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Numerical Aspects in Optimal Control of Elasticity Models with Large Deformations

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Numerical Aspects in Optimal Control of Elasticity Models with Large Deformations Numerical Aspects in Optimal Control of Elasticity Models with Large Deformations Andreas G¨unnel Dissertation submitted to the Faculty of Mathematics at Chemnitz University of Technology in accordance with the requirements for the degree Dr. rer. nat. Advisor: Prof. Dr. Roland Herzog Chemnitz, April 1, 2014 Gewidmet meinem Großvater Hans G¨unnel Contents Acknowledgments viii Abstract ix Chapter 1. Introduction1 1.1 Motivation1 1.2 Outline of the Thesis3 1.3 Notation5 1.4 Nomenclature6 Chapter 2. A Mathematical Model of Elasticity with Finite Deformations 11 2.1 Modeling of Finite Deformations 12 2.2 Models for Elastic Material Behavior 14 2.2.1. Balance-of-Forces Approach 14 2.2.2. Energy Minimization Approach 18 2.2.3. Loads (Volume, Boundary, Inner Pressure, Fiber Tension) 23 2.3 Comparison to Linear Elasticity 32 2.4 Parametrization of the Reference Configuration Ω 36 2.5 Hierarchical Plate Model 38 Chapter 3. Numerical Methods to Solve the Forward Problem 47 3.1 Newton's Method 48 3.2 Globalization by a Line Search 49 3.3 Krylov Subspace Methods 56 3.4 Discretization 61 3.5 Preconditioner and Multigrid Method 65 Chapter 4. Optimal Control Problems in Elasticity 72 4.1 Setting of the Optimal Control Problem 73 vi Contents 4.1.1. Quality functionals 74 4.1.2. Cost or Penalty Functionals 84 4.2 Lagrange-Newton: An All-at-once Approach 85 4.2.1. Solving the Lagrange Equation 86 4.2.2. Discretization 88 4.3 Quasi-Newton: A Reduced Formulation 89 4.3.1. Quasi-Newton Method 93 4.3.2. Broyden-Fletcher-Goldfarb-Shanno-Update 94 4.3.3. Simple Wolfe-Powell Line Search 97 4.3.4. Discretization 100 Chapter 5. Implementation in FEniCS 104 5.1 Introduction to FEniCS 105 5.2 Weak Formulations and UFL 109 5.2.1. Mutual Definitions 110 5.2.2. Objective Functions 113 5.2.3. Forward Problem 115 5.2.4. Lagrange Problem 118 5.2.5. Reduced Optimal Control Problem 121 5.2.6. Automatic Differentiation vs. Differentiation by Hand 124 5.2.7. Parametrization 126 5.2.8. Hierarchical Plate Model 131 5.3 Solver Routines 137 5.3.1. Multigrid Method 137 5.3.2. CG and MinRes Method 140 5.3.3. Forward Problem 141 5.3.4. Lagrange-Newton Problem 142 5.3.5. Reduced Problem 144 5.4 An Experimental Preconditioner 147 Chapter 6. Numerical Experiments 151 6.1 Experiments on the Forward Problem 153 6.1.1. Line Search and Iterates of Newton's Method 153 6.1.2. Plate Model 156 6.2 Experiments on the Optimal Control Problem 159 6.2.1. Elevated Bar with Fiber Tension Control 159 6.2.2. Regional Penalization 167 6.2.3. Flower Movement by Turgor Pressure 170 6.2.4. Enclosed Volume 173 vii Chapter 7. Conclusions and Perspectives 177 Appendix A. Appendix 181 A.1 Matrix Calculus 181 A.2 Derivatives of Energy Functionals 183 Appendix B. Theses 186 Appendix C. Curriculum Vitæ 188 Bibliography 191 Acknowledgments My deepest gratitude goes to my supervisor Roland Herzog. Not only for offering me the possibility to work at the TU Chemnitz and to write my thesis, but also for his support and patience with me. His advice and remarks were most helpful for my work and he taught me a lot in the last years. Arnd Meyer introduced me to the area of elasticity and plate theory, for which I am very grateful. Also, I thank Anton Schiela for the discussions on optimal control and his remarks. My position at the TU Chemnitz was partially funded by the BMBF project Qualit¨atspakt Lehre and the DFG Excellence Cluster MERGE, which is much appreciated. I want to express my gratitude for the people working at the faculty of mathematics at the TU Chemnitz for the enjoyable work climate. First of all, I thank my working group: Roland Herzog, Martin Bernauer, Frank Schmidt, Gerd Wachsmuth, Susann Mach, Ilka Riedel, Tommy Etling, Felix Ospald (especially for his help with FMG), Jan Blechschmidt and Ailyn Sch¨afer. Not only for the mathematical discussions but also for the social events. This also includes Arnd Meyer, Hansj¨orgSchmidt, Ren´eSchneider, Matthias Pester and Jens R¨uckert. Furthermore, I would like to thank the secretaries Anne-Kristin Glanzberg, Kerstin Seidel and Heike Weichelt as well as the people from the MRZ, Margit Matt, Elke Luschnat, Mathias Meier and Roman Unger. In the last years, I had the opportunity to teach several courses in mathematics and I am very thankful for this experience and the feedback my students gave me. My parents supported me throughout all my life and I am very thankful for all their care and nurture. This includes my dear brother Heiko as well, on whom I always can rely on. Finally, I thank Marcella in my heart of hearts for her constant encouragement, reading the manuscripts carefully and taking up with me in the last months when I