On the accuracy of adaptive quadratures in the numerical integration of singular Green's functions for layered media

Ana Carolina Azevedo Vasconcelos¹; Iago Cavalcante Geraldo¹; Josué Labaki¹

¹ Universidade Estadual de Campinas

doi:10.20906/CPS/CILAMCE2017-0063

Resumo

This work investigates methods for the numerical evaluation of definite integrals of singular functions. Such integrals arise in the solution of many problems by the boundary element method (BEM). Influence functions describing time-harmonic axisymmetric loads within half-spaces, for example, which occur in many soil-foundation interaction problems, can be derived in terms of Hankel integral transforms. These integrals contain a number of singularities corresponding to shear, pressure and Rayleigh wave numbers. This work investigates integration schemes to deal with singular integrands, taking as a representative example the aforementioned influence function. A normalization of the integration interval according to Hankel's wave number causes the singularities to fall within a predictable region, after which only an oscillatory-decaying behavior of the integrand remains. The interval where the singularities fall is divided at the singularities into a number of subintervals. The singularities were located by direct mathematical manipulation of the integrands for the case at hand. A technique of regularizing the singularities by the synthetic division of polynomials and its subsequent integration with a modified Legendre quadrature was used to solve the integral. A straightforward adaptive Gaussian quadrature was used for comparison. The results are compared with the analytical integration in representative cases. The findings in this article contribute to the accuracy in the numerical solution of integrals that are common in many branches of mathematical physics, remarkably in Green's functions and their applications in the BEM.

Palavras-chave: Numerical Integration; Singular Integrands; Adaptive Quadratures; Green's Functions