H2矩阵快速方法求解电磁散射问题研究

Fast Method for Solving Electromagnetic Scattering Problems Based on H^2-Matrix Algorithm

求解电大尺寸目标的电磁问题的关键是计算量和存储量。本文分析了共轭梯度法解基于分层基层次型(H2)矩阵快速算法的电磁散射问题时单步迭代的计算量。一方面临时存储部分矩阵向量积而避免层间及同层同一矩矢积的重复计算,以较小的存储增量为代价明显加快了计算速度;另一方面根据均匀Lagrange插值退化核函数矩阵的多层Toeplitz矩阵特性,压缩存储退化核函数矩阵,并用快速傅里叶变换加速该矩阵与向量的乘积。算例结果显示了本文方法在减少内存占用量和缩短单步迭代时间方面的有效性。

The critical issues of solving electrically large-scale electromagnetic problems are the computing time and memory requirement.In the frame of Hierarchical Basis H-matrix(H^2) fast algorithm,the computation cost of each single step of iteration in conjugate gradient method(CG) which is used to solve the linear algebraic equations is analyzed.On the one hand,a portion of matrix-vector products is stored temporarily which avoids double-count among different layers and the same layer.The computation cost reduces significantly at the cost of little increase in memory storage.On the other hand,the original kernel function matrix is degenerated by Lagrange interpolation and the degeneration kernel function matrix is stored in compression format according to the property of multi-layer Toeplitz matrix.Otherwise,the FFT is applied to speed up the product of matrix and vector.Examples are given to validate the effectiveness of the proposed method both in reducing the memory requirement and the computation time of each step of the iteration.