H-matrix Direct Method In The Analysis Of Electromagnetic Simulation

The development of computational electromagnetic requires that numerical methods can solve larger-scale problems. However, classical numerical computational methods are very difficult to meet with the memory and computational time requirements when solving these problems. Based on tangential vector-finite-element methods, fast-direct-solver techniques is concerned in this thesis. We develop an H-matrix-based direct finite-element solver for large-scale electromagnetic analysis.Firstly, we have an introduction to H-matrix and H-matrix arithmetic, including the concept of H-matrix and the operations defined in H-matrix arithmetic. The detailed numerical procedure of the H-matrix-based direct solver is given, and the complexity of the H-matrix-based direct solver is analyzed.Secondly, the vector FEM-based analysis of general electromagnetic problems is outlined. We proved that the sparse matrix resulting from a finite element based analysis can be represented by an H-matrix without any approximation, and the inverse or LU of this sparse matrix has a data-sparse H-matrix approximation with well error controlled. Based on this proof, we develop an H-matrix-based direct finite-element solver, which has O(N log N) memory complexity and O(N log2 N) time complexity. A method called iterative improvement of solution is proposed for the improvement of the approximate solution.Then, based on the sparse matrix direct solver, we develop a H-matrix preconditioner to accelerate the iterative solution of multilevel fast multipole approach (MLFMA). The numerical results can indicate that the significant convergence improvement is achieved.Finally, combined with the multilevel UV method, H-matrix-based direct solver is developed to solver the integral equations.