Cover image for Numerical methods for nonsmooth dynamical systems : applications in mechanics and electronics
Title:
Numerical methods for nonsmooth dynamical systems : applications in mechanics and electronics
Personal Author:
Series:
Lecture notes in applied and computational mechanics ; 35
Publication Information:
Berlin : Springer, 2008
Physical Description:
xx, 525 p. : ill ; 24 cm.
ISBN:
9783540753919

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30000010167374 TJ213 A22 2008 Open Access Book Book
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Summary

Summary

This book concerns the numerical simulation of dynamical systems whose trajec- ries may not be differentiable everywhere. They are named nonsmooth dynamical systems. They make an important class of systems, rst because of the many app- cations in which nonsmooth models are useful, secondly because they give rise to new problems in various elds of science. Usually nonsmooth dynamical systems are represented as differential inclusions, complementarity systems, evolution va- ational inequalities, each of these classes itself being split into several subclasses. The book is divided into four parts, the rst three parts being sketched in Fig. 0. 1. The aim of the rst part is to present the main tools from mechanics and applied mathematics which are necessary to understand how nonsmooth dynamical systems may be numerically simulated in a reliable way. Many examples illustrate the th- retical results, and an emphasis is put on mechanical systems, as well as on electrical circuits (the so-called Filippov's systems are also examined in some detail, due to their importance in control applications). The second and third parts are dedicated to a detailed presentation of the numerical schemes. A fourth part is devoted to the presentation of the software platform Siconos. This book is not a textbook on - merical analysis of nonsmooth systems, in the sense that despite the main results of numerical analysis (convergence, order of consistency, etc. ) being presented, their proofs are not provided.


Table of Contents

1 Nonsmooth Dynamical Systems: Motivating Examples and Basic Conceptsp. 1
1.1 Electrical Circuits with Ideal Diodesp. 1
1.1.1 Mathematical Modeling Issuesp. 2
1.1.2 Four Nonsmooth Electrical Circuitsp. 5
1.1.3 Continuous System (Ordinary Differential Equation)p. 7
1.1.4 Hints on the Numerical Simulation of Circuits (a) and (b)p. 9
1.1.5 Unilateral Differential Inclusionp. 12
1.1.6 Hints on the Numerical Simulation of Circuits (c) and (d)p. 14
1.1.7 Calculation of the Equilibrium Pointsp. 19
1.2 Electrical Circuits with Ideal Zener Diodesp. 21
1.2.1 The Zener Diodep. 21
1.2.2 The Dynamics of a Simple Circuitp. 23
1.2.3 Numerical Simulation by Means of Time-Stepping Schemesp. 28
1.2.4 Numerical Simulation by Means of Event-Driven Schemesp. 38
1.2.5 Conclusionsp. 40
1.3 Mechanical Systems with Coulomb Frictionp. 40
1.4 Mechanical Systems with Impacts: The Bouncing Ball Paradigmp. 41
1.4.1 The Dynamicsp. 41
1.4.2 A Measure Differential Inclusionp. 44
1.4.3 Hints on the Numerical Simulation of the Bouncing Ballp. 45
1.5 Stiff ODEs, Explicit and Implicit Methods, and the Sweeping Processp. 50
