Elementos Finitos

Finite Element Analysis with Error Estimators
An Introduction to the FEM and Adaptive Error Analysis for Engineering Students

Finite Element Analysis with Error Estimators
An Introduction to the FEM and Adaptive Error Analysis for Engineering Students
J. E. Akin

AMSTERDAM • BOSTON • HEIDELBERG • PARIS • SAN DIEGO • SAN FRANCISCO •

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NEW YORK • OXFORD • SYDNEY •TOKYO

Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published 2005 Copyright  2005, J.E. Akin. All rights reserved The right of J.E. Akin to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in anymaterial form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90Tottenham Court Road, London, England W1T 4LP. Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.co.uk. You may also complete yourrequest on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’ British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 6722 2 Forinformation on all Elsevier Butterworth-Heinemann publications visit our website at http://books.elsevier.com Printed and bound in Great Britain

Contents
Preface ..................................................................................................................... Notation..................................................................................................................
1. Introduction ..................................................................................................... 1.1 Finite element methods ......................................................................... 1.2 Capabilities of FEA ............................................................................... 1.3 Outline of finite element procedures..................................................... 1.4 Assembly into the system equations ..................................................... 1.5 Error concepts ....................................................................................... 1.6 Exercises ............................................................................................... 1.7 Bibliography......................................................................................... Mathematical preliminaries ........................................................................... 2.1 Introduction ........................................................................................... 2.2 Linear spaces and norms ....................................................................... 2.3 Sobolev norms *.................................................................................... 2.4 Dual problems, self-adjointness ............................................................ 2.5 Weighted residuals ................................................................................ 2.6 Boundary condition terms ..................................................................... 2.7 Adding more unknowns...