Plastic Analysis by Boundary Elements
In the traditional context of small deformation, quasti-static plasticity, we can distinguish the following kinds of problems (for details, see e.g. refs. [1] [2]).
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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.
This work is subject to copyright. All rights are reserved,
whether the whole or part of the material is concerned specifiCally those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. ISBN 978-3-7091-2826-8 (eBook) ISBN 978-3-211-82103-9 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..cpds=
{i)
•'H 1 (n)
is equivalent with the boundary condition
and ( ii) is a weak form of' A = -o u
I
0n
- b. u = f'
n
in
u = g
on
r '
together with
r .
on
Theorem 1.).1. The system {i), {ii} has a unique solution u ,
r.
Furtherly
for any pair
u,~
satisfying {i) and for any pair
v,A
satisfying {ii). Proof'. Uniqueness: From ( i) follows
u = g
01
on
r
{ii) holds in particular for all q>EH {0). Thus,
or u
u E Uad. is also
solution of' the variational equality (P)' and hence uniquely
A • Existence: As is easily
determined. The same is true for. seen, Let
u
and
r =-au/on
is a solution of' (i)' (ii).
satisfy (i) , i.e.
u,~
u = g
on
f
. Then
L(u,ll) = J{u)
Let
v,>..
satisfy (ii). For L(v,>..)=
q> = v
we find
-lvl!-2fAgds
On the other hand, introducing a function on
r '
we find from {ii) for
and hence
ql
u 0 EH 1 (n), u 0 =g
= u0
L(v,>..) ~ Jc(v) • Combining the results we find L(v,>..) = J c (v) ~ J(u) ~ J(u) = L(u,~)
16
W. Velte
Remark, Theorem 1,J,1 shows that as saddle point of the functional the functional
J(,)
and
L(.,.)
can be characterized
u,~
L(.,.) • As we have seen,
is c1osely related to the functionls
Jc(,) • The essential point is that the characteri-
zation by means of
L(.,}
and the system (i} , (ii}
yields,
when disretized, a different numerical scheme for approximating
-
the solution
u • Those schemes resting upon saddle point
characterizations of the solution are commonly used in finite elements.
1,J,2
COMPLEMENTARY PROB
