

[155]. Now, the Preisach operator for the electric polarization p computes as

p(t) =∫S

℘(α, β)Rβ,α(e(t)) dαdβ . (9.32)

In (9.32) ℘ denotes the Preisach function, which defines the shape of thehysteresis loops and fulfills the following properties [155]

℘(α, β)≥ 0 for (α, β) ∈ S= 0 for (α, β) /∈ S

(9.33)∫

S

℘(α, β) dαdβ = 1 (9.34)

℘(−β,−α) = ℘(α, β) . (9.35)

Now let us assume that the input e(t) increases monotonically up to a valueof e1 at t = t1. Thus, all Rβ,α operators with α less than e1 switch up, whichmeans that their outputs take on the value of +1. Within the set S of possible(α, β) values, we will obtain a straight line parallel to the β-axis with α = e1(see Fig. 9.10a). In the next step, we assume that the input e(t) starts todecrease monotonically to a value of e2 at t2. Now, all Rβ,α operators withdown-switching values β larger than e2 will turn back, so that their output

9.5 Hysteresis Model 255

takes on the value of −1. This leads to a straight line parallel to the α-axiswith value β = e2, which is illustrated in Fig. 9.10a. Therefore, as illustratedin Fig. 9.10a, we can subdivide the region S into S+ (p takes on the value of+1) and S− (p takes on the value of −1).

(a) e(t) increases till e1 and de-creases till e2

(b) Staircase line L(t)

Fig. 9.10. Decomposing into S+ and S−

For the general case, a staircase line L(t) will subdivide S into S+ and S−

(see Fig. 9.10b) according to the following two rules:

• A monotonically increasing input signal e(t) defines a straight line parallelto the β-axis with value e(t).

• A monotonically decreasing input signal e(t) defines a straight line parallelto the α-axis with value e(t).

Therefore, the horizontal lines represent relative maxima and the vertical linesrelative minima. In addition, by storing the local maxima and minima, thehysteresis can be uniquely constructed.

Due to the wiping-out property, not all relative maxima and minima haveto be stored. This property states that each local input maximum wipes outthe vertical L(t) whose α values are below this maximum, and each localminimum wipes out the vertices whose β values are above the minimum [155].The wiping out is best illustrated by an input signal e(t) as displayed in Fig.9.11. Only the relative maxima e1 and e3 as well as relative minima e2 and e4have to be stored. All other maxima (minima) will be intermediately storedduring the process in a list, but will be deleted due to the wiping-out property.

Furthermore, the Preisach model fulfills the congruence property [155],which states that all minor hysteresis loops corresponding to back-and-forthvariations of inputs between the same two consecutive extrema are congruent(see Fig. 9.12).


Fig. 9.11. Input signal e(t) for illustration of the wiping out property

Fig. 9.12. Congruency property of the hysteresis model

For the numerical computation of the Preisach operator, the followingefficient evaluation has been developed. With e1, ..., en those relative inputextrema that have not been wiped out yet at time t, the value of the outputat time t computes as

p(t) = E(−e1, e1) + 2n−1∑i=1

E(ei, ei+1) , (9.36)

with E(ei, ei+1) the Everett function (see Fig. 9.13)

E(e1, e2) =∫

T (e1,e2)

℘(α, β) dαdβ . (9.37)

For the simplest Preisach function ℘(α, β) = 1/2, the Everett function com-putes as

E(e1, e2) =14(e2 − e1)2sgn(e2 − e1) . (9.38)


Fig. 9.13. Computation of the Everett function E(e1, e2)

Thus, we have an efficient model for taking into account ferroelectric hysteresiswithin piezoelectric materials. For a detailed discussion concerning hysteresisoperators in PDEs, and especially their identification from measured data, werefer to [112].


In the following we will first derive the discrete form of the linear piezoelectricpartial differential equations applying the FE method. In the second step,we will investigate the numerical modelling of ferroelectric hysteresis withinpiezoelectric materials and finally, we will present simulation results.

9.6.1 Linear Case

Let us consider a simple setup (without loss of generality) as displayed in Fig.9.14. The strong formulation for the piezoelectric system reads as follows:

Fig. 9.14. Setup for the formulation


Given:u0 : Ω → IRd

u0 : Ω → IRd

V0 : Ω → IRρ, cij , eij , εij : Ω → IR .

Find: u(t), Ve(t) : Ω × [0, T ] → IRd

ρ u− BT([cE ]Bu− [e]T ∇Ve

)= 0 (9.39)

∇ ·([e]Bu− [εS ]∇Ve

)= 0 . (9.40)

Boundary conditionsu = 0 on Γ u

e × (0, T )Ve = 0 on Γ V1

e × (0, T )Ve = V0 on Γ V2

e × (0, T )n · [σ] = 0 on Γ σ

n × (0, T )

n · D = 0 on ΓDn × (0, T ) .

Initial conditionu(r, 0) = u0 , r ∈ Ωu(r, 0) = u0 , r ∈ Ω .

In this strong formulation of the piezoelectric partial differential equationsA denotes the surface area covered by the loaded electrode. The variationalformulation for this case with u′ and ψ defining appropriate test functions is∫

Ω

ρu′ · u dΩ +∫Ω

(Bu′)T [cE ]Bu dΩ +∫Ω

(Bu′)T [e]T BVe dΩ = 0

∫Ω

(Bψ)T e(Bu) dΩ −∫Ω

(Bψ)T [εS ] BVe dΩ = 0 (9.41)

with B = (∂/∂x, ∂/∂y, ∂/∂z)T . Now, using standard nodal finite elementsfor the mechanical displacement u and electric potential Ve (nn denotes thenumber of nodes with unknown displacement and unknown electric potential)

u ≈ uh =nd∑i=1

nn∑a=1

Nauiaei =nn∑

a=1

Naua ; Na =

⎛⎝Na 0 0

0 Na 00 0 Na

⎞⎠(9.42)

Ve ≈ V he =

nn∑a=1

NaVea (9.43)


as well as for the test functions u′ and ψ, we obtain ( [6, 146])

nn∑a=1

nn∑b=1

⎛⎝∫

Ω

ρNTa Nb dΩ ub +

∫Ω

BTa [cE ]Bb dΩ ub

+∫Ω

BTa [e]T Bb dΩ Veb

⎞⎠ = 0 (9.44)

nn∑a=1

nn∑b=1

⎛⎝∫

Ω

BaTeBb dΩ ub −

∫Ω

BTa [εS ] Bb dΩ Veb = 0 .

In (9.44) Ba computes as

Ba =

⎛⎜⎜⎝

∂Na

∂x

∂Na

∂y

∂Na

∂z

⎞⎟⎟⎠ . (9.45)

Introducing damping with a damping matrix Cu (see Sect. 3.7.2) we maywrite (9.44) and (9.45) in matrix form(

Mu 00 0

)(u

Ve

)+(

Cu 00 0

)(u

Ve

)+(

Ku KuV

KTuV −KV

)(uVe

)=(

0f

e

),

(9.46)with the matrices Mu, Cu, Ku as given in Sect. 3.7.1, KV in Sect. 4.6 andf

ea right hand side due to the electric Dirichlet boundary conditions. The

matrix KuV computes as

KuV =ne∧

e=1

keuV ; ke

uV = [kpq ] ; kpq =∫Ωe

BTp [e]T Bq dΩ

For the time discretization, a Newmark algorithm as described in Sect. 2.5.2is used.

9.6.2 Non-linear Case

Here, we will only consider the simplified non-linear model for piezoelectricitydescribed by the system of PDEs defined in (9.17), (9.18) and note that themore general model (see (9.28), (9.29)) can be treated in a similar manner. In astraightforward application of the FE method, one would treat the non-linearterm ∇ ·P = ∇ ·H[E] just as a right hand term within a fixed-point iteration.However, this approach will lead to a very poor convergence resulting in a largenumber of iteration steps or even nonconvergence. Therefore, we modify (9.18)


by introducing a differential permittivity tensor [εd]. We first decompose theelectric displacement vector D at time step n+ 1 to the value of the previoustime step n and an incremental displacement vector ∆D

Dn+1 = Dn +∆D . (9.47)

We now define the incremental displacement vector ∆D according to (9.16)by

∆D = [e][∆S] + [εd]∆E (9.48)

and use the differential permittivity tensor [εd] to represent ε0E+P . For thecomputation of Dn we use (9.16) in its original form

Dn = [e][Sn] + ε0En + Pn . (9.49)

We can now rewrite Maxwell’s equation for the electrostatic field in the caseof a piezoelectric material as follows

∇ ·Dn+1 = ∇ · (Dn +∆D) = 0 . (9.50)

By using (9.47) - (9.49) we obtain

∇ ·([e][Sn] + ε0En + Pn + [e]

([Sn+1] − [Sn]

)− [εd]

(∇V n+1e − ∇V n

e

))= 0

(9.51)which results in

∇ ·([e][Sn+1] − [εd]∇V n+1

e

)= ∇ ·

((ε0[I] − [εd])∇V n

e − Pn)

(9.52)

with [I] the identity tensor. The entries of the differential permittivity tensor[εd] = diag (εd1 , ε

d2 , ε

d3) are computed according to (9.48)

εdi =∆Di − eijl∆Sjl

∆Ei=

(Dn+1

i −Dni

)− eijl

(Sn+1

jl − Snjl

)En+1

i − Eni

. (9.53)

Now, the new system of coupled PDEs to be solved at time step n+ 1 readsas follows

ρ u− BT([cE ]Bu− [e]T ∇Ve

)= 0 (9.54)

∇ ·([e]Sn+1 − [εd]∇V n+1

e

)= ∇ ·

((ε0[I] − [εd])∇V n

e − Pn). (9.55)

Applying the FE method to the above equations and using the Newmarkscheme for time discretization will lead to the following non-linear algebraicsystem of equations(

K∗u KuV

KTuV −KNL1

V k

)(un+1

k+1

Ven+1k+1

)=(

0KNL2

V k Ven − f

k+1

), (9.56)


In (9.56) n denotes the time step counter, k the iteration counter and K∗u

computes as Ku + 1/(γH∆t2)Mu. The matrices Ku, Mu and KuV compute

as for the linear case, and the non-linear matrices KNL1V k , KNL2

V k as well as thenon-linear right-hand side f

k+1can be calculated as follows

KNL1V k =

ne∧e=1


BTp [εk

d]Bq dΩ

KNL2V k =

ne∧e=1


BTp

(ε0[I] − [εd

k])Bq dΩ

fk+1

=ne∧

e=1

fe ; fe = [fp] ; f

p=∫

Ωe

Np BTH[−BVen+1k

] dΩ

[εdk] = diag (εd1 , ε

d2 , ε

d3)

εdi =H[−BVe

n+1k

]i −H[−BVen]i − ε0

((BVe

n+1k

)i − (BVen)i

)−(BVe

n+1k

)i + (BVen)i

.


In the following, we will discuss in the first example the computation of theelectric impedance of a piezoelectric disc. In the second example we will per-form computations for large signal loading of a piezoelectric transducer usingthe nonlinear formulation as described in Sec. 9.6.2.

9.7.1 Computation of Impedance Curve

We consider a transducer as shown in Fig. 9.15, i.e., a rotational symmet-ric disc with electrodes on top and bottom. In order to obtain the wholeimpedance characteristics in one simulation run, a special technique must beused. The model of the transducer is excited by an electric voltage pulse andits charge response is computed by means of transient analysis. In general,the following equation for the impedance characteristics Z(ω) is valid

Z(ω) =U(ω)I(ω)

=U(ω)jωQ(ω)

, (9.57)

where U(ω), I(ω), and Q(ω) are Fourier spectra of the voltage, the current,and the charge time signal, respectively. One can see that the impedancecharacteristics can be calculated by dividing the spectrum of the voltage sig-nal (excitation) by the jω-multiple of the computed spectrum of the electric


Symmetry axis

Upper electrode

Lower electrode

u, q

Fig. 9.15. Piezoelectric transducer

charge. Within a postprocessing step we can compute the electric charge as afunction of time

q(t) =∫Γe

D · dΓ =∫Γe

([e]Bu− [εS ]BVe

)· dΓ . (9.58)

In (9.58) Γe denotes the electrode surface, e.g., of the loaded electrode and uand Ve the FE solution quantities. After performing the Fourier transforma-tion of q(t) we obtain Q(ω).

The voltage signal must obviously have a frequency spectrum that is ableto excite the transducer in the interesting frequency range. It is recommendedto choose a signal whose spectrum is nonzero up to a frequency of at least10fr (one order higher than the resonance frequency fr of the transducer). Dueto the rotational symmetry, the transducer is modelled by an axisymmetricformulation. In addition, the planar symmetry of the transducer is utilizedin order to reduce the size of the model (see Fig. 9.16). The geometricaldimensions of the electrode are neglected, i.e., it is modelled as an infinitelythin layer of surface nodes forming an equipotential area. The well-knownpiezoelectric ceramic material PZT-5A is assumed (see Table 9.1). Accordingto Fig. 9.16, the radius of the transducer is R = 10 mm and its thickness isD = 2 mm. Since the estimated resonance frequency can be computed by

fr =c

2D, (9.59)

we obtain, with c ≈ 4 000 m/s, a value of 1 MHz. Hence, the excitation voltagesignal should have a nonzero spectrum up to 10 MHz. The selected signal has aform of a triangular pulse (see Fig. 9.17). Due to the fact that a linear analysisis applied here, the magnitude of the pulse has no influence on the resultingimpedance characteristics. Therefore, a unit pulse is taken for simplicity.

The boundary conditions applied to the model are illustrated in Fig. 9.18.No displacement in the horizontal direction is allowed along the symmetry


Fig. 9.16. Model of the piezoelectric transducer

Fig. 9.17. Electric voltage excitation of the transducer

axis and, correspondingly, no displacement in the vertical direction is allowedat the symmetry plane. Furthermore, the symmetry plane serves as the ref-erence surface for electric potential, i.e., zero electric potential is prescribedthere. Finally, the upper surface, which represents the electrode, is an equipo-tential area, and the electric load is applied there. For the discretization of the

Fig. 9.18. Boundary conditions applied to the model of the transducer


simulation domain, linear finite elements have been used. Due to the simpledomain, a mapped meshing with five elements in thickness and 50 elementsin the radial direction is performed. The time step size ∆t is set to 20 ns anda total number of 8192 time steps have been computed. Therefore, we achievea frequency resolution of about 3 kHz and ten time samples for the triangularexcitation charge.After the simulation, a Fourier transformation for both the computed

Fig. 9.19. Computed impedance characteristics of the piezoelectric transducer

charge signal and the excitation voltage signal is performed. The computedimpedance according to (9.57) is displayed in Fig. 9.19. One can clearly seethe resonance and antiresonance points of the various vibration modes of thetransducer. Furthermore, the principal thickness mode of the transducer ap-pears at the frequency of about 1 MHz, but it is disturbed by additional modes.When designing such a piezoelectric transducer, this effect can be suppressedby increasing the diameter/thickness ratio, which leads to a better decouplingof the particular vibration modes.

9.7.2 Non-linear Case

In this section we will demonstrate measured and simulated data for large-signal loading of a piezoelectric disc (see Fig. 9.20), using for the numericalsimulation the scheme described in Sect. 9.6.2.

The disc transducer can be described by an axisymmetric model similar tothe previous example (see Fig. 9.16). The geometrical dimensions of the elec-


D

2R

u~

P

Fig. 9.20. Setup for the large-signal loading

trode are neglected. According to Fig. 9.20, the radius of the transducer isR = 5.5 mm and its thickness is D = 0.2 mm. The input voltage is a sinesignal with an amplitude of 50 V and a frequency of 1 MHz. According tomeasurements of the hysteresis curve, the following parameters are used forthe Preisach model:

Parameter Value

Saturation of electric field intensity Esat 2.0 MV/mSaturation of electric polarization Psat 0.04 C/m2

The displacement in the r-direction along the axis of symmetry and thedisplacement in the z-direction along the symmetry plane (z = 0) are setto zero. On the top electrode, the input signal is prescribed (inhomogeneousDirichlet boundary condition) and along the symmetry plane (z = 0) the elec-tric potential is set to zero. For the discretization of the simulation domainlinear finite elements have been used. Due to the simple domain, a mappedmeshing with four elements in thickness and 80 elements in the radial direc-tion is performed. The time step size ∆t is set to 25 ns (which means a totalof about 50 time samples per fundamental period).

Changes in the state of ferroelectric polarization are mainly responsiblefor the characteristics of the piezoceramic disc. In order to overcome withinthe experiement strong thermal influences due to the heat generation causedby internal friction in the piezoceramic material, a sine burst signal with asmall pulse/pause ratio is used for excitation. In Fig. 9.21(a) the time signalof the input current and voltage can be observed, whereas in Fig. 9.21(b)the spectral rates of the higher-order harmonics related to the fundamentalfrequency are plotted. These higher-order harmonics and the nonsymmetricresponse signal are caused by the ferroelectric hysteresis.


0 0.5 1 1.5 2-50 -50

-40 -40

-30 -30

-20 -20

-10 -10

0 0

10 10

20 20

30 30

40 40

50 50

U(V

)

I(m

A)

t (µs)

Voltage (measured)

Current (measured)

Current (simulated)

(a) Input voltage and current

(b) Higher-order frequency spectra

Fig. 9.21. Measured and simulated data for the dynamic load case

10

Computational Aeroacoustics

A large amount of the total noise in our daily lives is generated by turbulentflows (e.g., airplanes, cars, air conditioning systems, etc.). The physics behindthe generation process is quite complicated and still not fully understood.The use of numerical simulation tools is one important way to analyze thegeneration of flow-induced sound, e.g., [206].

In the following, we will first discuss the arising requirements for numer-ical schemes, when computing flow-induced noise. Then, we will focus onLighthill’s acoustic analogy and derive an FE formulation of the inhomoge-neous wave equation. Finally, we will present the numerical computation of aco-rotating vortex pair and the comparison to the analytical solution.

10.1 Requirements for Numerical Schemes

Since the beginning of computational aeroacoustics (CAA) several numericalmethodologies have been proposed, each of these trying to overcome the chal-lenges that the specific problems under investigation pose for an effective andaccurate computation of the radiated sound. The difficulties which have to beconsidered for the simulation of flow noise problems include [86, 87]:

• Energy disparity and acoustic inefficiency: There is a large disparity be-tween the energy of the flow in the non-linear field and the radiated acous-tic energy of an unsteady flow. In general, the total radiated power ofa turbulent jet scales with O(v8/c5) (v is the characteristic flow velocityand c the speed of sound), and for a dipole source arising from pressurefluctuations on surfaces inside the flow scales with O(v6/c3). This showsthat an aeroacoustics process at low Mach number is rather a poor soundemitter.

• Length scale disparity: Large disparity also occurs between the size of aneddy in the turbulent flow and the wavelength of the generated acoustic

268 10 Computational Aeroacoustics

noise. Low Mach number eddies have a characteristic length scale l, veloc-ity v, a life time l/v and a frequency ω. This eddy will then radiate acousticwaves of the same characteristic frequency, but with a much larger lengthscale, which scales as follows

λ ∝ cl

v=

l

Ma. (10.1)

In (10.1) Ma denotes the Mach number, which is defined by the ratio ofthe characteristic velocity v over the speed of sound c

Ma =v

c.

• Preservation of multipole character: The numerical analysis must preservethe multipole structure of the acoustic source in order to resolve the wholestructure of the source.

• Dispersion and dissipation: The discrete form of the acoustic wave equationcannot precisely represent the dispersion relation of the acoustic sound.Numerical discretization in space and time converts the original non-dispersive system into a dispersive discretized one, which exhibits wavephenomena of two kinds:1. Long wavelength components approaching the solution of the original

PDE as the grid is refined.2. Short wavelength components (spurious waves) without counterpart

in the original PDE evolving in the numerical scheme disturbing thesolution.

The wave equation shows a non-dissipative behavior; as such, dissipativeerrors must be avoided by a numerical implementation, in which both theamplitude and phase of the wave are of crucial importance.

• Difficulties in non-linear wave phenomena: In turbulent flows having ahigh speed, non-linear effects will play a role since the wave equation hasto be solved over a long range, which induces then dissipation of acousticenergy and refraction.

• Flows with high Mach and Reynolds number: Aeroacoustic problems ofteninvolve both high Mach and Reynolds numbers. Flows at a high Machnumber may induce new non-linear sources and convective effects whileflows at a high Reynolds number introduce multiple scale difficulties dueto the disparity between the acoustic wavelength λ and the size of theenergy dissipating eddies.

10.1 Requirements for Numerical Schemes 269

• Simulation of unbounded domains: As a main issue for the simulation ofunbounded domains using interior methods remains the boundary treat-ment which needs to be applied to avoid the reflection of the outgoingwaves on the truncating boundary of the computational domain (see Sect.5.5).

Currently available aeroacoustic methodologies overcome only some ofthese broad range of numerical and physical issues, which restricts their ap-plicability, making them, in many cases, problem dependent methodologies.The application of Direct Numerical Simulation (DNS) is becoming more fea-sible with the permanent advancement in computational resources. However,due to the large disparities of length and time scales between fluid and acous-tic fields, DNS remains restricted to low Reynolds number flows. In a DNS,all relevant scales of turbulence are resolved and no turbulence modelling isemployed. Therefore, although some promising work has been done in thisdirection [68], the simulation of practical problems involving high Reynoldsnumbers requires very high resolutions which are still far beyond the capa-bilities of current supercomputers [211]. Hence, hybrid methodologies haveestablished as the most practical methods for aeroacoustic computations, dueto the separate treatment of the fluid and the acoustic computations. In theseschemes, the computational domain is split into a non-linear source region anda wave propagation region, and different numerical schemes are used for theflow and acoustic computations. Herewith, first a turbulence model is used tocompute the unsteady flow in the source region. Secondly, from the fluid field,acoustic sources are evaluated which are then used as input for the computa-tion of the acoustic propagation. In these coupled simulations it is generallyassumed that no significant physical effects occur from the acoustic to thefluid field.

Figure 10.1 shows typical numerical methods which are employed when us-ing any of these hybrid methodologies. Among the group of hybrid approaches,integral methods remain widely used in CAA for solving open problems inlarge acoustic domains like airframe noise, landing gear noise simulation, fan(turbines) noise, rotor noise, etc. One reason which motivates the use of inte-gral formulations in such applications is that, in general, their acoustic sourcescan be considered to be compact and only an extension of the acoustic solutionat a few points in the far field is desired. Therefore, in such cases, integralmethods based on Lighthill’s acoustic analogy, Curle’s formulation, FfowcsWilliams and Hawkings (FW-H) formulation, Kirchhoff method or extensionthereof are computationally cheaper than interior methods where a wholediscretization of the acoustic domain is required (see e.g., [34, 50, 63, 150]).

On the other hand, for interior aeroacoustic problems, where non-compactsolid boundaries are present, or if structural/acoustic effects are considered,it is more appropriate to use an acoustic interior method to account for theinteractions between the solid surfaces and the flow-induced noise directly inthe acoustic simulation. In such cases, integral formulations would require a


priori knowledge of a hard-wall Green’s function that is not known for complexgeometries [165]. Furthermore, integral methods do not allow for a straightfor-ward inclusion of the elastic effects of structures in the flow. An additional ad-vantage of interior methods is that they can also be used to include the effectsof wave propagation in non-uniform background flows. Among the interiormethods we find those based on Linearized Euler Equations (LEE) [12, 55],Acoustic Perturbation Equation (APE) [62], FE formulations of Lighthill’sacoustic analogy [165], as well as the linearized perturbed compressible equa-tions (LPCE) [196] (for a discussion on these methods see [61]).

Fig. 10.1 depicts the general configuration when using these methods.Herewith, ΩF denotes the area where the flow field is firstly computed andwhere the acoustic sources are interpolated from the fluid mesh to the acousticmesh. In order to accurately resolve the source terms, unsteady computationalfluid dynamics (CFD) schemes are required. Mainly used turbulence modelsare LES (Large Eddy Simulation), DES (Detached Eddy Simulation) and SAS(Scale Adaptive Simulation). For a detailed discussion on these methods werefer to [35]. The acoustic propagation region is given by ΩF

⋃ΩA where the

acoustic field is computed in the second step by solving the inhomogeneouswave equation or a corresponding set of equations depending on the CAAmethodology followed. Since interior methods require the whole discretizationof the propagation domain, usually they are used to compute the radiatedsound until an intermediate region in the far field (i.e., until ΓA in Fig. 10.1),before moving to an integral formulation in which the acoustic solution fromthe interior method at the interface is used as input for computing pressurelevels at the far field. Such a combined scheme has been presented in [151]using LEE for the intermediate solution and a Kirchhoff method for the farfield noise.

10.2 Lighthill’s Analogy and its Extension

Let us start our derivation of Lighthill’s analogy by introducing the two mainequations of fluid dynamics: mass conservation and momentum conservation.The differential formulation of the mass conservation (continuity) equationreads as follows

∂ρ

∂t+∂ρvi

∂xi= 0 . (10.2)

In (10.2) ρ denotes the density of the fluid and vi the i-th component of theflow velocity vector v. The momentum conservation equation is expressed by

ρ∂vi

∂t+ ρvj

∂vi

∂xj= − ∂p

∂xi− ∂τij∂xj

= − ∂

∂xj(pδij + τij) , (10.3)

where p denotes the total pressure within the flow, [τ ] the viscous stress tensorand δ the Kronecker symbol. For Newtonian fluids, we can express [τ ] by

10.2 Lighthill’s Analogy and its Extension 271

Fig. 10.1. Schematic depicting some of the possible strategies when using an aeroa-coustic hybrid approach

τij = −µ(∂vi

∂xj+∂vj

∂xi

)+

23µδij

∂vk

∂xk(10.4)

with µ the dynamic viscosity.In the first step, we multiply (10.2) by vi (before we substitute the index

i by j) and add the so obtained equation to (10.3). In addition, we substitutep in (10.3) by p − p0 with p0 the mean pressure. This operation is allowed,since the relation ∂(p− p0)/∂xj = ∂p/∂xj holds. Therewith, we obtain

ρ∂vi

∂t+ vi

∂ρ

∂t︸︷︷︸∂ρvi

∂t

+ ρvj∂vi

∂xj+ vi

∂ρvj

∂xj︸︷︷︸∂

∂xj(ρvivj)

= − ∂

∂xj((p− p0)δij − τij) (10.5)

By introducing the momentum flux tensor [τ I] by

τ Iij = ρvivj + (p− p0)δij − τij (10.6)

we may write (10.5) as∂ρvi

∂t= −

∂τ Iij

∂xj. (10.7)


For linear acoustics, the momentum flux tensor [τ I] reduces to

τ0ij = (p− p0)δij (10.8)

and (10.7) reads as∂ρvi

∂t+

∂

∂xi(p− p0) = 0 . (10.9)

In addition, we define the acoustic pressure p′, the acoustic density ρ′ and itsrelation (just fulfilled for linear acoustics)

p′ = p− p0 ; ρ′ = ρ− ρ0 ;p′

ρ′= c2 (10.10)

with c the speed of sound. Substituting p− p0 in (10.9) by c2ρ′ and applying∂/∂xi to this equation, results in

∂

∂t

∂

∂xi(ρvi) +

∂2

∂x2i

(c2ρ′) = 0 . (10.11)

According to the mass conservation equation, we may rewrite (10.11) as

∂2ρ′

∂t2− ∂2

∂x2i

(c2ρ′) =(

1c2

∂2

∂t2− ∂2

∂x2i

)(c2ρ′) = 0 . (10.12)

Since we have neglected the flow and there are no other excitations, no acous-tic sound field will be generated, since the solution of (10.12) is ρ′ = ρ−ρ0 = 0.

Lighthill’s analogy is based on the fact that the sound generated by theflow in a real fluid is exactly equivalent to that produced in the ideal, linearacoustic field, forced by the stress distribution

Lij = τ Iij − τ0

ij

= ρvivj +((p− p0) − c2(ρ− ρ0)

)δij − τij , (10.13)

where [L] denotes the Lighthill stress tensor.We can now rewrite (10.7) as the momentum equation for the ideal, lin-

ear acoustic medium subjected to the externally applied stress according to(10.13)

∂ρvi

∂t+∂τ0

ij

∂xj= −

∂(τ Iij − τ0

ij)∂xj

(10.14)

Therewith, we obtain

∂ρvi

∂t+

∂

∂xi

(c2(ρ− ρ0)

)= −∂Lij

∂xj. (10.15)

Eliminating the momentum density ρvi similarly as for the linear case, weobtain

10.2 Lighthill’s Analogy and its Extension 273(1c2

∂2

∂t2− ∂2

∂x2i

)(c2(ρ− ρ0)) =

∂2Lij

∂xi∂xj. (10.16)

The integral formulation of (10.16) is given by

c2(ρ− ρ0)(x, t) =14π

∂2

∂xi∂xj

∫Ω1

∂Lij(ξ, t− r/c)r

dξ dη dζ (10.17)

with the source coordinates ξ = (ξ, η, ζ)T , the observation coordinates x =(x1, x2, x3)T and the distance vector r = (x1 − ξ, x2 − η, x3 − ζ)T .

The integral formulation given in (10.17) is no longer correct, if any solidand/or elastic bodies (in rest or moving) are present within the simulationdomain. For such situations, the acoustic field has to fulfill the correct bound-ary condition and we no longer can use Green’s function for free radiation.Now, the idea is, to derive an extended form of Lighthill’s equation, whichis defined in the whole simulation domain, and thus Green’s function for freeradiation can be applied to obtain an integral formulation [65].

We will assume a domain with a solid body, which is located within theconsidered domain Ω. The volume of the body is denoted by ΩS and itssurface by ΓS (see Fig. 10.2). We describe the surface of the body with a

Fig. 10.2. Computational domain Ω with a solid body ΩS

function f(x, t)

f(x, t) =

⎧⎪⎪⎨⎪⎪⎩f < 0 for x ∈ ΩS

f > 0 for x ∈ Ω \ΩS

f = 0 for x on ΓS

(10.18)

e.g., a breathing sphere: f(x, t) = |x−x0|2−R20(t). Additionally, we introduce

the Heaviside function H(f(x, t))

H(f(x, t)) =

1 for x ∈ Ω \ΩS

0 for x ∈ ΩS .(10.19)


First of all, we multiply the continuity equation (see (10.2)) with H(f)(∂ρ′

∂t+

∂

∂xi(ρvi)

)H(f) = 0 (10.20)

and rewrite it in the following form

∂

∂t(ρ′H(f)) − ρ′

∂H(f)∂t

+∂

∂xi((ρvi)H(f)) − (ρvi)

∂H(f)∂xi

= 0 . (10.21)

The Heaviside function H(f) fulfills two important properties [96]

∂H(f)∂xi

=∂f

∂xiδ(f) (10.22)

∂H(f)∂t

= δ(f)∂f

∂t= −vB

i

∂f

∂xiδ(f) (10.23)

with vBi the velocity of the solid body inΩ. Using (10.22) and (10.23) simplifies

(10.21) to

∂(ρ′H(f))∂t

+∂(ρviH(f))

∂xi=(ρ(vi − vB

i ) + ρ0vBi

) ∂f∂xi

δ(f) . (10.24)

This equation is an extension of the continuity equation, which is fulfilled allover Ω. As can be seen, the right-hand side of (10.24) is just on ΓS differentfrom zero and the left side is zero within ΩS. Applying similar operations tothe momentum conservation equation, we obtain

∂

∂t(ρviH(f)) +

∂

∂xj(ρvivj + p′δij − τij)H(f)

=(ρvi(vj − vB

j ) + p′δij − τij) ∂f∂xj

δ(f) . (10.25)

Using the definition of the Lighthill tensor Lij (see 10.13)) we may write(10.25) as

∂

∂t(ρviH(f)) +

∂

∂xi(c2ρ′H(f))

= −∂LijH(f)∂xj

+(ρvi(vj − vB

j ) + p′δij − τij) ∂f∂xj

δ(f) .(10.26)

Furthermore, we rewrite (10.24) in the following form

∂(ρviH(f))∂t

=(ρ(vi − vB

i ) + ρ0vBi

) ∂f∂xi

δ(f) − ∂(ρ′H(f))∂xi

. (10.27)


Applying ∂/∂xi to (10.26) and using the relation according to (10.27), wearrive at the extended form of Lighthill’s equation(

1c2

∂2

∂t2− ∂2

∂x2i

)(c2ρ′H(f)

)=

∂2

∂xi∂xj(LijH(f)) − ∂

∂xi

((ρvi(vj − vB

j ) + p′ δij − τij) ∂f∂xj

δ(f))

+∂

∂t

((ρ(vi − vB

i ) + ρ0vBi

) ∂f∂xi

δ(f)). (10.28)

According to the relations

∂f

∂xj= nj

∂f

∂xi= ni

the terms (vj −vBj ) and (vi −vB

i ) vanishes in (10.28). Since the extended formof Lighthill’s equation is valid throughout the whole domain Ω, we can useGreen’s function for free radiation to obtain a solution. Therefore, the integralform, which is known as the Ffowcs Williams and Hawkings equation [65],reads as

c2 ρ′H(f) =∂2

∂xi∂xj

∫Ω\ΩS

Lij

4π rdΩ − ∂

∂xi

∮ΓS

(p′ δij − τij)4π r

dΓ

+∂

∂t

∮ΓS

ρ0vBi

4π rdΓ . (10.29)

10.3 Finite Element Formulation

We will perform a volume discretization of Lighthill’s equation (see 10.16) byapplying the finite element method (FEM). Therewith, any solid–elastic bodywill be implicitly taken into account, and there is no need to use the extendedform of Lighthill’s equation as given by (10.28).

