Studer Numerics of Unilateral Contacts and Friction

Modeling and Numerical Time Integration in Non-Smooth Dynamics
1. Auflage 2009
ISBN: 978-3-642-01100-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 47, 182 Seiten

Reihe: Lecture Notes in Applied and Computational Mechanics

Mechanics provides the link between mathematics and practical engineering app- cations. It is one of the oldest sciences, and many famous scientists have left and will leave their mark in this fascinating ?eld of research. Perhaps one of the most prominentscientists in mechanics was Sir Isaac Newton, who with his 'laws of - tion' initiated the description of mechanical systems by differential equations. And still today, more than 300 years after Newton, this mathematical concept is more actual than ever. The rising computer power and the development of numerical solvers for diff- ential equations allowed engineersall over the world to predict the behavior of their physical systems fast and easy in an numerical way. And the trend to computational simulation methods is still further increasing, not only in mechanics, but practically in all branches of science. Numerical simulation will probablynot solve the world's engineering problems, but it will help for a better understanding of the mechanisms of our models.

Autoren/Hrsg.

Studer, Christian

Inhaltsverzeichnis

1;Title Page;2
2;Preface;6
3;Contents;8
4;Introduction;11
4.1;Aim and Scope;11
4.2;Literature Survey;14
4.2.1;Modeling of Unilateral Contacts and Friction;14
4.2.2;Time Discretization;15
4.2.3;Solvers for Inequality Problems;16
4.2.4;Application;17
4.3;Outline;17
5;Mathematical Preliminaries;19
5.1;Used Norms;19
5.2;Derivatives;20
5.3;Convex Analysis;20
5.4;Global and Local Representations;24
5.5;Differential Algebraic Equations;25
6;Non-Smooth Mechanics;27
6.1;Equations of Motion in the Smooth Case;27
6.2;Non-Impulsive Motion;29
6.2.1;Set-Valued Force Laws;29
6.2.2;Examples of Set-Valued Force Laws;31
6.2.3;Equations of Motion in the Non-Smooth Case;34
6.3;Impacts;36
6.4;Equality of Measures;37
7;Inclusion Problems;41
7.1;Linear Complementarity Problems;42
7.2;Augmented Lagrangian Approach;42
7.2.1;General Problem;43
7.2.2;Application to Non-Smooth Dynamics;45
7.2.3;Iterative Solution Process;46
7.2.4;Choice of ri and Convergence;49
7.2.5;Use of Non-Diagonal R’s;54
7.2.6;Successive Solution of the Set-Valued Force Laws;58
7.2.7;Examples;59
7.3;Alternative Approaches;63
7.3.1;Exact Regularization;65
7.4;Summary;67
8;Time-Stepping;69
8.1;Discretization of Differential Algebraic Equations;69
8.1.1;Index-1, Acceleration Level;71
8.1.2;Index-2, Velocity Level;72
8.1.3;Index-3, Displacement Level;74
8.1.4;Linearization of the Gap Function and Drift Stabilization;78
8.1.5;Conclusions;79
8.2;Time Evolution of Non-smooth Systems;81
8.2.1;Event-Driven Schemes;83
8.3;Time-Stepping Methods;84
8.3.1;General Discretization Technique;85
8.3.2;Moreau’s Midpoint Rule;86
8.3.3;The ModifiedT-Method;89
8.3.4;Time-Stepping by Paoli and Schatzman;90
8.3.5;Time-Stepping by Stiegelmeyr, Funk, Foerg, Pfeiffer et al.;91
8.3.6;Time-Stepping by Anitescu, Potra, Stewart, Trinkle et al.;92
8.3.7;Time-Stepping by GGL;98
8.3.8;Time-Stepping by Preconditioning;100
8.3.9;Discussion and Conclusions;102
9;Augmented Time-Stepping by Step Size Adjustement and Extrapolation;108
9.1;Integration Order of Moreau’s Midpoint Rule;108
9.1.1;Expansion of the Exact Solution q(t) and u(t);111
9.1.2;Expansion of Moreau’s Time-Stepping Scheme;114
9.1.3;Local Integration Order;116
9.1.4;Global Integration Order;117
9.2;Step Size Adjustment for Switching Points;118
9.2.1;Determining the Discrete State s;119
9.2.2;Localizing Switching Points;121
9.3;Higher Order Integration for Smooth Time Steps;123
9.3.1;Extrapolation;124
9.4;Overall Algorithm;126
9.4.1;Integration Order;126
9.4.2;Implementation;128
9.5;Examples;129
9.5.1;Point Mass Falling on a Table;129
9.5.2;Point Mass Sliding on a Table;129
9.5.3;Single DOF Impact Oscillator;132
9.5.4;The Woodpecker Toy;134
10;The dynamY Software;137
10.1;Overview;137
10.2;Modeling a Partial Mechanical System;143
10.3;Modeling a Non-Smooth Element;144
10.4;Modeling External Forces;146
10.5;One Step and Simulation Objects;147
10.6;Examples;147
10.6.1;Bodies in 3D Space;148
10.6.2;Non-Common Set-Valued Laws;155
10.6.3;Granular Media;162
10.6.4;Motorbike;163
10.6.5;Elevator;164
11;Summary;168
12;Appendix A;172
13;Appendix B;175
14;References;177
15;List of Figures;182
16;Index;184