master’s thesis - tum€¦ · this thesis would not have been possible without the help of many...

TECHNISCHE UNIVERSITÄT MÜNCHEN

Computational Science and Engineering(Int. Master’s Program)

Master’s Thesis

The Finite Cell Method for Linear

Thermoelasticity

Nils Zander

TECHNISCHE UNIVERSITÄT MÜNCHEN

Computational Science and Engineering(Int. Master’s Program)

Master’s Thesis

The Finite Cell Method for Linear

Thermoelasticity

Author: Nils Zander

Matriculation number: 3218999

1st examiner: Prof. Dr.rer.nat. Ernst Rank

2nd examiner: Prof. Dr.-Ing. André Borrmann

Assistant advisor: Dr.-Ing. Stefan Kollmannsberger

Thesis handed in on: March 15, 2011

I hereby declare that this thesis is entirely the result of my ownwork except whereotherwise indicated. I have only used the resources given in the list of references.

Munich, March 15, 2011 Nils Zander

Acknowledgments

This thesis would not have been possible without the help of many people and Iwant to use this chance to express my gratitude to them.

I would like to thank Dr.-Ing. Stefan Kollmannsberger explicitly for his great sup-port throughout the past five months. This thesis would not have been possiblewithout his help.

I’m thankful to Prof. Ernst Rank and Prof. Zohar Yosibash for their inspiringinput inmany discussions which gave my research the right direction and shapedthis thesis.

Further, I want to thank Prof. Ernst Rank and Prof. André Borrmann for examin-ing this work.

Thanks to all people from the chair for Computation in Engineering for their sup-port and the friendly atmosphere. It is a pleasure to work with them.

I thank Prof. Hans-Joachim Bungartz and all people from the chair of ScientificComputing for the great support during the past two years of the CSE Master’sProgramme.

Special thanks are due to my family. This thesis and my academic studies wouldnot have been possible without them. Especially, I want to thank my girlfriendSandra for her invaluable support and her lovely understanding in the past years.

vi

Abstract

This thesis presents an approach to approximate the phenomena of linear ther-moelasticity numerically in the framework of the Finite Cell Method (FCM). Forthis purpose, the governing equations of the thermal, elastic and linear thermoe-lastic system are derived in their strong and weak formulation. In particular,it is outlined how Dirichlet boundary conditions of the three problems can beimposed in the weak sense. In the second part of this work, the constrainedweak field equations are discretized in space utilizing the ideas of the Finite CellMethod. The performance of this approach is demonstrated through differentexamples.

vii

Contents

Acknowledgements vi

Abstract vii

1 Introduction 11.1 Motivation and goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Tensor notation . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Single field description 72.1 Generic field description . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Single field description . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Temperature field . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Displacement field . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Variational single field description 153.1 Introduction to variational calculus . . . . . . . . . . . . . . . . . . . 163.2 Governing potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Temperauture potential . . . . . . . . . . . . . . . . . . . . . 213.2.2 Elastic potential . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Constraint potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.1 Penalty method . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.2 Identified Lagrange multiplier method . . . . . . . . . . . . 343.3.3 Nitsche method . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Linear thermoelasticity 434.1 Linear thermoelasticity as a multi field problem . . . . . . . . . . . 43

4.1.1 Classification of multi field problems . . . . . . . . . . . . . 444.1.2 Assumptions on linear thermoelasticity . . . . . . . . . . . . 44

4.2 Formulation of linear thermoelasticity . . . . . . . . . . . . . . . . . 464.3 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 50

ix

Contents

4.3.1 Governing potential of linear thermoelasticity . . . . . . . . 504.3.2 Constraint potential of linear thermoelasticity . . . . . . . . 53

5 Numerical Methods 595.1 The Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . 59

5.1.1 FEM for the temperature problem . . . . . . . . . . . . . . . 625.1.2 FEM for the elastic problem . . . . . . . . . . . . . . . . . . . 63

5.2 The Finite Cell Method (FCM) . . . . . . . . . . . . . . . . . . . . . . 655.2.1 Concept of the Finite Cell Method . . . . . . . . . . . . . . . 655.2.2 FCM for the temperature problem . . . . . . . . . . . . . . . 675.2.3 FCM for the elastic problem . . . . . . . . . . . . . . . . . . . 69

5.3 FCM for linear thermoelasticity . . . . . . . . . . . . . . . . . . . . . 725.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6 Numerical examples 816.1 Weak Dirichlet boundary conditions for the FCM . . . . . . . . . . . 81

6.1.1 Elastic system . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.1.2 Thermoelastic system . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Bimetal strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Summary, conclusion and outlook 103

Bibliography 105

x

List of Figures

1.1 Numerical model of the electro-thermo-mechanical system . . . . . 21.2 Layout of the single field derivation . . . . . . . . . . . . . . . . . . 4

2.1 Generic Tonti diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Tonti diagram of the temperature field . . . . . . . . . . . . . . . . . 112.3 Tonti diagram of the elastic continuum . . . . . . . . . . . . . . . . . 14

3.1 Difference between variation and differential . . . . . . . . . . . . . 173.2 Tonti diagram of the weak thermal model . . . . . . . . . . . . . . . 243.3 Tonti diagram of the weak elastic model . . . . . . . . . . . . . . . . 293.4 Tonti diagram of the weakly formulated field descriptions . . . . . 31

4.1 Schematic illustration of multi field problem classes . . . . . . . . . 444.2 Tonti diagram of linear thermoelasticity . . . . . . . . . . . . . . . . 49

6.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 Approximation on conform mesh . . . . . . . . . . . . . . . . . . . . 856.3 Approximation on non-conform mesh with β 0 . . . . . . . . . . 866.4 Approximation on non-conform mesh with β 105 . . . . . . . . . 886.5 Approximation on non-conform mesh with exact integration (β 0) 896.6 Analytical temperature distribution . . . . . . . . . . . . . . . . . . 916.7 Temperature approximation on conform mesh . . . . . . . . . . . . 926.8 Displacement approximation on conform mesh . . . . . . . . . . . . 946.9 Simulation on non-conform mesh with β 0 . . . . . . . . . . . . . 956.10 Simulation on non-conform mesh with β 105 . . . . . . . . . . . . 976.11 Simulation on non-conform mesh with exact integration (β 0) . . 986.12 Setup of bimetal strip . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.13 Approximation of bimetal strip on conform mesh (FEM) . . . . . . 996.14 Approximation of bimetal strip on non-conform mesh (FCM) . . . 1006.15 Distribution of strain over beam height . . . . . . . . . . . . . . . . 1016.16 Distribution of vertical displacement over beam height . . . . . . . 102

xi

1Introduction

1.1 Motivation and goals

The idea of manufacturing tools form metallic powder without melting, nowa-days denoted as powder metallurgy or sintering, was already known to the an-cient civilizations of Egypt and Greece ([18]). During the industrialization of the19th and 20th century, this technology was enhanced and became part of modernmanufacturing processes. A relatively new development is the idea of a “FieldAssisted Sintering Technology” (FAST), in which an electric current is applied tothe metallic power while it is compressed with high pressure. This combinationhas the advantage of lower process temperatures and shorter holding times ([9]).

To optimize the quality of FAST products, the interaction between the powdercompression, the heat conduction and the applied electric field has to be under-stood. For this reason, the Deutsche Forschungsgemeinschaft (DFG) granted theresearch project “Electro-thermo-mechanical modeling of Field Assisted Sinter-ing Technology using high-order finite elements validated by experiments”, inwhich these phenomena shall be analysed ([9]). The aim of the project is to de-velop a thermo-viscoplasticity model, describing the behaviour of the powderunder large deformation. This constitutive equation will be used to simulate themulti-physical phenomena. For this purpose, an electro-thermo-mechanical cou-pling model will be formulated. Mathematically, this requires the descriptionof the three physical fields and their interaction. Due to the complexity of thismodel, an analytical solution can only be found for simple examples. Industrial

1

1 Introduction

applications typically require the use of numerical methods. To simulate the fieldinteraction, a partitioned approach is chosen, in which the three physical fieldsare separately discretized using the Finite Element and the Finite Cell Method([17], [6]) and the respective coupling data is exchanged. This ansatz is visual-ized in figure 1.1.

Temperature T Electric Potential: φ

Displacement u

u u

T

T

φ

φ

Punsh Die

Powder

Figure 1.1: Partitioned numerical model of the electro-thermo-mechanical sys-tem. ([9])

Excluding the influence of the electric field, a numerical model of the thermo-mechanical coupling shall be formulated and implemented in this thesis, us-ing the Finite Cell Method. To not exceed the scope of this work, the metallicpowder is modeled as an elastic continuum. With this assumption, the thermo-mechanical coupling yields the problem of thermoelasticity. Furthermore, the de-formations are assumed to be small such that the change of temperature causedby the deformation can be neglected. With this second assumption, the descrip-tion of thermoelasticity can be linearized. The numerical model, simulating theproblem of linear thermoelasticity in the framework of the Finite Cell Method,and its derivation is presented in the following chapters of this work.

2

1.2 Outline

1.2 Outline

To give an overview over the individual aspects that shall be discussed in thiswork, the content of the individual chapters and their structure will be brieflyoutlined in this section.

In the first part of this work, the mathematical descriptions of the thermal andelastic problem are derived in two different ways, not taking into account the fieldinteraction ( see figure 1.2). In chapter 2, conservation laws, governing the stateof the two systems, are introduced. From these, the individual field descriptioncan be derived in form of partial differential equations (PDEs). An alternativemodel, based on the idea that the equilibrium state of a system corresponds tothe extremum of a governing potential, is formulated in chapter 3. An alterna-tive field equation in form of a necessary extremal conditions is derived usingvariational calculus. Due to the lower smoothness requirements it imposes onthe solution function, it is denoted as the weak field equation, whereas the PDEs,derived in chapter 2, are the strong field equations. The extremal condition canbe transformed, yielding the Euler-Lagrange equation. Identifying this formu-lation as the strong field equations proves that the extremal conditions stratifiesthe conservation law in the weak sense. In section 3.2, Dirichlet boundary con-ditions of the weak field equation are addressed. For this propose, the penalty,the Lagrange multiplier and the Nitsche method are discussed. In particular it isshown how a mixed field formulation can be avoided by an identification of thethe Lagrange multiplier using the associated Lagrange-Euler equation.

Having derived the mathematical models of the thermal and elastic problem,these are combined to describe the phenomena of thermoelasticity in chapter 4.For this purpose, the description of the thermoelasticity is linearized and thecoupling conditions are introduced in section 4.1. The field equations are thenderived in the strong and weak form in section 4.2 and 4.3.1, respectively. Justas in the single field case, the Dirichlet boundary conditions can be imposed inthe weak sense and a mixed field formulation can be avoided by identifying theLagrange multiplier using the associated Lagrange-Euler equation. This is per-formed in section 4.3.1.

In chapter 5, the numerical model of linear thermoelasticity is derived. Forthis purpose the concept of the Finite Element Method (FEM) is outlined and theequation system, approximating the state of the thermal and elastic system, isderived in section 5.1. The same is carried out for the Finite Cell Method (FCM)in section 5.2, focusing on the Dirichlet boundary conditions. Coupling theseresults, the numerical model of linear thermoelasticity is formulated in section5.3.

To analyse the performance of the derived model, different examples are pre-

3

1 Introduction

Chapter 3VariationalFormulation

Section 3.2GoverningPotentials

Section 3.3ConstraintPotentials

PenaltyMethod

LagrangeMultiplierMethod

ConstraintPotential

ConstraintPotential

WeakDirichletBoundaryCondition

Euler-LagrangeEquation

WeakDirichletBoundaryCondition

define

find extremum

define

find extremum

reformulateidentifymultiplier

GoverningPotential

Weak FieldEquation

EulerLagrangeEquation

assume

find extremum

reformulate

--.Nitsche Method

Chapter 2Single FieldDescription

ConservationLaw

Strong FieldEquation

assume

derive

equals

----weakly

satisfies

Figure 1.2: Layout of the single field derivation in chapter 2 and 3. The shadeditems represent the major result of the respective sections.

4

1.3 Notation

sented in chapter 6.

The work concludes with a summary and an outlook in chapter 7.

1.3 Notation

Within this work, two different notations are used. To avoid confusion, theseshall be briefly outlined.

1.3.1 Tensor notation

The first way to express tensors is in the compact notation in which a tensor isrepresented by a underlined boldface symbol, e.g. σ. Thereby, the rank of thetensor is not indicated.

The alternative notation is the index notation, in which the an element of thetensor is represented by an index, e.g. σij . Thereby, the basis of the tensor is as-sumed to be Cartesian, so that co- and contra-variant tensors can both be notatedby sub-indices. To switch between both notations, the sign is used, e.g.

σ σij .

The contraction of tensors in the compact notation is given by

t σ n and W σ : ε.

In the index notation the repetition of an index implies its summation accordingto Einstein’s summation convention. Hence, the above example reads

ti σij nj and W σij εij.1.3.2 Matrix notation

Amatrix is represent by non-underlined boldface symbol, e.g. σ. The product ofmatrices or vectors is give by writing the expressions consecutively, e.g.

σ C ε.5

1 Introduction

6

2Single field description

As outlined in the introduction of this work, the phenomena of thermoelasticityarise from the interaction of the temperature in and the deformation of an elasticbody. For this reason, thermoelasticity is to be regarded as a multi field problem,governed by the interaction of a temperature and a deformation field. Therefore,the individual problems must be described independently of each other first, be-fore a mathematical model of a thermoelastic process can be formulated. Hence,the task, which will be addressed in the following sections, is to derive the re-spective field equations, with which the temperature and the displacement can becomputed at each point x of a domain Ω. To point out the similarities of the twomodels, a generic field description shall be derived in section 2.1, form which theindividual equations, governing the temperature and the elastic field, can theneasily be derived in section 2.2.

2.1 Generic field description

Most mathematical models of physical problems show major similarities in theirformulation. To point out these analogies, a generic field description shall bederived in this section. For this purpose, the classification scheme introduced byE. Tonti ([19], [20], [21]) is used, which divides the variables and equations usedto describe the physical field into groups depending on their characteristics.

Due to the complexity of most physical problems, their mathematical models

7

2 Single field description

involve the combination of several equations. To reveal the underlying structure,Tonti decomposes the fundamental equations into two basic sets of physical laws.On the one hand, equations whose validity is independent of any physical, geo-metrical or material properties are referred to as topological laws. Typically, theseequations contain algebraic sums, derivatives and integrals in space and timeand their linear combinations. On the other hand, equations which are of phe-nomenological kind are identified as constitutive laws. These equations typicallyinvolve material coefficients and physical or geometrical parameters. Commonexamples of topological laws are balance and conservation equations, whereasHooke’s law of elasticity, Ohm’s law of electric conduction and Fourier’s law ofheat conduction are typical examples of constitutive equations.

In addition to the structure of physical laws, Tonti points out that every physi-cal theory contains some variables which describe the configuration of the systemand some which describe the source of the phenomena. These variables are clas-sified as configuration and source variables, respectively. Common examples ofconfiguration variables are the displacement in continuummechanics or the elec-tromagnetic potential in electromagnetism, whereas forces, currents and the heatproduction density are examples of source variables ([19]).

Apart from configuration and source variables, the mathematical model of thephysical problem may contain additional unknowns. If these are connected to aconfiguration variable via a topological equation and hence independent of anyphysical, geometrical or material properties, they are regarded as configuration-type variables. Likewise, unknowns which are connected to a source variablevia a topological equation are source-type variables. A typical example of a con-figuration-type variable is the strain of a deformed body, whereas its stress is asource-type variable.

To complete the mathematical model of the physical process, the configura-tion variables must be linked to the source variables. This connection typicallyinvolves the specification of material coefficients and physical or geometricalparameters and is established by constitutive equations. Due to the physicalcomplexity however, it is not always possible to derive a direct connection be-tween the configuration and source variables but only between configuration-and source-type variables.

Once all equations are known, the configuration variable can be expressedas a function of the source variable. Commonly, this leads to one or more par-tial differential equations (PDEs), which must hold in each point of the domain.Additionally, boundary conditions (BCs) must be specified. In most continuumproblems, Dirichlet and/or Neumann boundary conditions are used. Dirichletboundary conditions specify the state of the field on the boundary by prescribingthe configuration variable. Neumann boundary conditions, on the other hand,specify the flux over the boundary, which is commonly related to the source-type

8

2.2 Single field description

terms.

With the derived classification scheme, the mathematical model of most con-tinuum mechanical problems can be visualized in a Tonti diagram as depictedin figure 2.1. The configuration and source variables c and s are placed on thebottom of the diagram. From these, the configuration- and source-type variablesε and σ are derived, using topological laws. The connection between the con-figuration and source side is established via a constitutive equation, connectingthe configuration- and source-type variables. With this set of equations, the PDE,directly connecting the configuration and source variables, can be derived. To-gether with its boundary conditions, the continuum mechanical problem if fullydescribed.

c

εσ

s c

σ

Dirichlet BC

Neumann BC

Topological Law

Constitutive Law

Topological Law

PDE

Constitutive variable

Constitutive-type variableSource-type variable

Source variable

Figure 2.1: Generic Tonti diagram. PDE partial differential equation; BC boundary condition


To derive the mathematical models describing the temperature and the displace-ment distribution in a domain Ω, the equations governing the physical problemshave to be formulated. This shall be done in the upcoming part of the presentwork, using the generic field description introduced in the previous section topoint out the analogies in both models.

2.2.1 Temperature field

In this section, a mathematical model describing the distribution of the thermalheat Q in a domain Ω shall be derived, using the temperature distribution T pxq

9


as the configuration variable. This description is based on the first law of thermo-dynamics. Form this, the law of thermal heat conservation can be derived ([16],[23], [13], [15]), stating that the change in time of the internal thermal heat Q ofthe system must be in balance with the internal heat production s and the flow ofheat q over the domain boundary Γ. Mathematically, this law can be formulatedas

dQ

dt »

Ω

s dΩ »Γ

q n dΓ,with n being the outward pointing normal vector of the domain boundary ([14]).Assuming the existence and continuity of the first partial derivative of q, the sur-face integral can be transformed into a volume integral using Green’s identity([26]). Additionally assuming the problem to be stationary, the conservation lawcan be rewritten as »

Ω

∇ q s

dΩ 0.

As this integral has to become zero for any arbitrary domain Ω, the integranditself must me zeros. Hence,

∇ q s 0 (2.1)

must hold in every point x in Ω. Since the derived local form of the stationaryheat conservation equation does neither involve any material coefficients, norany physical or geometrical parameters, it can be identified as a topological law,connecting the internal heat production s to the heat flux q. As the productionof heat inside the domain leads to a change of the temperature, s is the sourcevariable of the mathematical model. Following the Tonti schema introduced insection 2.1, the heat flux q is therefore a source-type variable.

Having derived the first topological equation, the second step is to find a con-nection between the source and the configuration variable. According to section2.1, this is achieved using a constitutive equation which expresses the influenceof the used material. For the materials under consideration in the present work,Fourier’s law of heat conduction is assumed to be valid. It states that the flowof heat q is always in the direction of lower temperature and proportional to thegradient of the temperature field, which shall be denoted by p. Mathematically,Fourier’s law reads

q κ p, (2.2)

with κ being the material’s heat conductivity.

As p is defined to be the gradient of the temperature field and hence is inde-pendent of any material or physical parameters, it is to be identified as a con-figuration-type variable, which is connected to the temperature by

p ∇T. (2.3)

10


This topological equation completes the mathematical model of the thermal field.

To fully define the temperature configuration, boundary conditions need to beintroduced. On the Dirichlet boundary ΓD, the configuration variable is set toa predefined temperature T , whereas on the Neumann boundary ΓN the heat,flowing out of the field, is specified using a given flux q. Hence, the field equa-tions describing the temperature distribution in a domain Ω can be summarizedas

∇ q s 0 in Ω (2.4a)

q κ p in Ω (2.4b)

p ∇T in Ω (2.4c)

T T on ΓD (2.4d)

q n q on ΓN . (2.4e)

With this, the task of finding the temperature distribution in a domain Ω cantherefore be formulated as

“Find the configuration function T pxq, fulfilling thefield equations (2.4a), (2.4b) and (2.4c) x P Ω andthe boundary conditions (2.4d) (2.4e) x P Γ.”

