adaptive isogeometric analysis of phase-field models

Paul Hennig

Adaptive Isogeometric Analysisof Phase-Field Models

Herausgeber

Prof. Dr.-Ing. Michael BeitelschmidtProf. Dr.-Ing. habil. Markus KästnerProf. Dr.-Ing. habil. Thomas Wallmersperger

Institut für FestkörpermechanikFakultät MaschinenwesenTechnische Universität Dresden01062 Dresdenhttps://tu-dresden.de/ing/maschinenwesen/ifkm

c©2020 Paul HennigPestalozziplatz 601127 [email protected]

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Preface

The content of this thesis was developed during my employment as a research associate at theInstitute of Solid Mechanics (IFKM) at the TU Dresden from 2015 until 2020. I gratefullyacknowledge the support from the German Research Foundation (DFG) that funded the corre-sponding project within the priority program 1748 as well as the University of Minnesota thatsupported me during my 4-month research stay at the Department of Civil, Environmental, andGeo- Engineering.

First of all I would like to express my gratitude to my supervisor Professor Markus Kästner,who supported me since my diploma studies. Due to his guidance and motivation I have notonly gained in-depth knowledge in the field of numerical solid mechanics but also developedmyself personally in various fields. I am sincerely grateful for the pleasant working atmosphere,the many interesting discussions and the nice time during our business trips.

Very special thanks are also directed to Professor Allesandro Reali for his interest in my researchtopic. It was a great honor for me that Professor Reali, who is a well known expert in the fieldof Isogeometric Analysis, agreed to act as an assessor for my thesis. Furthermore, I wouldlike to thank my project partners, colleagues and students for the helpful discussions and theircontribution to the success of this work, including among others: Marreddy Ambati, Prof.Laura De Lorenzis, Johann Duffek, Massimo Carraturo, Arne Hansen-Dörr, Leonhard Heindel,Thomas Linse, Roland Maier, Philipp Morgenstern, Sebastian Müller, Prof. Daniel Peterseimand Prof. Dominik Schillinger. In addition I would like to thank my friend Robert Müller forhis persistent support during my studies and his helpfulness in proofreading my work.

Of course I am also deeply grateful to all my colleagues who made my stay at the Institute anunforgettable time. The humorous but also harmonious atmosphere makes it hard for me toleave.

Last but certainly not least I would like to express my gratitude to my family and especially myparents Denise and Thomas who brought me with their continuous support, their example andtheir motivation to this point in my life. Finally, I would like to thank my beloved wife Irina forfilling my life with happiness and love.

Paul HennigDresden, December 2020

Vollständiger Abdruck der von der Fakultät Maschinenwesen der Technischen Universität Dres-den zur Erlangung des akademischen Grades Doktoringenieur (Dr.-Ing.) genehmigten Disser-tation.

Vorsitzender derPromotionskommission: Prof. Dr.-Ing. Jens Krzywinski

(Technische Universität Dresden)

Gutachter: Prof. Dr.-Ing. habil. Markus Kästner(Technische Universität Dresden)Professor Alessandro Reali(Università degli Studi di Pavia)

Mitglieder derPromotionskommission: Prof. Dr. Daniel Peterseim

(Universität Augsburg)Prof. Dr.-Ing. habil. Thomas Wallmersperger(Technische Universität Dresden)

Tag der Einreichung: 7. Mai 2020Tag der öffentlichen Verteidigung: 10. November 2020

Summary

In this thesis, a robust, reliable and efficient isogeometric analysis framework is presented thatallows for an adaptive spatial discretization of non-linear and time-dependent multi-field prob-lems. In detail, Bézier extraction of truncated hierarchical B-splines is proposed that allows fora strict element viewpoint, and in this way, for the application of standard finite element proce-dures. Furthermore, local mesh refinement and coarsening strategies are introduced to generategraded meshes that meet given minimum quality requirements. The different strategies areclassified in two groups and compared in the adaptive isogeometric analysis of two- and three-dimensional, singular and non-singular problems of elasticity and the Poisson equation. Sincea large class of boundary value problems is non-linear or time-dependent in nature and requiresincremental solution schemes, projection and transfer operators are needed to transfer all statevariables to the new locally refined or coarsened mesh. For field variables, two novel projectionmethods are proposed and compared to existing global and semi-local versions. For internalvariables, two different transfer operators are discussed and compared in numerical examples.

The developed adaptive isogeometric analysis framework is than combined with the phase-fieldmethod. Numerous phase-field models are discussed including the simulation of structural evo-lution processes to verify the stability and efficiency of the whole adaptive framework and tocompare the projection and transfer operators for the state variables. Furthermore, the phase-field method is used to develop an unified modelling approach for weak and strong disconti-nuities in solid mechanics as they arise in the numerical analysis of heterogeneous materialsdue to rapidly changing mechanical properties at material interfaces or due to propagation ofcracks if a specific failure load is exceeded. To avoid the time consuming mesh generation, adiffuse representation of the material interface is proposed by introducing a static phase-field.The material in the resulting transition region is recomputed by a homogenization of the ad-jacent material parameters. The extension of this approach by a phase-field model for crackpropagation that also accounts for interface failure allows for the computation of brittle fracturein heterogeneous materials using non-conforming meshes.

Contents

List of Figures IV

List of Symbols V

1 Introduction 1

2 Preliminaries to Isogeometric Analysis 72.1 B-Splines and NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Basis Functions and Geometric Map . . . . . . . . . . . . . . . . . . . 72.1.2 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Weak Form and Discretization . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Numerical Integration – Bézier Extraction . . . . . . . . . . . . . . . . 152.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Local Refinement with Truncated Hierarchical B-Splines 193.1 Local h-Refinement of Tensor Product B-splines . . . . . . . . . . . . . . . . 193.2 Bézier Extraction of Truncated Hierarchical B-splines . . . . . . . . . . . . . . 23

3.2.1 Active Elements and Basis Functions . . . . . . . . . . . . . . . . . . 243.2.2 Assembly of Global System of Equations . . . . . . . . . . . . . . . . 263.2.3 Equivalence to the Truncated Hierarchical Basis . . . . . . . . . . . . 29

3.3 Imposition of Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . 313.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Adaptive Isogeometric Analysis with Truncated Hierarchical B-Splinesand T-splines 354.1 Adaptive Mesh Refinement and Coarsening . . . . . . . . . . . . . . . . . . . 37

4.1.1 Greedy Refinement Strategy . . . . . . . . . . . . . . . . . . . . . . . 384.1.2 Safe Refinement Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.3 Coarsening Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Convergence, Condition and Sparsity . . . . . . . . . . . . . . . . . . . . . . . 424.3 Demonstration: Poisson Problem . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.1 Weak Form and Error Estimator . . . . . . . . . . . . . . . . . . . . . 444.3.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Demonstration: Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4.1 Weak Form and Error Estimator . . . . . . . . . . . . . . . . . . . . . 524.4.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Mesh Adaptivity for Incremental Solution Schemes – Projection and Trans-fer Operators 595.1 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

I

Contents

5.2 Mesh Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.1 Projection of Field Variables . . . . . . . . . . . . . . . . . . . . . . . 615.2.2 Comparison of Projection Methods . . . . . . . . . . . . . . . . . . . 64

5.3 Mesh Refinement Including History Variables . . . . . . . . . . . . . . . . . . 665.3.1 Transfer of History Variables . . . . . . . . . . . . . . . . . . . . . . . 675.3.2 Comparison of Transfer Operators . . . . . . . . . . . . . . . . . . . . 71

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Application to Phase-Field Models 756.1 Basic Concepts of Phase-Field Models . . . . . . . . . . . . . . . . . . . . . . 766.2 Weak and Strong Discontinuities in Solid Mechanics . . . . . . . . . . . . . . 79

6.2.1 Embedded Material Interfaces in Linear Elasticity . . . . . . . . . . . 796.2.2 Brittle and Ductile Fracture in Homogeneous and Heterogeneous Ma-

terials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.3 Evolving Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.3.1 Spinodal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 926.3.2 Topology Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7 Summary and Outlook 103

References 107

II

List of Figures

1.1 Phase-field approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Basis functions and geometric map . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Geometric map and knot insertion for bivariate B-spline . . . . . . . . . . . . . 92.3 Comparison of knot insertion algorithms . . . . . . . . . . . . . . . . . . . . . 102.4 B-spline refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Univariate Bernstein polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Element map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Bézier extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Two approaches to local mesh refinement . . . . . . . . . . . . . . . . . . . . 203.2 Multi-level basis and subdivision operator . . . . . . . . . . . . . . . . . . . . 213.3 Hierarchical and truncated hierarchical B-splines . . . . . . . . . . . . . . . . 223.4 Different sets of basis functions in the multi-level basis . . . . . . . . . . . . . 253.5 Sparsity patterns of stiffness matrices and hierarchical subdivision operator . . 273.6 Relation of hierarchical subdivision operator to truncated hierarchical basis . . 303.7 Projection of Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . 32

4.1 Refinement strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Example for the greedy THB-spline refinement . . . . . . . . . . . . . . . . . 384.3 Greedy THB-spline refinement in a worst case scenario . . . . . . . . . . . . . 384.4 Example for the safe THB-spline refinement . . . . . . . . . . . . . . . . . . . 394.5 Safe THB-spline refinement in a worst case scenario . . . . . . . . . . . . . . 394.6 Example for the greedy THB-spline coarsening . . . . . . . . . . . . . . . . . 404.7 Example for the safe THB-spline coarsening . . . . . . . . . . . . . . . . . . . 414.8 Worst case scenario – Condition number . . . . . . . . . . . . . . . . . . . . . 434.9 Worst case scenario – Sparsity pattern . . . . . . . . . . . . . . . . . . . . . . 444.10 L-shape and slit domain – Domain and boundary conditions . . . . . . . . . . 464.11 L-shape – Mesh and sparsity pattern . . . . . . . . . . . . . . . . . . . . . . . 474.12 L-shape – Convergence and condition number . . . . . . . . . . . . . . . . . . 484.13 Slit domain – Mesh and sparsity pattern . . . . . . . . . . . . . . . . . . . . . 494.14 Slit domain – Convergence and condition number . . . . . . . . . . . . . . . . 504.15 Fichera corner – Domain and boundary conditions . . . . . . . . . . . . . . . . 514.16 Fichera corner – Mesh and convergence . . . . . . . . . . . . . . . . . . . . . 524.17 Plate with hole – Domain and boundary conditions . . . . . . . . . . . . . . . 544.18 Plate with hole – Mesh and sparsity pattern . . . . . . . . . . . . . . . . . . . 554.19 Plate with hole – Convergence and condition number . . . . . . . . . . . . . . 564.20 Cook’s membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1 Uniform coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Local coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Transfer scheme for incremental computations including history variables . . . 67

III

List of Figures

5.4 Schematic illustration of transfer operators for history variables . . . . . . . . . 685.5 Shape function transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.6 Comparison of Transfer Operators . . . . . . . . . . . . . . . . . . . . . . . . 72

6.1 Double-well and one-well potentials . . . . . . . . . . . . . . . . . . . . . . . 766.2 Evolution of the order parameter . . . . . . . . . . . . . . . . . . . . . . . . . 786.3 Embedded material interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.4 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.5 Bi-material rod – Domain and boundary conditions . . . . . . . . . . . . . . . 826.6 Bi-material rod – Computational results . . . . . . . . . . . . . . . . . . . . . 836.7 Circular inclusion – Domain, boundary conditions, mesh and solution . . . . . 846.8 Circular inclusion – Computational results . . . . . . . . . . . . . . . . . . . . 856.9 Phase-field approach to fracture and numerical examples . . . . . . . . . . . . 866.10 Single edge notched shear test – Meshes and solution . . . . . . . . . . . . . . 876.11 Single edge notches shear test – Computational results . . . . . . . . . . . . . 886.12 Crack propagation in heterogeneous materials . . . . . . . . . . . . . . . . . . 896.13 Asymmetrically notched specimen – Meshes and solution . . . . . . . . . . . . 906.14 Asymmetrically notched specimen – Computational results . . . . . . . . . . . 916.15 Spinodal decomposition – Computational results . . . . . . . . . . . . . . . . 936.16 Spinodal decomposition – Meshes and solution . . . . . . . . . . . . . . . . . 946.17 Topology optimization – Domain and boundary condition . . . . . . . . . . . . 966.18 2D topology optimization – Computational results . . . . . . . . . . . . . . . . 976.19 2D topology optimization – Meshes and solution . . . . . . . . . . . . . . . . 986.20 3D topology optimization – Meshes and solution . . . . . . . . . . . . . . . . 996.21 3D topology optimization – Computational results . . . . . . . . . . . . . . . . 100

IV

List of SymbolsIn the following all abbreviations, operators and symbols that are used in this thesis are in-troduced. Bold italic symbols represent tensors of first or second order, whereas bold normalsymbols represent matrices and vectors of arbitrary dimension. The order or dimension of atensor or matrix, respectively, is mentioned in the text.

Abbreviations

BFT Basis function transferB-spline Basis splineBVP Boundary value problemCAD Computer aided designCPT Closest point transferDOF Degrees of freedomFE Finite elementFEM Finite element methodh-refinement Reduction of element sizeHB-splines Hierarchical B-splinesIGA Isogeometric analysisk-refinement Elevation of polynomial degree and continuityLLSQ Local discrete least squares methodLLSQ_w Weighted local discrete least squares methodLRB-splines Locally refined B-splinesLSQ Global discrete least squares methodNURBS Non-uniform rational B-splinesp-refinement Elevation of polynomial degreePDE Partial differential equationSPR Superconvergent patch recoverySLSQ Subspace discrete least squares methodT-splines B-spline that allows for T-junctionsTHB-splines Truncated hierarchical B-splinesWPLSQ Weighted patch based least squares fit

Operators Meaning

a(·,·) Bilinear formclosure Closure of elementD(·) Differential operatordet(·) DeterminantL(·) Linear formrefine Refine operatorsubdivide Subdivide elements in child elementssupp Parametric domain supported by a functionT Projection or transfer operator

V

List of Symbols

Operators Meaning

trunc Truncation of a basis functionTrunc Successive truncation with respect to all higher levels∇ Nabla operator∆ Laplace operator∈ Member of a set⊂ Subset∪ Set union∩ Set intersection\ Set difference∅ Empty set∃ There exists at least one∨ Logical or∧ Logical and

Latin Symbols

A Abbreviation for AHB or ATHB

A`a Active basis functions of level `AHB Hierarchical B-spline basis or spaceAi All active basis functions in the multi-level basis that are used

for integrationATHB Truncated hierarchical B-spline basis or spaceB Bernstein basis functionB Vector of Bernstein basis functionB Bézier element domain in physical spacec Scalar field variable or order parameterC ContinuityC Cauchy-Green tensorC Bézier extraction operatorC Fourth order material tensordP Dimension of parametric spacedS Dimension of physical/spatial spaceE Element boundary or Young’s modulusE Element domain in parameter spaceE Green-Lagrange strain tensorF Deformation gradientF Arbitrary function spacef , f Scalar or vector valued right hand side of PDEg Degradation functionG Gramian matrixGc Fracture toughnesshE Diameter of element EhE Diameter of element boundary Eh Mass flux vector

VI

Latin Symbols

H Neighbourhood of elementH1 Sobolev spaceI Identity tensorI Activity indicatorJ Jacobian determinantk Convergence rateK Set of edges of an elementK Stiffness matrix` Index of hierarchical level`c Length-scale parameter of diffuse interfaceL Number of levels in a hierarchical basisL2 Lebesque spacem Multiplicity of knot value or volume fractionM MobilityM Knot insertion or subdivision operatorMh Hierarchical subdivision operatorM Modified subdivision operatorM Element meshM`

a Mesh of active Elements on level `n Number of basis functions and corresponding control pointsnE Number of elementsnE Normal vector at element boundary EnN Normal vector at Neumann boundaryN , N (Vector of)B-spline basis functionN B-spline basis or spaceN Natural numbersp Polynomial degree of basis functionP Control point of dimension dPP Vector of control pointsQ Set of quadrature pointsr RadiusR NURBS basis functionRE Edge residual of element boundary ER Real numberss SkewnessS Marked elements for refinementt Timeu, u Solution of scalar or vector valued PDEu Discrete solution vectoruh, uh Numerical solution of scalar or vector valued PDEuD,uD Prescribed solution (vector) on Dirichlet boundaryv, v Scalar or vector valued weight functionvh, vh Scalar or vector valued discretized weight functionw Weight of a NURBS basis function

VII

List of Symbols

Latin Symbols

W NURBS weight functionxsd Signed distance functionx Spatial position vector (in current configuration)X Spatial position vector (in reference configuration)

Greek Symbols

α Coefficient, marking criterionβ AngleΓ Domain boundaryΓc Crack surfaceΓD Dirichlet boundaryΓN Neumann boundaryδu Virtual displacementε Linearised strain tensorη Error or thresholdηE Error per element Eθ Threshold valueκ Energy density parameterλ Lamé’s first parameterΛ State variablesΛF Field variablesΛH History variablesΛI Internal variablesµ Chemical potential or Lamé’s second parameterν Poisson’s ratioξ Position vector in parametric domainξ Position vector in unit domainΞ Knot vectorσ Stress tensorφ Angleψ Energy densityΨ EnergyΩ Physical domain (in current configuration)ΩP Parameter domainΩR Physical domain in reference configurationΩU Unit domain

VIII

1 IntroductionEnvironmental challenges demand the development of innovative, energy-efficient and re-source-saving products. Lightweight designs and multi-materials with complex characteristic microstructures are the key features for the development of the associated mechanical components.The computer aided engineering (CAE) of these sophisticated components requires a solution ofcoupled non-linear field problems and an appropriate discretization of weak discontinuities thatarise in the field variables due to rapidly changing mechanical properties at material interfaces.Furthermore, the fail-safe design of the components requires the understanding, constitutivemodelling and, in particular, the computational modelling of propagating cracks that lead tostrong discontinuities in the field variables. Robust, reliable and efficient non-standard numeri-cal methods are required to compute accurate primary field variables and their derivatives acrossthese weak and strong discontinuities.

Phase-field models are developed to solve interfacial problems and were introduced first inorder to model solidification processes by COLLINS & LEVINE [1]. In this approach differentphases of the system are identified by unique values of a continuous scalar field variable, theso-called order parameter. As illustrated in Fig. 1.1, sharp interfaces between the phases are reg-ularized by a smooth transition of the order parameter from one to another unique value. Thewidth of the resulting diffuse interface is governed by a length-scale parameter. The physicsof the system and interface are finally captured by a (higher-order) partial differential equa-tion (PDE) that describes the evolution of the order-parameter and thereby the evolution of thephases. In the limiting case when the length scale parameter tends to zero, the sharp interfacemodel should be obtained.

The phase-field method gained increased interest in the last decades and was therefore appliedto different physical problems. Beside the modelling of micro structure evolution processes,including solidification, solid-state phase transformations or martensitic grain growth [2, 3],the phase-field method is successfully applied to brittle [4–6], ductile [7, 8] and dynamic [9]fracture by coupling the order parameter to a mechanical field problem. In this way, complexcracking phenomena as crack deflection, branching and coalescence follow implicitly fromthe solution of the coupled field problem. Another possible field of application is topologyoptimization [10], where the optimal distribution of a given amount of material in a designdomain is sought, so that the compliance of the resulting mechanical structure under predefinedboundary conditions is minimized. The generalization to a multi phase-field model allows forthe consideration of heterogeneous or graded materials [11, 12] in the optimization process.

The implicit representation of interfaces in terms of the order parameter avoids a cumbersomenumerical tracking and re-discretization of the discontinuities and makes the phase-field methodto a suitable tool to solve moving interface problems numerically. However, in combinationwith finite element methods (FEM), the variability of the approach comes at the cost of highlyrefined meshes that are required along the discontinuities to properly resolve the gradients inthe order parameter field. Hence, adaptive local mesh refinement and coarsening is essential forefficient computations. Furthermore, the possibly higher-order character of the PDEs demandhigher continuous function spaces to spatially discretize the computational domain. Spline-

1

1 Introduction

x0

1

c `c

(b)(a)

Figure 1.1: Phase-field approach: (a) The different phases of the system, indicated by light gray and graycolour, are identified by the values 0 and 1 of the order parameter c. (b) The plot of theorder parameter along the dashed line in (a) illustrates the continuous transition of c across theinterfaces. The width of the resulting diffuse interface is governed by a length-scale parameter`c.

based approximations fulfil these requirements and are consequently an ideal discretizationtechnique for phase-field models.

Isogeometric analysis (IGA) was introduced by HUGHES et al. [13] as a novel discretiza-tion method and as a counterpart to finite element (FE) analysis. The main objective was toovercome the disjunction between a computational geometry model, commonly described byNon-Uniform Rational B-Splines (NURBS), and an FE model based on Lagrangian polyno-mial approximations of the geometry and the field variables. In the finite element method theNURBS geometry has to be meshed with finite elements, which can be a tedious, time consum-ing task that is hard to automate. In contrast, in IGA the same B-spline basis functions, whichrepresent the geometry are used to approximate the field variables in the numerical model. Inthis way, meshing of the geometry is no longer necessary and the total computing time of thenumerical analysis is reduced. In addition, the geometric discretization errors that result fromthe approximation of the computational geometry by Lagrangian polynomials are eliminated.This improves the quality of the solution, especially if thin-walled structures are modelled bysolids [14]. Furthermore, the higher continuity of the B-spline basis reduces the degrees of free-dom (DOF) compared to FEM if a certain error level has to be achieved and allows for a directdiscretization of higher-order PDEs. In this context, IGA was successfully used to discretizee.g. Kirchhoff-Love shells [15] or phase-field models for spinodal decomposition [16], crackpropagation [17] and topology optimization [18]. In order to enhance existing FEM codes withisogeometric analysis features, Bézier extraction can be used. It has been allready applied toe.g. B-splines and NURBS [19], T-splines [20] or hierarchical B-splines [21, 22] and thereforerepresents a canonical approach to the implementation of IGA.

Isogeometric analysis is also an ideal discretization technique to be combined with adaptivemesh refinement as already the coarsest mesh provides an exact computational geometry repre-sentation that is preserved during refinement. Tedious interactions with an underlying geometrymodel, as it is needed in FEM, are avoided. However, if B-splines or NURBS are consideredas a basis, their tensor product nature prohibits a truly local refinement within a single NURBSpatch. For that reason, various approaches were developed to overcome the restrictive tensor

2

product structure, including e.g. hierarchical (H)B-splines [23], truncated hierarchical (TH)B-splines[24], locally refined (LR) B-splines [25] and T-splines [26].

Objectives

This thesis has two main objectives. The first objective is to provide a robust, reliable andefficient isogeometric analysis framework that allows for an adaptive spatial discretization ofnon-linear and time-dependent multi-field problems.

To develop the adaptive isogeometric analysis framework,

1. Bézier extraction of truncated hierarchical B-splines is proposed,

2. local mesh refinement and coarsening strategies are introduced and applied to adaptiveIGA to generate graded meshes that meet given minimum quality requirements,

3. projection and transfer operators are proposed to transfer all state variables to the newmesh.

The development of Bézier extraction for truncated hierarchical B-splines allows for a strictelement viewpoint, and in this way, for the application of standard finite element procedures.Different from the direct implementation of the truncated hierarchical basis, standard Bézierextraction is used on different levels of the hierarchical mesh. It is shown that this approachproduces truncated hierarchical B-splines as proposed by GIANNELLI et al. [24] without theneed for an explicit truncation of basis functions.

To allow for an adaptive meshing, two different refinement strategies for THB-splines are in-troduced and compared against two different refinement strategies for T-spline. Following thestandard iterative procedure of adaptive finite element analysis, the refinement strategies areintegrated in an adaptive loop and applied to two- and three-dimensional, singular and non-singular problems of elasticity and the Poisson equation. The different strategies are classifiedin two groups: greedy and safe refinement strategies. It is shown that the save strategies leadto well graded meshes and optimal convergence rates in the benchmarks which verifies thenumerical implementation of the Bézier extraction framework and the refinement strategies.

A large class of boundary value problems is non-linear or time-dependent in nature and requiresincremental solution schemes. In this case, a projection or transfer of state variables is necessaryduring the mesh adaptive computations if a re-computation of the problem from the initial stateis to be avoided. For this reason, two novel projection methods for field variables are proposedand compared to existing global and semi-local versions. Some boundary value problems alsoinclude internal variables that are given at quadrature points and possess their own evolutionequation. For these variables two different transfer operators are discussed and compared innumerical examples. It is shown that isogeometric analysis improves the performance of theprojection and transfer operations as already the coarsest mesh represents the exact geometryand the hierarchical structure allows for quadrature free projection methods.

The second objective of this thesis is to combine the developed adaptive isogeometric analysisframework with the phase-field method. Numerous phase-field models are discussed includingthe simulation of structural evolution processes to verify the stability and efficiency of the wholeadaptive framework and to compare the projection and transfer operators for the state variables.

3

1 Introduction

Furthermore, the phase-field method is used to develop an unified modelling approach for weakand strong discontinuities in solid mechanics as they arise in the numerical analysis of hetero-geneous materials due to rapidly changing mechanical properties at material interfaces or due topropagation of cracks if a specific failure load is exceeded. The presented approach is especiallybeneficial if the conforming meshing of heterogeneities becomes time consuming or impossible.Typical examples are micro-structures that have a complex shape, that are strongly deformedduring the simulation or that are obtained from imaging techniques. To avoid the meshing, adiffuse representation of the material interface is proposed by introducing a static phase-field.The material in the resulting transition region is redefined in terms of the order parameter andrecomputed by a homogenization of the adjacent material parameters. To provide an appro-priate and efficient approximation of the diffuse interface an h`-adaptive refinement strategybased on the proposed adaptive IGA framework is applied. The extension of this approach bya phase-field model for crack propagation that also accounts for interface failure allows for thecomputation of brittle fracture in heterogeneous materials using non-conforming meshes.

It is shown that the combination of adaptive isogeometric analysis with the phase-field methodresults in efficient and straight-forward simulations of moving discontinuities and thereforemakes the phase-field method a powerful numerical tool. The use of IGA is especially beneficialif higher-order phase-field models are used or if the computational domain, on which the phase-field model is solved, is given by a computational geometry model.

