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Numerical Methods for Nonsmooth Dynamical Systems : Applications in Mechanics and Electronics - Vincent Acary

Numerical Methods for Nonsmooth Dynamical Systems

Applications in Mechanics and Electronics


Published: 19th February 2008
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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.

From the reviews:

"This monograph is a valuable and concise contribution to the numerics of nonsmooth ODEs ... . It is clearly designed for engineering applications in computational mechanics, and the main concern is the discussion of the construction of algorithms and their properties ... . written in a formal mathematical style with precise definitions, followed by results stated as Lemma, Proposition, or Theorem (and references to the literature for the proofs). This makes the overall work accessible for both, researchers in mechanics and in mathematics." (Christian Wieners, Zentralblatt fur Angewandte Mathemtik und Mechanik, Vol. 88 (7), 2008)

"The present book is a research monograph with numerous information and references to original publications which gives to the book an encyclopaedic nature. ... parallel presentation of nonsmooth models in mechanics and electronics indicates that the mentioned effects will also be of interest for people working in mechatronics, microelectromechanics and multiphysics. The book is intended for graduate students and scientists doing research and development in mechanics and electrical engineering, designers of modern electromechanical devices, as well as to researchers from other scientific communities ... ." (Georgios E. Stavroulakis, Zentralblatt MATH, Vol. 1173, 2009)

