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 Concepts | p. 1 |
Electrical Circuits with Ideal Diodes | p. 1 |
Mathematical Modeling Issues | p. 2 |
Four Nonsmooth Electrical Circuits | p. 5 |
Continuous System (Ordinary Differential Equation) | p. 7 |
Hints on the Numerical Simulation of Circuits (a) and (b) | p. 9 |
Unilateral Differential Inclusion | p. 12 |
Hints on the Numerical Simulation of Circuits (c) and (d) | p. 14 |
Calculation of the Equilibrium Points | p. 19 |
Electrical Circuits with Ideal Zener Diodes | p. 21 |
The Zener Diode | p. 21 |
The Dynamics of a Simple Circuit | p. 23 |
Numerical Simulation by Means of Time-Stepping Schemes | p. 28 |
Numerical Simulation by Means of Event-Driven Schemes | p. 38 |
Conclusions | p. 40 |
Mechanical Systems with Coulomb Friction | p. 40 |
Mechanical Systems with Impacts: The Bouncing Ball Paradigm | p. 41 |
The Dynamics | p. 41 |
A Measure Differential Inclusion | p. 44 |
Hints on the Numerical Simulation of the Bouncing Ball | p. 45 |
Stiff ODEs, Explicit and Implicit Methods, and the Sweeping Process | p. 50 |
Discretization of the Penalized System | p. 51 |
The Switching Conditions | p. 52 |
Discretization of the Relative Degree Two Complementarity System | p. 53 |
Summary of the Main Ideas | p. 53 |
Formulations of Nonsmooth Dynamical Systems | |
Nonsmooth Dynamical Systems: A Short Zoology | p. 57 |
Differential Inclusions | p. 57 |
Lipschitzian Differential Inclusions | p. 58 |
Upper Semi-continuous DIs and Discontinuous Differential Equations | p. 61 |
The One-Sided Lipschitz Condition | p. 68 |
Recapitulation of the Main Properties of DIs | p. 71 |
Some Hints About Uniqueness of Solutions | p. 73 |
Moreau's Sweeping Process and Unilateral DIs | p. 74 |
Moreau's Sweeping Process | p. 74 |
Unilateral DIs and Maximal Monotone Operators | p. 77 |
Equivalence Between UDIs and other Formalisms | p. 78 |
Evolution Variational Inequalities | p. 80 |
Differential Variational Inequalities | p. 82 |
Projected Dynamical Systems | p. 84 |
Dynamical Complementarity Systems | p. 85 |
Generalities | p. 85 |
Nonlinear Complementarity Systems | p. 88 |
Second-Order Moreau's Sweeping Process | p. 88 |
ODE with Discontinuities | p. 92 |
Order of Discontinuity | p. 92 |
Transversality Conditions | p. 93 |
Piecewise Affine and Piecewise Continuous Systems | p. 94 |
Switched Systems | p. 98 |
Impulsive Differential Equations | p. 100 |
Generalities and Well-Posedness | p. 100 |
An Aside to Time-Discretization and Approximation | p. 104 |
Summary | p. 104 |
Mechanical Systems with Unilateral Constraints and Friction | p. 107 |
Multibody Dynamics: The Lagrangian Formalism | p. 107 |
Perfect Bilateral Constraints | p. 109 |
Perfect Unilateral Constraints | p. 110 |
Smooth Dynamics as an Inclusion | p. 112 |
The Newton-Euler Formalism | p. 112 |
Kinematics | p. 112 |
Kinetics | p. 115 |
Dynamics | p. 117 |
Local Kinematics at the Contact Points | p. 123 |
Local Variables at Contact Points | p. 123 |
Back to Newton-Euler's Equations | p. 126 |
Collision Detection and the Gap Function Calculation | p. 128 |
The Smooth Dynamics of Continuum Media | p. 131 |
The Smooth Equations of Motion | p. 131 |
Summary of the Equations of Motion | p. 135 |
Nonsmooth Dynamics and Schatzman's Formulation | p. 135 |
Nonsmooth Dynamics and Moreau's Sweeping Process | p. 137 |
Measure Differential Inclusions | p. 137 |
