无界域电磁场问题的有限元-本征函数展开结合解法

Hybrid Finite Element/Eigenfunction Expansion Method for Unbounded Electromagnetic Field Problems

基于区域分裂的思想，通过引入一虚拟圆形边界，将整个无界域划分为圆内、外两部分．圆内有限区域采用通常的有限元法离散，而将整个圆外无限区域看作一个“无限大”的单元，采用以本征函数展开的Ｆｏｕｒｉｅｒ级数作为插值函数的离散方法．这种处理方法能够很好地把有限元法的求解区域推广至无穷远，实现了它在无界域电磁场数值分析中的应用，从而形成了一种新型的解法．文中以无界域中典型电磁场问题为例证，说明了有限元－本征函数展开结合解法简单易行、方便直观，具有实用价值．此外，场的离散化表示比较精确，保留了有限元法系数阵的特点，且具有变量较少和计算效率高等优点．

Based on the idea of region division, a novel technique known as the “Hybrid Finite Element/Eigenfunction Method” is developed for treating unbounded field problems in this paper. The general principle of the technique is first to introduce a fictitious circular boundary to enclose the structures with sources and inhomogeneous dielectrics.In the interior of the circular boundary, the finite element method is used to formulate the fields, whereas in the exterior region, the field is presented by an expansion of eigenfunction (in Fourier series). A macro element is employed for the interior which requires only potential continuity at the boundary nodes with the interior finite elements, so as to extend finite element method to unbounded field problems. The method has been applied to a variety of unbounded electromagnetic field problems, and shown to be accurate and powerful. The numerical examples of some canonical open region problems are included to demonstrate the accuracy, efficiency and capability of this method. The new hybrid scheme combining an analytical solution with the finite element method offers several advantages: (i) symmetric, sparse and banded matrices; (ii) much fewer elements and nodes for the same level of accuracy; (iii) easy means of calculating the external field, once the Fouriers coefficients are determined and the potential solution is obtained.