Outline

This thesis is organized as follows. Chapter 2 reviews the fundamentals of isogeometric analy-sis. For this purpose, the definition of NURBS geometries and corresponding refinement possi-bilities are introduced. Furthermore, the isogeometric analysis of partial differential equationsis discussed. This includes the integration of the system matrices in terms of Beziér extractionand the application of boundary conditions.

The following Chapter 3 is based on the work of HENNIG et al. [27] and presents isogeo-metric analysis with truncated hierarchical B-splines. To this end, basic concepts for localh-refinement of tensor product B-splines are reviewed. Subsequently, Bézier extraction of trun-cated hierarchical B-splines is proposed and the equivalence to the basis, introduced in [24], isdemonstrated.

Based on the work of HENNIG et al. [28], the isogeometric analysis framework is generalizedto adaptivity in Chapter 4. For this purpose two different refinement strategies and a coarseningstrategy are introduced for THB-splines and integrated into an adaptive meshing loop. Subse-quently, the methods are compared with two refinement strategies for T-splines. In numerousbenchmarks, the H1-errors of the discrete solutions, the degrees of freedom as well as sparsitypatterns and condition numbers of the discretized problems are compared.

Following the work of HENNIG et al. [29], mesh adaptivity for incremental solution schemesis considered in Chapter 5. In this case, a projection or transfer of state variables is neces-sary during the mesh adaptive computations. In order to develop appropriate operators, at firstthe different state variables are introduced and the incremental solution procedure is outlined.Subsequently, the different projection and transfer operators are proposed and compared.

4

In Chapter 6 the whole adaptive isogeometric analysis framework, introduced in the previousthree chapters, is applied to several phase-field models and therefore verified regarding its ac-curacy, reliability and efficiency. The numerical examples also serve the purpose of comparingthe introduced projection and transfer operators for field and internal variables. In detail, phase-field models for weak and strong discontinuities in solid mechanics and for structural evolutionprocesses are discussed. The diffuse modelling of the heterogeneous materials and its exten-sion to brittle fracture to simulate interface failure is based on the works of HENNIG et al. [30],HENNIG et al. [31] and HANSEN-DÖRR et al. [32]. Furthermore, this topic is influenced bythe findings of LINSE et al. [33] and KÄSTNER et al. [34].

The last Chapter 7 summarizes the content and findings of this thesis and gives an outlook onfuture work.

5

2 Preliminaries to IsogeometricAnalysis

In this chapter the fundamentals of isogeometric analysis are reviewed. For this purpose, thedefinition of NURBS geometries and corresponding refinement possibilities are introduced.Furthermore, the isogeometric analysis of partial differential equations is discussed. This in-cludes the integration of the system matrices in terms of Beziér extraction and the applicationof boundary conditions.

2.1 B-Splines and NURBSIn Computer Aided Design (CAD), NURBS geometries play an important role, due to theirflexibility to represent geometric shapes as spheres or ellipsoids. NURBS are based on B-splinebasis functions that are defined over so called patches in parametric space. The geometricmap transforms the patch from the parameter to the physical space. It can be understood asa parametric equation. To further increase the flexibility of design, individual patches can berefined, trimmed or combined with other patches. For a detailed discussion on B-splines andNURBS, the reader is referred to e.g. PIEGEL & TILLER [35].

2.1.1 Basis Functions and Geometric MapB-spline Basis Functions

In CAD the use of B-splines is favoured, due to the compact support of the correspondingB-spline basis functions that are only non-zero in a small domain. Univariate B-spline basisfunctions are defined by a knot vector Ξ = ξIn+p+1

I=1 that contains non-decreasing coordinatesξI ∈ R from the parameter domain ΩP = [ξ1, ξn+p+1]. The domain ΩP is also referred to aspatch. From the elements of the knot vector, n B-spline basis functions of order p that build theB-spline basisN = NI,pnI=1, are computed according to the Cox-de Boor recursion formula

NI,p(ξ) =ξ − ξI

ξI+p − ξINI,p−1(ξ) +

ξI+p+1 − ξξI+p+1 − ξI+1

NI+1,p−1(ξ) for p ≥ 1 , (2.1)

NI,0(ξ) =

1 if ξI ≤ ξ < ξI+1

0 otherwise. (2.2)

The B-Spline basis N spans the spline space that is also denoted by N . As illustrated in Fig.2.1a, the entities of the knot vector influence the spatial distribution, continuity and interpolationproperty of the individual basis functions. The continuity Cp−mI of the basis at a knot valueξI depends on the polynomial degree p and the multiplicity mI of ξI ∈ Ξ, e.g. C0 continuitymeans the basis is interpolating.

B-spline basis functions possess the following properties that have been shown to be essentialfor well-conditioned and sparse system matrices in isogeometric analysis [26]:

7

2 Preliminaries to Isogeometric Analysis

0 0.3 0.6 0.8 1ξ

0

0.5

1

NI,RI

R2

(a) Cubic B-spline/NURBS basis

0 0.5 1x1

-0.4

0

0.5

1

x2

P 2

(b) Geometric map x(ξ) : R1 → R2

Figure 2.1: Basis functions and geometric map: (a) The cubic B-spline basisN (solid blue lines) for knotvector Ξ = 0 0 0 0 0.3 0.6 0.6 0.8 0.8 0.8 1 1 1 1 has a reduced continuity of C1 at ξ =0.6 and is interpolating, i.e. C0-continuous at ξ = 0.8. The weight w2 of basis function R2

is increased in the corresponding NURBS basis (red dashed lines). (b) The B-spline/NURBScurve is created from the basis functions and corresponding control points P I (•). Due to theincreased weight w2, the NURBS curve (red dashed line) is closer to the control point P 2,in comparison to the B-spline curve (blue solid line). The black crosses (×) in (a) and (b)indicate the boundaries of the Bézier elements in both spaces.

1. Partition of unity:n∑I=1

NI,p(ξ) = 1 , (2.3)

2. Point-wise non-negativity:NI,p(ξ) ≥ 0 , (2.4)

3. Compact support:

NI,p(ξ)

6= 0 ξ ∈ [ξI ,ξI+p+1[

= 0 otherwise. (2.5)

The partition of unity property plays also an important role for the imposition of Dirichletboundary conditions in the numerical analysis. The domain of ΩP where NI does not vanish isdenoted by

suppNI,p = ξ : NI,p(ξ) 6= 0 ∧ ξ ∈ ΩP . (2.6)

Multivariate B-spline basis functions NI,p(ξ) of polynomial degree p = p1, . . . , pdPT anddimensions dP are constructed from tensor products

NI(I1,...,IdP ),p(ξ) =

dP∏k=1

NkIk,pk

(ξk) (2.7)

8

2.1 B-Splines and NURBS

x1

x2

ξ2

ξ1

x(ξ) = PTN(ξ)

(a) Geometric map x(ξ) : R2 → R2

x1

x2

(b) Bivariate knot insertion

Figure 2.2: Geometric map and knot insertion for bivariate B-splines: (a) The bivariate patch ΩP ⊂ R2 ismapped to the physical domain Ω ⊂ R2. The solid lines that subdivide Ω and ΩP, representthe knot lines of the tensor product of Ξ1 and Ξ2. (b) If e.g. the right corner of the specimenis refined by standard knot insertion, the tensor product structure of a multivariate B-splineprohibits a truly local refinement, as illustrated by the red lines.

of k = 1,...,dP univariate B-spline basis functions NkIk,pk

(ξk) that are defined by the knotvectors Ξk. The index map I is defined by I(I1,I2) = n1(I2 − 1) + I1 for dP = 2 andI(I1,I2,I3) = n1n2(I3 − 1) + n1(I2 − 1) + I1 for dP = 3. The knot vectors Ξk define thepatch ΩP =×dP

k=1[ξk1 , ξ

knk+pk+1] ⊂ RdP in the parameter space RdP . Finally, the multivariate

B-spline basis N = NI,pnI=1 is spanned by n = n1 · . . . · ndP multivariate basis functions,where nk is the number of univariate basis functions in parametric direction k.

B-spline and NURBS Geometries

To generate a geometric body in the physical space of dimension dS, the patch ΩP ⊂ RdP ismapped to the physical domain Ω ⊂ RdS by the geometric map x(ξ) : ΩP → Ω

x(ξ) =

n∑I=1

NI,p(ξ)P I , (2.8)

using control points P I ∈ RdS . The geometric map is illustrated in Fig. 2.1b for a univariateand in Fig. 2.2a for a bivariate basis, respectively.

To generate a multi-patch geometry, several patches and geometric maps have to be defined.The patches can be merged easily by connecting overlapping control points if the parametricdiscretizations along the patch-patch interfaces are equal. In case of differing discretizations, aweak coupling is employed using e.g. mortar or Nitsche’s method [36–38].

Because B-splines are piecewise polynomials, shapes like spheres and ellipsoids can not bemodelled exactly. This can be achieved by substituting NI,p by NURBS basis functions thatare weighted rational B-spline basis functions

RI,p(ξ) =NI,p(ξ)wIn∑J=1

NJ,p(ξ)wJ

=NI,p(ξ)wIW (ξ)

. (2.9)

9


102 104

number of inserted knots

10−4

10−2

100

runt

ime/s

OsloOsloVCasciola

(a) Runtime of knot insertion for p = 2

5 10 15polynomial degree

10−2

100

102

runt

ime/s

OsloOsloVCasciola

(b) Runtime of knot insertion for 214 knots

Figure 2.3: Comparison of knot insertion algorithms: (a) The runtime is measured for different amountsof knots that are inserted into an univariate basis of p = 2. (b) Always the same amount of 214

knots is inserted into a univariate basis of different polynomial degree p.

If the weight wI of a basis function is increased, the influence of the corresponding controlpoint in the geometric map is increased too, cf. Fig. 2.1b. If all weights are equal, the NURBSweight function W becomes W (ξ) = wI and RI,p(ξ) = NI,p(ξ).

For simplicity and better illustration, the explanations and definitions in this thesis are based onB-splines. However, the generalization to NURBS is highlighted where it is required. Further-more, it is assumed that pk is equal for all parametric directions. Therefore, it is omitted in theremainder.

Vector-matrix notation is used to shorten the equations. All basis function values at position ξof the B-spline basis N are stored in a row vector N(ξ) and corresponding control points in amatrix

P =

P1,1 P1,... P1,dS

P...,1 P...,... P...,dS

Pn,1 Pn,... Pn,dS

. (2.10)

That allows to rewrite equation (2.8) as

x(ξ) = PTN(ξ) . (2.11)

2.1.2 RefinementIn order to increase the flexibility of design or the approximation quality in the numerical analy-sis, the B-spline function spaceN can be enriched by the refinement techniques knot insertion,order elevation and k-refinement [13]. Knot insertion and order elevation are closely related tothe h- and p-refinement concepts in FEM. However, k-refinement is exclusive for IGA, wherein addition to the polynomial degree also the continuity of the basis is controlled. Order ele-vation and k-refinement techniques are not required for the derivations and explanations in thisthesis and are therefore only briefly presented for completeness.

10

2.1 B-Splines and NURBS

Knot Insertion

During knot insertion, a new knot ξ ∈ [ξI , ξI+1[ is inserted into the univariate knot vector Ξ

which is hence altered to Ξ = ξ1 . . . ξI ξ ξI+1 . . . ξn+p+1, (p < I < n+ 1) and whichresults in a set of new basis functions NJnJ=1. In order to retain the geometry while changingthe parametrisation of the basis, new control points P JnJ=1 have to be computed from theoriginal control points P JnJ=1 according to

P J =

P 1 J = 1αJP J + (1− αJ)P J−1 1 < J < nP n J = n

, (2.12)

αJ =

1 1 ≤ J ≤ I − pξ−ξJ

ξJ+p−ξJ I − p+ 1 ≤ J ≤ I0 J ≥ I + 1

. (2.13)

If several knots have to be inserted, the procedure above can be used to define a knot insertionoperator M. The new control points are then computed by

P = MTP . (2.14)

Due to the fact that the geometry is unchanged

x(ξ) = PTN(ξ) = PTN(ξ) = PTMN(ξ) , (2.15)

the relation between control points translates directly into a transformation rule for the basisfunctions

N = MN . (2.16)

Different methods exist to efficently compute M including the Oslo [39] and Casciola [40]algorithm, as well as the Boehm’s method [41]. The runtime and memory requirements of thesealgorithms were studied by D’ANGELLA et al. [22] and DUFFEK [42]. In Fig. 2.3a, the runtimeof the knot insertion algorithms is measured for different amounts of knots that are insertedinto an univariate knot vector with p = 2. In Fig. 2.3b, always the same amount of 214 knotsis inserted into an univariate knot vector with different polynomial degree p. It is shown thatthe fully vectorised version of the Oslo algorithm (OsloV) performs best, especially if a higheramount of knots has to be inserted or a lower order basis is used. For that reason the vectorisedversion of the Oslo algorithm is used for the computations in this thesis.

During knot insertion, knots can be inserted in between two existing knots in Ξ to increase thenumber of knot spans. This can be interpreted as an h-refinement strategy, where the elementsize of the finite element mesh is reduced, cf. Fig. 2.4b. But knots can also be inserted at anexisting knot position in Ξ. This increases the multiplicity of the knot and reduces the continuityof the basis, as outlined in the previous Section 2.1.1 and illustrated in Fig. 2.1. However, areduced continuity of the basis does not affect the continuity of the corresponding geometricmap.

11


0 0.5 1ξ

0

0.5

1

NI

(a) C1-continuous, quadratic B-spline basis

0 0.25 0.5 0.75 1ξ

0

0.5

1

NI

(b) C1-continuous, quadratic B-spline basisafter knot insertion

0 0.5 1ξ

0

0.5

1

NI

(c) C1-continuous, cubic B-spline basisafter order elevation

0 0.5 1ξ

0

0.5

1

NI

(d) C2-continuous, cubic B-spline basisafter k-refinement

Figure 2.4: B-spline refinement: Three possibilities exist to enrich the B-spline function space illustratedin (a). (b) Knot insertion increases the number of knot spans. (c) After order elevation, thepolynomial degree of the basis is increased, but the continuity keeps constant. (d) After k-refinement, the polynomial degree and the continuity of the basis are increased.

The generalisation to a multivariate refinement operator M of dimension dP is obtained fromthe Kronecker product M = M1 ⊗ ... ⊗MdP of univariate refinement operators MkdP

k=1.

Order elevation and k-refinement

Another possibility to enrich the spline space is order elevation and k-refinement. During orderelevation the polynomial degree of the basis is increased but its continuity remains constant. Anexample is given in Fig. 2.4c, where starting from a C1-continuous, quadratic B-spline basisthe polynomial degree is increased to p = 3 but the continuity remains C1. Efficient orderelevation algorithms are presented by PIEGEL & TILLER [35] and are not reviewed here.

As presented by HUGHES et al. [13] and COTTRELL et al. [14], k-refinement is an exclusiverefinement strategy for IGA, where the order and continuity of the basis are increased simul-taneously. As illustrated in Fig. 2.4d, the increased continuity leads to a reduced number ofbasis functions that are needed to span a spline space of same quality as a spline space afterorder elevation, cf. Fig. 2.4c. Due to the reduced number of corresponding degrees of freedom,k-refinement is also beneficial in a numerical analysis.

During all three refinement strategies the geometry remains unchanged. This is different froman FE model based on Lagrange polynomials, where the geometry is reapproximated after every

12

2.2 Numerical Analysis

refinement. Consequently, IGA is also an ideal discretization technique to be combined withadaptive h-refinement. However, if B-splines or NURBS are considered as a basis, their tensorproduct nature prohibit a truly local refinement within a single NURBS patch as illustrated inFig. 2.2b. For that reason different alternative refinement techniques exist as outlined in Chapter3.

2.2 Numerical AnalysisWith the definitions of the B-spline basis functions and the geometric map at hand, the isoge-ometric analysis of second-order elliptic boundary value problems (BVP) is discussed in thissection. Poisson’s equation is used for demonstration purposes, where u is sought such that

∆u+ f = 0 in Ω,∇u · nN = g on ΓN,

u|ΓD = uD on ΓD.

(2.17)

Here, ∆ is the Laplace operator,∇ the Nabla Operator, f a scalar valued function and Ω ⊂ RdS

the physical body with boundary ∂Ω = ΓD ∪ ΓN and ΓD ∩ ΓN = ∅. The function g is thegradient of u in the direction of the outer normal vector nN and prescribed on the Neumannboundary ΓN. The function uD is prescribed on the Dirichlet boundary ΓD.

The problem above can not be solved analytically on arbitrary domains, but different classesof numerical methods were developed to compute an approximation of u. In the following, theGalerkin method is shortly reviewed and implementation issues discussed.

2.2.1 Weak Form and DiscretizationBefore the weak form and its discretization are discussed, two function spaces are defined. TheL2 space is defined as the function space that contains all square integrable functions. Thismeans that for all u ∈ L2(Ω) is required that ||u||L2(Ω) < +∞, where

||u||L2(Ω) =

(∫Ω

u2 dΩ) 1

2

(2.18)

is the L2-norm of u. The H1 space is defined by

H1(Ω) =u ∈ L2(Ω) : Du ∈ L2(Ω)

(2.19)

where D =(

∂∂x1

. . . ∂∂xdP

). The space contains all functions u ∈ L2(Ω) that also have square

integrable first derivatives. The corresponding H1-norm is given by

||u||H1(Ω) =(||u||2L2(Ω) + ||Du||2L2(Ω)

) 12

. (2.20)

The method of weighted residuals is based on the requirement that a numerical solution doesnot have to satisfy (2.17) in every single point of Ω, but in a weighted and integral sense. For

13


this purpose, the PDE is multiplied by a weight function v and integrated over Ω. Integrationby parts and application of Gauss’s theorem result in the weak form of (2.17) that is defined byseeking u ∈ H1(Ω) such that∫

Ω

∇v · ∇u dΩ =

∫Ω

vf dΩ +

∫ΓN

vg dΓN for all v ∈ H10 (Ω),

u|ΓD = uD ,

(2.21)

where H10 (Ω) =

v ∈ H1(Ω) : v|ΓD = 0

. Following the basic idea of the Galerkin method,

the same function space is used for u and v. To ensure that ΓD ∩ ΓN = ∅, the weight function vis in H1 but has to vanish on ΓD. To shorten the notation in the following, the weak form (2.21)is rewritten in a more general way by a(u,v) = L(v).

Introducing a finite dimensional function space F ⊂ H1(Ω), the weak form is discretized andthe numerical solution uh ∈ F(Ω) sought, such that

a(uh,vh) = L(vh) for all vh ∈ F0(Ω),

uh|ΓD = uD ,(2.22)

where F0(Ω) = vh ∈ F(Ω) : vh|ΓD = 0.

In contrast to FEM, where Lagrange polynomials are used to span the function space F , in IGAthe B-spline basis N that also defines the the physical domain Ω (2.8) is used. Consequently,the solution u and the weight function v are approximated by

uh =∑I

NIuI and vh =∑I

NIvI . (2.23)

The control variables uI and control weights vI belong to the corresponding control points P I .Substituting (2.23) in (2.22) leads to

∑J

vJ

(∑I

a(NJ ,NI)uI − L(NJ)

)= 0 . (2.24)

The expression a(NJ ,NI) illustrates why the H1 space is the appropriate space for u and v,because it requires that the derivatives of N ∈ H1(Ω) are square integrable.

Recalling that (2.24) has to hold for all vh ∈ F0(Ω), it follows that the expression in parenthesishas to vanish. The resulting linear system of equations

Ku = F (2.25)

has no unique solution, as long as the Dirichlet boundary condition of (2.22) is not imposed.Equation (2.25) is written in vector-matrix notation, where K with KJI = a(NJ ,NI) is thecoefficient matrix, F with FJ = L(NJ) the right hand side vector and u the solution vectorcontaining all uI . In the following, K is denoted as the stiffness matrix. The computation of(2.25) and the imposition of the corresponding boundary conditions are outlined in the follow-ing two sections.

14


-1 0 1ξ

0

0.5

1

BI

(a) C1-continuous, quadratic Bernstein polynomials

-1 0 1ξ

0

0.5

1

BI

(b) C2-continuous, cubic Bernstein polynomials

Figure 2.5: Univariate Bernstein polynomials: Univariate Bernstein polynomials are commonly definedover the unit domain ΩU = [−1, 1] and are a fundamental ingredient of Bézier extraction.

2.2.2 Numerical Integration – Bézier ExtractionIn this section the computation of the stiffness matrix K is discussed. In detail, Bézier extractionis applied to enhance existing FE codes with isogeometric analysis features. The concept ofBézier extraction is based on Bernstein polynomials that are introduced in the following.

Bernstein Polynomials

Bernstein polynomials span the basis for Bézier curves and surfaces. Univariate Bernsteinpolynomials of order p are defined over the unit domain ΩU = [−1, 1] ⊂ R by

BI,p(ξ) =1

2p

(p

I − 1

)(1− ξ)p−(I−1)(1 + ξ)(I−1) (2.26)

using a dimensionless coordinate ξ ∈ ΩU, the binomial coefficient(pI−1

)= p!

(I−1)!(p+1−I)!and the index 1 ≤ I ≤ p + 1. As shown in Fig. 2.5, the polynomials are point-wise non-negative, build a partition of unity and are interpolating at the boundary of the unit domain.Comparing the cubic Bernstein polynomials in Fig. 2.5b with the B-spline basis in the intervalξ = [0.8, 1] of Fig. 2.1a, it can be seen that B-spline basis functions are a generalization ofBernstein polynomials.

Similar to (2.7), multivariate Bernstein polynomials BI,p(ξ) of dimension dP are computedfrom a tensor product of univariate polynomials over the unit domain ΩU =×dP

k=1[−1, 1] ⊂

RdP .

Bézier Extraction

In order to obtain the linear system (2.25), the domain is subdivided into elements and Gaussianquadrature is applied. As illustrated in Fig. 2.6, the elements are identified in the parameterspace by a subdivision of the patchΩP =M =

⋃nEE=1 EE in nE elements EE . Recalling Section

2.1.1, the patch is spanned by dP knot vectors Ξk. The element domain EE ⊂ ΩP is given by thenon-zero knot spans in Ξk by EE(E1,...,EdP ) =×dP

k=1

[ξkEk

, ξkEk+1

], whereE(E1, . . . ,EdP) is an

15


ξ1

ξ2

ΩU

-1 1-1

1

EE

ξ1

ξ2

ξ1I1

ξ1I1+1

ξ2I2

ξ2I2+1

x1

x2

x(ξ) = PTE

(CEB(ξ)

)

x(ξ|EE ) = PTENE(ξ|EE )

BE

ΩS

ΩP

Figure 2.6: Element map: The Bézier element BE ⊂ ΩS in physical space is either represented by ageometric map from the element EE ⊂ ΩP in parameter space using (2.27) or from the unitdomain ΩU using Bézier extraction and (2.31).

index map analogue to (2.7). Following (2.8), the element geometric map x(ξ|EE ) : EE → BEcreates a so called Bézier element BE ⊂ Ω in physical space:

x(ξ|EE ) = PTENE(ξ|EE ) . (2.27)

The notation ξ|EE restricts (2.27) to all ξ ∈ EE . Furthermore, NE is a row vector storing allbasis functions of NE = N ∈ N : suppN ∩ EE 6= ∅ and PE is the matrix of correspondingcontrol points.

In order to perform Gaussian quadrature in context of standard IGA, the elements are mappedto a parent element, see COTTRELL et al. [14] for details. But as shown in Fig. 2.7a for theunivariate case, the B-spline basis functions are defined globally over the entire patch and hence,are different in every element. As a result, basis function values have to be computed for everyquadrature point in every element.

Different from that, the standard finite element approach uses the same set of shape functions foreach element in the domain which requires to compute the basis function values at the quadra-ture points only once. The central idea of Bézier extraction is to adopt this idea and to representthe globally defined B-spline basis functions by locally defined Bernstein polynomials.

To obtain Bernstein polynomials BI,p(ξ), knot insertion is applied to all knots in ΞkdPk=1

until the multiplicity of each internal knot equals the polynomial degree p, as illustrated for anunivariate basis in Fig. 2.7b. Using the knot insertion operator (2.14) from 2.1.2 and settingC = M, the so called Bézier control points P are computed from the B-spline control points

P = CTP , (2.28)

16


0.75 10 0.25 0.5

(a)

(b)

(c)

(d)

0.75 10 0.25 0.5

Figure 2.7: Bézier extraction: B-spline curve x(ξ), corresponding control points (•) and element bound-aries (×) are represented in terms of a B-splines basis (a) as well as a Bernstein basis (b) usingB-spline (c) and Bézier control points (d).

and the B-spline basis N(ξ) from the Bernstein basis B(ξ)

N(ξ) = CB(ξ) . (2.29)

The linear operator C is called Bézier extraction operator and maps the piecewiseC0-continuousBernstein polynomials onto the B-spline basis of arbitrary continuity, cf. Fig. 2.7.

In order to allow for a strictly element-based implementation, Bézier extraction operators CEare extracted from C for each element EE ⊂ M. The element extraction operators are used tomap the element basis functions NE to the unit domain ΩU

NE(ξ) = CEB(ξ) , (2.30)

where B(ξ) is a row vector containing the multivariate Bernstein polynomials. With theseinformation at hand, the geometric map x(ξ) : ΩU → BE from the unit domain to the Bézierelement in physical space is given by

x(ξ) = PTE

(CEB(ξ)

)(2.31)

as illustrated in Fig. 2.6.

17


Similar to FEM, the stiffness matrix K, as introduced in Section 2.2.1 is now assembled ele-mentwise, performing Gaussian quadrature on the unit element. This requires the computationof the derivatives of the shape functions with respect to the physical coordinates

∂NE(ξ)

∂xi=

dP∑j=1

CE∂B(ξ)

∂ξj

∂ξj∂xi

, (2.32)

as well as the Jacobian determinant of the geometric map

JE =

∣∣∣∣∂x∂ξ∣∣∣∣ . (2.33)

The derivatives ∂ξj/∂xi are obtained by computing ∂xi/∂ξj from (2.31) and taking there in-verse. Finally, all expressions only depend on the same set of Bernstein polynomials B(ξ) thatare equal for every Bézier element.

The same Bézier extraction procedure can be applied to NURBS basis functions

RE(ξ) = WECEB(ξ)

WE(ξ), (2.34)

with the weight functionWE(ξ) = wT

ECEB(ξ) , (2.35)

the element weight vector wE , and the diagonal matrix of element weights WE . For moredetails on the method and an efficient algorithm to compute the element-local Bézier extractionoperators the reader is referred to BORDEN et al. [19].

