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

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• Mathematics

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• Publié par: technische_universitat_chemnitz
• Publié le: 01 janvier 2010
• Nombre de lectures: 8
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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
[e-mail]
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 p