Identification of material parameters in mechanical models [Elektronische Ressource] / eingereicht von Marcus Meyer
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Identification of material parameters in mechanical models [Elektronische Ressource] / eingereicht von Marcus Meyer

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Identification of material parametersin mechanical modelsDISSERTATIONzur Erlangung des akademischen GradesDoctor rerum naturalium(Dr. rer. nat.)TECHNISCHE UNIVERSITÄT CHEMNITZFakultät für Mathematikeingereicht von Dipl.-Math. techn. Marcus Meyergeboren am 28. Mai 1982 in SchlemaChemnitz, den 23. März 2010Betreuer: Prof. Dr. Bernd Hofmann (Chemnitz)Dr. Torsten Hein (Chemnitz)Gutachter: Prof. Dr. Bernd Hofmann (Chemnitz)Prof. Dr. Arnd Rösch (Duisburg/Essen)URL: http://archiv.tu-chemnitz.de/pub/2010/0052Author’s addressMarcus MeyerChemnitz University of TechnologyDepartment of MathematicsD-09126 Chemnitz, Germanymarcus.meyer@mathematik.tu-chemnitz.dehttp://www.tu-chemnitz.de/mathematik/inverse_problemeACKNOWLEDGEMENTSAcknowledgementsFinancial supportTheresearchpresentedinthisdissertationwassupportedbyDeutscheForschungs-gemeinschaft(DFG)withintheprojectNature of ill-posedness, approximate sourceconditions, and adapted regularization methods for identification problems (grantHO1454/7-1 and -2), which is embedded in the superior project Numerical sim-ulation of coupled problems in mechanics (grant PAK47/1).Personal thanksFirstofall,IwanttoexpressmygreatgratitudetomysupervisorBerndHofmann,who was during the last five years the most important person for the developmentof this dissertation.

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Identification of material parameters
in mechanical models
DISSERTATION
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
TECHNISCHE UNIVERSITÄT CHEMNITZ
Fakultät für Mathematik
eingereicht von Dipl.-Math. techn. Marcus Meyer
geboren am 28. Mai 1982 in Schlema
Chemnitz, den 23. März 2010
Betreuer: Prof. Dr. Bernd Hofmann (Chemnitz)
Dr. Torsten Hein (Chemnitz)
Gutachter: Prof. Dr. Bernd Hofmann (Chemnitz)
Prof. Dr. Arnd Rösch (Duisburg/Essen)
URL: http://archiv.tu-chemnitz.de/pub/2010/0052Author’s address
Marcus Meyer
Chemnitz University of Technology
Department of Mathematics
D-09126 Chemnitz, Germany
marcus.meyer@mathematik.tu-chemnitz.de
http://www.tu-chemnitz.de/mathematik/inverse_problemeACKNOWLEDGEMENTS
Acknowledgements
Financial
support
TheresearchpresentedinthisdissertationwassupportedbyDeutscheForschungsgemeinschaft(DFG)withintheprojectNature of ill-posedness, approximate source
conditions, and adapted regularization methods for identification problems (grant
HO1454/7-1 and -2), which is embedded in the superior project Numerical
simulation of coupled problems in mechanics (grant PAK47/1).
Personal thanks
Firstofall,IwanttoexpressmygreatgratitudetomysupervisorBerndHofmann,
who was during the last five years the most important person for the development
of this dissertation. When I was a student in Chemnitz, he already aroused
my interest for the theory of inverse problems and later he enabled me to be a
member of his research group for inverse in Chemnitz. During my time
as a PhD student he supported me in every conceivable way and offered me the
perfect research environment in Chemnitz. Due to his initiative, I was able to
participate in the above mentioned DFG research projects, which finally led to
most of the results of this dissertation. Furthermore, he offered me the chance
of visiting national and international conferences, whereby I got various exciting
and precious experiences. For all of this I am greatly grateful. I appreciate him
as a magnificent person.
Special thanks I want to express also to Torsten Hein, who was during my PhD
studies my mentor and colleague. Without his ingenious ideas it would have been
impossible for me to achieve the results of this dissertation in comparable quality
andtime. Inparticular, Iwanttoemphasizethatthecontentsofsection3–which
is in fact also the key to the main section 4 – originally base on his ideas. For
me it means great fortune that I am now able to benefit from his helpful advice.
Furthermore, I would like to thank Arnd Meyer for a lot of fruitful discussions
concerningthetheoryoflargedeformationmodelsandtheapplicationofadaptive
finite element methods. By his ideas and explanations, I was able to understand
many details of structural mechanics and the corresponding FEM methods.
In the end, I want to express my special gratitude to my family and to all of my
friends for their perpetual support in all imaginable challenges of a
mathematician’s life. Among all these magnificent persons I want to name only the two
most important - my mother and my father Anette and Matthias Meyer. They
are the best parents I could imagine and I know that they would do everything
for me. Without doubt, I would not be what I am now without them.
1CONTENTS
Contents
Notation index 4
1 Introduction 6
2 Parameter identification in elliptic differential equations 11
2.1 General framework of the inverse problem . . . . . . . . . . . . . 11
2.2 Nonlinear optimization methods . . . . . . . . . . . . . . . . . . . 14
2.3 Stochastic strategies . . . . . . . . . . . . . . . . . . 17
2.4 Ill-posedness and regularization approaches . . . . . . . . . . . . . 18
3 Identification of scalar and piecewise constant parameters 23
3.1 Identifying diffusion and reaction parameter . . . . . . . . . . . . 23
3.1.1 PDE model and inverse problem . . . . . . . . . . . . . . . 23
3.1.2 Discretization and solution of the inverse problem . . . . . 29
3.1.3 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Identifying Lamé’s constants in the small deformation model . . . 41
3.2.1 PDE model and inverse problem . . . . . . . . . . . . . . . 42
3.2.2 Discretization and solution of the inverse problem . . . . . 51
3.2.3 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . 58
4 Identifying parameter functions in large deformation models 64
4.1 The elasticity boundary value problem for large deformations . . . 64
4.1.1 Nonlinear PDE model . . . . . . . . . . . . . . . . . . . . 64
4.1.2 Material laws and the second Piola-Kirchhoff stress tensor 66
4.1.3 Solving the direct problem with an incremental method . . 72
4.2 Identification of material parameter functions . . . . . . . . . . . 76
4.2.1 The inverse problem as a constrained minimization problem 77
4.2.2 Solution via Newton-Lagrange methods . . . . . . . . . . . 81
4.2.3 Linearizing a(U;Vjp) for linear elastic material . . . . . . 91
2CONTENTS
4.3 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3.1 A two-dimensional model problem . . . . . . . . . . . . . . 93
4.3.2 Finite element discretization . . . . . . . . . . . . . . . . . 95
4.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 99
4.3.4 Alternative solution approaches . . . . . . . . . . . . . . . 102
5 Adaptive strategies for parameter identification 105
6 Open questions and future work 108
A Appendix: implementation of tensor calculus 110
A.1 Basic definitions in tensor calculus . . . . . . . . . . . . . . . . . . 110
A.2 Implementation of tensors in deformation theory . . . . . . . . . . 112
A.3 Linearizing the stored energy function . . . . . . . . . . . . . . 114
B Appendix: MATLAB implementation 116
References 118
Theses 125
List of figures 127
List of tables 128
3NOTATION INDEX
Notation index
Functions and constants
1 2; ; ; ; ; absolute and relative noise level1 2 rel rel rel
Kronecker symbol, see (3.16)ij
n number of measurement points for the statedata
n number of (material) parameterspar
n ;n ;n number of FE ansatz functions for state and parameteru U p
’;’~ FE ansatz functions for statei i
~ ; FE ansatz functions for parameteri i
characteristic function, cf. (3.29)

stored energy function
p2Q parameter to be identified
; density0
u;U2U state, solution of underlying differential equation
y;Y;y ;Y 2Y exact and noisy observation of state
Geometry objects
boundary of domain
; ; Dirichlet boundaryD D D0 t
; ; Neumann boundaryN N N0 t
T;T triangulations of domain2

;
;
bounded domain with Lipschitz boundary0 t
Matrix, tensor, and vector notations
C;C material tensor (A.6), (simplified) coordinate matrix (A.9)
E(U) Green-St.-Venant strain tensor (4.5),