1.5.1 Discretization of the Penalized Systemp. 51
1.5.2 The Switching Conditionsp. 52
1.5.3 Discretization of the Relative Degree Two Complementarity Systemp. 53
1.6 Summary of the Main Ideasp. 53
Part I Formulations of Nonsmooth Dynamical Systems
2 Nonsmooth Dynamical Systems: A Short Zoologyp. 57
2.1 Differential Inclusionsp. 57
2.1.1 Lipschitzian Differential Inclusionsp. 58
2.1.2 Upper Semi-continuous DIs and Discontinuous Differential Equationsp. 61
2.1.3 The One-Sided Lipschitz Conditionp. 68
2.1.4 Recapitulation of the Main Properties of DIsp. 71
2.1.5 Some Hints About Uniqueness of Solutionsp. 73
2.2 Moreau's Sweeping Process and Unilateral DIsp. 74
2.2.1 Moreau's Sweeping Processp. 74
2.2.2 Unilateral DIs and Maximal Monotone Operatorsp. 77
2.2.3 Equivalence Between UDIs and other Formalismsp. 78
2.3 Evolution Variational Inequalitiesp. 80
2.4 Differential Variational Inequalitiesp. 82
2.5 Projected Dynamical Systemsp. 84
2.6 Dynamical Complementarity Systemsp. 85
2.6.1 Generalitiesp. 85
2.6.2 Nonlinear Complementarity Systemsp. 88
2.7 Second-Order Moreau's Sweeping Processp. 88
2.8 ODE with Discontinuitiesp. 92
2.8.1 Order of Discontinuityp. 92
2.8.2 Transversality Conditionsp. 93
2.8.3 Piecewise Affine and Piecewise Continuous Systemsp. 94
2.9 Switched Systemsp. 98
2.10 Impulsive Differential Equationsp. 100
2.10.1 Generalities and Well-Posednessp. 100
2.10.2 An Aside to Time-Discretization and Approximationp. 104
2.11 Summaryp. 104
3 Mechanical Systems with Unilateral Constraints and Frictionp. 107
3.1 Multibody Dynamics: The Lagrangian Formalismp. 107
3.1.1 Perfect Bilateral Constraintsp. 109
3.1.2 Perfect Unilateral Constraintsp. 110
3.1.3 Smooth Dynamics as an Inclusionp. 112
3.2 The Newton-Euler Formalismp. 112
3.2.1 Kinematicsp. 112
3.2.2 Kineticsp. 115
3.2.3 Dynamicsp. 117
3.3 Local Kinematics at the Contact Pointsp. 123
3.3.1 Local Variables at Contact Pointsp. 123
3.3.2 Back to Newton-Euler's Equationsp. 126
3.3.3 Collision Detection and the Gap Function Calculationp. 128
3.4 The Smooth Dynamics of Continuum Mediap. 131
3.4.1 The Smooth Equations of Motionp. 131
3.4.2 Summary of the Equations of Motionp. 135
3.5 Nonsmooth Dynamics and Schatzman's Formulationp. 135
3.6 Nonsmooth Dynamics and Moreau's Sweeping Processp. 137
3.6.1 Measure Differential Inclusionsp. 137
3.6.2 Decomposition of the Nonsmooth Dynamicsp. 137
3.6.3 The Impact Equations and the Smooth Dynamicsp. 138
3.6.4 Moreau's Sweeping Processp. 139
3.6.5 Finitely Represented C and the Complementarity Formulationp. 141
3.7 Well-Posedness Resultsp. 143
3.8 Lagrangian Systems with Perfect Unilateral Constraints: Summaryp. 143
3.9 Contact Modelsp. 144
3.9.1 Coulomb's Frictionp. 145
3.9.2 De Saxce's Bipotential Functionp. 148
3.9.3 Impact with Frictionp. 151
3.9.4 Enhanced Contact Modelsp. 153
3.10 Lagrangian Systems with Frictional Unilateral Constraints and Newton's Impact Laws: Summaryp. 161
3.11 A Mechanical Filippov's Systemp. 162
4 Complementarity Systemsp. 165
4.1 Definitionsp. 165
4.2 Existence and Uniqueness of Solutionsp. 167
4.2.1 Passive LCSp. 168
4.2.2 Examples of LCSp. 169
4.2.3 Complementarity Systems and the Sweeping Processp. 170