Decomposition of the Nonsmooth Dynamics | p. 137 |
The Impact Equations and the Smooth Dynamics | p. 138 |
Moreau's Sweeping Process | p. 139 |
Finitely Represented C and the Complementarity Formulation | p. 141 |
Well-Posedness Results | p. 143 |
Lagrangian Systems with Perfect Unilateral Constraints: Summary | p. 143 |
Contact Models | p. 144 |
Coulomb's Friction | p. 145 |
De Saxce's Bipotential Function | p. 148 |
Impact with Friction | p. 151 |
Enhanced Contact Models | p. 153 |
Lagrangian Systems with Frictional Unilateral Constraints and Newton's Impact Laws: Summary | p. 161 |
A Mechanical Filippov's System | p. 162 |
Complementarity Systems | p. 165 |
Definitions | p. 165 |
Existence and Uniqueness of Solutions | p. 167 |
Passive LCS | p. 168 |
Examples of LCS | p. 169 |
Complementarity Systems and the Sweeping Process | p. 170 |
Nonlinear Complementarity Systems | p. 172 |
Relative Degree and the Completeness of the Formulation | p. 173 |
The Single Input/Single Output (SISO) Case | p. 174 |
The Multiple Input/Multiple Output (MIMO) Case | p. 175 |
The Solutions and the Relative Degree | p. 175 |
Higher Order Constrained Dynamical Systems | p. 177 |
Motivations | p. 177 |
A Canonical State Space Representation | p. 178 |
The Space of Solutions | p. 180 |
The Distribution DI and Its Properties | p. 180 |
Introduction | p. 180 |
The Inclusions for the Measures v[subscript i] | p. 182 |
Two Formalisms for the HOSP | p. 183 |
Some Qualitative Properties | p. 186 |
Well-Posedness of the HOSP | p. 187 |
Summary of the Main Ideas of Chapters 4 and 5 | p. 188 |
Specific Features of Nonsmooth Dynamical Systems | p. 189 |
Discontinuity with Respect to Initial Conditions | p. 189 |
Impact in a Corner | p. 189 |
A Theoretical Result | p. 190 |
A Physical Example | p. 191 |
Frictional Paroxysms (the Painleve Paradoxes) | p. 192 |
Infinity of Events in a Finite Time | p. 193 |
Accumulations of Impacts | p. 193 |
Infinitely Many Switchings in Filippov's Inclusions | p. 194 |
Limit of the Saw-Tooth Function in Filippov's Systems | p. 194 |
Time Integration of Nonsmooth Dynamical Systems | |
Event-Driven Schemes for Inclusions with AC Solutions | p. 203 |
Filippov's Inclusions | p. 203 |
Introduction | p. 203 |
Stewart's Method | p. 205 |
Why Is Stewart's Method Superior to Trivial Event-Driven Schemes? | p. 213 |
ODEs with Discontinuities with a Transversality Condition | p. 215 |
Position of the Problem | p. 215 |
Event-Driven Schemes | p. 215 |
Event-Driven Schemes for Lagrangian Systems | p. 219 |
Introduction | p. 219 |
The Smooth Dynamics and the Impact Equations | p. 221 |
Reformulations of the Unilateral Constraints at Different Kinematics Levels | p. 222 |
At the Position Level | p. 222 |
At the Velocity Level | p. 222 |
At the Acceleration Level | p. 223 |
The Smooth Dynamics | p. 224 |
The Case of a Single Contact | p. 225 |
Comments | p. 227 |
The Multi-contact Case and the Index Sets | p. 229 |
Index Sets | p. 229 |
Comments and Extensions | p. 230 |
Event-Driven Algorithms and Switching Diagrams | p. 230 |
Coulomb's Friction and Enhanced Set-Valued Force Laws | p. 231 |
Bilateral or Unilateral Dynamics? | p. 232 |
Event-Driven Schemes: Lotstedt's Algorithm | p. 232 |
Consistency and Order of Event-Driven Algorithms | p. 236 |
Linear Complementarity Systems | p. 240 |
Some Results | p. 241 |
Time-Stepping Schemes for Systems with AC Solutions | p. 243 |
ODEs with Discontinuities | p. 243 |
