Chin. Phys. B, 2009, Vol. 18(10): 4065    DOI: 10.1088/1674-1056/18/10/002
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An improved boundary element-free method (IBEFM) for two-dimensional potential problems

Ren Hong-Ping, Zhang Wu
School of Computer Engineering and Science, Shanghai University, Shanghai 200072, China
Abstract  The interpolating moving least-squares (IMLS) method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method (BEFM), combining the boundary integral equation (BIE) method with the IMLS method, the improved boundary element-free method (IBEFM) for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker δ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker δ function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.
Keywords:  moving least-squares approximation      improved boundary element-free method      interpolating moving least-squares method      potential problem      meshless method
Received:  07 October 2008      Revised:  01 February 2009      Accepted manuscript online:
 PACS: 02.70.Rr (General statistical methods) 02.30.Em (Potential theory) 02.30.Mv (Approximations and expansions) 02.30.Rz (Integral equations) 02.30.Sa (Functional analysis) 02.60.Nm (Integral and integrodifferential equations)
Fund: Project supported by the National Natural Science Foundation of China (Grant No 10871124), Innovation Program of Shanghai Municipal Education Commission (Grant No 09ZZ99) and Shanghai Leading Academic Discipline Project (Grant No J50103).