Fast Solution Techniques For Finite Element Equations In Electromagnetic Analysis

With the fleeting progress of computer technology, the finite element method(FEM) and FEM-based softwares such as ANSYS are becoming indispensable tools forthe design and analysis of electromagnetic problems. The key of the FEM inelectromagnetic problems is to solve sparse linear equations. Fast solution of sparselinear equations is one of the most important technologies in the FEM calculation, andis a hot issue in computational electromagnetics. With the increasing scale, addingcomplexity, and rising requirements of accuracy and speed in engineering and scientificresearch, computer performance still can not meet the needs of large-scale calculationeven though the peak processing power and memory bandwidth of Central ProcessingUnit (CPU) has constantly increased. Therefore, the research on how to further reducestorage requirements and improve the computational efficiency has importanttheoretical value and practical significance.This thesis builts the finite element model based on the ANSYS and the generalfinite element expression. Based on the analysis of exsting techniques for direct solvingsparse linear equations equations, we extend the Multifrontal (MF) algorithm,preconditioning techniques and Graphics Processing Unit (GPU) accelerate strategies(for the solution of precondition iterative methods) around the solution of sparse linearequations arising vector FEM. Main work and innovation of this thesis include:1. FEM models and general expression are established, and the direct methodsfor solving sparse linear equations equations are discussed. We present FEM modelingtechniques on the basis of ANSYS for multidielectric region, and the examples show thefeasibility of using our technology. Then we give general expressions of vector FEM forthe closed region and some open region problems. Based on the sparse linear equationsobtained by using the expression, we analyze and discuss the direct methods and theirkey technology. Comparison and analysis are made on the implementationcharacteristics of several Cholesky decomposition methods.2. Several solution methods for sparse linear equations are presented, and, apreconditioning technique is proposed and combined with iterative to solve theequations. Results show the proposed methods are correct and effective. Based on theCholesky-based MF method, we propose an expanded Cholesky method (ECM) andcombine it with MF for solving complex symmetric equations. Then we use LU-based MF method to solve unsymmetric equations and present its specific implementionmeasures and methods. Based on the study of the MF algorithm, we propose apreconditioning technique to improve the computation efficiency. And then, wecombine preconditioning technique and an iterative to solve the equations. Results showthe proposed methods are correct and effective.3. The memory access rules is presented and, based on the rules, a row-majorand three column-major compression storage formats are proposed. Performancecomparisons show the proposed formats can have fast and efficient computingperformance. We propose memory access rules on the basis of analysis about thememory models. Based on the rules, we study the compression storage formats used byGPU computation and group these formats into two main classes. Then we propose amore efficient row-major format called MCTO which can coalesce simultaneousaccessed addresses to memorys, and thus the MCTO satisfy the memory access rules.We propose three column-major formats including sliced EET, sliced EEV-T and slicedEEV-F which also satisfy the memory access rules. Performance comparisons show theproposed formats can have fast and efficient computing performance.4. The parallelization strategies for a row-major and three column-majorcompression storage formats. Performance evaluations verify the better performance ofthe parallel algorithms. Based on the study of row-major parallelization strategies, wepropose suitable parallelization strategy for MCTO and present the specificimplementation methods of addition, dot product and sparse matrix vector product(SMVP). Based on the study of column-major parallelization strategies, we proposesuitable parallelization strategies for sliced EET, sliced EEV-T and sliced EEV-Fformats. Performance evaluations verify the better performance of the parallelalgorithms.