Solving axisymmetric potential problems using the indirect boundary element method

M. A. Ponomareva ; E. A. Sobko ; V. A. Yakutenok

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 5 (2015), pp. 84-96

Résumé

The paper contains all necessary relations to implement the indirect boundary element method in order to solve axisymmetric potential problems. It includes fundamental solutions of Laplace’s equation for the potential and flux in the axisymmetric case. These solutions contain complete elliptic integrals of the first and second kinds. Based on this, boundary integral equations were written corresponding to the boundary value problem. The equations were quantized by means of constant elements. The approximate formula for the integral of the fundamental solution for the potential along the element with singularity was obtained using truncated Taylor series and complete elliptic integral of the first kind approximation by a polynomial. In a similar case for the flux, a value of 0.5 was used according to the theorem about the discontinuity in the derivative of the simple layer potential. Approximate convergence of the method was explored based on three test examples. Attained results demonstrate a good convergence of the method, except for the flux computation in a close proximity to the axis of symmetry and corner points, which have nonremovable singularities. It was also shown that using the Gauss quadrature formula with four points is sufficient to estimate the nonsingular integral along the elements with a sufficient level of accuracy.

Détail

Keywords: potential theory, axisymmetric problems, indirect boundary element method, singular integrals.
Mots-clés : Laplace’s equation