# Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications

Springer Science & Business Media, Aug 31, 1999 - Mathematics - 260 pages
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.

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### Contents

 MATHEMATICAL PRELIMINARIES 3 12 ELEMENTS OF NONSMOOTH ANALYSIS 18 13 EQUATIONS AND INEQUALITIES WITH MONOTONE OPERATORS 26 14 APPROXIMATION OF EQUATIONS AND INEQUALITIES OF MONOTONE TYPE 48 References 81 NONSMOOTH MECHANICS CONVEX AND NONCONVEX PROBLEMS 83 22 NONLINEAR ELASTOSTATICS 85 23 LITERATURE REVIEW 98
 TIME DEPENDENT CASE 163 41 DISCRETIZATION 166 42 CONVERGENCE ANALYSIS 170 43 ALGEBRAIC REPRESENTATION 194 44 CONSTRAINED HEMIVARIATIONAL INEQUALITIES 196 References 200 NONSMOOTH OPTIMIZATION METHODS 203 NONSMOOTH OPTIMIZATION METHODS 205

 Finite Element Approximation of Hemivariational Inequalities 101 APPROXIMATION OF ELLIPTIC HEMIVARIATIONAL INEQUALITIES 103 31 AUXILIARY RESULTS 107 32 DISCRETIZATION 110 33 CONVERGENCE ANALYSIS 115 34 CONSTRUCTION OF FINITE ELEMENT SPACES AND INTERPOLATION OPERATORS 121 35 ALGEBRAIC REPRESENTATION 134 36 CONSTRAINED HEMIVARIATIONAL INEQUALITIES 139 37 APPROXIMATION OF VECTORVALUED HEMIVARIATIONAL INEQUALITIES 151 References 161
 52 NONCONVEX CASE 217 References 225 Numerical Examples 229 NUMERICAL EXAMPLES 231 61 NONMONOTONE FRICTION AND CONTACT PROBLEMS 232 62 DELAMINATION PROBLEM 249 References 258 Index 259 Copyright