Narrow Results

The boundary element method (BEM), along with the finite element and finite difference methods, is commonly used to carry out numerical simulations in a wide variety of subjects in science and engineering. The BEM, rooted in classical mathematics of integral equations, started becoming a useful computational tool around 50 years ago. Many researchers have worked on computational aspects of this method during this time.

This paper presents an overview of the BEM and related methods. It has three sections. The first, relatively short section, presents the governing equations for classical applications of the BEM in potential theory, linear elasticity and acoustics. The second describes specialized applications in bodies with thin features including micro-electro-mechanical systems (MEMS). The final section addresses current research. It has three subsections that present the boundary contour, boundary node and fast multipole methods (BCM, BNM and FMM), respectively. Several numerical examples are included in the second and third sections of this paper.

The Boundary Node Method (BNM) is a meshfree scheme for solving Boundary Integral Equations (BIE). BNM simplifies the problem at two levels. Primarily, as BNM aims to solve the integral form of the governing equation, dimensionality of the problem is reduced by one order. Additionally, the mesh-free scheme eliminates the need for meshing, proving particularly beneficial for problems involving complex geometries. In this study, BNM is formulated using Element Free Galerkin (EFG) scheme to solve interior and exterior problems in acoustics. A 3-dimensional linear basis is used to construct moving least square shape functions. This eliminates the need to define additional local or parametric coordinate systems, making the method easily applicable to any arbitrary geometry. To illustrate the capabilities of the proposed method, four example problems are solved. An open pipe at resonance and a simple expansion muffler are analyzed to validate BNM’s performance in solving interior problems. Radiation from a pulsating sphere and radiation from a sphere with harmonic velocity excitation are analyzed as examples of exterior problems. Results from all four sample problems indicate that the BNM scheme proposed for solving acoustic problems is accurate and robust.

A new meshless method, called regular hybrid boundary node method (RHBNM) is developed by the authors, which combines the MLS interpolation scheme with the hybrid displacement variational formulation, and the source points of the fundamental solutions are located outside the domain. In this paper, the formulation of the RHBNM is presented briefly, taken the 2D potential problem as example. Then the numerical examples, not only for 2D potential problem, but also for 3D potential and 2D, 3D elasticity problems, are given to show the accuracy and applicability of this method.