Numerical Illustrations of Expected Troubles | p. 243 |
Consistent Time-Stepping Methods | p. 247 |
DIs with Absolutely Continuous Solutions | p. 251 |
Explicit Euler Algorithm | p. 252 |
Implicit [theta]-Method | p. 256 |
Multistep and Runge-Kutta Algorithms | p. 258 |
Computational Results and Comments | p. 263 |
The Special Case of the Filippov's Inclusions | p. 266 |
Smoothing Methods | p. 266 |
Switched Model and Explicit Schemes | p. 267 |
Implicit Schemes and Complementarity Formulation | p. 269 |
Comments | p. 271 |
Moreau's Catching-Up Algorithm for the First-Order Sweeping Process | p. 271 |
Mathematical Properties | p. 272 |
Practical Implementation of the Catching-up Algorithm | p. 273 |
Time-Independent Convex Set K | p. 274 |
Linear Complementarity Systems with r [less than or equal] 1 | p. 275 |
Differential Variational Inequalities | p. 279 |
The Initial Value Problem (IVP) | p. 280 |
The Boundary Value Problem | p. 281 |
Summary of the Main Ideas | p. 283 |
Time-Stepping Schemes for Mechanical Systems | p. 285 |
The Nonsmooth Contact Dynamics (NSCD) Method | p. 285 |
The Linear Time-Invariant Nonsmooth Lagrangian Dynamics | p. 286 |
The Nonlinear Nonsmooth Lagrangian Dynamics | p. 289 |
Discretization of Moreau's Inclusion | p. 293 |
Sweeping Process with Friction | p. 295 |
The One-Step Time-Discretized Nonsmooth Problem | p. 296 |
Convergence Properties | p. 303 |
Bilateral and Unilateral Constraints | p. 305 |
Some Numerical Illustrations of the NSCD Method | p. 307 |
Granular Material | p. 307 |
Deep Drawing | p. 309 |
Tensegrity Structures | p. 309 |
Masonry Structures | p. 309 |
Real-Time and Virtual Reality Simulations | p. 311 |
More Applications | p. 314 |
Moreau's Time-Stepping Method and Painleve Paradoxes | p. 315 |
Variants and Other Time-Stepping Schemes | p. 315 |
The Paoli-Schatzman Scheme | p. 315 |
The Stewart-Trinkle-Anitescu-Potra Scheme | p. 317 |
Time-Stepping Scheme for the HOSP | p. 319 |
Insufficiency of the Backward Euler Method | p. 319 |
Time-Discretization of the HOSP | p. 321 |
Principle of the Discretization | p. 321 |
Properties of the Discrete-Time Extended Sweeping Process | p. 322 |
Numerical Examples | p. 324 |
Synoptic Outline of the Algorithms | p. 325 |
Numerical Methods for the One-Step Nonsmooth Problems | |
Basics on Mathematical Programming Theory | p. 331 |
Introduction | p. 331 |
The Quadratic Program (QP) | p. 331 |
Definition and Basic Properties | p. 331 |
Equality-Constrained QP | p. 335 |
Inequality-Constrained QP | p. 338 |
Comments on Numerical Methods for QP | p. 344 |
Constrained Nonlinear Programming (NLP) | p. 345 |
Definition and Basic Properties | p. 345 |
Main Methods to Solve NLPs | p. 347 |
The Linear Complementarity Problem (LCP) | p. 351 |
Definition of the Standard Form | p. 351 |
Some Mathematical Properties | p. 352 |
Variants of the LCP | p. 355 |
Relation Between the Variants of the LCPs | p. 357 |
Links Between the LCP and the QP | p. 359 |
Splitting-Based Methods | p. 363 |
Pivoting-Based Methods | p. 367 |
Interior Point Methods | p. 374 |
How to chose a LCP solver? | p. 379 |
The Nonlinear Complementarity Problem (NCP) | p. 379 |
Definition and Basic Properties | p. 379 |
The Mixed Complementarity Problem (MCP) | p. 383 |
Newton-Josephy's and Linearization Methods | p. 384 |
Generalized or Semismooth Newton's Methods | p. 385 |
Interior Point Methods | p. 388 |
Effective Implementations and Comparison of the Numerical Methods for NCPs | p. 388 |
Variational and Quasi-Variational Inequalities | p. 389 |
Definition and Basic Properties | p. 389 |
Links with the Complementarity Problems | p. 390 |
Links with the Constrained Minimization Problem | p. 391 |
Merit and Gap Functions for VI | p. 392 |
Nonsmooth and Generalized equations | p. 396 |
Main Types of Algorithms for the VI and QVI | p. 398 |
Projection-Type and Splitting Methods | p. 398 |
Minimization of Merit Functions | p. 400 |
Generalized Newton Methods | p. 401 |
Interest from a Computational Point of View | p. 401 |
Summary of the Main Ideas | p. 401 |
Numerical Methods for the Frictional Contact Problem | p. 403 |
Introduction | p. 403 |
Summary of the Time-Discretized Equations | p. 403 |
The Index Set of Forecast Active Constraints | p. 403 |
Summary of the OSNSPs | p. 405 |
Formulations and Resolutions in LCP Forms | p. 407 |
The Frictionless Case with Newton's Impact Law | p. 407 |
The Frictionless Case with Newton's Impact and Linear Perfect Bilateral Constraints | p. 408 |
Two-Dimensional Frictional Case as an LCP | p. 409 |
Outer Faceting of the Coulomb's Cone | p. 410 |
Inner Faceting of the Coulomb's Cone | p. 414 |
Comments | p. 417 |
Weakness of the Faceting Process | p. 418 |
Formulation and Resolution in a Standard NCP Form | p. 419 |
The Frictionless Case | p. 419 |
A Direct MCP for the 3D Frictional Contact | p. 419 |
A Clever Formulation of the 3D Frictional Contact as an NCP | p. 420 |
Formulation and Resolution in QP and NLP Forms | p. 422 |
The Frictionless Case | p. 422 |
Minimization Principles and Coulomb's Friction | p. 423 |
Formulations and Resolution as Nonsmooth Equations | p. 424 |
Alart and Curnier's Formulation and Generalized Newton's Method | p. 424 |
Variants and Line-Search Procedure | p. 429 |
Other Direct Equation-Based Reformulations | p. 430 |
Formulation and Resolution as VI/CP | p. 432 |
VI/CP Reformulation | p. 432 |
Projection-type Methods | p. 433 |
Fixed-Point Iterations on the Friction Threshold and Ad Hoc Projection Methods | p. 434 |
A Clever Block Splitting: the Nonsmooth Gauss-Seidel (NSGS) Approach | p. 437 |
Newton's Method for VI | p. 440 |
The SICONOS Software: Implementation and Examples | |
The SICONOS Platform | p. 443 |
Introduction | p. 443 |
An Insight into Siconos | p. 443 |
Step 1. Building a Nonsmooth Dynamical System | p. 444 |
Step 2. Simulation Strategy Definition | p. 447 |
Siconos Software | p. 448 |
General Principles of Modeling and Simulation | p. 448 |
NSDS-Related Components | p. 451 |
Simulation-Related Components | p. 456 |
Siconos Software Design | p. 457 |
Examples | p. 460 |
The Bouncing Ball(s) | p. 460 |
The Woodpecker Toy | p. 463 |
MOS Transistors and Inverters | p. 464 |
Control of Lagrangian systems | p. 466 |
Convex, Nonsmooth, and Set-Valued Analysis | p. 475 |
Set-Valued Analysis | p. 475 |
Subdifferentiation | p. 475 |
Some Useful Equivalences | p. 476 |
Some Results of Complementarity Theory | p. 479 |
Some Facts in Real Analysis | p. 481 |
Functions of Bounded Variations in Time | p. 481 |
Multifunctions of Bounded Variation in Time | p. 482 |
Distributions Generated by RCLSBV Functions | p. 483 |
Differential Measures | p. 486 |
Bohl's Distributions | p. 487 |
Some Useful Results | p. 487 |
References | p. 489 |
Index | p. 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