Nonsmooth Dynamical Systems: Motivating Examples and Basic Conceptsp. 1
Electrical Circuits with Ideal Diodesp. 1
Mathematical Modeling Issuesp. 2
Four Nonsmooth Electrical Circuitsp. 5
Continuous System (Ordinary Differential Equation)p. 7
Hints on the Numerical Simulation of Circuits (a) and (b)p. 9
Unilateral Differential Inclusionp. 12
Hints on the Numerical Simulation of Circuits (c) and (d)p. 14
Calculation of the Equilibrium Pointsp. 19
Electrical Circuits with Ideal Zener Diodesp. 21
The Zener Diodep. 21
The Dynamics of a Simple Circuitp. 23
Numerical Simulation by Means of Time-Stepping Schemesp. 28
Numerical Simulation by Means of Event-Driven Schemesp. 38
Conclusionsp. 40
Mechanical Systems with Coulomb Frictionp. 40
Mechanical Systems with Impacts: The Bouncing Ball Paradigmp. 41
The Dynamicsp. 41
A Measure Differential Inclusionp. 44
Hints on the Numerical Simulation of the Bouncing Ballp. 45
Stiff ODEs, Explicit and Implicit Methods, and the Sweeping Processp. 50
Discretization of the Penalized Systemp. 51
The Switching Conditionsp. 52
Discretization of the Relative Degree Two Complementarity Systemp. 53
Summary of the Main Ideasp. 53
Formulations of Nonsmooth Dynamical Systems
Nonsmooth Dynamical Systems: A Short Zoologyp. 57
Differential Inclusionsp. 57
Lipschitzian Differential Inclusionsp. 58
Upper Semi-continuous DIs and Discontinuous Differential Equationsp. 61
The One-Sided Lipschitz Conditionp. 68
Recapitulation of the Main Properties of DIsp. 71
Some Hints About Uniqueness of Solutionsp. 73
Moreau's Sweeping Process and Unilateral DIsp. 74
Moreau's Sweeping Processp. 74
Unilateral DIs and Maximal Monotone Operatorsp. 77
Equivalence Between UDIs and other Formalismsp. 78
Evolution Variational Inequalitiesp. 80
Differential Variational Inequalitiesp. 82
Projected Dynamical Systemsp. 84
Dynamical Complementarity Systemsp. 85
Generalitiesp. 85
Nonlinear Complementarity Systemsp. 88
Second-Order Moreau's Sweeping Processp. 88
ODE with Discontinuitiesp. 92
Order of Discontinuityp. 92
Transversality Conditionsp. 93
Piecewise Affine and Piecewise Continuous Systemsp. 94
Switched Systemsp. 98
Impulsive Differential Equationsp. 100
Generalities and Well-Posednessp. 100
An Aside to Time-Discretization and Approximationp. 104
Summaryp. 104
Mechanical Systems with Unilateral Constraints and Frictionp. 107
Multibody Dynamics: The Lagrangian Formalismp. 107
Perfect Bilateral Constraintsp. 109
Perfect Unilateral Constraintsp. 110
Smooth Dynamics as an Inclusionp. 112
The Newton-Euler Formalismp. 112
Kinematicsp. 112
Kineticsp. 115
Dynamicsp. 117
Local Kinematics at the Contact Pointsp. 123
Local Variables at Contact Pointsp. 123
Back to Newton-Euler's Equationsp. 126
Collision Detection and the Gap Function Calculationp. 128
The Smooth Dynamics of Continuum Mediap. 131
The Smooth Equations of Motionp. 131
Summary of the Equations of Motionp. 135
Nonsmooth Dynamics and Schatzman's Formulationp. 135
Nonsmooth Dynamics and Moreau's Sweeping Processp. 137
Measure Differential Inclusionsp. 137
Decomposition of the Nonsmooth Dynamicsp. 137
The Impact Equations and the Smooth Dynamicsp. 138
Moreau's Sweeping Processp. 139
Finitely Represented C and the Complementarity Formulationp. 141
Well-Posedness Resultsp. 143
Lagrangian Systems with Perfect Unilateral Constraints: Summaryp. 143
Contact Modelsp. 144
Coulomb's Frictionp. 145
De Saxce's Bipotential Functionp. 148
Impact with Frictionp. 151
Enhanced Contact Modelsp. 153
Lagrangian Systems with Frictional Unilateral Constraints and Newton's Impact Laws: Summaryp. 161
A Mechanical Filippov's Systemp. 162
Complementarity Systemsp. 165
Definitionsp. 165
Existence and Uniqueness of Solutionsp. 167
Passive LCSp. 168
Examples of LCSp. 169
Complementarity Systems and the Sweeping Processp. 170
Nonlinear Complementarity Systemsp. 172
Relative Degree and the Completeness of the Formulationp. 173
The Single Input/Single Output (SISO) Casep. 174
The Multiple Input/Multiple Output (MIMO) Casep. 175
The Solutions and the Relative Degreep. 175
Higher Order Constrained Dynamical Systemsp. 177
Motivationsp. 177
A Canonical State Space Representationp. 178
