Numerical methods for nonsmooth dynamical systems. Applications in mechanics and electronics.

*(English)*Zbl 1173.74001
Lecture Notes in Applied and Computational Mechanics 35. Berlin: Springer (ISBN 978-3-540-75391-9/hbk). xxi, 525 p. (2008).

This book concerns the numerical simulation of dynamical systems whose trajectories may be not differentiable. These systems are called nonsmooth dynamical systems, and they represent an important class of systems, firstly because of many applications in which nonsmooth models are useful, secondly because they give rise to new problems in various fields of mathematics and computational mechanics.

The book is divided into three parts.

The first part presents the formulation of nonsmooth dynamical systems, with model problems from mechanics, electricity and control, as well as useful material from convex and nonsmooth analysis related to differential inclusions, variational inequalities, and complementarity systems. The model applications are taken from multibody systems with contact, impact and friction, and from electrical circuits with piecewise linear and ideal components.

The second part deals with numerical time-integration schemes, which can be divided into event-driven schemes and time-stepping schemes.

The third part is devoted to one-step nonsmooth problem solvers, and includes techniques suitable for the solution of variational inequalities, nonlinear programming problems and complementarity problems. The one-step problems include the nonlinear models known as holonomic models, which have a long tradition in engineering mechanics (cf. unilateral contact for static elastic bodies, the holonomic or Hencky model of plasticity). Nonsmooth modelling in mechanics is based on the seminal works of J. J. Moreau and T. Rockafellar on convex analysis and its usage in contact mechanics and elastoplasticity, as well as on later extensions to nonconvex problems by P. D. Panagiotopoulos. This development has been documented in a number of research monographs like [R. T. Rockafellar, Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press (1997; Zbl 0932.90001); P. D. Panagiotopoulos, Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Boston-Basel-Stuttgart: BirkhĂ¤user (1985; Zbl 0579.73014); J. J. Moreau, P. D. Panagiotopoulos, Nonsmooth mechanics and applications. CISM Courses and Lectures, 302. Wien etc.: Springer-Verlag (1988; Zbl 0652.00016)] and in many others.

Practical applications of nonsmooth mechanics are also presented in research monographs, see e.g. [F. Pfeiffer, C. Glocker, Multibody dynamics with unilateral contacts. CISM Courses and Lectures. 421. Wien etc.: Springer-Verlag (2000; Zbl 0960.00025)].

The present book is a research monograph with numerous information and references to original publications which gives to the book an encyclopaedic nature. The wealth of information and solution methods mentioned in this book certainly shows that the area of nonsmooth mechanics has arrived a level of maturity that allows for serious industrial applications, without loosing its attractiveness for research purposes. The 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 like applied mathematics, robotics, civil and mechanical engineering, mechatronics, virtual reality, etc.

The book is divided into three parts.

The first part presents the formulation of nonsmooth dynamical systems, with model problems from mechanics, electricity and control, as well as useful material from convex and nonsmooth analysis related to differential inclusions, variational inequalities, and complementarity systems. The model applications are taken from multibody systems with contact, impact and friction, and from electrical circuits with piecewise linear and ideal components.

The second part deals with numerical time-integration schemes, which can be divided into event-driven schemes and time-stepping schemes.

The third part is devoted to one-step nonsmooth problem solvers, and includes techniques suitable for the solution of variational inequalities, nonlinear programming problems and complementarity problems. The one-step problems include the nonlinear models known as holonomic models, which have a long tradition in engineering mechanics (cf. unilateral contact for static elastic bodies, the holonomic or Hencky model of plasticity). Nonsmooth modelling in mechanics is based on the seminal works of J. J. Moreau and T. Rockafellar on convex analysis and its usage in contact mechanics and elastoplasticity, as well as on later extensions to nonconvex problems by P. D. Panagiotopoulos. This development has been documented in a number of research monographs like [R. T. Rockafellar, Convex Analysis. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press (1997; Zbl 0932.90001); P. D. Panagiotopoulos, Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Boston-Basel-Stuttgart: BirkhĂ¤user (1985; Zbl 0579.73014); J. J. Moreau, P. D. Panagiotopoulos, Nonsmooth mechanics and applications. CISM Courses and Lectures, 302. Wien etc.: Springer-Verlag (1988; Zbl 0652.00016)] and in many others.

Practical applications of nonsmooth mechanics are also presented in research monographs, see e.g. [F. Pfeiffer, C. Glocker, Multibody dynamics with unilateral contacts. CISM Courses and Lectures. 421. Wien etc.: Springer-Verlag (2000; Zbl 0960.00025)].

The present book is a research monograph with numerous information and references to original publications which gives to the book an encyclopaedic nature. The wealth of information and solution methods mentioned in this book certainly shows that the area of nonsmooth mechanics has arrived a level of maturity that allows for serious industrial applications, without loosing its attractiveness for research purposes. The 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 like applied mathematics, robotics, civil and mechanical engineering, mechatronics, virtual reality, etc.

Reviewer: Georgios E. Stavroulakis (Chania)

##### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

74M15 | Contact in solid mechanics |

49J40 | Variational inequalities |

65Kxx | Numerical methods for mathematical programming, optimization and variational techniques |

74Sxx | Numerical and other methods in solid mechanics |