A meshless hybrid boundary-node method for Helmholtz problems

Abstract

By coupling the moving least squares (MLS) approximation with a modified functional, the hybrid boundary node-method (hybrid BNM) is a boundary-only, truly meshless method. Like boundary element method (BEM), an initial restriction of the present method is that non-homogeneous terms accounting for effects such as distributed loads are included in the formulation by means of domain integrals, and thus make the technique lose the attraction of its ‘boundary-only’ character. This paper presents a new boundary-type meshless method dual reciprocity-hybrid boundary node method (DR-HBNM), which is combined the hybrid BNM with the dual reciprocity method (DRM) for solving Helmholtz problems. In this method, the solution of Helmholtz problem is divided into two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by means of hybrid BNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of hybrid BNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The proposed method in the paper retains the characteristics of the meshless method and BEM, which only requires discrete nodes constructed on the boundary of a domain, several nodes in the domain are needed just for the RBF interpolation. The parameters that influence the performance of this method are studied through numerical examples and known analytical fields. Numerical results for the solution of Helmholtz equation show that high convergence rates and high accuracy are achievable.