Accurate evaluation of integrals present in reciprocity methods

Reciprocity methods generate boundary integrals of the form ∫Γh(x)f(r)g(R) dΓ, where f is singular, r and R are distances measured from a source point and a basis collocation point, respectively. This paper is concerned with the accurate numerical evaluation of integrals of this type. The approach adopted involves the approximation of g(R) by a polynomial p(r), obtained by truncating a Taylor series. The integral ∫Γh(x)f(r)g(R) dΓ is equal to ∫Γh(x)f(r)(g(R)−p(r)) dΓ+∫Γh(x)f(r)p(r) dΓ. The polynomial p(r) is designed to annihilate, where possible, the singularity in ∫Γh(x)f(r)(g(R)−p(r)) dΓ and thus facilitate evaluation using standard quadrature. The integral ∫Γh(x)f(r)p(r) dΓ is sufficiently simple to be transformed into a contour integral, which can be evaluated numerically using Gauss–Legendre quadrature. To demonstrate implementation of the scheme the thermoelastic BEM is considered. Numerical tests are performed on a simple test-problem for which a known analytical solution exists. The results obtained using the semi-analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods.