Reply to ‘Comments on ‘Boundary element‐free method (BEFM) and its application to two‐dimensional elasticity problems’’ by Zhigang Chen, International Journal for Numerical Methods in Engineering 2008; 74:347–348

Abstract

In our paper [1], using the Schmidt method, the orthogonal basis function set p = ( p i ) with the weight function can be formed as follows:

An interpolating boundary element-free method (IBEFM) for elasticity problems

“…The inner product (13) and the norm (14) are also taken as the inner product and the norm of the corresponding functions, respectively.…”

“…There are also fewer coefficients in the improved moving least-squares approximation than there are in the MLS approximation, and hence the computing speed and efficiency have increased. Combining the boundary integral equation method with the improved moving least-squares approximation, Cheng and Liew et al [9][10][11][12][13][14][15][16][17][18][19] come up with a direct meshless boundary integral equation method, called boundary element-free method (BEFM), to solve the problems, such as potential problems, elasticity, elastodynamics, and fracture. And the improved element-free Galerkin method based on the improved moving least-squares approximation was discussed by Zhang, Liew and Cheng [21][22][23].…”

“…The meshless method based on the MLS approximation can generate a solution possessing great precision. The meshless boundary integral equation methods are developed by combining the MLS approximation with boundary integral equation methods [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The boundary node method (BNM) is one of the meshless boundary integral equation methods, and Mukherjee et al [3][4][5] used it to solve potential problems and linear elasticity problems.…”

“…Combining different approximation functions in the meshless method with the boundary integral equation method, researchers have developed meshless boundary integral equation methods, such as the boundary node method (BNM) [3][4][5], the local boundary integral equation (LBIE) method [6][7][8], and the boundary element-free method (BEFM) [9][10][11][12][13][14][15][16][17][18][19].…”

An improved moving least-squares Ritz method for two-dimensional elasticity problems

“…The element-free or mesh-free methods have been extensively researched because of its important application for solving mathematical and physical problems [1][2][3][4][5][6][7][8][9][10]; especially when the traditional computational methods are not well suited for such problems that involved extremely large deformation, dynamic fracturing or explosion problems [11]. Based on different approximation functions, various element-free or mesh-free methods were proposed, including the element-free Galerkin method [12], the hp clouds method [13], the moving least-squares differential quadrature method [14,15], the reproducing kernel particle method [16], wavelet particle method [17], the radial point interpolation method [18][19][20], the complex variable meshless method [21,22] and the meshless boundary integral equation methods [23,24].…”