无条件稳定FETD方法中亚网格技术的研究

Research on Subgridding Technology in Unconditionally Stable FETD Method

本文在时域有限元(Finite-Element Time-Domain,FETD)方法中实现了一种新型亚网格技术,并通过空间滤模(Spatial Modes Filtering,SMF)发展为无条件稳定亚网格FETD(Subgridding SMF-FETD,SSMF-FETD)方法.本文给出了FETD中亚网格技术的具体实施方案,包括粗细网格交界处粗网格棱边的编号方案、系统矩阵的建立过程以及亚网格FETD(Subgridding FETD,S-FETD)方法的系统迭代方案.亚网格技术的引入在一定程度上会破坏系统矩阵的稀疏度,但基于有限元框架下建立的系统矩阵依然保持对称正定或半正定特性.因此,可以直接将SMF方法应用到S-FETD方法中.通过广义特征值分解获得S-FETD系统矩阵的不稳定模式,并修改其矩阵方程,进而得到SSMF-FETD方法.S-FETD方法能够有效减少未知量数目,在此基础上数值结果表明,SSMF-FETD可以有效扩大时间步长并保持结果准确,从而进一步提升计算效率.在面对含有复杂精细结构的问题时,所提方法具有较高的有效性和准确性.

A subgridding technology is implemented in the finite-element time-domain(FETD)method,and with spatial modes filtering(SMF)method further developed into the unconditionally-stable subgridding SMF-FETD(SSMF-FETD)method.The specific implementation scheme of the subgridding technology in FETD is given,including the numbering scheme of the edges of the coarse grid at the junction of coarse and fine grid regions,the establishment process of the system matrices,and the system iteration scheme of the subgridding FETD(S-FETD)method.The introduction of subgridding technology will destroy the sparsity of the system matrix to a certain extent,but the system matrices based on the finite element framework still maintain the symmetric positive definite or positive semi-definite characteristics.Therefore,the SMF method can be directly applied to the subgridding FETD method.The unstable modes of the subgridding FETD system matrices are obtained through generalized eigenvalue decomposition,and the subgridding FETD matrix equation is modified to obtain the SSMF-FETD method.The S-FETD method can effectively reduce the number of unknowns.On this basis,numerical results show that the SSMF-FETD method can effectively expand the time step and maintain the accuracy of the results,which further improves the calculation efficiency.The proposed method has high effectiveness and accuracy when facing the problems with complex and fine structures.