The derived mathematical model can be visualized as a Tonti diagram, shownin figure 2.2.

T

pq

s T

q

T T

q n q

p ∇T

q κ p∇ q s 0

PDE

in Ω

in Ω

in Ω

on ΓD

on ΓN

Temperature

Temperature gradientHeat flux

Heat source

Figure 2.2: Tonti diagram of the temperature field

11


2.2.2 Displacement field

In analogy to the temperature field description derived in the previous section, amathematical model of an elastic continuum shall be formulated, using the dis-placement upxq as the configuration variable. For this purpose, Newton’s secondlaw of motion is used, stating that the change in time of the momentum P of acontinuum is in balance with the sum of the body forces b and the traction forcest acting on it ([8], [13]). The mathematical formulation of this fundamental lawreads

dP

dt »

Ω

b dΩ »Γ

t dΓ.

According to Cauchy’s stress principle ([2], [13], [14]), there exists a rank-2-tensor σ, with which the traction force t, acting on the surface of the continuum,can be expressed as

t σ n,with n being the surface normal vector. Commonly, σ is referred to as the stresstensor. According to Newton, also the angular momentum must be conserved.Using this second conservation law, the stress tensor can be shown to be sym-metric ([13],[14]). Mathematically, this reads

σ σJ. (2.5)

Using the stress tensor, Newton’s second law can be expressed as

dP

dt »

Ω

b dΩ »Γ

σ n dΓ.Assuming the existence and continuity of the first partial derivatives of σ, thesurface integral can be transformed into a volume integral, using Green’s identity([26]). Additionally assuming the problem to be stationary, the equation reducesto »

Ω

p∇ σ bq dΩ 0.

As this integral has to become zero for any arbitrary domain Ω, the integranditself must be zeros. Hence,

∇ σ b 0 (2.6)

must hold in every point x of the domain Ω. Since the derived local form of thestationary momentum conservation equation does neither involve any materialcoefficients nor any physical or geometrical parameters, it can be identified as atopological law, connecting the body load b and the stress state σ. As the defor-mation of the continuum is caused by the body force acting on it, b is the sourcevariable. Therefore, according to section 2.1, the stress state σ is a source-type

12


variable.

In analogy to the derivation of the heat equation, the second step is to find aconnection between the source and the configuration variable. As outlined in sec-tion 2.1, this requires a description of the material. In the present work, only lin-ear elastic materials shall be considered. With this restriction, the internal stressstate σ is proportional to the normalized deformation of the continuum, denotedas the strain ε. The constitutive law hence reads

σ C : ε, (2.7)

using the rank-4 constitutive tensor C, describing material behaviour. For theclass of hyper elastic materials, which include the linear elastic case, the majorsymmetry of the constitutive tensor, reading

Cijkl Cklij, (2.8)

can be shown ([14]).

To complete the mathematical model of the elastic continuum, a description ofthe strain has to be found. Assuming the deformation of the body to be small, thelinear formulation can be used

ε 1

2 ∇u p∇uqJ . (2.9)

As the introduced equation does not involve any physical parameters, it is to beidentified as a topological law, linking the configuration variable u to the strain ε.According to the definition introduced in 2.1, the strain is hence a configuration-type variable. With this last equation, the mathematical model of the elastic con-tinuum is completed.

To fully describe the deformation of the continuum, the boundary conditionsmust be specified. On the Dirichlet boundary ΓD, the configuration variable uis set to the predefined displacement u, whereas on the Neumann boundary ΓN ,the stress state σ is specified using a predefined traction vector t. Hence, the fieldequations describing the deformed continuum can be summarized as

∇ σ b 0 in Ω (2.10a)

σ C : ε in Ω (2.10b)

ε 1

2 ∇u p∇uqJ in Ω (2.10c)

u u on ΓD (2.10d)

σ n t on ΓN . (2.10e)

13


In analogy to the temperature field, the task of finding the displacement distribu-tion of an elastic body can be formulated as

“Find the configuration function u pxq, fulfilling thefield equations (2.10a), (2.10b) and (2.10c) x P Ω

and the boundary conditions (2.10d) and (2.10e) x P Γ.”

A visualization of the derived mathematical model in form of a Tonti diagram isshown in figure 2.3.

u

εσ

b u

t

u uσ n t

ε 1

2 ∇u p∇uqJσ C : ε

∇ σ b 0

PDE

in Ω

in Ω

in Ω

on ΓD

on ΓN

Displacement

StrainStress

Body force

Figure 2.3: Tonti diagram of the elastic continuum

14

3Variational single field description

As shown in chapter 2, the mathematical models of the temperature and the dis-placement field can be derived by introducing conservation laws, governing therespective physical phenomena. With additional assumptions on the materialproperties, this approach leads to field descriptions in form of partial differentialequations.

An alternative to this approach can be derived from the idea, that the equilib-rium state of a system is associated with the extremum of a potential expression,which governs the physical phenomena. The task, which will be addressed inthis chapter, is hence to derive the respective potentials of the individual physicalfields and to formulate a way to determine the stationary point of these expres-sions. Since this can be done using the calculus of variations, this approach isdenoted as the variational formulation of the single filed equations. To brieflyoutline this field of mathematics, a short introduction to variational calculus isgiven in section 3.1, following the books of Betten ([3]) and Huebner et al. ([11]).From this, the individual equations of both fields can then be derived in part 3.2.The chapter will conclude with a description of how to impose Dirichlet bound-ary conditions in the variational form in section 3.3.

15

3 Variational single field description

3.1 Introduction to variational calculus

As stated by Huebner et al. , “the calculus of variations is in a sense an exten-sion of [classical] calculus” [11], since the aim of this mathematical field is to finda function for which a functional becomes stationary. This extends the field ofclassical calculus “in the sense” that here the stationary point of a function is de-termined. Closely following the books of Betten ([3]) and Huebner et al. ([11]),this section shall provide a brief introduction into this topic, by deriving the vari-ational formulation of a functional and the condition which a solution functionhas to fulfill to be its extremum. In addition, it shall be shown how this conditioncan be reformulated as an associated partial differential equation.

For this purpose, a generic functional Jψis defined as

Jψ »

Ω

Fx,ψ,∇ψ

dΩ,

withΩ being a physical domain and F being a generic function, depending on theposition vector x, a generic vector function ψ, describing the configuration of thegeneric field, and its spacial derivative∇ψ. To evaluate the above expression, thefunction F must be integrable. Typically, the definition of F involves the productof the argument function ψ with itself and/or its derivative∇ψ. It shall thereforebe assumed for the rest of this work, that F is defined such that it is integrable ifthe argument function ψ and its first partial derivatives ∇ψ are contained in thespace of square-integrable function, denoted by L2 pΩq. This is satisfied if ψ is anelement of the Sobolev space H1 pΩq ([7]).

Assuming that there exists a solution function ψ, for which J becomes ex-tremal, a mathematical description of ψ can be derived using the variation δψ

introduced by Lagrange ([3]). This virtual function measures the difference be-tween the solution function ψ and an arbitrary compare function ψ at the pointx. This must not be mistaken with the differential dψ BψBx dx, which, incontrast, measures the change of the function ψ, when advancing in the directionx. This distinction is displayed in figure 3.1.

Mathematically, the variation is defined as

δψ pxq ψ pxq ψ pxq . (3.1)

In analogy to the solution function, also the compare function ψ and the variationδψ are assumed to be an element of the the Sobolev space H1 pΩq. However,typically the set of possible compare functions is reduced to those having thesame values as the solution function on the domain boundary Γ. This reducedspace is denoted by H1

0. In this case, the variation satisfies the homogeneous

16


δψpxqsolution function ψpxqdψ

dx

ψ

x

homogeneous compare function ψpxqinhomogeneouscompare function

ψpxqFigure 3.1: Difference between variation δψ and differential dψ. ([3])

boundary condition reading

ψ, δψ P H1

0Ñ δψ pxq ψ pxq ψ pxq 0 on Γ. (3.2)

Likewise, the variation of the first derivative is defined as

δ∇ψ

∇ψ ∇ψ, (3.3)

which is commutative, since

∇δψ

(3.1) ∇

ψ pxq ψ pxq ∇ψ ∇ψ

(3.3) δ∇ψ

. (3.4)

Assuming that in the neighbourhood of ψ the variation δψ can be expressedas a single parameter function

δψ pxq ε η pxq , (3.5)

with ε being a scalar parameter and η being an arbitrary function, ψ becomes

ψ pxq ψ pxq ε η pxq . (3.6)

With this, the functional Jψ J

ψ pxq ε η pxq can be interpreted as a

scalar function J pεq, having the property

J pεq ε0

Jψ.

17


Therefore, in a small neighbourhood of ε 0, the function J has the same charac-teristic as the functional J . As ψ is assumed to be the solution function for whichthe functional J reaches its stationary point, this must also be true for J at ε 0.Hence,

dJ

dε

ε0

! 0 (3.7)

must hold as a necessary condition. Assuming that F has continuous first partialderivatives, the above condition yields

dJ

dε »

Ω

BFBψ dψdε BFB ∇ψ d∇ψdε

dΩ,

according to the product rule. Due to definition (3.6) it holds

dψ

dε d

dε

ψ ε η η and

d∇ψ

dε

∇η.

Furthermore, at ε 0 the derivative of F can be rewritten asBFBψ ε0

BFBψ and analogBFB ∇ψε0

BFB∇ψ .With this, equation (3.7) reads

dJ

dε

ε0

»Ω

BFBψ η BFB ∇ψ : ∇η

dΩ

! 0.

To express the derivative in terms of the variation δψ, the above equation is mul-tiplied with ε, yielding

ε dJdε

ε0

»Ω

BFBψ ε η BFB ∇ψ :ε ∇η dΩ

(3.5) »Ω

BFBψ δψ BFB ∇ψ : δ∇ψ

dΩ

(3.4) »Ω

BFBψ δψ BFB ∇ψ : ∇δψ

dΩ

! 0.

By defining the first variation of the functional J as

δJψ ε dJ

dε

ε0

,

18


the necessary condition (3.7) yields

δJψ »

Ω

BFBψ δψ BFB ∇ψ : ∇δψ

dΩ

! 0. (3.8)

Since δψ was said to measure the difference between the solution function andan arbitrary compare function, the above equation has to be fulfilled for any vari-ation δψ contained in H1 pΩq. With this result, the task of finding the stationarypoint of a functional J

ψcan hence be formulated as

“Find the configuration function ψ pxq P H1 pΩq which satisfiesthe equation (3.8) δψ P H1 pΩq.”

If it can be assumed that the function F also has continuous second partialderivatives, a second way of finding the stationary point can be derived by ap-plying Green’s identity to the second part of the integrand ([26]). With this, thegradient operator is shifted form the variation to the derivative of F and an addi-tional integral over the domain boundary Γ emerges. The transformed conditionreads

δJψ »

Ω

BFBψ ∇ BFB ∇ψ δψ dΩ »Γ

δψ BFB ∇ψ n dΓ ! 0, (3.9)

with n being the outward pointing normal vector of the domain boundary Γ.

The fundamental lemma of variational calculus states that the above equationcan only be fulfilled, if the bracket expressions inside the integrals are identicalzero. This can be shown easily by a counter example. Assuming these termswould not vanish, the variation δψ, as it is arbitrary, could always be chosen suchthat it has the same sign as the bracket expressions. Therefore, the integrandswould always be positive and therewith the integral were larger than zero, whichcontradicts equation (3.9). Therefore, the brackets expressions must be identicallyzero.

Applying this idea to the domain integral gives rise to the Euler-Lagrangeequation reading BFBψ ∇ BFB ∇ψ 0 in Ω. (3.10)

On the domain boundary Γ, the same approach can be used to derive the natural

19


boundary conditions of the Euler-Lagrange equation as followsBFB ∇ψ n 0 on Γ. (3.11)

If this behavior is not desired, homogeneous boundary conditions can be appliedinstead by restricting the space, fromwhich δψ can be chosen, toH1

0. With this re-

striction, the boundary integral in equation (3.9) vanishes and hence the conditionis fulfilled as well. Therefore, an alternative of the natural boundary conditions(3.11) are the homogeneous boundary conditions

δψ 0 on Γ. (3.12)

With the derivation of the Euler-Lagrange equation, the task of finding thestationary point of a functional J can be reformulated as

“Find the configuration function ψ pxq, fulfilling thefield equation (3.10) x P Ω and the boundary

condition (3.11) or(3.12) x P Γ.”

As shown, both formulations derived in this section describe the task of find-ing the stationary point of a functional. However, they are not fully equivalent,but only associated, as the equations (3.8) and (3.10) impose different smooth-ness requirements on their solution function. The approach of finding the zerocrossing of the first variation of the functional only requires the integrability ofthe solution function ψ and its first partial derivatives, whereas the divergenceoperator in the Euler-Lagrange equation will lead to terms depending on the ex-istence and continuity of the second partial derivatives of ψ. Due to this highersmoothness conditions, the Euler-Lagrange equation is commonly referred to asthe strong formulation of the task of finding the stationary point of a functionalJ . Likewise, the variational approach using equation (3.8) is denoted as the weakformulation.

3.2 Governing potentials

An alternative to the derivation of the mathematical model using conservationassumptions can be formulated by introducing a set of generalized coordinatesq1, . . . , qn, describing the configuration of a system. These span the so-called con-figuration space, in which every point refers to a specific system state. Parallelto the coordinates, generalized forces are introduced, which, when acting on the

20


system, lead to a change of state. Hence, if the generalized net force acting onthe system is zero, its state does not change and the system is said to be in equi-librium. The essential idea of this approach is, that the generalized forces areassumed to be conservative and can, therefore, be derived form a potential ex-pression by calculating its gradient in the configuration space ([8]). From this itfollows, that the equilibrium configuration of a system corresponds to the sta-tionary point of the governing potential and can hence be found using the rulesof variational calculus, derived in the previous section. This shall be done for thetemperature and elasticity field in the following sections.

3.2.1 Temperauture potential

Using the constitutive and topological equations (2.2) and (2.3), it can be shownthat the equilibrium of a thermal system is associated to the minimum of thepotential reading

Π pT q »Ω

1

2 κ p∇T q2 s T dΩ »

ΓN

q T dΓ, (3.13)

with the domain Ω, the Neumann boundary ΓN , the temperature T , the thermalconductivity κ, the internal heat source s and the heat flux q over the Neumannboundary ΓN ([14]). By grouping the expressions, the functional can be rewrittenas

Π pT q »Ω

F dΩ »ΓN

f dΓ,

with F 1

2 κ p∇T q2 s T

and f q T.According to equation (3.8) derived in section 3.1, the first variation of the poten-tial is computed by

δΠ pT q »Ω

BFBT δT BFB p∇T q ∇pδT q dΩ »Ω

BfBT δT BfB p∇T q ∇pδT q dΩ21


Assuming the existence and continuity of the first partial derivatives of F , theseyield BFBT BBT

1

2 κ p∇T q2 s T s

andBFB∇T BBT

1

2 κ p∇T q2 s T κ ∇T

Likewise, the derivatives of f readBfBT BBT pq T q q

andBfB∇T BB∇T pq T q 0.

Using this result, the necessary extremum condition yields»Ω

κ ∇T ∇pδT q dΩ »Ω

s δT dΩ »ΓN

q δT dΓ. (3.14)

In order to be able to evaluate the derived condition, is has to be assumed that itsindividual terms are square integrable. In particular, the temperature distributionT and its variation δT shall be contained in the Sobolev space H1 pΩq. As thederived condition must hold for any variation contained in this space, the taskof finding the minimum of the thermal potential and therefore a temperaturedistribution for which the thermal system is in equilibrium can be formulated as

“Find the configuration function T pxq P H1 pΩq, fulfilling the equation(3.14) δT P H1 pΩq.”

Assuming that, in addition to the first, also the continuous second partialderivatives of F and f exist, the associated Euler-Lagrange equation can be de-rived. For this purpose, the fist part of the first integral is rewritten in indexnotation, so that with the help of the product rule it follows

κ ∇T ∇pδT q κ BTBxi BδTBxi BBxi κ BTBxi δT BBxi κ BTBxi δT ∇ pκ ∇T δT q ∇ pκ ∇T q δT.22


With this transformation, the extremal condition can be rewritten as

δΠ pT q »Ω

∇ pκ ∇T δT q dΩ »Ω

r∇ pκ ∇T q ss δT dΩ »ΓN

rq δT s dΓ ! 0.

Applying the Gauss divergence theorem, the fist part of the volume integral canbe transformed into an integral over the domain boundary Γ. Hence, it holds

δΠ pT q »Ω

r∇ pκ ∇T q ss δT dΩ »Γ

rκ ∇T ns δT dΓ »ΓN

rq δT s dΓ ! 0.

Noting that the domain boundary Γ is composed of the Neumann and the Dirich-let part, the second integral can be split and rewritten as

δΠ pT q »Ω


rκ ∇T n qs δT dΓ »ΓD

rκ ∇T ns δT dΓ ! 0. (3.15)

According to the fundamental lemma of variational calculus stated in section 3.1,the derived condition can only be fulfilled if all bracket expression are identicallyzero. Hence, the Euler-Lagrange equation, associated to the minimization of thethermal potential, reads

∇ pκ ∇T q s 0 in Ω (3.16a)

κ ∇T n q 0 on ΓN (3.16b)

κ ∇T 0 or δT 0 on ΓD. (3.16c)

Therefore, the task of finding the equilibrium of the thermal potential can be re-formulated in the strong form:

“Find the configuration function T pxq, fulfilling thefield equation (3.16a) x P Ω and the boundary

condition (3.16b) and (3.16c) x P Γ.”

23


Comparing the above condition with equation (2.4), it can be observed, thatthe derived Euler-Lagrange equation is equal to the partial differential equationwhich follows from the law of heat conservation. Hence, the strong formulationof the task of finding the equilibrium of the temperature potential is equivalentto the task of finding a temperature distribution, conserving the heat energy ofthe system. The extremal condition (3.14) is therefore the weak formulation of thefield description (2.4), derived in section 2.2.1. For this reason, it is said to imposethe law of heat conservation in the weak sense. In figure 3.2, this alternative fielddescription, derived via the minimization principle, is visualized, using dottedlines to indicate the weak connections.

T

pq

s T

q

T T

q n q

p ∇T

q κ p∇ q s 0

PDE

in Ω

in Ω

in Ω

on ΓD

on ΓN

Temperature


Heat source

Figure 3.2: Tonti diagram of the thermal model derived by the minimization prin-ciple. The dotted line represent the connections, imposed in the weaksense.

In addition to the field equation, also the Neumann boundary condition is im-posed in the weak sense by the minimization condition. On the Dirichlet bound-ary however, homogeneous or natural conditions must be applied to satisfy equa-tion (3.15). Hence, no Dirichlet conditions in the from of (2.4d) can be imposedby minimizing the potential (3.13). This issue shall be discussed in section 3.3

3.2.2 Elastic potential

The potential energy of a deformed elastic continuum is composed of the strainenergy Πint, describing the internal potential energy, the external work Πext, per-formed by external forces acting on it, and the kinematic energy K. Mathemati-cally, this reads

Π Πint Πext K (3.17)

24


Assuming the body to be in rest, the kinematic energy is identically zero andhence must not be considered during the further derivation. Using the constitu-tive and topological equations (2.7) and (2.9), the individual potentials are givenby

Πint »Ω

1

2 σ : ε dΩ (3.18a)

Πext »Ω

u b dΩ »ΓN

u t dΓ, (3.18b)

with the displacement u, the strain ε, the stress σ, the body force b, the surfacetraction t, the domain Ω and the Neumann boundary ΓN ([26], [14]). To use thesame notation as in section 3.1, the potential is rewritten accordingly

Πint »Ω

Fint dΩ (3.19a)

Πext »Ω

Fext dΩ »ΓN

fext dΓ. (3.19b)

Thereby, the integrands are defined as

Fint pε,σq 1

2 σ : ε

Fext puq b ufext puq t u.

