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

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Springer Science & Business Media, Aug 31, 1999 - Mathematics - 260 pages
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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
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About the author (1999)

Markku Miettinen, MD, is Chairman and Distinguished Scientist in the Department of Soft Tissue and Orthopedic Pathology, Armed Forces Institute of Pathology, Washington, DC.

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