Meshless method based on collocation with consistent compactly supported radial basis functions

Based on our previously study, the accuracy of derivatives of interpolating functions are usually very poor near the boundary of domain when Compactly Supported Radial Basis Functions (CSRBFs) are used, so that it could result in significant error in solving partial differential equations with Neumann boundary conditions. To overcome this drawback, the Consistent Compactly Supported Radial Basis Functions (CCSRBFs) are developed, which satisfy the predetermined consistency conditions. Meshless method based on point collocation with CCSRBFs is developed for solving partial differential equations. Numerical studies show that the proposed method improves the accuracy of approximation significantly.

[1]  Ming-Wan Lu,et al.  Imposition of essential boundary conditions by displacement constraint equations in meshless methods , 2001 .

[2]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[3]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[4]  Satya N. Atluri,et al.  A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method , 1998 .

[5]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[6]  Satya N. Atluri,et al.  The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity , 2000 .

[7]  Xiong Zhang,et al.  Meshless methods based on collocation with radial basis functions , 2000 .

[8]  Wen Chen.  Meshfree boundary particle method applied to Helmholtz problems , 2002 .

[9]  M. Buhmann.  Radial functions on compact support , 1998 .

[10]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[11]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[12]  Ted Belytschko,et al.  Overview and applications of the reproducing Kernel Particle methods , 1996 .

[13]  Wen Chen,et al.  Symmetric boundary knot method , 2002, ArXiv.

[14]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[15]  Ming-Wan Lu,et al.  Least‐squares collocation meshless method , 2001 .

[16]  T. Liszka,et al.  hp-Meshless cloud method , 1996 .

[17]  Wing Kam Liu,et al.  Admissible approximations for essential boundary conditions in the reproducing kernel particle method , 1996 .

[18]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[19]  S. Timoshenko,et al.  Theory of Elasticity (3rd ed.) , 1970 .

[20]  E. Oñate,et al.  A FINITE POINT METHOD IN COMPUTATIONAL MECHANICS. APPLICATIONS TO CONVECTIVE TRANSPORT AND FLUID FLOW , 1996 .

[21]  Ted Belytschko,et al.  Enforcement of essential boundary conditions in meshless approximations using finite elements , 1996 .

[22]  Satya N. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics , 2000 .

[23]  Zongmin Wu,et al.  Compactly supported positive definite radial functions , 1995 .