多尺度径向基无单元法求解电磁对流扩散问题

Solving the problem of electromagnetic convection-diffusion using multiscale combined RBF collocation method

提出了多尺度径向基无单元法,该方法继承了径向基无单元法作为一种纯无单元方法的优势,选择主径向基函数和辅助径向基函数等含有多个尺度特征的径向基函数的组合形式替代单一尺度特征的径向基函数,然后通过辅径向基函数对主径向基参数的正交化过程形成所求解问题的系统矩阵,该替代过程克服了传统径向基无单元法使用单一尺度进行求解的局限性,能准确体现含多重变化特征的解,特别适合处理对流扩散方程中对流项占优的高速运动导体涡流问题等电磁场对流扩散问题.以对流项绝对占优的经典对流扩散方程为例,用有限元方法和传统径向基无单元法作为对比,证实了方法的有效性.结果表明,该方法可以准确求解电磁场对流扩散方程,而有限元方法和传统径向基无单元方法所得的数值解出现较大的数值震荡.

A novel multi-scale combined radial basis function （RBF） collocation method, as a truly mesh-less method, by combining two types of RBFs with different shape scales, namely the primary RBFs and the auxiliary RBFs, and orthogonalizing the auxiliary RBFs and the primary RBFs to form a systematical solving matrix instead of using a single RBF for collocation mesh-less techniques, is presented to overcome the shortcomings of general RBF collocation method and applied to analyzing eddy currents in a typical convection-diffusion problem with dominant convection terms, such as high speed moving conductors eddy current problems. A typical example is set to illustrate the accuracy and effectiveness of the proposed method, including a comparison with general RBF collocation method and finite element method （FEM）. It indicates that the numerical solution to the typical convection-diffusion problem from traditional RBF mesh-less method and FEM will cause to spurious oscillations, but the present method can get accurate solution. Therefore, the proposed method is greatly superior to FEM and the general RBF collocation methods in analyzing convection-diffusion problems.