Coupling projection domain decomposition method and Kansa's method in electrostatic problems
Introduction
In recent years, the theory of radial basis functions (RBFs) has undergone intensive research and enjoyed considerable success as a technique for interpolating multivariable data and functions [1]. Although most work to date on RBFs relates to scattered data approximation and in general to interpolation theory, there has recently been an increased interest in their use for solving PDEs. This approach, which approximates the whole solution of the PDE by a translates of RBFs, is very attractive due to the fact that it is a truly meshless method and spatial dimension independent, which can easily be extended to solve high dimensional problems. Furthermore, since the RBFs are smooth, it can easily be applied to solve high order differential equations. Unsymmetric collocation method to solve PDEs using radial basis functions was first proposed by Kansa [2] and is extensively studied by Schaback [4], [5], [6], [7]. It has been applied to solve some typical electromagnetic problems [8], [9].
In the view of parallelism, The major applied technique is the domain decomposition method (DDM). It is nowadays considered as one of the most popular technique that can be applied for numerical solution of partial differential equations. The idea behind the domain decomposition is to divide the considered domain into a number of subdomains and then try to solve the original problem as a series of subproblems that interact through an internal interfaces. The numerical solution can be computed either iteratively, using Schwarz method [3], by changing data interfaces between subproblems or by computing the interfaces data directly using Steklov technique and then use interface solution to solve each subproblem separately, namely, projection domain decomposition (PDM). Compared with another domain decomposition method, the PDM need neither the suitable choice of the acceleration parameters to ensure the convergent rate of the iteration arising in DDM nor the iteration at each subdomains. This property is meaningful for meshless method using radial basis function coupled with DDM because that the ill-conditioned linear system arising in the RBF-based meshless method.
The main objective of this paper is to couple the project domain decomposition method with asymmetric collocation method based on radial basis functions to solve electrostatic problem associated with an asymmetrical shielded stripline and shielded coupled-striplines. The paper is organized as follows. In Section 2, we use a general elliptic problem to illustrate the coupling method of PDM and RBF-based meshless collocation method. In Section 3, electrostatic problems associated with an asymmetrical shielded stripline and shielded coupled-striplines are solved by the proposed method. Conclusions are drawn in the final section, Section 4.
Section snippets
Meshless PDM using RBFs
As we known that the governing equation in magneto/electrostatic problems is an elliptic equation. Without the loss of generality, we use a general elliptic boundary value problem (BVP) to illustrate the meshless unsymmetric collocation method using RBFs, projection domain decomposition and the coupling algorithm.
Numerical examples
Example A Fig. 1 shows the cross-section of an asymmetric shielded stripline. The strip has a width of and a thickness of . The bisector in Fig. 1 of the thickness of the strip divides the height of the outer rectangular conductor into the lower and upper parts whose heights are and , respectively. Similarly, the bisector in Fig. 1 of the width of the strip divides the width of the outer conductor into the left and right parts with widths of
Conclusions
In this paper, a coupling algorithm of the RBFs-based unsymmetric method and projection domain decomposition for solving magneto/electrostatic problems is given. Unlike the FEM which interpolates the solution by using low order piecewise continuous polynomials or the FDM where the derivatives of the solution are approximated by finite quotients, the RBF-based meshless method provides a global interpolation formula not only for the solution but also for the derivatives of the solution. The
Acknowledgements
This work was supported by Research Foundation for Young scholar of UESTC under JX0646, 973 Programs (2008CB317110), NSFC (10771030), the Scientific and Technological Key Project of the Chinese Education Ministry (107098), the PhD. Programs Fund of Chinese Universities (20070614001), Sichuan Province Project for Applied Basic Research (2008JY0052) and the Project for Academic Leader and Group of UESTC.
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