INTERNATIONAl. CENTRE FOR MECHANICAL SCIENCES

COURSES AND LECTURES- No. 301

FINITE ELEMENT AND BOUNDARY ELEMENT TECHNIQUES

FROM MATHEMATICAL AND ENGINEERING

POINT OF VIEW

EDITED BY

E. STEIN UNIVERSITAT HANNOVER

W.WENDLAND UNIVERSITAT STUTTGART

Springer-Verlag Wien GmbH

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del Consiglio Nazionale delle Ricerche.

This volume contains 99 illustrations.

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ISBN 978-3-211-82103-9 ISBN 978-3-7091-2826-8 (eBook) DOI 10.1007/978-3-7091-2826-8

© 1988 by Springer-Verlag Wien

Originally published by CISM, Udine in 1988.

PREFACE

The finite element methods •· FEM'' and the more recent boundary element methods ''BEM" nowadays belong to the most popular numerical procedures in computational mechanics and in many engineering fields. Both methods have their merits and also their restrictions. Therefore, the combination of both methods becomes an impro­ved numerical tool. The development of these methods is closely related to the fast development of modern computers. As a result. one can see an impressive growth of FEM and BEM and also a rapid extension of the applicability to more and more complex problems. Nowadays, everywhere people are working on the improvement of the FEM and BEM methods which also requires detailed research of the mathema­ti<"al and engineering background and of the properties of these methods. Currently, the refinement and formulation of algorithms and also the error analysis are the main fields of mathematkal research in FEM and BEM in order to dassify reliability and performance. Existing algorithms in engineering are extended and modified and also new engineering problems are tat'kled with FEM and BEM. Clearly this type of work requires the dose interadion of engineers and mathematicians since sometimes even small improvements require rather deep knowledge of modern mathematical analysis and also the physical background as well as significant applications. The <"ombina­tion of mat.hemat.i<"s, mechanics aud numeri<"al methods makes this field particularly attrat'tJve and interesting. However, there are not" too many mathematicians and engineers who are willing to cooperate in this field. It was one of the goals of this course to encourage and initiate such a cooperation which requires much patience from both sides.

The aim of the course was to present significant basic formulations of FEM and BEM and to show their common practical and mathematical foundations, their dif­ferenct"s as well as possibilities for their t'ombination." These include:

1) Variational foundations in the general framework of complementary variational problems, whi<"h <"over dassical variational problems, variational inequalities as well as mixed and hybrid formulations.

2) A survey on non-linear finite element methods in continuum mechanics and on algorithms for the corresponding large systems of non-linear equations, furthermore the techniques of FEM in non-linear shell theory and in problems with elasto-plastir deformations and with constraints.

3) An introduction into the boundary element. mehtods for classical potential theory, the Laplace equation, the treatment of time-dependent problems and the coupling of boundary elements and finite elements in the domain. (This lecture is not in these Notes. See:[Brebbia C.A., Telles J.C.F. and Wrobel L.C. (1984), Boundary Element Techniques, Springer-Verlag Berlin] Chapters 2 and 13 and [Brebbia C.A. and Nar­dini D. (1986), Solution of Parabolic and hyperbolic time dependent problems using boundary elements, Comp. and Maths. with Appls. 12B, pp. 1061-1072].

4) An introduction to the boundary element method in elastostatics and linear frac­ture mechanics in problems with body forces and thermal loading.

5) The numerical treatment of corner and crack singularities is still a problem in both methods, FEM and BEM. In particular, the pollution effect, graded mesh re­finements, the singularity-subtraction method and the method of dual singularities and iteration procedures are presented. Some of these results are completely new providing significant improvements of FEM as well as BEM, in particular, for the efficit>nt and accurate computation of stress intensity factors.

6) The application of BEM to plastic analysis combines the linear BEM technique with increme-ntal and iteradtive methods involving the updating of field integrals. This method is based on variational BEM formulations leading to new algorithms.

7) The asymptotic error analysis of FEM as well as of BEM ran both be based on the Galerkin method for variational problems. This allows also the analysis of some roupled FEM and BEM methods. Some current results are surveyed in this general framework, some of them are new.

The course was an exciting experience, all participants tried hard to develop mu­tual understanding and cooperation which c.ulminated in a lively panel discussion on urgent open problems such as: Nonlinear BEM, Smoothing of singular boundaries, Time depending problems, Nonuniqueness of solutions, Symmetrization in BEM,

Coupling of FEM and BEM, Fundame-ntal solutions for dynamic problems, Nonhomogen~us bodies in the BEM, Numerical intt'gration in BEM.

After the course it took us almost two years to finish these Lecture Notes for two rt'asonli: 1. The authors tried again to close .the languagt' gap between mathematics and engineering. 2. FEM and BEM develop so fast that the authors were tempted to include very c."Urrent developments - a procedure without limits.

The more we are grateful to all the co-authors for their energetic engagement and the completion of these Lecture Notes. We also express our gratitude to all the par­ticipants in the course and to all the lecturers for the stimulating and encouraging discussions. We thank the International Center for Mechanical Science-. and its stafF for the excellent organization of the course, the lovely surrounding, good working conditions and cordial care. We are grateful to the city of Udine, the "Stiftung Volkswagenwerk" and the "German Research Foundation" for their support and last not )east to the Springer-Verlag for this splendid publication.

Erwin Stein and Wolfgang Wendland

CONTENTS

Page

Preface

Complementary Variational Principles by W. Velte .•.•.••.•••.•.••..•........•..•••..••••..•.•.•...•.•.••••.••••.••••• I

Five Lectures on Non~near Finite Element Methods by E. Stein, D. Bischoff, N. Miille.,..Hoeppe. W. Wagner, and P. Wriggers ....•••••••• 33

Boundary Element Technique in Elastostatics and Linear Fracture Mechanics by G. Kuhn •.•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 109

Numerical Treatment of Comer and Crack Singularities by H. Blum ...•••..•.•.••.•.•..••.••••••.•..•••••.•..•••. • •.•.•••••.•••••••.•• 171

Plastic Analyis by Boundary Elements by G. Maier, G. Novati, and U. Perego • •.•••.••••••••.•••.••••••.•••.•••••.••••• 213

On Asymptotic Error Estimates for Combined BEM and FEM by W.L Wendland .•••••..••..•••••.•.••••.••••••••••••••••••••••••••••••••••• 273