in section 3 used as linearized strain tensor (3.40)
E(U;V ) derivative (4.6) of the Green-St.-Venant strain tensor
F deformation gradient, see (4.3)
G right Cauchy-Green strain tensor (4.7)
GradU gradient tensor (A.2)
I identity
~n,~n ,~n outer normal vector0 t
p discrete parameter to be identified
P projection matrix
Cauchy stress tensor
1 2
T;T first and second Piola-Kirchhoff stress tensor
u;U discretized state
4NOTATION INDEX
Miscellaneous
[IP] identification problem
[IP-1] identifying diffusion and reaction parameter
[IP-2] iden Lamé’s constants
[IP-3] identifying material parameters in nonlinear elasticity
Norms and products
h:;:i scalar product with underlying Hilbert spaceX
h:;:i duality product in Hilbert spaceX and dual spaceXX ;X
1 1
1h:;:i ,k:k H -scalar product and H -norm, see (3.2)1H ( ) H ( )
2 2h:;:i 2 ,k:k L product and L see (3.3)2L ( ) L ( )
1k:k L -norm, see (3.30)1L ( )
k:k Euclidean vector norm, see (3.32)2
Operators and functionals
A :QU!Z implicit nonlinear forward operator, see (2.6) and (2.7)
F :Q!U explicit nonlinear forward op see (2.2)
I identity operator
L second order elliptic differential operator, see (2.8)-(2.11)
L :QUZ!R Lagrangian (2.20) of a constrained minimization problem
P :U!Y linear projection operator, see (2.3)
tr trace of a tensor or matrix
r nabla operator
Spaces
1 1H ( ) , H ( ) Sobolev spaces0
1 1 1L , L ( ) , L ( ) spaces of a.e. bounded functions
2 2 2L , L ( ) , L ( ) of quadratic integrable functions
nR n-dimensional Euclidean space
Q space of parameters
(n)Q n FE subspace ofQ
U space of states
(n) (n)(n)V ,V ,V n-dimensional FE subspaces ofU,Y, andZ0 D
Y space of observed data
Z,Z space of test functions and corresponding dual space
51 INTRODUCTION
1 Introduction
In almost every field of science and technology the simulation of mechanical or
physical problems with finite element software tools is extensively used. The
basis of the applicability of finite element methods is an appropriately chosen
mathematical model, which in our context refers to a partial differential
equation including a couple of model parameters. Those parameters characterize for
example material properties, physical constants, or other issues included in the
assumed model. Depending on the application context, the parameters can be
introduced as scalar parameters or parameter functions.
Naturally, the finite element simulation has to fit the underlying real world
problem and thus it is essential to know the correct model parameters. However,
mostly these parameters are not known a priori and, moreover, they cannot be
determined by direct measurements. In such situations one is interested in
estimations of the unknown parameters, derived from indirect measurements. These
problems are called parameter identification problems, representing a very
important class of inverse problems. Over the past decades, various parameter
identification problems were part of mathematical research and thus a wide range of
literature exists. For a survey of theory and applied identification problems in
differentialequations, werefere.g. to[4], [5], [8], [24], [38], [55], andthereferences
therein.
structural mechanics PDE model
direct problem
- -
FEM solver
displacematerial ment
parameter stress
inverse problem strain
FEM solver
optimization
regularization
Figure 1.1: Scheme of the direct problem and the inverse parameter identification
problem in a structural mechanics PDE model
Unfortunately, for a considerable number of identification problems the
phenomenon of ill-posedness arises, i.e., the identified parameters do not depend
continuously on the measured data. Thus, the solution of the identification
problem turns out to be instable for noisy data, which is in practice a common
situa61 INTRODUCTION
tion. In the context of inverse problems, the ill-posedness can be overcome by the
application of regularization methods. Here, the ill-posed original problem is
replaced by a well-posed neighboring problem yielding approximate solutions that
depend stably on the data. For an extensive survey of regularization methods we
refer e.g. to [29] (with emphasis on linear problems), [58] and [77].