For the external work, these first partial derivatives becomeBFextBu BBu pb uq band

BfextBu BBu pt uq t.According to equation (3.8), the first variation of the external potential then yields

δΠext puq »Ω

b δu dΩ »ΓN

t δu dΓ. (3.20)

25


In the same way, the derivatives of the internal work are expresses asBFintBε BBε 1

2 σ : ε

1

2 σ

andBfintBσ BBσ

1

2 σ : ε

1

2 ε.

With this, the variation of the internal potential yields

δΠint pε,σq »Ω

1

2 pσ : δε ε : δσq dΩ

Inserting the constitutive equation introduced in section 2.2, the variation of σcan be rewritten as

δσ(2.7) δ pC : εq C : δε.

Additionally using the major symmetry of the constitutive tensor, the equality ofboth integrands can be shown:

ε : δσ(2.7) ε : C : δε

(2.8) σ : δε.

With this, the variation of the internal potential yields

δΠint »Ω

1

2 pσ : δε σ : δεq dΩ »

Ω

σ : δε dΩ. (3.21)

As derived in section 3.1, the variation of the governing potential must vanishat its stationary point. Hence, it follows

δΠ δ pΠint Πextq ! 0 Ñ δΠint! δΠext.

Using the derived expressions (3.20) and (3.21), the necessary extremal conditionyields »

Ω

σ : δε dΩ »Ω

b δu dΩ »ΓN

t δu dΓ. (3.22)

Just as for the thermal field, is has to be assumed that the individual terms inthe above expression are square integrable to be able to evaluate the condition.In particular, the variations δu and δε shall be contained in the Sobolev spaceH1 pΩq. As the derived condition must hold for any variation contained in thisspace, the task of finding the stationary point of the elastic potential and thereforea displacement distribution for which the elastic system is in equilibrium can beformulated as

26


“Find the configuration function u pxq P H1 pΩq, fulfilling theequation (3.22) δu, δε P H1 pΩq.”

To derive the Euler-Lagrange equation associated with the elastic potential,the topological equation (2.9) is used to express δε in terms of the configurationfunction u. The transformed internal potential reads

δΠint »Ω

σ : δε dΩ(2.9) »

Ω

1

2 σ : ∇ pδuq σ : p∇ pδuqqJ dΩ.

Assuming existence and continuity of the second partial derivative ofu, the prod-uct rule can be applied, yielding

σ : ∇ pδuq σij BδuiBxj BBxj pσij δuiq BσijBxj δui(2.5) BBxj pσji δuiq δui BσijBxj ∇ pσ δuq δu p∇ σq (3.23a)

and σ : p∇ pδuqqJ σij BδujBxi BBxi pσij δujq BσijBxi δuj(2.5) BBxi pσij δujq δuj BσjiBxi ∇ pσ δuq δu p∇ σq , (3.23b)

and using the symmetry of σ, the equality of both expression can be shown.Therefore, the integrand can be rewritten as

σ : δε ∇ pσ δuq δu p∇ σq .With this, the variation of the internal potential yields

δΠint »Ω

σ : δε dΩ »Ω

∇ pσ δuq dΩ »Ω

δu p∇ σq dΩ.Using the Gauss divergence theorem, the fist integral can be transformed intoa surface integral over the domain boundary Γ, which is composed of the twoparts ΓD and ΓN . Hence, in a second step, the surface integral can be spilt and the

27


variation then reads

δΠint »Ω

δu p∇ σq dΩ »ΓN

δu σ n dΓ »ΓD

δu σ n dΓ. (3.24)

With this transformation, the minimization condition (3.22) yields

δΠ δΠint δΠext »Ω

δu p∇ σq dΩ »ΓN

δu σ n dΓ »ΓD

δu σ n dΓ »Ω

b δu dΩ »ΓN

t δu dΓ »Ω

δu rp∇ σq bs dΩ »ΓN

δu rσ n ts dΓ »ΓD

δu rσ ns dΓ! 0. (3.25)

According to the fundamental lemma of variational calculus, stated in section3.1, the derived condition can only be fulfilled if all bracket expression are iden-tically zero. Hence, using the constitutive equation (2.7) and the topological re-lation (2.9), the Euler-Lagrange equation associated to the elastic potential reads

∇ σ b 0 in Ω (3.26a)

σ n t 0 on ΓN (3.26b)

σ n 0 or δu 0 on ΓD. (3.26c)

Therefore, the task of finding the equilibrium of the elastic potential can be refor-mulated as

“Find the configuration function u pxq, fulfilling the fieldequation (3.26a),(2.7) and (2.9) x P Ω and the

boundary condition (3.26c) and (3.26b) x P Γ.”

Comparing the above condition with equation (2.10), it can be observed, thatthe derived Euler-Lagrange equation is equal to the partial differential equationwhich follows from Newton’s second law of motion. Hence, the strong formu-lation of the task of finding the equilibrium of the elastic potential is equivalentto the task of finding the displacement distribution which conserves the momen-tum of the elastic system. The extremal condition (3.22) is therefore the weak

28

3.3 Constraint potentials

formulation of the field description (2.10), derived in section 2.2.2, and is said toimpose the law of momentum conservation in the weak sense. In figure 3.3, thisalternative field description, derived via the variational principle, is visualized,using dotted lines to indicate the weak connections.

In addition to the field equation, also the Neumann boundary condition is im-posed in the weak sense by the minimization condition. On the Dirichlet bound-ary however, homogeneous or natural conditions must be applied to satisfy equa-tion (3.25). Hence, no Dirichlet conditions in the from of (2.10d) can be imposedby minimizing the potential (3.17). This issue shall be discussed in section 3.3.

u

εσ

b u

t

u uσ n t

ε 1


∇ σ b 0

PDE

in Ω

in Ω

in Ω

on ΓD

on ΓN

Displacement

StrainStress

Body force

Figure 3.3: Tonti diagram of the deformation of a linear elastic continuum, mod-eled via the minimization approach. The dotted lines represent theconnections, imposed in the weak sense.


As shown in the previous section, the minimization of the respective governingpotentials can be regarded as a weak formulation of the conservation laws, de-scribing the physical field. However, Dirichlet boundary conditions in the formof equation (2.4d) and (2.10d) can not be imposed by the minimization of theintroduced potentials. This issue shall be addressed in the following sections.

Constraining the state of the system in a certain subset Ω of the domain Ω

is associated with a change of the equilibrium point in the configuration space.Therefore, to fulfill the constraint in the weak sense, the stationary point of thegoverning potential has to be moved as well, to fit the new point of equilibrium.This is done by augmenting the original functionals, introduced in the previoussection, by a constraint potential.

29


In a generic description, the condition and the constraint potential are formu-lated as

Cψ

! 0 on Ω

ΠC

ψ »

Ω

FCψdΩ,

with the integrand F being a function specifying the way in which the conditionshall be imposed. With this, the augmented potential is written as

Πψ Π

ψ ΠC

ψ,

with Πψbeing the original potential introduced in the previous section. To

determine the configuration of the shifted equilibrium, the stationary point of thenew potential expression must be found using variational calculus. Mathemati-cally, this reads

δΠψ δ

Πψ ΠC

ψ δΠ

ψ δΠC

ψ

! 0. (3.27)

To impose the Dirichlet boundary conditions, the constraint is given as

Cψ ψ ψ ! 0 on ΓD,

with ψ begin the pre-described distribution on the boundary ΓD. With this, theconstraint potential, imposing the Dirichlet boundary conditions in the weaksense, yields

ΠC

ψ »

ΓD

Fψ ψ dΓ.

The task of finding a field distribution for which the system under considerationis in equilibrium and which satisfies the Dirichlet boundary conditions in theweak sense can therefore be formulated as

“Find the configuration function ψ pxq P H1 pΩq, fulfilling the equation (3.27) δψ P H1 pΩq.”Depending on the choice of the integrand function F , the boundary conditionscan be imposed differently. Three possible methods shall be outlined in the fol-lowing sections, following the books of Zienkiewicz et al. ([26],[25]) and the workof Zhuang ([24]). These extended mathematical models of the temperature andelasticity field are visualized as a Tonti diagram in figure 3.4.

30


T

pq

s T

q

T T

q n q

p ∇T

q κ p∇ q s 0

PDE

in Ω

in Ω

in Ω

on ΓD

on ΓN

Temperature


Heat source

(a) Temperature field

u

εσ

b u

t

u uσ n t

ε 1


∇ σ b 0

PDE

in Ω

in Ω

in Ω

on ΓD

on ΓN

Displacement

StrainStress

Body force

(b) Displacement field

Figure 3.4: Tonti diagram of the weakly formulated field descriptions. The dottedlines represent the connections, imposed in the weak sense.

3.3.1 Penalty method

In the penalty approach, the integrand function is defined as

Fψ 1

2 β C2,

with β being a scalar value. The Dirichlet boundary condition

C ψ ψ on ΓD

31


is hence enforced in the weak sense by a constraint potential reading

ΠP

ψ »

ΓD

1

2 β ψ ψ2

dΓ.

Assuming the existence and continuity of the derivative of F , the first variationof the potential is given by

δΠP

ψ »

ΓD

BFBψ δψ dΓ,with the derivative readingBFBψ BBψ

1

2 β ψ ψ2

β ψ ψ .Therefore, the variation yields

δΠP

ψ »

ΓD

β ψ ψ δψ dΓ (3.28)

As outlined in the introduction of this section, the constraint solution is foundby determining the stationary point of the augmented potential. In the case of thepenalty method this reads

Π Π ΠP Ñ stationaryÑ δΠ δ pΠ ΠP q δΠ δΠP! 0 (3.29)

To emphasize the fulfillment of the boundary condition, the constraint potentialcan be made dominant by increasing the size of β. This penalty strategy is givingthe method its name.

Temperature field

In the case of the temperature field, the configuration variable is the temperatureT . Hence, the boundary condition reads

C pT q T T! 0 on ΓD

and the constraint potential yields

ΠP pT q 1

2 β »

ΓD

T T

2

dΓ.

32


Hence, according to equation (3.28), the variation is given by

δΠP pT q β »ΓD

T T

δT dΓ (3.30)

As derived in section 3.2, the necessary extremal condition is given by equation(3.14). Hence, adding the above expression, according to (3.29), the augmentedextremal condition yields»

Ω

κ ∇T ∇pδT q dΩ β »ΓD

T δT dΓ! »

Ω

s δT dΩ »ΓN

q δT dΓ β »ΓD

T δT dΓ (3.31)

With this, the task of finding a temperature distribution for which the thermalsystem is in equilibrium and which satisfies the Dirichlet boundary conditions inthe weak sense can be formulated as

“Find the configuration function T pxq P H1 pΩq, fulfilling the equation(3.31) δT P H1 pΩq.”

Displacement field

According to (2.10d), the Dirichlet boundary condition of the elasticity field isgiven by

C puq u u ! 0 on ΓD.

With this, the constraint potential reads

ΠP puq 1

2 β »

ΓD

pu uq2 dΓand hence, according to (3.28), the variation yields

δΠP puq β »ΓD

pu uq δu dΓ. (3.32)

As the elastic potential, introduced in section 3.2, is composed of the inter-nal and external contribution, the extremal condition of the augmented potentialreads

δΠ δ pΠint Πext ΠP q δΠint δΠext δΠP! 0.

33


Inserting the equations (3.22) and (3.32) into the above expression yields the ex-tended extremal condition»

Ω

σ : δε dΩ β »ΓD

u δu dΓ! »

Ω

b δu dΩ »ΓN

t δu dΓ β »ΓD

u δu dΓ. (3.33)

With this, the task of finding a displacement distribution for which the elasticsystem is in equilibrium and which satisfies the Dirichlet boundary conditions inthe weak sense can be formulated as

“Find the configuration function u pxq, fulfilling theequation (3.33) δu, δε P H1 pΩq.”

3.3.2 Identified Lagrange multiplier method

An alternative to the penalty approach is the use of the Lagrange multipliermethod, in which an additional unknown λ is multiplied with the conditions.The integrand function is therefore defined as

Fψ,λ

λ C ψ.

In analogy to the penalty ansatz, the Dirichlet boundary condition is given by

C ψ ψ on ΓD,

from which the integrand function follows to be

Fψ,λ

λ ψ ψ .With this, the constraint potential of the Lagrange multiplier method yields

ΠL

ψ »

ΓD

λ ψ ψ dΓ.Just as the solution function, the Lagrange multiplier is assumed be an elementof the function space L2 to ensure the integrability of the constraint function.

Assuming the existence and continuity of the partial derivatives of F , the first

34


variation of the constraint potential is given by

δΠL

ψ »

ΓD

δλ BFBλ BFBψ δψ dΓ,

with the derivatives yieldingBFBλ BBλ λ ψ ψ ψ ψBFBψ BBψ λ ψ ψ λ

With this, the first variation of the Lagrange constraint potential follows to be

δΠL »ΓD

δλ ψ ψ dΓ »ΓD

λ δψ dΓ. (3.34)

To ensure the integrability of the derived condition, also the variation of the La-grange multiplier is assumed to be an element of the function space L2.

As for the penalty method, the shifted point of equilibrium is associated to thestationary point of the augmented potential. In case of the Lagrange ansatz, thisreads

Π Π ΠL Ñ stationary (3.35)Ñ δΠ δ pΠ ΠLq δΠ δΠL! 0 (3.36)

As the essential idea of the Lagrange ansatz is the multiplication by an ad-ditional unknown, the derived condition not only depends on the actual fieldconfiguration ψ but also on the additional variable λ. In its numerical implemen-tation, this will lead to a larger system of equations which may be ill conditioned.However, in some physical models, the Lagrange multiplier can be identified asa function of the state variable and hence eliminated as an unknown. In that case,this major disadvantage of the method can be overcome. This will be derived forthe temperature and displacement field in the following part of the work.

Temperature field

As discussed for the penalty method, the Dirichlet boundary condition of thetemperature field is given by

C T T

! 0 on ΓN ,

35


with which the constraint potential of the Lagrange method becomes

ΠL »ΓD

λ T TdΓ. (3.37)

Using equation (3.34), the first variation reads

δΠL »ΓD

δλ T TdΓ »

ΓD

λ δT dΓ. (3.38)

Augmenting the necessary extremal condition (3.14) derived in section 3.2with the above expression according to equation (3.36), the extended extremalcondition yields»

Ω

κ ∇T ∇pδT q dΩ »ΓD

δλ T dΓ »ΓD

λ δT dΓ »Ω

s δT dΩ »ΓN

q δT dΓ »ΓD

δλ T dΓ. (3.39)

As the derived condition must be satisfied for any variation δT and δλ containedin H1 pΩq, the task of finding a temperature distribution for which the thermalsystem is in equilibrium and which satisfies the Dirichlet boundary conditions inthe weak sense can be formulated as

“Find the configuration function T pxq P H1 pΩq and λ pxq P H1 pΩq, fulfillingthe equation (3.39) δT, P H1 pΩq , δλ P L2.”

As indicated in the introduction of this section, the extended extremal condi-tion does depend not only on the state T but also on the additional unknownλ. However, in the case of the temperature problem, the multiplier can be iden-tified as a function of the stated T and hence be eliminated. For this purpose,the extended Euler-Lagrange equation, associated to the augmented condition, isderived by extending (3.15) by the variation of the Lagrange potential, yielding

δΠ pT q »Ω


rκ ∇T n qs δT dΓ »ΓD

δλ T TdΓ »

ΓD

rκ ∇T n λs δT dΓ ! 0.

36


As this condition must hold for all variations δT and δλ, the extended Euler-Lagrange equation yields

∇ pκ ∇T q s 0 in Ω (3.40a)

κ ∇T q 0 on ΓN (3.40b)

T T 0 on ΓD (3.40c)

κ ∇T n λ 0 on ΓD. (3.40d)

The first two equations are equivalent to the Euler-Lagrange equation associatedto the original temperature potential and hence describe the temperature distri-bution in the domain Ω and its Neumann boundary conditions. The additionalequation (3.40c) shows that with the Lagrange multiplier method, the Dirichletboundary conditions in form of (2.4d) can be imposed in the weak sense. Withthe fourth equation, λ can be identified as

λ κ ∇T n (3.41)

and hence eliminated as an unknown.

By inserting this result into the definition of the Lagrange potential (3.37),the constraint potential of the identified Lagrange multiplier method, denotedas Πident

L , can be defined as

ΠidentL »

ΓD

κ ∇T n T TdΓ (3.42)

(3.43)

Applying the rules of variational calculus, the variation of this new expressionyields

δΠidentL »

ΓD

κ δ p∇T q n T TdΓ »

ΓD

κ ∇T n δT dΓ. (3.44)

Augmenting the extremal condition (3.14) of the original potential by the aboveexpression, the necessary extremal condition, fulfilling the Dirichlet boundaryconditions in the weak sense, yields»

Ω

κ ∇T ∇pδT q dΩ »ΓD

κ δ p∇T q n T dΓ »ΓD

κ ∇T n δT dΓ »Ω

s δT dΩ »ΓN

q δT dΓ »ΓD

κ δ p∇T q n T dΓ (3.45)

Therefore, using the identified Lagrange multiplier method, the task of findinga temperature distribution for which the thermal system is in equilibrium and

37


which satisfies the Dirichlet boundary conditions in the weak sense reads

“Find the configuration function T pxq P H1 pΩq, fulfillingthe equation (3.45) δT P H1 pΩq.”

Elastic field

As the Dirichlet boundary condition of the elastic field reads

C pu uq ! 0 on ΓN

the constraint potential yields

ΠL »ΓD

λ pu uq dΓ. (3.46)

According to equation (3.34), the first variation of the Lagrange potential is there-fore given by

δΠL »ΓD

δλ pu uq dΓ »ΓD

λ δu dΓ. (3.47)

Just as for to the temperature field, the extremal condition (3.22) of the originalelastic potential is extended by the above expression, yielding the augmentedextremal condition »

Ω

σ : δε dΩ »ΓD

δλ u dΓ »ΓD

λ δu dΓ »Ω

b δu dΩ »ΓN

t δu dΓ »ΓD

δλ u dΓ. (3.48)

As the derived condition must be satisfied for any variation δu and δλ containedin spacesH1 pΩq and L2, respectively, the task of finding a displacement distribu-tion for which the elastic system is in equilibrium andwhich satisfies the Dirichletboundary conditions in the weak sense can be formulated as

“Find the configuration function u pxq P H1 pΩq and λ pxq P H1 pΩq,fulfilling the equation (3.48) δu, δε P H1 pΩq , δλ P L2.”

As indicated in the introduction of this section, the extended extremal condi-tion does not only depend on the state u but also on the additional unknown λ.However, also in the case of the elastic problem, an identification of the multiplier

38


is possible by deriving the extended Euler-Lagrange equation. For this purpose,equation (3.25) is augmented by the variation of the Lagrange potential (3.47),yielding

δΠ puq »Ω

δu r∇ σ bs dΩ »ΓN

δu rσ n ts dΓ »ΓD


δu rσ n λs dΓ ! 0.

According to the fundamental lemma of variational calculus, the above expres-sion can only be fulfilled if all bracket expression are identically zero. This givesrise to the extended Euler-Lagrange equation reading

∇ σ b 0 in Ω (3.49a)

σ n t 0 on ΓN (3.49b)

u u 0 on ΓD (3.49c)

σ n λ 0 on ΓD. (3.49d)

Just as in the temperature case, the first two equations describe the displacementof an elastic continuum and its Neumann boundary conditions. The additionalthird equation shows that with the Lagrange multiplier method, the Dirichletboundary conditions in form of (2.10d) can be imposed in the weak sense. Withthe fourth equation, λ can be identified as

λ σ n (3.50)

and hence eliminated as an unknown.

By inserting this result into the definition of the Lagrange potential (3.46),the constraint potential of the identified Lagrange multiplier method, denotedas Πident

L , can be defined as

ΠidentL »

ΓD

pσ nq pu uq dΓ (3.51)

(3.52)

Applying the rules of variational calculus, the variation of this new expressionyields

δΠidentL »

ΓD

δ pσ nq pu uq dΓ »ΓD

δu σ n dΓ (3.53)

Augmenting the extremal condition (3.22) of the original potential by the above

39


expression, the necessary extremal condition yields»Ω

σ : δε dΩ »ΓD

δ pσ nq u dΓ »ΓD

pσ nq δu dΓ! »

Ω

b δu dΩ »ΓN

t δu dΓ »ΓD

δ pσ nq u dΓ. (3.54)

Therefore, using the identified Lagrange multiplier method, the task of findinga displacement distribution for which the elastic system is in equilibrium andwhich satisfies the Dirichlet boundary conditions in the weak sense can be for-mulated as

“Find the configuration function u pxq,fulfilling the equation (3.54) δu, δε P H1 pΩq.”

3.3.3 Nitsche method

As a last method to constrain the potential, the superposition of the penalty andLagrange approach shall be discussed. The integrand function of this ansatz isdefined as

Fψ,λ

1

2 β C2 λ C.

Just as for the previously discussed methods, the Dirichlet boundary conditionreads

C ψ ψ on ΓN ,

with which the constraint potential yields

ΠN

ψ »

ΓD

1

2 β ψ ψ2 λ ψ ψ dΓ ΠP Π

identL .

Using the derived expressions for the variation of the penalty and Lagrangepotential, the variation reads

δΠN δΠP δΠidentL »

ΓD

β ψ ψ δψ dΓ »ΓD

δλ ψ ψ dΓ »ΓD

λ δψ dΓ.Like for the penalty and Lagrange method, the shifted point of equilibrium isassociated to the stationary point of augmented potential. Mathematically, this

40


reads

Π Π ΠN Π ΠN ΠidentL Ñ stationary (3.55)

δΠ δ pΠ ΠP q δΠ δΠP δΠidentL

! 0 (3.56)

As shown in section 3.3.2, for some physical problems an identification of theLagrange multiplier is possible. In this case, the above ansatz is referred to as theNitsche method ([24]).

Temperature field

Extending the necessary extremal condition (3.14) of the original thermal poten-tial by the expressions (3.30) and (3.38), according to equation (3.56), the extremalcondition yields »

Ω

κ ∇T ∇pδT q dΩ β »ΓD

T δT dΓ »ΓD


κ ∇T n δT dΓ! »

Ω

s δT dΩ »ΓN

q δT dΓ β »ΓD

T δT dΓ »ΓD

κ δ p∇T q n T dΓ. (3.57)

Therefore, using the Nitsche method, the task of finding a temperature distri-bution for which the thermal system is in equilibrium and which satisfies theDirichlet boundary conditions in the weak sense can be formulated as

“Find the configuration function T pxq P H1 pΩq, fulfilling theequation (3.57) δT P H1 pΩq.”

Displacement field

The extremal condition of the Nitsche method for the elastic problem is derivedby extending the necessary extremal condition of the original elastic potential by

41


the expressions (3.32) and (3.47), according to equation (3.56). This yields»Ω

σ : δε dΩ β »ΓD

u δu dΓ »ΓD


pσ nq δu dΓ! »

Ω

b δu dΩ »ΓN

t δu dΓ β »ΓD

u δu dΓ »ΓD

δ pσ nq u dΓ. (3.58)

Therefore, using the Nitsche method, the task of finding a displacement distribu-tion for which the elastic system is in equilibrium andwhich satisfies the Dirichletboundary conditions in the weak sense can be formulated as

“Find the configuration function u pxq P H1 pΩq, fulfilling theequation (3.58) δu, δε P H1 pΩq.”

42

4Linear thermoelasticity

In the previous two chapters of this work, the mathematical models of the tem-perature and displacement field were derived, without taking into account theirinteraction. In this chapter, these models shall be coupled to describe the phe-nomena of thermoelasticity, using the temperature T and the displacement u asthe configuration variables. The task is therefore, to derive new field equations,which describe the temperature and displacement distribution on a shared do-main Ω, taking into account not only the source terms of the individual fields butalso the influence, resulting from the interaction. Just as for the single field de-scription, this model can be formulated in the strong and weak sense. This shallbe outlined in the sections 4.2 and 4.3, respectively. The chapter starts with abrief survey of multi field interaction and a description of linear thermoelasticityin section 4.1.

4.1 Linear thermoelasticity as a multi field problem

As outlined in the introduction of this work, thermoelasticity describes the inter-action of the temperature in and the deformation of a body. It can therefore beregarded as a multi field problem. This section shall give an brief overview overthis kind of problems and the essential assumptions leading to the linearized for-mulation of thermoelasticity.

43

4 Linear thermoelasticity

4.1.1 Classification of multi field problems

If the configuration of a physical system is governed by the interaction of twoor more physical fields, it is classified as a multi field problem. This group con-tains a wide range of application examples, which makes it hard to find a distinctclassification criterion. One differentiating factor, for example, is the strength ofthe field interaction which divides problems into strongly and weakly coupledsystems. Depending which category applies for a given application, differentsolution approaches may be chosen.

Another classification possibility is the distinction between surface and vol-ume coupled problems. In the first case, the different physical domains Ωi do notoverlap but only interact on a shared domain boundary (figure 4.1a). The govern-ing equations are therefore independently defined on the individual subdomainsand the interaction is induced by coupling the respective boundary conditions.Themost prominent example of this group of problems is the interaction betweenfluids and structures ([22]). Applications with overlapping physical domains, onthe other hand, are classified as volume coupled problems (figure 4.1b). In thiscase, the governing equations which model the different physical phenomena aredefined on a (partially) shared domain. Common examples of this kind are elec-tromechanical applications or the seepage through a porous medium ([22], [26]).Also, the field of thermoelasticity described in this chapter, is part of this group.

Ω1

Ω2

shared boundary

(a) Surface coupled problems

Ω1

Ω2

shared domain

(b) Volume coupled problems

Figure 4.1: Schematic illustration of multi field problem classes

4.1.2 Assumptions on linear thermoelasticity

In their most general formulation, multi field problems are mathematically de-scribed by the equations of the involved fields and their coupling conditions.Typically, all fields involved influence each other, leading to a fully coupled sys-tem of equations in which no sub problem can be solved independently of theother.

44

4.1 Linear thermoelasticity as a multi field problem

In the case of thermoelasticity, the systems interacting are a temperature and adisplacement field. As these distributions are defined on a shared domain, ther-moelasticity belongs to the class of volume coupled problems. Therefore, the fieldequation of the individual systems have to be extended by appropriate couplingconditions to model the interaction. This description of nonlinear thermoelastic-ity can be derived using the first and second law of thermodynamics ([1], [10],[16]). In this fully coupled formulation, the interaction in both directions is takeninto account.

However, if the displacement of the continuum is assumed to be small, thechange of temperature resulting from it can be neglected. With this simplifi-cation, the systems can be decoupled, as the temperature field does not longerdepend on the deformation of the continuum. In this linearized formulation ofthermoelasticity, the temperature distribution inside a body can be computed asa pre-step, using the single field descriptions derived in chapter 2 and 3. The re-sulting temperature can then, in a second step, be applied to the elastic body. Todescribe the resulting deformation, the material under consideration is assumedto expand isotropically when changing its temperature ([14]). Therefore, an in-crease in temperature relative to the reference temperature T0 yields an additionalthermal strain εth. With this second assumption, the coupling condition can beformulated as a constitutive law reading

εth γ pT T0q I, (4.1)

with γ being the thermal expansion coefficient of the material under considera-tion. As εth is connected to the configuration variable T via a constitutive equa-tion, it is to be identified as an additional source-type variable of the thermoelasticsystem. To simplify the further notation, the reference temperature T0 is assumedto be zero. With this, the above expression can be rewritten as

εth γ T I. (4.2)

The overall strain of a deformed continuum is therefore composed of the elasticstrain εel, resulting from the mechanical forces acting on the body, and the ther-mal strain εth. Mathematically, this reads

εtot εel εth, (4.3)

with εtot being the total strain, related to the displacement of the continuum.

In analogy to the strain, also the stress state of a deformed body is composed ofthe elastic stress σel, resulting from the mechanical forces, and the thermal stressσth, induced by the temperature:

σtot σel σth, (4.4)

45


with σtot being the total stress, related to the displacement of the body.

Using the derived coupling condition and the introduced terminology, themathematical model of linear thermoelasticity can be derived in the next sectionsof this chapter.

4.2 Formulation of linear thermoelasticity

As outlined in section 4.1, thermoelasticity can be regarded as a volume coupledmulti field problem, in which the governing equations are defined on a shareddomain. Form this it follows, that the mathematical model of thermoelasticityis to be derived by extending the individual field equations by the respectivecoupling conditions. This shall be outlined in the following section.

Since in the linearized description of thermoelasticity, the change of tempera-ture, caused by the deformation of the continuum, is neglected, the single fielddescription of the temperature, derived in section 2.2, can be reused. However,the model of the elastic field has to be extended to take into account the additionalstrain caused by the change of temperature. For this purpose, the equations de-rived in chapter 2 have to be reformulated in terms of the new strain and stressterminology, introduced in the previous section.

As the change of temperature leads to a deformation of the continuum, it canbe regarded as an additional load. Therefore, the total displacement u of an elas-tic continuum is caused by the sum of the body force b, the traction t and thetemperature load. The strain, related to this total displacement is the total strainεtot and can be expressed in terms of the configuration variable u, by reformulat-ing the topological equation (2.9) as

εtot puq 1

2 ∇u p∇uqJ .

The thermal strain, on the other hand, can be expressed as a function of the otherconfiguration variable T using the coupling condition

εth pT q γ T I,introduced in the previous section. In the case of the elastic strain, such a directconnection to the configuration variables is not possible. By rearranging equation(4.3) as

εel pu, T q εtot puq εth pT q , (4.5)

it can, however, be expressed as the difference between the first two strain types.The elastic strain, therefore, depends on both configuration variables and, hence,

46


represents the link between u and T .

In analogy to the strain, also the stress description has to take into accountthe temperature. According to the Tonti classification scheme, the stress and thestrain are connected via a constitutive equation, modelling the material behav-ior. In section 2.2, the material was assumed to be of linear elastic type, so thatthe elastic stress is proportional to the elastic strain. With this new terminology,equation (2.7) reads

σel C : εel.

Using equation (4.5), the elastic stress can be expressed in terms of the total andthermal strains

σel C :εtot εth C : εtot C : εth.

Comparing the above expression with equation (4.4), the total and thermal stresscan be identified as

σtot puq C : εtot puq (4.6a)

σth pT q C : εth pT q (4.6b)

and, therefore, expressed as a function of the configuration variables. In addition,the symmetry of both stress types can be observed.

In section 2.2, the balance equation, governing the elastic field, was derivedusing Newton’s second law of motion, stating that the total momentum of thesystem must be conserved. Assuming the body to be in rest, it followed that theforces acting on the body must be in balance with the internal stresses. As thesemechanical forces cause stresses of elastic type, the balance equation has to berewritten as

∇ σel b 0.

Using equation (4.4), the above expression can be formulated in terms of the totaland thermal stress, which, according to (4.6), directly depend on the configura-tion variables u and T , respectively. This transformation yields

∇ σtot puq σth pT q b 0.

Like the body forces, also the surface tractions cause an elastic stress. Therefore,also the Neumann boundary condition must be reformulated as

t σel n σtot n σth n.As described in section 4.1.2, in the linearized formulation of thermoelasticity, thetemperature distribution does not depend on the deformation of the body and,hence, can be computed in a pre-step. Therefore, when calculating the displace-ment, the temperature is given. With this, the divergence of the thermal stress

47


can be regarded as a known thermal force, yielding

∇ σtot b bth 0

with bth ∇ σth.

In the same way, the thermal stress on the Neumann boundary can be regardedas an additional traction force, yielding

t σtot n tthwith tth σth n.

With these results, the governing equations describing a thermoelastic contin-uum can be summarized as

∇ q s 0 inΩ (4.7a)

q κ p inΩ (4.7b)

p ∇T inΩ (4.7c)

T T on ΓD (4.7d)

q n q on ΓN (4.7e)

∇ σtot b bth 0 in Ω (4.7f)

σtot C : εtot in Ω (4.7g)

εtot 1

2 ∇u p∇uqJ in Ω (4.7h)

bth ∇ σth in Ω (4.7i)

σth C : εth in Ω (4.7j)

εth γ T I in Ω (4.7k)

u u on ΓD (4.7l)

σ n t tth on ΓN (4.7m)

tth σth n on ΓN . (4.7n)

Using these field equations, the task of finding a temperature and a displacementdistribution, conserving the heat energy and the momentum of a thermoelasticsystem, respectively, can be formulated as

48


“Find the configuration function T pxq, fulfilling the field equations (4.7a) to(4.7c) x P Ω and the boundary conditions (4.7d) and (4.7e) x P Γ,

and the configuration function u pxq, fulfilling the field equations (4.7f) to(4.7k) x P Ω and the boundary conditions (4.7l) to (4.7n) x P Γ.”

In figure 4.2, the derived model is visualized in form of a Tonti diagram.

T

pq

s T

q

T T

q n q

p ∇T

q κ p∇ q s 0

PDE

in Ω

in Ω

in Ω

on ΓD

on ΓN

Temperature


Heat source

u

εel

εtotεth

σel

b u

t

u uσel n t ε 1

2 ∇u p∇uqJσel C : εel

∇ σel b 0

PDE

εel εtot εthεth α T I

in Ω

in Ω

in Ω

on ΓD

on ΓN

in Ω

in Ω

Displacement

Elastic strain

Elastic stress

Body force

Total strainThermal strain

Figure 4.2: Tonti diagram of linear thermoelasticity

49


4.3 Variational formulation

In analogy to the single field description, also the mathematical model of ther-moelasticity can be derived in the variational formulation. For this purpose, thegoverning potentials, introduced in section 3.2, have to be extended to model thefield interaction. This will be outlined in part 4.3.1. In addition, also the con-straint potentials have to be modified to take into account the influence of thetemperature. This will be discussed in the second part of this section.

4.3.1 Governing potential of linear thermoelasticity

As described in section 4.1.2, the temperature change caused by the deformationof the continuum is neglected in the linearized description of thermoelasticity.Hence, the thermal potential introduced in 3.2.1 can be reused without any adop-tions. The elastic potential, however, has to be extended to model the thermalinfluence. This modification shall be derived in this part of the work.

In section 3.2.2, the potential energy of an elastic body, assumed to be at rest,was said to be composed of the internal and external energy. As the externalenergy describes the work done by the mechanical forces acting on the system,it is not influenced by the coupling to the temperature field and, hence, doesnot change. The internal energy however, describes the strain energy, caused bythe elastic deformation of the continuum. Using the terminology introduced insection 4.1.2, it must, therefore, be reformulated as

Πint »Ω

1

2 σel

: εel dΩ. (4.8)

Inserting equation (4.3), the above expression can be reformulated in terms of thetotal and thermal strains and stress. Additionally using the constitutive equation(4.6) and the major symmetry of the constitutive tensor, like in the derivation ofequation (3.21), this yields

Πint »Ω

1

2 σel

: εel dΩ »Ω

1

2 σtot σth

:εtot εth dΩ »

Ω

1

2 σtot

: εtot dΩ »Ω

1

2 σtot

: εth dΩ »Ω

1

2 σth

: εtot dΩ »Ω

1

2 σth

: εth dΩ

(2.8)(4.6)

»Ω

1

2 σtot

: εtot dΩ »Ω

σth: εtot dΩ »

Ω

1

2 σth

: εth dΩ.

50


Since the temperature distribution can be computed in a pre-step and hence is nounknown, no variation of the above expression with respect to T , but only withrespect to the deformation dependent terms must be computed. As only the firsttwo integrals depend on u, the variation of the internal potential becomes

δΠint »Ω

σtot: δεtot dΩ »

Ω

σth: δεtot dΩ.

Just as for the single field description, the point of equilibrium is associated tothe configuration, for which the variation of the internal and external potentialare in balance. Using the expression (3.20), the necessary extremal condition ofthe linearized thermoelastic continuum yields

δΠint! δΠext»

Ω


Ω

b δu dΩ »ΓN

t δu dΓ »Ω

σth: δεtot dΩ. (4.9)

To ensure the integrability of the derived condition, the individual terms are as-sumed to be square integrable. In particular, the variations δεtot and δu shall beelements of the Sobolev space H1 pΩq. As the derived condition must be satisfiedfor any variation contained in this space, the task of finding the extremum of thethermoelastic potential and therefore a displacement distribution, for which thethermoelastic system is in equilibrium with respect to a pre computed tempera-ture distribution, can be formulated as

“Find the configuration function u pxq P H1 pΩq, fulfillingthe equation (4.9) δu, δε P H1 pΩq.”

The reader might be interested to note, that, since the thermal stress is known,the third integral on the right hand side of equation (4.9), modelling the influenceof the temperature, can be regarded as an thermal body load. This correlates tothe description in the strong form, derived in the previous section. However, inthe weak formulation, no additional traction forces are present.

To show, that the above expression fulfills the partial differential equation, de-rived in 4.2 in the weak sense, the associated Euler-Lagrange equation has to bederived. For this purpose, the existence and continuity of the partial derivativesof the total and thermal stresses must be assumed. Using equation (3.24), the

51


variation of the internal potential can then be reformulated as

δΠint(3.24) »

Ω

δu ∇ σtotdΩ »

ΓN

δu σtot n dΓ »ΓD

δu σtot n dΓ »Ω

δu ∇ σthdΩ »

ΓN

δu σth n dΓ »ΓD

δu σth n dΓ.With this transformation, the condition of equilibrium yields

δΠ δΠint δΠext »Ω

δu ∇ σtotdΩ »

ΓN



δu ∇ σthdΩ »

ΓN


δu σth n dΓ »Ω

b δu dΩ »ΓN

t δu dΓ »Ω

δu ∇ σtot ∇ σth b dΩ »ΓN

δu σtot n σth n t dΓ »ΓD

δu σtot n σth n dΓ! 0. (4.10)

According to the fundamental lemma of variational calculus, introduced in sec-tion 3.1, the above condition can only be fulfilled if the bracket expression areidentically zero. Applying homogeneous boundary conditions, it hence follows

∇ σtot ∇ σth b 0 in Ω (4.11a)

σtot n σth n t 0 on ΓN (4.11b)

δu 0 on ΓD. (4.11c)

Therefore, the task of finding the displacement distribution, for which the ther-moelastic system is in equilibrium with respect to a pre computed temperaturedistribution, can be formulated as

“Find the configuration function u pxq, fulfilling thefield equation (4.11a) x P Ω and the boundary

conditions (4.11b) and (4.11c) x P Γ.”

Comparing the above expressions to (4.7) shows that the extremal conditionfulfills the partial differential equations, derived in the previous section, in theweak sense.

52


4.3.2 Constraint potential of linear thermoelasticity

As outlined in section 3.3, imposing Dirichlet boundary conditions on a singlefield system in the variational formulation requires the uses of an additional con-straint potential. The same idea can be appliedwhen describing the interaction oftemperature and deformation. Since in the linearized formulation of thermoelas-ticity, the change of temperature cause by the deformation is neglected, the con-straint potentials of the temperature field, derived in section 3.3, do not change.The same is true for the penalty and the Lagrange potential, constraining thedisplacement field. However, when identifying the Lagrange multiplier, the in-fluence of the temperature has to be taken into account. This shall be outlined inthe following section.

