The main focus of this book is to examine the convergence of the Euler scheme for various classes of nonsmooth mechanical problems, formulated via maximal monotone differential inclusions with sub-differential nonlinearities. Though the problems that need Moreau sweeping processes or complementarity approaches are not addressed, the book pays significant attention to stochastic forcing, a topic not considered in other books on nonsmooth mechanics.
Chapter 1 motivates the forthcoming material using dry-friction mass displacement and Kanai-Tajimi earthquake models. Here, the insufficiency of differential equations to model friction and impacts is addressed. The Euler numerical schemes for both deterministic and stochastic forcing are compared for varying step sizes. Instructive differences of the simulation results are briefly discussed.
The results on the existence, uniqueness and convergence of numerical schemes are then developed in Chapters 2-4, using a wide range of concepts from functional analysis, probability theory, and geometry. Those concepts are explained in a concise, but fully mathematically rigorous way. The theorems stated throughout the text either come with proofs or with references to proofs. The presentation of the more involved proofs is placed in the Appendix, where extended explanations of more mathematical concepts are also given.
Here is a brief outlook of the three theoretical chapters of the book. Chapter 2 introduces the concepts of maximal monotone operators and sub-differentials in the context of differential inclusions. Existence and uniqueness theorems are formulated in Euclidian and Hilbert spaces, and the results on the convergence of the Euler scheme are proposed. Uniqueness and existence results for multivalued stochastic differential equations are introduced in Chapter 3 along with the necessary probabilistic background, starting from basic definitions (probability space, random variable, etc.) and developing to theorems on strong and weak convergence of the numerical interpolation scheme. Beginning with the simple example of a unit mass moving on the unit sphere, Chapter 4 exposes the reader to Riemannian manifolds along with the required geometric theoretic concepts. Finally, solutions of multivalued stochastic differential equations on Riemannian manifolds are defined in the presence of stochastic forcing and the results on the existence and uniqueness of solutions are stated.
After this theoretical background, the authors embark on studying the existence, uniqueness and convergence of numerical schemes in specific models with friction, elastoplastic elements, and stochastic forcing. Various parallel and sequential assembles of elastoplastic elements with dry friction and applied load (constant, time-dependent or stochastic) connected to material points are considered in Chapter 5 (Saint-Venant elements and Prandtl models) and convergence of respective numerical schemes are established. Here the authors also establish the correspondence between the parameters of the generalized Prandtl model and the geometric parameters of the hysteresis loop by making various assumptions about the amplitude of the loading force. For the case of an infinite number of connected elastoplastic elements a result about the convergence of the discretized model to the continuous one is explained. Chapter 6 suggests a version of the numerical scheme for a forced mechanical oscillator with impacts that obey Newton’s restitution law.
Chapters 6-7 may serve as good training for readers interested in the modeling of elastoplastic media. The diverse examples considered help readers to create new models with the required dynamical properties. At the same time, the practical requirements of various models could have been explained better. For example, it would have been good to provide some justification for the proposed selection of models. In addition, it is not always clear why some elastoplastic elements (or their assembly) come with periodic forcing, while others come with stochastic forcing.
Chapter 7 considers several topics that do not require the theory of Chapters 2-4. These topics include a coupled oscillator with nonsmooth ingredients and a small parameter in front of highest derivative, a belt tensioner model with a view to industrial applications, a problem with delay and memory, a linear hardening model with a viscous damping regularization term, and ill-posed friction problems. This chapter can help the reader to see which extensions of the maximal monotone framework are possibly useful in more realistic applications.
All in all, the book is very useful for studying applications of numerical schemes to deterministic and stochastic models of nonsmooth mechanics. It is also a good reference point for the respective mathematical concepts, theorems and proofs. A little bit of explanation as to how the modeling choices were made would be a plus.