Treatment of singularities in Helmholtz-type equations using the boundary element method

Abstract

In many engineering problems, boundaries with sharp corners or abrupt changes in the boundary conditions and/or the material properties give rise to singularities of various types which tend to slow down the rate of convergence with respect to decreasing the mesh size of any standard numerical method used for obtaining the solution. In this paper, in order to develop a method which overcomes this difficulty, the singular solutions of isotropic and anisotropic Helmholtz-type equations with homogeneous Dirichlet and/or Neumann boundary conditions in the neighbourhood of a singular point are derived. The standard boundary element method (BEM) is then modified to take account of the form of singularity, without an appreciable increase in the computational effort and at the same time keeping a uniform discretization. Three examples are carefully investigated and the numerical results presented show an excellent performance of the approach developed.