Using the results derived in section 3.3.2, the variation of the Lagrange poten-tial, imposing the Dirichlet boundaries on the elastic continuum, can be givenas

δΠL puq (3.47) »ΓD


λ δu dΓ with λ, δλ P L2.

In analogy to the single field description, also the equilibrium of the thermoelas-tic system, fulfilling the Dirichlet boundary conditions, is described by the ex-tremum of the augmented potential:

Π Πint Πext ΠL Ñ stationaryÑ δΠ δΠint Πext ΠL

δΠint δΠext δΠL! 0.

Inserting the result derived in the previous section yields»Ω


ΓD

δλ u dΓ »ΓD

λ δu dΓ »Ω

b δu dΩ »ΓN

t δu dΓ »ΓD

δλ u dΓ »Ω

σth: δεtot dΩ. (4.12)

As the derived condition must be satisfied for any variation δu, δεtot and δλ con-tained in H1 pΩq, the task of finding the displacement distribution, for which thethermoelastic system is in equilibrium with respect to a pre computed tempera-ture distribution and which fulfills the Dirichlet boundary conditions in the weaksense, can be formulated as

“Find the configuration function u pxq P H1 pΩq, fulfilling theequation (4.12) δu, δε P H1 pΩq , δλ P L2.”

As outlined in section 3.3.2, the Lagrange multiplier can be identified by deriv-

53


ing the Euler-Lagrange equation, associated to the above expression. Assumingthe existence and continuity of the total and thermal stresses, equation (4.10) canbe reused and augmented by the variation of the Lagrange potential reading

δΠ puq »Ω

δu ∇ σtot ∇ σth b dΩ »ΓN

δu σtot n σth n t δu dΓ »ΓD


δu σtot n σth n λ dΓ ! 0.

By applying the fundamental lemma of variational calculus, the Euler-Lagrangeequation can be derived as

∇ σtot ∇ σtot b 0 in Ω (4.13a)

σtot n σth n t 0 on ΓN (4.13b)

u u 0 on ΓD (4.13c)

σtot n σth n λ 0 on ΓD. (4.13d)

Like in the single field description, the first three equations prove, that findingthe equilibrium point of the thermoelastic system, constrained by the Lagrangeansatz, is associated to solve the partial differential equations, derived in section4.2, in the weak sense. The fourth equation, however, shows that the Lagrangemultiplier has to be identified as

λ σth n σtot n. (4.14)

It is worth to point out, that, according to equation (4.4), the difference betweenthe total and thermal stress can be interpreted as the elastic stress. Hence, theidentification can be rewritten as

λ σtot σth

n (4.4) σel n,which correlates to the identification (3.50), derived in section 3.3.2, when onlythe elastic stress is considered. However, as only the total and thermal stressescan be directly expressed in terms of the configuration variables T and u, theidentification (4.14) will be used in the further derivation.

Using the expression (4.14) for λ, the Lagrange potential and its first variation

54


can be reformulated as

ΠidentL »

ΓD

σth n σtot n pu uq dΓ (4.15a)

δΠidentL »

ΓD

δσtot n pu uq dΓ »

ΓD


δu σtot n dΓ.(4.15b)

With this, the weak formulation of the thermoelastic problem yields»Ω


ΓD

u δσtot n dΓ »ΓD


b δu dΩ »ΓN

t δu dΓ »Ω

σth: δεtot dΩ »

ΓD


δu σth n dΓ. (4.16)

Therefore, the task of finding the a displacement distribution, for which the ther-moelastic system is in equilibrium with respect to a pre computed temperaturedistribution and which fulfills the Dirichlet boundary conditions in the weaksense, can be formulated as

“Find the configuration function u pxq P H1 pΩq, fulfillingthe equation (4.16) δu, δε P H1 pΩq.”

Remark:

Assuming the existence and continuity of first derivatives of σth, the derived con-ditions can be reformulated in terms of the thermal forces, derived in section 4.2.As this gives additional insight into the respective correlations, this will be out-lined in the last part of this section.

As the domain boundary Γ is composed of the Neumann and Dirichlet bound-ary, the last integral of equation (4.16) can be rewritten as»

ΓD

δu σth n dΓ »Γ

δu σth n dΓ »ΓN

δu σth n dΓAssuming the existence and continuity of the partial derivatives of the thermal

55


stress, the first integral on the right hand side can be rewritten as»Γ

δu σth n dΓ »Ω

∇ σth δu dΩ,using the divergence theorem. Applying the product rule in analogy to (3.23a),the integrand can be reformulated s

∇ σth δu (3.23a) σth: ∇ pδuq δu ∇ σth

.

As the first term on the right hand side involves the double contraction of tworank-2-tensors, it can be identified as a scalar s, defined as

s σth: ∇ pδuq . (4.17)

Since a scalar value is equal to its transposed, the expression can be transformedas

s sJ σth

: ∇ pδuqJ p∇ pδuqqJ :σth

J σth

J: p∇ pδuqqJ σth

: p∇ pδuqqJ ,with the last step holding due to the symmetry of the stress tensor. In addition,the scalar value can be rewritten as

s 1

2 ps sq 1

2 s sJ .

Inserting the above results yields

s 1

2 σth

: ∇ pδuq σth: p∇ pδuqqJ σth

:

1

2 ∇δu p∇ pδuqqJ σth

: δεtot.

With this transformation, the surface integral can be rewritten as»Γ

δu σth n dΓ »Ω


Ω

δu ∇ σthdΩ.

Inserting the above result in expression (4.17), the integral over the Dirichletboundary becomes»

ΓD

δu σth n dΓ »Ω


Ω

δu ∇ σthdΩ »

ΓN

δu σth n dΓ56


With this reformulation, the equilibrium condition yields»Ω


ΓD



b δu dΩ »ΓN

t δu dΓ

»Ω


ΓD

u δσtot n dΓ

»Ω


Ω

δu ∇ σthdΩ »

ΓN

δu σth n dΓand the domain integrals of the thermal stress cancel. By rearranging the integral,the condition can be reformulated as»

Ω


ΓD



b∇ σth

loooooomoooooonb bth

δu dΩ »ΓN

t σth nloooooomoooooon

t tth

δu dΓ »ΓD

u δσtot n dΓ (4.18)

and the thermal forces introduced in section 4.2 can be identified. Furthermore,comparing the above condition with equation (3.54) shows that the two expres-sion only differ by these pseudo forces. This reveals that the constrained weakfield equation of linear thermoelasticity can be derived in two ways.

The first possibility, outlined in the first part of this section, is to define theconstrained thermoelastic potential as

Π Πint Πext ΠidentL ,

with the individual terms being given by

Πint(4.8) »

Ω

1

2 σel

: εel dΩ,

Πext(3.19b) »

Ω

Fext dΩ »ΓN

fext dΓ

and ΠidentL

(4.15a) »ΓD

σth n σtot n pu uq dΓ.

As derived, the necessary extremal conditions of this potential and hence theconstrained weak field equation of linear thermoelasticity is given by equation(4.16).

An alternative derivation of the constrained weak field equation is possible by

57


defining the constrained thermoelastic potential as

Π Πint Πext ΠidentL ,

with the individual terms being given by

Πint(3.18a) »

Ω

1

2 σtot

: εtot dΩ,

Πext(4.18) »

Ω

b bth δu dΩ »

ΓN

t tth δu dΓ

and ΠidentL

(3.51) »ΓD

σtot pu uq dΓ.

Applying the rules of variational calculus to this potential yields the alternativeweak field equation of linear thermoelasticity (4.18). As shown in this remark,both descriptions can be transformed into one another. Therefore, the two weakfield equations are equivalent. However, as in the second formulation, the ther-mal body load bth is defined as the divergence of thermal stress, its numericalrepresentation might be discontinuous. As this can cause problems when numer-ically approximating the state of the thermoelastic system, the original formula-tion (4.16) will be used.

58

5Numerical Methods

In the first part of this work, the governing equations of the thermal, the elasticand the thermoelastic systems were derived. However, for non-trivial problemsetups, a solution function, satisfying the individual conditions, can not be foundeasily. For this reason, different numerical methods were developed to approxi-mate the analytical solution. One of the most prominent approaches is the FiniteElement Method (FEM), whose conceptual idea shall be outlined in section 5.1.In the second part of this chapter, the Finite Cell Method (FCM), which extendsthe idea of the FEM, will be discussed for the single field problems. In section5.3, these descriptions shall be combined to derive a numerical model of the ther-moelastic problem.

5.1 The Finite Element Method (FEM)

As outlined in the introduction to this chapter, the Finite Element Method is a nu-merical approach to approximate the analytical solution satisfying the respectivefield equations. Unlike other numerical methods, it is based on the weak formu-lation of the individual governing conditions. The fundamental concept of thisapproach shall be outlined in the following section.

In chapter 3, the weak formulation was derived by determining the stationarypoint of the governing potential. For a generic field ψ, the necessary extremal

59

5 Numerical Methods

condition reads »Ω

BFBψ δψ dΩ ! »Ω

BfBψ δψ dΩ. (5.1)

The task of finding the weak solution of a physical problem can, thus, be formu-lated as

“Find the solution function ψ pxq P H1 pΩq whichsatisfies condition (5.1) δψ P H1 pΩq.”

The essential idea of the FEM is to relax this formulation by restricting the func-tion space, in which the variations are contained, form the infinite Sobolev spaceH1 pΩq to a finite subspace V . With this, the validity of a solution candidate canbe tested with a finite number of variations δψ. These functions shall thereforebe denoted as test functions. Likewise, also the approximation to the solutionfunction shall be contained in the restricted space V . The task of finding the ap-proximation to the analytical solution function can, therefore, be formulated as

“Find the solution function ψ pxq P V whichsatisfies the necessary extremal condition δψ P V .”

The second idea of the FEM, giving the method its name, is to divide the do-main into a finite number of elements. With the decomposition, the domain in-tegral of the weak field equation can be decomposed into integrals over the indi-vidual element domains. Mathematically, this reads»

Ω

F dΩ i

»Ωi

F dΩ

,

with Ωi being the domain of the element i.

The third idea of the FEM is to express the numerical solution function and thetest function as a linear combination of ansatz functions Ni, spanning the spaceV . The two functions can therefore be rewritten as

ψ pxq i

Ni pxq ψi

δψ pxq

i

Ni pxq δψi

,

with ψi and δψi being the respective coefficients of the individual functions. Us-ing this idea, the task of finding the numerical approximation to the analyticalsolution can be reformulated as

60


“Find the coefficients ψi such that thenecessary extremal condition is satisfied δψi.”

For implementation reasons, the ansatz functions Ni are grouped in a matrixN . With this notation, the solution and test functions can be rewritten as

ψ pxq N pxq ψ and δψ pxq N pxq δψ, (5.2)

with ψ and δψ being the respective coefficient vectors. As derived in chapter3 and 4, the weak field equations typically involve differential operators, whichshall be denoted by L . As the solution and test functions are represented as alinear combination of the ansatz functions Ni, their spacial derivatives are com-puted by applying L on Ni. If the ansatz functions are grouped in the matrixN , also the differential operator has to be reformulated as a matrix L. This hencereads

ψ pxq i

LNi pxq ψi

LNloomoonB

ψ B ψ, (5.3)

with the matrix B being defined as the matrix product of L and N . Using thismatrix notation, the discretized weak field equation can be rewritten as a systemof linear equations

K ψ f , (5.4)

with the stiffness matrixK and the load vector f . The task of finding the numer-ical approximation to the analytical solution can hence be reformulated as

“Find the solution vector ψ satisfying the system of linear equations (5.4).”

Since the validity of a solution candidate is only enforced for a finite num-ber of test functions, the numerical result will differ from the analytical solution.This error in typically measured in the energy norm which compares the internalenergy of both solutions. Mathematically, it is defined as

eE d |Πexint Πnum

int |Πex

int

100 r%s , (5.5)

with Πexint and Πnum

int being the internal energies of the analytical and numerical so-lution, respectively ([6]). It can be shown that the solution obtained by applyingthe FEM, is the best approximation of the analytical solution with respect to theenergy norm which can be found in the function space V ([12]) . Due to this “bestapproximation property” of the FEM, the error of the numerical result can becontrolled by changing the characteristic of V . Since the ansatz functions Ni are

61

5 Numerical Methods

assumed to span this space, one possibility to increase the quality of the approx-imation is to enlarge this basis by using more of the same functions to representthe solution. As these are typically defined locally on each element, this ansatzcorrelates to a refinement of the finite element mesh and is therefore denoted ash-refinement. Alternatively, the space V can be modified by changing the typeof ansatz functions used. Since these are typically polynomials, the quality of thenumerical approximation can be controlled by increasing the order p ofNi, whichis denoted as p-refinement.

5.1.1 FEM for the temperature problem

To derive the finite element approximation of the temperature problem, the weakformulation of the heat equation will be used. As derived in section 3.2.1, it reads

δΠ(3.14) »

Ω

κ ∇T ∇pδT q dΩ »Ω

s δT dΩ »ΓN

q δT dΓ ! 0

By restricting the function space and using the matrix notation, introduced inthe previous section, the first term can be discretized as follows»

Ω

κ ∇T ∇pδT q dΩ (5.2)Ñ »Ω

LN δT

JκLN T

dΩ

(5.3) »Ω

δTJBJ κB T dΩ δT

J »Ω

BJ κB dΩlooooooomooooooonKT

T .

As shown in the last step, the coefficient vectors can be taken out of the integralsince they do not depend on the position x and the remaining expression can beidentified as the stiffness matrixKT of the temperature problem.

In the same way, the second and third term can be discretized, yielding»Ω

s δT dΩ »ΓN

q δT dΓ(5.2)Ñ »

Ω

N δT

Js dΩ »

ΓN

N δT

Jq dΓδT J »

Ω

NJ s dΩ »ΓN

NJq dΓlooooooooooooooooomooooooooooooooooonfT

.

Following the same line of arguments as in the first case, the coefficient vectorcan be taken out of the integral and the remaining expression can be identified as

62


the load vector fT .

With these transformations, the discretized weak field equation reads

δTJ KT T fT ! 0.

As the equation has to be fulfilled for any variation vector, the bracket expres-sion must be identically zero, which gives rise to the system of linear equationsreading

KT T fT . (5.6)

Hence, task of finding the numerical approximation to the analytical temperaturedistribution can be formulated as

“Find the solution vector T satisfying the system of linear equations (5.6).”

As derived in section 3.2.1, the weak field equation does not impose the Dirich-let boundary conditions. This would require the use of the additional constraintpotentials, derived in section 3.3. However, as the finite element mesh follows theshape of the physical domain Ω, the temperature values on the Dirichlet bound-ary directly correspond to the respective entries in the solution vector. Therefore,the Dirichlet boundary condition can alternatively be imposed by modifying theassociated rows of the stiffness matrixKT . Since, with this ansatz, the tempera-ture distribution on the Dirichlet boundary can be represented pointwise exactly,this approach is preferred. Mathematically, this corresponds to a restriction ofthe test function space, to those functions having homogeneous boundary condi-tions.

5.1.2 FEM for the elastic problem

In analogy to the temperature problem, also the weak formulation of the elasticityequations will be used to derive the finite element approximation of the elasticproblem. As derived in section 3.2.2, it reads

δΠint! δΠext

(3.22)Ñ »Ω

σ : δε dΩ »Ω

b δu dΩ »ΓN

t δu dΓ.To discretize this equation, the individual tensor expressions have to be refor-

mulated in matrix notation first. This shall be demonstrated in 2D. The tensors ε

63

5 Numerical Methods

and σ are transformed into vectors reading

εÑ ε ε11ε222ε12

and σ Ñ σ σ11σ22σ12

. (5.7)

In the same way, the material tensorC is transformed into a matrixC, so that theconstitutive equation can be reformulated as

σ C : ε C : pLuq Ñ σ C ε C LN u CB u. (5.8)

Using these results, the individual terms can be discretized as follows»Ω

σ : δε dΩ(5.7)Ñ(5.8)

»Ω

pLN δuqJ C pLN uq dΩ(5.3) »

Ω

δuJBJCB u dΩ δuJ »Ω

BJCB dΩlooooooomooooooonKu

u

and »Ω

b δu dΩ »ΓN

t δu dΓ (5.2)Ñ »Ω

pN δuqJ b dΩ »ΓN

pN δuqJ t dΓ δuJ »Ω

NJ b dΩ »ΓN

NJ t dΓloooooooooooooooooomoooooooooooooooooonfu

.

As before, the coefficient vectors do not depend on the position x and can there-fore be taken out of the integral. The remaining expression yield the stiffnessmatrixKu and the load vector fu of the elastic problem.

With these transformations, the discretized weak field equation reads

δuJ rKu u fus ! 0.

As the equation has to be fulfilled for any variation vector, the bracket expres-sion must be identically zero, which gives rise to the system of linear equationsreading

Ku u fu. (5.9)

Hence, the task of finding the numerical approximation to the analytical displace-ment distribution can be formulated as

64

5.2 The Finite Cell Method (FCM)

“Find the solution vector u satisfying the system of linear equations (5.9).”

In analogy to the temperature case, the Dirichlet boundary conditions are im-posed by modifying the respective rows of the stiffness matrix.


In the previous section, the Finite Element Method was introduced as an numer-ical approach to approximate the analytical solution of a physical problem. Al-though, over the past decades, finite elements became one of the most dominantnumerical methods on this field, they suffer the drawback that the mesh gener-ation for complicated domains is very time consuming. To overcome this issue,the Finite Cell Method (FCM) was developed as an extension of the FEM ([17]).This approach shall be outlined in the following sections, closely following thework of Parvizian et al. ([17]).

5.2.1 Concept of the Finite Cell Method

The essential idea of the FCM is to embed the physical domain Ω in an extendeddomain Ωe, which has a shape that can be meshed easily. On this extended do-main, a new field problem ψ is defined which shall satisfy an extended weakcondition reading »

Ωe

α BFBψ δψ dΩ ! »Ωe

α BfBψ δψ dΩ, (5.10)

with α being an additional scalar field defined as

α pxq 1.0 x P Ω

0.0 ¤ α pxq ¤ 1.0 x P ΩezΩ.When α is chosen to be zero outside of the physical domain, the weak conditionscan be rewritten as»Ω

1 BFBψ δψ dΩ

»ΩezΩ 0 BFBψ δψ dΩ ! »

Ω

1 BfBψ δψ dΩ

»ΩezΩ 0 BfBψ δψ dΩ,

yielding the weak field equation (5.1) of the original problem. Inside the physicaldomain Ω, the solution function, fulfilling the original field equation, will there-fore have the same values as the solution function of the extended field equation

65

5 Numerical Methods

(5.10). As the values outside the physical domain are of no interest, the originalproblem can be reformulated as

“Find the solution function ψ pxq P H1 pΩeq whichsatisfies condition (5.10) δψ P H1 pΩeq for α pxq 0 x P ΩezΩ.”

To solve this modified problem numerically, the ideas of the FEM are reused.However, instead of splitting the physical domain into elements, the extendeddomain is meshed. Due to the simpler geometry, this can be done easily. Toavoid confusions in the terminology, these segments shall be denoted as cells,giving the method its name. Using the same discretization techniques as for theFEM will again lead to a system of linear equations in the form of

K ψ f . (5.11)

The task of finding the numerical approximation to the analytical solution canhence be reformulated as

“Find the solution vector ψ satisfying the system of linear equations (5.11).”

Just as in the previous case, the weak field equation does not impose Dirichletboundary conditions. When using the Finite Element Method, this is no problemsince the values on the boundary directly correspond to the respective entries inthe solution vector and can hence be constrained easily by a modification of thestiffness matrix. In the case of the FCM, however, this is not possible since thecells discretize the extended domain without considering the shape of the phys-ical continuum. Therefore, the Dirichlet boundary is not resolved by the meshand, hence, there are no entries in the solution vector which directly correspondto the values on the boundary. To overcome this issue, Parvizian et al. suggestto induce the boundary conditions by a “stiff strip” of material, connecting theboundaries of the physical and the extended domains ([17]). The examples givenin their work show that this approach works well for homogeneous boundaryconditions, however, can cause mild singularities for unfavorable mesh configu-rations. An alternative approach to resolve this problem, which shall be followedin this work, is to extend the weak field equation by the constraint potentials,which impose the conditions in the weak sense. In the upcoming sections, thisidea of the weakly constrained Finite Cell Method shall be discussed for the tem-perature and the displacement field.

