Table Of ContentINTERNATIONAl. 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
Le spese di stampa di questo volume sono in parte coperte da contributi
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 impro
ved 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 mathema
ti<"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 <"ombina
tion 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 dif
ferenct"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 Nar
dini 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 frac
ture 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 re
finements, 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 mu
tual 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 par
ticipants 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
COMPLEMENTARY VARIATIONAL PRINCIPLES
W. Velte
Univesitit Wiirzburg, Wiirzburg, F.R.G.
INTRODUCTION
As is well known, variational methods belong to the fundamen
tal principles in mathematics and in mechanics. They are of
interest not only from the theoretical but also from the nu
merical point of view. Indeed, when discretized the variatio
nal principles immediately provide numerical schemes for sol
ving the underlying problem numerically. In particular, varia
tional principles in various forms are videly used in order to
establish Ritz - Galerkin schemes in finite element spaces.
In what follows, we will restrict ourselves to problems in-
volving linear elliptic partial differential operators. There
is now an abundance of papers on variational principles and
.finite element schemes, both, for classical boundary value
problems and also for problems involving inequality constraints
as, for instance, obstacle problems or problems involving
unilateral boundary conditions.
In order to establish variational characterizations of the
solution under consideration, various techniques have been
introduced. It is the aim of.these lectures to present, as
fare as linear symmetric differential operators are .concerned,
a unified and strai~htforward approach to some of these varia
tional characterizations and to exhibit the relations between
them. In addition, emphasis is given to a posteriori error
estimates in energy norm.
2 W. Velte
LINEAR BOUNDARY VALUE PROBLEMS
In this chapter, we will consider a certain class of linear
boundary value problems for elliptic partial differential
equations where the solution can be characterized by two
different extremal principles, known as dual or complementary
extremal principles.
In the applications we have in mind, the first of these two
principles will always be the principle of minimal potential
energy. Complementary principles, however, are not uniquely
determined. As we will see, there is often some freedom to
vary the complementary functional in a ~atural manner.
pairs of dual or complementary extremal principles have been
introduced long time ago: Dual to the principle of minimal
potential energy is, for instance, Thomson's principle in
electrostatics and Castigliano1s principle of complementary
work in linear elasticity. (See, for instance, Courant and
Hilbert [1l, vol.1.) The numerical methods of Ritz and of
Trefftz rest upon dual extremai characterizations for the
solution of the boundary value problem under consideration.
There are different techniques to establish pairs of dual
extremal principles. Widely used are methods of convex ana
lysis, in particular the theory of duality in convex optimi
zation. This approach is very general, but it requires some
knowledge in this field. (See, for instance, Ekeland and
Temam [JJ and the references given there.)
Another approach is due to Noble and Sewell. It applies to
linear boundary value problems that can be written in the form
T* Tu = f , where T and T* denote adjoint linear opera
tors defined in appropriate function spaces. In Noble's and
Sewell's approach, the problem is reformulated in terms of
a Hamiltonian or a·Lagrangian system of equations, respecti
vely. (See, for instance, [4], [5], [6], [7) and the refe
rences given there.)
Complementary Variational Principles 3
Here, we will present a simple and straightforward approach to
dual extremal principles starting from the principle of mini
mal potential energy and the related variational equation
(boundary value problem in weak form). This approach is very
natural by two reasons: Firstly, the principle of minimal po
tential energy is fundamental for many problems in mechanics
and in physics. Secondly, the related variational equation,
which has to be satisfied by the solution, is the basis for a
first important numerical method, namely the well known Ritz
Galerkin method. When appropriate finite element spaces are
introduced, this approach yields so called conforming finite
element methods.
From the variational equation we will construct in a straight
forward manner dual functionals and dual extremal principles.
Using both, the original and a dual principle, it will be
possible to establish a posteriori error estimates for
approximate solutions. (See also Velte (8].)
As we will see in chapter 2, the approach can be extended to
a certain class of problems involving unilateral constraints,
as obstacle problems or problems exhibiting unilateral boun
dary conditions. (In that case, hovewer, the solution has to
satisfy a variational inequality. Nevertheless, there is a
similar approach to dual extremal principles.) In this chap
ter , however, we will consider classical problems without
~nilateral constraints.
1.1 EXTREMAL FUNCTIONALS AND VARIATIONAL EQUATIONS
In this section, we will consider a certain class of linear
boundary value problems for (systems of) elliptic partial
differential equations where the solution can be characterized
by the principle ~f minimal potential energy. However, in or
der to exhibit the essential features we will introduce the
class of problems under consideration in the following manner:
