Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems

Abstract

SUMMARYIn this study, we first discuss the moving least-square approximation (MLS) method. In some cases, the MLS may form an ill-conditioned system of equations so that the solution cannot be correctly obtained. Hence, in this paper, we propose an improved moving least-square approximation (IMLS) method. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill-conditioned system of equations. Combining the boundary integral equation (BIE) method and the IMLS approximation method, a direct meshless BIE method, the boundary element-free method (BEFM), for two-dimensional elasticity is presented. Compared to other meshless BIE methods, BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied easily; hence, it has higher computational precision. For demonstration purpose, selected numerical examples are given.

A General 2D Meshless Interpolating Boundary Node Method Based on the Parameter Space

“…Meshless (mesh-free) methods have been developed in the past decades to reduce the required effort for mesh generation. Many meshless methods have been proposed thus far, including the smoothed particle hydrodynamics (SPH) method [6,7], the reproducing kernel particle methods (RKPM) [8], the hpclouds method [9], the element-free Galerkin method (EFG) [10], the meshless local Petrov-Galerkin (MLPG) approach [11], the boundary node method (BNM) [12][13][14], the boundary element-free method (BEFM) [15], the hybrid boundary node method (hybrid BNM) [16][17][18][19][20], the Galerkin boundary node method (GBNM) [21], the boundary face method (BFM) [22,23], and the boundary point interpolation method [24].…”

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

“…Based on the flexibility of the moving least-squares (MLS) approximation, the extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the MLS approximation. In deriving the MLS shape function, an orthogonal basis function set [22][23][24][25] is used to avoid ill conditioning and improve the accuracy of results. Because the MLS shape functions are unknown in a closed form, the singular integrals must be evaluated using a numerical scheme.…”

Comments on ‘Boundary element‐free method (BEFM) and its application to two‐dimensional elasticity problems’ by K. M. Liew, Y. Cheng and S. Kitipornchai, International Journal for Numerical Methods in Engineering 2006; 65:1310–1332

“…While using a set of orthogonal basis functions for the moving leastsquares approximation appears to be a good idea, the actual basis functions obtained using the Schmidt method are generally not orthogonal in the inner product space defined in [1]. For example, according to Liew et al [1], the first two orthogonal basis functions formed using the Schmidt method are as follows:…”

“…The reason behind this issue stems from the definition of the inner product space in [1]. In the inner product space defined in [1], the inner product is dependent on the position vector x, which is totally different from the conventional definition of inner product space. As a result, the basis functions obtained using the conventional Schmidt method [2] are generally not orthogonal.…”

“…Comments on 'Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems' The above paper [1] proposes using a set of orthogonal basis functions to improve the moving least-squares approximation. While using a set of orthogonal basis functions for the moving leastsquares approximation appears to be a good idea, the actual basis functions obtained using the Schmidt method are generally not orthogonal in the inner product space defined in [1].…”