Let us start at the strong formulation of (10.16), where we substitute ρ′

by p′/c2

Given: Lij : Ω × (0, T ) → IRc : Ω → IR

p′0 : Ω → IR

p′0 : Ω → IR

Find: p′ : Ω × [0, T ] → IR


1c2∂2p′

∂t2− ∂2p′

∂x2i

=∂2Lij

∂xi∂xj(10.30)

p′(r, 0) = p′0 , r ∈ Ωp′(r, 0) = p′

0, r ∈ Ω

In ( 10.30)Ω define the total computational domain, which consists ofΩF∪ΩA

as displayed in Fig. 10.1. Therewith, ΩF denotes the computational domainfor the flow computation, where we evaluate the acoustic source terms, andΩA the acoustic propagation domain.

At this stage we assume that the Lighthill tensor [L] is a known quantity,e.g., obtained by a flow computation using a large eddy simulation (LES).In the first step, we multiply (10.30) by an appropriate test function w andintegrate over the whole domain Ω (corresponding in Fig. 10.1 to ΩF ∪ΩA)∫

Ω

w

(1c2∂2p′

∂t2− ∂2p′

∂x2i

− ∂2Lij

∂xi∂xj

)dΩ = 0 . (10.31)

Now, we apply Green’s integral theorem to the second spatial derivative of p′

as well as Lij . This operation will result in the following relations∫Ω

w∂2p′

∂x2i

dΩ =∫

ΓS∪ΓA

w∂p′

∂ndΓ −

∫Ω

∂w

∂xi

∂p′

∂xidΩ (10.32)

∫Ω

w∂2Lij

∂xi∂xjdΩ =

∫ΓS

w∂Lij

∂xjni dΓ −

∫ΩF

∂w

∂xi

∂Lij

∂xjdΩ . (10.33)

We want to emphasize that the boundary integral (10.33) is just over thesurface ΓS of any solid–elastic body, whereas in (10.32) we have to integrateover ΓS as well as over ΓA, which limits the computational domain (see Fig.10.1).

Now we can substitute ∂Lij/∂xj within the first term on the right-handside of (10.33) by (10.15) and obtain∫

ΓS

w∂Lij

∂xjni dΓ =

∫ΓS

w

(−∂ρvi

∂t− ∂

∂xi(c2ρ′)

)ni dΓ . (10.34)

Since on a solid surface vini = 0 is fulfilled, we see that the surface integralterm over ΓS reduces to∫

ΓS

w∂Lij

∂xjni dΓ = −

∫ΓS

c2 w∂ρ′

∂ndΓ = −

∫ΓS

w∂p′

∂ndΓ . (10.35)

Therewith, we can rewrite (10.31) as

10.3 Finite Element Formulation 277∫Ω

1c2w∂2p′

∂t2+

∫Ω

∂w

∂xi

∂p′

∂xidΩ −

∫ΓS∪ΓA

w∂p′

∂ndΓ

= −∫

ΩF

∂w

∂xi

∂Lij

∂xjdΩ −

∫ΓS

w∂p′

∂ndΓ . (10.36)

Combining the surface integrals results in a single boundary integral justperformed over the outer boundary ΓI, on which we, e.g., apply absorbingboundary conditions of first order (see Sect. 5.5.1). Therewith, we utilize therelation

c∂p′

∂n= −∂p

′

∂t(10.37)

and arrive at the weak form of (10.30): Find p′ ∈ H1 such that∫Ω

1c2w∂2p′

∂t2+∫Ω

∂w

∂xi

∂p′

∂xidΩ +

∫ΓA

1cw∂p′

∂tdΓ = −

∫ΩF

∂w

∂xi

∂Lij

∂xjdΩ

for any w ∈ H1.Using standard nodal finite elements, we approximate the continuous

acoustic pressure p′ as well as the test function w by

p′ ≈ p′h =neq∑a=1

Nap′a (10.38)

w ≈ wh =neq∑a=1

Nawa . (10.39)

Thus, (10.38) is transformed to the following semidiscrete Galerkin formula-tion

Mp′n+1

+ Cp′n+1

+ Kp′n+1

= fn+1

(10.40)

with p′ = ∂2p′/∂t2, p′ = ∂p′/∂t, p′ the nodal unknowns of the acoustic pres-sure and n the time step counter. The matrices as well as right-hand sidevector compute as follows:

M =ne∧

e=1

me ; me = [mpq] ; mpq =∫Ωe

1c2NpNq dΩ

C =nΓI∧e=1

ce ; ce = [cpq] ; cpq =∫Γ e

I

1cNpNq dΓ


K =ne∧

e=1

ke ; ke = [kpq] ; kpq =∫

Ωe

∂Np

∂xi

∂Nq

∂xidΩ

fn+1

=ne∧

e=1


∂Np

∂xi

∂Ln+1ij

∂xjdΩ .

In the above equations ne is the number of finite elements, nΓI the numberof finite surface elements along the outer boundary ΓI and

∧the finite ele-

ment assembly operator. The time discretization is performed by applying astandard Newmark algorithm as described in Sec. 2.5.2.

By performing a harmonic analysis, it is possible to compute the soundradiation for specific frequency components present in the acoustic sources.In this way, we obtain the complex acoustic pressure at each node in thecomputational domain. For deriving the harmonic formulation of the imple-mentation, we can simply apply a Fourier-transformation to the semidiscreteGalerkin formulation from (10.40), since the matrices M, C and K are fre-quency independent. The resulting complex algebraic system of equations isgiven by (

−ω2M + iωC + K)p

n+1= f

n+1, (10.41)

where the source term f represents the complex nodal acoustic sources, whichare obtained by applying a Fourier transformation to the dataset of transientnodal sources interpolated from the fluid grid to the acoustic grid.

Great care has to be taken by the interpolation of the computed nodalsources from the fine flow grid to the coarser acoustic grid. Within the FEformulation we perform an integration over the volume (corresponds to thecomputational flow region) and project the results to the nodes of the fine flowgrid, which has to be interpolated to the coarser acoustic grid. Therewith, ourinterpolation has to be conservative in order to preserve the total acousticenergy. As illustrated in Fig. 10.3, we have to find for each nodal source fF

k

in which finite element of the acoustic grid it is located. Then, we computefrom the global position (xk, yk) its position (ξk, ηk) in the reference element.This is in the general case a non-linear mapping and is solved by a Newtonscheme. Now, with this data we can perform a bilinear interpolation and addthe contribution of fF

k to the nodes of the acoustic grid by using the standardfinite element basis functions Ni

fAi = Ni(ξk, ηk)fF

k .

Therewith, by this procedure the interpolation preserves the overall sum ofthe acoustic source.

10.4 Validation: Co-Rotating Vortex Pair 279

Fig. 10.3. Standard conservative interpolation on a quadrilateral mesh

10.4 Validation: Co-Rotating Vortex Pair

In this section we validate the FE implementation of Lighthill’s acoustic anal-ogy, by computing the far field caused by an incompressible, purely unsteadyvortical flow. The weak formulation of Lighthill’s inhomogeneous wave equa-tion is forced with the acoustic sources obtained from the hydrodynamic fieldinduced by a co-rotating vortex pair. The resulting acoustic field represents thebasic acoustic field generated by turbulent shear flows, jet flows, edge tones,etc. [139, 171]. The analytical solution for the acoustic far field, used for thevalidation of the acoustic results, has been obtained employing the methodof matched asymptotic expansion (MAE), first presented in [159]. The com-putation of flow-induced noise from a co-rotating vortex pair has been widelyused by other authors in the past as a benchmark for the validation of theirnumerical methods (see e.g., [54, 58, 62, 139,148]).

A schematic of the corotating vortex pair is presented in Fig. 10.4. Itconsists of two point vortices separated by a fixed distance of 2r0 with cir-culation intensity Γ . The vortices rotate around each other with a periodT = 8π2r20/Γ . Each vortex induces on the other a velocity vθ = Γ/(4πr0).The configuration results in a rotating speed ω = Γ/(4πr20), and rotatingMach numberMar = vθ/c = Γ/(4πr0c) = 2πr0/T c. The rotating non-circularstreamlines are directly associated with the hydrodynamic field of the rotat-ing quadrupole [139]. The incompressible, inviscid flow can be determinednumerically by the evaluation of a complex potential function Φ(z, t) [58,62]

Φ(z, t) =Γ

2πiln(z − b) +

Γ

2πiln(z + b) =

Γ

2πiln z2(1 − b2

z2) , (10.42)

where z = reiθ and b = r0eiωt.

The hydrodynamic velocity field required for the evaluation of the acousticsource term from (10.41), is obtained by differentiating (10.42) with respectto z as

ux − iuy =∂Φ(z, t)∂z

=Γ

iπ

z

z2 − b2. (10.43)


Fig. 10.4. Schematic diagram of corotating vortices

In the acoustic computation a linear propagation is assumed outside the fluidregion, governed by the homogeneous acoustic wave equation.

The analytical solution of the acoustic far-field pressure fluctuations, usedto validate the numerical results, is obtained using a matched asymptoticexpansion (MAE) [159], and computes as

p′ =ρ0Γ

4

64π3r40c2[J2(kr) cos(Ψ) − Y2(kr) sin(Ψ)] , (10.44)

where k = 2ω/c, J2(kr), Y2(kr) are the second-order Bessel functions of thefirst and second kind and Ψ = 2(ωt− θ).

For the validation, the flow field is evaluated in a numerical domain withdimensions 200 m × 200 m. This domain corresponds to the region where theacoustic sources for the inhomogeneous wave equation are computed. Theacoustic propagation is computed in a larger domain with dimensions 400 m× 400 m. For evaluating the complex potential function the spinning radiusis chosen to be r0 = 1 m, the circulation intensity Γ = 1.00531m2/s and thespeed of sound c = 1 m/s. This results in a wave length λ ≈ 39m and arotating Mach number Mar = 0.08.

In the two-step computation, the first step consists of evaluating the ve-locity field and the acoustic sources on the fine fluid grid. Secondly, afterinterpolation of the acoustic sources from the fine fluid grid to the coarseracoustic grid, we solve the inhomogeneous wave equation using the resultingsources to compute the acoustic propagation.

An additional issue for the computation of the noise generated by the co-rotating vortex pair is a singularity of the velocity field at the point vortices. Avortex core model of Scully type [62,139,192] has been applied to desingularizethe tangential velocity field, in order to allow the computation of the velocitygradient on these regions. In the region around the corotating vortices with

10.4 Validation: Co-Rotating Vortex Pair 281

dimensions 6 m × 6 m, the acoustic element size is chosen to be ha = 0.1m,which compared to the fluid discretization, corresponds to ha/hf = 5. Outsidethis area both fluid and acoustic meshes are correspondingly coarsened in theradial directions. A comparison of the numerical acoustic field obtained forthe main frequency component of the problem, f = 1/T = 0.026Hz, with theanalytical solution is presented in Fig. 10.6. Good agreement in both the spiralpattern as well as in amplitudes is found except at the center of the computa-tional domain, where the analytical solution has been truncated according tothe vortex core model. Figure 10.5 compares the decay of the acoustic pressurealong the positive x-axis between the numerical and analytical values.

Fig. 10.5. Decay of the acoustic pressure values along the x-axis


(a) Numerical computation

(b) Analytical solution

Fig. 10.6. Comparison of sound pressure field at frequency f = 0.026 Hz obtainednumerically using Lighthill’s acoustic analogy with analytical solution obtained usingMAE method. Distance scale in meters

11

Algebraic Solvers

In recent years, many different formulations using Lagrange (nodal) as wellas Nedelec (edge) finite elements for the numerical computation of Maxwell’sequations have been published, e.g., [23, 126]. The resulting algebraic systemof equations is mostly solved by applying the conjugate gradient method withincomplete Cholesky factorization as preconditioner (ICCG). However, thenumber of necessary iterations of ICCG increases strongly with the numberof unknowns. Recently, investigations have been done to adapt multigrid (MG)methods for the fast solution of 3D electromagnetic field problems (e.g., [9,92, 191]).

In this section, we will give a detailed discussion on geometric and al-gebraic multigrid methods specially adapted for Maxwell’s equations in thequasistatic case. Since a most robust solution strategy is a preconditionedconjugate gradient (PCG) solver with an appropriate multigrid method aspreconditioner, we will start with a brief description of the PCG method. Fora basic introduction into multigrid methods we refer to [36, 185, 186].

11.1 Preconditioned Conjugate Gradient (PCG) Method

Let us consider the algebraic system of equations of the form

Khuh = fh. (11.1)

Therein Kh ∈ IRnh×nh denotes the system matrix, fh

∈ IRnh the right-hand side and uh ∈ IRnh the solution vector of the unknown nodal (edge)quantity (usually the magnetic vector potential). The entries of Kh are givenby kij = (Kh)ij ∈ IRp×p with p defining the number of unknowns per node(edge). The number of unknowns nh is related to the usual discretizationparameter h by nh = O(h−d), with d = 2, 3 the spatial dimension. The systemmatrix Kh is supposed to be sparse and symmetric positive definite (SPD)as is, in fact, the case for the used discretization of Maxwell’s equations. In

284 11 Algebraic Solvers

general, nh is quite large and due to limited memory resources, iterative solvershave to be used instead of direct ones. However, the convergence of iterativesolvers strongly depends on the condition number κ of the system matrix Kh

κ(Kh) =λmax(Kh)λmin(Kh)

, (11.2)

with λmax and λmin the largest and the smallest eigenvalue of Kh, respec-tively. In general, the convergence rate decreases when κ gets large. Since Kh

stems from an FE discretization of a second-order partial differential equation(PDE), the condition number κ(Kh) typically behaves like O(h−2). In orderto cope with large condition numbers, we apply a symmetric preconditionerCh to (11.1), i.e.,

C−1h Kh uh = C−1

h fh, (11.3)

with the properties

• C−1h is an approximate inverse of Kh, and

• C−1h can be applied very fast

Consequently, the condition number of the preconditioned system is muchsmaller than the original one. Furthermore, the preconditioned system (11.3)is solved via a Krylov subspace method, i.e., conjugate gradient (CG) or quasi-minimal residual (QMR) method, see [187]. The standard method for solving(11.1) is to apply the preconditioned conjugate gadient (PCG) method as givenin Algorithm 1.

Algorithm 1 Preconditioned conjugate gradient (PCG) methodk = 0r0 = Khu0

h − fh

Solve Chd0 = −r0

s0 = −d0

r1 = r0

while ‖rk+1‖ > ε‖r0‖ do

αk = (rk)T sk

(dk)T Khdk

uk+1h = uk

h + αkdk

rk+1 = rk + αkKhdk

Solve Chsk+1 = rk+1

βk = (rk+1)T sk+1

(rk)T sk

dk+1 = −sk+1 + βkdk

k = k + 1end while

11.2 Multigrid (MG) Method 285

The following result gives a bound on the number of iterations that aresufficient for a prescribed desired error reduction ε:

Theorem 11.1. Let Kh ∈ IRnh×nh and Ch ∈ IRnh×nh be SPD with the rela-tion

γ1〈Chv, v〉 ≤ 〈Khv, v〉 ≤ γ2〈Chv, v〉 ∀v ∈ IRnh ,

and γ1, γ2 > 0. The starting solution error ||u− u0h||Kh

is reduced by a factorε applying

I(ε) =∣∣∣ln(ε−1 +

√(ε−2 + 1)

)/ ln ρ−1

∣∣∣PCG iterations with ρ =

(√γ2γ1

− 1)/(√

γ2γ1

+ 1).

In Theorem 11.1, || · ||Khdenotes the energy norm induced by the energy

inner product, computed as ||w||2Kh= 〈Khw,w〉 for w ∈ IRnh . According to

Theorem 11.1 the factor γ2/γ1, which is equivalent to κ(C−1h Kh) should be

as close as possible to 1 in order to obtain fast convergence. Of course thetheoretically best choice would be Ch = Kh, yielding γ2/γ1 = 1, but thiswould have the consequence to solve Khs

k+1 = rk+1 in the PCG method.Therefore, we have to find a preconditioner Ch with γ2/γ1 ≈ 1 such thatChs

k+1 = rk+1 can be solved in a very fast way. The conventional choiceis to use incomplete Cholesky (IC) factorization, i.e., Ch = RT R with Rcontaining the entries of the upper triangular matrix of the factorization ofKh but with the same structure as Kh (possible low fill in is allowed). In [29]it is shown that under special assumptions, the condition number κ(C−1

h Kh)when using IC as preconditioner behaves like O(h−1). However, κ still dependson the discretization parameter h. This fact leads us to MG methods for whichit can be shown that the number of necessary iterations does not depend onthe mesh parameter h [82]. Furthermore, it was shown in [111] that a mostrobust solution strategy (concerning the quality of the FE mesh) for (11.1) isPCG with a geometric multigrid preconditioner.

11.2 Multigrid (MG) Method

Multigrid methods improve the convergence by using information not only onthe computational grid on which the system of equations is supposed to besolved, but also on a (usually hierarchical) sequence of coarser grids. In orderto outline the construction of a MG preconditioner we explain this by meansof a two-grid method. The indices H and h are related to the coarse and finegrid of an FE discretization, respectively. The linear mappings (with nh > nH

and nh, nH the number of unknowns on the fine and coarse grid, respectively)

IHh : IRnh → IRnH and Ih

H : IRnH → IRnh (11.4)

are called restriction and prolongation operators. Therefore, the two-grid al-gorithm is performed as follows:


1. Smooth ν1 times on the fine grid Kh, uh, fh

2. Calculate the defectdh = f

h− Khuh

3. Restrict the defect dh onto the coarse grid

dH = IHh dh

4. Solve the coarse grid problem KHvH = dH

5. Prolongate the coarse grid correction vH to the fine grid

vh = IhHvH

6. Correct uh by vh, i.e., uh = uh + vh

7. Smooth ν2 times on the fine grid Kh, uh, fh

By replacing the exact solution of the coarse grid problem in step 4 itself bya two-grid approximation, we arrive at the recursive definition of a multigridcycle (see Fig. 11.1). The motivation for this approach comes from examining

Grid l

Presmoothing

Postsmoothing

Direct solver

Prolongation

Restriction

Grid -1l

Grid 2

Grid 1 (coarse)

Fig. 11.1. MG solution algorithm (V-Cycle)

the error of the numerical solution in the frequency domain. High-frequencyerrors, which include local variations in the solution, are well eliminated bysimple iterative smoothing methods (e.g., Gauss–Seidel smoother). Once thisis achieved, further fine-grid iterations would only result in a convergencedegradation. Therefore, the solution is transferred to a coarser grid by usingan appropriate projection operator IH

h . On this grid, the low-frequency errorsof the fine grid manifest themselves as relatively high-frequency errors, andare thus eliminated efficiently again using simple iterative smoothing methods.If the coarsest grid has been reached, the equation has to be solved exactly(e.g., direct solver). Consequently, each grid level is responsible for eliminatinga particular frequency bandwidth of the error.

11.3 Geometric MG Method 287

The MG iteration operator Mh mapping the k-th MG iteration errorek = uh − uk

h (uh being the exact solution of (11.1)) onto the (k + 1)-th MGiteration error ek+1 = uh − uk+1

h

ek = Mhek+1 (11.5)

is in the two-grid case given by

Mh =(Spost

h

)ν2 (Ih − IhHK−1

H IHh Kh

)(Spre

h )ν1 , (11.6)

provided that the coarse grid system (step 4) is solved exactly (see e.g., [82]).In (11.6) Ih ∈ IRnh×nh denotes the identity matrix, and Spre

h , Sposth the

smoothing operators, e.g., Gauss–Seidel forwards and backwards, respectively

Spreh = Ih − τh(Lh + Dh)−1Kh

Sposth = Ih − τh(Lh + Dh)−TKh ,

with Lh the lower triangular part of Kh, Dh = diag(Kh) and τh a relaxationfactor. In the general situation, the MG iteration operator Mh on the finestgrid can be defined in the following iterative process (by denoting the coarsestlevel by 1 and the finest by )

Mh = Ml

Mq =(Spost

q

)ν2 (Iq − Iqq−1 (Iq−1 −Mq−1)

K−1q−1Iq−1

q Kq

) (Spre

q

)ν1,

with q = 2, 3, ..., , M1 := 0.As already mentioned in the previous section, a most robust solution strat-

egy for (11.1) is PCG with a geometric MG preconditioner. This means thatsolving Chs

k+1 = rk+1 (see Algorithm 1) corresponds to applying pMG cyclesto Khs

k+1 = rk+1. Using the introduced iteration operator Mh and settingthe starting value sk+1

0 to zero, we obtain the iteration error according to(11.5)

K−1h rk+1 − sk+1

p = (Mh)p(K−1h rk+1 − sk+1

0 )

sk+1p = (Ih − (Mh)p)K−1

h rk+1 . (11.7)

By setting the so-obtained solution sk+1p equal to sk+1 = C−1

h rk+1, the pre-conditioner Ch takes the form

Ch = Kh (Ih − (Mh)p)−1. (11.8)

11.3 Geometric MG Method

In contrast to standard FE techniques, geometric MG methods are not basedon a fixed FE mesh that describes the unknown field variable accurately


enough. Geometric MG techniques start at a very coarse spatial discretizationT1 of the computational domain. By dissecting the elements of T1, a finer dis-cretization T2 is obtained as shown in Fig. 11.2. This refinement process can

Fig. 11.2. Adaptive refinement of an initial mesh

include either all elements (uniform refinement) or only an appropriately se-lected part of the elements (adaptive refinement). By repeating the refinement,we obtain a hierarchy of FE discretizations T1, ..., T for which the systems

Kquq = fqq = 1, 2, ..., (11.9)

are assembled and a MG cycle as described in Sect. 11.2 can be performed. Thegeneration of an appropriate coarse mesh and the adaptive refinement for thesubsequent finer FE discretizations levels is a challenging task, which needsthe full data exchange between the geometric modeler, the mesh generatorand the error estimator [194].

11.3.1 Geometric MG for Edge Elements

The two essential parts for a successful application of geometric MG toMaxwell’s equations are the choice of the prolongation and of the smoothingoperator. In order to determine the prolongation operator Iq+1

q , we considerthe refinement of a face Γ q on an edge tetrahedron element at level q. Dissect-ing the tetrahedron at level q into 8 tetrahedra at level (q+1), each face Γ q isdivided into 4 new faces Γ q+1

1 , .., Γ q+14 (Fig. 11.3). The prolongation operator

Iq+1q must guarantee that the magnetic flux across Γ q is equal to the one

across Γ q+11 + ..+ Γ q+1

4 [191]∫Γ q

(∇ × u) · dΓ =4∑

k=1

∫Γ q+1

k

(∇ × u) · dΓ . (11.10)


Fig. 11.3. Prolongation of the edge degrees of freedom from level q to level q + 1

By applying Stoke’s theorem (Appendix B.9), we obtain for each face Γ q+1k

the relation ∮Γ q+1

k

u · ds =14

∮Γ q

k

u · ds . (11.11)

By exploiting the degrees of freedom of the FE formulation (see Fig. 11.3),we obtain

uq+11 − uq+1

7 + uq+16 =

14

(uq1 + uq

2 + uq3) (11.12)

uq+12 + uq+1

3 − uq+18 =

14

(uq1 + uq

2 + uq3) (11.13)

uq+15 − uq+1

9 + uq+14 =

14

(uq1 + uq

2 + uq3) (11.14)

uq+17 + uq+1

8 + uq+19 =

14

(uq1 + uq

2 + uq3) . (11.15)

In addition, since the magnetic vector potential u is constant along the edge,we may write

uq+11 = uq+1

2 =12uq

1 (11.16)

uq+13 = uq+1

4 =12uq

2 (11.17)

uq+15 = uq+1

6 =12uq

3 . (11.18)

Combining the above results allows us to define the transfer operator as follows

Iq+1q =

⎛⎝0.5 0.5 0.0 0.0 0.0 0.0 0.25 0.25 −0.25

0.0 0.0 0.5 0.5 0.0 0.0 −0.25 0.25 0.250.0 0.0 0.0 0.0 0.5 0.5 0.25 −0.25 0.25

⎞⎠T

,

which fulfills the requirements of flux conservation. The restriction operatorIq

q+1 is chosen to be that transposed to Iq+1q , i.e.,


Iqq+1 = (Iq+1

q )T . (11.19)

For the construction of the smoothing operator, we have to consider thefact that the Sobolev space H0(curl) has a Helmholtz decomposition of theform

H0(curl ) = N (curl ) + N (curl )⊥, (11.20)

with N (curl ), N (curl )⊥ the kernel of the curl operator and its orthogonalcomplement. The smoothing operator has to damp out errors in both spacesefficiently, see [92]. In [9] it has been shown that overlapping block-smoothers,which collect all edges sharing a common vertex in a block, have this propertyand show a convergence rate independent of the error reduction value ε.

Therefore, a simple Gauss–Seidel method does not behave well for Maxwellproblems, but due to [9] and [92] it is known that properly designed block-Gauss–Seidel iterations do the job. Each block is assigned to a vertex of themesh and connects all edges sharing this node. Since each of the edges of themesh is associated to more than one node, no standard block-Gauss–Seidelsmoother but an overlapping technique must be applied. Therefore, all degreesof freedoms belonging to edges with a common node are smoothed together.This can be achieved by first introducing a connectivity matrix Rj , whoseentries are zeros and ones that allows the corresponding subblock Kj

q to bepicked out of our system matrix Kq (q denotes the MG level). The dimensionof Rj is (nj × ne) with nj the number of edges belonging to node j and ne

the total number of edges (unknowns) in the mesh. Using this matrix, we canpick out the quadratic sub-blocks Kj

q of the matrix Kq as follows

Kjq = RjKqRT

j . (11.21)

Each of these small matrices has to be inverted in the preparation phase of themultigrid method. One step of the block-Gauss–Seidel iteration with initialapproximation uj

q,i is defined as

ujq,i+1 = uj

q,i + RTj (Kj

q)−1Rj(f q

− Kqujq,i) j = 1, ..., n , (11.22)

with i the iteration counter. It has to be mentioned that not the whole residualf

q− Kqu

jq,i has to be computed at each step, but only the few components

picked out with Rj . Therefore, one block-Gauss–Seidel step is not much moreexpensive than a simple Gauss–Seidel step.

11.3.2 Case Study

In order to demonstrate the advantages of the presented scheme, TEAM (Test-ing Electromagnetic Analysis Methods) Workshop problem 20 is considered.Therefore, the convergence behavior of the multigrid solvers is compared tostandard approaches. The TEAM Workshop problem 20 is a 3D, non-linear,and static magnetic field problem [161]. The structure of this problem, con-sisting of a center pole, a yoke, and a coil, is displayed in Fig. 11.4.


Nested Multigrid

First, the structure is discretized with a coarse mesh T1 of linear edge tetra-hedron elements, which is shown in Fig. 11.4. Thereby, the symmetries of the

Fig. 11.4. Coarse FE discretization of TEAM problem 20 (without air)

problem in the xz-plane and the yz-plane are exploited. By dissecting eachedge tetrahedron element of level 1 into 8 elements of level 2, a new refinedmesh T2 is obtained. This procedure is repeated until a mesh T4 is producedthat is accurate enough to describe the magnetic field. In Table 11.1 the gen-erated hierarchy of FE meshes is shown. In order to achieve a good initial

Table 11.1. FE hierarchy of TEAM problem 20

Grid level Edge elements Edges (dofs)

1 3 050 3 8002 24 500 30 5003 196 000 236 0004 1 570 000 1 900 000

guess for the non-linear iteration procedure on the finer levels, the problem isfirst solved on the coarser grids and the solution is prolongated to the finermeshes and used as a start approximation for the non-linear iteration process.By this nested-multigrid approach, the number of costly iterations at the finergrids is considerably reduced [90].

Convergence of the MG-PCG solverThe most time-consuming part of the computation process is the solution ofthe matrix equation system at the finest mesh T4. Thereby, the MG solver


is compared to a standard solution technique, based on a CG method withadapted block preconditioning (PCCG). In Fig. 11.5 the convergence behav-ior of both methods for a matrix equation system at the finest level with1 900 000 edges is displayed. The MG solver achieves the requested normal-

Fig. 11.5. Number of iterations versus normalized residual

ized residual of 10−6 after 13 iterations and 890 s, whereas the PCCG withblock preconditioning needs 180 iterations and 9860 s. Thereby, an SGI ORI-GIN with a RS12000 processor (300 MHz) is used. If applying a conventionalCG solver with incomplete Cholesky preconditioning (ICCG) to the matrixequation, more than 1000 iterations would be necessary and therefore muchhigher simulation times arise [70].

Optimal complexityTo show the optimal complexity of the MG solver, the number of necessaryiterations and the solution time to achieve a normalized residual of 10−6 ateach discretization level are displayed in Table 11.2. Since the number of MGiterations is almost independent of the size of the FE meshes and the timefor a single MG iteration increases linearly with the number of unknowns, alinear dependency between the degrees of freedom and the solution time canbe detected.

Accumulated solution times of nested MG and a conventional ap-proachIn Table 11.3 the convergence behavior of the non-linear iteration process iscompared for different excitations Θ (given in Ampere-turns). For small Θ the

11.4 Algebraic MG Method 293

Table 11.2. Necessary MG-PCCG iterations and simulation times to reduce thenormalized residuum of algebraic system of equations to 10−6

Level Iterations Time (s)

1 10 1.52 12 113 12 1054 13 890

number of necessary iterations at the coarser levels is low, but if the excitationincreases, a higher number of iterations is necessary. On the other hand, dueto the nested MG approach, the number of iterations on the finer levels is al-most independent of the strength of the non-linearity. Since the iterations onthe finer grids are the most time-consuming parts of the computation process,

Table 11.3. Necessary non-linear iterations for different excitations Θ (Computer:SGI, RS12000 processor 300 MHz)

Θ Iterat. at level Accum. time Accum. time(Ampere-turns) 1 2 3 4 nested MG (s) conventional (s)

1 000 4 3 3 2 8 300 45 4003 000 11 6 5 3 9 570 114 5004 500 15 6 5 3 9 600 154 0005 000 17 6 5 3 9 600 173 600

also the accumulated simulation time is almost independent of the strength ofthe magnetic non-linearity. In the seventh column of Table 11.3 the accumu-lated simulation time of a conventional approach, which means PCCG solversfor the resultant matrix equation systems, without exploiting the coarse gridinformation at the finer grids, is displayed. Thereby, a considerable advantageof the nested MG technique can be clearly seen.

