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Numerics of Unilateral Contacts and Friction - Modeling and Numerical Time Integration in Non-Smooth Dynamics
of: Christian Studer
Springer-Verlag, 2009
ISBN: 9783642011009 , 182 Pages
Format: PDF
Copy protection: DRM
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Numerics of Unilateral Contacts and Friction - Modeling and Numerical Time Integration in Non-Smooth Dynamics
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
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