Numerics of Unilateral Contacts and Friction (eBook)

Title Page 2
Preface 6
Contents 8
Introduction 11
Aim and Scope 11
Literature Survey 14
Modeling of Unilateral Contacts and Friction 14
Time Discretization 15
Solvers for Inequality Problems 16
Application 17
Outline 17
Mathematical Preliminaries 19
Used Norms 19
Derivatives 20
Convex Analysis 20
Global and Local Representations 24
Differential Algebraic Equations 25
Non-Smooth Mechanics 27
Equations of Motion in the Smooth Case 27
Non-Impulsive Motion 29
Set-Valued Force Laws 29
Examples of Set-Valued Force Laws 31
Equations of Motion in the Non-Smooth Case 34
Impacts 36
Equality of Measures 37
Inclusion Problems 41
Linear Complementarity Problems 42
Augmented Lagrangian Approach 42
General Problem 43
Application to Non-Smooth Dynamics 45
Iterative Solution Process 46
Choice of ri and Convergence 49
Use of Non-Diagonal R’s 54
Successive Solution of the Set-Valued Force Laws 58
Examples 59
Alternative Approaches 63
Exact Regularization 65
Summary 67
Time-Stepping 69
Discretization of Differential Algebraic Equations 69
Index-1, Acceleration Level 71
Index-2, Velocity Level 72
Index-3, Displacement Level 74
Linearization of the Gap Function and Drift Stabilization 78
Conclusions 79
Time Evolution of Non-smooth Systems 81
Event-Driven Schemes 83
Time-Stepping Methods 84
General Discretization Technique 85
Moreau’s Midpoint Rule 86
The ModifiedT-Method 89
Time-Stepping by Paoli and Schatzman 90
Time-Stepping by Stiegelmeyr, Funk, Foerg, Pfeiffer et al. 91
Time-Stepping by Anitescu, Potra, Stewart, Trinkle et al. 92
Time-Stepping by GGL 98
Time-Stepping by Preconditioning 100
Discussion and Conclusions 102
Augmented Time-Stepping by Step Size Adjustement and Extrapolation 108
Integration Order of Moreau’s Midpoint Rule 108
Expansion of the Exact Solution q(t) and u(t) 111
Expansion of Moreau’s Time-Stepping Scheme 114
Local Integration Order 116
Global Integration Order 117
Step Size Adjustment for Switching Points 118
Determining the Discrete State s 119
Localizing Switching Points 121
Higher Order Integration for Smooth Time Steps 123
Extrapolation 124
Overall Algorithm 126
Integration Order 126
Implementation 128
Examples 129
Point Mass Falling on a Table 129
Point Mass Sliding on a Table 129
Single DOF Impact Oscillator 132
The Woodpecker Toy 134
The dynamY Software 137
Overview 137
Modeling a Partial Mechanical System 143
Modeling a Non-Smooth Element 144
Modeling External Forces 146
One Step and Simulation Objects 147
Examples 147
Bodies in 3D Space 148
Non-Common Set-Valued Laws 155
Granular Media 162
Motorbike 163
Elevator 164
Summary 168
Appendix A 172
Appendix B 175
References 177
List of Figures 182
Index 184