11.4 Algebraic MG Method

In contrast to geometric MG, algebraic multigrid (AMG) needs no FE dis-cretization with hierarchical grids. The matrices Kq with q = 1, .., on thedifferent levels are set up only by knowledge of the matrix Kh = Kl obtainedfrom the FE discretization. In recent years, a lot of different approaches werepublished, e.g., [27, 131, 185], which mostly concern the scalar case of matrixequation systems arising from a nodal finite element discretization.

Geometric MG methods suffer from the inherent need of a hierarchical FEmesh (see [82]), and thus algebraic multigrid (AMG) methods are of specialinterest, if at least one of the following cases arises:


• The discretization provides no hierarchy of FE meshes, which would beessential for the geometric MG method. This is the case for many FEcodes, especially commercial ones.

• The coarsest grid of a geometric multigrid method is too large to be solvedefficiently by a direct or classical iterative solver.

• Classical iterative solvers are not efficient enough.

AMG methods try to mimic their geometric counterpart, but only rely on theinformation available on a given single grid (for the pioneering work on AMGsee [185]). While within a geometric MG solver the construction of a matrixhierarchy is rather simple if a hierarchy of grids is available (see e.g., [82]),this task is not as easy if either the matrix only or the information on thefinest grid is available. The classical AMG approach assumes an SPD systemmatrix that is additionally an M matrix [185]. For such matrix classes a matrixhierarchy can be constructed, imitating the geometric counterpart well. It canbe easily shown that the information of an SPD system matrix is not enoughin order to construct an efficient and robust AMG method. Therefore, weassume the knowledge of the underlying PDE, the FE discretization schemeand additional information on the given FE mesh. Therefore, such enhancedAMG methods are able to reproduce the behavior of geometric MG methodseven for Maxwell’s equation, although the system matrices are not M matriceshere. For a detailed discussion we refer to [179].

First, we have to perform the coarsening process to extract from the givensystem matrix (arising from the FE discretization) matrices with decreasingdimension. The key point of the coarsening process is to construct an auxiliarymatrix on which the coarsening is performed. Therefore, we can always guar-antee a coarsening that is appropriate and, in addition, very fast. Furthermore,we have to define the smoothing operator and the restriction (prolongation)operator for the transfer of data between the different hierarchy levels basedon the auxiliary matrix.

11.4.1 Auxiliary Matrix

Let us assume that the system matrix Kh stems from an FE discretization onthe FE mesh ωh = (ωn

h , ωeh), with ωn

h , |ωnh | = Mh being the set of nodes and

ωeh being the set of edges (see Fig. 11.6). An edge is defined as a pair of indices

for which the connection of the two corresponding points is a geometric edge.For instance, let i, j ∈ ωn

h be the indices of the nodes xi, xj ∈ IRd then theedge is given by

eij = (i, j) ∈ ωeh ,

and the corresponding geometric edge vector can be expressed by

aij = xi − xj ∈ IRd . (11.23)


Fig. 11.6. Clipping of an FE mesh in 2D

The first task we are concerned with is the construction of an auxiliary matrixBh ∈ IRMh×Mh with the following properties

(Bh)ij =bij ≤ 0 if i = j,1 −

∑j =i bij ≥ 0 if i = j .

(11.24)

The entries of Bh should be defined in such a way that the distance andparameter jumps of the variational forms are reflected. The matrix patternof Bh can be constructed via different objectives: Bh reflects the geometricFE mesh, which is of importance for an edge element discretization, or Bh

reflects the matrix pattern of the system matrix Kh, which is useful for nodalFE discretization.

11.4.2 Coarsening Process

The auxiliary matrix Bh is a sparse M matrix and therefore the coarseningprocess for Bh is straightforward and can be done in a robust way. We knowthat Bh represents a virtual FE mesh ωh = (ωn

h , ωeh). Such a virtual FE mesh

can be split into two disjoint sets of nodes, i.e.,

ωnh = ωn

C ∪ ωnF , ω

nC ∩ ωn

F = ∅ ,

with sets of coarse grid nodes ωnC and fine grid nodes ωn

F . The splitting isusually performed such that no coarse grid nodes are connected directly andthat the number of coarse grid nodes is as large as possible (see Fig. 11.7).

In order to perform a coarsening algorithm, let us introduce the followingsets

N ih = j ∈ ωn

h : |bij | = 0 , i = j ,Si

h = j ∈ N ih : |bij | > coarse (Bh, i, j) , i = j ,

Si,Th = j ∈ N i

h : i ∈ Sjh ,

where N ih is the set of neighbors for node i, Si

h denotes the set of strong con-nections and Si,T

h is related to the set of nodes that have a strong connectionto node i, respectively. The cutoff (coarsening) function is chosen as, e.g.,


Fig. 11.7. Illustration of coarsening

coarse (Bh, i, j) =

⎧⎨⎩θ ·√|bii||bjj | , see [212] ,

θ · maxl=i |bil| , see [185] ,θ , see [130] ,

(11.25)

with an appropriate θ ∈ [0, 1]. In addition, we define the local sets

ωiC = ωn

C ∩N ih , ωi

F = ωnF ∩N i

h (11.26)

andEi

h = (i, j) ∈ ωeh : j ∈ N i

h . (11.27)

The coarsening algorithm is described in Algorithm 2.

Algorithm 2 Coarsening phaseωn

C ← ∅, ωnF ← ∅

while ωnC ∪ ωn

F = ωnh do

i ← Pick(ωnh \ (ωn

C ∪ ωnF ))

if |Si,Th | + |Si,T

h ∩ ωnF | = 0 then

ωnF ← ωn

h \ ωnC

elseωn

C ← ωnC ∪ i

ωnF ← ωn

F ∪ (Si,Th \ ωn

C)end if

end while

Therein the function

i← Pick(ωnh \ (ωn

C ∪ ωnF ))

returns a node i for which the number |Si,Th | + |Si,T

h ∩ ωnF | is maximal.


Fig. 11.8. Example with anisotropic mesh and parameter jump (material 1 inelements with number 1 and 2; material 2 in elements with numbers 3 and 4)

Example: Let us consider the FE mesh of Fig. 11.8. The auxiliary matrix isdefined on an finite element r by the setting

brij =νr

‖aij‖2i = j ,

with νr the material parameter and aij the geometric edge vector (see (11.23)).Let us assume the following entries for row 5 of the assembled auxiliary matrix

b51 = −202 b54 = −400 b57 = −100b52 = −2 b55 = 10n09 b58 = −101b53 = −200 b56 = −2 b59 = −1 .

Using the coarsening function of [185] (see (11.25)) with θ = 0.25, we obtain

N5h = 1, . . . , 4, 6, . . . , 9S5

h = 1, 3, 4, 7, 8S5,T

h = 4, 6, 7 .

For the construction of set S5,Th we assumed

S1h = 3 S6

h = 1, 2, 5, 8S2

h = 1 S7h = 4, 5, 8

S3h = 1 S8

h = 7, 9S4

h = 1, 3, 5, 7, 8 S9h = 7, 8 .

A special coarsening algorithm is the agglomeration technique, where θ isset to 0. Consequently, N i

h = Sih = Si,T

h for all i = 1, . . . ,Mh. Furthermore,we call (Ii

h)MH

i=1 (|ωnC | = MH < Mh) a disjoint splitting for the agglomeration

method if

Iih ∩ Ij

h = ∅,MH⋃i=1

Iih = ωn

h ,

is valid, see Fig. 11.9.


Fig. 11.9. Virtual FE mesh with a feasible agglomeration

If an appropriate prolongation Qh for Bh is defined then a coarse auxiliarymatrix is computed by

BH = (Qh)TBhQh ,

and BH represents again a virtual FE mesh ωH = (ωnH , ω

eH), with ωn

H = ωnC .

It can be shown that BH is again an M-matrix if the prolongation operatorQh fulfills certain criteria [185]. Thus the coarsening process can be appliedrecursively. Finally, it is assumed that the degrees of freedom on the coarsegrid are numbered first. For instance, the nodes are reordered like

ωnh = (ωn

C , ωnF )

(similarly for edges) and as a consequence the system matrix can be writtenas

Kh =(KCC KCF

KTCF KFF

).

11.4.3 Prolongation Operators

For a given splitting ωnh = ωn

C ∪ωnF the optimal prolongation operator is given

by the Schur complement, i.e.,

KH = KCC − KCFK−1FF KFC = PT

h KhPh

withPh = (IH ,−KCF K−1

FF )T .

The prolongation operator Ph can hardly be realized in practice since theexpression −KCFK−1

FF involves the inverse of KFF , which in turn impliesa global transport of information. In addition, the coarse grid matrix KH

becomes dense. The goal of an AMG method is to approximate Ph by someprolongation operator Ph, which acts only locally and therefore produces asparse coarse-grid matrix.


11.4.4 Smoother and Coarse-grid Operator

An essential point in MG methods is the smoothing operator Sh ∈ IRNh×Nh

that reduces the high-frequency error components. Typically, a particularsmoother works for certain classes of matrices. It is shown in [32] that a pointGauss–Seidel or point Jacobi smoother is appropriate for FE discretizationswith Lagrange FE functions for scalar elliptic PDEs of second order. Analo-gously, the block-Gauss–Seidel and block Jacobi smoother work well for theblock counterpart, e.g., discretization of Maxwell’s equation with nodal finiteelements. For the edge FE discretization we use a patch smoother.

The coarse grid matrix KH is usually constructed by Galerkin’s method,i.e.,

KH = PTh KhPh . (11.28)

After a successful setup, an AMG-cycle can be performed as usual (seee.g., [82]). For instance in Algorithm 3 a V (νF , νB)-cycle with variable pre-and post-smoothing steps is described. The variable CoarseLevel storesthe number of levels generated by the coarsening process until the size of thesystem is smaller than CoarseGrid.

Algorithm 3 V(νF , νB)-cycle AMGStep(K,u, f , )K ← K, f

← f , u ← u

if = CoarseLevel thenu ← CoarseGridSolver (LL

T , f

)

Returnelse

d ← 0, w+1 ← 0u ← SνF

(u, f )

d ← f− Ku

d+1 ← (P)T d

AMGStep(K+1, w+1, d+1, + 1)w ← Pw+1

u ← u + w

u ← SνB (u, f

)end if

In the following subsections we specialize the abstract algorithms, definethe components for nodal and edge FE discretization and additionally proposea method for complex symmetric systems. Prior to that we mention that static,transient, and non-linear analysis of a given problem results in the solutionof linear systems (11.1). Therefore, we restrict the discussion to the linearanalysis. Other applications can be found in [116, 117, 119, 120, 182].


11.4.5 AMG for Nodal Elements

First we consider (4.120) and use nodal elements for discretization. Note thatthe following approach includes the scalar case (p = 1; e.g., scalar potentialequation).

Construction of virtual FE meshes: The definition of the auxiliarymatrix Bh plays an important role for this problem class. The classical ap-proach uses

(Bh)ij = −‖kij‖∞ for i = j ,

with ‖ ‖∞ the maximum norm. The diagonal entries of Bh are computedaccording to (11.24). Now the degrees of freedom per node of the systemmatrix have to be related to an entry in the auxiliary matrix, which in turnimplies that the matrix pattern of Kh and of Bh has to be equal, i.e.,

‖kij‖∞ = 0 ⇔ |bij | = 0 .

Construction of coarse FE spaces: The simplest prolongation operator isgiven by

(Ph)ij =

⎧⎪⎨⎪⎩

Ip if i = j ∈ ωnC ,

1

|Si,Th

∩ωnC | · Ip if i ∈ ωn

F , j ∈ Si,Th ∩ ωn

C ,

0 else ,(11.29)

with Ip ∈ IRp×p the p-dimensional identity matrix. The AMG method shows abetter convergence behavior as compared to (11.29) with the following discreteharmonic extension

(Ph)ij =

⎧⎨⎩

Ip if i = j ∈ ωnC ,

−k−1ii

(kij + cij

)if i ∈ ωn

F , j ∈ ωiC ,

0 else ,(11.30)

withcij =

∑p∈ωi

F

( ∑q∈ωi

C

kpq

)−1kipkpj .

However, the increasing memory requirement and the slower application com-pared to the prolongation (11.29) is the major drawback of the discrete har-monic extension. Note that the entries of the prolongation operators are ma-trix valued, e.g., (Ph)ij ∈ IRp×p, like the entries of the system matrix Kh.

Smoothing operator: We use a block-Gauss–Seidel method as smooth-ing operator, e.g., [32], or a patch-block Gauss–Seidel method, e.g., [131]. Thelatter should be used for anisotropic problems.


11.4.6 AMG for Edge Elements

The second class originates from an FE discretization with edge FE functionsof the variational form (4.112).

Construction of virtual FE-meshes: According to [92], the refinementof the FE mesh can be performed on the nodes as is usually done for LagrangeFE functions. We use this fact and base our coarsening on an auxiliary matrixBh, which is constructed for instance by the finite element wise setting

brij = − νr

‖aij‖2i = j and (i, j) ∈ ωe

h ,

with νr the reluctivity of the material. Again the diagonal elements are com-puted via (11.24).

Example: Let us consider the FE mesh of Fig. 11.8 and choose element r = 1.We get the following element matrix

B1h = 100 ·

⎛⎜⎜⎝

2.5 −0.5 −1 0−0.5 2.5 0 −1−1 0 2.5 −0.50 −1 −0.5 2.5

⎞⎟⎟⎠ .

The entries (B1h)14, (B1

h)23, (B1h)41, and (B1

h)32 are zero, i.e., there is no di-agonal edge in the virtual FE mesh.

Let us recall that an FE mesh is represented by

ωh = (ωnh , ω

eh) ,

i.e., the set of nodes ωnh and the set of edges ωe

h. The coarse grid is definedby identifying each coarse grid node j ∈ ωn

C with an index k ∈ ωnH . This is

expressed by the index map ind (.) as

ωnH = ind (ωn

C) .

A useful set of coarse grid edges ωeH can be constructed if we invest in a special

prolongation operator Qh for the auxiliary matrix Bh. The prolongation op-erator Qh is constructed such that each fine grid node prolongs exactly fromone coarse grid node, so that one arrives at a partition of ωn

h into clusters,each of them being represented by a coarse grid variable. We extend the indexmap ind : ωn

C → ωnH defined above onto the whole fine set ωn

h by assigningto all fine grid nodes of a cluster the coarse grid index of the representative

ind : ωnh → ωn

H .

A consequence is that ind (i) = ind (j) iff i, j ∈ ωnh prolongate from the same

coarse grid variable. We define an agglomerate (cluster) Iih of a grid point

i ∈ ωnh by (see Fig. 11.10)


Iih = j ∈ ωn

h | ind (j) = ind (i) ⊂ N ih ,

and hence the set of coarse grid nodes can be written as

ωnH = ind (i) | i ∈ ωn

h .

The prolongation operator Qh has only 0 and 1 entries by construction, i.e.,

Fig. 11.10. Virtual FE mesh with a feasible agglomeration and coarse-grid edges

(Qh)ij =

1 i ∈ ωn

h , j = ind (i)0 otherwise .

(11.31)

Now, a coarse-grid edge only exists if there is at least one fine edge con-necting the agglomerates Ii

h and Ikh with i = k (see Fig. 11.10), i.e.,

∃r ∈ Iih, ∃s ∈ Ik

h such that (r, s) ∈ ωeh .

Note that a decrease of the number of edges in the coarsening process is notproved in general, but a decrease is heuristically given, if the average numberof nonzero entries of Bh does not grow too fast.

Construction of coarse FE spaces: The construction of the pro-longation operator Ph : IRNH → IRNh , is delicate because of the ker-nel of the curl -operator consisting of all gradient fields. Ph is defined fori = (i1, i2) ∈ ωe

h, j = (j1, j2) ∈ ωeH as

(Ph)ij =

⎧⎨⎩

1 if j = (ind (i1), ind (i2)),−1 if j = (ind (i2), ind (i1)),0 otherwise ,

(11.32)


by assuming a positive orientation of an edge j = (j1, j2) from j1 to j2 if j1 <j2 holds. The constructed prolongation operator Ph has full rank, because thecoarse grid edges prolongate to NH distinct fine-grid edges by construction.For a detailed discussion see [180].

Smoothing operator: To complete the components for an AMG methodfor edge element FE discretizations, we need an appropriate smoother. Weconsider two different types of smoothers for Kh. The first one was suggestedin [9]. This is a block-Gauss–Seidel smoother where all edges that belong toEi

h (see (11.27)) are smoothed simultaneously for all i ∈ ωnh (see Fig. 11.11).

Fig. 11.11. Detail view of a virtual FE mesh

Another kind of smoother was suggested in [92]. A mathematically equiv-alent formulation is outlined in Algorithm 4. Therein the vector ge,i

h∈ IRNh

Algorithm 4 Hybrid smootheruh ← GaussSeidel(Kh, f

h, uh)

for all i ∈ ωnh do

uh ← uh +((fh

−Khuh),ge,ih )

(Khge,ih

,ge,ih )

· ge,i

h

end for

is defined by

ge,ih

= grad hgn,ih

=

⎧⎨⎩

1 if j < i (i, j) ∈ Eih ,

−1 if j > i (i, j) ∈ Eih ,

0 otherwise ,

with a vector gn,ih

∈ IRMh , (gn,ih

)j = δij .


11.4.7 AMG for Time-harmonic Case

In the harmonic case the time derivative of the magnetic vector potential issubstituted by

∂A∂t

→ jωA ,

with j the complex number, ω the angular frequency and A the complexmagnetic vector potential. Therefore, we have to apply the AMG method toa complex valued and symmetric algebraic system of equations with systemmatrix

Kh = Kreh + jKim

h . (11.33)

In (11.33) Kreh denotes the real part and Kim

h the imaginary part of the systemmatrix.

The application to scalar potential equations has been presented in [183],and adaption to the magnetic vector potential formulation is straightforwardas shown below.

Construction of virtual FE-meshes: The auxiliary matrix is defined to bereal valued. This means that we set up Bh for an edge element discretizationas defined in Sect. 11.4.6. For a nodal element discretization we can use theprocedure described in Sect. 11.4.5 for Kre

h .

Construction of coarse FE spaces: For the construction of a coarse-gridoperator KH we define the system prolongation to be real valued and com-puted as defined in Sect. 11.4.5 as well as Sect. 11.4.6. Therefore, we get

KH = PTh KhPh = PT

h Kreh Ph + jPT

h Kimh Ph = Kre

H + jKimH .

The prolongation Qh is also taken from the real-valued realization correspond-ingly.

Smoothing operator: In the case of an algebraic system of equations aris-ing from a nodal FE discretization we apply a block Jacobi or Gauss–Seidelsmoother in the complex variant. The complex version of the smoother pro-posed in [9] is used for an edge FE discretization.

11.4.8 Case Studies

In order to gain robustness and efficiency, the proposed AMG methods wereused as a preconditioner in the conjugate gradient (CG) method for the staticand eddy current case (in the time domain) and in the quasi-minimal residual(QMR) method for the time-harmonic case. The iteration was stopped as soonas the error in the preconditioner energy norm has been reduced by a factor10−6 for the PCG method. In the time-harmonic case (QMR solver) we usethe stopping criterion as follows


‖fh− Khuh‖2 ≤ 10−6 ‖f

h‖2 .

For all calculations, a V (2, 2)-cycle has been applied and the coarsest matrixequation is solved by a Cholesky factorization, if the degrees of freedom lessthan 500. All computations were done on a PC with a Pentium 1.7 GHz chip.

A good measure for the speed of coarsening is the so-called grid com-plexity, which is given by

GC(Kh) =

L∑i=1

Mi

M1, (11.34)

with L the number of levels and Mi the number of nodes (edges) for level i.This number is close to 1, if the reduction of unknowns is done very fast. Ifthe number is very large then the coarsening is usually very slow. A secondmeasure that is more related to the memory consumption and arithmetic costsis the operator complexity, i.e.,

OC(Kh) =

L∑i=1

NMEi ·Ni

NME1 ·N1, (11.35)

where NMEi denotes the average number of nonzero system matrix entrieson level i and Ni the number of unknowns on this level. This number gives anidea of how much memory is used with respect to the finest grid. The sameapplies for the arithmetic costs. Again this number is close to 1 if only a smallamount of memory is required. The abbreviation MB denotes the amount ofmemory used.

The computations with Lagrange and Nedelec FE functions were alwaysdone on the same FE mesh. We want to emphasize that Nh = p |ωn

h | for node(static case: p = 3; eddy current case: p = 4) and Nh = |ωe

h| for edge FEdiscretization.

Static Analysis

For the computational domain we consider the geometry of TEAM 20 (seeFig. 11.12), which has been discretized by tetrahedron elements.

Table 11.4 displays evaluated grid complexity GC and operator complexityOC as well as the required memory for the nodal and edge case. Nh definesthe number of unknowns. It can be clearly seen that the required memoryscales optimally with the number of unknowns and the values for OC andGC are close to 1. The number of iterations as well as elapsed CPU timesare shown in Table 11.5. The short time for performing the setup makes theAMG solvers very attractive for non-linear problems.


Fig. 11.12. FE mesh of TEAM 20 (without air region)

Table 11.4. TEAM 20: Complexities and memory requirement

Nh GC OC MBNodes Edges Nodes Edges Nodes Edges Nodes Edges

1.263 2.253 1.4 1.2 1.07 1.02 1.5 38.022 16.217 1.3 1.2 1.06 1.02 10 2456.673 122.762 1.3 1.2 1.07 1.03 65 173

Table 11.5. TEAM 20: CPU times and number of iterations

Nh Setup (s) Solve (s) IterNodes Edges Nodes Edges Nodes Edges Nodes Edges

1.263 2.253 0.2 0.2 0.2 0.2 18 98.022 16.217 0.4 0.5 2.0 2.8 27 1656.673 122.762 1.7 2.9 30.7 35.3 48 24

Transient Analysis

In order to show the performance of the proposed enhanced AMG methods foran eddy current problem, we present results of 3D magnetic field computationsfor a simplified MRI scanner with a z-gradient coil, as shown in Fig. 11.13[175]. Here, gradient and magnet coils are assumed as smeared cylindricalcoils. Furthermore, only the three inner cryostat cylinders are modeled. Table11.6 displays the values for the grid complexity GC, the operator complexity


Fig. 11.13. FE mesh of a simplified MRI scanner (not the full air region is displayed)

OC and the required memory. Again a very good performance with optimalmemory requirement can be found.

Table 11.6. MRI scanner: Complexities and memory requirement

Nh GC OC MBNodes Edges Nodes Edges Nodes Edges Nodes Edges

27.834 61.342 1.2 1.2 1.06 1.03 75 9488.053 197.375 1.2 1.2 1.06 1.03 250 330162.882 368.131 1.2 1.2 1.05 1.03 510 639

The performance of the proposed AMG solvers concerning the number ofiterations and CPU time is shown in Table 11.7. Since in this example we

Table 11.7. MRI scanner: CPU times and number of iterations

Nh Setup (s) Solve (s) IterNodes Edges Nodes Edges Nodes Edges Nodes Edges

27.834 61.342 2.5 1.5 10.3 7.6 15 1088.053 197.375 7.7 5.8 31.2 38.2 15 15162.882 368.131 14.6 10.1 64.1 75.2 16 15

have no parameter jump in the reluctivity, the number of iterations remainsquite constant, which results in an optimal convergence rate.


Time-harmonic Analysis

Finally, we show the performance of the AMG method for the harmonic anal-ysis. As a case study, we have chosen a configuration of a coil and a centerediron core surrounded by air. The established mesh can be seen in Fig. 11.14.

Fig. 11.14. FE mesh of an iron core and surrounding air (broken open at yz-planeelement boundaries, coil not displayed)

The core diameter was fixed at 2 mm and the coil thickness was set to 1mm.At an excitation frequency f of 500Hz, a relative permeability µr = 1000and a conductivity of γ = 107 S/m one calculates a penetration depth ofδ = 0.22 mm, which corresponds to approximately 1/5 of the core radius. Fora different conductivity of γ = 105 S/m one gets an eddy current penetrationdepth of δ = 2.2 mm, which is equal to a full penetration of the core. For thediscretization, we have used at least ten finite elements per penetration depth.

In the first step we performed calculations on varying grids and have listedthe evaluated characteristic solver data in Table 11.8. Therein, the operatorand grid complexity are close to 1, which indicates a fast coarsening, and inaddition small memory requirements for the AMG preconditioner. Further-more, it can be noticed that the solution time scales approximately linearlywith the number of edges Nh.

In the second step we performed computations for different material pa-rameters to investigate on the robustness of our new solver. In Figs. 11.15and 11.16 the convergence behavior of our QMR-AMG solver is displayed.The typical QMR behavior—a short remaining at a constant relative errorover a few iteration steps—can be detected clearly. Summarizing, we can notethat the proposed QMR-AMG solver is very robust against parameter jumps.


Table 11.8. Case study: Performance and complexity for different FE meshes

Nh Iter Setup (s) Solver (s) OC GC

18.708 30 0.39 4.81 1.026 1.216107.793 30 2.55 31.33 1.031 1.263245.418 32 6.04 81.35 1.029 1.211

Fig. 11.15. Convergence behavior with different conductivities γ and fixed relativepermeability µr = 1000 for the iron core

As a practical example, the results of a 3D magnetic field computation foran electric transformer are shown. In Fig. 11.17 the model including the finiteelement discretization is displayed.

Fig. 11.16. Convergence behavior with different conductivities γ and fixed relativepermeability µr = 1 for the iron core


Fig. 11.17. FE mesh of an electric transformer (no air region is displayed)

Due to symmetries, it is only necessary to simulate one quarter of the fullconfiguration by applying proper boundary conditions. The core is made ofiron (µr = 1000) and has a conductivity of γ = 106 S/m. The number ofcoil windings, respectively, the size of the current have been chosen in wayto ensure a maximum current density of 4A/mm2 to avoid an unnecessaryheating of the coil. Therefore, with 100 windings one gets a current amplitudeof 30A, considering an inner coil radius of 55mm, an outer coil radius of60mm and a coil length of 150mm.

In Table 11.9 one can see the characteristic solver data, e.g., setup and so-lution times as well as operator and grid complexity for different FE meshes.Again, the grid and operator complexity values are close to 1. However, itshould be mentioned that for an optimal multigrid method, the iterationnumbers are independent of the number of unknowns. In our case, a smallincrease of the iterations can be determined.

Table 11.9. Electric transformer: Performance and complexity for different FEmeshes

Nh Iter Setup (s) Solver (s) OC GC

9.115 22 0.24 1.38 1.017 1.19617.977 24 0.34 3.2 1.020 1.20632.835 30 0.63 7.47 1.021 1.195485.451 52 11.35 274.23 1.024 1.181


Fig. 11.18. Main magnetic induction in the iron core of the electric transformer

The presented AMG solvers are well suited for the efficient solution—both concerning CPU time and memory requirements—of algebraic systemsof equations arising from nodal as well as edge FE discretizations of Maxwell’sequations. In particular, the presented algorithms for the coarsening strategymake the solvers very attractive also for non-linear electromagnetic field prob-lems, since the setup time can be kept very small.

Further research is concentrated on improvements of the prolongation op-erators to obtain even better convergence rates. One possible method wasproposed in [212] (so-called smoothed aggregation), which could be appliedfor our problem classes.

If even more speedup is required for practical applications, the presentedAMG methods can be parallelized on distributed-memory computers. Thefirst promising results can be found in [80, 81].

12

Industrial Applications

12.1 Electrodynamic Loudspeaker

The electrodynamic loudspeaker to be investigated is shown in Fig. 12.1. Acylindrical, small, light, voice coil is suspended freely in a strong radial mag-netic field, generated by a permanent magnet. The magnet assembly, consist-ing of pole plate and magnet pot, helps to concentrate most of the magneticflux within the magnet structure and, therefore, into the narrow radial airgap. When the coil is loaded by an electric voltage, the interaction betweenthe magnetic field of the permanent magnet and the current in the voice coilresults in an axial Lorentz force. The voice coil is wound onto a former, whichis attached to the rigid, light, cone diaphragm in order to couple the forcesmore effectively to the air and, hence, to permit acoustic power to be radi-ated from the assembly. The main function of the spider and the surround isto allow free axial movement of the moving coil driver, while nonaxial move-ments are suppressed almost completely. Since in the case of a loudspeaker

Fig. 12.1. Schematic of an electrodynamic cone loudspeaker

314 12 Industrial Applications

the interaction with the ambient fluid must not be neglected, the electrody-namic loudspeaker represents a typical coupled magnetomechanical systemimmersed in an acoustic fluid. This is, why for the detailed finite elementmodelling of these moving-coil drivers the magnetic, the mechanical as wellas the acoustic fields including their couplings have to be considered as onesystem, which cannot be separated. Furthermore, electrodynamic loudspeak-ers in the low-frequency range under large-signal conditions show a stronglynon-linear behavior, which is caused mostly by two factors—the inhomogene-ity of the magnetic field in the air gap, i.e., magnetic non-linearities, and thenon-linearity of the suspension stiffness, i.e., mechanical non-linearities. Thesenon-linearities are caused by the large vibration amplitudes, especially at lowfrequencies. For large input powers the distortions increase rapidly and reachthe same order of magnitude as the fundamental.

To reduce the efforts in the development of electrodynamic loudspeakers,precise and efficient computer modelling tools have to be used. With thesecomputer simulations, the costly and lengthy fabrication of a large numberof prototypes, required in optimization studies by conventional experimentaldesign, can be reduced tremendously. For many applications an equivalentelectromechanical circuit model has been developed (see e.g., [209]). However,the main drawback of these simulation models is that the circuit-element pa-rameters have to be determined empirically by measurements on a prototype.Therefore, we will demonstrate that a model based on the partial differen-tial equations including all coupling terms and solved by an appropriate FEmethod (see Sects. 7 and 8) can totally fulfill the needs of an engineer. Sucha method just requires the geometry of the loudspeaker as well as materialdata of each part.

In the following, we will first discuss finite element models for the small-and large-signal behavior. Later, comparisons between simulation results andaccordingly measured data are shown for verification purposes. The main focuswill be on the demonstration of the practical usability of this scheme withinthe industrial computer-aided engineering of electrodynamic loudspeakers.

12.1.1 Finite Element Models

Small-Signal Computer Model

The finite element discretization of the electrodynamic loudspeaker undersmall-signal conditions is shown in Fig. 12.2. Here, the voice coil is discretizedby so-called magnetomechanical coil elements based on the motional emf-termmethod (see Sect. 7.3.4), which solves the equations governing the electromag-netic and mechanical field quantities and takes account of the full couplingbetween these fields. Due to the concentration of the magnetic flux withinthe magnet assembly, the magnet structure and only a small ambient regionhave to be discretized by magnetic finite elements. Furthermore, the surround,

12.1 Electrodynamic Loudspeaker 315

spider, diaphragm, and former are modelled by mechanical finite elements. Fi-nally, the surrounding fluid region in front of the loudspeaker is discretized byacoustic finite elements. The fluid region is surrounded by infinite elements forallowing open-domain computation (see Sect. 5.5.1), which have to be locatedin the far field of the moving-coil driver in order to work correctly. The inputlevel of these simulations is 1W referred to 4Ω.

Fig. 12.2. Small-signal finite element model of an electrodynamic loudspeaker

Large-Signal Computer Model

The finite element discretization of the electrodynamic loudspeaker underlarge-signal conditions is shown in Fig. 12.3. The following modifications havebeen performed in comparison to the above-explained small-signal computermodel:

1. To take into account the variation of the force factor for a coil underlarge excursions (i.e., magnetic non-linearities), the magnetomechanicalcoil elements discretizing the voice coil of the loudspeaker are based onthe moving material method (see Sect. 7.3.4). The force factor is definedby

αf = BlN ,

with B the magnetic induction in the air gap, l the length of one windingand N the number of turns of the voice coil, which are located in thehomogeneous magnetic field.


