Abstract
A localized virtual boundary element–meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex two-dimensional (2D) geometries. “Localized” refers to employing the moving least square method to locally approximate the physical quantities of the computational domain after introducing the traditional virtual boundary element method. The LVBE-MCM is a semi-analytical and domain-type meshless collocation method that is based on the fundamental solution of the governing equation, which is different from the traditional virtual boundary element method. When it comes to 2D problems, the LVBE-MCM only needs to calculate the numerical integration on the circular virtual boundary. It avoids the evaluation of singular/strong singular/hypersingular integrals seen in the boundary element method. Compared to the difficulty of selecting the virtual boundary and evaluating singular integrals, the LVBE-MCM is simple and straightforward. Numerical experiments, including irregular and doubly connected domains, demonstrate that the LVBE-MCM is accurate, stable, and convergent for solving both Laplace and Helmholtz equations.
Highlights
The boundary element method (BEM) [1,2] is a well-known numerical method that has become an alternative to domain methods such as the finite element method (FEM) [3,4]for the simulation of certain physical problems
The fundamental solutions are introduced into the traditional moving least squares method, and we developed the augmented moving least squares approximation, which is similar to the one outlined in [35]
The traditional virtual boundary element method with a global approximation was modified to a local approximation approach by introducing the moving least square method and local approximation theory
Summary
The boundary element method (BEM) [1,2] is a well-known numerical method that has become an alternative to domain methods such as the finite element method (FEM) [3,4]. Similar to the fundamental solution method, the VBEM uses fundamental solutions as the basis functions and requires a virtual boundary outside of the physical domain to avoid source singularity. The selection of this artificial boundary is still a well-known tricky issue in spite of the great deal of effort that has been made to address this problem [32,33], especially in terms of complex geometries. The resulting LVBE-MCM system is sparse and can be solved using an ordinary computer This means that the method has certain application prospects for solving large-scale problems.
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