Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry

Abstract

SUMMARYThe boundary knot method (BKM) of very recent origin is an inherently meshless, integration-free, boundary-type, radial basis function collocation technique for the numerical discretization of general partial di erential equation systems. Unlike the method of fundamental solutions, the use of non-singular general solution in the BKM avoids the unnecessary requirement of constructing a controversial artiÿcial boundary outside the physical domain. The purpose of this paper is to extend the BKM to solve 2D Helmholtz and convection-di usion problems under rather complicated irregular geometry. The method is also ÿrst applied to 3D problems. Numerical experiments validate that the BKM can produce highly accurate solutions using a relatively small number of knots. For inhomogeneous cases, some inner knots are found necessary to guarantee accuracy and stability. The stability and convergence of the BKM are numerically illustrated and the completeness issue is also discussed.

Symmetric boundary knot method

“…Unlike the DR-BEM [1,13] and MFS [3] using the singular fundamental solution, the BKM [6][7][8]10] approximates u h by means of the nonsingular general solution, namely,…”

Application of radial basis meshless methods to direct and inverse biharmonic boundary value problems

“…Detailed references can be found in References [25,[31][32][33]. Finally, we want to mention that there are other variations of RBF collocation methods [34][35][36][37], which achieves similar accuracy as Kansa's method but the implementation is a little bit complicated.…”

Solving Helmholtz Equation by Meshless Radial Basis Functions Method

“…For example, hybrid boundary-node method [7], boundary knot method [8], radial point interpolation method (RPIM) [9],meshfree least square-based finite difference method [10], and Element-free Galerkin (EFG) method [11], have been applied to the Helmholtz equation. EFG was also applied to the time-domain field problem [12].…”

“…EFG was also applied to the time-domain field problem [12]. Some of these meshless methods were reported to have some advantages: Significantly lower dispersion than the FEM [9], high convergence rates and high accuracy [7], and suitable for complicate geometry simulation [8].…”