2. Furthermore, first simulation results showed that the mechanical non-linearities, i.e., the geometric non-linearity as a result of large displace-ments and the material non-linearity due to a non-linear stress–strain rela-tionship have to be taken into consideration only for the spider. Therefore,to allow a more efficient computation of the large-signal behavior the di-aphragm and the surround are discretized by finite elements solving linearmechanics.

3. Finally, measurements have shown that the distortion factors of the nearfield and diaphragm acceleration are in excellent agreement. Due to thiscorrelation a modified axisymmetric finite element model has been ap-plied, in which acoustic elements were eliminated completely (see Fig.12.3). The influence of the surrounding air, which consists of mass-loadingeffects and damping due to the sound emission, is now realized by so-calledspring elements. These elements have been located on the outside bound-ary of the surround and diaphragm.

Fig. 12.3. Large-signal finite element model of an electrodynamic loudspeaker

12.1.2 Verification of Computer Models

The verification of the computer models described above has been performedby comparing simulation results with corresponding measured data. In thefirst step, the most important small-signal results (frequency dependencies ofthe electrical input impedance, diaphragm acceleration and axial sound pres-sure levels as well as Thiele-Small parameters [174]) were considered. As canbe seen in Fig. 12.4, good agreement between simulation results and measureddata was achieved.


Next, the force–displacement characteristics were measured and comparedwith simulations (see Fig. 12.5). After this basic validation of the large-signalcomputer model, the total harmonic distortion (THD) factors of the voicecoil currents and diaphragm accelerations at large-signal conditions have beencalculated, which compute as

THD =

√p22 + p2

3 + ...√p21 + p2

2 + p23 + ...

, (12.1)

with pi the amplitude of the i-th harmonic. In addition, we define the

k2 =p2√

p21 + p2

2 + p23 + ...

(12.2)

k3 =p3√

p21 + p2

2 + p23 + ...

. (12.3)

The input level of these simulations was 32W referred to 4 Ω. As can beseen in Fig. 12.5, the good agreement of measured and simulated results overa wide frequency range validates the large-signal model depicted in Fig. 12.3.

Fig. 12.4. Comparison of simulated and measured small-signal results: (a) Fre-quency dependency of electrical input impedance Z, (b) Axial small-signal soundresponse level SPL at 1m distance

12.1.3 Numerical Analysis of the Non-linear Loudspeaker Behavior

Measurements as well as simulation results show that at frequencies f < 60Hzthe odd-order harmonics and at higher frequencies the even-order harmonicsdominate. The large advantage of computer modelling is the separation of thedifferent non-linearities for the different components of the loudspeaker. Inthis way, the influence of the different non-linear effects on the loudspeaker


Fig. 12.5. Comparison of simulated and measured large-signal results: (a) Force-displacement characteristic of the loudspeaker, (b) Total harmonic distortion (THD)of diaphragm acceleration at an input power of 32 W

behavior can be very efficiently extracted and researched in the simulation. Forexample, simulations showed that the magnetic non-linearities cause notablequadratic distortion factors at frequencies f > 60Hz, whereas the mechan-ical non-linearities cause the rapid increase of the even-order harmonics inthe lower-frequency range (see Fig. 12.6). Furthermore, both mechanical andmagnetic non-linearities are responsible for the cubic distortion factor.

Fig. 12.6. Numerical investigation of distortion factors of diaphragm accelerationat an input power of 32 W: (a) Quadratic distortion factor k2, (b) Cubic distortionfactor k3

In the next step of the numerical analysis, the influence of design param-eters of the magnet system on the distortion factors has been investigated.


Simulations considering only magnetic non-linearities showed that large coilflux variations result in notable odd-order harmonics. On the other hand, anonsymmetric magnetic field in the air gap causes large coil offsets resultingin significant even-order harmonics. Further computations showed that theposition of the permanent magnet has a big influence on the symmetry of themagnetic field in the air gap and therefore can be used in the optimization ofthe system (see Fig. 12.7b). Furthermore, it could be shown that the transient

Fig. 12.7. Finite element models: (a) Original magnet system, (b) Optimized mag-net system, (c) Original and optimized spider

magnetic field of the current-carrying voice coil must not be neglected at large-signal conditions. To reduce the influence of the coil field under large-signalconditions on the symmetry of the force factor, the whole magnet pot has tobe saturated and the upper air gap above the pole plate has to be increased(see Fig. 12.7b). These design modifications result in a much more symmetricdecrease of the force factor (see Fig. 12.8a). Furthermore, to minimize thevariation of the force factor, i.e., to raise the so-called jump-out excursion,the thickness of the pole plate has to be reduced (see Fig. 12.7b and Fig.12.8a). Finally, since this design modification results in a smaller efficiency ofthe loudspeaker, the width of the permanent magnet has to be increased.

After the above-explained numerical analysis of magnetic non-linearities,the influence of design parameters of the spider on the distortion factors causedby the mechanical non-linearities has been investigated. Simulations consider-ing magnetic and mechanical non-linearities showed that a larger spider heightresults in a more linear force–displacement characteristic and significantlysmaller odd-order harmonics Furthermore, a continuous displacement of eachmidpoint of the spider grooves causes a more symmetric force–displacementcharacteristic resulting in smaller even-order harmonics. For a more detaileddiscussion we refer to [174].


-20 -10 0-8

-4

0

Displacement (mm)

Force (N)10 20

4

8

Optimized

Original

b)-10 -4 0 10

Axial coil displacement (mm)

Normalized force - factor (%)Bl

4a)

0

-10

-20

-30

-40

-50

-60-6 6

Optimized

Original

Fig. 12.8. Comparison of the original and optimized loudspeaker: (a) Simulatedcoil flux variation (normalized to the original small-signal value), (b) Simulatedforce-displacement characteristic

12.1.4 Computer Optimization of the Non-linear LoudspeakerBehavior

In the course of this computer optimization, the knowledge of the sensitivitystudies explained in the previous section was put into a new prototype toreduce the even- and odd-order harmonics under large-signal conditions. Ascan be seen in Fig. 12.9, significant smaller distortion factors were achieved.In particular, cubic distortion factors could be reduced tremendously. For ex-ample, at a frequency of 20Hz the improvement is 70% with respect to theoriginal loudspeaker. This significant reduction of cubic harmonics is in ac-cordance with studies concerning the subjective perception of low-frequencydistortions [72]. According to [72], odd-order harmonics are above all respon-sible for the deterioration of the sound quality. Furthermore, the importantancillary condition of a similar small-signal behavior with respect to the orig-inal loudspeaker must be fulfilled. Small-signal simulations resulted in an ac-ceptable reduction in efficiency of 0.5 dB.

Furthermore, the numerically predicted improvements in the large-signalbehavior of the loudspeaker could be successfully confirmed by measurementson the new prototype (see Fig. 12.9). Therefore, it can be stated that thepresented simulation scheme is well suited to the industrial computer-aideddesign of electrodynamic loudspeakers, since an optimization with a signifi-cantly reduced number of prototypes can be achieved.

12.2 Noise Computation of Power Transformers

The sound emission of power transformers conflicts more and more with tight-ened low emission standards, which must be fulfilled, especially at night.

12.2 Noise Computation of Power Transformers 321

Fig. 12.9. Comparison of simulated and measured distortion factors of the op-timized loudspeaker (at an input power of 16 W): (a) Total harmonic distortion(THD), (b) Quadratic distortion factor k2, (c) Cubic distortion factor k3

Therefore, the prediction and reduction of these sound emissions is of in-creasing interest for the electrical power industry. The transformer noise ismainly caused by the following sources [51, 88]:

1. The no-load noise caused by magnetostrictive strain of core laminations.

2. The noise produced by fans or oil pumps.

3. The load-controlled noise caused by Lorentz forces resulting from the in-teraction between the magnetic stray field of one current-carrying wind-ing and the total electric currents in the conductors of the other winding.These forces cause vibrations of the winding and result in acoustic radia-tions with twice the line frequency (100 Hz or 120 Hz).

During recent decades the magnetic noise caused by magnetostrictivestrain of the core laminations and the noise of fans have been investigated andconsiderably decreased [101, 125, 178]. Therefore, the coil-emitted noise (seeitem 3 above) is of increasing interest. At the moment, approximate empiricalprediction formulas, which primarily depend only on the rated power of thetransformer, represent the state-of-the-art. However, the main disadvantage ofthese prediction formulas is that accurate parameters on the load-controllednoise are not available.

12.2.1 Finite Element Models

The goal of the numerical simulation is to predict very precisely the emittedsound of loaded transformers within a test hall. In the first step of the mod-elling scheme the outermost winding surface displacements are calculated byan acoustic-magnetomechanical finite element model of one winding of the


Fig. 12.10. Overview of the developed calculation scheme

oil-filled transformer. Due to rotational symmetry of the winding and sym-metric load (as a result of the ideal measurement condition in the factory testfield), a 2D finite element model based on axisymmetric elements can be used(see Fig. 12.11). In the finite element model, the voltage-loaded conductorsof the winding are discretized using magnetomechanical coil elements. Theseelements solve the equations governing the electric circuit, the magnetic aswell as the mechanical field equations, and take account of the full couplingbetween these fields (see Sect. 7). All winding clamping and insulation mate-rials between each coil as well as the winding support platforms are modelledusing magnetomechanical finite elements, which solve the coupled magneticand mechanical field equations. Instead of a complete model of the highly per-meable core by magnetic finite elements, this computer model was simplifiedby applying Neumann boundary conditions at the boundary of the windingwindow (see Fig. 12.11). Furthermore, the surrounding oil within the tankis discretized using purely acoustic finite elements and magnetic-acoustic fi-nite elements (solving the magnetic as well as the acoustic partial differentialequation without any coupling). Finally, the tank is modelled using standardmechanical finite elements.


Fig. 12.11. Axisymmetric acoustic-magnetomechanical finite element model of onewinding of the oil-filled power transformer

Additionally, the following aspects have to be considered for the precisecomputer simulation of the winding vibrations of loaded power transformers:

• To measure the load-controlled noise in a factory test field, the transformerhas to be operated at short-circuit and at rated currents. In this case, dueto the small voltage during the short-circuit test, the core-emitted noisecan be neglected and, therefore, a clear distinction between the no-loadnoise and the load-controlled noise can be achieved. In the simulations,this effect was taken into account by modelling the innermost low-voltage(LV) winding as a voltage-loaded coil with an external voltage of zero. Thehigh-voltage (HV) winding and both in-series connected, outermost tap-ping windings, however, are loaded with the measured short-circuit voltagein a star connection.

• Furthermore, the core clamping supports have been ignored in the finiteelement model to reduce the effort. Therefore, the influence of these sup-ports, which consists of an additional axial stiffness of the winding, isrealized by so-called spring elements. As shown in Fig. 12.11, these springelements have been located at the outside boundary of the upper and lowerwinding support platform. In the simulations, a stiffness of 85 MN/m hasbeen used, which is in accordance with the experience of the transformer


manufacturers.

• Finally, since measurement results revealed a big influence of the tap-changer position on the measured vibrations and sound pressure levels, thefollowing simulations have been performed for three nominal positions:– Tap-changer position 1: The HV winding and both tapping windings

are connected in series.

– Tap-changer position 2: The HV winding and the coarse tapping wind-ing are connected in series.

– Tap-changer position 3: Only the HV winding is connected.

For the computation of the amplitude of the winding surface displace-ments, a dynamic analysis using a sinusoidal 50 Hz (or 60 Hz) excitationsignal for the voltage between the two supply terminals of the high-voltagewinding was performed. It should be noted that further input parameters arethe geometry of the power transformer, the density, modulus of elasticity,Poisson’s ratio and loss factor for the mechanical materials (tank, conduc-tors, insulation and clamping materials), the electrical conductivity for theconductors as well as the density and bulk modulus for the surrounding oil.After the computation of the response signals (current in the conductors ofboth windings as well as mechanical displacements of the outermost winding),the Fourier transform of the output signals has to be calculated. Finally, the100-Hz (or 120-Hz) component of the spectrum must be extracted.

In the second step, the previously calculated winding surface displace-ments are now taken as mechanical excitation in a 3D acoustic-mechanicalfinite element model of the complete oil-filled tank (see Fig. 12.12). Further-more, in the 3D finite element model the 120 degree phase shift between thethree windings is taken into account. In this model the tank, the core clamp-ing as well as the connections between core clamping and top of the tank,are modelled using mechanical finite elements. Furthermore, the surroundingoil within the tank is discretized using acoustic finite elements. Finally, theadditional stiffness of the core clamping supports is again implemented usingspring elements. Since the load-controlled noise is primarily a simple 100-Hz(or 120-Hz) tone, a harmonic analysis has been performed.

In the last step the previously calculated tank-surface vibrations are nowapplied as mechanical excitation in the final acoustic simulations to calculatethe radiated transformer noise. Here, the radiation within closed rooms suchas a high-voltage laboratory are calculated (see Fig. 12.10). For the compu-tation of the sound radiation within a high-voltage laboratory, a 3D acoustic-mechanical finite element model has been set up (see Fig. 12.13). Here, thetank of the transformer is discretized using mechanical finite elements. Fur-thermore, the surrounding air within the test hall is modelled using acousticfinite elements. The walls of a typical high-voltage laboratory are not covered


Fig. 12.12. Three-dimensional acoustic-mechanical finite element model of an oil-filled transformer

with any absorbing material. Therefore, in these simulations, the transformerwas assumed to be positioned within a hall with ideally reflecting walls. Inthis computer simulation 640 000 3D finite elements have been used.

Fig. 12.13. Three-dimensional acoustic-mechanical finite element model of thetransformer tank and the high-voltage laboratory

12.2.2 Verification of the Computer Models

The verification of the computer models described above has been performedby comparing simulation results with corresponding measured data. It shouldbe noted that due to the complexity of the sound emission of the loaded


power transformer, analytic calculations are unavailable and therefore cannotbe used for verification purposes.

12.2.3 Verification of the Calculated Winding and Tank-surfaceVibrations

In the first step, the axisymmetric finite element model has been verified bycomparing measured and calculated short-circuit currents, mechanical eigen-frequencies as well as winding surface accelerations.

In Table 12.1, the short-circuit currents obtained by measurements as wellas simulations are shown for two tap-changer positions. The good agreementbetween measured and calculated values (the deviation is within 1.25%) vali-dates again the developed coil-modelling scheme.

Table 12.1. Measured and simulated short-circuit currents (LV Low voltage; HVHigh voltage)

Measurement Simulation(A) (A)

Current in LV winding, at tap-changer position 1 825 832Current in HV winding, at tap-changer position 1 159 161Current in LV winding, at tap-changer position 2 825 825Current in HV winding, at tap-changer position 2 180 181

Next, the measured and calculated transfer functions and mechanicaleigenfrequencies of the complete winding system mounted on the core havebeen compared (see Fig. 12.14). The deviation at the second eigenfrequency isdue to the fact that the actual winding does not show an exact axisymmetricconstruction. Furthermore, it should be noted that the winding was axiallyexcited at the upper winding support platform and the resulting radial coilacceleration of the outermost winding was measured as well as simulated.

In a final verification of the axisymmetric finite element model, the simula-tion results have been compared with corresponding measured winding surfaceaccelerations. Here, the axial accelerations on the upper winding support plat-form and the radial vibrations on the outermost fine tapping winding weremeasured using an oil-resistant piezoelectric accelerometer. The measurementsshowed that the sensitivity of these sensors against electromagnetic interfer-ences of the high-voltage winding is negligible. Furthermore, the deviationsof subsequent measurements were within a range of ±1.5%. In Table 12.2,the winding vibrations of the transformer without a tank, in Table 12.3, thenormalized accelerations of the transformer with an oil-filled tank are com-pared, respectively. In the case of the transformer with an oil-filled tank (see


Fig. 12.14. Comparison of measured and simulated transfer functions and mechan-ical eigenfrequencies of the winding mounted on the core

Table 12.3), measurements as well as simulations reveal that the surroundingoil does not influence the axial accelerations of the winding support platform.However, due to the mass-loading effect of the surrounding oil, the radial coilacceleration amplitudes are nearly halved when compared to the vibrationsignoring the oil-filled tank. In summary, it can be stated that an axisymmet-ric finite element model precisely predicts the winding surface accelerationsof a loaded power transformer for both configurations, with and without anoil-filled tank.

Table 12.2. Transformer without oil-filled tank: Measured and simulated windingaccelerations

(m/s2) (m/s2)

Radial coil acceleration at tap-changer position 1 0.047 0.046Axial clamping acceleration at tap-changer position 1 0.036 0.037Radial coil acceleration at tap-changer position 2 0.021 0.019Axial clamping acceleration at tap-changer position 2 0.031 0.032

Finally, the 3D acoustic-mechanical finite element model of the oil-filledtank has been verified by comparing the calculated tank side wall accelerationswith values measured at nine different positions (see Table 12.4).


Table 12.3. Transformer with oil-filled tank: Measured and simulated winding ac-celerations (normalized to the corresponding result without an oil-filled tank (seeTable 12.2))

Measurement Simulation

Radial coil acceleration at tap-changer pos. 1 0.59 0.62Axial clamping acceleration at tap-changer pos. 1 0.96 0.99

Table 12.4. Measured and simulated tank accelerations

Position Measurement Simulation Position Measurement Simulation(m/s2) (m/s2) (m/s2) (m/s2)

1 0.16 0.13 6 0.029 0.032 0.063 0.075 7 0.01 0.0153 0.14 0.1 8 0.02 0.0254 0.055 0.07 9 0.04 0.0455 0.04 0.05

12.2.4 Verification of the Sound-field Calculations

After these basic validations of the computational models, the A-weightedsound-power level [145] of the short-circuited transformer was measured inaccordance with the European standard EN 60551 [204] and compared withthe corresponding acoustic simulations. This standard requires that the A-weighted sound-pressure levels around the transformer have to be measuredat a distance of 0.3 m from the tank surface and at half the tank height. Fur-thermore, these measurements have to be performed in a typical high-voltagelaboratory at a transformer manufacturer. Due to this fact, the 3D finite el-ement model for the calculation of the radiated noise within closed rooms,as shown in Fig. 12.13, has been used. An A-weighted sound power level of66.5 dB(A) was calculated from the simulated sound-pressure levels. Consid-ering the fact that the reproducibility of the sound-pressure measurementslies within a range of ±1 dB, a good agreement between measurement andsimulation was achieved (see Table 12.5).

Furthermore, as can be seen from Table 12.5, in this case the deviation be-tween measured and calculated sound power level is considerably smaller thanthose resulting from the current-prediction formulas for the load-controllednoise. Therefore, it can be concluded that this pure finite element scheme iswell suited to the computation of the load-controlled noise of oil-filled powertransformers, which are operated within a typical high-voltage laboratory.

12.2.5 Influence of Tap-changer Position

As a first application, the influence of the tap-changer position on the windingand tank surface vibrations as well as on the A-weighted sound-power level


Table 12.5. Radiation within closed rooms: Measured, simulated, and predictedsound-power levels

Sound-power level(dB(A))

Sound-pressure measurement according to [204] 68Finite element simulation 66.5Empirical prediction formula according to [204] 63.5Empirical prediction formula according to [177] 60.5

has been investigated. Table 12.6 shows the simulation results at tap-changerposition 1 (HV winding and both tapping windings are connected in series)and at tap-changer position 3 (only the HV winding is connected). Simulationsas well as measurements reveal that the radial vibrations of the outermost, finetapping winding are greatly decreased at tap-changer position 3 as comparedto position 1.

Table 12.6. Influence of the tap-changer position obtained by simulation

Tap-changer Tap-changerposition 1 position 3

Radial coil acceleration (m/s2) 0.033 0.009Axial winding clamping acceleration(m/s2) 0.044 0.033Tank side wall acceleration (m/s2) 0.025 0.009A-weighted sound power level (dB(A)) 59 45

With these simulations it was found that the radial magnetic volume forcesacting on the innermost LV and HV winding are almost independent of thetap-changer position. Therefore, this reduction of the radial coil vibrations attap-changer position 3 is based on the noncurrent-carrying, outermost, finetapping winding.

On the other hand, simulation results revealed that the axial magneticvolume forces acting on the LV and HV winding are larger at tap-changerposition 3 when compared to position 1. This is caused by the edge-fringingeffect and is responsible for the fact that the axial winding clamping vibrationsare almost independent of the tap-changer position (see Table 12.6). Further-more, due to the decrease of the radial surface accelerations of the outermostwinding at tap-changer position 3, the tank-surface vibrations and, therefore,the calculated A-weighted sound-power level are greatly reduced.

Therefore, in contrast to [177], where it is assumed that only axial wind-ing vibrations are responsible for the load-controlled noise, these simulationsclearly show that the radial coil vibrations also have a significant influence onthe coil-emitted noise.


12.2.6 Influence of Stiffness of Winding Supports

In a second application, the influence of the stiffness of the winding and coresupports on the load-controlled noise has been investigated. This stiffness hasbeen modelled in the finite element simulations by applying mechanical springelements, which were located at the outside boundary of the upper and lowerwinding support platform (see Fig. 12.11). As expected, the simulations revealthat neglecting this axial stiffness causes significantly increased axial clamp-ing accelerations and slightly decreased radial coil accelerations. Therefore,greatly increased sound-pressure levels result (see Table 12.7). These resultsindicate that this stiffness has a strong influence on the radiated transformernoise effect, which can be used in the optimization of the system.

Table 12.7. Influence of stiffness of winding supports obtained by simulation (SPLSound-pressure level)

Radial coil Axial winding SPL inacceleration acceleration 0.3 m

(m/s2) (m/s2) (dB)

With stiffness of winding support 85 MN/m 0.046 0.037 56.8Without stiffness of winding support 0.044 0.62 82.3

12.3 Fast-switching Electromagnetic Valves

Modern fast-acting solenoidal valve applications demand further improve-ments of the operation speed and reproducibility of the opening and closingphase. The development goes towards lightweight construction in combinationwith sophisticated energizing concepts. This gives rise to structural vibrationas well as sound-emission problems whose elimination by means of passivedamping is not sufficient. A dynamic analysis of the switching behavior withsufficient precision taking into consideration all significant physical effects canbe done only by a numerical analysis.

The reproduction of the dynamics of fast-switching solenoidal valves provesto be highly non-linear. Both the magnetization state of the ferromagneticarmature material and the induced eddy current distribution inside the ar-mature during its accelerated motion within an inhomogeneous magnetic fieldare accounted for. In addition to the numerical calculation scheme presentedin Sect. 7.3, we have to take impact dynamics into account to fully model avalve-switching cycle. Impact/contact problems are different from Neumannand Dirichlet boundary problems since the contact constraints are unknownin time and space and have to be determined as a part of the solution.

12.3 Fast-switching Electromagnetic Valves 331

Fig. 12.15. Magnetic field region of the actuator including impact region

12.3.1 Modelling and Solution Strategy

To precisely compute the dynamic behavior of a solenoidal actuator a numer-ical calculation scheme has to be able to handle the electromagnetic field, themechanical field as well as the coupling terms. The strong coupling is nec-essary because of the interdependency between the position and velocity ofmoving mechanical parts and the inductivity of the solenoid as well as eddycurrent induction. Therefore, we are able to reproduce the dominant influ-ence of eddy currents on the dynamics as well as the damping effect duringthe bouncing period at valve opening and closing.

Magnetic Field

The governing equation describing the magnetic field can be derived fromMaxwell’s equations for the quasistatic case (neglecting the displacement cur-rents) using the magnetic vector potential as state variable (see Sect. 4.2.1).To precisely compute the magnetic field within a solenoidal actuator, the fol-lowing physical phenomena have to be taken into account:

• B–H curve:The measured B–H curve data are approximated by an enhanced smooth-ing spline technique to guarantee a smooth approximation of the curveas well as of its derivative. The resulting non-linear magnetic equation issolved by a Newton method with a line search algorithm (see Sect. 4.7.5).

• Voltage-loaded coil:For a voltage-loaded coil the additional circuit equation is simultaneouslysolved with the partial differential equation of the magnetic field (see Sect.7.3.1).


Mechanical Field and Contact Mechanics

For the mechanical field, Navier’s equation is solved with the mechanical dis-placement as state variable (Sect. 3.7). The impact of moving parts on thestatic parts of the valves at the switching of the valve state results in a noisyand unreliable bouncing, known as hard landing. A correct physical mod-elling of the impact as well as the overcoming of the inherent convergenceproblem of dynamic contact problems can be solved only by using a contactalgorithm that satisfies all kinematic and kinetic conservation laws as can befound, e.g., in [8]. Including frictionless contact mechanics into the systemstarts with a contact-detection algorithm based on the normal distance of thetwo contacting bodies. In the case of satisfied contact conditions, the contactpressure in the normal direction is applied by using an exponential contactpressure–displacement relation

pc(u) = p0(l − g(u))m . (12.4)

Here, p is the contact pressure and g the gap length, both in the normal di-rection and is dependent on the nodal displacements u within the discretizedsystem. The surface hardness p0 can be considered as a penalty parameter toincorporate the contact constraints. The exponent m can be derived analyt-ically using a statistical treatment of Hertzian microcontacts and is verifiedby measurements within a range of 2.0 to 3.3 [215]. The constant l can beinterpreted as the surface roughness and can be used to realize contact ata finite gap length as described later. Applying now the FE method to thecontinuum mechanics of contact, we have to add the contact force vector

fc(u) =

ne∧e=1

fe(u) with f

e=∫Ωe

pc(u)N dΩ (12.5)

to the right-hand side of the algebraic system of equations. The nodal contactforce vector f

cis given by the assembly of the element contact force vectors

fe

over all ne contact elements currently in closed contact. With pc we denotethe contact pressure determined in (12.4). To achieve quadratic convergence insolving the non-linear mechanical system with the Newton method we haveto add further to the linear stiffness matrix (see Sect. 3.7.1) the tangentialcontact stiffness matrix

Kc =f

c(un+1)

∂u

∣∣∣∣∣un+1

, (12.6)

which is obtained by an exact and complete linearization of the contact forcevector using a directional derivative with respect to the nodal displacementsu at time step (n+ 1).


Fig. 12.16. Eddy-current-induced hysteresis of magnetic flux, magnetic force, andcoil inductivity

12.3.2 Actuator Characteristics

The direct acting and normally closed solenoid valve investigated can be char-acterized as a short-stroke valve with a compact design at a volume of 28 cm3

and an air-gap cross-sectional area of 61 mm2. Valve opening is forced by theelectromagnet against the force of the return spring with a spring constantof 6.7 kN/m and a pretensioning of 18 N. The limited stroke of max. 45µmcauses only a small dependency of the static actuator properties on the arma-ture position in the complete operation range of the actuator. With increasingcoil current and magnetic saturation, the inductivity of the electromagnet isdecreasing in compliance with the decreasing permeability. At the same time,the gain in magnetic force diminishes due to the saturation. Investigating,furthermore, the actuator dynamics, the eddy-current-induced hysteresis canbe acquired by using a high-level signal harmonic excitation at different fre-


quencies (Fig. 12.16). The B–H magnetization curves of ferromagnetic valvecomponents used for the numerical analysis have been adapted to the hystere-sis measurement results. This procedure is necessary since some componentsare manufactured with particular mechanical and heat treatment wherebychanging their original magnetic properties and a direct measurement of thesaturation curves of small parts is full of uncertainty.

Comparing simulation results using the adapted B–H magnetization curveswith measurement results at higher frequencies, it can be shown that magnetichysteresis effects are negligible in comparison to the eddy-current-inducedhysteresis (Fig. 12.16).

Fig. 12.17. Effect of eddy current induction on the frequency response

Eddy current induction increases with rising frequency and affects the dif-fusion rate and extension of the magnetic field inside the armature as well as itstemporal expansion, known as the skin effect (see Sect.4.2.2). Eddy currentsare therefore a critical factor concerning the dynamic behavior of electromag-netic actuators since they constrain the operation range of the actuator. Themagnitude and phase of the actuator frequency response (magnetic force overapplied coil voltage, respectively current) is shown in Fig. 12.17. The transferfunction shows a cutoff frequency of 860 Hz for the current-fed coil, which isbeyond the operation range of the valve. In summary, the dynamic magneticforce is a complex function of the system’s operation point defined by theparameters:


• Armature position and velocity• Applied coil current• Magnetization state of the ferromagnetic material• Induced eddy current distribution inside the armature

12.3.3 Actuator Dynamics

Investigating the actuator dynamics, it is instructive to consider first twotypes of coil energizing: A current-fed coil and a voltage-fed coil. Connectinga current source to the coil results in the shortest pull-in time since, dueto the applied current, no voltage feedback effect can become active and theelectromagnetic compensation process can be neglected. This forces the fastestmovement of the armature and can be approximately realized by a digitalcontroller unit. On the other hand, the use of a voltage source leads to muchlonger time constants due to the delayed rise of the coil current. As a result,applying a current, respectively voltage, Heaviside function to the coil, thecurrent (voltage)-fed coil has minimal pull-in time of 0.16 ms (0.35 ms) and aminimum response delay time of 0.08 ms (0.26 ms) as shown in Fig. 12.18.

Fig. 12.18. Simulation results: Valve-opening times and strike velocity

No further reduction of the pull-in time can be achieved without using amore sophisticated energizing concept or reduction of eddy current inductionthrough design features. But although eddy currents are undesired, since theyincrease the switching times, they are desired to damp the bouncing of thearmature. A short pull-in time comes with high strike velocity at the end


of the valve-opening process (Fig. 12.18). Therefore, there is an optimizationconflict between obtaining short valve-opening times and reducing bouncingeffects caused by high impact velocities to achieve a reproducible valve func-tion without reopening.

12.3.4 Dynamics Optimization I: Electrical Premagnetization

The actuator dynamic can be affected basically by

• Forcing the mechanical compensation process by enhancement of the dy-namic magnetic force

• Forcing the electrical compensation process by enhancement of the ener-gizing power during the pull-in time

Fig. 12.19. Stroke and velocity profile for several levels of electrical premagnetiza-tion

One method to accelerate the pull-in time of the valve is to premagne-tize the actuator by a permanent magnet or electrically using a coil-currentcontrol unit. Premagnetization causes a magnetization state inside the arma-ture at increased magnetic field intensity and reduced permeability. The lower


Fig. 12.20. Effect of electrical premagnetization on the valve dynamics

permeability again leads to an increased magnetic field diffusion velocity andtherefore to an accelerated rise of the magnetic force at a following coil ex-citation spike. The magnetic force caused by premagnetization itself as wellas its accelerated rise will exceed the level of the return spring force at anearlier time stage (Fig. 12.19), resulting in a reduced response delay time.In contrast, the actual time of movement (valve needle flight time) itself isalmost not influenced (Fig. 12.20).

12.3.5 Dynamics Optimization II: Overexcitation

Further improvements in the valve dynamics can be achieved by applying ahigh current peak to the coil at the start of the valve-opening phase, knownas overexcitation. The current spike acts until the end of the opening process.Afterwards, the coil current is reduced and controlled to the nominal current.The profit in higher magnetic force is low, since we get into the region ofmagnetic saturation, but due to the fact that the permeability decreases inthis region, the diffusion velocity of the magnetic field into the armature canbe increased. As a consequence thereof we obtain an accelerated rise of themagnetic force, which results in shorter pull-in times (Fig. 12.21).

Applying different levels of overexcitation, the accelerated magnetic fielddiffusion into the armature material can be made visible by simulation (Fig.12.22). The level of overexcitation is mainly limited by the thermal capacityof the actuator system.

12.3.6 Switching Cycle

Modern applications of solenoidal actuators use a digital controller, whichexecutes complex algorithms to provide a wide variation of the switching


Fig. 12.21. Effect of coil overexcitation on the valve dynamics

Fig. 12.22. Magnetic field diffusion inside the armature for different levels of overex-citations at one point in time

times and precise actuator control. The modulation of the current accordingto a predefined profile permits control of the electromagnetic force and thus ofthe valve state. In the dosing valve system under investigation, the objectiveis to shorten the opening and closing time, since in these periods an undefinedfluid dosage takes place. The coil current is controlled in an ON/OFF modeby a pulsewidth modulation generated by the controller using a so-called softswitching technique where the lower coil voltage level is zero (Fig. 12.23). Softswitching gives the coil current lower ripple and achieves a smoother outputprofile of the magnetic force.