2.2.3 Boundary ConditionsSimilar to FEM, Neumann boundary conditions are imposed in IGA by integrating over theNeumann boundary ΓN using e.g. Bézier extraction. However, if Dirichlet boundary conditionsare imposed strongly, e.g. by a penalyty method, the non-interpolatory character of the B-spline basis functions has to be considered. If uD is homogeneous or a constant function, thevalue of uD is imposed directly to all control values that are located along the boundary ΓD,similar as in FEM. In case uD is a non-constant function, uD has to be projected onto all basisfunctions that are non-zero along ΓD. The projection can be performed in the euclidian norm,as e.g. in WANG et al. [43] or in the L2-norm as outlined by e.g. GOVINDJEE et al. [44] andTHOMAS et al. [45].

More details about the projections are given in the next chapter where global and local L2

projections are introduced and numerically compared to impose Dirichlet boundary conditionson a locally refined B-spline space.

However, the solution of the projection is only an approximation of uD ∈ F , as long as thefunction space F 6⊂ N is not part of the B-spline function space defined along the Dirichletboundary. An alternative to the projection is the weak imposition of Dirichlet conditions asoutlined by COTTRELL et al. [14].

18

3 Local Refinement with TruncatedHierarchical B-Splines

This chapter is based on the work in HENNIG et al. [27] and presents a modelling approachfor local refinement with truncated hierarchical B-splines. For this purpose, basic concepts forlocal h-refinement of tensor product B-splines are reviewed. Subsequently, Bézier extractionof truncated hierarchical B-splines is proposed and the equivalence to the basis defined in [24]demonstrated. The strict use of an element viewpoint allows for the application of standardfinite element procedures.

3.1 Local h-Refinement of Tensor Product B-splinesIn the previous Chapter 2 basic concepts of B-splines were introduced. In Fig. 2.2b it was shownthat the tensor product structure of the B-spline basis prevents a truly local refinement within asingle patch. Local refinement with these bases requires the subdivision of the analysis domaininto several patches which then are refined uniformly [14, 46]. However, a weak coupling of thepatches is needed that comes at the cost of additional integrals, which have to be solved alongthe patch boundaries. Furthermore, such approaches would lead to a C0-continuity between thepatches which is not sufficient for the direct discretization of higher-order differential equations.

For that reason, various approaches were developed to overcome the restrictive tensor productstructure. In this thesis Truncated Hierarchical (TH) B-splines, which are a generalised versionof Hierarchical (H) B-splines, are used. They are introduced in detail subsequently. BesideHB- and THB-splines other commonly used refinement techniques exist:

1. T-splines introduced by SEDERBERG et al. [47, 48] result from the insertion of extravertices into the tensor product mesh. This produces so called T-junctions which arecomparable to hanging nodes in the standard finite element method. T-splines enable therepresentation of complex geometries in a single continuous patch and local refinement.BAZILEVS et al. [26] introduced T-splines into isogeometric analysis. Further contri-butions have focussed on the linear independence of the basis [20, 49–52] and on localrefinement strategies [53, 54].

2. Locally Refined (LR) B-splines were introduced by DOKKEN et al. [55]. They are in someway dual to T-splines, i.e. tensor product B-splines are locally refined by the insertionof knot line segments instead of extra vertices. The properties of LR-B-splines wereanalysed by BRESSAN et al. [56] and first applications to adaptive IGA are given byJOHANNESSEN et al. [25].

3. Hierarchical T-splines were introduced by EVANS et al. [57] and combine the strictlylocalised refinement possibilities of hierarchical B-splines with the geometrical represen-tation capabilities of T-splines.

Local mesh refinement using T-splines and HB-splines is illustrated in Fig. 3.1 for the bench-mark case of a sharp internal layer. It can be seen that the T-spline-based refinement is not aslocal as with hierarchical B-splines [53, 57]. The refinement propagates along the parametric

19

3 Local Refinement with Truncated Hierarchical B-Splines

(a) (b) (c)

Sharp internal layer

Figure 3.1: Two approaches to local mesh refinement: (a) Initial mesh with a cubic B-spline basis, degreesof freedom DOF = 76 and an sharp internal layer. (b) Local refinement (four levels) usingHB-splines (DOF = 1444) and (c) T-splines (DOF = 2058).

directions as a consequence of the restrictions imposed by the required analysis suitability, i.e.the linear independence of the basis. This behaviour is further analysed in Section 4.2, whereTHB-splines are compared against T-splines in several benchmark problems.

Multi-Level Basis and Subdivision Operator

The hierarchical approach to local refinement is based on a nested sequence of L multivariateB-spline spaces that are defined over the domain ΩP

N 0 ⊂ N 1 ⊂ ... ⊂ NL−1 , (3.1)

as illustrated in Fig. 3.2 for a univariate basis. In the following, this sequence of spline spacesis referred to as the multi-level basis. The spaces are spanned by the corresponding B-splinebases N ` = N `

In`

I=1 of different levels ` = 0,...,L − 1. The basis function sets are definedby nested knot vectors Ξk,` ⊂ Ξk,`+1, created by successive uniform knot insertion in everyparametric direction k = 1, . . . ,dP.

As a consequence, any B-spline function N `I ∈ N ` can be represented as a linear combination

of basis functions N `+1J ∈ N `+1 of the next finer level

N `I =

∑N`+1

J ∈N `+1

αIJN`+1J , (3.2)

where the coefficients αIJ are computed using the knot insertion algorithm of Section 2.1.2.The relation (3.2) can be re-written in vector-matrix notation

N` = M`,`+1N`+1 (3.3)

with the subdivision operator M and MIJ = αIJ , cf. Fig. 3.2. Following Section 2.1.2, control

values P`+1 of the next finer level are computed by P`+1 =(

M`,`+1)T

P`.

20

3.1 Local h-Refinement of Tensor Product B-splines

0

0.5

1N

0 I

0

0.5

1

N1 I

0

0.5

1

N2 I

N1 = M1,2N2

N0 = M1,2N1

Figure 3.2: Multi-level basis and subdivision operator: Three levels of nested B-spline bases N ` are gen-erated by successive uniform knot insertion. The coarse scale basis functions are representedby the fine scale basis functions in terms of the subdivision operator M`,`+1.

Subdivision operators for two non-consecutive levels are obtained from

NI = MI,JNJ =

J−1∏`=I

M`,`+1NJ . (3.4)

For a generalisation to NURBS, the subdivision operator has to be modified to interact betweenrational basis functions RÌn

`

I=1, and R`+1J n`+1

J=1 of two consecutive levels. The denominatorof R`+1

J is unchanged during the refinement process

R`+1J (ξ) =

w`+1J N `+1

J (ξ)

W `+1(ξ)=w`+1J N `+1

J (ξ)

W `(ξ)(3.5)

for a fixed geometry [58]. Substituting Equation (3.3) inRÌ(ξ), and replacingN `+1J (ξ) in terms

of Equation (3.5), a two-scale relation for rational B-splines is given by

RÌ(ξ) =wÌN

Ì (ξ)

W `(ξ)=wÌ∑JM

`,`+1IJ N `+1

J (ξ)

W `(ξ)= wÌ

∑J

M `,`+1IJ

w`+1J

R`+1J (ξ) . (3.6)

For NURBS, the rational subdivision operator

M `,`+1IJ = wÌ

M `,`+1IJ

w`+1J

(3.7)

replaces the standard subdivision operator M`,`+1.

21


0

0.5

1

N0 I

0 0.2 0.4 0.6 0.8 1ξ

0

0.5

1

NI

0

0.5

1

N1 I

0

0.5

1

N2 I

Ω0P

Ω1P

Ω2P

(a)

(b)

Figure 3.3: Hierarchical and truncated hierarchical B-splines: (a) The area of refinement is defined byΩ`P .(b) Compared to HB splines (solid and dashes lines), THB-splines (only solid lines) reducethe interaction between basis functions of different levels and recover the partition of unityproperty.

Hierarchical B-splines

Hierarchical B-splines has benn introduced by FORSEY & BARTELS [59] already in 1988 andwere further developed [21, 23, 58, 60, 61] to meet the requirements of IGA. To create a locallyrefined basis, refinement areasΩ`P are defined on each level of the multi-level basis as illustratedin Fig. 3.3a. The parameter domains Ω`P have to be nested

ΩP = Ω0P ⊇ Ω1

P ⊇ ... ⊇ ΩLP , (3.8)

and the boundaries of Ω`P have to be aligned with the knots in Ξk,`.

In order to span a hierarchical B-spline basisAHB, basis functions from finer hierarchical levelsare not simply added to the coarse scale basis. This would result in the loss of linear inde-pendence, due to the nestedness of the spline spaces. To avoid linear dependencies, only basis

22

3.2 Bézier Extraction of Truncated Hierarchical B-splines

functionsN ∈ N `, with a support entirely located inΩ`P but not inΩ`+1P , are combined. Hence,

the hierarchical spline basis reads

AHB =

L−1⋃`=0

N ∈ N ` : suppN ⊆ Ω`P and suppN 6⊆ Ω`+1

P

. (3.9)

A univariate HB-spline basis is illustrated by the solid and dashed lines in Fig. 3.3b.

Truncated Hierarchical B-splines

The hierarchical basis inherits the properties from the basis functions it is created from, butdoes not satisfy the partition of unity property. Truncated hierarchical B-splines, introducedby GIANNELLI et al. [24], recover this property by an alternative definition of the basis. Thecoresponding basis functions span the same space AHB but have reduced overlap and henceprovide lower condition numbers and sparser system matrices when applied to the discretizationof a PDE [58, 62]. Properties of THB-splines are studied by GIANNELLI et al. [63] and apossible data structure for the construction of THB-splines based on quad-trees is provided byKISS et al. [64].

The basic idea in the construction of a THB-spline basis is to truncate the basis functionsN ∈ N ` that have support in Ω`+1

P . In the example in Fig. 3.3b all dashed basis functionsare truncated which leads to a reduced interaction with basis functions from adjacent levels.Recalling equation (3.2), the level to level truncation operation for a B-spline basis N `

I ∈ N ` isdefined as

trunc`N `I =

∑N`+1

J ∈(N `+1\AHB)

αIJN`+1J , (3.10)

where only basis functions N `+1J ∈ N `+1 are considered that not belong to AHB. Finally, the

truncated hierarchical B-spline basis is defined as

ATHB = TruncN : N ∈ AHB , (3.11)

whereTruncN = truncL−1(. . . (trunc`+1(trunc`N)) . . .) (3.12)

is the successive truncation with respect to all higher levels of a basis function N ∈ N `. Inequation (3.12), the truncation operation (3.10) is applied successively starting with the trun-cation of basis N ∈ N ` and proceeding with the truncation of the next finer levels until levelL− 1 is reached. This is necessary, if basis function of non-consecutive levels interact.

3.2 Bézier Extraction of Truncated HierarchicalB-splines

This section provides a framework that allows to use the THB-spline basis ATHB to discretizeboundary value problems with a Galerkin method as introduced in Section 2.2.1. Using anelement based approach similar to [21, 22, 27], Bézier extraction of truncated hierarchical B-splines is proposed. In this way, the truncation operation (3.12) does not have to be performed

23


during the integration of the system matrices and standard FE procedures can be seamlesslytransferred to the framework of isogeometric analysis.

In detail, standard Bézier extraction as introduced in Section 2.2.2 is used on different levelsof the hierarchical mesh. The resulting global system of equations is referred to as multi-levelsystem because entries resulting from different refinement levels do not communicate with eachother. The introduction of a hierarchical subdivision operator recovers this communication anda matrix multiplication is used to obtain the final system of equations. It is shown that this novelprocedure produces the truncated hierarchical basis presented by Giannelli et al. [24].

The following explanations are valid for B-spline and NURBS bases of arbitrary dimension.For a better understanding, the framework is illustrated on an univariate B-spline basis.

3.2.1 Active Elements and Basis FunctionsTo define the hierarchical basis in an element-based framework, a multi-level mesh and basisare required. The multi-level basis consists of a hierarchy of ` = 0 . . . L − 1 nested splinespaces as introduced in Section 3.1 and illustrated in Fig. 3.4 for a univariate basis with p = 2.The total number of basis functions in the multi-level basis is denoted by n =

∑` n

`.

According to the framework of Bézier extraction, cf. Section 2.2.2, a number of nÈ elementswith domain EÈ is defined on every level by the non-zero intervals of the patch spanned byknot vectors Ξk,`dP

k=1. These elements define the sets M` =⋃E EÈ . Assuming that two

consecutive levels result from uniform subdivision of each element in two (dP = 1), four (dP =2) or eight (dP = 3) subdomains, a hierarchy of nested elements is obtained that is called themultilevel mesh. In this tree structure each parent element on level ` possesses e.g. two childrenon level `+1 for the univariate case, cf. Fig. 3.4. The total number of elements in the multi-levelmesh is denoted by nE =

∑` n

È.

To generate a locally refined mesh and the corresponding hierarchical spline space, the follow-ing definitions are needed:

1. Active elements: By some marking criterion elements of different hierarchy levels haveto be chosen to discretize the analysis domain. These so called active elements EÈ ∈M`

define subsets Mà =

⋃E EÈ ⊂ M` and cover the domain ΩP = M =

⋃L−1`=0 M`

awithout any overlap. This is shown in Fig. 3.4 where active elements are indicatedin green. Additionally, the sets M`

− =⋃`−1I=0MI

a that contain all active elements oncoarser hierarchy levels andM`

+ =⋃L−1I=`+1MI

a that contain all active elements on finerhierarchy levels than ` are defined.

2. Active basis functions: Every active element is associated to a number of (p+ 1)dP basisfunctions in the multi-level basis. The union of these basis functions on each level ` isdenoted by

A` = N ∈ N ` : suppN ∩Mà 6= ∅ (3.13)

and plotted in colour in Fig. 3.4. However, to ensure a linearly independent basis not allof these functions can contribute to the hierarchical approximation, i.e. attention has tobe paid to basis functions whose support overlaps with the domains of active elements on

24


0 0.2 0.4 0.6 0.8 1ξ

0

0.5

1N

0 I(ξ

)

0

0.5

1

NI(ξ

)

0

0.5

1

N1 I(ξ

)

0

0.5

1

N2 I(ξ

)

Figure 3.4: Different sets of basis functions in the multi-level basis: Based on the basis function sets A`– basis functions belonging to active elements (all coloured lines), A`− – basis functions withsupport inM`

− (dotted lines) and A`+ – basis functions with support inM`+ (dashed lines),

the hierarchical basis AHB, is defined.

finer or coarser hierarchy levels. In this context, the definition of two subsets is conve-nient:

a) Basis function set A`+ : Basis functions in A` with support in the domains of activeelements on finer hierarchy levels form the set

A`+ = N ∈ A` : suppN ∩M`+ 6= ∅ (3.14)

and contribute actively to the hierarchical approximation. Corresponding functionsare indicated in Fig. 3.4 by dashed coloured lines.

b) Basis function set A`− : To ensure a linearly independent hierarchical approxima-tion, basis functions ofA` with support in the domain of active elements on coarserlevels cannot contribute to the hierarchical basis. Functions belonging to these sets

A`− = N ∈ A` : suppN ∩M`− 6= ∅ (3.15)

25


are plotted by dotted coloured lines in Fig. 3.4.

With these definitions at hand, the hierarchical basis AHB (3.9) can be defined from anelement viewpoint

A = AHB =

L−1⋃`=0

Aà . (3.16)

Here, Aà = A` \A`− are referred to as active basis functions of level `. Furthermore, theset of basis functions that are later used in the integration is defined by

Ai =

L−1⋃`=0

A` (3.17)

and contains all basis functions that belong to active elements.

To keep a clear notation in the definitions below, vectors and matrices are amended by thecorresponding function sets given in brackets. For instance, the expression u

[A`]

reduces thefull vector u of size n` to

uK[A`]

=

uI if

(N Ì ∈ A`

)∅ else , (3.18)

where I counts over the basis functions NI ∈ N ` of the corresponding level ` and K = f(I) isan index map from the full to the reduced vector. Analogously the expression M [A,Ai] reducesa full matrix M of size n× n to

MKL [A,Ai] =

MIJ if (NI ∈ A) ∧ (NJ ∈ Ai)∅ else , (3.19)

where I and J count over the basis functions NI ∈⋃L−1`=0 N ` and K = f(I) and L = g(J) are

the index maps from the full to the reduced matrix.

3.2.2 Assembly of Global System of EquationsIn the preceding section, procedures for selecting the elements and basis functions that con-tribute to the hierarchical approximation AHB were introduced. In the following, a linear sys-tem of equations according to (2.25) is assembled for a THB-spline basis ATHB, where thetruncation operation (3.12) has not to be carried out explicitly. Instead, the system matricesare obtained from Bézier extraction performed for all active elementsM` of each level `. Therequired truncation is eventually achieved in terms of a hierarchical subdivision operator.

The procedure consists of three steps:

1. At first the element matrices of all active elements inM` are computed using the basisA` on each individual level ` and hence without considering information on whether thebasis function contributes to the hierarchical basis or not. This ensures the applicabilityof standard Bézier extraction as introduced in Section 2.2.2.

26


(a) (b)

(c) (d)

K0

K1

K2 I2

I1

I0 M0,1 M0,2

M1,2

Figure 3.5: Sparsity patterns of the stiffness matrices and the hierarchical subdivision operator for thehierarchical basis of Fig. 3.4: (a) The coefficient matrix K [Ai,Ai] with clear separation ofthe different levels ` is illustrated. (b) In the hierarchical subdivision operator Mh [A,Ai], theentries on the diagonal result from the activity indicators I` and off-diagonal elements fromMI,J

. (c) The hierarchical coefficient matrix K [A,A] is computed from (3.22). In (a), (b)and (c), the full matrices with all the zero entries are plotted. A sparsity pattern of a moreefficiently stored matrix K [A,A] is illustrated in (d).

2. The matrices of active elements of each hierarchy level ` are assembled into sub-systemsof equations

K[A`,A`

]u[A`]

= F[A`]. (3.20)

The sub-systems are combined to form the global system of equations:K0 0 . . . 00 K1

.... . .

0 KL−1

︸︷︷︸

K[Ai,Ai]

ΛF[A0]

ΛF[A1]

...ΛF[AL−1

]

︸︷︷︸ΛF[Ai]

=

F[A0]

F[A1]

...F[AL−1

]

︸︷︷︸F[Ai]

, (3.21)

where K` = K[A`,A`

].

27


3. In the multi-level system of equations (3.21) there is no communication between indi-vidual levels. This interconnection is introduced in terms of the hierarchical subdivisionoperator Mh. It acts as a transformation matrix on the multi-level system by transferringthe contributions from all basis functions in Ai to the hierarchical basis A

K [A,A] = Mh [A,Ai] K [Ai,Ai] MTh [A,Ai] , (3.22)

F [A] = Mh [A,Ai] F [Ai] , (3.23)

which yields the hierarchical system of equations

K [A,A] u [A] = F [A] . (3.24)

It is shown below that the basis used in (3.24) corresponds to the truncated hierarchicalbasis, A = ATHB. Hence, the numerical solution of the underlying boundary valueproblem is approximated by truncated hierarchical B-splines uh = u [A]

T N [A], whereN is a row vector containing all N ∈ ATHB. The THB-spline basis is denoted byA in thefollowing.

For demonstration purposes, in Fig. 3.5 the sparsity patterns of the stiffness matrices and thehierarchical subdivision operator are illustrated for the hierarchical basis of Fig. 3.4. In Fig.3.5a, b and c, the full matrices with all the zero entries are plotted. The empty spaces in thecharacteristic band structure of the stiffness matrix K [Ai,Ai] result from inactive elements oneach level and illustrate the missing connection between the individual levels. It is noted that inorder to reduce memory requirements, efficient matrix libraries with adjusted indexing are usedto avoid zero rows and columns in the multi-level and hierarchical system matrices as well asin the subdivision operator. The sparsity pattern of such a reduced coefficient matrix K [A,A]is illustrated in Fig. 3.5d.

Hierarchical Subdivision Operator

The hierarchical subdivision operator Mh is an upper triangular matrix with elements

Mh =

I0 M

0,1M

0,2. . . M

0,L−1

I1 M1,2

. . . M1,L−1

I2 . . . M2,L−1

0. . .

IL−1

. (3.25)

It has to transfer the contributions from all active basis functions of the multi-level to the hier-archical system of equations. This is accomplished in terms of an activity indicator matrix I`

for each level

I`(II) =

1 for NI ∈ A`a0 else and I`IJ = 0 for I 6= J . (3.26)

In addition, basis functions belonging to active elements but not to the hierarchical basis, i.e.the functions in the sets Ak−, were used during the integration of the coefficient matrices of

28


hierarchy levels k > 0. Hence, contributions of basis functions of Ak− have to be transferredto active basis functions of level ` < k, particularly to those in the set A`+. The relationbetween these basis functions is given in terms of the subdivision operator according to equation(3.3) and (3.4), respectively. Since these are the only transformations required, a modifiedsubdivision operator

M `,kIJ =

M `,kIJ for N `

I ∈ A`+ ∧NkJ ∈ Ak−

0 else(3.27)

is used in Mh.

Non-Linear Problems

Non-linear problems require iterative solution procedures as the Newton method. The compu-tation of the stiffness matrix iK[Ai,Ai] and the right hand side vector iF[Ai] is than required ineach iteration i. However, the matrices depend on the field variables i−1u [Ai] of the previousiteration i− 1. In order to compute the involved integrals, the solution of the previous iterationi−1u[A], cf. (3.24), has to be transferred from the basis A to the basis Ai solving

i−1u [Ai] = MTh [A,Ai]

i−1u [A]. (3.28)

3.2.3 Equivalence to the Truncated Hierarchical BasisAlthough the hierarchical basis is never computed explicitly in the present approach, it is shownthat the resulting numerical approximation that is computed from (3.24) is identical to an ap-proximation that is computed directly from a truncated hierarchical basis ATHB.

The correspondence of the present approach and the truncated basis results from the procedureused to compute the coefficient matrices and from the mapping of contributions of basis func-tions in A`+1

− to those in A`+. In order to understand this effect, the basis functions N03 and N0

4

according to Fig. 3.4 are considered as an illustrative example in Fig. 3.6.

During the multi-level Bézier extraction, standard extraction operators are computed for eachactive element and numerical integration is carried out subsequently. This has two major impli-cations:

1. All basis functions supported in an active element are considered during quadrature irre-spectively of the fact if they are part of the hierarchical basis.

2. The contribution of basis functions on level ` is limited to the domains of active elements,i.e. the integrated domain.

The basis functions N01 to N0

4 fully add to the coefficient matrix K0 during quadrature of theactive elements with domains E0

1 and E02 . However, the remainder of the support of N0

3 and N04

on the coarsest level ` = 0 is not considered during numerical integration as it is not coveredby an active element, cf. Fig. 3.6. Instead, integration continues with active elements withdomains E1

5 and E16 . In these two elements, the basis functions N1

5 and N16 , which are not part

of the hierarchical basis, contribute to the matrix K1.

29


0

0.5

1

N0 I(ξ

)

0

0.5

1

N1 I(ξ

)

0

0.5

1

N2 I(ξ

)

0 0.2 0.4 0.6 0.8 1ξ

0

0.5

1

NI(ξ

)

N03 N0

4

N15 N1

6

truncN04

E01 E0

2

E15 E1

6

Figure 3.6: Relation of the hierarchical subdivision operator to the truncated hierarchical basis: The re-mainder of the active functions N0

3 and N04 is cut off during multi-level Bézier extraction. On

the other hand, the non-active basis functionsN15 andN1

6 that are considered during numericalintegration of E1

5 and E16 . The hierarchical subdivision operator Mh maps the matrix elements

in the stiffness matrix resulting from N15 and N1

6 to those related to N03 and N0

4 that leadsautomatically to truncation of these functions. As a result, the whole domain is integratedimplicitly, using truncated hierarchical B-splines. The effect of Mh is indicated by the greydomains.

Eventually, the hierarchical subdivision operator accounts for the activity of the basis functionsand recover the correct support of active basis functions. The matrix elements resulting fromN1

5 and N16 are mapped onto those related to N0

3 and N04 . In this way the contribution of

the active basis functions N03 and N0

4 in the domains E15 and E1

6 is taken into account. Asthe support of N1

5 and N16 is limited to these two active elements on level ` = 1, the support

domain of N04 is truncated according to (3.2). The resulting basis is identical to the proposal

of GIANNELLI et al. [24]. Therefore, the presented approach provides approximations thatpossess the partition-of-unity property and a reduced interaction of basis functions betweendifferent refinement levels.

30

3.3 Imposition of Dirichlet Boundary Conditions

3.3 Imposition of Dirichlet Boundary ConditionsAs already mentioned in Section 2.2.1, the computed linear system of equations (3.24) does notpossess a unique solution as long as the Dirichlet boundary condition u|ΓD = uD is not imposed.Due to the non-interpolatory character of the B-spline basis functions, a projection is requiredif the Dirichlet boundary condition uD is neither zero nor constant.

The goal of the projection is an approximation of the function uD on the Dirichlet boundary ΓD

T (uD) = uT [A]N[A]. (3.29)

Here, T is the projection operator, N a row vector containing all basis functions N ∈ A andu a row vector that contains the desired control variables of the Dirichlet boundary conditionthat are imposed strongly into (3.24). The THB-spline basis A is defined by all basis functionsN ∈ A that does not vanish at the boundary ΓD

A = N ∈ A : suppN ∩ ΓD 6= ∅ . (3.30)

The L2-projection is obtained by minimising the error of the approximation in the L2-norm,

minu[A]‖uD − T (uD)‖2L2(ΓD) . (3.31)

The minimisation leads to a linear system of equations∫ΓD

NT [A]N[A]

dΓD︸︷︷︸G[A,A]

u[A]

=

∫ΓD

uDN[A]

dΓD︸︷︷︸F[A]

, (3.32)

where G is denoted as the Gramian matrix and F is the right hand side.

