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Digitale Dissertation

Rolf H. Krause :
Monotone Multigrid Methods for Signorini's Problem with Friction
Monotone Mehrgitterverfahren fuer Signorinis Problem mit Reibung

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Abstract

In this work, we consider the numerical simulation of contact problems. Since the numerical realization of contact problems is of high importance in many application areas, there is a strong demand for fast and reliable simulation method. We introduce and analyze a new nonlinear multigrid method for solving contact problems with and without friction. As it turns out, by means of our new method nonlinear contact problems can be solved with a computational amount comparable that of linear problems. In particular, in our numerical experiments we observe our method to be of optimal complexity. Moreover, since we do not use any regularization techniques, the computed discrete boundary stresses as well as the computed displacements turn out to be highly accurate. The new method is based on the succesive minimization of the associated energy functional in direction of properly choosen functions. We show the global convergence of our method and give several numerical examples in two and three space dimensions, illustrating the robustness and the performance of the method. In addition to the theoretical analysis, the method has been implemented in an object oriented way. We explain the concepts of our implementation and show the flexibility of our approach by deriving a nonlinear algebraic multigrid method. To include frictional effects, we use a discrete fixed point iteration. As a faster alternative, also a Gauss-Seidel like iteration scheme is proposed. Both methods are compared in numerical examples. The resulting nonlinear algorithm turns out to be fast and reliable. Finally, we consider the case of contact between elastic bodies. Here, the information transfer at the interface is realized by means of non conforming domain decomposition methods (mortar methods). This gives rise to a non-linear Dirichlet Neumann Algorithm.

Table of Contents

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Title 4
Table of contents 4
Introduction 4
1. Concepts of Linear Elasticity 7
1.1 Kinematics and Strain 7
1.2 Stress and the Equilibrium conditions 9
1.3 Constitutive Equations 12
1.4 The Equations of Linear Elasticity 15
1.5 Hyperelastic Materials 16
2. Signorini's Problem 20
2.1 Linearized Contact 20
2.2 Strong Formulation 22
2.3 Weak formulation 24
2.4 Discretization and Error Estimates 29
3. Monotone Multigrid Methods 31
3.1 Minimization of Energy 33
3.2 Truncated Coarse Grid Functions 40
3.3 Monotone Restrictions 44
3.4 Algebraic Formulation 47
4. Software Concept and Implementation 51
4.1 The Obstacle Problem Class 53
4.2 Basic User Interface 58
4.3 Abstract Nonlinear Gauss--Seidel 60
4.4 Modified Restriction 61
4.5 Fast Mofification of the Coarse Grid Matrices 61
4.6 An Application of the Concept: Algebraic Multigrid 64
5. Numerical Results 67
5.1 Hertzian Contact Problem 68
5.2 An Unphysical Example 71
5.3 Elastic Cylinder and Two Rigid Rods 73
5.4 Comparison with Standard Multigrid 77
5.5 Influence of the Start Iterate 79
5.6 Truncated Nodal Basis versus Standard Nodal Basis 82
5.7 Performance of the Parallel Monotone Multigrid Method 85
6. Frictional Contact Problems 88
6.1 Weak Formulation 89
6.2 Fixed Point Iteration 91
6.3 Numerical Results 100
7. Elastic Contact 105
7.1 Nonlinear Dirichlet-Neumann Algorithm 109
7.2 Numerical Results 110
7.3 Elastic Contact with Coulomb Friction 113
7.4 Numerical Results with Coulomb Friction 115
References
List of Figures 118
List of Tables 119
References 119

More Information:

Online available: http://www.diss.fu-berlin.de/2001/240/indexe.html
Language of PhDThesis: english
Keywords: multigrid methods,contact problems, linear elasticity,friction, Signorini's problem
DNB-Sachgruppe: 27 Mathematik
Classification MSC: 65N30, 65N55,73T05
Date of disputation: 18-Jul-2001
PhDThesis from: Fachbereich Mathematik u. Informatik, Freie Universität Berlin
First Referee: Prof. Dr. Ralf Kornhuber
Second Referee: Prof. Dr. Ronald W. Hoppe
Contact (Author): krause@math.fu-berlin.de
Contact (Advisor): kornhube@math.fu-berlin.de
Date created:28-Nov-2001
Date available:04-Dec-2001

 

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