The Space of Solutionsp. 180
The Distribution DI and Its Propertiesp. 180
Introductionp. 180
The Inclusions for the Measures v[subscript i]p. 182
Two Formalisms for the HOSPp. 183
Some Qualitative Propertiesp. 186
Well-Posedness of the HOSPp. 187
Summary of the Main Ideas of Chapters 4 and 5p. 188
Specific Features of Nonsmooth Dynamical Systemsp. 189
Discontinuity with Respect to Initial Conditionsp. 189
Impact in a Cornerp. 189
A Theoretical Resultp. 190
A Physical Examplep. 191
Frictional Paroxysms (the Painleve Paradoxes)p. 192
Infinity of Events in a Finite Timep. 193
Accumulations of Impactsp. 193
Infinitely Many Switchings in Filippov's Inclusionsp. 194
Limit of the Saw-Tooth Function in Filippov's Systemsp. 194
Time Integration of Nonsmooth Dynamical Systems
Event-Driven Schemes for Inclusions with AC Solutionsp. 203
Filippov's Inclusionsp. 203
Introductionp. 203
Stewart's Methodp. 205
Why Is Stewart's Method Superior to Trivial Event-Driven Schemes?p. 213
ODEs with Discontinuities with a Transversality Conditionp. 215
Position of the Problemp. 215
Event-Driven Schemesp. 215
Event-Driven Schemes for Lagrangian Systemsp. 219
Introductionp. 219
The Smooth Dynamics and the Impact Equationsp. 221
Reformulations of the Unilateral Constraints at Different Kinematics Levelsp. 222
At the Position Levelp. 222
At the Velocity Levelp. 222
At the Acceleration Levelp. 223
The Smooth Dynamicsp. 224
The Case of a Single Contactp. 225
Commentsp. 227
The Multi-contact Case and the Index Setsp. 229
Index Setsp. 229
Comments and Extensionsp. 230
Event-Driven Algorithms and Switching Diagramsp. 230
Coulomb's Friction and Enhanced Set-Valued Force Lawsp. 231
Bilateral or Unilateral Dynamics?p. 232
Event-Driven Schemes: Lotstedt's Algorithmp. 232
Consistency and Order of Event-Driven Algorithmsp. 236
Linear Complementarity Systemsp. 240
Some Resultsp. 241
Time-Stepping Schemes for Systems with AC Solutionsp. 243
ODEs with Discontinuitiesp. 243
Numerical Illustrations of Expected Troublesp. 243
Consistent Time-Stepping Methodsp. 247
DIs with Absolutely Continuous Solutionsp. 251
Explicit Euler Algorithmp. 252
Implicit [theta]-Methodp. 256
Multistep and Runge-Kutta Algorithmsp. 258
Computational Results and Commentsp. 263
The Special Case of the Filippov's Inclusionsp. 266
Smoothing Methodsp. 266
Switched Model and Explicit Schemesp. 267
Implicit Schemes and Complementarity Formulationp. 269
Commentsp. 271
Moreau's Catching-Up Algorithm for the First-Order Sweeping Processp. 271
Mathematical Propertiesp. 272
Practical Implementation of the Catching-up Algorithmp. 273
Time-Independent Convex Set Kp. 274
Linear Complementarity Systems with r [less than or equal] 1p. 275
Differential Variational Inequalitiesp. 279
The Initial Value Problem (IVP)p. 280
The Boundary Value Problemp. 281
Summary of the Main Ideasp. 283
Time-Stepping Schemes for Mechanical Systemsp. 285
The Nonsmooth Contact Dynamics (NSCD) Methodp. 285
The Linear Time-Invariant Nonsmooth Lagrangian Dynamicsp. 286
The Nonlinear Nonsmooth Lagrangian Dynamicsp. 289
Discretization of Moreau's Inclusionp. 293
Sweeping Process with Frictionp. 295
The One-Step Time-Discretized Nonsmooth Problemp. 296
Convergence Propertiesp. 303
Bilateral and Unilateral Constraintsp. 305
Some Numerical Illustrations of the NSCD Methodp. 307
Granular Materialp. 307
Deep Drawingp. 309
Tensegrity Structuresp. 309
Masonry Structuresp. 309
Real-Time and Virtual Reality Simulationsp. 311
More Applicationsp. 314
Moreau's Time-Stepping Method and Painleve Paradoxesp. 315
Variants and Other Time-Stepping Schemesp. 315
The Paoli-Schatzman Schemep. 315
The Stewart-Trinkle-Anitescu-Potra Schemep. 317
Time-Stepping Scheme for the HOSPp. 319
Insufficiency of the Backward Euler Methodp. 319
Time-Discretization of the HOSPp. 321
Principle of the Discretizationp. 321
Properties of the Discrete-Time Extended Sweeping Processp. 322
Numerical Examplesp. 324
Synoptic Outline of the Algorithmsp. 325
Numerical Methods for the One-Step Nonsmooth Problems
Basics on Mathematical Programming Theoryp. 331
Introductionp. 331
The Quadratic Program (QP)p. 331
Definition and Basic Propertiesp. 331