4.2.4 Nonlinear Complementarity Systemsp. 172
4.3 Relative Degree and the Completeness of the Formulationp. 173
4.3.1 The Single Input/Single Output (SISO) Casep. 174
4.3.2 The Multiple Input/Multiple Output (MIMO) Casep. 175
4.3.3 The Solutions and the Relative Degreep. 175
5 Higher Order Constrained Dynamical Systemsp. 177
5.1 Motivationsp. 177
5.2 A Canonical State Space Representationp. 178
5.3 The Space of Solutionsp. 180
5.4 The Distribution DI and Its Propertiesp. 180
5.4.1 Introductionp. 180
5.4.2 The Inclusions for the Measures v[subscript i]p. 182
5.4.3 Two Formalisms for the HOSPp. 183
5.4.4 Some Qualitative Propertiesp. 186
5.5 Well-Posedness of the HOSPp. 187
5.6 Summary of the Main Ideas of Chapters 4 and 5p. 188
6 Specific Features of Nonsmooth Dynamical Systemsp. 189
6.1 Discontinuity with Respect to Initial Conditionsp. 189
6.1.1 Impact in a Cornerp. 189
6.1.2 A Theoretical Resultp. 190
6.1.3 A Physical Examplep. 191
6.2 Frictional Paroxysms (the Painleve Paradoxes)p. 192
6.3 Infinity of Events in a Finite Timep. 193
6.3.1 Accumulations of Impactsp. 193
6.3.2 Infinitely Many Switchings in Filippov's Inclusionsp. 194
6.3.3 Limit of the Saw-Tooth Function in Filippov's Systemsp. 194
Part II Time Integration of Nonsmooth Dynamical Systems
7 Event-Driven Schemes for Inclusions with AC Solutionsp. 203
7.1 Filippov's Inclusionsp. 203
7.1.1 Introductionp. 203
7.1.2 Stewart's Methodp. 205
7.1.3 Why Is Stewart's Method Superior to Trivial Event-Driven Schemes?p. 213
7.2 ODEs with Discontinuities with a Transversality Conditionp. 215
7.2.1 Position of the Problemp. 215
7.2.2 Event-Driven Schemesp. 215
8 Event-Driven Schemes for Lagrangian Systemsp. 219
8.1 Introductionp. 219
8.2 The Smooth Dynamics and the Impact Equationsp. 221
8.3 Reformulations of the Unilateral Constraints at Different Kinematics Levelsp. 222
8.3.1 At the Position Levelp. 222
8.3.2 At the Velocity Levelp. 222
8.3.3 At the Acceleration Levelp. 223
8.3.4 The Smooth Dynamicsp. 224
8.4 The Case of a Single Contactp. 225
8.4.1 Commentsp. 227
8.5 The Multi-contact Case and the Index Setsp. 229
8.5.1 Index Setsp. 229
8.6 Comments and Extensionsp. 230
8.6.1 Event-Driven Algorithms and Switching Diagramsp. 230
8.6.2 Coulomb's Friction and Enhanced Set-Valued Force Lawsp. 231
8.6.3 Bilateral or Unilateral Dynamics?p. 232
8.6.4 Event-Driven Schemes: Lotstedt's Algorithmp. 232
8.6.5 Consistency and Order of Event-Driven Algorithmsp. 236
8.7 Linear Complementarity Systemsp. 240
8.8 Some Resultsp. 241
9 Time-Stepping Schemes for Systems with AC Solutionsp. 243
9.1 ODEs with Discontinuitiesp. 243
9.1.1 Numerical Illustrations of Expected Troublesp. 243
9.1.2 Consistent Time-Stepping Methodsp. 247
9.2 DIs with Absolutely Continuous Solutionsp. 251
9.2.1 Explicit Euler Algorithmp. 252
9.2.2 Implicit [theta]-Methodp. 256
9.2.3 Multistep and Runge-Kutta Algorithmsp. 258
9.2.4 Computational Results and Commentsp. 263
9.3 The Special Case of the Filippov's Inclusionsp. 266
9.3.1 Smoothing Methodsp. 266
9.3.2 Switched Model and Explicit Schemesp. 267
9.3.3 Implicit Schemes and Complementarity Formulationp. 269
9.3.4 Commentsp. 271