The complete control profile for a valve operation cycle can be subdividedas follows:


Fig. 12.23. Valve-switching cycle: Control profile and system response

1. Electrical premagnetization:Using the concept of electrical premagnetization as presented in the previ-


ous section with a coil current of 1 A, the valve-opening process can startfrom a higher magnetization level.

2. Valve opening:The control accomplishes the valve opening by applying a momentaryhigh-voltage overexcitation spike to build a high-density magnetic fluxfield almost instantaneously, resulting in a steeper magnetic force gradi-ent to shorten the needle flight time.

3. Hold phase:During the hold phase—the actual dosing period—the nominal coil cur-rent of 3 A creates the holding force at the end of the stroke.

4. Valve closing:After turning off the driving coil current with a damping negative voltagespike, the spring force returns the armature to its initial position.

In a final simulation, the complete switching cycle of the valve was reproduced(Fig. 12.23). Combining the concepts of premagnetization as well as overex-citation to optimize the actuator dynamics, the pure valve needle flight timeat valve opening can be reduced to 200µs.

12.4 Cofired Piezoceramic Multilayer Actuators

Piezoceramic actuators are widely used for high-precision positioning systems.Their almost infinite resolution (in the nanometer range) and their very goodrepeatability predestine these actuators for the usage in linear stages, camerashutters or printer heads [210]. Rapid improvements in the performance of theceramic materials used, make new smart designs possible and offer new ap-plication fields for piezoceramic actuators. The advantage of the piezoelectricactuator lies in the enormous force density and the high dynamics. In order toachieve improved deflections, hundreds of thin piezoceramic layers are stacked.Therefore, the mechanical displacement of the individual layers sum up, whilethe electric driving levels can be reduced due to a parallel switching of theceramic sheets. The setup is similar to that used for multifoil-capacitors, asshown in Fig. 12.24. The highly dynamic deflection of these complex struc-tures necessitates a sophisticated design in order to guarantee an effectiveoperation and a large number of duty cycles. Due to the thin ceramic layers,with a thickness of about 100µm, strong electric fields are established in theceramic material at typical driving voltages of about 200 V. Therefore, theseactuators show a strongly non-linear response mainly caused by the ferro-electric nature of the ceramic materials. These effects, which are responsiblefor the actuator’s performance, have to be considered during the design pro-cess. Therefore, non-linear material relations as well as ferroelectric hysteresiseffects have to be considered.

12.4 Cofired Piezoceramic Multilayer Actuators 341

Fig. 12.24. Piezoceramic multilayer stack actuator

12.4.1 Setup of Multilayer Stack Actuators

The thin ceramic layers used for the multilayer actuators are fabricated bytape casting using the so-called doctor blade technique (see [158]) in order toachieve a certain thickness of the layers. On these green sheets the internalelectrodes consisting of silver (Ag) and palladium (Pd) are applied by screenprinting leading to layers of a few micrometers thickness. After the cuttingprocess the stacks are assembled and sintered together into a monolithic block.Therefore, these structures are called cofired. Due to the absence of adhesivelayers the actuators provide a high stiffness and a fast response time. Theinternal electrodes of the actuator are arranged interdigitally, so that insu-lation gaps towards the outer side walls can be formed. External electrodesare placed at those sides where only internal electrodes of a single polarityare conducted to each actuator surface. After the manufacturing process thedipole orientations of the grains are statistically distributed, so no piezoelec-tric effect can be detected. By applying a strong electric field to the externalelectrodes a poling of the domains occurs, leading to a remanent polarizationwithin the structure, so that the material becomes piezoelectric. Therefore, thepiezoelectric effect in these materials is based on the ferroelectric hysteresiseffect. The best performance of these actuators can be achieved if the polariza-tion vector is homogeneous throughout the whole transducer, oriented in theprincipal working direction. However, due to the interdigitally arranged elec-trodes a homogeneous electric-field distribution cannot be achieved. Regionsthat are unpolarized, and therefore inactive, as well as regions with polar-ization vectors deviating strongly from the working axis occur, thus leadingto strong mechanical stresses within the structure. While piezoceramic ma-terials are able to handle large compressive stresses up to 300 MPa, they arerather sensitive concerning tensile stresses. This problem can be overcome pri-marily by simply applying a prestressing to the actuator, thus reducing the


tensile stresses during the load cycle. On the other hand, locally restricted ten-sile stress peaks caused by an inhomogeneous electric field distribution maylead to cracks and a breakdown of the whole structure. The fatigue behaviorof the ferroelectric material is very complicated, especially under highly dy-namic driving conditions. A variety of models have been developed allowingthe fatigue behavior of these structures to be analyzed for quasistatic loadingconditions from a micro-mechanical point of view [3]. These models are verycomplex and their numerical simulation would require an extremely high com-putational effort, thus not allowing the whole complex multilayer structure tobe handled. Furthermore, due to the fact that they are restricted to staticanalysis, dynamic effects that influence the stress distribution and materialbehavior dramatically, cannot be taken into account.

12.4.2 Finite Element Model

The numerical analysis has been performed for a multilayer stack actuatorconsisting of 18 active ceramic layers, as shown in Fig. 12.25. Inactive zonesare placed at the top and bottom of the structure. The symmetry of the setuphas been used in our model by applying appropriate boundary conditions,therefore reducing the number of elements. The multilayer stack actuatorhas been modelled using up to 120 000 hexahedron elements, thus leading to450 000 unknowns that have to be solved numerically. For this model linearas well as non-linear calculations have been performed to analyze the stressand electric field distribution at the tips of the internal electrodes.

The FE formulation used is based on a direct coupling of the electric andmechanical field equations as discussed in Sect. 9. For the numerical analysisthe influence of ferroelectric hysteresis has been studied. Therefore, typicalvalues for the saturation field strength and polarization, as well as for thecoercitive field intensity have been used. Due to the fact that the problem ishighly non-linear a harmonic or modal analysis cannot be employed, even ifthe input signals are sinusoidal. Instead, the calculations are performed in thetime domain using a Newmark time stepping algorithm (see Sect. 9.6). Thenon-linear iterative calculation scheme has to be executed for each time stepusing the solution of the previous time step as the initial guess. Therefore, thedynamic analysis of a complete piezoceramic stack actuator is made available.By using a sinusoidal input voltage with a peak value of 200 V and a frequencyof 300 Hz we observed the displacement signal of the actuator. For this setup,measurements and numerical simulations have been performed. The excitationof the actuator has been realized using a high-power voltage amplifier allowinglarge capacitive loads to be driven. The displacements have been obtainedoffline by integration of the velocity signal, which was measured using a laservibrometer.

12.4 Cofired Piezoceramic Multilayer Actuators 343

Fig. 12.25. Finite element model (dimensions in mm)

12.4.3 Measured and Simulated Results

In Figs. 12.26 – 12.29 the measured and simulated results for the displace-ments are plotted showing the time signals as well as the frequency spectra.Due to the ferroelectric hysteresis higher-order harmonics are showing up.

Fig. 12.26. Measured displacement of the actuator

By comparing the measured and simulated spectral rates, higher-harmonicpeaks of comparable magnitude can be detected. The results retrieved fromnumerical analysis show a slightly stronger influence of the ferroelectric ef-fect. This is mainly caused by the used model parameters. The polarizationhysteresis loop is normally measured in a quasistatic case. Due to the higherfrequency applied, a reduction in hysteresis takes place due to the fact thatfrictional effects slightly suppress the local depolarization effects.


Fig. 12.27. Simulated displacement of the actuator

Fig. 12.28. Frequency spectra of measured displacement

However, as can be seen from the results, the exactness of the simulationsdepends primary on the exactness of the material parameters, which have tobe determined by appropriate measurements. Especially in the case of piezoce-ramic layers used for multilayer actuators that are fabricated by tape casting,process restrictions limit the sheet’s thickness. Thus, the manufacturing ofprobes with idealized geometries as required for measurements according tothe standards [38, 105] becomes almost impossible. Therefore, we have devel-oped a new method using a parameter-optimization technique for determiningthe linear parameters from measurements on samples with no special geome-tries [114]. Furthermore, we are going to improve this method to allow thelarge-signal parameter identification using inverse techniques. Combining bothtechniques, measurement and simulation, makes further improvements in theperformance and duration of life of multilayer actuators possible.

12.5 Capacitive Micromachined Ultrasound Transducers 345

Fig. 12.29. Frequency spectra of simulated displacement

12.5 Capacitive Micromachined Ultrasound Transducers

3D ultrasound imaging is an area of extensive research and of great inter-est for many industrial applications ranging from nondestructive testing tomedical-imaging systems (see e.g., [75]). To obtain a 3D ultrasound image, a2D array of transducer elements with electronic focusing and beam steeringis needed. Using standard piezoelectric transducer elements will result in dif-ficulties concerning fabrication as well as electronic connections. Capacitivemicromachined ultrasound transducers (CMUTs) may overcome many of thedrawbacks of piezoelectric transducers, since CMUTs can be fabricated byadding a few technological steps to a standard CMOS process [2,57,128]. Dueto small size and the possibility of integrating signal-processing electronics ona chip [129], these transducers may be an attractive alternative to standardultrasound transducers. Figure 12.30 shows the top view of a CMOS test-chip with 4 transducer arrays, each containing 19 capacitive transducers. Thetransducers are used in the transmitting as well as receiving mode, so thata classical pulse-echo mode can be performed. The principle setup of suchan array is shown in Fig. 12.31 and the detail of one cell in Fig. 12.32. Byapplying a short voltage signal to the electrodes of each transducer cell, themembranes are deflected by the resulting electrostatic force, and an acousticpulse is generated that propagates into the fluid. Now, the same transducercells are used for measuring the reflected acoustic pulse, since the membraneswill be deflected according to the fluid loading, resulting in a change of theircapacitances.


Fig. 12.30. Top view of a CMOS chip with four arrays, each containing 19 capacitivetransducers

Fig. 12.31. Principle setup of cells Fig. 12.32. Detailed view toa capacitive cell

12.5.1 Requirements to Numerical Simulation Scheme

The design of such micromachined transducers strongly depends on the avail-ability of appropriate computer-aided engineering (CAE) tools, since the fab-rication of each prototype is quite costly. In addition, the signal-to-noise ratiois still too low compared to piezoelectric transducers, and a lot of research hasstill to be done [147]. Therefore, precise computer simulations are needed toanalyze and, furthermore, optimize the dynamic behavior as well as efficiencyof such transducers. However, the precise numerical computation of CMUTsis a quite complicated task, since one has to deal with several challengingproblems, which can be summarized as follows:

• Multifield-Problem: We have to deal with a multifield problem con-sisting of the electrostatic, mechanical and acoustic field including theircouplings (see Sect. 6 and Sec. 8).

• Geometric aspect-ratio:A typical electrostatic cell has the following dimensions:


Thickness of cell-membrane : 400 nm - 1 µmSide-length of cell-membrane : 20µm - 40µmAir-gap : 200 nm - 1µm

Assuming an air-gap of 200 nm and the wavelength λ in water at f =1 MHz ( λ = c/f = 1.5 mm), we achieve at an aspect-ratio of

wavelengthair-gap

=1.5 10−3

200 10−9= 7500 ,

which clearly results in a big problem concerning the computational gridgeneration. An enhanced solution approach for this problem will be theuse of nonmatching grids as described in Sect. 8.3.2.

• Non-linearities:According to the setup, different non-linearities of the involved physicalfields have to be considered. Furthermore, the coupling mechanisms be-tween the physical fields have to be taken into account, which are highlynon-linear in practice.

– Large deflections:If the deflections are in the range of the thickness of the structure,we have to consider the geometric non-linear case for the mechanicalcomputation (see example in Sect. 3.7.4).

– Stress-Stiffening effectThe fabrication process of CMUTs leads to a large prestressing withinthe structure, which results in a stiffer structure behavior. This effectis fully considered within the geometric non-linear case, see Sect. 3.7.3.

– Electric fieldThe electrostatic force leads to a deformation of the electrode-membranestructure, and therefore, introduces a kind of geometric non-linearityin the electrostatic field PDE. This means that we always have to com-pute the electrostatic field on the actual configuration by performingan updated Lagrangian formulation (see Sect. 6).

– Electrostatic forceIn a first case, let us consider a parallel plate capacitor, for which theelectrostatic force computes as follows

FCel =

ε0A

2U2

d2. (12.7)


In (12.7) U denotes the applied electric voltage,A the electrode surface,d the distance between the electrodes and ε0 the permittivity of the air.In a second case, we consider a parallel plate capacitor with insulationlayer having a relative permittivity of εr. The electrostatic force at theinterface between the insulation layer and air calculates by

F Iel =

ε0A

2U2

(d2 + d1/εr)2(εr−1) .

(12.8)

Therewith, the ratio of the two forces is

F Iel

FCel

≈ (d1 + d2)2

d22

εr ≈ εr

and we clearly see the importance of considering both parts of the elec-trostatic force, the force due to the electric charges on the electrodesas well as the interface force at changing permittivity. As describedin Sect. 6.1, we will apply the method of virtual work, which consid-ers the overall electrostatic force by evaluating (6.24) within an FEformulation.

In the following, we will analyze the dynamic behavior of such transducers,especially their problems concerning crosstalk.

12.5.2 Single CMUT Cell

The first numerical case study concentrates on a single CMUT cell. The com-putational domain consisting of the cell, the silicon wafer and the fluid domainas displayed in Fig. 12.33. The focus of the investigation in this section is thesnap-in curve of an individual cell and its sensitivity to the variation of thegeometric dimensions. Within a static analysis, we obtain a snap-in voltage of194 V (see Fig. 12.34) compared to a measured snap-in voltage of 192 V [37].However, the measurements of the geometric dimensions are not very accurateand in addition due to the fabrication process the dimensions will vary fromcell to cell. In order to obtain the above mentioned agreement between mea-sured and simulated snap-in voltage, we had to adapt the air-gap size as wellas the membrane radius in the range of about 8 %. The change of the snap-involtage as a function of the air-gap size as well as membrane radius is dis-played in Fig. 12.35 and Fig. 12.36. Later we will emphasize that the materialparameters will also have a strong influence on the computed results.

Applying a transient analysis to our FE model (now also taking into ac-count the fluid region) by exciting the CMUT cell by a short pulse u(t), wecan compute the electric input impedance as follows


Fig. 12.33. Setup of the CMUT cell

0 20 40 60 80 100 120 140180

160

140

120

100

80

60

40

20

0

Bias voltage (V)

Cen

ter

disp

lace

men

t ( n

m)

Fig. 12.34. Center displacement of membrane

Z(ω) =FFT(u(t))

j ω FFT(Q(t)). (12.9)

In (12.9) ω denotes the angular frequency, j the imaginary unit, FFT() thefast Fourier transformation and Q(t) the obtained total charge at the topelectrode. Figure 12.37 shows the amplitude of the electric input impedanceas a function of frequency. It can be clearly seen that the immersion of theCMUT cell into water strongly damps the resonance and antiresonance peaksand furthermore shifts them to lower frequencies due to the large mass loadingof the water as compared to air.


Fig. 12.35. Center displacement of membrane: variation of air-gap

Fig. 12.36. Center displacement of membrane: variation of membrane radius

12.5.3 CMUT Array

In a second numerical case study we consider a CMUT array consisting of 9cells. Due to symmetry, we just have modelled 4.5 cells (see Fig. 12.38). Eachindividual cell has the geometric setup as described in the previous section(see Fig. 12.33). The main focus of this investigation is the analysis of thecross-talk between the individual cells. Therefore, we just excite the centercell (cell 1) with a short sine-burst and compute the mechanical vibration atthe center of cells 2 to cell 5. With this result, we compute the cross-talk asfollows


Fig. 12.37. Input impedance in air and water

Fig. 12.38. Setup of the CMUT with nine cells

cross-talk = 20 log10

uirms

u1rms

i ∈ [2, 3, 4, 5] . (12.10)

In (12.10) urms denotes the root mean square of the mechanical displacementu(t). By neglecting the fluid domain, the occurring cross-talk must be purelymechanical, and as demonstrated in Table 12.8, the resulting values are quitesmall.However, if we additionally consider the fluid domain (CMUT array immersedin water) the cross-talk increases tremendously (see Table 12.9).


Table 12.8. Mechanical cross-talk

Cell 2 Cell 3 Cell 4 Cell 5

Level (dB) -60.1 -63.2 -66.5 -70.8

Table 12.9. Total cross-talk

Cell 2 Cell 3 Cell 4 Cell 5

Level (dB) -20.2 -22.1 -23.3 -23.8

Therefore, we can conclude that the dominant cross-talk between the individ-ual cells is due to the acoustic coupling.

12.5.4 Controlled CMUT Array

For the 3D simulation one of the four arrays (each consisting of 19 capacitivetransducer cells), as shown in Fig.12.30, was considered. The used finite ele-

Fig. 12.39. Detail of the finite element model; the membranes are marked by thenumbers 1–7

ment model consists of a quarter of one array (see Fig. 12.39). The thicknessof the membranes is 1µm and the gap between the electrodes has a size of500 nm. We apply a dc voltage of 10 V to the electrodes and for excitation asingle period of a sine burst with a frequency of 5 MHz and amplitude of 10 V.

To study the acoustic crosstalk between the individual membranes, onlythe center membrane is excited and the deformations of the neighboring mem-branes are computed. As an example of the simulation results the centerdisplacements of membranes 4 and 7 are shown in Fig. 12.40 together withthe dynamic response of the driving membrane 1. The observed significantcrosstalk between the individual elements leads to a reduction of the effectivemembrane deflections when all membranes are driven in parallel. This can be


Fig. 12.40. Center displacements of driving membrane 1 (solid line), membrane 4(dashed line) and membrane 7 (dotted line)

seen from Fig. 12.41, where the corresponding center displacements of mem-brane 1 are shown, and is in agreement with the results obtained for the 2Dsimulations.

Fig. 12.41. Center displacements of membrane 1: all seven membranes (solid line)and just membrane 1 excited (dashed line)


In the next step, the radiated pressure signal of the array is investigated.Of course, the long ring-down time of the membranes also shows up in thepressure signal. In the case that all membranes are driven in parallel, thisresults in a main, high-amplitude signal, which is followed by a secondarysignal of lower amplitude (Fig. 12.42).

Fig. 12.42. Pressure signal of the array: all membranes (solid line) and just mem-brane 1 excited (dashed line)

When only the center membrane is excited, however, this secondary sig-nal is of the same order in amplitude as the primary signal (Fig. 12.42). Asa consequence, the bandwidth is significantly reduced for the single drivenmembrane.

In order to decrease the ring-down time of the membranes, we have de-signed and applied controllers to each capacitive transducer of the array. Dueto the quadratic dependency of the electrostatic force on the deflection of themembranes, no standard PID controller can be used. Instead, a non-linear con-troller has been designed. In each outer iteration step k (see Sect. 6.2.1) thechange of the capacitance of each transducer is computed from the mechani-cal displacements un+1

k and used as the input of the controller. The controlleralgorithm then calculates the voltage for each transducer, which is a directinput value for the electric source vector f

u. In Fig. 12.43 the dynamic re-

sponse of the center membrane, when all membranes are driven in parallel, isdepicted. As can be seen, the controller strongly decreases the ring-down timeof the membrane. Furthermore, the use of the controller strongly decreasesthe acoustic crosstalk between the membrane elements.


Fig. 12.43. Center displacements of uncontrolled (solid line) and controlled (dashedline) membrane 1, when all membranes are excited

This can be seen in Fig. 12.44, where the center deflection of membrane 2 isshown, in the case that only membrane 1 is excited. Therefore, the secondary

Fig. 12.44. Center displacements of uncontrolled (solid line) and controlled (dashedline) membrane 2, when just membrane 1 is excited

signal in the acoustic pressure, as observed for the uncontrolled case, is nolonger present for the controlled membrane array. This is shown in Fig. 12.45for the case that all membranes are driven in parallel. As a consequence a


Fig. 12.45. Pressure signal of uncontrolled (solid line) and controlled (dashed line)array, when all membranes are excited

smoothing effect of the controller is also observed in the frequency spectrum(Fig. 12.46).

Fig. 12.46. Frequency spectrum of uncontrolled (solid line) and controlled (dashedline) array, when all membranes are excited

12.6 High-Intensity Focused Ultrasound 357

12.6 High-Intensity Focused Ultrasound

High-power ultrasound sources have found their way into a wide variety of ap-plications, ranging from medical ultrasound, like lithotripsy or HIFU-therapy(High-Intensity Focused Ultrasound), to ultrasonic cleaning or welding andsonochemistry [115]. In contrast to ultrasonic applications with sources whichradiate low amplitude pressure waves, the appearance of non-linear effects likesawtooth and shock formation is observed at high-power ultrasonic sound gen-erators. In order to speed up the design process of these high-power sourcesappropriate numerical simulation tools are required. These have to take intoaccount not only the electro-mechanical and the fluid–solid coupling, but alsomust be able to simulate the propagation of finite amplitude waves throughlossy and compressible media.

Previous investigations on the numerical simulation of non-linear wavepropagation problems are mainly based on the KZK (Khokhlov-Zabolotskaya-Kuznetsov), NPE (non-linear progressive wave equation), or the Burgers equa-tion, to mention only the most popular ones. Various methods for the solutionof the above equations can be found in the literature, ranging from spectral [1]to time domain algorithms [141], finite difference schemes (FD) and finite el-ement approaches (FEM) [219] as well as operator splitting methods [41].We will apply the computational scheme as described in Sect. 5.4.2, whichnumerically solves Kuznetsov’s equation by an enhanced FE formulation.

12.6.1 Piezoelectric Transducer and Input Impedance

The principle setup of the acoustic power transducer is shown in Fig. 12.47.Due to the geometric focusing of the lens, high acoustic intensity can beachieved in the focus region. The piezoelectric transducer has a diameter of60 mm and the radius of curvature of the lense is 55 mm, which results ina focal distance of 70 mm. The operating frequency of the transducer is at1.7 MHz.

As the starting point in our investigations impedance calculations havebeen performed for the piezoelectric disc transducer. This was mainly per-formed to establish all necessary and unknown material parameters. Thepiezoelectric disc has a thickness of 1.2 mm and a diameter of 60 mm. Thesimulated and the measured electric impedance of the piezoelectric transduceris displayed in Fig. 12.48. Due to good agreement over a wide frequency range,we can trust our used material parameters. Next, the impedance of the wholeHIFU source was simulated with water load. The finite element model usedin the impedance simulation of the complete HIFU source is shown in Fig.12.49. The computed electric impedance in water is displayed in Fig. 12.50.

12.6.2 Pressure Pulse Computation

In the numerical simulation of the HIFU antenna, the piezoelectric and thefluid–solid coupling as well as the non-linear wave propagation in the fluid


Fig. 12.47. Principle setup of high-power ultrasound source

Fig. 12.48. Impedance of the M453 piezo transducer

must be taken into account. Therefore, a complete finite element model hasbeen setup in which the piezoelectric transducer, the lens, the matching layerand the water has been discretized (see Fig. 12.51). The FE model consisted ofabout one million elements for both linear and non-linear acoustic simulations.Near the sound source 20 elements per wavelength have been used. To accountfor the generation of higher harmonics, the spatial discretization was increasedin propagation direction to 40 elements per wavelength in the focus region.Therewith, at least 8 elements per wavelength λ are guaranteed for the first


Fig. 12.49. FE model of the HIFU source used for the impedance calculations

Fig. 12.50. Electrical input impedance of the HIFU source loaded with water

Fig. 12.51. Finite element mesh (for display reasons, just a coarse mesh is shown)

4 harmonics. For the excitation of the piezoelectric transducer a sine burstat 1.7 MHz and varying amplitude, as shown in Fig. 12.52, has been used.The simulation results were observed at several points on the rotational axisbetween the source and the focus region. Transient analyses were performedwith 13500 time steps with a time step size of 4 ns.


0 0.5 1 1.5 2 2.5 3

60

40

20

0

20

40

60

U (

V)

t ( s)

Fig. 12.52. Excitation voltage for pressure measurements

46 47 48 49 50 51 52 530.5

0

0.5

t ( s)

p (M

Pa)

lin. simulationmeasurement

Fig. 12.53. Pressure pulse signal for low intensity measurement and linear simula-tion at the focal distance

First, we considered pressure pulses due to a low excitation voltage ofU = 9Vpp. As can be seen from Figs. 12.53 and 12.54, linear simulation resultsand measurements compare very well. For the high-intensity measurement, anexcitation voltage of U = 133Vpp has been used. It should be noted that alsofor the higher excitation voltage the piezoelectric transducer still behavedtotally linear. Therefore, any distortions in the pressure signal stem fromnon-linear effects in the fluid only. The comparison of measurements with


1 2 3 4 5 670

60

50

40

30

20

10

0

f (MHz)

A (

dB)

lin. simulationmeasurement

Fig. 12.54. Frequency spectra of pressure pulse for low intensity measurement andlinear simulation at the focal distance

non-linear simulation results is shown in Figs. 12.55 and 12.56. The non-linear distortion of the sine wave due to the generation of higher harmonicsis clearly visible. In the simulations a damping value of 0.22 dB/MHz2m hasbeen used.

12.6.3 High-Power Pulse Sources for Lithotripsy

In this section we will discuss the numerical computation of two high-powerpulse sources used for lithotripsy application: a piezoelectric driven pulsesource and an electromagnetic pulse source. In such applications we haveup to 80 MPa in the focus, and we need a quite fine grid within the acousticdomain in order to resolve the higher harmonics forming the high-pressurepulse.

We will start with the piezoelectric high power pulse source as shownin Fig. 12.57 (for a discussion on the physics we refer to [53]). In parallelconnected piezoceramic discs are placed on a spherical surface. The drivingvoltage is provided by decharging a capacitor. Due to geometrical focusing,high-power ultrasound is achieved in the focal region of the source. A simula-tion of the non-linear sound field was performed for the whole fluid domain.To account for the higher harmonics, we have chosen about 200 finite el-ements per fundamental wavelength, resulting in about 2.9 million bilinearquadrilateral elements for the axisymmetric setup. The sound pressure sig-nal at the surface of the source is found by measuring the radiated pressure


46 47 48 49 50 51 52 534

2

0

2

4

6

8

t ( s)

p (M

Pa)

nlin. simulation, B/A=5measurement

Fig. 12.55. Pressure pulse signal for high-intensity measurement and non-linearsimulation at the focal distance

signal near the surface of the source, and is used for the simulation (see Fig.

1 2 3 4 5 670

60

50

40

30

20

10

0

f (MHz)

A (

dB)

nlin. simulationmeasurement

Fig. 12.56. Frequency spectra of pressure pulse for high-intensity measurement andnon-linear simulation at the focal distance


Fig. 12.57. Piezoelectric high-power pulse source DL100.

12.58). Measured, linear and non-linear simulated pressure signals in the focus

Fig. 12.58. Sound pressure signal at the surface of the source used in the simulation.

region of the source are compared in Fig. 12.59. The maximum sound pres-sure in the measurement is 75.6 MPa, the maximum sound pressure in thenon-linear simulation is 69.7 MPa. In comparison to the non-linear result, thelinear simulation has been performed using the same simulation parameters.The maximum sound pressure here is 46.2 MPa. The sound pressure is there-fore raised by a factor of 33.7% due to the non-linear behavior of the soundwave in the fluid domain.


38 39 40 41 42 43 44 45 46 4720

0

20

40

60

80

t ( s)

p (M

Pa)

measurementnonlinear simulationlinear simulation

Fig. 12.59. Comparison between measured, non-linear- and linear-simulated soundpressure level in the focal region of the source

The second power source for lithotripsy is based on an electromagneticprinciple, and its schematic setup is displayed in Fig. 12.60. When the slab

Fig. 12.60. Schematic of an electromagnetic driven acoustic power source

coil is loaded by a capacitor discharge, eddy currents are induced in the metal-lic membrane. The interaction between these eddy currents and the overallmagnetic field results in a magnetic volume force (Lorentz force) acting on


the membrane. Therewith, the membrane-rubber structure is deformed andan acoustic pulse is radiated into the fluid and focused by the lens. For thenumerical simulation a finite element grid width of 90µm (corresponds toabout 70 finite elements per fundamental wavelength) was used for the acous-tic domain. Since in this case, we have to consider the non-linearities withinthe electromagnetic transducer, we perform the numerical simulation in twosteps:

1. Transducer Computation:Since the non-linearities of the acoustic field near the transducer can beneglected, we compute the acoustic pressure using the linear acoustic waveequation. Therewith, we fully take into account the fluid loading of thetransducer. For modelling the electromagnetic transducer we consider allrelevant non-linearities ( updated Lagrangian formulation for the mag-netic field, geometric non-linearity for the aluminum membrane and thenon-linear electromagnetic force term, see Sect. 7).

2. Non-linear Wave Propagation Computation:In a second run, we fully solve Kuznetsov’s non-linear wave equation us-ing the computed pressure near the transducer obtained from the firstsimulation step.

The measured and simulated pressure signals in the focus region of thesource are shown in Fig. 12.61. The dispersion at the beginning and the de-

Fig. 12.61. Comparison between measured and simulated sound pressure level inthe focal region of the electromagnetic pulse source

creasing slope of the simulated pressure pulse as compared to the measuredone indicates that the mesh size and the time step have to be further reducedfor a more precise computer simulation.


12.7 Noise Generated from a Flow around a SquareCylinder

In the following, we will demonstrate the applicability of the developed calcu-lation scheme for computational aeroacoustics as described in Sect. 10. There-with, we will investigate in the noise generated by a flow around a square cylin-der as displayed in Fig. 12.62. We denote with ΩF the fluid domain, where wewill evaluate the acoustic sources on the fine flow grid and transform them tothe coarser acoustic grid.

In order to analyze our scheme especially concerning the transfer of theacoustic sources on the much finer flow grid to the coarser acoustic grid, wewill start with 2D transient as well as harmonic acoustic computations. In thesecond part we will also perform full 3D computation of the acoustic field. Inboth cases, the acoustic computations are based on 3D large eddy simulations(LES) of the flow.

Fig. 12.62. Principle setup for the computation of the flow around a square cylinder

12.7.1 3D Flow and 2D Acoustic Computations

As explained, in the first step we will perform a 2D acoustic computation ata cross section and investigate in the different simulation parameters. Fig-ure 12.63 represents the configuration chosen for the simulations, where ΩF

denotes the 3D domain for the flow computation and ΩAc defines the crosssection at z = 2D (D denotes the side length of the cylinder) for the 2Dacoustic simulation. The square cylinder has a side-length D of 20 mm and aheight of 4D. This cylinder is positioned in a box with Lx = 40D, Ly = 11Dand Lz = 4D as displayed in Fig. 12.63 for the flow computation (correspondsto ΩF).

12.7 Noise Generated from a Flow around a Square Cylinder 367

Fig. 12.63. Principle setup for the aeroacoustic computation

Flow Field computation

In the present case study, a large eddy simulation based on Smagorinsky modelwas performed with the code FASTEST-3D [56] to resolve the flow field. Sincein this case the height of the square cylinder is equal to the height of the overallflow domain, we achieve a turbulent flow, which do not vary strongly in z-direction. The numerical domain for the problem is displayed in Fig. 12.63and the boundary conditions are listed in Table 12.10. Spatial discretizationof the numerical grid is Lx×Ly×Lz = 192×96×128, with a stretching factorof 1.05 downstream from the cylinder, 1.35 upstream from the cylinder and1.18 in span wise direction from the cylinder. This results in a total numberof 2.3 million control volumes. For the case, where inlet velocity ux = 10m/s,D = 20mm and fluid air at 25 oC we achieve a Reynolds number around13000. Second-order spatial discretization was used with Crank-Nicolson timestepping scheme and a step size of tf = 10µs. For the Smagorinsky constantCs we choose a value of 0.065.

