Astudy On Hierarchical-Matrix-Based Algorithm For Solving Electromagnetic Problems Of The Volume Integral Equation

In the field of computational electromagnetism, fast algorithms for accelerating numerical method are main techniques for solving problems. Hierarchical Matrix (short for H-matrix) method is a typical method. Essentially, in H-Matrix-Based methods, the source variables and field variables of the integral kernel functions are separated to obtain a degenerated kernel, to reduce the computational operations of iteration procedures and to save the storage space for the coefficient matrix.In this thesis, firstly, an algorithm for the fundamental relationships between nodes, lines, facets, and tetrahedra of basic elements of discretized scatterer is presented and implemented for common facets between two adjacent tetrahedral and outer facets of tetrahedral on the surface, which the complexity is linear order. Secondly, Octree theory is combined with the algorithm to reorder all the common and non-common facets for H-matrix structures, which can order the triangular cells number in a hierarchical way. Thirdly, the Lagrange interpolation polynomial to make degenerate the kernel function of integral equation is studied, and the MOM matrix can be decomposed to a sparse matrix. Lastly, dielectric cube, sphere and cylinder with a finite length are analyzed using the proposed methods. The numerical results validates the correctness and validity of the algorithm raised. The numerical examples show that the required memory space and the CPU time are proportional toΟ? NlogN? by using the H-matrix-based method. H-matrix-based method with proper degenerated kernel may be more explored for algorithms with higher efficiency.