扩散过程数值仿真的径向基函数方法

Numerical Simulation of Diffusion Processes Based on Radial Basis Function Method

扩散问题仿真的传统网格方法在求解中常面临网格剖分和近似精度较低的问题,因而将径向基函数方法引入进来,作为一类配点型无网格方法,它不再需要网格剖分,而且基函数光滑性好,近似精度高。详细阐述了径向基函数方法和差分法结合求解扩散方程的原理,给出方法的具体实施方案和离散求解模型,并以热传导和涡流模型为例进行仿真和分析,结果表明该方法在仿真扩散过程中不仅实施简单,而且计算精度较高,即使在较大的时间步距下也能得到较高的精度。

Traditional mesh - based methods, such as finite difference method and finite element method face two big challenges in solving diffusion problems. One comes from the mesh generation, which is a big difficulty especially in complex geometry. The other originates from the local linear approximation, whose approximation accuracy is poor. So a radial basis function method was introduced in this paper in order to solve the drawbacks. As a meshless collocation type method, RBF needn＇ t mesh generation, and can gain high approximation accuracy because of good smooth- ness of basis function. The basic idea of this method is using RBF to approximate the part of spatial function whereas difference method for the part of temporal function. Principles of the combination method of RBF and difference method are presented in detail. Then practical implementation scheme and discrete model were constructed. Two simulation instances of heat conduction and eddy current model were carried out. The results show that the RBF method can not only acquire high accuracy even though the time interval is set long enough in numerical simulations, but also can be implemented easily.