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REGULAR BOUNDARY INTEGRAL EQUATIONS FOR STRESS ANALYSIS

c.

Patterson and M.A. Sheikh

Dept. of Mechanical Eng., University of Sheffield, U.K. INTRODUCTION THE boundary integral method is now well established as a general numerical technique available for the solution of field problems. In contrast with the finite element method freedoms need only.be defined on the boundary of the domain of the problem. Once these are determined the solution within the domain is obtained using appropriate surface integrals of the boundary solution. Central to the method is the generation of Boundary Integral Equations which properly state the problem to be solved in terms of unknown field functions on the boundary only. These equations are usually obtained using the Fundamental Solution of the given problem with the singular point located on the boundary. (The equations for the interior solution are obtained similarly, by locating the singular point within the domain of the problem). There ensues an infinite system of singular surface equations, one for each boundary point (being generated by moving the singularity around the boundary) . The system is discretized by defining boundary elements, after the manner of finite elements, and the resulting finite system of singular integrals are evaluated, thereby giving a system of algebraic equations. Two discomforting features are apparent in this, conventional, approach. Firstly, accurate evaluation of the singular integrals requires special and careful treatment in the neighbourhood of the singular point. Secondly, the class of problems for which the method is well defined may be unduly resttrictive because of divergence of the equations. In this paper it is shown that 'Regular Boundary Integral Equations can quite readily be derived which also properly state the given problem. These are obtained by the simple device of moving the singularity of the fundamental solution

C. A. Brebbia (ed.), Boundary Element Methods © Springer-Verlag Berlin Heidelberg 1981

86 outside the domain of the problem. The resulting system of equations tolerates higher order singularities in the solution than previously and requires no special attention to a singular integrand. The practicality of the method is demonstrated in twodimensional elastostatics. A critical comparison is made of the results obtained using the new and conventional approaches for constant and linear elements. THEORY The governing equations of elastostatics in terms of the stress field and in the absence of body forces can be written as:

aa ..

3

l:

~J

i

0

'1~ J= J

1, 21 3

(1)

The stresses and strains are related by the constitutive relation for an isotropic body as:

+ 2]..1£ij

A.o ..

£ ..

J ox. + ox.

~J

~J

where