Finite Element Method for Hemivariational Inequalities

by Haslinger, J.; Miettinen, Markku; Panagiotopoulos, P. D.

• ISBN13: 9780792359517
• ISBN10: 0792359518
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: Kluwer Academic Pub

Summary

Hemivariational inequalities represent an important class of problems in nonsmooth and nonconvex mechanics. By means of them, problems with nonmonotone, possibly multivalued, constitutive laws can be formulated, mathematically analyzed and finally numerically solved. The present book gives a rigorous analysis of finite element approximation for a class of hemivariational inequalities of elliptic and parabolic type. Finite element models are described and their convergence properties are established. Discretized models are numerically treated as nonconvex and nonsmooth optimization problems. The book includes a comprehensive description of typical representants of nonsmooth optimization methods. Basic knowledge of finite element mathematics, functional and nonsmooth analysis is needed. The book is self-contained, and all necessary results from these disciplines are summarized in the introductory chapter. Audience: Engineers and applied mathematicians at universities and working in industry. Also graduate-level students in advanced nonlinear computational mechanics, mathematics of finite elements and approximation theory. Chapter 1 includes the necessary prerequisite materials.

Table of Contents

Preface	ix
List of Notations	xiii
Introduction	xix
Part I Introductory Topics
Mathematical Preliminaries	3	(80)
Functional spaces and their properties	3	(15)
Elements of nonsmooth analysis	18	(8)
Equations and inequalities with monotone operators	26	(22)
Approximation of equations and inequalities of monotone type	48	(35)
Chapter References	81	(2)
Nonsmooth Mechanics	83	(20)
Introduction	83	(2)
Nonlinear elastostatics	85	(13)
Literature review	98	(5)
Chapter References	98	(5)
Part II Finite Element Approximation of Hemivariational Inequalities
Approximation of Elliptic Hemivariational Inequalities	103	(60)
Auxiliary results	107	(3)
Discretization	110	(5)
Convergence analysis	115	(6)
Construction of finite element spaces and interpolation operators	121	(13)
Algebraic representation	134	(5)
Constrained hemivariational inequalities	139	(12)
Approximation of vector-valued hemivariational inequalities	151	(12)
Chapter References	161	(2)
Time Dependent Case	163	(42)
Discretization	166	(4)
Convergence analysis	170	(24)
Algebraic representation	194	(2)
Constrained hemivariational inequalities	196	(9)
Chapter References	200	(5)
Part III Nonsmooth Optimization Methods
Nonsmooth Optimization Methods	205	(26)
Convex case	205	(12)
Nonconvex case	217	(14)
Chapter References	225	(6)
Part IV Numerical Examples
Numerical Examples	231	(28)
Nonmonotone friction and contact problems	232	(17)
Delamination problem	249	(10)
Chapter References	258	(1)
Index	259