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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.
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.
This book addresses the formulation, approximation and numerical
solution of optimal shape design problems: from the continuous
model through its discretization and approximation results, to
sensitivity analysis and numerical realization. Shape optimization
of structures is addressed in the first part, using variational
inequalities of elliptic type. New results, such as contact shape
optimization for bodies made of non-linear material, sensitivity
analysis based on isoparametric technique, and analysis of cost
functionals related to contact stress distribution are included.
The second part presents new concepts of shape optimization based
on a fictitious domain approach. Finally, the application of the
shape optimization methodology in the material design is discussed.
This second edition is a fully revised and up-dated version of
Finite Element Method for Optimal Shape Design. Numerous numerical
examples illustrate the theoretical results, and industrial
applications are given.
The efficiency and reliability of manufactured products depend on,
among other things, geometrical aspects; it is therefore not
surprising that optimal shape design problems have attracted the
interest of applied mathematicians and engineers. This
self-contained, elementary introduction to the mathematical and
computational aspects of sizing and shape optimization enables
readers to gain a firm understanding of the theoretical and
practical aspects so they may confidently enter this field. In
contrast to existing texts on structural optimization, Introduction
to Shape Optimization: Theory, Approximation, and Computation
treats sizing and shape optimization in a comprehensive way,
covering everything from mathematical theory (existence analysis,
discretizations, and convergence analysis for discretized problems)
through computational aspects (sensitivity analysis, numerical
minimization methods) to industrial applications. Some of the
applications included are contact stress minimization for
elasto-plastic bodies, multidisciplinary optimization of an
airfoil, and shape optimization of a dividing tube. By presenting
sizing and shape optimization in an abstract way, the authors are
able to use a unified approach in the mathematical analysis for a
large class of optimization problems in various fields of physics.
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