9.4 Moreau's Catching-Up Algorithm for the First-Order Sweeping Processp. 271
9.4.1 Mathematical Propertiesp. 272
9.4.2 Practical Implementation of the Catching-up Algorithmp. 273
9.4.3 Time-Independent Convex Set Kp. 274
9.5 Linear Complementarity Systems with r [less than or equal] 1p. 275
9.6 Differential Variational Inequalitiesp. 279
9.6.1 The Initial Value Problem (IVP)p. 280
9.6.2 The Boundary Value Problemp. 281
9.7 Summary of the Main Ideasp. 283
10 Time-Stepping Schemes for Mechanical Systemsp. 285
10.1 The Nonsmooth Contact Dynamics (NSCD) Methodp. 285
10.1.1 The Linear Time-Invariant Nonsmooth Lagrangian Dynamicsp. 286
10.1.2 The Nonlinear Nonsmooth Lagrangian Dynamicsp. 289
10.1.3 Discretization of Moreau's Inclusionp. 293
10.1.4 Sweeping Process with Frictionp. 295
10.1.5 The One-Step Time-Discretized Nonsmooth Problemp. 296
10.1.6 Convergence Propertiesp. 303
10.1.7 Bilateral and Unilateral Constraintsp. 305
10.2 Some Numerical Illustrations of the NSCD Methodp. 307
10.2.1 Granular Materialp. 307
10.2.2 Deep Drawingp. 309
10.2.3 Tensegrity Structuresp. 309
10.2.4 Masonry Structuresp. 309
10.2.5 Real-Time and Virtual Reality Simulationsp. 311
10.2.6 More Applicationsp. 314
10.2.7 Moreau's Time-Stepping Method and Painleve Paradoxesp. 315
10.3 Variants and Other Time-Stepping Schemesp. 315
10.3.1 The Paoli-Schatzman Schemep. 315
10.3.2 The Stewart-Trinkle-Anitescu-Potra Schemep. 317
11 Time-Stepping Scheme for the HOSPp. 319
11.1 Insufficiency of the Backward Euler Methodp. 319
11.2 Time-Discretization of the HOSPp. 321
11.2.1 Principle of the Discretizationp. 321
11.2.2 Properties of the Discrete-Time Extended Sweeping Processp. 322
11.2.3 Numerical Examplesp. 324
11.3 Synoptic Outline of the Algorithmsp. 325
Part III Numerical Methods for the One-Step Nonsmooth Problems
12 Basics on Mathematical Programming Theoryp. 331
12.1 Introductionp. 331
12.2 The Quadratic Program (QP)p. 331
12.2.1 Definition and Basic Propertiesp. 331
12.2.2 Equality-Constrained QPp. 335
12.2.3 Inequality-Constrained QPp. 338
12.2.4 Comments on Numerical Methods for QPp. 344
12.3 Constrained Nonlinear Programming (NLP)p. 345
12.3.1 Definition and Basic Propertiesp. 345
12.3.2 Main Methods to Solve NLPsp. 347
12.4 The Linear Complementarity Problem (LCP)p. 351
12.4.1 Definition of the Standard Formp. 351
12.4.2 Some Mathematical Propertiesp. 352
12.4.3 Variants of the LCPp. 355
12.4.4 Relation Between the Variants of the LCPsp. 357
12.4.5 Links Between the LCP and the QPp. 359
12.4.6 Splitting-Based Methodsp. 363
12.4.7 Pivoting-Based Methodsp. 367
12.4.8 Interior Point Methodsp. 374
12.4.9 How to chose a LCP solver?p. 379
12.5 The Nonlinear Complementarity Problem (NCP)p. 379
12.5.1 Definition and Basic Propertiesp. 379
12.5.2 The Mixed Complementarity Problem (MCP)p. 383
12.5.3 Newton-Josephy's and Linearization Methodsp. 384
12.5.4 Generalized or Semismooth Newton's Methodsp. 385
12.5.5 Interior Point Methodsp. 388
12.5.6 Effective Implementations and Comparison of the Numerical Methods for NCPsp. 388
12.6 Variational and Quasi-Variational Inequalitiesp. 389
12.6.1 Definition and Basic Propertiesp. 389
12.6.2 Links with the Complementarity Problemsp. 390