66


5.2.2 FCM for the temperature problem

To approximate the temperature distributionwith the help of the Finite CellMethod,the governing potential of the thermal problem has to be reformulated on the ex-tended domain as follows

Πe pT q »

Ωe

α 12 κ p∇T q2 s T dΩ »

ΓN

q T dΓ.Applying the rules of variational calculus, the extendedweak field equation yields

δΠe »Ωe

α κ ∇T ∇pδT q dΩ »Ωe

α s δT dΩ »ΓN

q δT dΓ ! 0.

Applying the same discretization techniques as in the case of the FEM to the in-dividual terms gives rise to the stiffness matrix and the load vector, reading

KT »Ωe

αBJ κB dΩ and fT »Ωe

αNJ s dΩ »ΓN

NJq dΓ.Following the same argumentation as before, the, weak field equation, discretizedfollowing the Finite Cell Method, yields

KT T fT . (5.12)

Unfortunately, the stiffness matrix may become ill-conditioned if α is chosen tobe zero outside of the physical domain, in which case, the equation system cannot be solved numerically. For this reason, α must be chosen large enough toavoid numerical problems but sufficiently small to prevent the fictitious domainto become dominant. Hence, the task of finding the numerical approximation tothe analytical temperature distribution can be formulated as

“Find the solution vector T satisfying the system oflinear equations (5.12) for α pxq Ñ 0 x P ΩezΩ.”

As outlined in the previous section, the Dirichlet boundary conditions can notbe imposed by a modification of the stiffness matrix. Instead, the constraint po-tential, derived in section 3.3, have to be used to impose the conditions in theweak sense. Following the Nitsche approach, introduced in section 3.3.3, the con-strained weak field equation yields

δΠe δΠidentL δΠP

! 0,

67

5 Numerical Methods

with ΠP and ΠidentL being the constraint potentials of the penalty and Lagrange

multiplier method, respectively. In analogy to the above derivation, these expres-sions have to be discretized.

In section 3.3.2, the constraint potential of the identified Lagrange multipliermethod, imposing the Dirichlet boundary conditions of the temperature distri-bution in the weak sense, and its variation were derived as

ΠidentL

(3.42) »ΓD

κ ∇T n T TdΓ

δΠidentL

(3.44) »ΓD


κ ∇T n δT dΓ »ΓD

κ δ p∇T q n T dΓ.By restricting the function space and using the matrix notation, introduced insection 5.1, the fist term can be discretized as»

ΓD

κ δ p∇T q n T dΓ (5.2)Ñ »ΓD

LN δT

Jκ n

N T

dΓ

(5.3) δTJ »

ΓD

BJ κ nN dΓloooooooooomoooooooooonGT

T .

Following the same argumentation as in section 5.1, the coefficient vectors can betaken out of the integral and the remaining expression is denoted as the matrixGT . The same can be done with the second term, yielding»

ΓD

κ ∇T n δT dΓ (5.2)Ñ »ΓD

N δT

Jκ nJ

LN TdΓ

(5.3) δTJ »

ΓD

NJ κ nJB dΓloooooooooomoooooooooonGT

J T ,

and the remaining integral can be identified as the transposed of the matrix GT .Applying the same transformation to the third term leads to»

ΓD

κ δ p∇T q n T dΓ (5.2)Ñ »ΓD

LN δT

Jκ n T dΓ

(5.3) δTJ »

ΓD

BJ κ n T dΓlooooooooomooooooooongT

,

68


giving rise to a new vector, which shall be denoted as gT .

In analogy to the Lagrange expressions, the constraining terms of the penaltymethod have to be discretized. As derived in section 3.3, in case of the tempera-ture problem, the variation of the potential reads

δΠP pT q (3.30) β »ΓD

T δT dΓ β »ΓD

T δT dΓ.Following the same approach as in the Lagrange case, the two expression can bediscretized, yielding»

ΓD

T T

δT dΓ (5.2)Ñ »ΓD

N δT

J N T

dΓ »

ΓD

N δT

JT dΓ

(5.3) δTJ »

ΓD

NJN dΓlooooooomooooooonMT

T δTJ »

ΓD

NJ T dΓloooooomoooooonmT

,

giving rise to a matrix and a vector, denoted byMT andmT , respectively.

Using these five new terms, the discretized weak field equation (3.57), con-strained by the Nitsche method, reads

δTJ KT T GT T GT

J T MT T fT gT mT

! 0.

Since the derived equation must be satisfied for any variation, the bracket expres-sion must be identically zero, which gives rise to the system of linear equationsreading

KT GT GTJ β MT

T fT gT β mT . (5.13)

Hence, the task of finding the numerical approximation to the analytical tempera-ture distribution, satisfying the Dirichlet boundary conditions in the weak sense,can be formulated as

“Find the solution vector T satisfying the system oflinear equations (5.13) for α pxq Ñ 0 x P ΩezΩ.”

5.2.3 FCM for the elastic problem

Just as in the temperature case, also the elastic potential has to be redefined on theextended domain in order to apply the Finite Cell Method. Following the sameidea as in the previous section, the augmented weak field equation of the elastic

69

5 Numerical Methods

system yields

δΠeint

! δΠeext

(3.22)Ñ »Ωe

α σ : δε dΩ »Ωe

α b δu dΩ »ΓN

t δu dΓ. (5.14)

Applying the same transformations as in the case of the FEM, the individualterms can be discretized, yielding

Ku »Ωe

αBJCB dΩ and fu »Ωe

αNJ b dΩ »ΓN

NJ t dΓ. (5.15)

As before, the discretization of the weak field equation yields

Ku u fu. (5.16)

With this result, the task of finding the numerical approximation to the analyticaldisplacement distribution can be formulated as

“Find the solution vector u satisfying the system oflinear equations (5.16) for α pxq Ñ 0 x P ΩezΩ.”

To impose the Dirichlet boundary conditions in the weak sense, again theNitsche method is applied. The constrained weak field equation hence yields

δΠeint δΠe

ext δΠidentL δΠP

! 0,


multiplier method, respectively. In analogy to the above derivation, these expres-sions have to be discretized.

As derived in section 3.3.2, the constraint potential of the identified Lagrangemultiplier method, imposing the Dirichlet boundary conditions of the displace-ment distribution in the weak sense, and its variation read

ΠidentL

(3.51) »ΓD

pσ nq pu uq dΓδΠident

L

(3.53) »ΓD


δu σ n dΓ »ΓD

δ pσ nq u dΓ70


Before the individual terms can be discretized, the contraction of the tensors σand nmust be reformulated as a matrix product, yielding

σ nÑ nJ σ with n n1 0

0 n2

n2 n1

in 2D. (5.17)

By restricting the function space, the fist term can then be discretized as follows»ΓD

δ pσ nq u dΓ (5.17)Ñ(5.7)

»ΓD

δnJ σJ pN uq dΓ

(5.8) »ΓD

nJCLN δu

J pN uq dΓ(5.3) δuJ »

ΓD

BJCJ nN dΓlooooooooooomooooooooooonGu

u.

Following the same line of arguments as before, the coefficient vectors can betaken out of the integral and the remaining expression can be denoted as thematrixGu. The second term is discretized accordingly»

ΓD

δu pσ nq dΓ (5.17)Ñ(5.7)

»ΓD

δ pN uqJ nJ σ dΓ(5.8) »

ΓD

pN δuqJ nJC LN u

dΓ

(5.3) δuJ »ΓD

NJ nJCB dΓlooooooooooomooooooooooonGu

J u,

and the remaining integral can be identified as the transposed of the matrix Gu.Applying the same transformation to the third term leads to»

ΓD

δ pσ nq u dΓ (5.17)Ñ(5.7)

»ΓD

δnJ σJu dΓ

(5.8) »ΓD

nJC LN δu

Ju dΓ

(5.3) δuJ »ΓD

BJCJ n u dΓloooooooooomoooooooooongu

,

giving rise to a new vector, denoted as gu.

Like the Lagrange expressions (3.53), the constraining terms of the penaltymethod have to be approximated. As derived in section 3.3, in case of the elastic

71

5 Numerical Methods

problem, the variation of the potential reads

δΠP puq (3.32) »ΓD

δu pu uq dΓ.Following the same approach as in the Lagrange case, the two expression can bediscretized as»

ΓD

δu pu uq dΓ (5.2)Ñ »ΓD

pN δuqJ pN uq dΓ »ΓD

pN δuqJ u dΓ(5.3) δuJ »

ΓD

NJN dΓlooooooomooooooonMu

u δuJ »ΓD

NJ u dΓloooooomoooooonmu

,

which gives rise to a matrix and a vector, denoted asMu andmu, respectively.

Using these five new terms, the discretized weak field equation (3.58), con-strained by the Nitsche method, reads

δuJ Ku uGu uGu

J uMu u fu gu mu

! 0.

Since the derived equation must be satisfied for any variation, the bracket expres-sion must be identically zero, which gives rise to the system of linear equationsreading

Ku Gu GuJ β Mu

u fu gu β mu. (5.18)

Hence, the task of finding the numerical approximation to the analytical displace-ment distribution, satisfying the Dirichlet boundary conditions in the weak sense,can be formulated as

“Find the solution vector u satisfying the system oflinear equations (5.18) for α pxq Ñ 0 x P ΩezΩ.”

5.3 FCM for linear thermoelasticity

In chapter 4, the field equations governing linear thermoelasticity were derivedin their strong andweak formulation. However, as in the single field case, an ana-lytical solution, fulfilling these conditions can only be found for simple, academicexamples. Real-world problems typically require the use of numerical methods.In this work, the Finite Cell Method shall be used to approximate the analyticalsolution of a thermodynamic problem. For this purpose, the same transforma-tions as in the single field case have to be applied to the weak field equation of

72


thermoelasticity. These steps shall be outlined in the following sections.

5.3.1 Discretization

As discussed in chapter 4, the mathematical model of thermoelasticity is com-posed of the field equations describing the temperature distribution in and thedeformation of an elastic continuum. Since in the linearized formulation of thismodel the change of temperature resulting from the deformation is neglected, thefield equations describing the thermal system do not change. Therefore, also theoriginal numerical model can be used to approximate the temperature distribu-tion without any adoptions. As derived in section 5.2.2, the numerical solutioncan be computed by solving the equation system

KT T fT .However, in the case of the displacement field, the influence of the temperature

has to be taken into account. Using equation (4.9), the weak field equation ofthermoelasticity can be reformulated for the extended domain, yielding

δΠeint

! δΠeext»

Ωe

α σtot: δεtot dΩ »

Ωe

α b δu dΩ »ΓN

t δu dΓ »Ωe

α σth: δεtot dΩ.

Following the same idea as in the single field case, the left hand side expressionand the first and second integral on the right can be discretized by the stiffnessmatrixKu and the load vector fu. As derived in section 4.2, the total strain andstress are directly associated to the displacement (see equation (4.1) and (4.6)).Therefore, the discretization of these terms yield

Ku »Ωe

αBJCB dΩ and fu »Ωe

αNJ b dΩ »ΓN

NJ t dΓ.and hence do not differ form the pure elastic case. To discretize the third term onthe right hand side, the constitutive and coupling equations, derived in chapter4, are used to express the thermal stress as a function of the temperature. Thistransformation reads

σth (4.6) C : εth(4.2) C : pγ T Iq . (5.19)

By replacing the solution function and the variation with their numerical approx-

73

5 Numerical Methods

imations, the expression can be discretized as follows»Ωe

α σth: δεtot dΩ

(5.19)Ñ(5.2)

»Ωe

α pLN δuqJ C γN T

dΩ

(5.3) »Ωe

α δuJBJC γN T dΩ δuJ »Ωe

αBJC γN dΩlooooooooooomooooooooooonCΩ

Tu

T .

Following the same argument as before, the coefficient vectors can be taken outof the integral, giving rise to an additional matrix. As this term expresses theconnection between the temperature and the displacement field on the domainΩ, it is denoted as the coupling matrix CΩ

Tu.

With this result, the discretized weak field equation of the thermoelastic sys-tem can be written as

δuJ Ku u fu CΩ

TuT

! 0.

Just as in the single field case, the derived equation must be satisfied for anyvariation and hence the bracket expression must be identically zero, which givesrise to a equations system reading

Ku uCΩ

Tu T fu.By combing the derived expressionwith the condition of the temperature prob-

lem, the equation system determining the state of thermoelastic system can bewritten as

KT 0

Ku CΩ

Tu

T

u

fTfu

. (5.20)

Hence, the task of finding the numerical approximation to the analytical tempera-ture and displacement distribution of an thermoelastic system can be formulatedas

“Find the solution vector

u

T

satisfying the system of

linear equations (5.20) for α pxq Ñ 0 x P ΩezΩ.”As outlined in the single field discussion, when applying the Finite CellMethod,

the Dirichlet boundary conditions can not be imposed by a modification of thesystem matrix. Instead, the constraint potentials must be used, which imposethe conditions in the weak sense. Following the Nitsche approach, the weak fieldequations are augmented by the constraint potentials of the penalty and Lagrange

74


multiplier method. In analogy to the above derivation, these expressions have tobe discretized.

Since the temperature change, caused by the deformation of the elastic bodyis neglected in the linearized description of thermoelasticity, also the constrainedfield equation needs no modification and hence is given by

δΠe δΠidentL δΠP

! 0,


multiplier method, respectively. As derived in section 5.2.2, the discretization ofthese terms give rise to thematricesGT andMT and the vectors gT andmT . Withthese expressions, the temperature distribution of a thermoelastic system, satis-fying the Dirichlet boundary conditions in the weak sense, can be approximatedby solving

KT GT GTJ β MT

T fT gT β mT .

In case of the thermoelastic system, the constrained field equation reads

δΠeint δΠe

ext δΠidentL δΠP

! 0.

As before, ΠP is the constraint potential of the penalty method. As this approachdoes not take into account the influence of the temperature, no modification hasto be made at this point. As shown in section 5.2.3, the discretization of theseexpressions give rise to the matrixMu and the vectormu.

However, in contrast to the penalty method, the influence of the temperatureis taken into account in the Lagrange approach since the multiplier needs to bere-identified. As derived in section 4.3.2, the constraint potential and its variationyield

ΠidentL

(4.15a) »ΓD

σth n σtot n pu uq dΓ

δΠidentL

(4.15b) »ΓD

δσtot n u dΓ »

ΓD


δσtot n u dΓ »

ΓD

δu σth n dΓAccording to equation (4.6), the total stress can be expressed as a function of thedisplacement. Therefore, the same transformations as in the pure elastic case can

75

5 Numerical Methods

be applied and the first three integrals can be discretized as follows»ΓD

δσtot n u dΓ Ñ δuJ »

ΓD

BJCJ nN dΓlooooooooooomooooooooooonGu

u,»ΓD

δu σtot n dΓ Ñ δuJ »ΓD

NJ nJCB dΓlooooooooooomooooooooooonGu

J u,»ΓD

δσtot n u dΓ Ñ δuJ »

ΓD

BJCJ n u dΓloooooooooomoooooooooongu

.

As shown in (5.19), the thermal stress can be expressed as a function of the tem-perature. Hence, the fourth term of condition can be discretized by»

ΓD

δu σth n dΓ (5.17)Ñ(5.19)

»Ω

pN δuqJ C nJσth

dΩ

(5.3) »Ω

δuJBJC nJγN T dΩ δuJ »Ω

BJC nJγN dΩlooooooooooomooooooooooonCΓ

Tu

T ,

giving rise to a second matrix, expressing the coupling of the temperature andthe displacement field on the Dirichlet boundary ΓD. It hence shall be denoted asCΓ

Tu.

Using the five constraining terms and the additionalmatrixCΓ

Tu, the discretizedweak field equation of the thermoelastic system, constrained by theNitschemethod,is given by

δuJ Ku uGu uGu

J uMu u fu gu mu CΩ

TuT CΓ

TuT

! 0.

Just as in the single field case, the derived condition must be satisfies for anyvariation. Therefore, the bracket expression must be identically zero. This givesrise to the equation system reading

Ku Gu GuJ β Mu

u

CΩ

Tu CΓ

Tu

T fu gu β mu.

By combing the derived expressionwith the condition of the temperature prob-lem, the equation system determining the state of thermoelastic system can be

76


written asKT GT GT

J β MT 0

Ku Gu GuJ β Mu CΓ

TuCΩ

Tu

T

u

fT gT β mT

fu gu β mu

, (5.21)

with the respective terms being given by

KT »Ωe

αBJ κB dΩ fT »Ωe

αNJ s dΩ »ΓN

NJq dΓGT »

ΓD

BJ κ nN dΓ gT »ΓD

BJ κ n T dΓMT »

ΓD

NJN dΓ mT »ΓD

NJ T dΓand

Ku »Ωe

αBJCB dΩ fu »Ωe

αNJ b dΩ »ΓN

NJ t dΓGu »

ΓD

BJCJ nN dΓ gu »ΓD

BJCJ n u dΓMu »

ΓD

NJN dΓ mu »ΓD

NJ u dΓCΩ

Tu »

Ωe

αBJC γN dΩ CΓ

Tu »

Ω

BJC nJγN dΩ.

Hence, the task of finding the numerical approximation to the analytical tem-perature and displacement distribution of an thermoelastic system, satisfying theDirichlet boundary conditions in the weak sense, can be reformulated as

“Find the solution vector

u

T

satisfying the system of

linear equations (5.21) for α pxq Ñ 0 x P ΩezΩ.”Since the temperature is not influenced by the displacement, an alternative to

this monolithic formulation can be found by decoupling the equation system. Inthis partitioned approach, the problem is solved in two steps. First, a numericalapproximation to the analytical temperature distribution is computed by solvingthe equations system

KT GT GTJ β MT

T fT gT β mT . (5.23)

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5 Numerical Methods

With this result, the coupling matrices CΩ

TuandCΓ

Tucan be computed, yielding

CΩ

Tu Ñ fΩ

th »Ωe

αBJC γ T dΩ and CΓ

Tu Ñ fΩ

th »Ω

BJC nJγ T dΩ.Since these two thermal load vectors contain no unknowns, they can be shifted tothe right hand side of the displacement equation. With this transformation, thenumerical approximation to the analytical displacement distribution is computedin a second step, by solving the equations system

Ku Gu GuJ β Mu

u fu gu β mu fΩ

th fΓ

th. (5.24)

Hence, following this partitioned approach the task of finding the numerical ap-proximation of the analytical temperature and displacement distribution of anthermoelastic system, satisfying the Dirichlet boundary conditions in the weaksense, can be reformulated as

“Find the solution vector T satisfying the system oflinear equations (5.23) and the solution vector u satisfying the system of

linear equations (5.24) for α pxq Ñ 0 x P ΩezΩ.”5.3.2 Implementation

In the partitioned ansatz, formulated in the previous section, the problem of ap-proximating the state of a thermoelastic system is decomposed into two smallerparts. This offers the advantage, that the equation systems of the individual sub-problems can be setup and solved in two separate programs. By this, major partsof the code, approximating the state of the single field systems, can be reused.Due to these advantages, this partitioned approach shall be followed to numeri-cally solve the problem of thermoelasticity.

Over the past years, the chair for Computation in Engineering (CiE) of theTechnische Universität München (TUM) developed the program AdhoC ([5]),which shall be used for this purpose. AdhoC applies the Finite Element Methodto approximate the state of a temperature and an elastic system. As discussed insection 5.2, the FCM can be regarded as an extension of the FEM. For this rea-son, the implementations of both methods are very similar and major parts of theAdhoC code can be reused, when implementing the FCM for thermoelasticity.The points, in which the implementation of the FCM and the FEM differ, shall bebriefly outlined in this section. Furthermore, the implementation of the couplingterms will be discussed.

