Numerical solution of Variational Inequalities by Adaptive Finite Elements
Franz-Theo Suttmeier describes a general approach to a posteriori error estimation and adaptive mesh design for finite element models where the solution is subjected to inequality constraints. This is an extension to variational inequalities of the so-cal
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VIEWEG+TEUBNER RESEARCH Advances in Numerical Mathematics Herausgeber | Editors: Prof. Dr. Dr. h. c. Hans Georg Bock Prof. Dr. Dr. h. c. Wolfgang Hackbusch Prof. Mitchell Luskin Prof. Dr. Rolf Rannacher
Franz-Theo Suttmeier
Numerical solution of Variational Inequalities by Adaptive Finite Elements
VIEWEG+TEUBNER RESEARCH
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1st Edition 2008 All rights reserved © Vieweg+Teubner | GWV Fachverlage GmbH, Wiesbaden 2008 Readers: Christel A. Roß Vieweg+Teubner is part of the specialist publishing group Springer Science+Business Media www.viewegteubner.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: KünkelLopka Medienentwicklung, Heidelberg Printing company: Strauss Offsetdruck, Mörlenbach Printed on acid-free paper Printed in Germany ISBN 978-3-8348-0664-2
Summary This work describes a general approach to a posteriori error estimation and adaptive mesh design for finite element models where the solution is subjected to inequality constraints. This is an extension to variational inequalities of the so-called Dual-Weighted-Residual method (DWR method) which is based on a variational formulation of the problem and uses global duality arguments for deriving weighted a posteriori error estimates with respect to arbitrary functionals of the error. In these estimates local residuals of the computed solution are multiplied by sensitivity factors which are obtained from a numerically computed dual solution. The resulting local error indicators are used in a feed-back process for generating economical meshes which are tailored according to the particular goal of the computation. This method is developed here for several model problems. Based on these examples, a general concept is proposed, which provides a systematic way of adaptive error control for problems stated in form of variational inequalities.
F¨ ur Alexandra, Katharina und Merle
Contents
1 Introduction
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2 Models in elasto-plasticity
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2.1
Governing equations . . . . . . . . . . . . . . . . . . . . . . . .
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2.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 The dual-weighted-residual method
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3.1
A model situation in plasticity . . . . . . . . . . . . . . . . . .
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3.2
A posteriori error estimate . . . . . . . . . . . . . . . . . . . . .
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3.3
Evaluation of a posteriori error bounds . . . . . . . . . . . . . .
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3.4
St
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