Table 12.10. Boundary condition for square cylinder simulation

Position Boundary Condition

x = 0 inlet, ux = 10 m/sx = 40D convective exity = 0, y = 11D symmetryz = 0, z = 4D periodiccylinder no slip

Investigation of the interpolated acoustic nodal sources

Initially, before computing the radiated sound field, several characteristicpoints within the turbulent flow region have been chosen to investigate theproperties of the interpolated acoustic sources. Herewith, we have analyzedthe interpolated acoustic nodal sources at locations depicted in Fig. 12.64.


Fig. 12.64. Schematic representation of source region depicting the square cylinderwith selected points (distance scale in millimeters)

Initially, analyzes are carried out at two different characteristic points,p5 = (x, y) = (0.1 m , 0.01 m) and p6 = (x, y) = (0.1 m , 0.0 m), located at5D in the downstream direction along the fringe of the cylinder and alongthe x-axis, respectively. Figure 12.65 presents the results for these two points.Their characteristics confirm the measurements from a turbulent flow arounda square cylinder [5]. For the point located along the fringe, among other fre-quency modes, we find the 65Hz component, which is the actual main compo-nent for this tonal problem, observed in the vortical structures in the fluid com-putation. Therewith, we have a fundamental wavelength of 343/65 ≈ 5.3 m.This aspect has also been mentioned in the 2D investigation presented in [149],concerning the sound generation in a laminar flow past an elongated squarecylinder. However, for the point located exactly along the x-axis we find, asmain frequency component, a value which is twice higher than the main fre-quency of 65Hz. This fact can be associated with the combination at thecenter line in the downstream flow of the upper and lower main vortices, eachhaving a frequency of 65Hz. Significant differences in the source values arefound at a point p7 = (x, y) = (0.2 m , 0.06 m) located outside the centralregion of the wake, as presented in Fig. 12.66. In this case several other fre-quency components are found, but still the first two dominant componentsare present.

2D Acoustic Field Computations

In the following, transient and harmonic investigations concerning the flow-induced noise propagation are presented. The acoustic field is computed inthe cross section ΩAc located at z = 2D (see Fig. 12.63).

In the first step, we will perform a transient analysis of the flow inducedacoustic field. Figure 12.67, depicts two different spatial discretizations ofthe source region (corresponds to the total computational flow region) of theacoustic domain from which results are evaluated. The mapped mesh has an


0.06 0.07 0.08 0.09 0.1 0.11 0.128

7

6

5

4

3

2

1

0

1

2

t (s)

RH

S (

kg /

s2 )

along fringes of cylinderin the center of wake

(a) Time domain

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f (Hz)

norm

aliz

ed s

ourc

e va

lues

along the fringes of cylinderIn center of wake

(b) Frequency spectrum

Fig. 12.65. Acoustic source values as a function of time as well as frequency atpoints p5 and p6 located at 5×D in the downstream direction along the fringe ofthe cylinder and along the x-axis, respectively

element size h = 10 mm. For the second mesh, we set the element size nearthe cylinder equal to the ones used in the flow computation (h = 1 mm) andcoarse it until reaching an element size of h = 10 mm (see Fig. 12.67 (b)).

For the transient case, the radius of the complete acoustic domain hasbeen chosen to be r = 40m. Hereby, it is possible to investigate the acoustic


(a) Time domain

(b) Frequency spectrum

Fig. 12.66. Acoustic source values in time and frequency domain at point p7 =(x, y) = (0.2 m , 0.06 m) located outside the central region of the wake

solution during several periods, without any influence from spurious reflectionsof waves impinging not orthogonally on the acoustic boundary (we apply firstorder absorbing boundary conditions). The discretization of the farthermostelements in the acoustic region ΩAc , located near the absorbing boundary,corresponds to about 7 elements per fundamental acoustic wave length, whichis for the main frequency component of 65 Hz about 5.3 m. The total number


(a) Mapped mesh

(b) Progressively coarsened mesh

Fig. 12.67. Two different acoustic coupled regions used for the evaluation in tran-sient computations

of second-order quadrilateral elements in the cross sectional domain ΩAc is41.584. The time step size for the acoustic computation is chosen to be ta =100µs, which corresponds to about 12 time samples for the largest frequencyoccurring in the source region (f ≈ 400 Hz).

Figure 12.68 presents the results at a point located 5 m away from thecylinder using the two different discretizations in the source region (see Fig.12.67). Simulations performed with the mapped mesh in the source region,produce a slight better quality in the solution when compared with the resultsobtained using the progressively coarsened mesh.

Contour plots of the acoustic pressure in the near and the far field arepresented in Fig. 12.69. In the near field a strong radiation in the upstream andin the downstream is observed. Here it is important to mention that althoughthis does not seem to correspond to the expected dipole radiation in thecross-flow direction, it is from the reciprocal oscillation of these two radiationpatterns that the far field dipole characteristic originates. The contour far fieldpressure from Fig. 12.69b) at time t = 140ms helps to understand this soundgeneration mechanism. Near the square cylinder the acoustic field shows thestrong radiation in the upstream and downstream direction whereas in the farfield the expected dipole radiation for this problem dominates.

In the second step, we perform a harmonic analysis using a computationaldomain as depicted in Fig. 12.70. The dimensions of the complete acousticdomain without considering the PML region are Lx ×Ly = 4.4m×3.3m withthe cylinder located at (x, y) = (2 m, 0 m). The PML region and its discretiza-tion is also shown in Fig. 12.70, to emphasize that a small and coarse meshwith 612 elements suffices for this region to obtain accurate results. For directcomparison with the transient results, an initial harmonic computation hasbeen performed using a numerical domain with the same discretization in the


(a) Using progressively coarsened mesh

(b) Using mapped mesh

Fig. 12.68. Acoustic pressure values in time for a point (x, y) = (0m, 5m) located5m away along the y-axis

source region as in the transient case (see Fig. 12.67 (a)). The computationalmesh exhibits a total number of 8.488 second-order quadrilateral finite ele-ments, including the PML region. Figure 12.71 presents the amplitude andphase values of the acoustic pressure field in the whole acoustic domain, com-puted for the main frequency component, f = 65Hz. Similar to the transientresults, we find the expected dipole sound radiation in the acoustic field. Di-


(a) Acoustic near-field around the square cylinder (nor-malized pressure).

(b) Acoustic pressure distribution at time t = 140 ms (nor-malized). Distance scale in m.

Fig. 12.69. Acoustic pressure field obtained from transient analysis

rectivity patterns obtained from the transient and harmonic analyses at aradius r = 1m away from the cylinder are directly compared in Fig. 12.72. Inthis comparison, only small numerical differences are noticeable, which can bedue to the evaluation of the amplitudes from the transient pressure signals.Moreover, these results have demonstrated that the excellent performance ofthe PML allows to use a small numerical domain, which in all directions en-


compasses just a fraction of the acoustic wave length for the main frequencycomponent (about λ/3, since for f = 65Hz, we obtain for the wavelengthλ ≈ 5.3m).

Fig. 12.70. Schematic of numerical domain used in harmonic computations. Dis-tance scale in meters

Fig. 12.71. Contour visualization of harmonic results for frequency f = 65Hz


Fig. 12.72. Comparison of directivity patterns at radius r = 1.0 m between har-monic and transient computations

In the following, additional harmonic acoustic computations are performedusing several spatial discretizations to evaluate the robustness of the conser-vative interpolation scheme and its influence on the results. Hereby, we inter-polate the computed acoustic nodal sources from the fine flow grid containing91.492 cell points, to four different acoustic meshes with a total number ofnodes in the source region ranging from 10.307 to 590. As previously men-tioned, all computations are performed using second-order quadrilateral ele-ments ( about 8 finite elements per element). Figure 12.73 presents the fourdifferent discretization of the acoustic source region used for the investigation,where Fig. 12.73 (b) corresponds to the mesh previously used for comparisonwith the transient results. The finest numerical grid corresponds to a ratio be-tween the acoustic and fluid discretization sizes, ha/hf ≈ 5. Figure 12.73 (d)is a extremely coarse mesh containing 590 nodes, which corresponds to a ratioha/hf ≈ 20 in the region directly around the cylinder and ha/hf ≈ 60 at theoutermost regions. The aim for using this latter numerical grid is mainly toestimate the limits of the interpolation for achieving acceptable results. Fig-ure 12.73 also presents the discretization ratios for the different meshes withrespect to the wave length λmin obtained for the 400Hz component found inthe acoustic nodal sources (see Fig. 12.66).

Figure 12.74 presents the results from the flow-induced noise computationsfor the four numerical grids. Except for the coarsest computation, where am-plitudes are significantly smaller, all other results are in very good agreementwith each other. Even for the coarse discretization from Fig. 12.73 (c), whereha/hf is around ten near the cylinder and 50 at outer regions, results still showvery similar amplitudes in comparison to the results from the finest numeri-


(a) 10307 nodes, λmin/h ≈ 187 (b) 5480 nodes, λmin/h ≈ 84

(c) 1616 nodes, 22 < λmin/h < 84 (d) 590 nodes, 20 < λmin/h < 40

Fig. 12.73. Different spatial discretizations of acoustic coupled region as used forsensitivity analysis

cal grid. For this flow-induced noise computation, this fact demonstrates thatthe conservative interpolation scheme makes the implementation a robust andefficient approach, producing good results even for very coarse acoustic reso-lutions in comparison to the CFD discretization, which significantly reducesthe computational cost for the acoustic simulation.

Fig. 12.74. Directivity patterns from flow-induced noise harmonic computationusing different grids. Amplitudes 1m away from cylinder


12.7.2 3D Flow and 3D Acoustic Computations

In this section, we will present the noise generated from the flow around awall mounted square Cylinder.

Flow Field Computation

The flow field computations are performed with the commercial CFD codeANSYS-CFX. In this case, the height of the square cylinder is as in the previouscomputation equal to 4D, but now the height of the overall flow computationaldomain is extended to 11D. The same inlet velocity profile with ux = 10m/sis applied, which results in a Reynolds number Re of about 13000. We haveapplied for the modelling of the turbulence the SAS (Scale Adaptive Simu-lation) model [156]. This method allows coarser numerical grids than thoseused in LES computations, causing a shorter computation time. The bound-ary conditions used in the simulation are presented in Table 12.11. The spatialdiscretization of the computational flow domain results in about 1.1 millioncells, and the time step size has been set to 10 µs.

Figure 12.75 displays results for the velocity, pressure and eddy viscosityfields for the flow around the wall mounted cylinder.

Table 12.11. Boundary conditions used for fluid computation

Position Boundary Condition

x = 0 block inlet profile, 10 m/sx = 40D convective exit boundaryy = 0, y = 11D symmetry boundary conditionz = 11D symmetry boundary conditionwall no slip boundary condition

Acoustic Field Computation

The acoustic field has just been computed for the main frequency compo-nent within a time-harmonic analysis. In the source region, the acoustic meshconsists of about 10000 3D finite elements and the overall acoustic mesh in-cluding the PML layer has 200000 3D finite elements. Figure 12.76 presentsthe configuration of the simulation domain showing the monitoring points and12.77 the iso-surfaces of the acoustic pressure for f = 58Hz, which was themain frequency component found in the simulation. The outermost iso-surfacecorresponds to 44dB. This result shows the typical dipole characteristic withopposite phases expected for this tonal noise problem.

Further directivity analysis has been done on the principal planes at severalradii from the square cylinder. Figure 12.77 a) depicts the position of the


Fig. 12.75. Transient flow field around wall-mounted cylinder

Fig. 12.76. Schematic drawing of the acoustic domain used for the harmonic com-putation showing points used for directivity analysis. Distance scale in meters

monitoring points at radius r = 1.0m. Directivity plots from these planesat two different distances are shown in Fig. 12.78. The directivity patternsat the bottom XZ-plane as well as those on the cross-flow YZ-plane showthe dipole characteristic of the problem. From the cross-flow plots it can be


Fig. 12.77. Iso-surface of acoustic pressure at frequency f = 58Hz normalized withphase and clipped through YZ-plane. Outer iso-surface corresponds to p′ = 3mPa.PML region is shown dotted

Fig. 12.78. Directivity patterns at radii r = 1.0 m and r = 0.75 m on the threeprincipal planes. Left : XZ-plane (bottom plane). Right : YZ and XY planes (cross-flow and stream-wise planes)

observed that the higher amplitudes (48 dB at r = 0.75m) are located atthe opposite points near the bottom plane and the lower one right above thecylinder (11 dB). On the other hand, the directivity plot on the stream-wiseplane presents much lower sound pressure levels in comparison to the other


two plots. For the bottom XZ-plane plot at r = 0.75m a strong influence ofthe turbulent field is observed in the downstream direction, where the acousticfield is not yet homogeneous.

A

Norms

A.1 Vector Norms

Definition A.1. Vector norm: A vector norm on IRn is a function || || :IRn → IR, which fulfills the following properties:

(i) ||x|| ≥ 0 for all x ∈ IRn

(ii) ||x|| = 0 iff x = 0(iii) ||x + y|| ≤ ||x|| + ||y||(iv) ||αx|| = |α| ||x||

We use the Holder or p-norms, which are defined by

||x||p =

(n∑

i=1

|xi|p)1/p

with p ≥ 1 . (A.1)

Therefore, we compute, e.g., the 1-norm, 2-norm, and ∞-norm as follows

||x||1 =n∑

i=1

|xi| (A.2)

||x||2 =

(n∑

i=1

|xi|2)1/2

(A.3)

||x||∞ = maxi|xi| . (A.4)

The p-norms have the following useful properties

• |xTy| ≤ ||x||p||y||q, 1p + 1

q = 1 (Holder inequality)• |xTy| ≤ ||x||2 ||y||2 (Cauchy–Schwarz inequality)• ||x||2 ≤ ||x||1 ≤

√n||x||2

382 A Norms

• ||x||∞ ≤ ||x||2 ≤√n||x||∞

• ||x||∞ ≤ ||x||1 ≤ n||x||∞With the help of norms we can define a distance on a vector space, and

furthermore, we call a vector space with a norm a normed space.

A.2 Matrix Norms

Definition A.2. Matrix norm: A matrix norm on IRn×m is a function || || :IRn×m → IR, which fulfills the following properties:

(i) ||A|| ≥ 0 for all A ∈ IRn×m

(ii) ||A|| = 0 iff A = 0(iii) ||A + B|| ≤ ||A|| + ||B|| for all A ,B ∈ IRn×m

(iv) ||αA|| = |α| ||A|| for all α ∈ IR and A ∈ IRn×m

The matrix norm associated to the vector p-norm is defined by the operatornorm

||A||p = supx =0

||Ax||p||x||p

. (A.5)

Other matrix norms are

||A||F =

⎛⎝ n∑

i=1

m∑j=1

|aij |2⎞⎠1/2

Frobenius or F-norm (A.6)

||A||1 = maxj

n∑i=1

|aij | column sum norm (A.7)

||A||∞ = maxi

m∑j=1

|aij | row sum norm . (A.8)

B

Scalar and Vector Fields

Definition B.1. (Scalar field) If we assign to each point in IR3 defined bythe vector r a scalar quantity V (r) (e.g., electric potential, temperature, acous-tic velocity potential), then V is called a scalar field.

For the illustration of scalar fields we use iso-lines in the 2D case and iso-surfaces in the 3D case, where the scalar quantity V (r) is constant (Fig. B.1).

Fig. B.1. Illustration of a scalar field V with the help of equipotential surfaces

Definition B.2. (Vector field) If we assign to each point IR3 defined by thevector r a vector quantity F(r) (e.g., electric field, magnetic field, mechanicaldeformation), then F is called a vector field.

384 B Scalar and Vector Fields

Vector fields are divided into irrotational vector fields (e.g., electrostatic field)and solenoidal vector fields (e.g., magnetic field) as shown in Fig. B.2 (see alsoSects. B.12 and B.13).

Fig. B.2. (a) Solenoidal vector field; (b) Irrotational vector field

Fig. B.3. Lines of force for the vector field F(r)

The lines of force (see Fig. B.3) are defined by

F(r) × dr = 0 , (B.1)

which means that in each point of the lines the field vector F is parallel tothe tangential vector.

In the following, we try to compute the lines of force for the vector field

F(r) =rr3

(B.2)

with the help of (B.1). Since we are just interested in the direction, we haveto solve

r × dr = 0 . (B.3)

By using a Cartesian coordinate system, we obtain

B Scalar and Vector Fields 385

r = xex + yey + zez (B.4)dr = dxex + dyey + dzez , (B.5)

(B.6)

and

r × dr =

∣∣∣∣∣∣ex ey ez

x y zdx dy dz

∣∣∣∣∣∣ =

⎛⎝ y dz − z dyz dx− xdzxdy − y dx

⎞⎠ . (B.7)

Therefore, we can formulate the following three relations

y dz = z dy (B.8)z dx = xdz (B.9)xdy = y dx . (B.10)

We now search for the line of force including point P0(x0, y0, z0). Integrationof (B.8) results in

lny

y0= ln

z

z0(B.11)

andz0y = y0z . (B.12)

Analogously, we can compute the solutions of the other two differential equa-tions

z0x = x0z (B.13)y0x = x0y . (B.14)

From (B.13) a plane through point P0 containing the y-axis, and from (B.14)a plane through point P0 containing the z-axis is defined. The intersectionof the two planes leads to a straight line through the origin, and thereforewe obtain the vector field drawn in Fig. B.4, which corresponds, e.g., to thevector field of an electric charge.

Fig. B.4. Lines of force of the vector field F(r) = r/r3


B.1 The Nabla (∇) Operator

First, we recall that a scalar function may depend on one or more variables,e.g., using Cartesian coordinates, a function can be denoted by

f = f(x, y, z) .

The partial derivatives read as

∂f

∂x,∂f

∂y,∂f

∂z.

The nabla operator ∇ is defined in Cartesian coordinates by

∇ =∂

∂xex +

∂

∂yey +

∂

∂zez =

⎛⎜⎜⎝

∂∂x

∂∂y

∂∂z

⎞⎟⎟⎠ , (B.15)

where ex, ey and ez are the unit vectors in x-, y-, and z-directions. Theinteraction between the nabla operator and a scalar or a vector field yields itsgeometric significance.

B.2 Definition of Gradient, Divergence, and Curl

We introduce a scalar function V with nonzero first-order partial derivativeswith respect to the coordinates x, y, and z, and a vector field F with compo-nents Fx, Fy, and Fz . Then, the following operations are defined:

1. Gradient of a scalar:

grad V = ∇V =

⎛⎜⎜⎝

∂V∂x

∂V∂y

∂V∂z

⎞⎟⎟⎠ .

As can be seen, the result of this operation is a vector.

2. Divergence of a vector:

divF = ∇ · F =∂Fx

∂x+∂Fy

∂y+∂Fz

∂z.

Therefore, the result of this operation is a scalar value.

B.3 The Gradient 387

3. Curl of a vector:

curl F = ∇× F =

∣∣∣∣∣∣∣∣ex ey ez

∂/∂x ∂/∂y ∂/∂z

Fx Fy Fz

∣∣∣∣∣∣∣∣ =

⎛⎜⎜⎝

∂Fz

∂y − ∂Fy

∂z

∂Fx

∂z − ∂Fz

∂x

∂Fy

∂x − ∂Fx

∂y

⎞⎟⎟⎠ .

The result of taking the curl of a vector is again a vector.

B.3 The Gradient

We will consider the scalar function V (x, y, z) with its partial derivatives∂V/∂x, ∂V/∂y, ∂V/∂z and dependent on a point P = (x, y, z). In the firststep we calculate the total differential of V

dV =∂V

∂xdx+

∂V

∂ydy +

∂V

∂zdz . (B.16)

Now, we define a point P′ infinitely close to P by P′ = (x+ dx, y+ dy, z+dz).By calculating the vector dP = P′ − P, which has the components dP =(dx, dy, dz)T , we can write (B.16) as

dV =(∂V

∂xex +

∂V

∂yey +

∂V

∂zez

)· ( dxex + dyey + dzez) (B.17)

= ∇V · dP . (B.18)

For the geometrical illustration of the gradient, consider an equipotential sur-face, i.e., a surface with V = const. (see Fig. B.5). Hence, for all differential

Fig. B.5. The gradient is orthogonal to a constant potential surface

displacements from P to P′ on this surface dV = 0 holds, and therefore,

∇V · dP = 0 . (B.19)


From the definition of the scalar product it is clear that ∇V and dP areorthogonal. In this situation the displacement from P to P′ points into thedirection of increasing V , as shown in Fig. B.6, and the scalar product ∇V · dPis positive.

Fig. B.6. Geometrical representation of the gradient

From the foregoing arguments, we conclude that ∇V is a vector, perpen-dicular to the surface on which V is constant and that it points in the directionof increasing V .

As an example we consider a function r(x, y, z), which defines the distanceof a point P from the origin (0, 0, 0). The surface r = const. is a sphere ofradius r with center (0, 0, 0), whose equation is given by

r =√x2 + y2 + z2 .

Therefore, the gradient calculates as

∂r

∂x=

x√x2 + y2 + z2

=x

r

∂r

∂y=y

r

∂r

∂z=z

r

∇r =xex + yey + zez

r=

rr.

Geometrically speaking, ∇r points in the direction of increasing r, or towardsspheres with radii larger than r.

B.4 The Flux 389

B.4 The Flux

Definition B.3. (Flux) The vector field F(r) and a corresponding surface Γas shown in Fig. B.7 are given. The vector n denotes the normal unit vectorof the differential surface dΓ. Therefore, the differential flux dψ through dΓis defined by

dψ = F · dΓ = F · n dΓ . (B.20)

The total flux ψ computes as

ψ =∫Γ

F · dΓ . (B.21)

In the following, we want to compute the flux ψ of the vector field F(r) = r

Fig. B.7. Flux through the surface Γ

through the square Γ with side length h according to Fig. B.8. With thenormal unit vector n = ex and dΓ = dy dzex we obtain

ψ =

h∫0

h∫0

(−hex + yey + zez) · ex dy dz

= −hh∫

0

h∫0

dydz

= −h3 . (B.22)

The total flux ψ through a closed surface S is given by


Fig. B.8. Flux ψ through the square with area h2

ψ =∮

Γ

F · dΓ (B.23)

and defines whether we have sources (ψ > 0) or sinks (ψ < 0) within Γ .A very important property of the flux ψ defined by a closed surface is

given by (see Fig. B.9)∮Γ1∪Γ0

F · dΓ +∮

Γ2∪Γ0

F · dΓ =∮

Γ1∪Γ2

F · dΓ . (B.24)

Fig. B.9. Flux through the closed surface Γ1 ∪ Γ2

B.5 Divergence

Definition B.4. (Divergence) The vector field F(r) is given. If we dividethe flux ψ, defined by a closed surface Γ , by the corresponding volume Ω andlet the volume Ω tend to zero, then the obtained value is called the divergence(source density)

B.5 Divergence 391

divF = limΩ→0

1Ω

∮Γ

F · dΓ =dψdΩ

∣∣∣∣r

. (B.25)

Let us now consider the closed surface of a differential cube (see Fig. B.10)and the general vector field F(r) = Fxex + Fyey + Fzez. In the first step, let

Fig. B.10. Flux through a cube

us compute the differential flux through the hatched surfaces

F · dΓ = [F(x+ dx/2, y, z)− F(x− dx/2, y, z)] · ex dy dz

≈[Fx(x, y, z) +

∂Fx

∂x

dx2

−(Fx(x, y, z) − ∂Fx

∂x

dx2

)]dy dz

=∂Fx

∂xdxdy dz . (B.26)

Analogously, we obtain the contribution of the other two directions, and thus,the differential flux

dψ =(∂Fx

∂x+∂Fy

∂y+∂Fz

∂z

)dxdy dz . (B.27)

Since the differential volume dΩ is equal to dx dy dz, we end up with thefollowing expression for the divergence of a vector field in Cartesian coordi-nates

divF =∂Fx

∂x+∂Fy

∂y+∂Fz

∂z, (B.28)

or, by using the nabla operator,

divF = ∇ ·F . (B.29)


B.6 Divergence Theorem (Gauss Theorem)

By the definition of the divergence (see B.25) we get

dψ = ∇ · F dΩ (B.30)

ψ =∫

Ω

∇ · F dΩ . (B.31)

On the other hand, we have the relation for the flux ψ according to (B.23).Combining these two expressions for the flux results in

ψ =∫

Ω

∇ ·F dΩ =∮

Γ (Ω)

F · dΓ . (B.32)

This equality between the two integrals tells us that the flux of the vector Fthrough the closed surface Γ is equal to the volume integral of the divergenceof F over the volume Ω enclosed by the surface Γ .

Fig. B.11. Radial vector field

Consider a radial vector field F as shown in Fig. B.11, and assume thatthe magnitude of F is constant in all points on a sphere centered at P . Tocompute the flux of the vector field F through a spherical shell of radius R,we note that ds and F are colinear and in the same direction

ψ =∮

S

F · dΓ = F

∮Γ

dΓ = 4πR2F .

From the divergence theorem, (the flux is nonzero) we conclude

∇ · F = 0 .

B.7 The Circulation

The circulation of a vector field F(r) along a closed contour C is given by theclosed-line integral

B.8 The Curl 393

Z =∮

C

F · ds . (B.33)

Therefore, the important property follows (see Fig. B.12)∮C1∪C0

F · dr +∮

C2∪C0

F · dr =∮

C1∪C2

F · dr . (B.34)

If the circulation along a closed curve C is not equal to zero, then we say the

Fig. B.12. Circulation along the closed line C1 ∪ C2

closed line contains eddies.

B.8 The Curl

Definition B.5. (Curl) We consider a point defined by r (Fig. B.13), inwhich the curl of the vector field F has to be computed. Furthermore, wedefine a closed line C enclosing the area Γ and consider the circulation alongC. If the area Γ tends to zero, we obtain the definition of the curl by

n · curl F = limΓ→0

∮C

F · dsΓ

=dZdΓ

. (B.35)

The vector curlF is obtained by a separation in the three directions of theunit vectors ex, ey, and ez

curl F = (ex · curl F)ex + (ey · curl F)ey + (ez · curl F)ez . (B.36)

The circulation for the differential square in Fig. B.14 is given by

dZx = ( F(x, y, z − dz/2)− F(x, y, z + dz/2) ) · ey dy+ ( F(x, y + dy/2, z)− F(x, y − dy/2, z) ) · ez dz

≈(∂Fz

∂y− ∂Fy

∂z

)dydz . (B.37)


Fig. B.13. Curl in a point defined by r

Therefore, we obtain the x-component of curlF with dΓ = dy dz

ex · curl F =∂Fz

∂y− ∂Fy

∂z. (B.38)

Analogously, the y- and z-component of curlF can be computed, and the full

Fig. B.14. x-component of curlF

vector in Cartesian coordinates reads as

curl F =

⎛⎜⎜⎝

∂Fz

∂y − ∂Fy

∂z

∂Fx

∂z − ∂Fz

∂x

∂Fy

∂x − ∂Fx

∂y

⎞⎟⎟⎠ , (B.39)

B.9 Stoke’s Theorem 395

or with the help of the nabla operator

curl F = ∇ × F =

∣∣∣∣∣∣∣∣ex ey ez

∂∂x

∂∂y

∂∂z

Fx Fy Fz

∣∣∣∣∣∣∣∣ . (B.40)

B.9 Stoke’s Theorem

We consider the vector field F on the surface Γ with fixed oriented contourC as shown in Fig. B.15. For a differential surface dΓν , we obtain according

Fig. B.15. Vector field F on the surface Γ with fixed oriented contour C

to (B.35)

dZν = n(rν) · curl F(r) dΓν

Zν =∫Γν

curl F(r) · dΓν , (B.41)

andZ =

∫Γ

curl F · dΓ . (B.42)

Furthermore, according to the definition of the circulation Z (see B.33),we get the following relation

Z =∮

C

F · dr =∫

Γ

curl F · dΓ . (B.43)


For a radial vector field F as shown in Fig. B.11, the closed-line integralalong a circle C of constant radius∮

C

F · ds

is zero, and therefore, the curl of this vector field ∇ × F is zero, too.

B.10 Green’s Integral Theorems

The integral theorems of Green can be derived from the divergence theorem.For this purpose, we first introduce the Laplace operator by

∆ = ∇ · ∇ =∂2

∂x2+

∂2

∂y2+

∂2

∂z2. (B.44)

This differential operator can be applied to scalar as well as vector quantities

∆V = div (grad V ) (B.45)∆F = (∆Fx)ex + (∆Fy)ey + (∆Fz)ez . (B.46)

Setting a vector F equal to V1∇V2 and using the divergence theorem, weobtain according to (B.32)∫

Ω

∇ · (V1∇V2) dΩ =∮

Γ

(V1∇V2) · dΓ . (B.47)

Since the term ∇ · (V1∇V2) can be expressed by (see B.54 below)

∇ · (V1∇V2) = V1∆V2 + ∇V1 · ∇V2 , (B.48)

we get the following integral theorem, called Green’s first integral theorem∫Ω

V1∆V2 dΩ +∫

Ω

∇V1 · ∇V2 dΩ =∮

Γ

V1∂V2

∂ndΓ . (B.49)

By substituting V1 with V2 and vice versa in (B.47) and subtracting the re-sulting equation from (B.47), we achieve Green’s second integral theorem∫

Ω

V1∆V2 dΩ −∫

V

V2∆V1 dΩ =∮

Γ

(V1∂V2

∂n− V2

∂V1

∂n

)dΓ . (B.50)

In addition, Green’s first integral theorem in vector form is∫Ω

(∇ × u · ∇ × v − u · ∇ × ∇ × v) dΩ

=∫Γ

(u × ∇ × v) · n dΓ , (B.51)

B.12 Irrotational Vector Fields 397

and Green’s second integral theorem in vector form reads as∫Ω

(u · ∇ × ∇ × v − v · ∇ × ∇ × u) dΩ

=∫Γ

(v × ∇ × u − u × ∇ × v) · n dΓ . (B.52)

B.11 Application of the Operators

By using the definitions of gradient, divergence, and curl in Cartesian coordi-nates, the following relations hold:

∇(V1V2) = V1∇V2 + V2∇V1 (B.53)∇ · (V F) = V∇ ·F + F · ∇V (B.54)

∇ · (F1 × F2) = F2 · ∇ × F1 − F1 · ∇ × F2 (B.55)∇ × (V F) = V∇ × F− F × ∇V (B.56)

∆F = ∇(∇ ·F) − ∇ × (∇ × F) . (B.57)

These relations combine the essential differential operators and build up abasis for the description of physical fields.

B.12 Irrotational Vector Fields

We consider a vector field F, which is given as the gradient of a scalar potentialF = ∇V . The computation of a line integral from point A to point B yields

B∫A

(∇V ) · dr = V (B) − V (A) . (B.58)

Therefore, for any closed contour within this vector field, the following relationholds ∮

(∇V ) · dr = 0 . (B.59)

This result proves that any vector field that can be expressed by the gradientof a scalar potential is irrotational. Furthermore, the local quantity, given bythe curl of the vector field, is zero

∇ × ∇V = 0 . (B.60)


Fig. B.16. Domain for solenoidal vector field

B.13 Solenoidal Vector Fields

We will consider the solenoidal vector field ∇ × F for a domain as displayedin Fig. B.16. This domain shall consist of two subdomains defined by theirsurfaces Γ1 and Γ2 with their related contours C1 and C2. By using Stoke’stheorem, we obtain the following relation∮

Γ

(∇ × F) · dΓ =∮

Γ1

(∇ × F) · n1 dΓ +∮

Γ2

(∇ × F) · n2 dΓ

=∮

C1

F · dr +∮

C2

F · dr

= 0 . (B.61)

Thus, the total flux (global quantity) is zero, and, furthermore, the localsolenoidality, too

∇ · (∇ × F) = 0 . (B.62)

C

Appropriate Function Spaces

Let us define the derivative of order α with respect to the multi-index α, with|α| =

∑i αi and αi ∈ IN, as follows

Dαv :=∂|α|v

∂xα11 · · · xαn

n. (C.1)

For example, the partial derivatives of order 2 in IR2 can be written as Dαvwith α = (2, 0), α = (1, 1) or α = (0, 2), since |α| = α1 +α2 = 2 is fulfilled forall three cases

α = (2, 0) Dαv =∂2v

∂x21

α = (1, 1) Dαv =∂2v

∂x1∂x2

α = (0, 2) Dαv =∂2v

∂x22

.