78


As illustrated in the section 5.2, the essential idea of the FCM is to embed thephysical domain in an extended domain and to define a scalar distribution α pxqon this extended domain. It could be shown, that the original problem is recov-ered if the value of α is equal to one inside the physical domain and zero outside.Therefore, the α distribution and hence also the integrand of the stiffness matrixand the load vector are discontinuous functions. Since in the original FEM theGaussian quadrature rule is used ([26], [12]), this imposes some difficulties onthe numerical integration as this method loses its accuracy qualities if the inte-grand is no polynomial function.

One possibility to overcome this problem is to increase the number of evalua-tion points, to reduce the error when approximating the integral of a discontinu-ous function. With this straight forward approach, the implementation of AdhoCmust not be modified since it already offers Gauss integration schemes up to anorder of 50. However, to assign the correct value to α, it is necessary to deter-mine the domain in which the Gauss integration point lies. This can be doneusing the tree-based library, developed at Technische Universität Braunschweig([4]). Knowing whether the integration point lies inside or outside the physi-cal domain, the value of α can be assigned and the stiffness matrix and the loadvector of the Finite Cell Method can be computed. An alternative to the high or-der Gauss integration is the use of a possible better performing sub-integrationscheme as suggested by Parvizian et al. ([17]) and Düster et al. ([6]).

The second major difference of the FEM and the FCM, regarding their imple-mentation, is that, in latter case, the Dirichlet boundary conditions must be im-posed in the weak sense. As discussed in section 3.3 and 5.2, this requires theuse of constraint potentials, which, when discretized, give rise to expressionsGu,Mu, gu andmu. To use the FCM in AdhoC, the program needs therefore to beextended to compute these expressions. Since the constraint potentials are de-fined as integrals over the Dirichlet boundary, this requires the implementationof a quadrature scheme to numerically approximate these matrices. For this pur-pose, the contour of the boundary is represented by line segments and the Gaussintegration method is used. With these extensions, the AdhoC program can beused to approximate the state of a thermal and an elastic system, following thefinite cell approach.

In order to use AdhoC to solve the thermoelastic problem, the setup of thethermal load vectors fΩ

thand fΓ

thmust be implemented. AdhoC already includes

a routing for a pre-strain load vector, which can be used to compute fΩ

th. As

fΓ

th is defined as an integral over the Dirichlet boundary, the same quadraturescheme as for the constraint expressions can be used to setup this second thermalload vector. With these extensions, the displacement of an elastic body, result-ing from mechanical forces and a change of temperature, can be approximatedby AdhoC, using the Finite Cell Method. Therefore, the thermoelastic problem

79

5 Numerical Methods

can be solved numerically, following the partitioned approach, by coupling twoinstances of AdhoC, the first one computing the temperature in and the secondone computing the displacement of the thermoelastic body. In the next chapter,some numerical examples shall be presented which employ this approach.

80

6Numerical examples

In the previous chapter, the Finite Cell Method was introduced as an approach tonumerically determine the state of a physical system. As the analytical field dis-tribution can only be approximated, the numerical result typically deviates fromthe correct solution. This error is a measurable and essential quality criterium ofa numerical method. In this chapter, the performance of the FCM with respect tothis aspect shall be discussed.

As shown by Düster et al. and Parvizian et al. , the deformation of an elasticcontinuum can be approximated verywell using the Finite CellMethod ([6], [17]).Especially, exponential convergence rates can be achieved when applying a p-refinement. Therefore, this aspect shall not be discussed in more detail in thefollowing chapter. Instead, the analysis will focus on two different characteristics.First, the compliance of the Dirichlet boundary conditions, imposed in the weaksense, shall be analysed in section 6.1. Additionally, the idea of the Finite CellMethod shall be applied to resolve discontinuous material properties. For thispurpose, the deflection of a bimetal strip is approximated in part 6.2.

6.1 Weak Dirichlet boundary conditions for the FCM

As suggested by Parvizian et al. , one possibility to impose homogeneous Dirich-let boundary is to assume a “stiff strip” of material between the Dirichlet bound-aries of the physical and extended domain ([17]). As shown in section 5.2, an

81

6 Numerical examples

alternative to this approach is the use of the constraint potential, derived in sec-tion 3.3. Since this has not been carried out in the context of thermoelasticity,the study in this section shall focus on the performance of these weak Dirichletboundary conditions in thermoelasticity.

For this purpose, a reverse engineering approach is chosen by defining an an-alytical solution function on a physical domain. Using the strong field equationsderived in section 2.2, the source terms and the boundary conditions, associatedto this distribution, can be computed. With this data, the setup of a numericalsimulation is possible and, since the analytical distribution is known, the error ofthe approximation can be analysed.

To get an impression of the characteristics of these weak boundary conditions,this is first performed for an elastic system in section 6.1.1. In part 6.1.2, the exam-ple is then combined with a temperature field to yield a thermoelastic problem.

6.1.1 Elastic system

As outlined in the introduction to this section, the idea of the reverse engineeringapproach is to analyse the error of a numerical simulation, approximating a pre-defined solution. In this section, the quality of the weak boundary conditions ofa displacement field shall be analysed accordingly.

For this purpose, an elastic continuum is assumed to be deformed in x- andy-direction equally by a displacement u, defined as

u ux uy 1

4 p1 xq p1 yq . (6.1)

The out-of-plane deformation uz is set to zero. Additionally assuming the strainin z-direction to vanish, the three dimensional problem can be reduced to a twodimensional plane strain problem. The shape of the body is defined as a squareform 1 to 1 in x- and y-direction, shown as physical domain Ω in figure 6.1a.The thickness of this plate is set to 1. The distribution of the displacement u onthis domain is displayed in figure 6.1b. Thereby, the lower surface represents thephysical domain and the zero level, and the vertical position of the upper surfacecorrelates to the disablement value.

To setup a numerical simulation, the volume load and the boundary condi-tions, corresponding to distribution 6.1, have to be specified. As derived in sec-tion 2.2.2, the volume load, acting on an elastic body, has to be in balance with itsinternal stresses. Using the constitutive and topological equations, it can there-

82


Ω

1

1.5

11.5

1 1.511.5 x

y

Ωe

Γ

E 1ν 0

α 1012

t 1

α 1

(a) Domain setup. Ω: physical domain,Ωe: extended domain, Γ: Dirichletboundary, E: Young’s modulus, ν:Poisson’s ration, t: thickness, α: FCMparameter

(b) Analytical displacement distribution.

Figure 6.1: Problem setup

fore be expressed as a function of the displacement as follows

b(2.10a) ∇ σ (2.10b) ∇ pC : εq (2.10c) ∇ C :

1

2 ∇u p∇uqJ .

Since the displacement uz is assumed to be zero, also no forces can act in thisdirection. Thus,

bz 0.

Hence, only the forces in x- and y-direction have to be determined and the aboveexpression can be rewritten in the matrix notation as follows

b Lσ L pC εq . (6.2)

In the plane strain case, the constitutive matrix of an isotropic elastic material isgive by

C Ep1 νq p1 2νq 1 ν ν 0

1 ν ν

sym. 12ν2

,with the Young’s modulus E and the Poisson’s ratio ν ([11]). In this example,these values shall be give as

E 1 and ν 0, (6.3)

83


so that the material matrix yields

C 1 0 0

1 0

sym. 1

2

. (6.4)

With this result, the volume load, corresponding to the displacement distribution(6.1), is given as

bxby

Lσ L pC εq (6.1)(6.3)

1

8

1

1

. (6.5)

In addition to the volume load, the Dirichlet boundary conditions have to bespecified by evaluating the displacement distribution u on the domain boundaryΓ, yielding

u pxq y1

1

2 p1 xq u pyq

x1

0

u pyq x1

1

2 p1 yq u pxq

y1

0. (6.6)

Using the derived values, the stiffness matrixKu and the load vector fu can becomputed. Also the setup of the constraint expression Gu,Mu, gu andmu, im-posing the Dirichlet boundary conditions in the weak sense, is possible. With this,the analytical solution can be approximated numerically by solving the equationsystem (5.18).

For a first numerical simulation, the physical domain is divided into a conformmesh of 4 equal cells and the polynomial degree p of the ansatz functions Ni is setto two. As outlined in section 5.2.3, the boundary conditions are imposed in theweak sense following the approach of Nitsche, which combines the penalty andidentified Lagrange multiplier method. In this first example, only the Lagrangeansatz shall be used, which is achieved by choosing the penalty value β as zero.The result of this numerical computation is displayed in figure 6.2a.

As in figure 6.1b, the lower surface represents the physical domain Ω and thezero level and the vertical position of the upper surface correlates to the numer-ically computed displacement value. Above that, the arrows on the outer edgeof the cells represent the normal vector n of the boundary domain Γ and theirvertical positions follow the analytical boundary values (6.6).

By comparing the shape of the numerical and analytical solution (figure 6.1b),no deviation can be observed. In particular, no error of the boundary conditionsappears, although they are imposed in the weak sense only. The numerical ap-proximation represents the analytical solution therefore correctly.

84


(a) Numerical approximation. (p 2, β 0) (b) Error of displacement approximation

Figure 6.2: Numerical approximation on a conform mesh. Dirichlet boundaryconditions are imposed in the weak sense by the identified Lagrangemultiplier method. p: polynomial degree of ansatz functions; β:penalty value of Nitsche method.

β integrationcells1

Πint eE r%sanalytical solution 6.250 101

conform mesh 0 1 6.250 101 0.0

non-conform mesh

01 5.809 101 26.6

3 6.250 101 0.0

1051 6.271 101 5.8

3 6.250 101 0.0

Table 6.1: Global error of the displacement approximation measured in the en-ergy norm.

To quantitatively analyse the numerical result, the error of the displacementapproximation is displayed in figure 6.2b. As it lies in the order of 108, it hasno major significance. Especially, no increase of the error on the boundary can beobserved. Therefore, the Dirichlet boundary conditions are imposed to numericalaccuracy by the identified Lagrange multiplier method. Also the error in theenergy norm, measuring the global deviation of the numerical solution, is zero,as shown in table 6.1. Hence, also in quantitative sense, the numerical resultrepresents the analytical solution. The reason for these good results is the fact

1Despite the denoted number of sub-integration cells, the strain energywas always computedexactly. It is worth to note that use of an inexact integration scheme in the post-processing, yieldsa significant increase in the global error.

85


that the analytical solution is a bi-linear function and hence an element of thechosen ansatz space. Therefore, the numerical approximation is only perturbedby minor numerical inaccuracies.

In a second numerical simulation, the physical domain is embedded in an ex-tended domain, defined as a square from 1.5 to 1.5 in both directions (see figure6.1a). Like in the first example, this extended domain is meshed by 4 equal cells.Since in this case, the mesh does not follow the geometry of the physical domain,it is denoted as non-conform. In order to still approximate the original problem,the FCM parameter α has to be set to zero outside of the physical domain. How-ever, as outlined in section 5.2, this causes problems when solving the equationsystem numerically. Therefore, a value of α 1012 is chosen instead. 50 Gaussintegration points in each spatial direction are used per cell to approximate thediscontinuity in the integral. The result of this numerical computation is dis-played in figure 6.3a.

(a) Numerical approximation.(p 2, α 1012, 50 Gauss points,β 0)

(b) Error of displacement approximation.Vertical position 50 times magnified

Figure 6.3: Numerical approximation on a non-conform mesh. Dirichlet bound-ary conditions are imposed in the weak sense by the identified La-grange multiplier method. p: polynomial degree of ansatz functions;β: penalty value of Nitsche method; α: FCM parameter

Just as in the conform case, the lower surface represents the physical domainΩ,the vertical position of the upper surface correlates to the numerically computeddisplacement value, the arrows represent the normal vector n of the boundarydomain Γ and their vertical position follows the analytical boundary values (6.6).Above that, the extended domain Ωe and the extended solution function are rep-resented as transparent surfaces.

By comparing the second numerical result with the analytical distribution (fig-ure 6.1b)), the same characteristic of the displacement inside Ω can be observed.

86


Furthermore, the solution function is smoothly extended beyond the physicaldomain. This correlates to the results presented by Parvizian et al. and showsthat the concept of the Finite Cell Method works. Also on the boundary, the nu-merical result approximates the analytical distribution very well, although someminor offset between the solution surface and the vertical position of the nor-mal vectors can be observed. This indicates, that the boundary conditions are notimposed exactly by the identified Lagrange multiplier method in this second sim-ulation. Yet, the overall characteristic of the analytical solution is approximatedcorrectly and hence qualitatively, the numerical solution is correct.

This, however, does not hold for the quantitative consideration. As shown infigure 6.3b, the error of the displacement approximation is in the order of 102

and is hence six magnitudes higher than in the conform simulation. This lossof quality is caused by the inexact numerical integration scheme. Although 50

Gauss integration points are used in each spacial direction per cell to approxi-mate the integral, the discontinuous integrand function can not be fully resolved.Therefore, the original problem can not be represented exactly and hence the so-lution of a perturbed problem is approximated by simulation instead. As shownin table 6.1, this inaccuracy in the integration yields a global error, measured inthe energy norm, of 26.6%.

Analysing the compliance of the boundary condition, a major deviation can beobserved. This is due to the fact that in the Lagrange ansatz, the elastic and con-straint potentials are weighted equally. Therefore, a deviation on the boundaryhas the same significance as an error inside the domain. As outlined in section5.1, the result obtained by the Finite Element Method minimizes the error in en-ergy norm. As shown by Parvizian et al. , this “best approximation property” canalso be proven for the Finite Cell Method. Therefore, the result of the numericalsimulation is a compromise between the compliance of the weak field equations,perturbed by the inexact integration, and the weak boundary conditions, whichminimizes this global error.

If this characteristic of the numerical approximation is not desired, the compli-ance of the boundary conditions can be emphasized by combining the Lagrangeapproach with the penalty ansatz, yielding the Nitsche method. In figure 6.4a,the result of a third numerical computation with a penalty value of β 105 isshown.

As before, the qualitative behavior of the numerical approximation correspondsto the analytical distribution and the solution function is smoothly extended be-yond the physical domain. Above that, the distribution of the error, displayed infigure 6.4b, shows that the deviation on the boundary is drastically reduced byincreasing the penalty value. However, the error inside the domain is increased.This is to be explained with the shifted weighting of the elastic and constraintpotentials. Due to the high value of β, a deviation on the boundary has more

87


(a) Numerical approximation.(p 2, α 1012, 50 Gauss points,β 105)

(b) Error of displacement approximation.Vertical position 50 times magnified

Figure 6.4: Numerical approximation on a non-conform mesh. Dirichlet bound-ary conditions are imposed in the weak sense by the Nitsche method.p: polynomial degree of ansatz functions; β: penalty value of Nitschemethod; α: FCM parameter

significance than an error inside the domain. Therefore, when determining theapproximation that minimizes the overall error, the penalty approach forces thenumerical solution to satisfy the boundary conditions and shifts the deviation,caused by the inexact integration, into the domain. As shown in table 6.1, thisalso significantly reduces the global error, measured in the energy norm, from26.6% to 5.8%.

This is explained by the fact that the boundary integral of the constrainingexpression Gu,Mu, gu andmu can be integrated exactly while the domain inte-gral suffers form the inexact integration. Since the numerical solution is forcedto satisfy the boundary conditions, the compliance of the weak field equation,perturbed by the inexact integration, becomes secondary.

To further improve the the quality of the numerical approximation, the discon-tinuity of the integrand has to be captured correctly when calculating the stiffnessmatrix. This requires a more sophisticated quadrature method. As shown byParvizian et al. and Düster et al. , very good results can be achieved when apply-ing a sub-integration scheme ([17], [6]). This idea shall be utilized in a furthersimulation.

In the present case, the geometry of the physical domain can be representedexactly if the four cells discretizing Ωe are subdivided three times in each spatialdirection. Since none of these sub-integration cells is intersected by the domainboundary, the order of the Gaussian quadrature scheme can be reduced. In fig-ure 6.5a the numerical result of this setup is depicted. The Dirichlet boundary

88


conditions are induced by the identified Lagrange multiplier method.

(a) Numerical approximation.(p 2, α 1012, β 0, 3 3 sub-integration cells)

(b) Error of displacement approximation.

Figure 6.5: Numerical approximation on a non-conform mesh with sub-cell inte-gration. Dirichlet boundary conditions are imposed in the weak senseby the identified Lagrange multiplier method. p: polynomial degreeof ansatz functions; β: penalty value of Nitsche method; α: FCM pa-rameter

As before, the distribution of the numerical solution corresponds to the ana-lytical displacement inside Ω and is smoothly extended beyond the physical do-main. However, in contrast to figure 6.3a, no deviation on the boundary can beobserved although no penalty terms are used to emphasize the compliance of theboundary conditions. Furthermore, the approximation error, displayed in figure6.5b, is reduced by six magnitudes compared to figure 6.3b and is, hence, in thesame order as for the conform discretization of the physical domain (see 6.2b).Also the global error of the approximation, measured in the energy norm, is re-duced to zero as presented in table 6.1. Hence, the numerical solution representsthe displacement correctly.

These results demonstrate that, if the discontinuous integrands of the stiffnessmatrix are captured, the analytical solution can be approximated exactly on anon-conform mesh by utilizing the identified Lagrange multiplier approach andthe Finite Cell Method. However, in a real-world application, the geometry of acomplex physical domain may not be resolved precisely by this sub-integrationapproach. Therefore, it is interesting to note that the quality of the numericalapproximation can be increased significantly when emphasizing the complianceof the boundary conditions by the penalty method.

89


6.1.2 Thermoelastic system

In this section, the quality of the weak boundary conditions of a thermoelasticsystem shall be analysed. For this purpose, the same reverse engineering ap-proach as before is applied by determining the temperature problem correspond-ing to the elastic example, introduced in the previous section. In analogy to thesingle field problem, the dimensionality is thereby reduced to two and the phys-ical and extended domains, displayed in figure 6.1a, are re-used.

As shown in section 4.2, the strong field equations of the elastic problem canbe augmented for the thermoelastic case as follows

∇ σ b 0(4.7f)Ñ ∇ σtot b bth 0.

In order to yield the original problem, the mechanical and thermal loads, actingon the thermoelastic continuum, have to sum up to the original body load of theelastic problem. Assuming that no mechanical forces act on the thermoelasticsystem, the thermal load can be identified as

bth b (6.5) 1

8

1

1

. (6.7)

Using the field equations of linear thermoelasticity, the thermal load can be ex-pressed in terms of the thermal strain, yielding

bth(4.7i) ∇ σth (4.7j) ∇ C : εth

.

As outlined in section 4.1.2, the thermal strain is connected to the temperature viathe coupling condition and can be rewritten as

εth(4.7k) γ T I γ T 1 0

0 1

.

Switching to the matrix notation, the thermal strain hence yields

εth γ Tγ T0

.Using the constitutive matrix, introduced in the previous section, the thermalstress, written in matrix notation, becomes

σth σ11σ22σ12

C εth (6.4) 1 0 0

1 0

sym. 1

2

γ Tγ T0

γ Tγ T0

.90


In the tensor notation, the thermal stress therefore reads

σth γ T 0

0 γ T . (6.8)

With these results, the thermal load vector can be expressed as a function of thetemperature distribution as follows

bth ∇ σth Bσ11Bx1

Bσ12Bx2Bσ21Bx1

Bσ22Bx2

(6.8) BBx1

pγ T qBBx2

pγ T q ∇ pγ T q !(6.7)

1

8

1

1

.