12.6.3 Links with the Constrained Minimization Problemp. 391
12.6.4 Merit and Gap Functions for VIp. 392
12.6.5 Nonsmooth and Generalized equationsp. 396
12.6.6 Main Types of Algorithms for the VI and QVIp. 398
12.6.7 Projection-Type and Splitting Methodsp. 398
12.6.8 Minimization of Merit Functionsp. 400
12.6.9 Generalized Newton Methodsp. 401
12.6.10 Interest from a Computational Point of Viewp. 401
12.7 Summary of the Main Ideasp. 401
13 Numerical Methods for the Frictional Contact Problemp. 403
13.1 Introductionp. 403
13.2 Summary of the Time-Discretized Equationsp. 403
13.2.1 The Index Set of Forecast Active Constraintsp. 403
13.2.2 Summary of the OSNSPsp. 405
13.3 Formulations and Resolutions in LCP Formsp. 407
13.3.1 The Frictionless Case with Newton's Impact Lawp. 407
13.3.2 The Frictionless Case with Newton's Impact and Linear Perfect Bilateral Constraintsp. 408
13.3.3 Two-Dimensional Frictional Case as an LCPp. 409
13.3.4 Outer Faceting of the Coulomb's Conep. 410
13.3.5 Inner Faceting of the Coulomb's Conep. 414
13.3.6 Commentsp. 417
13.3.7 Weakness of the Faceting Processp. 418
13.4 Formulation and Resolution in a Standard NCP Formp. 419
13.4.1 The Frictionless Casep. 419
13.4.2 A Direct MCP for the 3D Frictional Contactp. 419
13.4.3 A Clever Formulation of the 3D Frictional Contact as an NCPp. 420
13.5 Formulation and Resolution in QP and NLP Formsp. 422
13.5.1 The Frictionless Casep. 422
13.5.2 Minimization Principles and Coulomb's Frictionp. 423
13.6 Formulations and Resolution as Nonsmooth Equationsp. 424
13.6.1 Alart and Curnier's Formulation and Generalized Newton's Methodp. 424
13.6.2 Variants and Line-Search Procedurep. 429
13.6.3 Other Direct Equation-Based Reformulationsp. 430
13.7 Formulation and Resolution as VI/CPp. 432
13.7.1 VI/CP Reformulationp. 432
13.7.2 Projection-type Methodsp. 433
13.7.3 Fixed-Point Iterations on the Friction Threshold and Ad Hoc Projection Methodsp. 434
13.7.4 A Clever Block Splitting: the Nonsmooth Gauss-Seidel (NSGS) Approachp. 437
13.7.5 Newton's Method for VIp. 440
Part IV The SICONOS Software: Implementation and Examples
14 The SICONOS Platformp. 443
14.1 Introductionp. 443
14.2 An Insight into Siconosp. 443
14.2.1 Step 1. Building a Nonsmooth Dynamical Systemp. 444
14.2.2 Step 2. Simulation Strategy Definitionp. 447
14.3 Siconos Softwarep. 448
14.3.1 General Principles of Modeling and Simulationp. 448
14.3.2 NSDS-Related Componentsp. 451
14.3.3 Simulation-Related Componentsp. 456
14.3.4 Siconos Software Designp. 457
14.4 Examplesp. 460
14.4.1 The Bouncing Ball(s)p. 460
14.4.2 The Woodpecker Toyp. 463
14.4.3 MOS Transistors and Invertersp. 464
14.4.4 Control of Lagrangian systemsp. 466
A Convex, Nonsmooth, and Set-Valued Analysisp. 475
A.1 Set-Valued Analysisp. 475
A.2 Subdifferentiationp. 475
A.3 Some Useful Equivalencesp. 476
B Some Results of Complementarity Theoryp. 479
C Some Facts in Real Analysisp. 481
C.1 Functions of Bounded Variations in Timep. 481
C.2 Multifunctions of Bounded Variation in Timep. 482
C.3 Distributions Generated by RCLSBV Functionsp. 483
C.4 Differential Measuresp. 486
C.5 Bohl's Distributionsp. 487
C.6 Some Useful Resultsp. 487
Referencesp. 489
Indexp. 519