To compute (3.32), the Bézier extraction framework and the hierarchical subdivision operatorof the previous section are used

Mh[A,Ai

]G[Ai,Ai

]MT

h

[A,Ai

]u[A]

= Mh[A,Ai

]F[Ai], (3.33)

that allows to compute G[Ai,Ai

]and F

[Ai]

on the multi level basis using standard Bézierextraction. Here, the function set Ai is given by

Ai = N ∈ Ai : suppN ∩ ΓD 6= ∅ . (3.34)

Assuming that the function uD is in some function space F , the projection T (uD) is exact ifF ⊂ A. Otherwise, T (uD) is only an approximation of uD.

The linear system of equation (3.32) grows with the problem size. To avoid the solution ofoversized linear systems of equations, GOVINDJEE et al. [44] proposed a projection for non-hierarchical B-splines, where the function is projected elementwise and an average is applied tosmooth the discontinuous solution on the control variables subsequently. THOMAS et al. [45]proposed so called Bézier projection, where the inversion of the Gramian matrix is prevented,

31


100 105

Degrees of freedom

10−5

100

rel.

erro

rinL

2

globallocallocal+w 1

3

(a) univariate, quadratic THB-spline basis

100 105

Degrees of freedom

10−10

10−5

100

rel.

erro

rinL

2

globallocallocal+w 1

4

(b) univariate, cubic THB-spline basis

Figure 3.7: Projection of Dirichlet boundary conditions for uniformly (green) and locally (red) refinedbasis: All local and global projection methods converge with the optimal rate of p+ 1.

but an inverted Bézier extraction operator is needed. The main improvement of the latter projec-tion is the advanced weighting scheme. To the best of author’s knowledge, the local projectionmethods have not applied to a locally refined THB-spline basis, yet. A straight forward andeasy implemented way is a local projection not onto the hierarchical basis A but onto the multilevel basis Ai as performed and analysed in the following example.

The global (3.32) and the two local projection methods, motivated by GOVINDJEE et al. [44]and THOMAS et al. [45], are compared by projecting the function uD = 1/(3(2.003π − x))with x ∈ [0, 2π] onto an uniformly and a locally refined univariate THB-spline bases. Asshown in Fig. 3.7, all local and global projection methods converge with the optimal rate ofp + 1 if the error is measured in L2-norm. Due to the higher gradients of uD around x = 2π,the locally refined bases, improve the convergence rate in the preasymptotic regime. The errorof the projection increases from the global to Bézier to local projection, especially for p = 3.

3.4 ConclusionsIn this chapter local refinement with truncated hierarchical B-splines was proposed, includingthe Bézier extraction of THB-splines and the imposition of Dirichlet boundary conditions.

While Bézier extraction provides a multi-level system of equations with independent levels,the communication between different levels of the hierarchy was introduced in terms of a hi-erarchical subdivision operator for B-splines and NURBS. It was shown that a simple matrixmultiplication produces a hierarchical system of equations which is identical to the use of thetruncated hierarchical basis. The explicit truncation of the basis is avoided.

The use of the element viewpoint and the application of Bézier extraction facilitate the imple-mentation of the proposed framework into any existing finite element code. Moreover, standardprocedures of adaptive finite element analysis are directly applicable as it is demonstrated in thenext Chapter 4. Also numerical examples are provided in this chapter to verify the approach.

32

3.4 Conclusions

The efficiency of the framework in terms of reduced computation time is demonstrated in theapplication on different phase-field models in Chapter 6.

To impose Dirichlet boundary conditions on a THB-spline basis by an L2-projection, the Bézierextraction framework can be used, too. It was shown that the projection leads to optimal con-vergence rates for the global and element-wise projections. The global projection method isused for the computations in this thesis.

33

4 Adaptive Isogeometric Analysiswith Truncated HierarchicalB-Splines and T-splines

This chapter is based on the work in HENNIG et al. [28] and presents adaptive isogeometricanalysis based on truncated hierarchical B-splines. For this purpose, two different refinementstrategies and a coarsening strategy are introduced for THB-splines and integrated into an adap-tive meshing loop.

The two different refinement strategies are:

1. refine+: A refinement based on THB-splines as proposed by e.g. SCOTT et al. [21]and HENNIG et al. [27], where the mesh allows only for a one-level difference betweenneighbouring mesh elements.

2. refine++: A refinement based on THB-splines, where only 2-admissible meshes are al-lowed. As introduced by BUFFA & GIANNELLI [65], in m-admissible meshes basisfunctions of up to m different levels are allowed to interact in an element.

As stated in the introduction of Chapter 3, a widely used and common alternative to THB-splines are T-splines. For that reason, this chapter will not only focus on a verification andcomparison of the THB-spline refinement strategies above, but also on a comparison betweenTHB-splines and T-splines in general. The following two different refinement strategies forT-splines will be considered:

3. refine_tspline+: A T-spline refinement as introduced by SCOTT et al. [54], where therefinement process is divided into two steps. At first, marked elements are refined andsecondly, an additional refinement is processed to recover the linear independence of theT-spline functions, called analysis-suitability.

4. refine_tspline++: A T-spline refinement as introduced by MORGENSTERN & PETER-SEIM [51] and MORGENSTERN [52], where also the vicinity of the marked element isconsidered. By defining a class of admissible T-meshes, the proposed refinement pre-serves the analysis-suitability of the T-splines directly.

The methods above are referred to as greedy refinement (method 1 and 3) and safe refinement(method 2 and 4). Differences between this two classes are illustrated in Fig. 4.1, where aninitial square mesh with 64 elements is locally refined in the lower left corner.

The safe refinement strategies allow for a mathematical proof of linear complexity[51, 66].Together with results on the convergence of the adaptive algorithm [65], this facilitates a math-ematical proof of optimal convergence rates [67]. In this chapter these findings are verifiednumerically with a focus on the influence of the different mesh classes.

To obtain suitable graded THB-spline meshes the here presented coarsening strategy is splitinto two steps. At first, marked elements are coarsened and secondly, some elements are re-

35

4 Adaptive Isogeometric Analysis with THB-Splines and T-splines

T-sp

lines

safe refinementgreedy refinementT

HB

-spl

ines

Figure 4.1: Refinement strategies: An initial square mesh with 64 elements is locally refined in the lowerleft corner using THB- and T-splines with a greedy and a safe refinement strategy. The illus-trated meshes are the Bézier meshes.

fined again to meet the requirements outlined in strategy 1 or 2 above. Please note that CAR-RATURO et al. [12] presented an alternative one step strategy to obtain the admissible meshdirectly. LORENZO et al. [68] developed a so called function support balancing to guarantywell graded meshes after coarsening.

In the following section the two different refinement strategies and the coarsening strategy forTHB-splines are outlined. Subsequently, the computational comparison between THB- andT-splines is conducted. In numerous two and three-dimensional examples, including singularand non-singular problems of elasticity and the Poisson problem, the H1-errors of the discretesolutions, the degrees of freedom as well as the sparsity patterns and condition numbers of thediscretized problems are compared. The examples are well-established benchmark problems inthe context of adaptive isogeometric analysis [23, 27, 53, 69].

For more information regarding the T-spline refinement strategies, the reader is referred to HEN-NIG et al. [28].

36

4.1 Adaptive Mesh Refinement and Coarsening

4.1 Adaptive Mesh Refinement and CoarseningDue to the use of the Bézier extraction framework of the previous chapter and the correspond-ing element viewpoint, the application of the THB-spline basis to adaptive IGA, follows thestandard iterative procedure of adaptive finite element analysis. This procedure consist of thesteps

SOLVE→ ESTIMATE→ MARK→ REFINE/COARSEN

which are described in the following.

Solve: A numerical solution is computed for the given field problem on a given mesh hM.

Estimate: The error of the solution ηE is estimated for every active element E ∈ hM. Inthe numerical studies in Sections 4.3 and 4.4 below, a residual based a posteriori errorestimator is used. If no error estimator is available, other significant quantities can becomputed as it is demonstrated in Chapter 6.

Mark: Given the results of the previous step and a suitable threshold value θ, elements S ⊆hM are marked for refinement or coarsening. In the numerical studies in Sections 4.3and 4.4 below, elements are marked for refinement based on the estimated element errorsηE : E ∈ hM ⊂ R and the marking parameter α ∈ [0,1], which is chosen manually. Indetail, a quantile marking is used where α percent of the elements with the highest errorare marked: Let hM = E1, . . . ,EK and ηE1 ≥ · · · ≥ ηEK , then S = E1, . . . ,Ek withk ≈ αK. Further alternative marking strategies are presented by e.g. HENNIG et al. [28].

Refine/Coarsen: Based on the individual refinement and coarsening strategies, the markedelements are refined or coarsened and a new mesh h+1M is generated. Subsequently, anew iteration is started.

The refinement/coarsening strategies below are described as

h+1M = refine(hM,S) , (4.1)h+1M = coarsen(hM,S) , (4.2)

where for refinement, the new mesh is obtained by

h+1M = hM\ S ∪ subdivide(S) (4.3)

and for coarsening byh+1M = hM\ S ∪ unite(S) . (4.4)

The subdivide operation in (4.3) is performed by deactivating the elements in S and activatingall corresponding child elements in the multi-level mesh, cf. Section 3.2.1. However, the uniteoperation in (4.4) is performed by deactivating the elements in S and activating all correspond-ing parent elements in the multi-level mesh. Note that the corresponding parent element is onlyactivated if all its child elements are marked for coarsening. Furthermore, the level k of anactive element E ∈ hM is denoted by `(E) = k.

37


→ →

Figure 4.2: Example for the greedy THB-spline refinement: First, an element E ∈ hM is marked (high-lighted in blue), hence S = E. Second, closure+(S,hM) is computed (highlighted in blue).Third, all elements in closure+(S,hM) are subdivided.

→ →

Figure 4.3: Greedy THB-spline refinement in a worst case scenario: First, an element E ∈ hM is marked(highlighted in blue), hence S = E. Second, closure+(S,hM) is computed, which nowcoincides with the actually marked element. Third, all elements in closure+(S,hM) are sub-divided, which is only E .

4.1.1 Greedy Refinement StrategyThe greedy THB-spline refinement allows only for a one-level difference between neighbouringmesh elements. To ensure this restriction, first for each E ∈ hM the coarse neighbourhood

H+(E ,hM) =E ′ ∈ hM : `(E ′) < `(E), E ∩ E ′ 6= ∅

(4.5)

is defined that contains all directly neighboured coarser active elements. Equation (4.5) is gen-eralized byH+(S) =

⋃E∈S H+(E) and

Hk+

(S) = H+(. . .H+(︸︷︷︸k times

S) . . . ) . (4.6)

In (4.6) the coarse neighbourhood is computed again for every element that is in the coarseneighbourhood of a previously identified element. This procedure is repeated until no furtherelements have a coarse neighbourhood. With the definition of the closure

closure+(S,hM) = S ∪ Hmax `(S)+

(S), (4.7)

the extended refinement procedure is defined

refine+(hM,S) = refine(hM, closure+(S,hM)

), (4.8)

where not only S but the closure of S is refined.

38


→ →

Figure 4.4: Example for the safe THB-spline refinement: First, an element E ∈ hM is marked (high-lighted in blue), hence S = E. Second, closure+(S,hM) is computed (highlighted inblue). Third, all elements in closure+(S,hM) are subdivided.

→ →

Figure 4.5: Safe THB-spline refinement in a worst case scenario: First, an element E ∈ hM is marked(highlighted in blue), hence S = E. Second, closure+(S,hM) is computed. Third, allelements in closure+(S,hM) are subdivided.

The greedy refinement strategy is illustrated in two examples in Fig. 4.2 and Fig. 4.3 for a bi-variate cubic spline basis. While in Fig. 4.2 some further elements are marked for refinement,in Fig. 4.3, the greedy refinement refines only the initially marked element that leads to a steepgrading in the mesh.

4.1.2 Safe Refinement StrategyThe safe refinement for THB-splines defined below is conceptionally similar to the refinementprocedure described by BUFFA & GIANNELLI [65] and BUFFA et al. [66]. It only differs in theconstruction of the neighbourhood H++ . The resulting meshes are 2-admissible, meaning thatinteracting basis functions in an element belong to at most two different levels. To ensure the 2-admissibility during refinement, we first define for each active element E ∈ hM the same-levelneighbourhood

HSL(E) =E ′ ∈M`(E) : ∃N ∈ A`(E) : E ⊂ suppN ⊃ E ′

, (4.9)

that consists of all elements that are of the same level as E and that share the support of at leastone basis function with E . The coarse neighbourhood is then defined by

H++(E ,hM) =E ′ ∈ hM : `(E ′) < `(E), ∃E ′′ ∈ HSL(E) : E ′′ ⊂ E ′

, (4.10)

that consists of all active elements that are also parent elements of the elements in the same-levelneighbourhood of E .

39


→ →

Figure 4.6: Example for the greedy THB-spline coarsening: In the first step, an intermediate mesh isgenerated by coarsening the subset S of the marked elements S (highlighted in red). In thesecond step, the coarse neighbourhood of the parent elements of E ∈ h+1/2M is computed(the blue domain indicates the coarse neighbourhood of the parent element of the green region)and elements refined again if necessary.

As for the greedy refinement (4.10) is generalized toH++(S) =⋃E∈S H++(E) andHk

++(S) =

H++

(. . .H++

(︸︷︷︸k times

S) . . . ) . With the definition of the closure

closure++(S,hM) = S ∪ Hmax `(S)++

(S), (4.11)

the extended refinement procedure is defined

refine++(hM,S) = refine(hM, closure++(S,hM)

), (4.12)

where not only S but the closure of S is refined.

Examples for the safe refinement strategy are given in Fig. 4.4 and Fig. 4.5 for a bivarite cubicspline basis. It is clearly visible that the safe refinement strategy leads to a smoother mesh grad-ing compared to the greedy refinement. Due to the definition of the same-level neighbourhood(4.9), the mesh grading also depends on the polynomial degree. The higher the polynomialdegree, the smoother is the mesh grading.

4.1.3 Coarsening StrategyTo obtain also well graded meshes that satisfy the requirements defined for the greedy and saferefinement after coarsening, a two step strategy is applied:

1. An intermediate mesh h+1/2M is generated

h+1/2M = coarse(hM,S) (4.13)

by coarsening only a subset of marked elements S = E ∈ S : P(E ,S) 6= ∅. The set Scontains only elements that have parent elements

P(E ,S) =E ′ ∈M`(E)−1 : E ⊂ E ′ ⊂ S

(4.14)

that are fully located in S .

40


→ →

Figure 4.7: Example for the safe THB-spline coarsening: In the first step, an intermediate mesh is gener-ated by coarsening the subset S of the marked elements S (highlighted in red). In the secondstep, the coarse neighbourhood of the parent elements of E ∈ h+1/2M is computed (the bluedomain indicates the coarse neighbourhood of the parent element of the green region) andelements refined again if necessary.

2. For all elements E ∈ h+1/2M corresponding parent elements P(E ,h+1/2M) are iden-tified and their coarse neighbourhood with respect to the mesh h+1/2M computed. Therefinement of this coarse neighbourhood generates the admissible mesh

h+1M = refine

h+1/2M,⋃

E∈h+1/2M

Hmax `(h+1/2M)+(+)

(P(E ,h+1/2M),h+1/2M

)with respect to the requirements of the safe or greedy refinement.

Examples for the greedy and safe coarsening strategy are given in Fig. 4.6 and Fig. 4.7 fora bivarite cubic spline basis. It is clearly visible that the safe coarsening strategy leads to asmoother mesh grading compared to the greedy coarsening.

41


4.2 Convergence, Condition and SparsityIn this and the following two sections, numerical examples are exploited to verify the Bézierextraction framework of the previous Chapter 3 and to compare the mesh refinement strategiesfor THB-splines of the previous section with each other and against T-splines. Hence, T-splinesare compared with THB-splines and greedy with safe refinement. In addition to achievableconvergence rates and the mesh grading, the comparison includes numerical properties of thestiffness matrix, i.e. sparsity and condition number. The coarsening strategy is verified inChapter 6, where adaptive isogeometric analysis is applied to phase-field models.

Convergence

In the presented convergence studies, the error of the numerical solution is measured in theH1-norm (2.20). During h-refinement the error ηM =

∑E∈M ηE decreases according to

ηM = CD−k , (4.15)

where C is a constant, D the number of degrees of freedom and k the convergence rate. Ifthe solution of the problem is smooth enough and uniform refinement is applied, the optimalconvergence rate k = p/dP depends on the polynomial degree p and the dimension of theparameter space dp. If instead the solution has a singularity, classical convergence theory doesnot hold.

For the case of dP = 2 the order of convergence is then governed by the angle β of the singu-larity [70] which leads for uniform refinement to

k = 12 min

(p, π

2π−β). (4.16)

Only for local refinement and if all steps of the adaptive loop are correctly implemented, optimalconvergence rates of k = p/2 are obtained.

Condition and Sparsity

The condition number and sparsity of the system matrix influences the efficency of the solverand the quality of the solution. For this reason the condition number is measured with respectto the degrees of freedom and the numerical error. The sparsity is represented by plotting thesparsity pattern of the system matrix.

To clearly point out differences with respect to condition and sparsity between the refinementstrategies, the following example is designed as a worst case scenario and does not correspondto a physical problem. An initial square mesh with 64 elements is locally refined in the lowerleft corner, where only one element is marked for refinement in each refinement loop. Theresulting Bézier meshes are presented in Fig. 4.1. It can be seen that the greedy THB-splinerefinement does only refine the marked element whereas the safe refinement routines extend therefinement region. Also the greedy T-spline refinement has to insert additional control points toensure analysis-suitability. The degrees of freedom are plotted against the refinement steps inFig. 4.8a to illustrate this behaviour.

42

4.2 Convergence, Condition and Sparsity

(a) (b)

(c)

THB-splines ++THB-splines +

T-splines ++T-splines +

uniform

Figure 4.8: Worst case scenario: The relations between (a) the degrees of freedom, (b) the condition num-ber of the stiffness matrix and (c) the refinement steps are illustrated.

The locality of the refinement comes at the cost of an increased interaction between differentlevels (cf. Section 3.2) in the case of greedy THB-spline refinement. In this example, basisfunctions from the coarsest level interact with basis functions of the finest level. This leadsto the occurrence of quasi-dense rows and columns and the loss of any band structure in thestiffness matrix, as it can be seen in Fig. 4.9. The other refinement routines do not produceanomalies in their sparsity patterns.

The local mesh refinement also influences the behaviour of the condition number of the stiffnessmatrix. GAHALAUT et al. [71] analysed these condition numbers for NURBS-based isogeomet-ric discretizations, showing that the condition number increases linearly with respect to degreesof freedom. This is also reflected in all the experiments. As expected, for all kinds of localrefinement the condition numbers grow at higher rates, see Fig. 4.8b. The rate is apparently in-dependent of the type (T- or THB-splines) but does depend on the locality of refinement (greedyor safe ), and thus on the grading of the mesh. However, if the condition numbers are comparedwith respect to the refinement step (cf. Fig. 4.8c), the safe THB-spline refinement produceshigher condition numbers than the greedy one, and the T-splines higher condition numbers thanthe THB-splines. This shows that the number of additional DOF per refinement step can hasa dominant influence on the condition number. Hence, for a clear comparison, the condition

43


T-sp

lines

safe refinementgreedy refinementT

HB

-spl

ines

Figure 4.9: Worst case scenario: The sparsity patterns of the stiffness matrices after six refinement stepsare illustrated. Especially the greedy THB-spline refinement results in a dense stiffness matrix.

number has to be compared with respect to a quantity of main interest. For this reason thenumerical error of the solution is plotted over the condition number in the following examples.

4.3 Demonstration: Poisson ProblemIn this section, adaptive IGA is applied to the Poisson problem. After the introduction of themodel problem, its weak form and the corresponding error estimator, two and three-dimensionalbenchmark problems are solved. For the comparison of the refinement strategies, cubic B-splinebasis functions are used due to the limitations of T-splines. To verify the Bézier extractionframework of the previous chapter also examples with different polynomial degrees are carriedout for THB-splines.

4.3.1 Weak Form and Error EstimatorThe strong and weak form (2.17)-(2.21) of the Poisson problem and its discretization by B-spline basis functions (2.22) as well as the resulting linear system of equations were introduced

44

4.3 Demonstration: Poisson Problem

in Section 2.2.1. From the linear system of equations, the numerical solution uh ∈ N is com-puted.

Error Estimator

With the numerical solution at hand, the residual based error estimator [72] for an elementE ∈ M is defined by

ηE =(h2E ‖∆uh + f‖2L2(E) +

∑E∈K(E)

hE ‖RE(uh)‖2L2(E)

)1/2

, (4.17)

where K(E) is the set of element boundaries of E , hE the diameter of E , hE the diameter ofthe element boundary E and ‖•‖L2 the L2-norm as defined in (2.18). The boundary residualRE(uh) is defined by

RE(uh) =

12

[[∇uh · nE

]]E

if E is an element boundary,

g −∇uh · nE if E is a outer element boundary,(4.18)

where nE is the normal vector on E. For any interior element boundary E = E ∩ E ′, thenotation [[•]]E = •|E − •|E′ describes the jump across E. In case the spline basis is globallyhigher than C0-continuous and that the Neumann boundary condition is met exactly (e.g. in thecase g = 0), the above error estimator reduces to

ηE = hE ‖∆uh + f‖L2(E) . (4.19)

4.3.2 ApplicationL-Shape

In this example, the Poisson problem with f = 0 is solved on the two-dimensional domainΩ = (−1,1)× (−1,1) \ (0,1)× (0,1), referred to as the L-Shape that is characterized bya re-entrant corner with an opening angle of β = 90 and a given exact solution

u = r23 sin 2φ−π

3 (4.20)

in polar coordinates (r,φ), cf. Fig. 4.10. The boundary conditions are applied by setting uD = 0at the Dirichlet boundary ΓD and g = ∇u · νN at the Neumann boundary ΓN. The L-Shape ismodelled by a single C1-continuous B-spline patch. The geometry leads to a singularity of thesolution at the re-entrant corner.

The initial mesh of the L-shape problem consists of 16 elements. Fig. 4.11 shows the Béziermeshes after L refinement steps, as well as the marking parameters α. For the adaptive localrefinement, the error in the H1-norm is plotted over the degrees of freedom in Fig. 4.12a. Allrefinement strategies recover the optimal order of convergence in the asymptotic range. Dueto the coarse initial mesh, the safe refinements produce a greater amount of DOF in the pre-asymptotic range which is in particular observed for the safe T-spline refinement. As a result,the safe refinements are not as local as the greedy refinements but create more smoothly graded

45


0

1.25

(a) (b) (c)

Figure 4.10: Domain and boundary conditions for (a) the L-shape and (c) the slit domain as well as (b) thecorresponding analytical solutions.

meshes. To counteract the non-local refinements, the marking parameter for safe refinements ischosen higher. Especially for the greedy THB-spline refinement, the computed stiffness matrixhas a bad sparsity. For all other refinement strategies no clear tendency is visible in the sparsitypatterns in Fig. 4.11.

The condition number is plotted over the DOF in Fig. 4.12b. Due to the geometric map of the L-shape, a rate higher than one is reached for uniform refinement. Regarding the local refinement,results similar to the previous example are obtained. However, the differences between thegreedy and safe refinements are not as large as in the first experiment.

As mentioned above, also the error of the numerical solution with respect to the conditionnumber (cf. Fig. 4.12c) is of interest. It can be seen that for the same order of accuracy, alllocal refinement techniques produce smaller condition numbers compared to the uniform case.This means that for local refinement, the error of the solution decreases faster per DOF thanthe condition number increases per DOF. This is an important result, because it illustrates thatthe negative influence of a locally refined mesh on the condition number does not predominatethe benefits of local refinement regarding the error level. The refinement strategies comparedamong themselves show similar results.

Slit Domain

This example differs from the previous one only by a modified opening angle of β = 0. Thereduced opening angle leads to a more pronounced singularity and a given exact solution of

u = r12 sin φ

2 . (4.21)

The boundary value problem is illustrated in Fig. 4.10.

The initial mesh of the slit domain consists of 64 elements. The Bézier meshes after L refine-ment steps, as well as the marking parameters α are illustrated in Fig. 4.13. As expected, themeshes of the safe refinement routines propagate the refinement area but produce well gradedmeshes. On the other hand, the greedy T-spline refinement leads to a mesh with little structureand badly shaped elements with aspect ratios up to 64. Concerning the sparsity patterns of the

46


(b) THB-splines:

- safe refinement

(c) T-splines:

- greedy refinement

(d) T-splines:

- safe refinement

(a) THB-splines:

- greedy refinement- L = 7- α = 0.9

- L = 6- α = 0.925

- L = 7- α = 0.95

- L = 7- α = 0.975

Figure 4.11: L-shape: The marking parameters α, the Bézier meshes and the sparsity patterns of the stiff-ness matrices after L refinement steps for all (a)-(d) refinement strategies. The safe refine-ment strategies result in well graded meshes, the greedy refinement strategies in more un-structured meshes. Again, the greedy THB-spline refinement creates the stiffness matrixwith the highest density and interaction.

47


(c)

(b)(a)



uniform

Figure 4.12: L-shape: (a) The convergence rates as well as the relations between (b)-(c) the conditionnumber of the stiffness matrix, the numerical error of the solution and the degrees of freedomare illustrated. (a) - All refinement strategies converge with the expected convergence ratek = 1.5 in the asymptotic range.

stiffness matrix, only the greedy THB-spline refinement creates matrices with a higher density,due to the increased interaction between the basis functions.

For the adaptive local refinement, the error in the H1-norm is plotted over the degrees of free-dom in Fig. 4.14a. It can be seen that the error of the greedy refinement routines appear toconverge with a higher rate in the pre-asymptotic range and later approach the theoreticallypredicted rate of k = 1.5. The safe refinement routines have a lower convergence rate in thepre-asymptotic range, but then also converge with the theoretical rate of k = 1.5. A reason forthis behaviour is found again in the relatively coarse initial mesh, which forces the safe T-splinerefinement to refine almost the whole domain in the first refinement steps. As a result, thesafe T-spline refinement requires six times more degrees of freedom than the greedy T-splinerefinement for the same error level.