For simplicity, the thermal expansion coefficient is set to one. Therefore, in orderto induce the same thermal load as in the previous example, the spacial gradientof the temperature distribution must yield 18 in x- and y-direction. Hence, it canbe defined as

T px, yq 1

8 px yq 1

4. (6.9)

In figure 6.6, this distribution is shown in a 3D plot. As in the previous example,the lower surface thereby represents the physical domain and the zero level, andthe vertical position of the upper surface correlates to the temperature value.

Figure 6.6: Analytical temperature distribution

Just as for the elastic problem, the source terms and the boundary conditions,corresponding to this distribution, have to be specified. Using the strong fieldequations and assuming the heat conductivity κ to be constant, the heat source scan be expressed as a function of the temperature as follows

0(2.4a) ∇ q s

(2.4b) ∇ κ p s(2.4c) ∇ pκ ∇T q s s κ ∆T.

Inserting the defined temperature distribution, the heat source yields

s κ ∆T κ B2TBx2 B2TBy2 (6.9) 0.

Therefore, no heat sources are present inside the field and the temperature prob-

91


lem is fully governed by the Dirichlet boundary conditions which yield

T pxq y1

1

8 px 1q T pyq

x1

1

8 px 3q

T pyq x1

1

8 py 1q T pxq

y1

1

8 py 3q . (6.10)

Using the derived values, the stiffness matrix KT and the load vector fT canbe computed. Also the setup of the constraint expressions GT ,MT , gT andmT ,imposing the Dirichlet boundary conditions in the weak sense, is possible. Withthis, the analytical temperature distribution can be approximated numerically bysolving the equation system (5.13).

As for the elastic case, in the first numerical simulation, the physical domain isdivided into a conform mesh of 4 equal cells and the polynomial degree p of theansatz functions Ni is set to two. Also, the identified Lagrange multiplier methodis used to impose the boundary conditions in the weak sense. The result of thisnumerical computation is displayed in figure 6.7a.

(a) Numerical approximation. (p 2, β 0) (b) Error of temperature approximation. Ver-tical position 50 times magnified

Figure 6.7: Numerical temperature approximation on a conform mesh. Dirichletboundary conditions are imposed in the weak sense by the identifiedLagrange multiplier method. p : polynomial degree of ansatz func-tions, β : penalty value of Nitsche method.

In analogy to the previous section, the lower surface represents the physicaldomain Ω and the zero level and the vertical position of the upper surface corre-lates to the numerically computed temperature value. Above that, the arrows onthe outer edge of the cells represent the normal vector n of the boundary domainΓ and their vertical position follows the analytical boundary values (6.10).

92


Just as in the elastic case, no deviation can be observed when comparing thenumerical and analytical solution (figure 6.6). In particular, no error of the bound-ary conditions appears. Qualitatively, the numerical approximation representsthe analytical solution therefore correctly.

To quantitatively analyse the result, the error of the temperature approxima-tion is displayed in figure 6.7b. As it lies in the order of 109, it has no major sig-nificance. Especially, no increase of the error on the boundary can be observed.Therefore, also the Dirichlet boundary conditions of the temperature problem areimposed exactly by the identified Lagrange multiplier method. Also the errorin the energy norm, measuring the global deviation of the numerical solution, iszero, as shown in table 6.2. Hence, the numerical result represents the analyti-cal solution exactly also in the quantitative sense. Just as before, the reason forthese good results is the fact that the analytical solution is an element of the cho-sen ansatz space. Therefore, the numerical approximation is only perturbed bynegligible numerical inaccuracies.

β integrationcells 2

Problem Πint eE r%sanalytical solution

temperature 6.250 102

displacement 6.250 101

conform mesh 0 1temperature 6.250 102 0.0

displacement 6.250 101 0.0

non-conform mesh

0

1temperature 5.969 102 21.2




105





Table 6.2: Global error of the thermoelastic simulations, measured in the energynorm.

2Despite the denoted number of sub-integration cells, the strain energywas always computedexactly. It is worth to note that use of an inexact integration scheme in the post-processing, yieldsa significant increase in the global error.

93


With the numerical approximation of the temperature distribution, the setupof the thermal load vectors fΩ

thand fΓ

this possible and the displacement of the

thermoelastic continuum, induced by this temperature distribution, can be ap-proximated numerically. The result of this numerical computation is displayedin figure 6.8a.

(a) Numerical approximation. (p 2, β 0) (b) Error of displacement approximation.Vertical position 50 times magnified

Figure 6.8: Numerical approximation of displacement induced by temperature ona conform mesh. Dirichlet boundary conditions are imposed in theweak sense by the identified Lagrange multiplier method. p : polyno-mial degree of ansatz functions, β : penalty value of Nitsche method.

By comparing the numerically computed displacement, induced by the changeof temperature, with the analytical distribution, introduced in the previous sec-tion (figure 6.1b), no major deviation can be observed. Hence, the approximationrepresents the analytical solution qualitatively correctly. Moreover, the error dis-tribution, displayed in figure 6.8b, and the global error, listed in table 6.2, confirmthat this is also true in a quantitative sense.

This shows three things. First, it proves that the temperature distribution wasderived correctly and is in fact associated to the displacement field (6.1). Sec-ondly, it additionally demonstrates that the thermal load vector fΩ

thcouples the

two systems correctly inside to domain Ω. Finally, the results also prove that theinteraction of the two fields on the boundary is correctly represented by fΓ

thand

that the boundary conditions are imposed exactly.

Like for the elastic problem, the same numerical simulation can be performedon the extended domain. Again, the value of α is set to 1012 outside of thephysical domain and 50 Gauss integration points in each spatial direction areused per cell to approximate the discontinuous domain integral. The result ofthis numerical computation is displayed in figure 6.9a and 6.9c.

94


(a) Numerical approximation of temperaturedistribution. (p 2, α 1012, 50 Gausspoints, β 0)

(b) Error of temperature approximation. Ver-tical position 50 times magnified

(c) Numerical approximation of displace-ment distribution. (p 2, α 10

12,50 Gauss points, β 0)

(d) Error of displacement approximation.Vertical position 50 times magnified

Figure 6.9: Numerical approximation of temperature and displacement distribu-tion on a non-conform mesh. Dirichlet boundary conditions are im-posed in the weak sense by the identified Lagrangemultiplier method.p: polynomial degree of ansatz functions, β: penalty value of Nitschemethod, α: FCM parameter.

Just as in the single field case, the characteristic of the temperature and dis-placement approximation inside the physical domain Ω corresponds to the re-spective analytical distributions. Additionally, both functions are smoothly ex-tended beyond the physical domain and also the boundary conditions are ap-proximated very well. This shows that the concept of the Finite Cell Method notonly works for single field problems but also for their interaction.

However, a qualitative analysis of the error distribution, displayed in figure

95


6.9b and 6.9d, reveals deviations in the order of 103 and 102, respectively. Inanalogy to the elastic problem, this is caused by the inability of the integrationscheme to resolve the original problem exactly. In the energy norm, this corre-sponds to an error of about 30%. Comparing the deviations of the elastic andthermoelastic problem, a small increase in the error can be observed (see table6.1 and 6.2). This is to be explained by the fact that, in the thermoelastic case,the body load is replaced by a temperature distribution, which is approximtednumerically. The error of this first step is passed on to the elastic problem, whichitself is again perturbed by the inexact integration. These two error sources addup, yielding a larger deviation.

Analysing the compliance of the boundary condition, similar deviations as forthe elastic problem can be observed for both fields. As explained in the previoussection, these errors are caused by the equal weighting of the constraint potential.To emphasize the compliance of the boundary conditions, the Lagrange approachcan be combined with the penalty ansatz, yielding the Nitsche method. In figure6.10a and 6.10c, the result of a third numerical computation with a penalty valueof 105 is shown.

As before, the qualitative behavior of both numerical approximations corre-sponds to the analytical distributions and the solution functions are smoothlyextended beyond the physical domain. Moreover, the distribution of the error,displayed in figure 6.10b and 6.10d, shows that the deviations on the boundaryare drastically reduced by increasing the penalty value. However, the error insidethe domain is increased. Just as in the single field case, this is to be explained bythe higher weighting of the constraint potentials which increases the significanceof a deviation on the boundary. Also the global error, measured in the energynorm, is reduced significantly for both systems (see table 6.2). In analogy to theelastic problem, this is to be explained by the fact, that the the compliance of theweak field equation, perturbed by the inexact integration, becomes insignificantdue to the emphasis of the boundary conditions.

As discussed in the previous section, to further improve the quality of the nu-merical approximation, the discontinuity of the integrand has to be captured cor-rectly. Again, the sub-integration scheme shall be utilized to demonstrate thisaspect in the context of linear thermoelasticity. For this purpose, the four cellsdiscretizing Ωe are subdivided three times in each spatial direction. In figure6.11a and 6.11c, the numerical results of this setup are depicted. The Dirichletboundary conditions are induced by the identified Lagrange multiplier method.

As before, the numerical solutions correspond to the analytical displacementand temperature distribution inside Ω and are smoothly extended beyond thephysical domain. However, in analogy to the single field case, no deviation on theboundary can be observed although no penalty terms are used to emphasize thecompliance of the boundary conditions. Furthermore, the approximation errors,

96


(a) Numerical approximation of temperaturedistribution. (p 2, α 1012, 50 Gausspoints, β 105)

(b) Error of temperature approximation. Ver-tical position 50 times magnified

(c) Numerical approximation of displace-ment distribution. (p 2, α 10

12,50 Gauss points, β 105)

(d) Error of displacement approximation.Vertical position 50 times magnified

Figure 6.10: Numerical approximation of temperature and displacement distri-bution on a non-conform mesh. Dirichlet boundary conditions areimposed in the weak sense by the Nitsche method.p : polynomialdegree of ansatz functions, β : penalty value of Nitsche method, α:FCM parameter.

displayed in figure 6.11b and 6.11d, are reduced by six magnitudes compared tothe inexact integration and is, hence, in the same order as for the conform dis-cretization of the physical domain. Also the global error of both approximations,measured in the energy norm, is reduced to zero as presented in table 6.2. Hence,the numerical solution represents the state of the thermoelastic system correctly.

These results demonstrate that, if all discontinuities are captured correctly, alsothe analytical solution of a thermoelastic problem can be approximated exactly on

97


(a) Numerical approximation of temperaturedistribution. (p 2, α 1012, β 105,3 3 sub-integration cells)

(b) Error of temperature approximation.

(c) Numerical approximation of displace-ment distribution. (p 2, α 1012,β 105, 3 3 sub-integration cells)

(d) Error of displacement approximation.

Figure 6.11: Numerical approximation of temperature and displacement distri-bution on a non-conform mesh with sub-cell integration. Dirichletboundary conditions are imposed in the weak sense by the identifiedLagrange multiplier method.p : polynomial degree of ansatz func-tions, β : penalty value of Nitsche method, α: FCM parameter.

a non-conformmesh by utilizing the identified Lagrangemultiplier approach andthe Finite Cell Method. However, as outlined in the previous section, this mightnot be possible for complex physical domains. Therefore, it is interesting to notethat, also in the thermoelastic case, the quality of the numerical approximationcan be increased significantly when emphasizing the compliance of the boundaryconditions by the penalty method.

98

6.2 Bimetal strip

6.2 Bimetal strip

In the fist part of this chapter, the performance of the Finite Cell Method and ofthe weak boundary conditions was analysed using an simple analytical example.In this second part of the chapter, this study shall be extended onto a more prac-tical example. For this purpose, two different elastic materials are joined togetherto form a bimetal strip.

Ω1: E 2.1 105, ν 0, t 1, γ 102

10Ω2: E 2.1 105, ν 0, t 1, γ 103

1110.375

0.375

1.5

Ωe

Figure 6.12: Setup of bimetal strip. Ω1,2: physical domains, Ωe: extended domain,E: Young’s modulus, ν: Poisson’s ration, t: thickness, γ: thermalexpansion coefficient

As shown in figure 6.12, the physical domain, defined as a beam with a sidelength of 2 and 10, is split horizontally into two subdomains with different ma-terial properties. The strip is supported by two bearings at the left hand side. Toinduce a deflection, the beam is heated uniformly by a temperature of 1 degree.Due to the higher thermal expansion coefficient of the uppermaterial, the bimetalstrip will bend downwards. Note that, the material discontinuity will induce akink in the displacement distribution and a jump of the stress and strain valuesat the internal boundary.

A simulation of this process with the Finite Element Method is possible by aconform discretization of both domains. In this case, the internal boundary isresolved by the mesh and distinct material properties can be assigned to everyelement. In figure 6.13 the deflected bimetal strip, approximated by this ansatz,is displayed.

Figure 6.13: FEM approximation of bimetal strip on conform mesh. (p 2)

99


As expected, the higher expansion coefficient of the upper material causes thebimetal strip to bend downwards when the beam is heated. Furthermore, thecolor distribution, representing the strain in y-direction, points out the predicteddiscontinuity. Therefore, the numerical result represents the problem characteris-tics correctly in a qualitative sense.

An alternative to this approach is to resolve the material transition when ap-proximating the integrals of the stiffness matrix. In order to thereby assign thecorrect material properties to the individual Gauss integration points, their corre-sponding domain has to be determined. For this purpose, the same library, usedto assign the correct value of α, can be be applied ([4]). Hence, with this FCMidea, the material discontinuity can be resolved on the Gauss point level and theinternal boundary has not to be considered when generating the finite cell mesh.This shall be demonstrated in a second simulation.

To demonstrate all features of the Finite Cell Method, the physical domain isembedded in an extended domain as shown in figure 6.12. This setup, however,prohibits a correct positioning of the vertical bearing since the lower left corner ofthe beam is no longer resolved by the mesh. Therefore, the absolute displacementvalues of the numerical solutions will not coincide. Nevertheless, the character-istics of their distribution can be compared. The horizontal bearing does not facethis problem as it can be induced in the weak sense by the Nitsche method. Infigure 6.14, the deflected bimetal strip, approximated by this alternative ansatz,is displayed. Like in the previous sections, the opaque and transparent parts ofthe solution thereby represent the physical and extended domains, respectively.

(a) p 2, β 1050 (b) p 5, β 10

50

Figure 6.14: FCM approximation of bimetal strip on non-conform mesh.

As depicted in figure 6.14, the cells used to discretize the beam neither re-solve its geometry nor the internal boundary. Yet, just as in the conform case,the higher thermal expansion coefficient of the upper material causes the beamto bend downwards. The color distribution reveals that also the discontinuity ofthe strain can be approximated even with low polynomials. Therefore, when ap-plying the ideas of the Finite Cell Method, the essential solutions characteristicscan be approximated correctly in a qualitative sense even if neither the geometrynor the material transition is resolved by the mesh.

To analyse the numerical results quantitatively, the values of the strain in y-

100

6.2 Bimetal strip

direction is plotted over the beam height in figure 6.15. In case of a conformdiscretization, the discontinuous change of material can be represented exactlysince the elements resolve the internal boundary. Furthermore, the strain valuesin both subdomains correlate to the different thermal expansion coefficients of thetwo materials. In case of non-conform meshing, this characteristic is representedcorrectly in the outer parts of the mesh. In the middle cells, which contain thechange of material, the discontinuity of the strain is approximated by the polyno-mial ansatz functions. This causes oscillations in the solution function. However,as shown in the plot, these can be damped by increasing the polynomial degree ofthe ansatz functions. Above that, they do not radiate into the whole domain butonly effect the distribution within the central element. Therefore, the deviation islimited locally and can, hence, be reduced by refining the cells, cut by the internalboundary.

y

ε yy

n 4, p 2

n 3, p 2

n 3, p 5

n 3, p 10

-1 -0.5 0 0.5 1-2

01

5.5

10

14 103

Figure 6.15: Distribution of strain in y-direction over beam height at x 5. n:number of cells over the beam height, p: polynomial degree of ansatzfunctions

In analogy to the strain, the values of the vertical displacement uy are plottedover the beam height in figure 6.16. As outlined before, the absolute values of thedisplacements do not coincide since the vertical bearing can not be imposed cor-rectly. Nevertheless, the results can be analysed relatively to each other. Thereby,it can be observed that the predicted kink in the displacement distribution canonly be represented correctly if the internal boundary is resolved by the mesh(figure 6.16a). If instead a non-conform discretization is used, this transition cannot be represented exactly by the polynomial ansatz functions. As shown in the

101


plots, the quality of this approximation can be raised effectively by increasing thepolynomial degree of the ansatz functions.

y

uy

-1.5 -1 -0.5 0 0.5 1 1.5-0.085

-0.08

-0.075

-0.07

(a) Conform mesh (FEM). p 2

y

uy

-1.5 -1 -0.5 0 0.5 1 1.5-0.075

-0.07

-0.065

-0.06

-0.055

-0.05

(b) Non-conform mesh (FCM). p 2

y

uy

-1.5 -1 -0.5 0 0.5 1 1.5-0.075

-0.07

-0.065

-0.06

-0.055

-0.05

(c) Non-conform mesh (FCM). p 5

y

uy

-1.5 -1 -0.5 0 0.5 1 1.5-0.12

-0.115

-0.11

-0.105

-0.1

-0.095

(d) Non-conform mesh (FCM). p 10

Figure 6.16: Distribution of vertical displacement uy over beam height at x 5. 3

p: polynomial degree of ansatz functions

It is interesting to note that the internal change of material causes perturba-tions of the strain and displacement solutions, whereas both functions are ex-tended smoothly beyond the physical domain without any effects, even thoughthis domain transition represents a similar discontinuity. The difference in thesolution characteristics is to be explained by the fact that two materials with thesame stiffness face the internal boundary, whereas the extended domain is highlyflexible due to the lower value of α. Therefore, the values on this void domaincan be adapted to best fit the solution the physical domain which is not possible ifthe discontinuity lies within the material itself. This clearly demonstrates underwhich circumstances the Finite Cell Method works best.

3Due to the different positioning of the vertical bearing, the absolute values of the displace-ments do not coincide.

102

7Summary, conclusion and outlook

In this master’s thesis, a partitioned algorithm was developed in the frameworkof the Finite Cell Method (FCM) to approximate the phenomena of linear ther-moelasticity numerically. For this purpose, the governing equations of the ther-mal, elastic and linear thermoelastic field were derived in their strong and weakformulation. In particular, it was shown how Dirichlet boundary conditions canbe imposed in the weak sense via the penalty, identified Lagrange multiplier andNitsche method. In the second part of the work, the Finite Cell Method was in-troduced as an embedded domain approach which resolves the geometry of thephysical domain not on the mesh- but on the integration-level. This idea wasutilized to discretize the derived weak field equations in space.

The examples presented in a first study demonstrate that this method workswell not only for the single field applications but also for linear thermoelasticity.In particular, excellent performance of the weakly imposed Dirichlet boundaryconditions was achieved. However, the results also showed that the essentialrequirement for a low approximation error is the correct capturing of disconti-nuities when integrating the stiffness matrix. Nevertheless, it could be observedthat, if the compliance of the boundary conditions is emphasized by the penaltymethod, the negative effects of the inexact integration can be reduced.

Similar results could be achieved in a second study when applying the idea ofthe Finite Cell Method to resolve the internal boundary of a bimetal strip. How-ever, the discontinuous material transition causes a kink in the analytical dis-placement distribution within the physical domain itself which can not be rep-

103

7 Summary, conclusion and outlook

resented exactly. Likewise, the associated jump of the strain values is only ap-proximated by the polynomial ansatz functions which induces oscillations. Theanalytical solution can, hence, only be represented correctly if the change of ma-terial is resolved by the spacial discretization. These results show that the ideaof the FCM to resolve discontinuities on the integration level is best applicable insituations where the solution function can extend smoothly into a void domain ifno further enhancements are applied.

As outlined in the introduction of the work, this thesis presents a pre-step of aresearch project granted by the Deutsche Forschungsgemeinschaft (DFG) whichaims to numerically simulate field assisted sintering processes ([9]). For this pur-pose, the interaction of a thermal, electric andmechanical field has to bemodeled.Although the work done in this thesis was limited to the analysis of linear ther-moelasticity, the gained insight can be used for the accomplishment these tasks.

104

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