Equality-Constrained QPp. 335
Inequality-Constrained QPp. 338
Comments on Numerical Methods for QPp. 344
Constrained Nonlinear Programming (NLP)p. 345
Definition and Basic Propertiesp. 345
Main Methods to Solve NLPsp. 347
The Linear Complementarity Problem (LCP)p. 351
Definition of the Standard Formp. 351
Some Mathematical Propertiesp. 352
Variants of the LCPp. 355
Relation Between the Variants of the LCPsp. 357
Links Between the LCP and the QPp. 359
Splitting-Based Methodsp. 363
Pivoting-Based Methodsp. 367
Interior Point Methodsp. 374
How to chose a LCP solver?p. 379
The Nonlinear Complementarity Problem (NCP)p. 379
Definition and Basic Propertiesp. 379
The Mixed Complementarity Problem (MCP)p. 383
Newton-Josephy's and Linearization Methodsp. 384
Generalized or Semismooth Newton's Methodsp. 385
Interior Point Methodsp. 388
Effective Implementations and Comparison of the Numerical Methods for NCPsp. 388
Variational and Quasi-Variational Inequalitiesp. 389
Definition and Basic Propertiesp. 389
Links with the Complementarity Problemsp. 390
Links with the Constrained Minimization Problemp. 391
Merit and Gap Functions for VIp. 392
Nonsmooth and Generalized equationsp. 396
Main Types of Algorithms for the VI and QVIp. 398
Projection-Type and Splitting Methodsp. 398
Minimization of Merit Functionsp. 400
Generalized Newton Methodsp. 401
Interest from a Computational Point of Viewp. 401
Summary of the Main Ideasp. 401
Numerical Methods for the Frictional Contact Problemp. 403
Introductionp. 403
Summary of the Time-Discretized Equationsp. 403
The Index Set of Forecast Active Constraintsp. 403
Summary of the OSNSPsp. 405
Formulations and Resolutions in LCP Formsp. 407
The Frictionless Case with Newton's Impact Lawp. 407
The Frictionless Case with Newton's Impact and Linear Perfect Bilateral Constraintsp. 408
Two-Dimensional Frictional Case as an LCPp. 409
Outer Faceting of the Coulomb's Conep. 410
Inner Faceting of the Coulomb's Conep. 414
Commentsp. 417
Weakness of the Faceting Processp. 418
Formulation and Resolution in a Standard NCP Formp. 419
The Frictionless Casep. 419
A Direct MCP for the 3D Frictional Contactp. 419
A Clever Formulation of the 3D Frictional Contact as an NCPp. 420
Formulation and Resolution in QP and NLP Formsp. 422
The Frictionless Casep. 422
Minimization Principles and Coulomb's Frictionp. 423
Formulations and Resolution as Nonsmooth Equationsp. 424
Alart and Curnier's Formulation and Generalized Newton's Methodp. 424
Variants and Line-Search Procedurep. 429
Other Direct Equation-Based Reformulationsp. 430
Formulation and Resolution as VI/CPp. 432
VI/CP Reformulationp. 432
Projection-type Methodsp. 433
Fixed-Point Iterations on the Friction Threshold and Ad Hoc Projection Methodsp. 434
A Clever Block Splitting: the Nonsmooth Gauss-Seidel (NSGS) Approachp. 437
Newton's Method for VIp. 440
The SICONOS Software: Implementation and Examples
The SICONOS Platformp. 443
Introductionp. 443
An Insight into Siconosp. 443
Step 1. Building a Nonsmooth Dynamical Systemp. 444
Step 2. Simulation Strategy Definitionp. 447
Siconos Softwarep. 448
General Principles of Modeling and Simulationp. 448
NSDS-Related Componentsp. 451
Simulation-Related Componentsp. 456
Siconos Software Designp. 457
Examplesp. 460
The Bouncing Ball(s)p. 460
The Woodpecker Toyp. 463
MOS Transistors and Invertersp. 464
Control of Lagrangian systemsp. 466
Convex, Nonsmooth, and Set-Valued Analysisp. 475
Set-Valued Analysisp. 475
Subdifferentiationp. 475
Some Useful Equivalencesp. 476
Some Results of Complementarity Theoryp. 479
Some Facts in Real Analysisp. 481
Functions of Bounded Variations in Timep. 481
Multifunctions of Bounded Variation in Timep. 482
Distributions Generated by RCLSBV Functionsp. 483
Differential Measuresp. 486
Bohl's Distributionsp. 487
Some Useful Resultsp. 487
Referencesp. 489
Indexp. 519
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783540753919
ISBN-10: 3540753915
Series: Lecture Notes in Applied and Computational Mechanics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 525
Published: 19th February 2008
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.6  x 3.81
Weight (kg): 2.07