The condition number is plotted over the DOF in Fig. 4.14 (b). Due to the badly shapedelements, the condition number for the greedy T-spline refinement increases fastest. The THB-spline refinements instead seem to benefit from their hierarchical structure together with theabsence of a deforming geometry mapping. At a certain stage of refinement, the condition num-

48


(b) THB-splines:

- safe refinement

(c) T-splines:

- greedy refinement

(d) T-splines:

- safe refinement

(a) THB-splines:


- L = 7- α = 0.975

- L = 7- α = 0.975

- L = 7- α = 0.975

Figure 4.13: Slit domain: The marking parameters α, the Bézier meshes and the sparsity patterns of thestiffness matrices after L refinement steps for all (a)-(d) refinement strategies. The safe re-finement strategies result in well graded meshes. Especially the greedy T-spline refinementcreates an unstructured mesh with badly shaped elements. Again, the greedy THB-splinerefinement creates the stiffness matrix with the highest density and interaction.

49


(c)

(a) (b)



uniform

Figure 4.14: Slit domain: (a) The convergence rates as well as (b)-(c) the relations between the conditionnumber of the stiffness matrix, the numerical error of the solution and the degrees of freedomare illustrated. (a) - All refinement strategies converge with the expected convergence ratek = 1.5 in the asymptotic range.

ber does not increase further. This behaviour was also observed by JOHANNESSEN et al. [62]where HB-splines are compared against THB- and L-RB-splines. In the context of hierarchi-cal finite elements [73], it is known and even proven that the condition number of the stiffnessmatrix scales with O(log(DOF)) instead of O(DOF), due to orthogonalities w.r.t. the energyproduct between basis functions of different levels. In 1D, this leads to block-diagonal stiffnessmatrices; in higher dimensions, this effect is milder (see e.g. Fig. 4.13a), but still yields goodconditioning. It seems that (Truncated) Hierarchical B-splines share these benefits, howeverfurther investigation is needed in future.

Due to this effect, the greedy THB-spline refinement performs best if the numerical error isplotted over the condition number (cf. Fig. 4.14c). Since only a small amount of DOF is addedduring the refinement and due to the fact that the condition number grows slowly per DOF, anincreased level of accuracy is reached without increasing the condition number. But comparedto the uniform refinement, also the T-spline refinements produce smaller condition numbers.

50

4.4 Demonstration: Elasticity

0

1.25

(a) (b)

Figure 4.15: Fichera corner: (a) Domain and boundary conditions as well as (b) the corresponding analyt-ical solution are illustrated.

Fichera Corner

In this example, the safe THB-spline refinement strategy is applied to a three-dimensionalPoisson problem to verify the Bézier extraction framework of Chapter 3 also for higher di-mension. In Fig. 4.15a the Fichera Corner model problem is illustrated. The domain Ω =(−1,1)× (−1,1)× (−1,1) \ (0,1)× (0,1)× (0,1) is characterised by a re-entrant cornerthat is located in the origin of the coordinate system. The source term of the Poisson problem

f = −3

4

(x2

1 + x22 + x2

3

)−3/4(4.22)

is manufactured in such a way that the exact solution of the model problem reads

u = (x21 + x2

2 + x23)1/4 , (4.23)

as motivated by SCHILLINGER et al. [69] and illustrated in Fig. 4.15b. The boundary conditionsare applied by setting uD = u at the Dirichlet boundary ΓD and g = 0 at the Neumann boundaryΓN. The domain is modelled by seven tri-variate and Cp−1-continuous B-spline patches.

The initial mesh consists of 512 elements. Fig. 4.16a shows the Bézier mesh after five re-finement steps for the use of a tri-quadratic basis. Due to the strong singularity, the markingparameter is set to α = 0.975. For uniform and adaptive local mesh refinement, the error in theH1-norm is plotted over the degrees of freedom in Fig. 4.16b. Contrary to the uniform refine-ment, the safe refinement strategy recovers the optimal convergence in the asymptotic regimefor the tri-quadratic and tri-cubic THB-spline basis.

4.4 Demonstration: ElasticityIn this section the different refinement strategies are applied to a linear elastic benchmark prob-lem. Furthermore, the safe THB-spline refinement is applied to a problem of finite elasticityto verify the Bézier extraction framework of Chapter 3 also for non-linear problems. After theintroduction of the model problem, its weak form and the corresponding error estimator, thetwo two-dimensional benchmark problems are solved.

51


10−4

10−3

10−2

10−1

103 104 105 106

rel.

erro

rinH

1Degrees of freedom

p = 2, adaptivep = 2, uniformp = 3, adaptivep = 3, uniform

32

11

(a) (b)

Figure 4.16: Fichera corner: (a) The mesh after five refinement steps is illustrated for a tri-quadratic ba-sis. (b) Due to the singularity, optimal convergence rates are obtained for the adaptive meshrefinement only.

4.4.1 Weak Form and Error EstimatorTo describe the deformation of a homogeneous, isotropic, elastic body, the corresponding phys-ical domain is given in a reference configuration ΩR ⊂ RdS and in a current configurationΩ ⊂ RdS . The boundary in the current configuration is subdivided by ∂Ω = ΓD ∪ ΓN un-der the condition ΓD ∩ ΓN = ∅, analogue to the Poisson problem. Every material point ofthe body is identified by a position vector X ∈ ΩR in the reference configuration. Aftera displacement process u(X,t) at time t, the location of every material point is defined byx(X,t) = X + u(X,t) ∈ Ω in the current configuration. The corresponding deformationgradient is given by F = ∂x/∂X and the corresponding Green-Lagrange strain tensor byE = 1

2 (C − I). Here, C = F T · F is the right Cauchy-Green and I the identity tensor.

If inertia terms are neglected, the equilibrium requirement in the current configuration definesthe field problem by seeking u such that

−∇ · σ(u) = f in Ωσ(u) · nN = g on ΓN

u|ΓD= uD on ΓD .

(4.24)

The symbol ∇ is the Nabla operator with respect to the current configuration, nN is the outernormal vector at the Neumann boundary ΓN, f is the volume force vector, g is the prescribedtraction vector and uD is the prescribed displacement vector. The Cauchy stress tensor

σ =2

JF · ∂Ψ(C)

∂C· F T (4.25)

is given by a material specific strain energy function Ψ(C) and the Jacobian J = det(F ).

52


Following the Galerkin method as introduced in Section 2.2.1 and choosing the virtual dis-placement as the weight function v = δu, the discretized weak form is defined by seekinguh ∈ F ⊂ H1(Ω) such that∫

Ω

σ(uh) : ∇δuh dΩ =

∫Ω

f · δuh dΩ +

∫ΓN

g · δuh dΓN for all δuh ∈ F0 ,

uh|ΓD = uD,

(4.26)

where F is a finite dimensional function space and F0 = δuh ∈ F : δuh|ΓD = 0. Followingthe idea of IGA,F is spanned by a spline basis. Due to the non-linearity of (4.26), a linearisationis required to obtain a linear system of equations that is solved in an iterative solution procedureuntil equilibrium is reached.

Linear Elasticity

If only small strains are considered, the distinction in a reference and in a current configurationis not necessary and the Green-Lagrange strain tensor reduces to the linearised strain tensorε(u) = 1

2

(∇u+ (∇u)T

). The constitutive relation (4.25) reduces to linear-elastic material

behaviour given by σ = C : ε(u), where σ is the stress tensor and C the fourth order elasticitytensor.

As a result of the assumptions above, (4.26) is linear in uh and leads to a linear system ofequations.

Error Estimator

With the numerical solution uh ∈ N , the residual based error estimator [74] is defined similarto (4.17) by

ηE =(h2E ‖divσ(uh) + f‖2L2(E) +

∑E∈K(E)

hE ‖RE(uh)‖2L2(E)

)1/2

, (4.27)

where K(E) is the set of element boundaries of E , hE the diameter of E , hE the diameter ofthe element boundary E and ‖•‖L2 the L2-norm as defined in (2.18). In addition to the L2-error of the strong form of the field problem, the error estimator (4.27) also accounts for jumpsin the normal stress across elements boundaries. They are measured by the boundary residualRE(uh), defined by

RE(uh) =

12 [[σ(uh) · nE]]E if E is an interior element boundary,g − σ(uh) · nE if E is an ounter element boundary,

(4.28)

where nE is the normal vector on E. For any interior element boundary E = E ∩ E ′, thenotation [[•]]E = •|E − •|E′ describes the jump across E. In case the spline basis is globallyhigher than C0-continuous and that the Neumann boundary condition is met exactly (e.g. in thecase g = 0), the above error estimator reduces to

ηE = hE ‖divσ(uh) + f‖L2(E) . (4.29)

53


Figure 4.17: Infinite plate with a circular hole: (a) numerical analysis domain and boundary conditions,and (b) solution for σ11.

4.4.2 ApplicationInfinite Plate with a Circular Hole

As an example of linear elasticity, an infinite plate with a circular hole under uniaxial in-planetension σ0 and vanishing volume force f = 0 is considered according to Fig. 4.17a. Theanalytical solution is given by TIMOSHENKO [75] in polar coordinates (r,φ)

σr =σ0

2

[1− r2

i

r2+

(1− 4

r2i

r2+ 3

r4i

r4

)cos(2ϕ)

], (4.30)

σϕ =σ0

2

[1 +

r2i

r2−(

1 + 3r4

i

r4

)cos(2ϕ)

], (4.31)

σrϕ =σ0

2

(−1− 2

r2i

r2+ 3

r4i

r4

)sin(2ϕ) , (4.32)

where ri = 1 mm is the radius of the hole. A numerical solution is conveniently obtained onthe quarter of an annulus with Dirichlet boundaries to enforce the symmetry conditions, anda Neumann boundary ΓN at the outer radius to enforce the exact normal stress. The uniaxialtensile stress σ0 = 1 MPa is applied in the x1-direction and material parameters E = 105 Paand ν = 0.3 are used. The computational domain is modelled by a singleC1-continous NURBSpatch with an outer radius ro = 8 mm. The model is solved assuming plane stress conditions.

The exact solution features a stress concentration at (x,y) = (0,ri) of σ11 = 3σ0 as illustratedin Fig. 4.17b. Due to the lack of a singularity, optimal convergence rates k = −p/2 are obtainedby uniform h-refinement. Local refinement does not improve this rate in the asymptotic limit[27, 53]. There is however a benefit of the adaptive refinement which increases with the localityof the stress concentration. That is, if the outer radius ro is larger, the stress concentration ismore localised in the computational domain, cf. Fig. 4.18a, and an improved convergence rateis achieved in the pre-asymptotic region.

This improvement can be obtained for all refinement techniques by setting the marking param-eter around α = 0.5 to generate a more extensive refinement. For this example the greedy and

54


- safe refinement- L = 6- α = 0.55


- both refinements- L = 6- α = 0.45

(a) THB-splines: (b) T-splines: (c) T-splines:

Figure 4.18: Infinite plate with a circular hole: The marking parameters α, the Bézier meshes and thesparsity patterns of the stiffness matrices after L refinement steps for all (a)-(d) refinementstrategies. The greedy and safe THB-spline refinement show an identical refinement be-haviour. Neither in the Bézier meshes, nor in the sparsity patterns, clear differences betweenthe refinement strategies are visible.

safe THB-spline refinement produce same results. The meshes after L refinement steps and themarking parameters α are illustrated in Fig. 4.18. All refinement techniques lead to similarmeshes. As a result, also the sparsity patterns are similar and do not show any tendency. If thecondition number is plotted over DOF (cf. Fig. 4.19b), no differences in the rate are visiblebetween local and uniform refinement. In general it may be said that no numerical differencesoccur between the refinement techniques if the refinement area is extensive.

Cooks Membrane

In this example, the safe THB-spline refinement is applied to a two-dimensional problem offinite elasticity. A benchmark provided by DÜSTER & SCHRÖDER [76] is used here, wherethe Cook’s membrane under combined bending and shear loading is considered for a vanishingvolume force f = 0. The specimen, illustrated in Fig. 4.20a, is clamped at the left and aconstant shear load p0 = 20 MPa is applied on the right edge. The set-up of the model leads

55


(a) (b)

THB-splines ++

THB-splines +

T-splines ++

T-splines +uniform

Figure 4.19: Infinite plate with a circular hole: The convergence rates as well as the condition numberover the degrees of freedom are plotted. (a) - The local refinement strategies improve theconvergence rate in the pre-asymptotic regime, but reduce to k = 1.5 in the asymptoticregion.

to a singularity in the upper left corner of the specimen. A hyperelastic material model with astrain-energy function according to (4.25)

Ψ =µ

2I1 +

λ

4J2 −

(λ

2+ µ

)lnJ (4.33)

is used. Here, λ = 432.009 MPa and µ = 185.185 MPa constitute the Lamé parameters andI1 = trC is the first principle invariant of the right Cauchy Green tensor. The model is solvedassuming plane strain conditions.

To compare the solution with alternative discretization methods, the convergence behaviouris analysed by computing the displacement uy in y-direction of the upper right corner of thespecimen, cf. point A in Fig. 4.20. The problem was solved for different polynomial degreescombined with local and global refinement. It turned out that increasing the polynomial orderhas a greater impact on the solution than a local refinement towards the singularity. A purelylocal refinement at lower polynomial order (p = 2,3,4,...) improves the solution only slightlycompared to a global refinement. For that reason an hp-refinement is applied, which meansthat starting from an initial mesh with h = 3 and p = 1, in each computational step one morehierarchical level h is added and the polynomial degree is increased by one up to the final meshwith h = 10 and p = 8. The meshes for three selected constellations as well as the convergencebehaviour are plotted in Fig. 4.20a and b, respectively. Due to the safe refinement strategy forTHB-splines, the grading of the meshes is lower for higher polynomial degree.

We compare our formulation with the formulations used in the benchmark, where based ontriangular elements, a pure displacement formulation with quadratic shape functions (denoted asT2) and a mixed formulation with an additional constant field for the pressure (denoted as T2P0)

56

4.5 Conclusions

y

A

10410210

11

(a) (b)x

p0

uy/

mm

Degrees of freedom

- h = 5- p = 3

- h = 7- p = 5

- h = 10- p = 8

Figure 4.20: Cook’s membrane: (a) The domain and boundary conditions are visualised on the left.Meshes for different computational steps are plotted. (b) The displacement uy at point Ais plotted over the degrees of freedom.

as well as a p-FEM discretization with hexahedral elements and an initial mesh refinementtowards the singularity are used.

The convergence behaviour of the presented hp-IGA approach is similar to the p-FEM ap-proach. The shift of the curve to the left results from the spacious support of the B-spline basisfunctions and the local refinement.

4.5 ConclusionsIn this chapter two different refinement techniques for THB-splines were introduced, applied toadaptive IGA and compared against T-splines regarding their numerical properties. For this pur-pose, numerous numerical examples were studied. Furthermore, the Bézier extraction frame-work, introduced in the previous chapter, was verified for two- and three-dimensional as wellas linear and non-linear problems.

The successive use of an elementary refinement routine would cause uncontrolled function over-lap and dense stiffness matrices in the case of THB-splines or would yield non-nested discretespaces in the case of T-splines. The greedy refinement routines refine+ and refine_tspline+

eliminate these major drawbacks from a practical approach. The safe refinement routinesrefine++ and refine_tspline++ have shown the expected optimal asymptotical behaviour and re-sult in sparser stiffness matrices as well as in lower condition numbers compared to the creedyrefinements. However, they did not outperform the greedy refinement routines in every experi-ment.

In general, concerning the mesh grading and the numerical properties of the stiffness matrix,obvious differences increase with the locality of the problem. Whereas the refinement rou-tines produce similar results for extensive refinements, the differences between the comparedmethods become more visible for localized refinements. For those problems in which very localfeatures are to be resolved, the greedy refinement routines show a clear increase in the conditionnumber per degrees of freedom and especially for the refinement routine refine+ dense stiffnessmatrices arise. Furthermore, the greedy refinement routines lead to unstructured meshes around

57


the refinement area. For the refinement routine refine_tspline+ this can lead to badly shapedelements with ever-growing aspect ratios, here, up to 64.

From the subjective view of the author the implementation effort increases from the refinementroutine refine+ to refine++ to refine_tspline++ to refine_tspline+. The effort grows further forT-splines in general if a polynomial degree distinct from three is used, or for the greedy T-splinerefinement, if a generalisation to three dimensions is desired.

Summarizing these findings, the readers are advised to chose the refinement method whichbest fits the requirements of their application. However, especially for local refinement areas itmight be wise to use one of the safe refinements to avoid unpredictable numerical inaccuracies.Furthermore, the safe refinement strategies simplify the projection methods for field variablesas shown in the following chapter were only the safe THB-spline refinement is applied.

58

5 Mesh Adaptivity for IncrementalSolution Schemes – Projection andTransfer Operators

In the previous two chapters the implementation of THB-splines in adaptive IGA and corre-sponding mesh refinement and coarsening strategies were introduced. In contrast to the modelproblems of the previous chapter, a large class of boundary value problems is non-linear ortime-dependent in nature and requires incremental solution schemes. In this case, a projec-tion or transfer of state variables is necessary during the mesh adaptive computations if a re-computation of the problem from the initial state is to be avoided.

In the following, the different state variables are introduced and the incremental solution proce-dure is outlined. In the subsequent sections different projection and transfer operators are pro-posed and compared regarding their application to adaptive isogeometric analysis. It is shownthat isogeometric analysis improves the performance of transfer operations as already the coars-est mesh represents the exact geometry and the hierarchical structure allows for quadrature freeprojection methods.

This chapter is based on the work in HENNIG et al. [29]

State Variables

Three different sets of state variables, stored in state vector Λ = ΛF,ΛH,ΛI, are consid-ered. The first subset ΛF contains field variables, e.g. displacements or temperatures, whichare stored at element nodes/control points, respectively. The second and third subset containvariables which are only given at integration points. Variables of the second kind ΛH possessown evolution equations at integration point level, e.g. plastic strains or hardening variables,and are referred to as history variables. Variables of the third kind ΛI, such as stresses or strains,are computed from ΛF and ΛH and are referred to as internal variables.

During the transfer operation of all state variables

• equilibrium,

• the consistency with constitutive equations, and

• the minimisation of numerical diffusion of history variables

have to be ensured [77]. The consistency with constitutive equations might be lost if non-linear relations between the state variables are transferred by linear operators. To reduce thiseffect, LEE et al. [78] suggest to limit the number of transferred variables to a minimum. AsΛI can be computed from ΛF,ΛH, the transfer operation should be restricted to the field andhistory variables. For problems without history variables this conserves the equilibrium of thesystem. In the other case, interaction between the transfer of field and history variables has tobe considered to recover equilibrium [77].

59

5 Mesh Adaptivity for Incremental Solution Schemes

Incremental Solution Procedure

For an incremental solution procedure the four steps, introduced in Section 4.1 (solve, estimate,mark and refine) have to be applied between increment n and the following increment n + 1.Recapitulating these steps, the solution of the boundary value problem is computed on a givenmesh hM with basis hA. Based on an estimated error or another significant quantity, elementsare marked and subsequently refined or coarsened to generate the new mesh h+1M with basish+1A.

In contrast to the procedure in Section 4.1, in incremental solution schemes the stiffness matrixK[h+1A,h+1A

]and the right hand side vector F

[h+1A

]for the next increment n+ 1 depend

on the state variables hΛn from the current increment n. For that reason, the state variableshave to be transferred from the old to the new mesh by a suitable transfer operation

h+1Λn = T (hΛn). (5.1)

Different transfer operations T are proposed in the subsequent sections. Three different scenar-ios are investigated: mesh refinement without history variables ΛH, mesh coarsening withouthistory variables ΛH and mesh refinement including history variables ΛH.

The detailed procedures are restricted to 2-admissible THB-splines that result from the appli-cation of the safe refinement and coarsening strategy of the previous chapter. In this way, thetransfer operations are limited to adjacent levels of the hierarchical mesh, i.e. communicationacross several levels is avoided. However, the basic concepts of the presented transfer opera-tions can be applied to arbitrary hierarchical bases.

5.1 Mesh RefinementWithout history variables the set of state variables consists of field variables ΛF and internalvariables ΛI. As mentioned above, a transfer of both sets of variables could lead to a datasetthat is not self-consistent. For this reason and because ΛI can be computed from ΛF, the transferoperation (5.1) is restricted to ΛF that are given or required at the control points indicated bythe function set in the bracket

ΛF[h+1Ai

]= T (ΛF

[hAi]) . (5.2)

Remember that ΛF[hAi]

can be computed from (3.28).

Projection of Field Variables

For the safe refinement strategy, introduced in Section 4.1.2, the projection of field variablesreduces to a projection between two nested basis function spaces with known relation. Unlike instandard FEM, the solution of non-linear systems of equations to compute the position of a newcontrol point in the parameter domain of the old element [77] using e.g. inverse isoparametricmapping techniques [78] is not required. In IGA, an error free matrix-vector multiplicationis sufficient for refinement, because the fine scale basis is able to represent the coarse scalesolution [27, 68, 79, 80].

60

5.2 Mesh Coarsening

In order to increase efficiency, the transfer operation is limited to the newly created elementsM = h+1M\ hM. Hence, field variables have to be transferred only to a subspace h+1Ai ⊂h+1Ai defined by

h+1Ai =

L−1⋃`=0

h+1A` and h+1A` = h+1A` \ hA` , (5.3)

yielding the reduced transfer operation

ΛF

[h+1Ai

]= T (ΛF

[hAi]) . (5.4)

Taking advantage of the hierarchical basis, the subdivision operator according to equation (3.3)is used as a transfer operator for each hierarchical level starting from ` = 0 to ` = L− 2

ΛF

[h+1A`+1

]=(

M`,`+1)T [

hA`,h+1A`+1]

ΛF[hA`

]. (5.5)

Similar and generalized versions of this operator have benn already proposed in [27, 68, 79,80].

To shorten the notation of matrix vector equations such as (5.5), where the information about thefunction sets of the matrix are defined by the function sets of the vectors, the argument after thematrix is omitted in the following, yielding e.g. for (5.5) ΛF[h+1A`+1] = (M`,`+1)TΛF[hA`].

5.2 Mesh CoarseningIn certain applications adaptive refinement and coarsening are required to increase the efficiencyof the computation. This is the case if, e.g., strong gradients are moving with time as in phase-field or contact modelling. Since the problem is assumed to be independent of history variables,(5.2) still holds.

In the following, four different projection methods for coarsening are outlined and compared.The comparison of all four methods is done in quantitative benchmark tests. In Chapter 6 theyare applied to phase-field models of spinodal decomposition processes and shape optimization.

5.2.1 Projection of Field VariablesUnlike in refinement, the field variables have to be transferred not onto the next finer level` + 1 but onto the next coarser level ` − 1. In this case the projection causes an error sinceN ` 6⊂ N `−1.

This error can be minimized in theL2-norm leading to a projection between two function spaces

minΛF[N `−1]

∥∥∥ΛF − N`−1ΛF[N `−1

]∥∥∥2

L2(5.6)

where ΛF = N`ΛF[N `]

is the known solution field of level ` and ‖v‖L2is the L2-norm of v as

defined in Section 2.2.1. The minimization leads to a system of equations that is solved for thesolution ΛF

[N `−1

]but requires an integration over the computational domain Ω.

61


Alternatively, dealing with a hierarchical basis, again the subdivision operator according toequation (3.3) can be used

ΛF[N `]

=(

M`−1,`)T

ΛF[N `−1

], (5.7)

and the solution ΛF[N `−1

]is approximated by least squares

minΛF[N `−1]

∥∥∥∥ΛF[N `]−(

M`−1,`)T

ΛF[N `−1

]∥∥∥∥2

2

, (5.8)

where the error is minimized in the euclidean norm defined by ‖v‖2 =(∑n

i=1 v2i

)1/2for a

vector v of size n. The minimization results in a linear system of equations

M`−1,`(

M`−1,`)T

ΛF[N `−1

]= M`−1,`ΛF

[N `], (5.9)

orΛF[N `−1

]= M`−1,`

ΛF[N `], (5.10)

where the pseudo-inverse of the subdivision operator

M`−1,`=

(M`−1,`

(M`−1,`

)T)−1

M`−1,` (5.11)

is identified as the required transfer operator. Note that this operation is quadrature free.

In the following paragraphs, four different projection techniques are outlined. In addition to thefirst two, already existing approaches, two novel methods are proposed.

Global Discrete Least Squares Method (LSQ)

Equation (5.9) is referred to as the global discrete least squares method.

Note that the computation of ΛF[N `−1

]in (5.9) requires the solution of a linear system that

grows with the spline space N `−1. Also the pseudo-inverse in (5.10) is dense and expensive tocompute.

For this reason, three possible alternatives are proposed and compared in the following. Eachof these methods has to be executed level-wise starting from ` = L− 1 to ` = 1.

Subspace Discrete Least Squares Method (SLSQ)

The computational effort is reduced by limiting the transfer operation to the newly activatedelements M. Hence, the least squares fit is performed onto the subspace h+1Ai, cf. equation(5.3), and the transfer operation between two consecutive levels reduces to

62

5.2 Mesh Coarsening

ΛF

[h+1A`−1

]= M`−1,`

ΛF[hA`

]. (5.12)

This method has been already proposed by JIANG et al. [79]. However, for three-dimensionalproblems the transfer operation can still be large and in the limiting case, where all elementsare selected for coarsening, it corresponds to the LSQ.

Local Discrete Least Squares Method (LLSQ)

GOVINDJEE et al. [44] have proposed a localL2-projection for an efficient imposition of Dirich-let boundary conditions in IGA, cf. Section 3.3. Instead of inverting the whole Gram matrixonly finite element based sub-matrices of the Gram matrix are inverted.

This idea is adapted and applied to the subdivision operator that results in a quadrature freelocal discrete least squares method. The transfer operation is split into two parts

ΛF

[h+1A`−1

]= T s

⋃E`−1e ∈M

T pe (ΛF

[hA`

])

, (5.13)

starting with a transfer operation on element level T pe and a subsequent smoothing operation

T s:

1. Transfer operation T pe : For each element E`−1

e ∈ M with basis functions h+1A`−1e ,

the corresponding child elements in the multi level basis are identified and their basisfunctions stored in hA`C =

⋃chA`c, where c runs over the indices of the child elements

on level `. Subsequently, the pseudo-inverse has to be computed only for the reducedsubdivision operator yielding the elements-wise transfer operation T p

e

ΛF[h+1A`−1

e

]= M`−1,`

ΛF[hA`C

]. (5.14)

2. Smoothing operation T s: Due to the element-wise projection, the approximated solutionis discontinuous at control points and has to be smoothed by a weighting scheme. Eachfield variable λI ∈ ΛF

[h+1A`−1

]is computed from a weighted sum

λI =∑E`−1e ∈E

weIλeI , (5.15)

where E contains all elements E`−1e in the support of the basis function NI ∈ h+1A`−1

and λeI ∈⋃E`−1e ∈MΛF

[h+1A`−1

e

]are the field variables of basis functions NI com-

puted for elements E`−1e from (5.14). Here the simple averaging proposed by GOVIND-

JEE et al. [44] is used, which yields λI = 1nI

∑E`−1e ∈E λ

eI , where nI is the number of

elements that are under the support of basis function NI .