was busy writing this thesis. I can only hope that I will make it up to her someday. Abstract This thesis addresses optimal control problems with elasticity for large deformations. A hyperelastic model with a polyconvex energy density is employed to describe the elastic behavior of a body. The two approaches to derive the nonlinear partial differential equation, a balance of forces and an energy minimization, are compared. Besides the conventional volume and boundary loads, two novel internal loads are presented. Furthermore, curvilinear coordinates and a hierarchical plate model can be incorporated into the formulation of the elastic forward problem. The forward problem can be solved with Newton's method, though a globalization technique should be used to avoid divergence of Newton's method. The repeated solution of the Newton system is done by a CG or MinRes method with a multigrid V-cycle as a preconditioner. The optimal control problem consists of the displacement (as the state) and a load (as the control). Besides the standard tracking-type objective, alternative objective functionals are presented for problems where a reasonable desired state cannot be provided. Two methods are proposed to solve the optimal control problem: an all- at-once approach by a Lagrange-Newton method and a reduced formulation by a quasi-Newton method with an inverse limited-memory BFGS update. The algorithms for the solution of the forward problem and the optimal control prob- lem are implemented in the finite-element software FEniCS, with the geometrical multigrid extension FMG. Numerical experiments are performed to demonstrate the mesh independence of the algorithms and both optimization methods. 1 Introduction 1.1 Motivation Elastic deformations are a common phenomenon in nature: for example, large parts of the tissues of animals are elastic, otherwise movements would be impossible or would cause damage due to material fatigue. But also in modern technology, elastic parts are often components of machines and devices, which allow a flexible shape or movement. There are cases where the shape of the deformed elastic part is of interest, and the question arises how a certain shape can be attained. We consider the case where an exterior or interior load acts on the body and we try to find a load that deforms the body in a desired way. This will be formulated as an optimal control problem. As we seek a noticeable change in the shape of the body, we expect the deformation to be large, which requires an elasticity model for large (non-infinitesimal) defor- mations. These models are inherently nonlinear and challenging to solve. In this thesis, we focus on polyconvex hyperelasticity models, which can be formulated as an energy minimization problem. This will constitute our forward problem: the computation of the deformation of an elastic body for a given load. This forward problem is the basis for the optimal control problem, where we seek to find a load such that the deformation is close to the desired one. One possible application in medicine of implant design is investigated by Lars Lubkoll, Anton Schiela and Martin Weiser, see Lubkoll et al.(2012). Consider a patient who lost a part of his jaw bone along with the skin due to an accident. The bone could be replaced by an implant that is reconstructed from the undamaged part of the intact jaw bone. The skin can be transplanted from another part of the body, but the thickness and elastic behavior is likely to be different than that of the original facial skin. As a consequence, the facial reconstruction might be unsatisfactory to the patient as symmetry in facial structures is a key aspect to appearance. In order to improve this procedure, the implant could be modeled in such a way that the resulting facial structure matches the undamaged right half of the jaw. The authors have shown that the optimal control problem with a contact between the rigid implant and the elastic skin can be reformulated as an optimal control problem with a boundary load. In this setting, the deformation of the skin is the state and the boundary load on the inside is the control. Other applications can be found in industrial manufacturing. 2 1 Introduction The elasticity model which is used in this thesis is fundamental to extensions like elasto-plastic material models for large deformations. Several mathematicians in Germany are working on the optimal control of elasto-plasticity. The Sonder- forschungsbereich (SFB) 666: Integral Sheet Metal Design with Higher Order Bifur- cations has a project on optimal control of deep-drawing processes, investigated by Stefan Ulbrich and Daniela Bratzke from the TU Darmstadt. Furthermore, Michael Stingl and Stefan Werner from the Friedrich-Alexander University Erlangen-N¨urn- berg worked on the optimal control of hot rolling processes, in which friction plays an important role.
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