The Coupling of Boundary and Finite Element Methods for Infinite Domain Problems in Elasto- Plasticity
The implementation of a coupled analysis capability into an existing Finite Element computer program is discussed. The coupled analysis is then applied to a circular excavation in a infinite domain where the region of plasticity is confined to the Finite
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THE COUPLING OF BOUNDARY AND FINITE ELEMENT METHODS FOR INFINITE DOMAIN PROBLEMS IN ELASTO- PLASTICITY G. Beer and J.L. Meek Dept. of Civil Engineering, University of Queensland, Australia ABSTRACT The implementation of a coupled analysis capability into an existing Finite Element computer program is discussed. The coupled analysis is then applied to a circular excavation in a infinite domain where the region of plasticity is confined to the Finite Element mesh. Further potential usage of the coupled analysis is then discussed in relation to mine design. INTRODUCTION The coupling of Boundary Element and Finite Element methods was first discussed in a general context by Zienkiewicz et al., (1977) although some more specific applications appeared earlier (Chen 1974). The method was applied to a number of field problems and problems in fluid mechanics (Kelly 1979). Application to problems in elasto-statics have appeared more recently (Brebbia 1979 and Mustoe 1980). None of these, however, deal with the 'exterior' problem i.e. one involving an infinite domain. The principle approach of the above methods is to obtain a stiffness matrix of the region which is bounded by Boundary Elements. The stiffness matrix is then made symmetric by an energy approach or by minimising the errors in the non-symmetric terms. This method will be called the Symmetric Dire~t Boundary Element method. At the same time as the above developments took place, a completely different approach was presented by Ungless for elasto-static problems and by Bettess for field problems and problems in fluid mechanics. This approach used special Finite Elements which extended to infinity in one or more directions. The functions or displacements were assumed to decay in the infinite direction exponentially or inversely proportional to the distance (l) from a decay origin. The authors have later R
C. A. Brebbia (ed.), Boundary Element Methods © Springer-Verlag Berlin Heidelberg 1981
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shown that a decay of R gives good results for underground excavation problems. In contrast with the Boundary Elements, these Elemen~still require a volume integration which is carried out numerically in a finite mapped region of the infinite domain. These two approaches will be examined herein in more detail. Both methods have been programmed, incorporated in an existing Finite Element program, and run on a PDP-11 mini computer at the University of Queensland. SYMMETRIC DIRECT BOUNDARY ELEMENT METHOD Boundary integral equations The derivation of Boundary Integral equations for elastostatics is well known (Rizzo 1967) and will not be repeated here. This paper follows the notation by Watson. For the exterior problem, the following integral equation is obtained (see Fig. 1) : c .. (x)u.(x) ~J
J
+
s
I
T .. (x,y)u.(y) dSy J
~J
=I u .. (x,y)t.(y) s J ~J
dSy
(1)
where lim e:-+0
I
(2)
Ti.(x,y) dSy J
S(x,e:)
INFINITE
t. S(X,E)
Figure 1.
DOMAIN
x2
L
Opening in an infinite domain
x1
577
In the above equation, x is a point on the surface and y the integratio
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