Definition C.1. Continuously differentiable functions: Let Ω be a closeddomain in IRn and let C(Ω) denote the space of continuous functions on Ω.Now, the space of up to order m continuously differentiable functions is givenby

Cm(Ω) = v : Ω → IR | Dαv ∈ C(Ω), |α| ≤ m . (C.2)

If the function v is infinitely often continuously differentiable on Ω, we writev ∈ C∞(Ω).

For the function u(x) shown Fig. C.1 the following inclusions hold (withv(x) = u′(x))

v ∈ C0

u ∈ C1 .

400 C Appropriate Function Spaces

Fig. C.1. Example of a C1 function

Definition C.2. Square integrable functions: Let Ω be a closed domain inIRn. Then, the function u is called square integrable, if it fulfills the followingrelation ∫

Ω

|u(x)|2 dx <∞ . (C.3)

We denoteL2(Ω) = u : Ω → IR |

∫Ω

| u(x) |2 dx <∞ . (C.4)

For example, the function f(t) with the definition

f(t) =

⎧⎨⎩

1 for 0 < x < 20 for x = 0

−1 for −2 ≤ x < 0(C.5)

belongs to the space L2(−2, 2) (see Fig. C.2).

1

2

u x( )

x

-20

-1

Fig. C.2. Function u(x) = sgn(x) in the interval (–2,2)

Analogously to the above definition, we obtain the definition for Lp(Ω)-spaces for p ∈ [1,∞).

C Appropriate Function Spaces 401

Definition C.3. Lp(Ω)-spaces: Let Ω be a closed domain in IRn. Then, thespace of p-integrable functions is given by

Lp(Ω) = u : Ω → IR|∫Ω

|u(x)|p dx <∞ . (C.6)

Let us assume that the function u has a continuous derivative u′. According tothe formula for partial integration, we have for each continuously differentiablefunction ϕ with ϕ(a) = ϕ(b) = 0 the following relation

b∫a

u(x)ϕ′(x) dx = −b∫

a

u′(x)ϕ(x) dx . (C.7)

With the help of (C.7), we can define the derivative of functions, which have nofinite derivative in the classical sense. If u and w denote integrable functionsthat fulfill the following relation

b∫a

u(x)ϕ′(x) dx = −b∫

a

w(x)ϕ(x) dx (C.8)

for all differentiable functions ϕ with ϕ(a) = ϕ(b) = 0, then the function w iscalled the derivative of u in the weak sense (with respect to x). The function

Fig. C.3. Example of a function in H1(a, b)

u defined by (see Fig. C.3)

u(x) =x+ 1 for −1 ≤ x ≤ 01 − x for 0 < x ≤ 1

will have no derivative in the classical sense at x = 0. Applying partial inte-gration for differentiable functions ϕ(x) with ϕ(−1) = ϕ(1) = 0, we obtain

402 C Appropriate Function Spaces

1∫−1

u(x)ϕ′(x) dx =

0∫−1

(x+ 1)ϕ′(x) dx+

1∫0

(1 − x)ϕ′(x) dx

= −0∫

−1

ϕ(x) dx+ (x+ 1)ϕ(x)|0−1

−1∫

0

(−1)ϕ(x) dx+ (1 − x)ϕ(x) |10

= −

⎡⎣ 0∫−1

ϕ(x) dx+

1∫0

(−1)ϕ(x) dx

⎤⎦+ ϕ(0) − ϕ(0)︸︷︷︸

=0

.

Therefore, in the weak sense of differentiation we obtain

u′(x) =

1 for −1 ≤ x < 0−1 for 0 < x ≤ 1

with an arbitrary value for u′(0).

Definition C.4. Sobolev space: Let Ω be a domain in IRn. The functionalspace

Wmp (Ω) = u ∈ Lp(Ω)|Dαu ∈ Lp, |α| ≤ m (C.9)

is called Sobolev space Wmp (Ω). The partial derivatives of u are defined in the

weak sense.

The appropriate norms on Sobolev spaces are defined by

||u||W mp (Ω) =

⎛⎝∫

Ω

∑|α|≤m

|Dαu|p dx

⎞⎠1/p

(C.10)

and its semi-norm by

|u|W mp (Ω) =

⎛⎝∫

Ω

∑|α|=m

|Dαu|p dx

⎞⎠1/p

. (C.11)

If we restrict p to two, then we obtain a Hilbert space (Wm2 (Ω) = Hm(Ω))

with the scalar product

(u, v) =∫Ω

⎛⎝ ∑

|α|≤m

DαuDαv

⎞⎠ dx . (C.12)

C Appropriate Function Spaces 403

For example, the function u(x) is in the space H1(a, b), if u′(x) exists and iswithin the space L2(a, b). The norm is computed via

||u||H1(a,b) =

√√√√√ b∫a

(u(x))2 dx+

b∫a

(u′(x))2 dx , (C.13)

its semi-norm by

|u|H1(a,b) =

√√√√√ b∫a

(u′(x))2 dx , (C.14)

and the scalar product as follows

(u, v)H1(a,b) =

b∫a

u(x)v(x) dx+

b∫a

u′(x)v′(x) dx . (C.15)

Definition C.5. Let Ω be a domain in IRn and denote by C∞0 (Ω) the space

of infinitely often differentiable functions with zero boundary values. Then wewrite for the closure of C∞

0 (Ω) with respect to the H1 norm

H10 (Ω) = C∞

0 (Ω)H1(Ω) ⊂ H1(Ω) . (C.16)

Definition C.6. Partial Integration: Let Ω ⊂ IRn, n = 2, 3 be a domainwith smooth boundary Γ . Then, for any u, v ∈ H1(Ω) the following relationholds ∫

Ω

∂u

∂xiv dx =

∫Γ

uvn · ei ds−∫Ω

u∂v

∂xidx . (C.17)

In (C.17) n denotes the outer normal and Ω the considered domain Ω withboundary Γ .

By a multiple application of (C.17), we arrive at Green’s formula∫Ω

∆u v dx =∫Γ

∂u

∂nv ds−

∫Ω

(∇u)T ∇v dx (C.18)

for all u ∈ H2(Ω) and v ∈ H1(Ω).

D

Solution of Nonlinear Equations

In this section we are concerned with the solution of systems of nonlinearequations. As an example, we will consider the nonlinear Poisson equation,given as follows

−∇ · ε(|∇u|)∇u− f = 0 (D.1)u = 0 on Γ . (D.2)

This defines a nonlinear operator F that allows us to rewrite (D.1) and (D.2)as

F(u) = 0 . (D.3)

The weak formulation of (D.1) and (D.2) for all test functions v ∈ H10 reads

as ∫Ω

ε(|∇u|)∇v · ∇u dΩ −∫Ω

vf dΩ = 0 . (D.4)

By applying the finite element method, we arrive at the following algebraicsystem

K(u)u = f , (D.5)

with the matrix K ∈ IRn×n, f ∈ IRn, u ∈ IRn and n the number of unknowns.Since we cannot solve (D.5) explicitly, we have to establish an approximatesolution by setting up a series uk (k = 0, 1, 2, 3, ..) that is supposed to convergeto the correct solution. Concerning the rate of convergence, we will restrictthe discussion to the following types:

Definition D.1. Convergence: Let u∗ ∈ IRn be the exact solution. Then

• uk converges towards u∗ q-quadratically (q stands for quotient), if thereexists a C such that

||uk+1 − u∗|| ≤ C||uk − u∗||2 . (D.6)

406 D Solution of Nonlinear Equations

• uk converges towards u∗ q-linearly with the q-factor σ ∈ (0, 1), if

||uk+1 − u∗|| ≤ σ||uk − u∗|| . (D.7)

In general, a q-quadratically convergent algorithm is preferable to a q-linearly convergent one. However, we always have to take into account thenumerical cost for one iteration. Therefore, in some cases the method withthe slower convergence rate can even be faster.

Since we solve (D.5) numerically by computing a series of approximatingsolutions uk, the question of the stopping criterion is of great importance. Ingeneral, we distinguish between the following two types of stopping criteria:

1) Error criterion:We take the solutions of two successive iteration steps and define an ab-solute accuracy εabs by

||uk+1 − uk||2 < εabs , (D.8)

and a relative accuracy εrel by

||uk+1 − uk||2 < εrel||uk+1||2 , (D.9)

which has to be achieved. However, in some analysis the true solution maystill be far away, although the above-defined stopping criteria are fulfilled.This may particularly occur in the solution methods that have to use a linesearch (see Sect. D.1) to avoid possible divergence during early steps ofthe iteration process or due to nonmonotonic material relations. Then, itcan happen that the control parameter becomes very small, which resultsin almost no difference between uk+1 and uk.

2) Residual criterion:By computing the residual of the obtained solution, we can define an ab-solute accuracy εabs

res by

||K(uk+1)uk+1 − f ||2 < εabsres , (D.10)

as well as a relative accuracy εrelres by

||K(uk+1)uk+1 − f ||2 < εrelres||f ||2 . (D.11)

As shown in Fig. D.1, according to the problem type, this stopping criterionmay also be reached too early.

As a consequence of the above discussion, it is preferable to check bothstopping criteria.

D.1 Fixed-point Iteration 407

u*

eabs

res

uk+1

|| ( ) -K ||u u fh h h

2

Fig. D.1. Obtained solution uk+1 is still far away from the true solution u∗

D.1 Fixed-point Iteration

The simplest method of solving (D.5) is to rewrite it as a fixed-point equation

u = K−1(u)f . (D.12)

This will result in the following sequence

uk+1 = K−1(uk)f (D.13)K(uk)uk+1 = f . (D.14)

Thus, we can write the damped fixed-point iteration method as follows

K(uk)∆u = f − K(uk)uk = r(uk) (D.15)uk+1 = uk + η∆u . (D.16)

The nodal vector r(u) is known as the residual of the problem and a solutionis given by the set of nodal values u, for which the residual is zero. The scalarparameter η ∈ [0, 1] is introduced to control the possible divergence during theearly steps of the iteration process or due to nonmonotonic material relations.A common algorithm to compute η is a line search (see [227]) defined by

|G(η)| → min , (D.17)

withG(η) = ∆u · r(uk + η∆u) . (D.18)

One simple method of approximating the optimal η is as follows

408 D Solution of Nonlinear Equations

1. Evaluate g1 = G(0.1) and g2 = G(1.0)

2. Calculate the straight line l(g1, g2) between g1 and g2

3. Calculate the value η = 10g1−g210·(g2−g1) for which l(g1, g2) = 0 holds

A graphical interpretation of the fixed-point method is given in Fig. D.2.

Fig. D.2. Graphical interpretation for solving a nonlinear equation using the fixed-point method

D.2 Newton’s Method

Let us introduce the following linearization of the nonlinear operator F(u) atuk

F(u) ≈ F(uk) + F ′(uk)[s] (D.19)

with uk+1 = uk + s. The term F ′(uk)[s] denotes the Frechet - derivative ofthe nonlinear operator F at uk in the direction of s and is defined as follows

Definition D.2. Frechet - derivative: Let X and Y be two normed vectorspaces and D ⊂ X an open domain. The operator F : D → Y is differentiablein the sense of Frechet at x, iff there exists an operator A : X → Y, so thatfor all y ∈ D

F(y) = F(x) + A(y − x) + R(x, y) ,

D.2 Newton’s Method 409

with

limy→x

||R(x, y)||||y − x|| = 0

is fulfilled. A is the Frechet derivative F ′(x).

Therefore, Newton’s method reads as

F ′(uk)[s] = −F(uk) (D.20)uk+1 = uk + s . (D.21)

Analogously to the fixed-point method, a line-search parameter may acceleratethe convergence, and in addition may guarantee a global convergence of theNewton method. A graphical interpretation of Newton’s method is displayedin Fig. D.3.

Fig. D.3. Graphical interpretation for solving a nonlinear equation using Newton’smethod

To derive the Frechet derivative F ′ and Newton’s method for the nonlinearPoisson equation given in (D.1), we first compute the difference between F(u+s) and F(u) in the weak formulation for arbitrary test functions v ∈ H1

0∫Ω

ε(|∇(u+ s)|)∇v · ∇(u+ s) dΩ −∫Ω

ε(|∇u|)∇v · ∇u dΩ . (D.22)

Now, we will add to and at the same time subtract from (D.22) the term∫Ω

ε(|∇(u)|)∇v · ∇(u + s) dΩ, and obtain

410 D Solution of Nonlinear Equations∫Ω

(ε(|∇(u+ s)|) − ε(|∇u|))∇v · ∇(u + s) dΩ +∫Ω

ε(|∇u|)∇v · ∇s dΩ .

The term ε(|∇(u+ s)|) − ε(|∇u|) can be approximated as follows

ε(|∇(u + s)|) − ε(|∇u|) ≈ ε′(|∇u|) (|∇(u + s)| − |∇u|) . (D.23)

Now, let us investigate the term (|∇(u + s)| − |∇u|)

|∇(u+ s)| − |∇u| =|∇(u+ s)|2 − |∇u|2|∇(u+ s)| + |∇u| (D.24)

=∇u · ∇u+ ∇s · ∇s+ 2∇u · ∇s− ∇u · ∇u

|∇(u+ s)| + |∇u| (D.25)

≈ ∇u · ∇s

|∇u| . (D.26)

With this result, we can write∫Ω

(ε(|∇(u+ s)|) − ε(|∇u|))∇v · ∇(u+ s) dΩ

≈∫Ω

ε′(|∇u|)∇u · ∇s

|∇u| ∇v · ∇u dΩ . (D.27)

Summarizing the above results, we conclude that the Frechet derivativeF ′(uk)[s] in the weak formulation of the PDE for a test function v is given by∫

Ω

ε(|∇uk|)∇v · ∇s dΩ +∫Ω

ε′(|∇uk|)∇uk · ∇s

|∇uk|∇v · ∇uk dΩ . (D.28)

Therefore, by using (D.20) as well as (D.21), we obtain Newton’s method forthe nonlinear Poisson equation∫

Ω

ε(|∇uk|)∇v · ∇s dΩ +∫Ω

ε′(|∇uk|)∇uk · ∇s

|∇uk|∇v · ∇uk dΩ

=∫vf dΩ

−∫Ω

ε(|∇uk|)∇v · ∇uk dΩ ∀v ∈ H10 (Ω)

uk+1 = uk + s . (D.29)

By apply the finite element method to the above equation, we will arriveat the appropriate algebraic system of equations.

References

1. S. I. Aanonsen, T. Barkvek, J. N. Tjotta, and S. Tjotta, Distortion and har-monic generation in the near field of a finite amplitude sound beam, J. Acoust.Soc. Am. 75 (1984), 749 – 768.

2. O. Ahrens, D. Hohlfeld, A. Buhrdof, O. Glitza, and J. Bimder, A New Classof Capacitive Micromachined Ultrasonic Transducres, Proceedings of the IEEEUltrasonics Symposium, vol. 2, IEEE, 2000.

3. M. Ainsworth, Discrete Dispersion Relation for hp-Version Finite Element Ap-proximation at High Wave Number, SIAM J. Numer. Anal. 42 (2004), no. 2,553–575.

4. M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite ElementAnalysis, John Wiley & Sons Inc., 2000.

5. I. Ali, M. Escobar, C. Hahn, M. Kaltenbacher, and S. Becker, Numerical andExperimental Investigations of Flow Induced Noise around a Square Cylinder,10th AIAA/CEAS Aeroacoustics Conference, no. 2004-3004, 2004, Manchester,UK.

6. H. Allik and T. J. R. Hughes, Finite Element Method for Piezoelectric Vibra-tion, International Journal for Numerical Methods in Engineering 2 (1970),151–157.

7. D. Andelfinger and E. Ramm, EAS-elements for two-dimensional, three-dimensional, plate and schell structures and their equivalence to HR-elements,Int. J. Numer. Meth. Engng. 36 (1993), 1311–1337.

8. F. Armero and E. Petocz, Formulation and analysis of conserving algorithmsfor dynamic contact/impact problems, Computer Methods in Applied Mechan-ics and Engineering 158 (1998), 269–300.

9. D. Arnold, R. Falk, and R. Winther, Multigrid in H(div) and H(curl), Numer.Math. 85 (2000), 197–218.

10. D.N. Arnold and F. Brezzi, Some new elements for the Reissner-Mindlin platemodel, Boundary Value Problems for Partial Differential Equations and Appli-cations (J.-L. Lions and C. Baiocchi, eds.), Masson, Paris, 1993, pp. 287–292.

11. A. Arockiarajan, B. Delibas, A. Menzel, and W. Seemann, Studies on nonlin-ear electromechanical behavior of piezoelectric materials using finite elementmodeling, International Workshop on Piezoelectric Materials and Applicationsin Actuators, IWPMA, Heinz Nixdorf Institute, University of Paderborn, May2005.

412 References

12. C. Bailly and D. Juve, Numerical solution of acoustic propagation problemsusing linearized euler equations, AIAA Journal 38 (2000), no. 1, 22–29.

13. Alain Bamberger, Roland Glowinski, and Quang Huy Tran, A domain de-composition method for the acoustic wave equation with discontinuous coeffi-cients and grid change, SIAM J. Numer. Anal. 34 (1997), no. 2, 603–639. MRMR1442931 (98c:65161)

14. M.L. Barton and Z.J. Cendes, New vector finite elements for three-dimensionalmagnetic field computation, J. Appl. Phys. 61 (1987), no. 8, 3919–3921.

15. E. Bassiouny and A. F. Ghaleb, Thermodynamical formulation for coupledelectromechanical hysteresis effects: Combined electromechanical loading, In-ternational Journal of Engineering Science 27 (1989), no. 8, 989–1000.

16. K.J. Bathe, Finite Element Procedures, Prentice Hall, 1996.17. E. Becache, S. Fauqueux, and P. Joly, Stability of matched layers, group veloc-

ities and anisotropic waves, J. Comp. Phys. 188 (2003), 399–433.18. T. Belytschko, K Lui, W, and B. Moran, Nonlinear Finite Elements for Con-

tinua and Structures, Wiley, 2000.19. B. Bendjima, K. Srairi, and M. Feliachi, A coupling model for analysing dynam-

ical behaviours of an electromagnetic forming system, IEEE Trans. on Magnet-ics 33 (1997), no. 2, 1638–1641.

20. J.P. Berenger, A perfectly matched layer for the absorption of electromagneticwaves, J. Comp. Phys. 114 (1994).

21. C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming approach todomain decomposition: the mortar element method, Nonlinear partial differ-ential equations and their applications. College de France Seminar, Vol. XI(Paris, 1989–1991), Pitman Res. Notes Math. Ser., vol. 299, Longman Sci.Tech., Harlow, 1994, pp. 13–51. MR MR1268898 (95a:65201)

22. K.J. Binns, P.J. Lawrenson, and C.W. Trowbridge (eds.), The Analytic andNumerical Solution of Electric and Magnetic Fields, Wiley, 1992.

23. O. Biro and K. Preis, On the Use of the Magnetic Vector Potential in theFinite Element Analysis of Three-Dimensional Eddy Currents, IEEE Trans.on Magnetics 25 (July 1989), no. 4, 3145–3159.

24. M. Bischoff and I. Romero, A generalization of the method of incompatiblemodes, Int. J. Numer. Meth. Engng. 69 (2007), 1851–1868.

25. D.T. Blackstock, Connection between the Fay and Fubini Solutions for PlaneSound Waves of Finite Amplitude, J. Acoust. Soc. Am. 39 (1966), no. 6, 1019–1026.

26. A. Bossavit and J.C. Verite, A mixed FEM-BIEM method to solve 3-D eddycurrent problems, IEEE Trans. on Magnetics 18 (1982), 431–435.

27. D. Braess, Towards algebraic multigrid for elliptic problems of second order,Computing 55 (1995), 379–393.

28. , Finite elements, Cambridge University Press, 2001.29. , Finite Elemente, Springer Verlag, Berlin, Heidelberg, New York, 2003,

(3. korregierte Auflage).30. , Nonstandard mixed methods, 2005, Lecture at Radon Institute of Com-

putational and Applied Mathematics, Linz, Austria.31. J.H. Bramble and J.E. Pasciak, Analysis of a finite pml approximation for the

three dimensional time-harmonic maxwell and acacoustic scattering problems,Mathematics of Computation (2006).

32. A. Brandt, Algebraic multigrid theory: The symmetric case, Appl. Math. Com-put. 19 (1986), 23–56.

References 413

33. C.A. Brebbia and J. Dominguez, Boundary Elements, An Introductory Course,McGraw-Hill, 1992.

34. K. Brentner and F. Farassat, An analytical comparison of the acoustic analogyand kirchhoff formulation for moving surfaces, AIAA Journal 36 (1998), no. 8,1379–1386.

35. M. Breuer, Direkte Numerische Simulation und Large-Eddy Simulation turbu-lenter Stromungen auf Hochleistungsrechnern, Shaker-Verlag, 2002.

36. W.L. Briggs, A Multigrid Tutorial, SIAM, 1987.37. G. Caliano, R. Carotenuto, A. Caronti, M. Pappalardo, V. Foglietti, E. Cianci,

L. Visigalli, and I. Persi, Cmut echographic probes: design and fabrication pro-cess, Proceedings of the IEEE Ultrasonics Symposium, IEEE, 2002, pp. 1040–1043.

38. CENELEC, Piezoelektrische Eigenschaften keramischer Werkstoffe undBauteile, 1998, Teil 1-2, prEN 50324-1, 2.

39. J.M. Cesar de Sa and R.M. Naytal Jorge, New enhanced strain elements forincompressible problems, Int. J. Numer. Methods Eng. 44 (1999), 229–248.

40. D. Chapelle and R. Stenberg, An optimal low-order locking-free finite elementmethod for mindlin–reissner plates, Math. Models and Methods in Appl. Sci.8 (1998), 407–430.

41. P. T. Christopher and K. J. Parker, New approaches to nonlinear diffractivefield propagation, J. Acoust. Soc. Am. 90 (1991), 488 – 499.

42. R. Clayton and B. Engquist, Absorbing boundary conditions for acoustic andelastic wave equations, Bulletin of the Seismological Society of America 67(1977), 1529–1540.

43. G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations,Springer, 2002.

44. P. C. Coles, D. Rodger, P. J. Leonard, and H. C. Lai, Finite element mod-elling of 3d moving conductor devices with low conductivity, IEEE Trans. onMagnetics 32 (1996), no. 3, 753–755.

45. F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates,SIAM J. Sci. Comp. 19 (1998), 2061–2090.

46. M. Costabel and M. Dauge, Weighted regularization of Maxwell equations inpolyhedral domains, Tech. Report IRMAR 01-26, IRMAR, Institut Mathema-tique Universite de Rennes 1, 2001.

47. J.L. Coulomb and G. Meunier, Finite Element Implementation of Virtual WorkPrinciple for Magnetic and Electric Force and Torque Computation, IEEETrans. on Magnetics 20 (1984), no. 5, 1894–1896.

48. O. Craiu, N. Dan, and E. A. Badea, Numerical analysis of permanent magnetdc motor performances, IEEE Trans. on Magnetics 31 (1995), no. 6, 3500–3502.

49. C. de Boor, A Practical Guide to Splines, Springer, 1987.50. P. Di Francescantonio, A new boundary integral formulation for the prediction

of sound radiation, Journal of Sound and Vibration 202 (1997), no. 4, 491–509.51. W. Dietrich, Transformatoren: Stand der Technik und Tendenzen, VDE-Verlag,

Berlin, 1986.52. D. Dreyer and O. von Estorff, Improved conditioning of infinite elements for

exterior acoustics, Int. J. Numer. Meth. Engng 58 (2003), 933–953.53. T. Dreyer, W. Kraus, E. Bauer, and R. E. Riedlinger, Investigations of Compact

Focusing Transducers using Stacked Piezoelectric Elements for Strong SoundPulses in Therapy, Proceedings of the IEEE Ultrasonics Symposium, IEEE,2000, pp. 1239–1242.

414 References

54. M. Dumbser, Arbitrary High Order Schemes for the Solution of HyperbolicConservation Laws in Complex Domains, Ph.D. thesis, University of Stuttgart,2005.

55. Michael Dumbser and Claus-Dieter Munz, ADER discontinuous galerkinschemes for aeroacoustics, Comptes Rendus Mecanique 333 (2005), no. 9, 683–687.

56. F. Durst and M. Schafer, A Parallel Block-Structured Multigrid Method forthe Prediction of Incompressible Flows, Int. J. Num. Meth. Fluids 22 (1996),549–565.

57. C. Eccardt, K. Niederer., T. Scheiter, and C. Hierold., Surface micromachinedultrasound trasnducers in CMOS technology, Proceedings of the IEEE Ultra-sonics Symposium, IEEE, 1996, pp. 959–962.

58. J. Ekaterinaris, New Formulation of Hardin-Pope Equations for Aeroacoustics,AIAA Journal 37 (1999), 1033–1039.

59. B.O. Enflo and C.M. Hedberg, Theory of Nonlinear Acoustic Fluids, KluwerAcademic Publisher, 2002.

60. B. Engquist and A. Majda, Absorbing Boundary Conditions for the NumericalSimulation of Waves, Mathematics of Computation 31 (1977), 629–651.

61. M. Escobar, Finite Element Simulation of Flow-Induced Noise ApplyingLighthill’s Acoustic Analogy, Ph.D., Universitat Erlangen-Nurnberg, 2007.

62. R. Ewert and W. Schroder, Acoustic perturbation equations based on flow de-composition via source filtering, J. Comp. Phys. 188 (2003), 365–398.

63. F. Farassat, Acoustic radiation from rotating blades - The Kirchhoff method inaeroacoustics, Journal of Sound and Vibration 239 (2001), no. 4, 785 – 800.

64. J.H. Ferzinger and M. Peric, Computational Methods for Fluid Dynamics,Springer, 2002.

65. J.E. Ffowcs-Williams and D.L. Hawkings, Sound radiation from turbulence andsurfaces in arbitrary motion, Phil. Trans. Roy. Soc. A 264 (1969), 321–342.

66. M. Fischer and L. Gaul, Fast BEM-FEM Mortar Coupling for Acoustic-Structure Interaction, Int. J. Numer. Meth. Engng. 62 (2005), no. 12, 677–1690.

67. B. Flemisch, Non-matching triangulations of curvilinear interfaces applied toelectr-mechanics and elasto-mechanics, Ph.D. thesis, University of Stuttgart,2006.

68. J. B. Freund, S. K. Lele, and P. Moin, Direct numerical simulation of a mach1.92 turbulent jet and its sound field, AIAA Journal 38 (2000), 2023–2031.

69. A. Frohlich, Mikromechanisches Modell zur Ermittlung effektiver Materi-aleigenschaften von piezoelektrischen Polykristallen, Dissertation, UniversitatKarlsruhe (TH), Forschungszentrum Karlsruhe, 2001.

70. K. Fujiwara, T. Nakata, and H. Ohashi, Improvement for Convergence Char-acteristic of ICCG Method for the A-Φ Method Using Edge Elements, IEEETrans. on Magnetics 32 (May 1996), no. 3, 804–807.

71. T. Furukawa, K. Komiya, and I. Muta, An Upwind Galerkin Finite ElementAnalysis of Linear Induction Motors, IEEE Trans. on Magnetics 26 (1990),no. 2, 662–665.

72. Krump. G., Concerning the Perception of Low-frequency Distortions,Fortschritte der Akustik DAGA, Oldenburg, 2000.

73. B. Gaspalou, F. Colamartino, C. Marchland, and Z. Ren, Simulation of anelectromagnetic actuator by a coupled magnetomechanical modelling, COMPEL

References 415

- The International Journal for Computation and Mathematics in Electricaland Electronic Engineering 14 (1995), no. 4, 203–206.

74. V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer Verlag, Berlin Heidelberg New York, 1979.

75. B. Girod, G. Greiner, and H. Niemann, Principles of 3D Image Analysis andSynthesis, Kluwer Academic Publishers, 2000.

76. D. Givoli, T. Hagstrom, and I. Patlashenko, Finite element formulation withhigher-order absorbing boundary conditions for time-dependent waves, Compu-tational methods in applied mechanics and engineering 195 (2006), 3666–3690.

77. C. Golovanov, J.-L. Coulomb, Y. Marechal, and G. Meunier, 3d mesh connec-tion techniques applied to movement simulation, IEEE Trans. on Magnetics 34(1998), no. 5, 3359–3362.

78. Ch. Großmann and H.G. Roos, Numerik Partieller Differentialgleichungen,Teubner, 1994.

79. Whitney. H., Geometric Integration Theory, Princeton Univ. Press, 1957.80. G. Haase, M. Kuhn, U. Langer, S. Reitzinger, and J. Schoberl, Parallel Maxwell

Solvers, Scientific Computing in Electrical Engineering (U. Van Rienen,M. Gunther, and D. Hecht, eds.), Lecture Notes in Computational Scienceand Engineering, vol. 18, Springer, 2000.

81. G. Haase, M. Kuhn, and S. Reitzinger, Parallel AMG on Distributed MemoryComputers, SIAM J. Sci. Comput. 24 (2002), no. 2, 410–427.

82. W. Hackbusch, Multigrid Methods and Application, Springer Verlag, Berlin,Heidelberg, New York, 1985.

83. S.-Y. Hahn, J. Bigeon, and J.-C. Sabonnadiere, An upwind finite elementmethod for electromagnetic field problems in moving media, Int. J. Numer.Meth. Engng. 24 (1987), 2071–2086.

84. M.F. Hamilton and D.T. Blackstock, Nonlinear Acoustics, Academic Press,1998.

85. I. Harari, M. Slavutin, and E. Turkel, Analytical and numerical studies of afinite element pml for the helmholtz equation, J. Comput. Acoust. 8 (2000),121–127.

86. J. C. Hardin and M. Y. Hussaini (eds.), Computational aeroacoustics, ch. Com-putational aeroacoustics for low Mach number flows, pp. 50–68, Springer-Verlag, 1992, New York.

87. J. C. Hardin and M. Y. Hussaini (eds.), Computational aeroacoustics, ch. Re-garding numerical considerations for computational aeroacoustics, pp. 216–228,Springer-Verlag, 1992, New York.

88. M. J. Heathcote, The J & P Transformer Book, Newnes, 1998.89. T. Hegewald, B. Kaltenbacher, M. Kaltenbacher, and R. Lerch, Efficient Model-

ing of Ferroelectric Behavior for the Analysis of Piezoceramic Actuators, Jour-nal of Intelligent Material Systems and Structures (2007), submitted.

90. B. Heise, Mehrgitter-Newton-Verfahren zur Berechnung nichtlinearer magnetis-cher Felder, Dissertation, TU Chemnitz, 1994.

91. F. Henrotte, A. Nicolet, H. Hedia, A. Genon, and W. Legros, Modelling ofelectromechanical relays taking into account movement and electric circuits,IEEE Trans. on Magnetics 30 (1994), no. 5, 3236–3239.

92. R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal.36 (1999), no. 1, 204–225.

93. M. Hofer, Finite-Elemente-Berechnung von periodischen Oberflachenwellen-Strukturen, Dissertation, Universitat Erlangen, 2003.

416 References

94. J. Hoffelner, Simulation, Erzeugung und Anwendung von hochintensivem Ul-traschall, Dissertation, Universitat Erlangen, 2003.

95. H. Hofmann, Das elektromagnetische Feld, Springer, 1986.96. M.S. Howe, Theory of vortex sound, Cambridge Univ. Press, 2003.97. Fang Q. Hu, Absorbing boundary conditions, Int. J. Comp. Fluid Dyn. 18

(2004), 513–522.98. D. C. Hughes and J. T. Wen, Preisach modeling and compensation for smart

material hysteresis, Proceedings: Active Materials and Smart Structures, vol.2427, February 1995, pp. 50–64.