The focal point of this work is on parameter identification problems in mechanical
applications, which arise in the context of stress and strain simulation for elastic
materials. According to the scheme in figure 1.1, the simulation refers to the
direct problem, whereas for given geometry and loads the displacement of a body
d
R , d = 2; 3, is calculated as the solution of an elliptic system of partial
differential equations. Here, the crucial point is the appropriate choice of an
adequate material law. In the material law the material properties are included
as a number of material parameters. Consequently, the related inverse problem
consists in the identification of the material parameters from given displacement
or force data.
In the context of stress and strain theory, one has to distinguish linear and
nonlinear theory. For small deformations with linear elastic material behavior, the
elasticity problem approximately satisfies a linear partial differential equation
model. So far, the corresponding identification problems with material
parameters as scalars and functions are widely understood. We refer in the linear case
for example to the extensive survey [13] and the references therein. Note that
independent from linearity or nonlinearity of the forward problem, the inverse
problem is in general nonlinear.
At the other hand, this dissertation will mainly be devoted to large deformation
theory, which refers to a nonlinear differential equation even for linear material
laws (see e.g. [20], [67]). Additionally, nonlinearity arises from nonlinear
laws, which are applied for instance in the case of hyperelastic materials.
Consequently, sophisticated methods from nonlinear optimization and nonlinearinverse
problems theory have to be introduced for the solution of the considered
parameter identification problems. In the large deformation context the identification
of scalar material parameters is already solved satisfactorily in several numerical
studies, see e.g. [39]. Currently, the identification of parameter functions with
large deformation theory is an issue of growing interest. Such problems arise
for example in the field of elastic imaging, whereas properties of typically
inhomogeneous biologic materials and soft tissues are identified by the deformation
behavior. The paper [36] denotes one of the less existing studies on this topic.
Therefore, in the major part of our considerations we discuss the estimation of
material parameter functions, which in particular is correlated with ill-posedness
and instability of the identification problem. At this, the major questions we are
interested in are the following:
1. Is it possible to estimate the unknown parameter such that the underlying
71 INTRODUCTION
PDE model fits the measured data?
2. What kind of measured data has to be available such that the unknown
parameter can be estimated? Under what conditions exists a unique solution
of the identification problem? How does experimental design influence the
identifiability of a parameter?
3. How strong is the influence of data noise on the quality of the identification
results? Is the application of regularization methods necessary?
4. What methods provide an efficient numerical solution of the parameter
identification problem?
The last question is of special interest since the solution of an inverse problem
in general contains the solution of many forward (FEM) problems. Note that a
growing nonlinearity increases the number of forward operator evaluations
additionally. Thus, in sophisticated models with a demand of high accuracy and
fine discretization the numerical costs for solving an identification problem may
explode.
In recent times, adaptive FE methods were developed, which enable a fast
solution of partial differential equations (see e.g. [7] or [14]). Hence, the embedding
of existing high-efficient adaptive FEM tools in the parameter identification
software is essential. For the solution of large deformation PDE problems, e.g. the
application of the adaptive software package SPC (developed in the group for
numerical analysis at the Chemnitz University) may be promising. We refer to [67]
for an illustrated application of the SPC software in structural mechanics with
large deformations. However, in this dissertation we will not consider the
practicalimplementationofadaptiveFEsoftwaresincethisdenotesaquitecrucialtopic
for its own. Nevertheless, basing on a discussion of several solution approaches
we will outline options for a useful application of adaptive methods independent
from the particular framework of the considered identification problem.
Finally, the aim of this work is to give a survey on the solution of practical
motivated identification problems – beginning at the underlying model and ending
in the numerical solution. Thus, we have to combine three elementary topics:
first the formulation of the structural mechanics model as a partial differential
equation, second the inverse theory coinciding with the implementation of
nonlinear optimization and regularization methods, and finally the numerical solution
aspects involving a finite element discretization. We consider the practical
implementation of several identification algorithms and compare these methods with
respect to applicability, results quality, and efficiency. Questions on the
experimental design are discussed as well. Extensive numerical studies are presented.
The dissertation is organized in the following manner. In section 2 a general
framework of a nonlinear identification problem is presented. We discuss the
8