Please note that there exists also a basis function wise local discrete least square fit, which ispreferable if isogeometric analysis is implemented from a basis function viewpoint [81].

63


Weighted Local Least Squares Method (LLSQ_w)

The final proposed transfer operation is similar to the previous one but uses other weights in thesmoothing operation. Here, the weights introduced by THOMAS et al. [45] are used to smootha discontinuous approximation created by so called Bézier projection. The idea is to weigh thefield variable λeI with respect to the impact of basis function NI on element E`−1

e , which resultsin

weI =

∫ΩeNIdΩ∑

E`−1e ∈E

∫ΩeNIdΩ

. (5.16)

The integrals in (5.16) can be also approximated by the element Bézier extraction operator, asproposed by THOMAS et al. [45]. This technique can be adapted due to the used approach totruncated hierarchical B-splines, cf. Chapter 3 that is also based on Bézier extraction.

5.2.2 Comparison of Projection MethodsBelow, the projection methods outlined in the previous section are compared regarding theirconvergence and error level. Furthermore, they are compared against the corresponding localand semi-local L2-projections that can be created using the minimization (5.6) instead of (5.8).For this purpose, a continuous analytical function describing a regularized circular inclusion ofradius r and centre (xm,ym)

c(x,y) = 1− tanh

(√(x− xm)2 + (y − y2

m)− r√2ε

)(5.17)

is projected by a least squares fit onto the finest level of the multi level basis on the domainΩ = [0, 1] × [0, 1]. The parameter ε describes the size of the transition zone between c = 1(inside of the inclusion) and c = 0 (outside of the inclusion). The analytical function is plottedin Fig. 5.1a for r = 0.4 and ε = 0.01.

Starting from the finest approximation, the whole domain is coarsened in each step and theinitial field variables are projected onto the new, uniformly coarsened meshes using the methodsabove. In Fig. 5.1b-d the errors of the projected solutions with respect to the function (5.17) arecompared in the L2-norm for basis functions of different polynomials degree p. The solid linesrepresent the L2-projections, the dashed lines the discrete least square fits.

Since the whole domain is coarsened, the results of the SLSQ coincide with the LSQ results.All variants show the expected convergence rate of (p+1)/2. The discrete least square fits, pre-sented here, produce similar results as the L2-projections, where the error level increases fromLSQ to LLSQw to LLSQ for both variants. Especially for the L2-projections, LLSQ produces astronger deviation for higher polynomial degrees that can be eliminated using LLSQw which isin agreement with the findings of GOVINDJEE et al. [44] and THOMAS et al. [45]. Interestingly,the local discrete least square fit LLSQ produces smaller errors than its L2-version, althoughthe L2-projections are expected to be more accurate in general. The differences between theprojection techniques increase with the polynomial degree.

In the next analysis, the initial mesh is coarsened locally. Coarsening is controlled adaptively.Instead of an error estimator, the field variable c itself is used as a significant measure. An

64

5.2 Mesh Coarsening

0

1

Figure 5.1: Uniform coarsening: The analytical function (5.17) is initially projected onto the finest meshwhich is then coarsened uniformly. The error of the projections for global, uniform coarseningmeasured in the L2-norm is plotted vs. the degrees of freedom for (b) bi-quadratic, (c) bi-cubic, and (d) bi-quartic bases. The solid lines represent the L2-projections, the dashed linesthe discrete least square fits. The blue dashed line nearly coincides with the blue solid line andis therefore not visible.

element is marked for coarsening if the magnitude of the gradient of c, ‖∇c‖ is lower than aspecified threshold θ at one of the integration points of this element. In Fig. 5.2, the error inL2-norm for the different local and semi-local methods is plotted vs. the degrees of freedomfor threshold values θ = 1 (Fig. 5.2a-b) and θ = 6 (Fig. 5.2 c-d) and compared to uniformcoarsening using LSQ (black solid line). Depending on the threshold, the adaptive coarseningresults either in a lower reduction of DOF but a constant error level or in a higher reduction ofDOF but a slight increase of the error level. Furthermore, it can be observed that the accuracyof the method increases again from LLSQ to LLSQw to SLSQ. As for uniform coarsening, theL2-projections produce better results than the discrete least square fits, except for the LLSQ.

It is concluded that in case of local coarsening, all methods are able to reduce the degrees offreedom by up to 86% without a significant increase of error if the threshold for coarsening ischosen appropriately. The SLSQ methods performs best but can lead to large systems that wouldbe expensive to solve. Comparing the local methods LLSQ and LLSQw, the latter producessmaller errors but requires the computation of the enhanced weights. Also the L2-versionsof SLSQ and LLSQw perform better than their corresponding discrete least square fits, but

65


Figure 5.2: Local coarsening: For threshold values of θ = 1 and θ = 6 the error in L2-norm is plottedwith respect to the degrees of freedom in (a) and (c), respectively. The local and semi-localmethods are compared against uniform coarsening (LSQ - black solid line). The correspondingmeshes after the last coarsening steps (b) and (d) are illustrated.

require an integration over all coarsened elements. Depending on the application, the user hasto balance the requirements regarding accuracy and computational effort.

5.3 Mesh Refinement Including History VariablesIn this section, local mesh refinement is considered for computations where the state vector Λincludes field variables ΛF, internal variables ΛI as well as history variables ΛH, e.g. the plasticstrain in mechanical computations with inelastic material behaviour.

Since field variables ΛF are given at control points whereas history variables ΛH are given atquadrature points, the transfer operation (5.1) has to be split into a projection of ΛF

ΛF[h+1Ai

]= TF(ΛF

[hAi]) (5.18)

using the procedures of Section 5.1, and a transfer operation for ΛH

ΛH[h+1Q

]= TH(ΛH

[hQ]) . (5.19)

66

5.3 Mesh Refinement Including History Variables

Figure 5.3: Transfer scheme for incremental computations including history variables [77]: The transferoperation for computations including history variables is split in TF and TH which are executedin different increments to facilitate the computation of the equilibrium state on the new meshin the new increment.

Here h+1Q =⋃Eeh+1Qe contains all quadrature points h+1Qe of the newly activated elements

Ee ∈ M = h+1M\ hM whereas hQ contains the quadrature points of the parent elements Mof M. As mentioned before, ΛI should be computed after the transfer operation from ΛF andΛH to avoid inconsistencies with the constitutive model.

To increase the stability and efficiency of the incremental solution, PERIC et al. [77] propose notto apply transfer operations TF and TH simultaneously if elements are selected for refinementin increment n, as illustrated in Fig. 5.3. In detail, the new increment n + 1 is at first solvedon the old mesh resulting in field variables hΛn+1

F . Subsequently, the transfer operation TF isapplied to generate h+1Λn+1

F,trial which provide a trial solution for the Newton-Raphson scheme.Now, TH is executed to generate the history variables h+1Λn

H at the new integration points andthe increment n+ 1 is solved on the fine mesh, using h+1Λn

H and h+1Λn+1F,trial. This results in the

required solution h+1Λn+1 for increment n+ 1 on mesh h+ 1.

The projection of primary field variables TF was already discussed in Section 5.1. Therefore,the following section deals with the transfer operator TH for history variables only.

5.3.1 Transfer of History VariablesOne possibility to transfer history variables is to use interpolation techniques. The closest pointtransfer is the oldest and simplest transfer method for integration point quantities. With thismethod, the history variable at a new integration point is obtained from the closest integrationpoint in the previously used mesh. This method can be enhanced by considering all old integra-tion points within a given distance and averaging them with specific weights, e.g. a normalizeddistance [82]. Similarly, the least squares projection transfer [83] incorporates the surroundingold integration points by computing the new history variable from a least squares fit of the oldhistory variables.

A more sophisticated transfer method, which is referred to as the shape function transfer, wasintroduced by LEE et al. [78] and further enhanced by PERIC et al. [77]. Here, the historyvariables are first projected to the control points of the old mesh using a recovery method basedon averaging [84], followed by a projection onto the control points of the new mesh which issimilar to the projection of field variables. Eventually, the history variables at the quadraturepoints of the new mesh are computed using the element shape functions of the new mesh. To

67


Figure 5.4: Schematic illustration of transfer operators for history variables on a bi-quadratic B-splinepatch: (a) Closest point transfer (CPT); (b) Basis function transfer (BFT); (c) Weighted patchbased least squares fit (WPLSQ).

improve the projection from integration to control points during the shape function transfer, thesuperconvergent patch recovery (SPR) method, introduced by ZIENKIEWICZ & ZHU [85] asan error estimator, can be used [86]. Furthermore, the numerical diffusion of material historycan be reduced by projecting the variables directly from the old control points onto the newintegration points after the recovery fields are computed [87]. However, attention has to bepaid to possible inconsistencies as shown by PAVANACHAND & KRISHNAKUMAR [88]. Itis noted that, due to the existence of history variables, the integration points have to be usedinstead of the superconvergent points in the recovery process [89]. The cost for an additionalsolution of the evolution equations at the superconvergent points would not be in proportion toa possibly increased accuracy of the recovered history field. Interested readers are referred toKUMAR et al. [90] for a detailed review and comparison of various transfer operators and toBUCHER et al. [91] for transfer operators in hierarchical FEM.

In the following, the easily and fast implemented closest point transfer is shortly reviewed. Sub-sequently two possible transfer operators suitable for IGA are introduced. In the first approach,

68


Figure 5.5: Shape function transfer: (a) - The transfer operation TH is split in three parts. (b) - The transferoperation TF is equal to the projection in 5.1.

the combination of hierarchical B-splines with the shape function transfer method is proposed.In the second method, the SPR method is adopted as an enhanced interpolation technique totransfer history variables directly from old to new integration points. A schematic illustrationof the three different operators is given in Fig. 5.4.

Closest Point Transfer (CPT)

This operator represents the simplest possibility and transfers the history variables directly fromhQ to h+1Q. The history variables for a quadrature point of the newly activated elements istaken from the closest quadrature point of the corresponding parent element, cf. Fig. 5.4a.

Basis Function Transfer (BFT)

This operator combines IGA with the shape function transfer, where control points and corre-sponding basis functions are used in an intermediate step. Therefore, the operator is split inthree parts as illustrated in Fig. 5.4b and Fig. 5.5a:

1. TH,1: In a first step, the history variables at the integration points ΛH[hQ]

are projectedlevel-wise onto the basis hA`, i.e. the corresponding control values are computed. Similarto Section 5.2, a discrete least squares fit is used for this purpose

minΛH[hA`]

∥∥ΛH[hQ]− TΛH

[hA`

]∥∥2

2, (5.20)

where the error is minimized in the euclidean norm. This leads to the linear system ofequations

TTTΛH[hA`

]= TTΛH

[hQ], (5.21)

where TIJ = NJ(xI) with NJ ∈ hA` and xI the physical coordinate of the quadraturepoint QI ∈ hQ.

To increase the efficiency by avoiding a solution of the whole linear system, the globaloperation is split into local element-wise operations as illustrated in Fig. 5.4b. Theprocedure, including the projection and the subsequent weighting, is similar to the one inSection 5.2 where projection is applied between two function spaces, with the differencethat here a data set ΛH is projected onto a given function space. Hence, T is used insteadof the subdivision operator M.

69


Note that fully integrated elements have to be used with the local version to provide anadequate number of sampling points for the least squares fit. Alternatively patches ofelements have to be selected.

Furthermore, instead of a discrete least squares fit (5.20), an L2-projection could be used

minΛH[hA`]

∥∥ΛH − TΛH[hA`

]∥∥2

L2 , (5.22)

where the error is minimized in the L2-norm to compute the control values ΛH[hA`

].

Here, ΛH is the history field and TI ∈ hA`. This could increase the accuracy of theprojection, but would require an additional integration over the whole domain Ω∫

Ω

TTTdΩΛH[hA`

]=

∫Ω

TTΛHdΩ (5.23)

in the global or over each element domain Ωe in the local version, respectively. Thenumerical integration of the right hand side of equation (5.23) is possible, because thediscrete history field ΛH is already given at the quadrature points.

2. TH,2: In the second step, control values of the history variables of all levels ΛH[hAi]

are projected similarly to the field variables onto the refined basis ΛH[h+1Ai

]using the

transfer operation TF (5.18), cf. Fig. 5.5. Remember that this projection is exact in thecase of refinement.

3. TH,3: In the last step, the history variables at the quadrature points

ΛH[h+1Q

]= TΛH

[h+1Ai

](5.24)

are interpolated from the control points ΛH[h+1Ai

]using the shape function matrix T,

where TIJ = NJ(xI) withNJ ∈ h+1Ai and xI the physical coordinate of the quadraturepoint QI ∈ h+1Q.

By PERIC et al. [77] the operation TH,2 is mentioned as the most complex of the three steps.With the use of structured and nested hierarchical meshes, this step is trivial, efficient and free oferrors in the presented approach using IGA. The only error occurs in TH,1 during the projectionof ΛH

[hQ]

onto hA`. Since this process is similar to the transfer operation in Section 5.2, afurther improvement is expected by using the basis function based weighting scheme from theLLSQ_w method.

Weighted Patch Based Least Squares Fit (WPLSQ)

The use of the SPR method is proposed in the following to transfer the history variables, andimprove the quality of the transferred field by a weighted patch based least squares fit. Thismethodology is adapted from KUMAR et al. [92], where the SPR technique [85] was extendedfrom standard FEM to IGA to develop a recovery based error estimator for LRB-splines. There,internal variables are transferred from the superconvergent points to the integration points.Hence, this method is adapted to transfer directly from hQ to h+1Q. Note that these integration

70


points result from standard quadrature rules and are not superconvergent points regarding thehistory variables. The operation is split in three steps as illustrated in Fig. 5.4c:

1. At first, all basis functions with support on M are stored in a set AH = NJ ∈ hA :suppNJ ∩M 6= ∅ and the elements Ee ∈ hM that support a basis function NJ ∈ AH inthe corresponding patch of elements PJ = Ee ∈ hM : suppNJ ∩ Ee 6= ∅. Rememberthat M contains the parent elements of M. In Fig. 5.4c, the element patches of threecontrol points that belong to the element selected for refinement are exemplarily indicatedby colour. All integration points located in patch PJ are stored in the set QJ .

2. Subsequently, the discrete history variables are smoothed by a continuous field Λ∗J = PaJfor each patch PJ . Following the SPR technique, the matrix P consists of monomials ofthe same order as the basis functions of the field variables, e.g. P = [1,x,y,x2,xy,y2,x2y,yx2,x2y2] for a bivariate quadratic B-spline basis. To compute the coefficients aJfor each patch PJ , a least squares fit is used

n∑K=1

(PT(xK)P(xK)

)aJ =

n∑K=1

(PT(xK)

)ΛH [QK ] , (5.25)

where xK are the physical coordinates of the n quadrature points QK ∈ QJ .

Note that boundary element patches have to be extended by surrounded elements to pro-vide a sufficient number of sampling points if a reduced integration scheme is used. Anal-ogous to the BFT, also here a L2-projection could be used instead of the least square fit toincrease the accuracy, but would require an additional integration over the patch domain.It is referred to KUMAR et al. [92] for a discussion on patch configurations and possibleprojection methods.

3. With the smoothed history fields Λ∗J at hand, the history variables ΛH at new quadra-ture points QK ∈ h+1Q with physical coordinates xK is computed by ΛH [QK ] =∑J NJ(xK)Λ∗J(xK). Here, NJ(xK) ∈ AH acts as a weight, to ensure that a patch

closer to the quadrature point QK has a higher impact than those far away. Note that thisweighting scheme is only applicable if the locally refined basis has the partition of unityproperty, as given for THB-splines.

In Fig. 5.4c one integration point QK ∈ h+1Q of the four newly activated elements isillustrated by the red square.

5.3.2 Comparison of Transfer OperatorsIn the following, the transfer operators introduced in the previous section are compared androbustness towards mesh distortion is checked. For this purpose, an analytical function

Λ = sin(2πx) cos(2πy) (5.26)

is used as a history variable on the parallelogram Ω characterized by the angle φ, cf. Fig. 5.6a.The skewness s is defined by the normalized angle deviation method where s = 1 correspondsto a rectangle and s = 0 to a fully degenerated element. In our case, s = φ/90, that results

71


Figure 5.6: Comparison of Transfer Operators: (a) The analytical trigonometric function is computed onthe integration points of a distorted domain defined by skewness s. (b)-(d) For uniform refine-ment, the error in L2-norm after the transfer to the new integration points is plotted vs. thedegrees of freedom for the different transfer operators for bi-quadratic (solid line) or bi-cubic(dashed line) basis functions and for differently distorted meshes.

for Fig. 5.6a, where φ = 22.5 in s = 0.25. To measure the quality of the transfer operatorsquantitatively, the error is computed in L2-norm with respect to the analytical function.

In the numerical study, the behaviour for uniform refinement is analysed. Starting from a meshwith four elements, in each step, the analytical values of Λ are computed and stored at thequadrature points of the current mesh. Subsequently, the whole mesh is refined and the historyvariable is transferred using the operators CPT, BFT, and WPLSQ. The error is computed atthe quadrature points of the refined mesh with respect to the analytical function values (5.26).This procedure is repeated until the finest mesh of 65536 elements is created. The resultsfor bi-quadratic (solid line) and bi-cubic (dashed line) approximations are visualised for s =0, 0.5, 0.75 in Fig. 5.6b-d. For the bi-quadratic basis, nine integration points per element,and for the bi-cubic basis 16 integration points per element were used.

For all methods, the error of the transfer operation decreases as the mesh is refined. As expected,the simple CPT operator produces large errors independently of the polynomial degree. TheBFT and the WPLSQ operators lead to a similar convergence rate, however the BFT producessmaller errors in general. While an increased polynomial degree further reduces the error in the

72

5.4 Conclusions

case of BFT, the opposite is found for WPLSQ. It is assumed that the larger errors especiallyfor coarse meshes are induced by the enlarged patches of four times four elements used forbi-cubic basis. In this case, the bi-cubic fit ΛI is not able to represent the analytical function,since a single patch covers a large area of the analysis domain.

Considering mesh distortion, the accuracy of the methods is deteriorated with an increasedskewness except for CPT that always produces similar results. For slightly distorted elementsof s = 0.5 the BFT and WPLSQ clearly improve the quality of the transfer operation. However,the WPLSQ with cubic polynomials is only efficient for fine meshes. For strongly distortedelements with s = 0.75, the BFT and WPLSQ only slightly improve the mapping process forfine meshes.

This study only measures the quality of the transfer operation itself and hence give an answer onhow strong the numerical diffusion of the history variable might be during the refinement. Asalready stated in the introduction of this chapter, it is also important that the transfer operatorpreserves the consistency with the constitutive equations and the equilibrium of the system.To investigate this influence, in the following Chapter 6, the operators are applied to adaptivephase-field modelling of ductile fracture.

5.4 ConclusionsIn this chapter projection methods and transfer operators were analysed and discussed. Theseare required if adaptive mesh refinement/ coarsening is applied to problems that are based onincremental solutions and that may include internal variables with an evolution equation solvedat integration points.

For the transfer of field variables, the use of IGA leads to an error-free projection in case ofrefinement and simplifies the projection in case of coarsening to a quadrature free discrete leastsquare fit between two function spaces. To avoid the solution of a global system of equation toproject the primary field variables, two local projection methods were introduced and comparedto existing semi-local and global versions. In the benchmark problems all of them provide goodresults, whereas SLSQ and LLSQ_w achieve the lowest error levels. However, for SLSQ andthree-dimensional problems the system can still be large. Comparing the local methods LLSQand LLSQw, the latter produces smaller errors but requires the computation of the enhancedweights. For this reasons the local discrete least squares fit LLSQ and the corresponding L2-version are applied to two diffrent phase-field methods in the following Chapter 6.

The transfer of history variables requires more sophisticated transfer operators and was studiedfor the case of adaptive refinement. It was shown that the application of IGA simplifies theprocedures for shape function transfer compared to standard finite elements. Furthermore, atransfer operator based on the superconvergent patch recovery method were introduced, whichresults in a weighted patch based least square fit. The benchmark studies concerning differentlydistorted meshes showed that both BFT and WPLSQ provide much better results than the simpleCPT for slightly distorted elements. However for strongly distorted elements, BFT and WPLSQimprove the solution only for fine meshes. Furthermore BFT outperforms WPLSQ if cubicpolynomials are used.

73

6 Application to Phase-Field ModelsAs stated in the introduction of this thesis, the phase-field method is a powerful tool to solveproblems that include (evolving) interfaces numerically. The method avoids the cumbersomenumerical tracking and discretization of discontinuities. However, the variability of the ap-proach comes at the cost of highly refined meshes that are required along the interfaces toproperly resolve the gradients in the order parameter field, cf. Fig. 1.1. Hence, adaptive localmesh refinement and coarsening are essential for efficient computations.

In this chapter the adaptive isogeometric analysis framework, introduced in the previous threechapters, is applied to several phase-field models and verified regarding its accuracy, reliabilityand efficiency. The numerical examples are also used to compare the introduced projection andtransfer operators. In detail, the transfer of history variables is considered during the simulationof ductile fracture in Section 6.2.2 and the projection of field variables on coarsened meshesduring the simulation of structural evolution processes in Section 6.3.

Furthermore, this chapter presents a unified phase-field modelling approach for weak and strongdiscontinuities in solid mechanics as they arise in the numerical analysis of heterogeneous mate-rials due to rapidly changing mechanical properties at material interfaces or due to propagationof cracks if a specific failure load is exceeded. In a standard FE analysis of such problems,the mesh is either aligned with the material interfaces or decoupled along the cracks to ob-tain a C0 or C−1-continuous basis. But for some applications, the meshing process is costlyand the numerical tracking of the discontinuities cumbersome. To overcome these drawbacks,non-standard discretization methods exist as meshless methods, where the nodes are not con-nected [93], or the extended finite element method [94, 95], where a local enrichment of theapproximation space by discontinuous functions is combined with adapted integration tech-niques. The embedded domain method avoids the meshing processes by an implicit represen-tation of the physical domain in a regular background mesh [96, 97]. While good convergencerates are achieved for homogeneous materials, stress oscillations occur in the heterogeneouscase because weak discontinuities are modelled in terms of a continuous basis. For that reason,ELHADDAD et al. [98] propose the use of a separate background mesh for every heterogeneity.

In this work, a generalized Ginzburg-Landau phase-field model [4, 6] is used to simulate prop-agating strong discontinuities, whereas a static phase-field is introduced to regularize weakdiscontinuities. The regularization leads to a diffuse interface region of finite width ì betweenthe heterogenities, where the material properties are not defined. Instead of a simple interpo-lation, a homogenization is applied to compute the effective material parameters in the diffuseinterface region, similar to [99, 100]. To provide an appropriate and efficient approximation anh/ì-adaptive refinement strategy based on the proposed adaptive IGA framework is applied,where mesh size h and interface width ì are reduced simultaneously. Finally, both methods forweak and strong discontinuities are combined to simulate crack propagation in heterogeneousmaterials.

In the following section, preliminaries to the phase-field method are given. Subsequently, inSection 6.2 the modelling approach for weak and strong discontinuities in solid mechanics is

75

6 Application to Phase-Field Models

0 0.5 1c

0

0.1

fh(c)

(a) Double-well potential

0 0.5 1c

0

1

fh(c)

(b) One-well potential

Figure 6.1: Double-well and one-well potentials: (a) The double-well potential of the form fh = c2(1−c)2

has a minimum at c = 0 and c = 1 (b) The one-well potential of the form fh = (1 − c)2 hasonly one minimum at c = 1.

presented. Phase-field models for structural evolution processes are discussed in Section 6.3followed by conclusions in Section 6.4.

This chapter is based on the work of HENNIG et al. [29], HENNIG et al. [30] and HEN-NIG et al. [31].

6.1 Basic Concepts of Phase-Field ModelsConsidering only two-phase problems, an order parameter c = [0, 1] is introduced that takes thevalue c = 0 in the one phase and the value c = 1 in the other phase, cf. Fig. 1.1. Following thework of CAHN & HILLIARD [101], the local free energy of such a non-homogeneous systemdoes not only depend on the order parameter it-self, but also on its immediate environment.Consequently, the local free energy of the system

ψ(c,∇c) = fh(c) + αfg(∇c) (6.1)

consist of the sum of the free energy of the homogeneous phase fh(c) and a gradient termfg(∇c) that considers the local composition of the order parameter and that is scaled by aconstant α. Equation (6.1) is also referred to as the Ginzburg-Landau free energy in case fh(c)is given by a double-well potential as illustrated in Fig. 6.1a and fg = 1

2 |∇c|2 [102].

In the phase-field method a spatial distribution of the order parameter c(x), x ∈ RdS , is soughtthat minimizes the total free energy

Ψ(c,∇c) =

∫Ω

ψ(c,∇c) dΩ , (6.2)

in a given physical domain Ω ⊂ RdS .

76

6.1 Basic Concepts of Phase-Field Models

In the following, two different types of phase-field models are shortly reviewed. In the firstmodel, the volume fraction m ∈ [0, 1] of the phase c in the domain Ω, defined by∫

Ω

c dΩ = m

∫Ω

dΩ , (6.3)

is not constant. Therefore, one phase can grow within another homogeneous phase as in case ofa crack that propagates through an intact specimen. In the second model, the order parameter isconserved and m = const., as for example during the separation of binodal mixtures.

Ginzburg-Landau Phase-Field Model

In order to find a minimum of (6.2), the functional derivative δΨδc has to vanish. The Ginzburg-

Landau evolution equation

c =∂c

∂t= −M δΨ

δc(6.4)

relaxes this condition by the introduction of a pseudo time t and a mobility constant M . Areduced mobility delays the evolution of the order parameter towards the steady state. In thelimiting case where M → ∞, the original condition is restored. During the evolution, thevolume fraction m can vary.

If Ψ is given by the Ginzburg-Landau free energy, (6.4) is also referred to as the Allen-Cahnequation [102].