99. T.J.R. Hughes, The Finite Element Method, 1 ed., Prentice-Hall, New Jersey,1987.

100. T.J.R. Hughes and T.E. Tezduyar, Finite elements based upon Mindlin platetheory with particular reference to the four-node bilinear isoparametric eleemnt,J. Appl. Mech., ASME (1981), 587–596b.

101. H. Huttner, Noise and vibration of transformers, their origin and reduction,ELIN (1983), 41–46.

102. A. Ibrahimbegovic and E.L. Wilson, A modified method of incompatible modes,Communication in Applied Numerical Methods 7 (1991), 187–194.

103. N. Ida, Modeling of velocity effects in eddy current applications, J. of Appl.Phys. 63 (1988), no. 8, 3007–3009.

104. , Engineering Electromagnetics, Springer, 2004.105. IEEE, IEEE Standard on Piezoelectricity, 1987, Std 176.106. F. Ihlenburg and I. Babuska, Finite Element Solution of the Helmholz Equation

with High Wave Number Part I: The h Version of the Finite Element Method,Comp. Math. Appl. 30 (1995), 9–37.

107. , Finite Element Solution of the Helmholz Equation with High WaveNumber Part II: The h-p Version of the FEM, SIAM J. Numer. Anal. 34(1997), no. 1, 315–358.

108. D.-H. Im and C.-E. Kim, Finite element force calculation of a linear inductionmotor taking account of the movement, IEEE Trans. on Magnetics 30 (1994),no. 5, 3495–3498.

109. M. Ito, T. Takahashi, and M. Odamura, Up-wind finite element solution oftravelling magnetic field problems, IEEE Trans. on Magnetics 28 (1992), no. 2,1605–1610.

110. M. Jung and U. Langer, Methode der finiten Elemente fur Ingenieure, Teubner,2001.

111. M. Jung, U. Langer, A. Meyer, W. Queck, and M. Schneider, Multigrid precon-ditioners and their application, Proceedings of the 3rd GDR Multigrid Sem-inar held at Biesenthal, Karl-Weierstraß-Institut fur Mathematik, May 1989,pp. 11–52.

112. B. Kaltenbacher and M. Kaltenbacher, Modelling and iterative identificationof hysteresis via preisach operators in pdes, Radon Series Comp. Appl. Math.(2007).

113. M. Kaltenbacher, Numerische Simulation Magnetomechanischer Transducermit Fluidankopplung, Dissertation, Universitat Linz, 1996.

114. M. Kaltenbacher, B. Kaltenbacher, R. Simkovics, and R. Lerch, Determina-tion of piezoelectric material parameters using a combined measurement andsimulation technique, Proceedings of the IEEE Ultrasonics Symposium, IEEE,2002, pp. 1023–1026.

References 417

115. M. Kaltenbacher, H. Landes, J. Hoffelner, and R. Simkovics, Use of ModernSimulation for Industrial Applications of High Power Ultrasonics, Proceed-ings of the IEEE Ultrasonics Symposium, CD-ROM Proceedings, IEEE, 2002,pp. 673–678.

116. M. Kaltenbacher, H. Landes, S. Reitzinger, and R. Peipp, 3D Simulation ofElectrostatic-Mechanical Transducers using Algebraic Multigrid, IEEE Trans-actions on Magnetics 38 (2002), no. 2, 985–988.

117. M. Kaltenbacher and S. Reitzinger, Algebraic Multigrid for Solving Electrome-chanical Problems, Lecture Notes in Computational Science and Engineering,Multigrid Methods VI, Springer-Verlag, September 1999, pp. 129–135.

118. , Appropriate Finite Element Formulations for 3D ElectromagneticField Problems, IEEE Transactions on Magnetics 38 (2002), no. 2, 513–516.

119. , Nonlinear 3D Magnetic Field Computations using Lagrange FE-functions and Algebraic Multigrid, IEEE Transaction on Magnetics 38 (2002),no. 2, 1489–1496.

120. M. Kaltenbacher, S. Reitzinger, and J. Schoberl, Algebraic Multigrid Methodfor Solving 3D Nonlinear Electrostatic and Magnetostatic Field Problems, IEEETransactions on Magnetics 36 (2000), no. 4, 1561–1564.

121. A. Kameari, Three dimensional eddy current calculation using edge elementsfor magnetic vector potential, Applied Electromagnetics in Materials (1986),225–236.

122. M. Kamlah, Ferroelectric and ferroelastic piezoceramics - modeling of elec-tromechanical hysteresis phenomena, Continuum Mech. Thermodyn. 13(2001), 219–268.

123. M. Kamlah and U. Bohle, Finite element analysis of piezoceramic componentstaking into account ferroelectric hysteresis behavior, International Journal ofSolids and Structures 38 (2001), 605–633.

124. M. Kamlah and C. Tsakmakis, Phenomenological modeling of the non-linearelectro-mechanical coupling in ferroelectrics, International Journal of Solids andStructures 36 (1999), 669–695.

125. M. Kanoi, Y. Hori, M. Maejima, and T. Obata, Transformer noise reductionwith new sound insulation panel, IEEE Trans. on Power Apparatus and Sys-tems (1983), 2817–2825.

126. Y. Kawase, O. Miyatani, and T. Yamaguchi, Numerical analysis of dynamiccharacteristics of electromagnets using 3-d finite element method with edge el-ements, IEEE Trans. on Magnetics 30 (September 1994), no. 5, 3248–3251.

127. C.T. Kelly, Iterative Methods for Linear and Nonlinear Equations, SIAM, 1995.128. B.T. Khuri-Yakub, C.H. Cheng, F.L. Degertekin, and S. Ergun, Silicon Micro-

machined Ultrasonic Transducers, Jap. J. Appl. Phys. 39 (2000), 2883–2887.129. B.T. Khuri-Yakub, F.L. Degertekin, X-C. Jin, S. Calmes, I. Ladabaum,

S. Hansen, and X.J. Zang, Silicon micromachined ultrasonic transducers, Pro-ceedings of the IEEE Ultrasonics Symposium, IEEE, 1998.

130. F. Kickinger, Algebraic multigrid for discrete elliptic second-order problems,Multigrid Methods V. Proceedings of the 5th European Multigrid conference(W. Hackbusch, ed.), Springer Lecture Notes in Computational Science andEngineering, vol. 3, 1998, pp. 157–172.

131. F. Kickinger and U. Langer, A note on the global extraction element-by-elementmethod, ZAMM (1998), no. 78, 965–966.

132. M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis, Tech. report,Nauka, 1983.

418 References

133. K. Kuhnen, Inverse Steuerung piezoelektrischer Aktoren mit Hysterese-,Kriech- und Superpositionsoperatoren, Dissertation, Universitat des Saarlan-des, Saarbrucken, 2001.

134. S. Kurz, Die numerische Behandlung elektromechanischer Systeme mit Hilfeder Kopplung der Methode der finiten Elemente und der Randelementmethode,Dissertation, Universitat Stuttgart, 1998.

135. V.P. Kuznetsov, Equations of Nonlinear Acoustics, Soviet Physics - Acoustics16 (1971), no. 4, 467–470.

136. D. J. P. Lahaye, F. Maggio, and A. Quarteroni, Hybrid finite element–spectralelement approximation of wave propagation problems, East-West J. Numer.Math. 5 (1997), no. 4, 265–289. MR MR1604852 (99d:65263)

137. H. C. Lai, D. Rodger, and P. J. Leonard, Coupling meshes in 3d problemsinvolving movements, IEEE Trans. on Magnetics 28 (1992), no. 2, 1732–1734.

138. C. M. Landis, Non-linear constitutive modeling of ferroelectrics, Current Opin-ion in Solid State and Materials Science 8 (2004), 59–69.

139. Duck Joo Lee and Sam Ok Koo, Numerical Study of Sound Generation due toa Spinning Vortex Pair, AIAA Journal 33 (1995), 20–26.

140. J. H. Lee, J. C. Kim, and D. S. Hyun, Dynamic characteristics analysis of syn-chronous reluctance motor considering saturation and iron loss by fem, IEEETrans. on Magnetics 34 (1998), no. 5, 2629–2632.

141. Y. S. Lee and M. F. Hamilton, Time-domain modeling of pulsed finite amplitudesound beams, J. Acoust. Soc. Am. 97 (1995), 906 – 917.

142. G. Lehner, Elektromagnetische Feldtheorie, Springer, 1996.143. P.J. Leonard and D. Rodger, Comparision of methods for modelling jumps in

conductivity using magnetic vector potential based formulations, IEEE Trans.on Magnetics 33 (March 1997), no. 2, 1295–1298.

144. R. Lerch, Sensorik und Prozessmesstechnik, Vorlesungsskript, UniversitatErlangen-Nurnberg.

145. , Technische Akustik / Akustische Sensoren, Vorlesungsskript, Univer-sitat Erlangen-Nurnberg.

146. , Simulation of Piezoelectric Devices by Two- and Three-DimensionalFinite Elements, IEEE Transactions on UFFC 37 (1990), no. 3, 233–247, ISSN0885-3010.

147. R. Lerch, M. Kaltenbacher, A. Hauck, G. Link, and M. Hofer, Accurate Model-ing of CMUTs, Proceedings of the IEEE Ultrasonics Symposium, IEEE, 2004,pp. 264–269.

148. K. Liow, M.C. Thompson, and K. Hourigan, Computation of Acoustic Wavesgenerated by a co-rotating Vortex Pair, 14th Australasian Fluid MechanicsConference, Adelaide University, December 2001, Adelaide, Australia.

149. Y. S. K. Liow, B. T. Tan, M. C. Thompson, and K. Hourigan, Sound generatedin laminar flow past a two-dimensional rectangular cylinder, Journal of Soundand Vibration 295 (2006), 407–427.

150. D. Lockard and J. Casper, Permeable Surface Corrections for Ffwocs Williamsand Hawkings Integrals, Proceedings of 11th AIAA/CEAS Aeroacoustics Con-ference, May 2005, Monterey, USA.

151. E. Manoha, C. Herrero, P. Sagaut, and S. Redonnet, Numerical Predictionof Airfoil Aerodynamic Noise, Proceedings of 8th AIAA/CEAS AeroacousticsConference, June 2002, Breckenridge, USA.

References 419

152. Y. Marechal, G. Meunier, J. L. Coulomb, and H. Magnin, A general pur-pose tool for restoring inter-element continuity, IEEE Trans. on Magnetics 28(1992), no. 2, 1728–1731.

153. J.C. Maxwell, A Treatise on Electricity and Magnetism, vol. I, Dover, 1954.154. , A Treatise on Electricity and Magnetism, vol. II, Dover, 1954.155. Mayergoyz, I.D., Mathematical Models of Hysteresis, Springer Verlag, New

York, 1991.156. F. Menter and Y. Egorov, A scale adaptive simulation model using two-equation

models, 43rd AIAA Aerospace Sciences Meeting and Exhibit, no. AIAA-2005-1095, 2005.

157. K. Meyberg and P. Vachenauer, Hohere Mathematik 1, Springer, 1993.158. A.J. Moulson and J.M. Herbert, Electroceramics, Chapman and Hall, London,

1990.159. E-A. Muller and F. Obermeier, The Spinning Vortices as a Source of Sound,

Fluids dynamics of Rotor and Fan supported Aircraft at Subsonic Speeds 22(1967), 1–8.

160. G. Mur and A.T. Hoop, A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media, IEEE Trans. onMagnetics 21 (1985), 2188–2191.

161. T. Nakata, N. Takahashi, H. Morishige, J.L. Coulomb, and C. Sabonnadiere,Analysis of 3D static force problem, Proceedings of the TEAM Workshop onComputation of Applied Electromagnetics in Materials, 1993, pp. 73–79.

162. J. Nedelec, A new family of mixed finite elements in R3, Numer. Math. 50(1986), 57–81.

163. J.C. Nedelec, Mixed Finite Elements in R3, Numer. Meth. 35 (1980), 315–341.164. A. Nysveen and R. Nilssen, Time domain simulation of magnetic systems with

a general moving geometry, IEEE Trans. on Magnetics 33 (1997), no. 2, 1394–1397.

165. A. Oberai, F. Ronaldkin, and T. Hughes, Computational procedures for de-termining structural-acoustic response due to hydrodynamic sources, Comp.Methods Appl. Mech. Engineering 190 (2000), 345–361.

166. L. Olson and K. Bathe, An infinite element for analysis of transient fluid-structure interactions, Eng. Comput. 2 (1985), 319–329.

167. T. Onuki, S. Wakao, and T. Yoshizawa, Eddy Current Computations in Mov-ing Conductors by the Hybrid FE-BE Method, IEEE Trans. on Magnetics 31(1995), no. 3, 1436–1439.

168. H. Parkus, Mechanik der Festen Korper, Springer, 1986.169. C. Pechstein and B. Juttler, Monotonicity-preserving Interapproximation of B-

H Curves, Sfb report, Johannes Kepler University Linz, SFB ”Numerical andSymbolic Scientific Computing”, 2004.

170. A.D. Pierce, Acoustics, An Introduction to its Physical Principles and Appli-cations, Acoustical Society of America, 1991.

171. A Powell, Theory of vortex sound, Journal of the Acoustical Society of America36 (1964), 177–195.

172. K. Preis, O. Biro, and I. Ticar, Gauged Current Vector Potential and ReentrantCorners in the FEM Analysis of 3D Eddy Currents, IEEE Trans. on Magnetics36 (July 2000), no. 4, 840–843.

173. E. Preisach, Uber die magnetische Nachwirkung, Z. Phys. (1935), no. 94, 277–302.

420 References

174. M. Rausch, Numerische Analyse und Computeroptimierung von elektrody-namischen Aktoren - am Beispiel eines elektrodynamischen Lautsprechers, Dis-sertation, Universitat Erlangen-Nurnberg, 2001.

175. M. Rausch, M. Gebhardt, M. Kaltenbacher, and H. Landes, MagnetomechanicalField Computation of a Clinical Magnetic Resonance Imaging (MRI) Scanner,COMPEL - The International Journal for Computation and Mathematics inElectrical and Electronic Engineering 22 (2003), no. 3, 576–588.

176. S. Reese, A large deformation solid-shell concept based on reduced integrationwith hourglass instability, Int. J. Numer. Meth. Engng. 69 (2006), 1671–1716.

177. E. Reiplinger, Lastabhangige Transformatorgerausche, etz 3 (1989), 106–109.178. E. Reiplinger and H. Stelter, Gerauschprobleme, etz-a 3 (1977), 224–228.179. S. Reitzinger, Algebraic Multigrid Methods for Large Scale Finite Element

Equations, Reihe C - Technik und Naturwissenschaften, no. 36, Univer-sitatsverlag Rudolf Trauner, 2001.

180. S. Reitzinger and J.Schoberl, Algebraic Multigrid for Edge Elements, NumericalLinear Algebra with Applications 9 (2002), 223–238.

181. S. Reitzinger, B. Kaltenbacher, and M. Kaltenbacher, A note on the approx-imation of B-H curves for nonlinear computations, Technical Report, SFBF013: Numerical and Symbolic Scientific Computing, September 2002, Linz,Osterreich.

182. S. Reitzinger and M. Kaltenbacher, Algebraic Multigrid Methods for Magne-tostatic Field Problems, IEEE Transactions on Magnetics 38 (2002), no. 2,477–480.

183. S. Reitzinger, U. Schreiber, and U. van Rienen, Algebraic Multigrid for ComplexSymmetric Matrices: Numerical Study, Tech. Report 02-01, Johannes KeplerUniversity Linz, SFB: Numerical and Symbolic Scientific Computing, 2002.

184. D. Rodger, H. C. Lai, and P. J. Leonard, Coupled elements for problems in-volving movement, IEEE Trans. on Magnetics 26 (1990), no. 2, 548–550.

185. J. W. Ruge and K. Stuben, Algebraic multigrid (AMG), Multigrid Methods(S. McCormick, ed.), Frontiers in Applied Mathematics, vol. 5, SIAM, Philadel-phia, 1986, pp. 73–130.

186. U. Rude, Mathematical and Computational Techniques for Multilevel AdaptiveMethods, SIAM, 1993.

187. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publ. Company,1996.

188. N. Sadowski, Y. Lefevre, M. Lajoie-Mazenc, and J. Cros, Finite element torquecalculation in electrical machines while considering the movement, IEEE Trans.on Magnetics 28 (1992), no. 2, 1410–1413.

189. P. S. Sangha and D. Rodger, Design and analysis of voltage fed axisymmetricactuators, IEEE Trans. on Magnetics 30 (1994), no. 5, 3240–3243.

190. M. Schinnerl, M. Kaltenbacher, U. Langer, and R.Lerch, An Efficient Methodfor the Numerical Simulation of Magneto-Mechanical Sensors and Actuators,European Journal for Applied Mathematics (2007), (to be publisched).

191. M. Schinnerl, J. Schoberl, and M. Kaltenbacher, Nested multigrid methods forthe fast numerical computation of 3D magnetic fields, IEEE Transactions onMagnetics 36 (2000), no. 4, 1557–1560.

192. C. Schram, J. Anthoine, and A. Hirschberg, Calculation of Sound Scatter-ing Using Curle’s Analogy for Non-Compact Bodies, 11th AIAA AeroacousticsConference, no. 2005-2836, 2005, Monterey, USA.

References 421

193. J. Schroder and H. Romanowski, A simple coordinate invariant thermody-namic consistent model for nonlinear electro-mechanical coupled ferroelectrica,ECCOMAS 2004 Proceedings (Jyvaskyla, Finnland), vol. 2, The EuropeanCommunity on Computational Methods in Applied Sciences, University ofJyvaskyla, Department of Mathematical Information Technology, 2004.

194. J. Schoberl, An advancing front 2D/3D-mesh generator based on abstract rules,Comput. Visual. Sci. (1997), no. 1, 41–52.

195. W. Seemann, A. Arockiarajan, and B. Delibas, Modeling and simulation ofpiezoceramic materials using micromechanical approach, European Congresson Computaional Methods in Applied Sciences and Engineering (ECCOMAS)(Jyvaskyla), July 2004.

196. J. Seo and Y. Moon, Linearized perturbed compressible equations for low machnumber aeroacoustics, J. Comp. Phys. 218 (2006), no. 2, 702–719.

197. P. Silvester and R. Ferrari, Finite Elements for Electrical Engineers, Cam-bridge, 1996.

198. R. Simkovics, Nichtlineares piezoelektrisches Finite-Elemente-Verfahren zurModellierung piezokeramischer Aktoren, Dissertation, Universitat Erlangen-Nurnberg, 2002.

199. J.C. Simo and M.S. Rifai, A class of mixed assumed strain methods and themethod of incompatible modes, Int. J. Numer. Meth. Engng. 29 (1990), 1595–1638.

200. I. Singer and E. Turkel, A perfectly matched layer for helmholz equation in asemi-infinite strip, J. Comp. Phys. 201 (2004), 439–465.

201. W.R. Smythe, Static and Dynamic Electricity, Taylor & Francis, 1989.202. K. Srairi, M. Feliachi, and Z. Ren, Electromagnetic actuator behavior analysis

using finite element and parametrization methods, IEEE Trans. on Magnetics31 (1995), no. 6, 3497–3499.

203. P. N. Sreeram, G. Salvady, and N. G. Naganathan, Hysteresis prediction for apiezoceramic material system, The ASME Winter Annual Meeting (New Or-leans, Louisiana), vol. 35, ASME, 1993.

204. European Standard, EN 60551 / A1 1997 on Determination of transformerand reactor sound levels.

205. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, Inc., 1941.206. C. K. W. Tam, Computational aeroacoustics, AIAA Journal 33 (1995), 1788–

1796.207. R.L. Taylor, P.J. Beresford, and E.L. Wilson, A non-conforming element for

stress analysis, Int. J. Numer. Meth. Engng. 10 (1976), 1211–1219.208. F.L. Teixeira and W.C. Chew, Complex space approach to perfectly layers: a

review and some developments, Int. J. Numerical Modelling 13 (2000), 441–455.

209. H. A. C. Tilmans, Equivalent circuit representation of electromechanical trans-ducers: I. lumped-parameter systems, Journal of Micromechanics and Micro-engineering 6 (1996), 157–176.

210. K. Uchino, Recent developments in ceramic actuators, Proc. SPIE Smart Struc-tures and Materials, vol. 3321, 1998, pp. 46–57.

211. A. Uzun, A. S. Lyrintzis, and G. A. Blaisdell, Coupling of Integral AcousticsMethods with LES for Jet Noise Prediction, Proceedings of AIAA AerospaceSciences Meeting and Exhibit, no. 2004-0517, January 2004, Reno, NV. USA.

422 References

212. P. Vanek, J. Mandel, and M. Brezina, Algebraic multigrid by smoothed aggre-gation for second and fourth order elliptic problems, Computing 56 (1996),179–196.

213. T. Weiland, Time Domain Electromagnmetic Field Computation, Int. J. Num.Modeling 9 (1996), 295–319.

214. J. S. Welij, Calculation of eddy currrent in terms of H on hexahedra, IEEETrans. on Magnetics 21 (1985), 2239–2241.

215. K. Wilner, Ein statistisches Modell fur den Kontakt metallischer Korper, Dis-sertation, Universitat der Bundeswehr Hamburg, 1995.

216. E.L. Wilson, R.L. Taylor, W.P. Doherty, and J. Ghaboussi, Incompatible dis-placement modes, Numerical and Computer Methods in Structural Mechanics(1973), 43–57.

217. Barbara I. Wohlmuth, A mortar finite element method using dual spaces forthe Lagrange multiplier, SIAM J. Numer. Anal. 38 (2000), no. 3, 989–1012.MR MR1781212 (2001h:65132)

218. , A comparison of dual Lagrange multiplier spaces for mortar finiteelement discretizations, M2AN Math. Model. Numer. Anal. 36 (2002), no. 6,995–1012. MR MR1958655 (2004b:65193)

219. G. Wojcik, T. Szabo, J. Mould, L. Carcione, and F. Clougherty, Nonlinearpulse calculations and data in water and tissue mimic, Proc. IEEE UltrasonicsSymposium (1999), 1521–1526.

220. P. Wriggers, Nichtlineare Finite-Element-Methoden, Springer, 2001.221. G. Wunsch, Feldtheorie, 2, Verlag Technik, Berlin, 1975.222. G. Wunsch and H.G. Schulz, Elektromagnetische Felder, Verlag Technik Berlin,

1995.223. www.compumag.co.uk, International compumag society.224. R. Zengerle, S. Kluge, M. Richter, and R. Richter, A bidirectional silicon mi-

cropump, Proceedings of Sensor95 (Nuremberg), 1995, pp. 727–732.225. Ziegler, F., Mechanics of Solids and Fluids, Springer, New York, 1995.226. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, vol. 1, Butter-

worth - Heinemann, 2003.227. , The Finite Element Method, vol. 2, Butterworth - Heinemann, 2003.228. E. Zwicker and H. Fastl, Psychoacoustics, Springer, 1999.

Index

absorbing boundary condition 175acoustic

averaged energy density 147averaged intensity 147

averaged power 147density 140

energy density 146energy flux 146

field 139impedance 148

intensity 146

linear wave equation 145non-linear wave equation 156

overall sound pressure level (OSPL)155

particle velocity 140pressure 140

quantities 146sound-field impedance 148

sound-intensity level 154sound-power level 154

sound-pressure level (SPL) 154spherical spreading law 151

velocity potential 145actuator see mechatronic

adiabatic 143bulk modulus 144

compressibility 144

aeroacoustics 267co-rotating vortex pair 279

Ffowcs Williams and Hawkingsequation 275

Lighthill’s analogy 270

agglomeration technique see coarsen-ing

algebraic multigrid see multigridAmpere 95approximation

B–H curve 129auxiliary matrix 294

B–H curve see approximationbalanced reduced and selective

integration 85Biot–Savart’s law 108boundary condition 9

Dirichlet 9essential 10natural 10Neumann 9

bulk viscosity 156Burger’s equation 159butterfly curve 248

circulation 392CMUT see micromachinedco-rotating vortex pair 279coarse grid operator see gridcoarsening

agglomeration technique 297function 295process 295

coilcurrent-loaded 131voltage-loaded 211

computational aeroacoustics 267condition number 284

424 Index

configurationdeformed 55initial 55

congruency 255conjugate gradient (PCG) method see

preconditionedconservation

of mass 140of momentum 141

contact mechanicscondition 332pressure-displacement relation 332tangent stiffness matrix 332

continuity equation 140convergence 405Coulomb-gauge see gaugecoupling

aeroacoustics 267electromagnetics-mechanics 207electrostatics-mechanics 195mechanics-acoustics 229piezoelectrics 243

coupling mechanisms 4Courant-Friedrich-Levi condition 40Crank-Nicolson scheme 39curl 387, 393

damping 69modal 69Rayleigh model 69

density 54derivative

Frechet 408global/local 29weak sense 401

design process 3CAE-based 3experimental-based 3

diamagnetic 105dielectric remanent 109differential operator 54diffusion equation 103diffusivity of sound 159displacement current density 96divergence 386, 390

theorem 392

elasticity modulus 60electric

charge density 94conductivity 94, 108current 95current density 94field intensity 94flux density 94permittivity 94polarization 94, 245scalar potential 104specific resistivity 108

electrodynamic loudspeaker seeloudspeaker

electromagneticenergy 210field 93force 209interface conditions 109quasistatic field 101

electromagnetic-mechanical system207

calculation scheme 216electromotive force 98electrostatic

energy 196field 104force 196

electrostatic-mechanical system 195calculation scheme 202

Enhanced assumed strain method 83error

a posteriori 46a priori 46discretization 45dispersion 170interpolation 170pollution 170

Euler equation 141Eulerian coordinate 55Everett function 256

Faraday 96Fay solution 192ferroelectricity 109, 245ferromagnetic 105Ffowcs Williams and Hawkings equation

275filter

1/3 octave 152octave 152

Index 425

finite element 7assembling procedure 32compatible 23conforming 23edge 43formulation 9hexahedral 27infinite element 174isoparametric 21Lagrangian 20method 7mortar 166Nedelec 43nodal 20quadrilateral 23tetrahedral 26triangular 26

finite element/boundary elementmethod 219

fixed-point iteration 407flux 389force

electromagnetic 209electrostatic 196

formulationstrong 9variational 10weak 10

Fubini solution 191functional spaces 399

Lp 400continuously differentiable 399Hilbert 402Sobolev 402square integrable 399weighted Sobolev 120

Galerkin 10method 10semidiscrete formulation 12

gauge 102Gauss 99Gauss theorem see divergencegeometric multigrid see muiltigridgradient 387

deformation 55displacement 56of a scalar 386

Green’s integral theorem 396

scalar form 396vector form 396

gridcoarse 295coarse-grid operator 299complexity 305fine 295

harmonic distortion 317Helmholtz decomposition 63, 290Hook’s law 60Hu-Washizu principle 83hysteresis 253

Preisach model 253

incompatible modes method 81induced electric voltage 134inductance see magneticinfinite finite elements 174interpolation

conservativ 278function 22

irrotational 99, 384vector field 397

isothermal process 143

Jacobi 29matrix 29

Khokhlov–Zabolotskaya–Kuznetsov(KZK) equation 160

Kuznetsov’s equation 156

Lagrange multiplier 220Lagrangian

coordinate 55updated formulation 208

Lame parameters 60Lighthill’s analogy 270line search 407litotripsy 361local support 21locking 78

effect 77membrane 79Poisson 78shear 78

Lorentz force 4, 95loss factor 69

426 Index

loudspeaker 4, 218, 313

magneticfield intensity 94flux 96, 132hard material 105hysteresis 106inductance 132induction 94permeability 94, 104reluctivity 105remanent field 105scalar potential 107soft material 105vector potential 101

magnetic valve 207, 330overexcitation 337premagnetization 336switching cycle 337

magnetization 94magnetomechanical system see

electromagnetic-mechanicalsystem

Maxwell’s equations 93mechanical

acceleration 54axisymmetric stress–strain 63contact 332damping see dampingfield 51plane strain 61plane stress 62strain 56stress 51stress-stiffening effect 347yield stress 66

mechanical-acoustic system 229calculation scheme 233

mechatronic 1actuator 1sensor 1

micromachinedcapacitive ultrasound array (CMUT)

345motional electromotive force 99, 207

method 219, 222, 314moving body

electric field 203magnetic field 207

moving coilcurrent-loaded 218voltage-loaded 218

moving-material method 221, 223, 315moving-mesh method 203, 221multigrid 283

algebraic 293geometric 287method 285nested 291

multilayer actuator see piezoelectric

nabla operator 386Navier’s equations 54Newmark scheme 41Newton method 408

electromagnetics 125mechanics 72

non-matching gridsacoustics 166mechanics-acoustics 234

norms 381Holder 381matrix 382p-norms 381vector 381

numerical computationelectromagnetics 114electromagnetics-mechanics 216electrostatics 113electrostatics-mechanics 202geometric non-linear case 70linear acoustics 161linear elasticity 66mechanics-acoustics 233non-linear acoustics 163non-linear electromagnetics 125piezoelectrics 257

numerical integration 30Gaussian quadrature 30

operatorcomplexity 305nonlinear 405

paramagnetic 105parameter of non-linearity 157partial differential equation 9

hyperbolic 40

Index 427

parabolic 36patch test 82penalty formulation 119penetration depth see skin depthperfectly matched layer (PML) 176permeability see magnetic permeabil-

itypiezoelectric 243

ceramics 245cofired multilayer 340direct effect 243inverse effect 243systems 243

Poisson ratio 60polarization

irreversibel 247permanent 247saturation 247

poling 247polymers 245power transformer 320preconditioned conjugate gradient

(PCG) method 283predictor-corrector algorithm 38, 42Preisach

function 254model 253operator 254

prestressing 205principle of virtual work 196, 199, 209,

213process

adiabatic 143isothermal 143

prolongation 285operator 285, 298

Rayleigh damping model see dampingremanent magnetic field see magneticrestriction 285

operator 285

saturation strain 248scalar

acoustic velocity potential 145electric potential 104field 383magnetic potential 107

sensor see mechatronic

shape function see interpolationshear modulus 60shear viscosity 156shock-formation distance 192single crystals 245skin

depth 103effect 102

smoothingoverlapping block-smoothers 290block-Gauss-Seidel 290Gauss-Seidel backforward 287Gauss-Seidel forward 287hybrid 303operator 299post 286pre 286

Sobolev space see functional spacessolenoidal 100, 384

vector field 398solid/fluid interface 230sound velocity 139spherical spreading law see acousticSPL see acousticstack actuator see piezoelectricstate equation 143Stoke’s theorem 395stopping criterion 406

error 406residual 406

strain see mechanicalstrain tensor

Green–Lagrangian 59linear 59

stress tensor1st Piola–Kirchhoff 712nd Piola–Kirchhoff 71Cauchy 53

stress-stiffening effect 347surface integration 43

TEAM (Testing ElectromagneticAnalysis Methods) 290

tensorof dielectric constants 244of elasticity moduli 60, 244of piezoelectric moduli 244

test function 9thermal strain 66

428 Index

time discretization 35effective mass formulation (hyper-

bolic) 42effective mass formulation (parabolic)

38effective stiffness formulation

(hyperbolic) 42effective stiffness formulation

(parabolic) 38explicit (hyperbolic) 41explicit (parabolic) 39implicit (hyperbolic) 41implicit (parabolic) 39

transducing mechanisms 3transformer see power transformertrapezoidal difference scheme 37

ultrasoundhigh intensity focused (HIFU) 357

litotripsy 361

vector field 383irrotational 384solenoidal 384

virtual work see principleVoigt notation 54, 59

wavelongitudinal 65, 139number 148plane 148shear 65spherical 150

weighted regularization 120Westervelt equation 160wiping-out 255

yield stress 66

manfredkaltenbacher - wordpress.com · chap. 12 demonstrates the applicability to real-life...

Documents