Accurate evaluation of integrals present in reciprocity methods

K. Davey, M. T Alonso Rasgado

    Research output: Contribution to journalArticlepeer-review


    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. © 2001 Elsevier Science Ltd. All rights reserved.
    Original languageEnglish
    Pages (from-to)2511-2526
    Number of pages15
    JournalComputers and Structures
    Issue number29-30
    Publication statusPublished - 2001


    • Boundary elements
    • Domain integral
    • Numerical integration
    • Reciprocity method


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