Cahn-Hilliard Phase-Field Model

In contrast to the Ginzburg-Landau phase-field model, in the Cahn-Hilliard phase-field model,the order parameter is conserved and m = const.. Consequently, interpreting the order param-eter as a concentration, it has to fulfil the continuity equation

c = −∇ · h . (6.5)

The mass flux vector h is related to the chemical potential µ

h = −M(c)∇µ (6.6)

by a concentration dependent mobility M . The chemical potential is usually defined as thepartial derivative of the free energy with respect to c. But due to the existence of densitygradients in (6.1), it is defined as the functional derivative here µ = δΨ

δc [102].

Substituting (6.6) in (6.5), the evolution equation for the order parameter is given by

c = ∇ ·(M(c)∇δΨ

δc

), (6.7)

referred to as the Cahn-Hilliard equation.

77


0 0.5 1x

0

0.5

1

c

t

(a) Ψ =∫Ω

(κ(

1`cfh +`cfg

)+ h)

dΩ

0 0.5 1x

0

0.5

1

c

t

(b) Ψ =∫Ω

(κ(1

`cfh + `cfg

)+ h)

dΩ

0 0.5 1x

0

0.5

1

c

t

(c) Ψ =∫Ω

(κ(

1`cfh + `cfg

)+ h)

dΩ

0 0.5 1x

0

0.5

1

c t

(d) Ψ =∫Ω

(κ(

1`cfh + `cfg

)+ h)

dΩ

Figure 6.2: The influence of the different terms in Ψ =∫Ω

(κ(

1`cfh + `cfg

)+ h)

dΩ on the evolutionprocess of the order parameter c over the pseudo time t is schematically illustrated: (a) Startingfrom a linear distribution of c, the minimization of the first term with the double-well potentialfh = c2(1 − c)2 results in a sharp jump in c. (b) Starting from a sharp jump in c, the mini-mization of the second term with fg = `c|∇c|2 results in a liner distribution of c. (c) Startingfrom a linear distribution of c, the minimization of the sum 1

`cfh +`cfg results in homogeneous

phases with a diffuse interface between them. (d) The additional term h(c,u) acts as a drivingforce for the already developed phase-field.

Generalized Ginzburg-Landau Phase-Field Model

The two presented phase-field models can be generalized to represent different interfacial prob-lems by modifying the potential fh and gradient fg terms in (6.1) as well as by extending thefree energy by additional terms as e.g. the elastic strain energy density.

In the following models, extended free energies of the form

Ψ(c,u) =

∫Ω

[κ

(1

`cfh(c) + `cfg(∇c)

)+ h(c,u)

]dΩ (6.8)

are used, where κ is the interface parameter, `c the length-scale parameter and h(c,u) an ad-ditional term that can include additional state variables u. In order to find a minimum of theextended free energy with respect to c and u, the condition

δΨ

δu= 0 (6.9)

78

6.2 Weak and Strong Discontinuities in Solid Mechanics

has to be fulfilled during the evolution of the order parameter, given by equation (6.4). As aresult, a coupled system of PDEs is obtained.

The influence of the individual terms in (6.8) on the evolution process (6.4) are illustrated in Fig.6.2. Using a double-well potential for fh(c), mixed states of the order parameter are penalized,cf. Fig. 6.2a. However, the term fg(∇c) acts as diffusion equation and penalizes the gradientsbetween the phases, cf. Fig. 6.2b. As a result, the minimization of the sum fh + fg, resultsin homogeneous phases with a diffuse interface between them, cf Fig. 6.2c. The parameter `cweights the influence of the two terms and thus determines the width of the interface. As shownin Fig. 6.2d, the additional third term h(c,u) acts as a driving force for the phase-field.

6.2 Weak and Strong Discontinuities in SolidMechanics

In this section, a unified phase-field modelling approach for weak and strong discontinuities insolid mechanics is presented. For this purpose, in the following section a static phase-field isintroduced to regularize material interfaces that are embedded in a regular background mesh.Furthermore, a homogenization is applied to compute effective material parameters in the re-sulting transition zone. One- and two-dimensional numerical benchmarks are performed tostudy the properties and efficiency of the approach.

Subsequently, phase-field models for brittle and ductile fracture are reviewed in Section 6.2.2.To simulate crack propagation in heterogeneous materials, the phase-field model for brittlefracture is combined with the modelling approach for the embedded material interfaces.

In all examples, the safe mesh refinement strategy of Section 4.1 is used to improve the ef-ficiency of the computations. Due to the occurrence of field and history variables, also thetransfer operators introduced in Section 5.3 have to be applied. The influence of the differ-ent operators on the quality of the solution is investigated for the phase-field model of ductilefracture in Section 6.2.2.

6.2.1 Embedded Material Interfaces in Linear ElasticityThe problem of linear elasticity was introduced in Section 4.4.1 and is now extended to het-erogeneous materials. As illustrated in Fig. 6.3, the heterogeneity of a body is modelled bya subdivision of its physical domain Ω in n smaller subdomains Ω =

⋃ni Ω(i) with differ-

ent elasticity tensors C(i). Material interfaces are represented by internal boundaries Γij be-tween subdomains Ω(i) and Ω(j). For any function f in Ω, the jump of f on Γij is defined byJfK := f (i)|Γij − f (j)|Γij , where f (k) is the restriction of f to Ω(k).

Assuming a perfect bonding between the different material phases Ω(i), the displacement vectorhas to be continuous JuK = 0 and the traction vector in normal direction νΓ to the interface Γijhas to fulfil

JσK · νΓ = 0 . (6.10)

79


`

Ω(3)

Ω(1)

Ω(2)

νΓ

c = 0 c ∈ (0,1)

c = 1

Γ12µΓ

δ

x

y

Figure 6.3: Embedded material interfaces: The heterogeneities are embedded into a non-conforming finiteelement mesh. The material interface Γ12 is regularized by the order parameter c over a finitelength ì. The near field is defined as the domain that surrounds the interface Γ12 up to adistance of δ/2 from the interface.

However, based on the Hadamard jump condition, the strain is allowed to jump perpendicularto the interface but has to be continuous in tangential direction

JεK = a⊗ νΓ , (6.11)

where a is an arbitrary vector. This results in a weak discontinuity in the solution field u acrossthe interface.

To spatially discretize the problem above, a C0-continuous basis has to be used to approximatethe weak discontinuity in the displacement field. In the context of an finite element analysis,this can be achieved by aligning the computational mesh to the material interface. Here, thisrestriction is relaxed by the introduction of a static phase-field, thus avoiding the associatedmeshing process.

For this purpose, an order parameter c is introduced that takes the value c = 0 in Ω(1) andc = 1 in Ω(2). The resulting sharp interface problem can be approximated by means of theModica-Mortola energy [99]

Ψ(c,∇c) =

∫Ω

(6

ìc2(1− c2) +

3

2ì|∇c|2

)dΩ , (6.12)

that is a special version of the Ginzburg-Landau energy (6.1). The minimization of that energyleads to an order parameter field given by

c =1

2

(1 + tanh

xsd

ì

). (6.13)

Here, xsd is the signed-distance function that describes Γ12 with xsd = 0 and ì is a length scaleparameter that controls the size of the regularized interface. In situations where the geometrydescription is obtained from imaging techniques, the regularized interface can be also deriveddirectly from gray scale values.

80


c

1− c Ω(2)

Ω(1)

x

y

〈σxx〉 = σ(1)xx = σ

(2)xx = 0 〈σxx〉 = cσ

(1)xx + (1− c)σ(2)

xx 〈σxy〉 = σ(1)xy = σ

(2)xy = const

〈σyy〉 = σ(1)yy = σ

(2)yy = const 〈σyy〉 = σ

(1)yy = σ

(2)yy = 0 〈εxy〉 = cε

(1)xy + (1− c)ε(2)

xy

〈εxx〉 = ε(1)xx = ε

(2)xx = const 〈εxx〉 = ε

(1)xx = ε

(2)xx = const

〈εyy〉 = cε(1)yy + (1− c)ε(2)

yy 〈εyy〉 = cε(1)yy + (1− c)ε(2)

yy

Figure 6.4: Homogenization: A representative volume element with straight interface is subjected to threedifferent deformation states that result in constant stress and strain fields σ(i) and ε(i). Theorder parameter c and 1 − c is interpreted as the dimension in y direction of the individualphases. The components of the homogenized material tensor are obtained from the effectivestresses 〈σ〉 and strains 〈ε〉 and Hooke’s law.

The material tensor C is now defined in terms of the static phase-field to provide a smoothtransition from the material tensor C(1) to C(2). For this purpose, homogenization theory isapplied at material points where both phases are present, c ∈ (0,1), cf. Fig. 6.3. As illustratedin Fig. 6.4, a representative volume element with a straight interface between the phases Ω(1)

and Ω(2) is defined in a reference coordinate system (x,y,z). The order parameter c and 1 − care interpreted as the dimension in y direction of the individual phases. Effective stresses andstrains are obtained by volume average

〈σ〉 = cσ(1) + (1− c)σ(2) ,

〈ε〉 = cε(1) + (1− c)ε(2) .

In both phases, Hooke’s law for plane strain with individual Young’s moduli E(1) and E(2) aswell as Poisson’s ratios ν(1) and ν(2) is applied. The effective material tensor follows from theconstitutive equation

〈σ〉 = C(c) : 〈ε〉. (6.14)

A illustrated in Fig. 6.4, six deformation states for three-dimensional and three deformationstates for two-dimensional problems have to be considered to compute the components of C(c).The applied deformation states result in constant stresses and strains in the individual phases.

To take the orientation of the material interface into account, the homogenized fourth-order ma-terial tensor C(c) is transformed from the reference coordinate system with basis ex,ey,ez to

81


xΩδ

Ω(1) Ω(2)

f

uD

Figure 6.5: Bi-material rod: Domain with material E(1) = 100 MPa, E(2) = 0.1E(1) and boundaryconditions f = 12x2 kN mm−1 and uD = 0.01 mm.

the material coordinate system with basis µΓ ,νΓ ,ξΓ by an orthogonal second-order rotationtensor Q(∇c). The rotation reads in Einstein notation:

Cijkl(c,∇c) = QimQjnQkoQlpCmnop(c) . (6.15)

For the illustrated two dimensional case in Fig. 6.3, the rotation matrix

Q(∇c) =

(cosα − sinαsinα cosα

)depends on the angle α = arccos(ey · νΓ ), where the normalized normal vector νΓ of theinterface is computed from the order parameter νΓ = ∇c

||∇c|| .

The resulting homogenized material tensor C(c,∇c) satisfies the static equilibrium (6.10) at theinterface and the kinematic compatibility across the interface (6.11). Furthermore it contains theReuss and Voigt type homogenizations as limiting cases and coincides with the homogenizedmaterial tensors proposed in [99, 100].

Numerical Example – Bi-Material Rod

The modeling approach above is tested and verified in the following numerical example of abi-material, uni-axial rod under volume load f , cf. Fig. 6.5. The solution is compared againstthe embedded sharp interface representation under local h-refinement similar to [97]. For abetter evaluation, the physical domain Ω is divided in a near field Ωδ and a far field Ω \ Ωδ .As illustrated in Fig. 6.3, the near field is the domain that surrounds all interfaces Γij up to adistance of δ/2 from the interface.

The rod geometry with length 1 mm, boundary conditions and material parameters are givenin Fig. 6.5. The volume force f does not act in the near field Ωδ that is located in a distanceδ/2 = 0.125 mm around the material interface at x = 0.5 mm. The domain is discretizedwith THB-splines that are C0 continuous at the boundary of Ωδ to allow for the jump in thevolume force f and Cp−1-continuous across element boundaries in general. The numericalperformance of the modelling approach is examined for an hì-adaptive refinement strategy,where mesh size h and length scale ì are reduced simultaneously at a fixed ratio ì/h = 2.The convergence plot in Fig. 6.6a shows that optimal convergence rates are obtained in thefar field (solid lines) for different polynomial degrees only if an appropriate homogenizationis applied in the diffuse interface region. In case of the embedded sharp interface, the error ofthe near field spreads out to the far field and no optimal rates are obtained. The error in the

82


10−1110−1010−910−810−710−610−510−410−310−210−1

101 102 103 104

rel.

erro

rinH

1

Degrees of freedom

sharp, p2sharp, p3

diffuse, p2diffuse, p3

1

12

3

1 0.5

(a) Convergence in H1-norm

-2

-1

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

Stre

ssσ/

MPa

x/mm

referencesharp, C1

diffuse, C1

(b) Stress along the axes of the rod

Figure 6.6: Bi-material rod under local h`-refinement: (a) Optimal convergence rates are obtained in thefar field (solid lines) for different polynomial degrees if an appropriate homogenization is ap-plied, whereas the error in the whole domain (dashed lines) converges with a rate independentof the polynomial degree. (b) The regularization of the interface avoids stress oscillations.

whole domain Ω (dashed lines) is governed by the error in the near field and converges with arate independent of the polynomial degree. However, the application of the local hì-refinementallows to reduce the error in the near field to 10−3. The proposed approach also avoids stressoscillations that occur for sharp interface representations as shown in Fig. 6.6b, where the exactsolution is overestimated by more than 100% in a significant width around the interface.

It can be summarized that the diffuse interface representation bounds the main error to the nearfield and in this way allows for optimal convergence rates in the far field. This behaviour isalso mathematically justified by HENNIG et al. [30]. The error in the near field can be furtherreduced by the application of the hì-refinement strategy to an error range that is typical forembedded methods and suitable for engineering applications. Furthermore, stress oscillationsare avoided, which would be disadvantageous if the method is combined with a phase-fieldmodel for interface failure.

Numerical Example – Inclusion Problem

In the next example, illustrated in Fig. 6.7a, an inclusion problem is analysed, where a smallercylinder with domain Ω(1), radius a = 3 mm and material parameters E(1) = 105 MPa andν(1) = 0.3 is embedded in a larger cylinder with domain Ω(2), radius b = 15 mm and materialparameters E(2) = 0.1E(1) and ν(2) = ν(1). The outer boundary of the larger cylinder issubjected to a constant displacement load uD perpendicular to the interface. But only a squarewith length c = 8 mm is simulated and the exact solution [103] is applied to its boundary.Again, hì-adaptive refinement is applied, cf. Fig. 6.7b.

In contrast to the one-dimensional example, no optimal convergence rates are obtained for arbi-trary shaped interfaces in multi-dimensional problems. It is assumed that this results from theviolation of the homogenization assumptions made above to derive the homogenized materialtensor. There, constant stress and strain states in the two different phases along the material

83


sym

met

ryy

uD

symmetrya b x,rc

θ

Ω(1)

Ω(2)

(a) Domain and boundary conditions

σyy

1.3 ∗ 104

2.7 ∗ 104

(b) Adaptively refined mesh and corresponding stress σyy

Figure 6.7: Circular inclusion problem: Model problem as well as mesh and stress solution for the problemwith diffuse interface representation, p = 2 and C1-continuous basis after local h`-refinementare illustrated.

interface are assumed. However, as shown in Fig. 6.8a, the total error is clearly reduced usinglocal h`i-refinement. Furthermore, the diffuse modelling approach is able to improve the con-vergence rates compared to the embedded sharp interface using local h-refinement. Anotherimportant property of the approach is illustrated in Fig. 6.8b, where the stress σyy is plottedalong the x-axis. While the jumps in both fields are smoothed due to the diffuse interfacerepresentation, high oscillations occur for the sharp interface.

6.2.2 Brittle and Ductile Fracture in Homogeneous andHeterogeneous Materials

In contrast to standard sharp crack models, in the phase-field approach to fracture the discretecrack is regularized using an order parameter c. The transition zone between the fully brokenmaterial c = 0 and the undamaged material c = 1 is characterised by an internal length scaleparameter `c. By coupling the scalar order parameter to a mechanical boundary value problem,the initiation and propagation of the sharp crack Γc are described by the evolution of the crackphase-field, cf. Fig. 6.9a.

In the following, phase-field models for brittle and ductile fracture are reviewed and applied towell known benchmark problems. To simulate local damage and failure in heterogeneous ma-terials, the modelling approach for embedded interfaces from the previous section is combinedwith a generalized phase-field model for brittle fracture that also acounts for interface failurebetween two materials [32].

Brittle Fracture in Homogeneous Materials

The variational formulation of the Griffith theory for brittle fracture, initially introduced byFRANCFORT & MARIGO [104], is based on the energy functional

Ψ =

∫Γc

GcdΓc +

∫Ω

ψel(ε(u))dΩ , (6.16)

84


10−6

10−5

10−4

10−3

10−2

10−1

102 103 104 105

rel.

erro

rin

ener

gyno

rm

Degrees of freedom

sharpdiffuse

(a) Convergence in H1-norm

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8

Stre

ssσ

yy/10

4∗

MPa

x/mm

referencesharp

diffuse

(b) Stress along the bottom edge of the specimen

Figure 6.8: Circular inclusion: The convergence behaviour in Ω (solid lines) and Ω \ Ωδ (dashed lines)using local h`i-refinement is illustrated on the left. The stress σyy is plotted along the x-axisfor the sharp and diffuse interface representation with a C1-continuous basis, p = 2 and` = 0.0624 mm on the right.

where Ω is the physical domain of the body that has an outer boundary Γ and an internaldiscontinuous boundary Γc, cf. Fig. 6.9a. Interpreting Γc as a crack, the fracture toughness Gcof the material is integrated over the corresponding crack surface. Following Section 4.4.1, thelinear elastic strain energy density is given by ψel = 1

2ε(u) : C : ε(u).

To avoid the integration over the crack surface, a regularized version of (6.16) was presentedby BOURDIN et al. [4]. The resulting formulation can be interpreted as a generalized Ginzburg-Landau phase-field model [6]. According to Section 6.1, the extended free energy (6.8) is givenby

Ψ(u,c) =

∫Ω

Gc

((1− c)2

4lc+ lc|∇c|2

)dΩ︸︷︷︸

Ψsurf

+

∫Ω

g(c)ψel(ε(u))dΩ︸︷︷︸Ψel

, (6.17)

where a one-well potential fh(c) = (1 − c)2/4, cf. Fig. 6.1b, is used in combination with agradient term fg(∇c) = |∇c|2. The use of a one-well potential ensures that in the limiting casewhere `c → 0, Ψsurf converges to the surface energy

∫ΓcGcdΓc of the variational formulation

(6.16). The strain energy Ψel acts as a driving force for the crack phase-field. The functiong(c) = c2 degrades the elastic energy density ψel where the material is broken.

To prevent crack propagation under compressive loads, the strain energy can be split in a de-graded tensile and a non-degraded compressive part [5, 105, 106]. To avoid crack heeling,MIEHE et al. [5] introduced a strain energy history function that can be interpreted as a damage-like irreversibility condition [33].

The approach leads to a fully coupled two-field problem for the displacement field u and thephase-field c, given by the Ginzburg-Landau evolution equation (6.4) c = −M δΨ

δc and theequilibrium condition (6.9) δΨ

δu = 0. The latter leads to the mechanical momentum balance.The quasi static solution is found by choosing M → ∞ in combination with a staggered

85


Figure 6.9: Phase-field approach to fracture and numerical examples: (a) Displacement field u and frac-ture phase-field c are coupled on the solid domain Ω with boundary Γ . The sharp crack Γc

is regularized over the length `c. (b) Numerical model and boundary conditions for the singleedge notched shear test. (c) Numerical model and boundary conditions for the asymmetricallynotched specimen. All dimensions are given in mm.

[107] or monolithic [9] solution scheme. A review of different formulations is presented byAMBATI et al. [106] while a critical discussion of the solution procedures and the choice ofboundary conditions is given by MAY et al. [108].

In the following, the phase-field model of brittle fracture is solved for the problem illustratedin Fig. 6.9b. It considers a square specimen that is notched on the left edge and clamped atthe bottom edge while a given displacement is applied on the top edge. The initial notch isprescribed as a discrete crack in terms of a C−1-continuity line between two patches. TheLamé parameters for the material are λ = 121 150 MPa and µ = 80 770 MPa and the phase-field parameters are given by the fracture toughness Gc = 2.7 MPa mm and the characteristiclength lc = 0.0075 mm. These parameters are identical to the work of MIEHE et al. [107] whichallows for a quantitative comparison. A staggered, quasi static solution scheme is used with adisplacement increment of ∆u = 10−5 mm per time step.

A fine mesh is required in the vicinity of the crack to properly resolve the gradients in the orderparameter. Here, a THB-spline basis with six hierarchical levels is used, leading to an elementsize for the finest level of hE = 0.266`c. The initial mesh is pre-refined in the vicinity of theinitial crack tip, cf. Fig. 6.10a bottom. For adaptive refinement, the safe refinement strategyof Section 4.1.2 is used in combination with a threshold value of c = 0.5, i.e. elements aremarked for refinement if the phase-field parameter at any quadrature point of the element fallsbelow 0.5. The non-linearity of the problem requires a mapping of the solution according to thealgorithm presented in Section 5.1.

In Fig. 6.10 the adaptively refined mesh and the order parameter c are illustrated for dif-ferent prescribed displacements. The solution produces the same crack pattern as given by

86


u = 0.014 mm

(a) (b) 10c

u = 0.01 mm

Figure 6.10: Meshes and solution of the single edge notched shear test: (a) Initial mesh for the adaptive(bottom) and the pre-refined mesh for the reference solution (top) are illustrated. (b) Theadaptively refined meshes and the solution of the phase-field are shown for two differentdisplacement steps.

MIEHE et al. [107]. Furthermore, it can be seen that the adaptive algorithm resolves the crackpath correctly.

To measure the efficiency of the adaptive approach, the solution is compared to a referencesolution, obtained on a locally pre-refined mesh, cf. Fig. 6.10a top. As illustrated in Fig. 6.11a-b, the force-displacement curves obtained with both solutions are identical but the adaptiveapproach significantly reduces the number of elements. This improves the efficiency comparedto uniform refinement and reduces the computation time by up to 76%, cf. Fig. 6.11c. Notealso that knowledge on the expected crack path was used to generate the uniformly pre-refinedmesh which is impossible for problems of arbitrary complexity.

Brittle Fracture in Heterogeneous Materials

To model brittle fracture in heterogeneous materials, the phase-field model for brittle fracture ofthe previous paragraph is combined with the diffuse modelling approach for embedded materialinterfaces from Section 6.2.1. The free energy for a brittle, heterogeneous, linear elastic materialthan reads

Ψ(u,c) =

∫Ω

[Gc(xsd,`i)

((1− c)2

4lc+ lc|∇c|2

)+ g(c)ψel (ε(u),s,∇s)

]dΩ . (6.18)

87


(a)

(b)

0 15000

20000

Increment

Elements

1

0

rel.Time

(c)

adaptivereference

adaptivereference

adaptivereference

Figure 6.11: Computational results of the single edge notches shear test: The adaptive computation is con-sistent with the uniformly refined reference computation (a), reduces the number of elements(b) and hence the summarized computation time (c).

Here, c is still the order parameter of the fracture phase-field but s is now the static order pa-rameter that describes the pre-defined and unchangeable heterogeneity as introduced in Section6.2.1. Consequently, (6.18) has to be minimized only with respect to u and c. The linear elasticstrain energy is redefined in terms of the homogenized material tensor C(s,∇s) (6.15), by

ψel(ε,s,∇s) =1

2ε : C(s,∇s) : ε . (6.19)

Furthermore, to account for interface failure between two materials the fracture toughnessGc(xsd,ì) is locally reduced to the interface fracture toughness Gint in the area of the regu-larized material interface and therefore depends on the signed distance function xsd and thewidth of the diffuse material interface ì. Due to the use of a diffuse crack and a diffuse inter-face model, an interaction between the length-scale parameters `c and ì can occur. Followingthe approach of HANSEN-DÖRR et al. [32], a scaling factor is computed that further lowersGint and therefore compensate the influence of the fracture toughness of the bulk material onthe crack propagation along the interface.

To illustrate the capabilities of this approach, the problem shown in Fig. 6.9b is extendedby using a heterogeneous material , cf. 6.12. Three inclusions (s = 1) with glass fibre likematerial properties λgf = 17 392 MPa, µgf = 33 761 MPa and Ggf

c = 2 MPa mm are embeddedin a matrix (c = 0) with thermoset like material properties λts = 2299 MPa, µts = 1082 MPaand Gts

c = 1 MPa mm. The material interface has a reduced fracture toughness of Gint =0.5 MPa mm. The internal length scales are set to ` = 2ì = 4`c = 0.03 mm.

The adaptive discretization equals the discretization of the previous homogeneous example ex-cept that the mesh is also pre-refined along the material interfaces. Again the safe refinementstrategy of Section 4.1.2 and the projection operator of Section 5.1 is used.

88


u

s

0

1c

0

1

u = 0.0058 mm u = 0.0073 mm u = 0.0076 mm

Figure 6.12: Crack propagation in heterogeneous materials: Three stiffer inclusions are embedded in a ho-mogeneous matrix using the order parameter s. The specimen has an initial crack, indicatedby the black solid line, and is subjected to a shear load. In contrast to the fracture process in ahomogeneous body, the crack initiates at the material interface. To increase the efficiency ofthe computation, the mesh is pre-refined along the material interfaces and adaptively refinedwhere the crack phase-field c is smaller 0.5.

As illustrated in Fig. 6.12, after loading, two cracks initiate at the weaker material interfacesof two of the inclusions. Subsequently, they propagate into the matrix material, join each otherand form a larger crack with an orientation of around 45 with respect to the loading direction.This is in contrast to the homogeneous case, where the phase-field crack initiates at the stresssingularity at the initial crack tip as shown in Fig. 6.10.

Please note that in the presented approach no split of the strain energy is considered, whichwould lead to wrong results under mixed loads. Hence, further investigations are required, tocombine the diffuse material interface representation with a split of the strain energy (6.19)in a tensile and compressive part. A first approach in this direction is proposed by HANSEN-DÖRR et al. [109].

Ductile Fracture in Homogeneous Materials

Phase-field modelling of ductile fracture has been the subject of a few investigations in the past,see ALESSI et al. [110]. Here, the model of AMBATI et al. [8] is adopted, where the free energyfunctional (6.17) is extended by a plastic energy density ψp

Ψ(u,c) =

∫Ω

[Gc

((1− c)2

4lc+ lc|∇c|2

)+ g(c,p)ψel(ε(u)) + ψp

]dΩ . (6.20)

The coupling between plasticity and damage is realized through a modified degradation function

g(c,p) = c2p with p =εp

eq

εpeq,crit

89


Figure 6.13: Phase-field modelling of ductile fracture in an asymmetrically notched specimen: (a) Initialmesh for the adaptive (left) and the pre-refined mesh of the reference solution (right) areillustrated. The adaptive refinement of the mesh is controlled in terms of the equivalent plasticstrain εp

eq (b). The adaptively refined meshes are illustrated in (c), and the order parameterc (c = 1: intact material, c = 0: fully cracked material) is shown in (d) for three differentstages of crack initiation and propagation.

that depends on the von Mises equivalent plastic strain εpeq and a corresponding user defined

threshold value εpeq,crit. In this way, the evolution of the phase-field (and thus the occurrence of

fracture) is driven by the accumulated plastic strain.

In the following an asymmetrically notched specimen is considered. The problem and boundaryconditions are illustrated in Fig. 6.9c. The domain is discretized by 2-admissible, bi-quadraticTHB-splines with maximum four levels. The parameters are as follows: elastic constants λ =53 473 MPa and µ = 27 280 MPa, yield strength σy = 345 MPa, hardening modulus h =250 MPa, fracture toughness Gc = 9.31 MPa mm, length scale `c = 0.08 mm and equivalentplastic strain threshold εp

eq,crit = 0.1. An incremental displacement is applied on the top edgeand the solution is computed using a staggered, quasi static solution scheme.

The initial mesh for adaptive computations as well as the mesh for the non-adaptive referencecomputation are shown in Fig. 6.13a. The resolution of the initial mesh is chosen to capture theplastic effects in the pre-cracked state appropriately. The refinement of the mesh is necessary tocapture the localisation of the plastic strain and to properly approximate the crack phase-field.

90


Figure 6.14: Phase-field modelling of ductile fracture in an asymmetrically notched specimen: (a) Theforce-displacement curves are illustrated for the different transfer operators. (b) The com-putation time required to solve the system (blue) and to adaptively refine the mesh (red) areplotted with respect to the normalized time needed for the reference computation. (c) Com-pared to the number of elements required in the computation BFT has the best quality-costrelationship.

Since cracking is expected if the equivalent plastic strain exceeds a critical value, the elementsare marked for refinement if εp

eq > εpeq,crit at any quadrature point. The elements are refined

until the finest hierarchical level is reached. Due to the elastic-plastic material model, the meshrefinement strategy from Section 5.3 that also includes history variables, is used. All threeintroduced transfer operators are applied and compared.

Contour plots of the results for the adaptive computation using the BFT operator are shown inFig. 6.13b-d. The mesh as well as the distribution of the phase-field variable and the equivalentplastic strain are shown for three different stages of the displacement-controlled loading. It canbe seen that the mesh refinement follows the area with the largest εp

eq before the crack starts togrow along this path.

A quantitative comparison of the performance of the proposed transfer operators from Section5.3 is possible using the force-displacement curve, cf. Fig. 6.14a. The computation with theBFT operator is almost identical to the non-adaptive reference solution, while the CPT andWPLSQ operators lead to a delayed fracture process. Higher numerical diffusion of plasticstrain during the data transfer could be a reason for that. Furthermore, the CPT operator leadsto instable computations which could result from the violation of constitutive equations and theequilibrium of the system after the transfer.

In Fig. 6.14b the computation times required to solve the system (blue) and to adaptivelyrefine the mesh (red) are plotted relative to the time needed for the reference computation. All

91


adaptive schemes reduce the time by around 30%. Note that in computations with an unknowncrack path the refinement area has to be chosen much larger than in the current example, cf Fig.6.13a (right). Relative to the number of elements required in the computation, cf. Fig. 6.14c,the CPT requires the lowest effort followed by BFT and WPLSQ. Hence, it can be concludedthat BFT has the best quality/cost ratio.

6.3 Evolving StructuresIn this section, two different phase-field models are considered that represent structural evolu-tion processes. In the first example, the classical Cahn-Hilliard model is used to simulate thetwo-dimensional spinodal decomposition of binary mixtures. In the second example a gener-alized Ginzburg-Landau phase-field model is used to optimize the topology of two and three-dimensional mechanical structures.

In both examples, mesh refinement and coarsening are applied using the safe strategies of Sec-tion 4.1 in combination with the projection operators for field variables from Section 5.1 and5.2. For the case of mesh coarsening, the local discrete least squares method (LLSQ) and itsL2-version are applied. A transfer of history variables is not required.

6.3.1 Spinodal DecompositionBinary mixtures of fluids can become unstable if e.g. the temperature falls below a specificthreshold. In the subsequent spinodal decomposition, the two components separate to reacha stable energetic state again. The dynamic of this process is governed by the Cahn-Hilliardequation.

Following Section 6.1, the concentration c of one of the two components is taken as the orderparameter and the total free energy is given by

Ψ = Ψbulk + Ψint =

∫Ω

(f(c) +

α

2|∇c|2

)dΩ . (6.21)

The bulk free energy Ψbulk drives the nucleation of the decomposition which dominates the earlystages of the evolution process. The interfacial free energy Ψint accounts for the influence of theindividual interfaces on the system. The constant α is the interface parameter that governs theinterface width and f(c) is chosen here as the logarithmic double well potential [111], given by

f(c) = A[cln(c) + (1− c)ln(1− c) +Bc(1− c)] (6.22)

with constants A and B.

Substituting (6.21) in (6.7), the evolution equation is given by the fourth order PDE

c = ∇ ·[M(c)∇

(∂f(c)

∂c− α∆c

)], (6.23)

where M(c) = Dc(1 − c) is the concentration dependent mobility with scaling parameter D.The symbol ∆ is the Laplace operator. For a detailed description of the model, the reader isreferred to [112] where a convergence study of the Cahn-Hilliard equation was performed.

92

6.3 Evolving Structures

10

18000

Elements

10

rel.Time

adaptivereference

adaptivereference

Figure 6.15: Spinodal decomposition: The total Ψ, interface Ψint, and bulk Ψbulk energies as well as theaverage concentration c are plotted versus time (a) for a reference computation on a uniformfine mesh, and (b) compared to the mesh adaptive computation. (c) The required number ofelements and computation time is significantly reduced using adaptive meshing.

For spatial discretization, a C2-continuous THB-spline basis with four hierarchical levels andp = 3 and for temporal discretization a generalised-α method in combination with an adaptivetime stepping scheme is used. Homogeneous Neumann boundary conditions are applied on thewhole boundary of the domain. In order to accurately capture the nucleation process, a veryfine mesh has to be used in the early stages of the simulation.

For the computation the following set of parameters is assumed: A = 3000 J, B = 3, D =1 m3 s kg−1, λ = 1 J m−1. The averaged concentration in the domain (6.3) is set to m = 0.5.The initial concentration is linearly distributed along the horizontal direction of the specimen.A random perturbation in the magnitude of 10−3 is introduced to promote the evolution of thesystem.

In Fig. 6.15 (a), the total free energy and its individual contributions are plotted. In the be-ginning of the computation the interface energy is zero. When separation starts, interfaces arecreated and Ψint increases. Subsequently, the system reduces the surface energy and the inclu-sions coarsen. At this point, at t = 1× 10−4 s, adaptive refinement and coarsening are activated

93


Figure 6.16: Spinodal decomposition process: The concentration and computational mesh are plotted forthree different instants of time.

to track the evolving interfaces. The euclidean norm of the gradient of the order parameter isused as marking criterion

‖∇c‖

> θ refineelse coarsen , (6.24)

where the threshold is set to θ = 0.5. To project the field variables onto coarsened elements,the LLSQ method of Section 5.2 is applied.

To measure the accuracy of the approach, the relative errors in the individual energies of theadaptive computation with respect to the reference solution are plotted in Fig. 6.15 (b). Theerrors range from 10−5 to 10−2 in the beginning when the inclusions coarsen fast but drop toaround 10−7 in the end. Furthermore, the error in the average concentration m only increasesslightly up to 10−4, which indicates that the LLSQ projection did not introduce significantartificial diffusion of the transferred field variable c. The number of elements is reduced clearly,and the computation time is cut to 74% of the reference solution, cf. Fig. 6.15 (c).

Different states of the computation are illustrated in Fig. 6.16. It can be seen that the refinedmesh adaptively follows the interfaces and that areas with c ≈ const. are coarsened.

94


6.3.2 Topology OptimizationTopology optimization aims to find the design of a structure that fulfils special criteria. Here, theoptimal distribution of a given amount of material in a design domain is sought that minimizesthe compliance of the resulting mechanical structure under predefined boundary conditions.Numerous numerical methods exist to solve such a problem, including the homogenizationor the solid isotropic material penalization method, see e.g. CARRATURO et al. [113] for anoverview. First introduced by BOURDIN & CHAMBOLLE [10], also different phase-field modelswere developed, based on Cahn-Hilliard or Ginzburg-Landau equations. It was shown thatdue to the diffuse interface representation the occurence of checkerboard patterns is prevented[114]. Isogeometric analysis was first applied to a phase-field model for topology optimizationby DEDÈ et al. [18], using uniform B-splines.

Following the phase-field approach, the distribution of the order parameter c defines areas ofmaterial c = 1 and areas of void c = 0. The volume fraction of distributed material m in thegiven design domain Ω is defined by (6.3) and has to be conserved during the optimizationprocess.

Similar to the diffuse representation of material interfaces in Section 6.2.1, the order parameterallows for an interpolation/homogenization of the elastic properties

C(c) = Cbcq + Cv(1− c)q (6.25)

between the bulk material tensor Cb and the void material tensor Cv = εCb that is set to asmall value choosing ε 1. The arbitrary constant q ∈ N is set to q = 3, as proposed byBENDSØE et al. [115]. Based on the results of section 6.2.1, the specific choice of (6.25) isof particular interest when heterogeneous materials are used in the optimization process andtherefore subject of current research.

With the homogenized material tensor at hand, a displacement field u ∈ H1(Ω) can be com-puted that fulfils the weak form of the linear elastic problem∫

Ω

ε(u) : C(c) : ε(v) dΩ =

∫ΓN

g · v dΓN for all v ∈ H10(Ω) ,

u|ΓD = uD ,

(6.26)

as defined in Section 4.4.1.

To obtain the spatial distribution of the order parameter that minimizes the compliance Ψcompof the system, a generalized Ginzburg-Landau phase-field model is used.

According to Section 6.1, the extended free energy (6.8) is given by

Ψ(c,u(c)) =

∫Ω

[κ

(1

`cc2(1− c)2 + `c|∇c|2

)]dΩ︸︷︷︸

Ψint

+

∫ΓN

g · u(c) dΓN︸︷︷︸Ψcomp

, (6.27)

95


x

y

x

y

z

(a) (b)

gg

ΩΓD

ΓN

Figure 6.17: Topology optimization: Domain and boundary conditions are illustrated for the two- andthree dimensional beam problem. Dimensions are given in mm.

where a double well potential fh = c2(1 − c)2 is used in combination with a gradient termfg(∇c) = |∇c|2 in the interface energy Ψint. But in contrast to all previous introduced phase-field models, (6.27) has to be minimized under the constraint that (6.3) and (6.26) holds.

Following the work of BLANK et al. [114] and CARRATURO et al. [113], the constrained opti-mization problem can be rewritten in terms of the Lagrangian function

L(c,u,p,λ) =

∫Ω

[κ

(1

`cc2(1− c)2 + `c|∇c|2

)]dΩ +

∫ΓN

g · u dΓN

+

∫ΓN

g · p dΓN −∫Ω

ε(u) : C(c) : ε(p) dΩ

+ λ

∫Ω

c−m dΩ ,

(6.28)

where p is the adjoint variable and λ the Lagrange multiplier.

Searching the steady state of L with respect to (c,u,p,λ) shows that both u and p solve (6.26).Hence, the approach leads to a fully coupled three-field problem given by the equilibrium con-dition δL

δu = 0 and δLδλ = 0 and the Ginzburg-Landau evolution equation c = −M δL

δc .

In the following, the model problem above is solved monolithically using C1-continuous THB-splines with p = 2 for spatial and a backward Euler method for temporal discretization. Anadaptive time stepping scheme is applied that considers the number of Newton iterations, re-quired to solve a time increment. Adaptive mesh refinement and coarsening are activated fromthe beginning of the computation. As in the previous section, the gradient of the order parameteris used as a heuristic marking criterion with θ = 2, cf. equation (6.24).

In the first example, illustrated in Fig. 6.17a, a cantilever beam is modelled as a disc un-der plane strain condition. The specimen is clamped on the left and a line load with g =[0,−104]T MPa mm is applied on the right end of the lower edge. The following set of pa-rameters is assumed: λ = 121 153 MPa, µ = 80 769 MPa, `c = 0.02 mm, m = 0.5 andκ = 103 MPa mm. The initial value of the order parameter is set everywhere in Ω to c = 0.5.

96


10−7

10−6

10−5

10−4

10−3

10−2

10−1

0 0.2 0.4 0.6 0.8 1

rel.

Err

or

t/s

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1

Ψ/J

t/s0

10000

30000

0 0.2 0.4 0.6 0.8 1

Ele

men

ts

0

1

0 0.2 0.4 0.6 0.8 1

rel.

Tim

e

t/s

(a)

(b)

(c)

LLSQLLSQ-L2

uniform

LLSQLLSQ-L2

uniform

LLSQLLSQ-L2

ΨintΨcomp

Figure 6.18: 2D topology optimization : The compliance and interface energy are plotted versus time (a)for a reference computation on a uniform fine mesh. (b) The relative error in the energiesof the mesh adaptive computations are given for the LLSQ projection and the correspondingL2-version. (c) The required number of elements is reduced significantly for both meshadaptive computations but only the L2-version of the LLSQ is able to outperform the uniformreference computation.

In the computation, THB-splines with five hierarchical levels are used. The element size ofthe finest level is hE = 0.39`c. To project the field variables onto the coarsened elements, theLLSQ method from Section 5.2 and its L2-version are applied.

As shown in Fig. 6.18a for the uniformly pre-refined reference computation, the compliance andthe interface energy of the system drop steeply in the beginning of the computation and convergeto a steady state afterwards. The evolution of the order parameter and the locally refined meshesare illustrated in Fig. 6.19 for the mesh adaptive computation. It can be seen that the meshrefines if the gradient of the order parameter increases, and that it coarsens again if the gradientdecreases. In Fig. 6.18b, the corresponding relative errors in the different energies are plotted.They never exceed 10−1 for LLSQ and 10−2 for its L2-version. The relative errors in the finalstate of the computation are only around 10−4. As shown in Fig. 6.18c, using the adaptivemeshing in combination with the L2-version of the LLSQ method, up to 70% of elements and75% of computation time is saved. However, using the LLSQ method, the projection errors leadto insufficient trial solutions for the following increments. As a result, much more iterations are

97


Figure 6.19: 2D topology optimization: The concentration and computational mesh are plotted for threedifferent instants of time.

required per increment and only small time steps can be applied. This leads to an increasedcomputation time and in the end, to a similar performance as the reference computation. Incontrast to the Cahn-Hilliard phase-field model of the previous section where good results withthe LLSQ method could be obtained, it seems that the topology optimization method is moreprone to incorrect trial solutions. Another possible explanation is that in the Cahn-Hilliardphase-field model a uniform fine mesh was used in the beginning of the computation and theadaptive meshing started later after the first interfaces had already been developed.

In the second example, illustrated in Fig. 6.17b, a solid beam is considered. The model set upis similar to the problem presented by DEDÈ et al. [18]. The beam is clamped on the back anda surface load g = [0,−104, 0]T N mm−2 is applied on a square domain on the front face of thespecimen. The material and model parameters are identical to thre previous example, exceptthe volume fraction that is set to m = 0.35 here. The initial value of the order parameter is seteverywhere in Ω to c = 0.35.

In the computation, THB-splines with three hierarchical levels are used. The element size ofthe finest level is hE = 0.625`c. Based on the results above, the L2-version of LLSQ methodfrom Section 5.2 is used to project the field variables onto the coarsened elements. Due to thesymmetry of the problem, only the half of the domain is modelled.

98


Figure 6.20: 3D topology optimization: The order parameter and the computational mesh are plotted forthree different instants of time.

In Fig. 6.20 the evolution of the order parameter as well as the locally refined mesh are il-lustrated for three different instants of time. Due to the cantilevered structure of the designdomain, a slender support structure is automatically generated. The mesh automatically fol-lows the gradients of the order parameter. The total compliance and interface energy of thesystem are plotted in Fig. 6.21b. As in the 2D example, the energies drop steeply in the begin-ning of the computation and converge to a steady state afterwards. Interestingly, the complianceΨcomp reaches a local minimum first, increases again and then converges to a second minimum.However, the total energy Ψ that is minimized in the simulation converges monotonically. The

99


(a)

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Ψ/J

t/s

ΨΨint

Ψcomp

(b)

Figure 6.21: 3D topology optimization : (a) The structure that results from the topology optimization isillustrated by a contour plot of the order parameter with c = 0.5. (a) The total energy Ψ aswell as the compliance Ψcomp and interface Ψint energies are plotted versus time for the meshadaptive computation.

resulting structure of the optimization problem is illustrated in Fig. 6.20a by a contour plot ofthe order parameter with c = 0.5. Unfortunately, a comparison with a uniformly pre-refined ref-erence computation is not possible, due to the memory limitations of computing facilities. Butnote that a uniform fine mesh would consist of 524 288 elements, whereas in the mesh-adaptivecomputation only around 150 000 elements were used.

6.4 ConclusionsIn this chapter the adaptive isogeometric analysis framework, introduced in chapters 3-5, wasapplied to the Cahn-Hilliard and several generalized Ginzburg-Landau phase-field models. Itwas shown that the diffuse representation of the discontinuities avoids a cumbersome numericaltracking and re-discretization of interfaces. Furthermore, the application of the adaptive IGAframework led in combination with appropriate projection and transfer operators to accurate,reliable and more efficient computations compared to the corresponding non-adaptive referencesolutions. This is especially of high importance if three-dimensional problems are considered.

The different transfer operators for history variables, proposed in Section 5.3, were applied tothe simulation of ductile fracture. The comparison showed that only the BFT method producessimilar results to the reference computation but provides a significant reduction of computationtime.

Following the results of Section 5.2, the local least squares method (LLSQ) was applied tosimulate the process of spinodal decomposition. While producing results comparable to thereference computation, the efficiency could be increased clearly. However, the application ofthe LLSQ method to the topology optimization problem led to insufficient trial solutions for thefollowing increment. As a result, the computing time was comparable to that of the referencesolution. If in contrast the L2-version of the LLSQ was applied, the computation time could bereduced by up to 75%. This leads to the conclusion that the L2-projection itself is more timeconsuming because of the required integration over the elements, but in the end leads to more

100

6.4 Conclusions

efficient computations due to improved trial solutions and a faster convergence of the Newtonscheme.

In the context of solid mechanics, this chapter presented a unified phase-field modelling ap-proach for weak and strong discontinuities. For this purpose, phase-field models for brittleand ductile fracture where reviewed and a static phase-field introduced to regularize materialinterfaces. To consider the mechanical jump conditions at the interface, a homogenization ofadjacent material parameters was introduced. Furthermore, an h`-adaptive refinement strategywas applied, to provide an appropriate and efficient approximation of the diffuse interface re-gion. It could be shown that the presented approach avoids stress oscillations at the interfaceand results in an error range that is typical for embedded methods and suitable for engineeringapplications. The extension of this approach with a modified phase-field model that also ac-counts for interface failure allowed for the computation of fracture in heterogeneous materialsusing non-conforming meshes.

101

7 Summary and OutlookThe objective of this thesis was to provide a robust and reliable isogeometric analysis frameworkthat allows for an adaptive spatial discretization of non-linear and time-dependent multi-fieldproblems. The application to several phase-field models allowed for a verification of the sta-bility and efficiency of the whole framework and a comparison of the presented projection andtransfer operators. Furthermore, the phase-field method was used to develop a unified modellingapproach for weak and strong discontinuities in solid mechanics as they arise in the numericalanalysis of fracture in heterogeneous materials.

In detail, Bézier extraction of THB-splines was proposed in Chapter 3. While Bézier extrac-tion provides a multi-level system of equations with independent levels, the communicationbetween different levels of the hierarchy was introduced in terms of a hierarchical subdivisionoperator for B-splines and NURBS. It was shown that a simple matrix multiplication producesa hierarchical system of equations, which is identical to the use of the truncated hierarchicalbasis, avoiding the explicit truncation of individual basis functions. The use of an elementviewpoint and the application of Bézier extraction facilitate the implementation of the proposedframework into any existing finite element code. Moreover, standard procedures of adaptivefinite element analysis are directly applicable.

The Bézier extraction framework can be used also, to impose Dirichlet boundary conditionson a THB-spline basis by an L2-projection. It was shown that the projection leads to optimalconvergence rates for the global and element-wise projections.

In Chapter 4 two different refinement strategies and one coarsening strategy for THB-splineswere introduced and implemented in an adaptive framework. Based on numerous numericalbenchmarks they were analysed and compared against two different refinement strategies forT-splines. It was shown that the different strategies can be classified in two groups: safe andgreedy refinement strategies. Between these groups obvious differences in the mesh grading andthe numerical properties of the stiffness matrix increase with the locality of the problem. Forthose problems in which very local features are to be resolved, the greedy refinement routinesshowed a clear increase in the condition number per degrees of freedom and led to unstructuredmeshes around the refinement area. However, for extensive refinements, all refinement routinesproduce similar results. Consequently, the readers are advised to chose the refinement methodwhich best fits the requirements of their application, but especially for local refinement areas itmight be wise to use the safe refinement to avoid unpredictable numerical inaccuracies. Fur-thermore, the safe refinement strategies simplify the projection and transfer operators for statevariables due to the reduced interaction of basis functions from different levels.

The application of the adaptive framework to two- and three-dimensional singular and non-singular problems of elasticity and the Poisson equation led to optimal convergence rates andverified in this way the numerical implementation of the Bézier extraction framework and therefinement strategies.

Since a large class of boundary value problems is non-linear or time-dependent in nature andrequires incremental solution schemes, projection and transfer operator for field and history

103

7 Summary and Outlook

variables were introduced in Chapter 5. To avoid the solution of a global system of equationsduring the projection of primary field variables on coarsened meshes, two local projection meth-ods were proposed and compared to existing semilocal and global versions. In addition to thediscrete least square methods, also corresponding L2-projections were analysed. Based on thebenchmark results, the local least squares method showed the best benefit-cost ratio. For his-tory variables, two different transfer operators were discussed and compared against the simpleclosest point transfer (CPT). The benchmark studies concerning differently distorted meshesshowed that both methods provide much better results than the CPT.

In general it can be summarized that isogeometric analysis improves the performance of theprojection and transfer operations as already the coarsest mesh represents the exact geometryand the hierarchical structure allows for quadrature free projection methods.

In Chapter 6 different phase-field models were analysed to verify the stability and efficiencyof the adaptive isogeometric framework and to compare the projection and transfer operatorsfor the state variables in more complex examples. The local least squares method (LLSQ) thatprojects field variables onto coarsened elements was applied to simulate the process of spin-odal decomposition. While producing results comparable to the reference computation, theefficiency could be increased clearly. However, the application of LLSQ to the topology op-timization method led to insufficient trial solutions for the following increment and hence toa worse convergence of the Newton scheme. The use of the L2-version of the LLSQ couldresolve this problem and resulted in efficient and accurate computations. The different transferoperators for history variables were applied to the simulation of ductile fracture. The compari-son showed that in particular the basis function transfer produces similar results to the referencecomputation and provides a significant reduction of computation time.

Furthermore, in Chapter 6 a unified phase-field modelling approach for weak and strong discon-tinuities in solid mechanics was proposed. To avoid costly meshing processes, a static phase-field was introduced to represent material interfaces in a non-conforming mesh. In the resultingtransition region, a homogenization of adjacent material parameters was applied to consider themechanical jump conditions at the interface. It was shown that for one-dimensional problemsthe error is bounded to the near field and that optimal convergence rates can be obtained in thefar field. The error in the near field could be further decreased by the application of an hl-adaptive refinement strategy. For arbitrarily shaped interfaces in multi-dimensional problemsthe presented approach avoids stress oscillations at the interface and results in an error range thatis typical for embedded methods and suitable for engineering applications. The extension of theapproach with a modified phase field model that also accounts for interface failure allowed forthe computation of fracture in heterogeneous materials using non-conforming meshes.

Summarizing all findings, it was shown that the application of adaptive IGA in combinationwith appropriate refinement and coarsening strategies as well as appropriate projection andtransfer operators, leads to stable simulations and a clear reduction of computation time withoutthe loss of accuracy. This is of particular importance if it is unknown where in the computationaldomain the local refinement/coarsening is required, or in general if large three dimensionalsystems are considered. Furthermore, under special circumstances, it turned out that methodswhich first seem to be more time-consuming, in the end lead to more efficient simulations sinceiterative solvers converge better.

104

In combination with the phase-field method the adaptive meshing eliminates the critical point ofinefficient computations and allows for a straight-forward simulation of moving discontinuitiesby the solution of an (additional) scalar field problem. This makes the phase-field method incombination with the adaptive discretization a powerful numerical tool. The application ofadaptive isogeometric analysis is especially beneficial if higher-order phase-field models areused or if the computational domain, on which the phase-field model is solved, is described byNURBS-geometries.

Regarding the presented adaptive isogeometric analysis framework, future work is required toimprove the coarsening strategy that is currently based on a less efficient two-step algorithm.Furthermore, the weighted patch based least square method for the transfer of history variablesneeds further investigations. The application of this method is to be preferred, especially ifcombined with a recovery based error estimator. However, the numerical tests did not lead tosatisfactory results in its current form of implementation.

Regarding the presented phase-field models many open topics exist. This concerns in particu-lar the phase-field modelling of fracture in heterogeneous materials. In the presented approachno split of the strain energy is considered, which would lead to wrong results under mixedloads. Hence, further investigations are required in order to combine the diffuse material in-terface representation that considers the mechanical jump conditions with a spilt of the strainenergy (6.19) in a tensile and compressive part. A first approach in this direction is proposedby HANSEN-DÖRR et al. [109]. The consideration of jump conditions at regularized materialinterfaces also plays a role in the topology optimization of multi-materials. The influence ofdifferent interpolation/homogenization schemes on the optimization process is already subjectof current research.

105

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