电磁问题分析中的有限元方程组的快速求解技术
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摘要
随着计算机技术的飞速发展，有限元计算方法以及基于该方法的ANSYS等软件正成为工程和科学研究不可或缺的电磁问题分析工具。采用有限元法分析电磁问题，均可以归结为求解稀疏线性方程组。因此，稀疏线性方程组的快速解法成为有限元计算的关键技术之一，也是计算电磁学中的热点问题。尽管计算机CPU峰值速度和内存存贮容量在不断的提高，但随着工程和科学研究规模增大、复杂程度增加以及对分析精度和速度要求的提高，计算机性能仍然不能满足大规模电磁计算的需要。因此，研究如何进一步减少存贮需求及提高计算效率具有重要的理论和实践价值。
本文构建了基于ANSYS的有限元模型及矢量有限元通用计算公式，针对该技术得到的稀疏线性方程组，在分析现有稀疏线性方程组求解技术的基础上，围绕如何减少存贮需求和提高计算效率的问题，拓展了多波前法、预条件技术以及基于预条件迭代求解的GPU加速技术，主要工作和创新点如下：
1.构建了基于ANSYS的有限元模型及矢量有限元通用计算公式，并分析了稀疏线性方程组求解技术。对基于ANSYS的有限元建模技术进行了研究，给出了多介质区域的模型建立方法，结果验证了该建模技术的可行性；针对闭域和部分开域问题分别推导了有限元基本公式，并给出通用计算表达式；针对基于通用计算公式得到的稀疏线性方程组，分析并讨论了稀疏线性方程组的直接求解方法及其关键技术，特别就稀疏矩阵的Cholesky分解法的几种执行方式的特征进行了对比和分析。
2.提出了有限元稀疏线性方程组的求解方法，并基于该方法提出了预条件技术，将其与迭代法结合进行求解，结果验证了所提方法的正确性和有效性。基于Cholesky分解方法提出了适用于复对称矩阵的拓展的Cholesky分解法，并将之与多波前法结合以用于对称问题的计算；针对非对称问题，采用基于LU分解的多波前法进行求解，并给出具体实施策略；基于对两种多波前方法的研究，论文提出了预条件技术以提高计算效率，并将其与迭代法结合求解有限元方程组。结果验证了所提方法的正确性和有效性。
3.提出了GPU的内存访问准则，并基于该准则，提出一种行排序和三种列排序GPU压缩存贮格式，通过性能比较表明提出的存贮格式能够快速有效地提高计算性能。在研究中，首先分析了GPU的内存模型，并基于不同内存模型的特性提出了相应的内存访问准则；进而，基于该准则研究了GPU计算采用的压缩存贮格式，将所有格式分为行排序和列排序两大类；随后，提出更高效率的MCTO行存贮格式，该格式能够合并同时访问同一内存的地址，从而满足内存访问准则；最后，提出sliced EET, sliced EEV-T and sliced EEV-F三种列存贮格式，三种格式均满足内存访问准则。性能比较表明提出的格式能够具有快速有效的计算性能。
4.针对本文提出的四种压缩存贮格式，提出了相应的GPU并行计算策略。基于对行排序并行策略的研究，提出了适合MCTO格式的并行计算方法，并分别给出该算法在加法、点乘和稀疏矩阵矢量乘中的具体实施策略；基于对列排序并行策略的研究，提出了适合sliced EET、sliced EEV-T以及sliced EEV-F格式的并行计算方法。基于两种压缩存贮格式的并行算法性能评估的结果表明，并行策略能得到更好的计算性能。
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.
本文构建了基于ANSYS的有限元模型及矢量有限元通用计算公式，针对该技术得到的稀疏线性方程组，在分析现有稀疏线性方程组求解技术的基础上，围绕如何减少存贮需求和提高计算效率的问题，拓展了多波前法、预条件技术以及基于预条件迭代求解的GPU加速技术，主要工作和创新点如下：
1.构建了基于ANSYS的有限元模型及矢量有限元通用计算公式，并分析了稀疏线性方程组求解技术。对基于ANSYS的有限元建模技术进行了研究，给出了多介质区域的模型建立方法，结果验证了该建模技术的可行性；针对闭域和部分开域问题分别推导了有限元基本公式，并给出通用计算表达式；针对基于通用计算公式得到的稀疏线性方程组，分析并讨论了稀疏线性方程组的直接求解方法及其关键技术，特别就稀疏矩阵的Cholesky分解法的几种执行方式的特征进行了对比和分析。
2.提出了有限元稀疏线性方程组的求解方法，并基于该方法提出了预条件技术，将其与迭代法结合进行求解，结果验证了所提方法的正确性和有效性。基于Cholesky分解方法提出了适用于复对称矩阵的拓展的Cholesky分解法，并将之与多波前法结合以用于对称问题的计算；针对非对称问题，采用基于LU分解的多波前法进行求解，并给出具体实施策略；基于对两种多波前方法的研究，论文提出了预条件技术以提高计算效率，并将其与迭代法结合求解有限元方程组。结果验证了所提方法的正确性和有效性。
3.提出了GPU的内存访问准则，并基于该准则，提出一种行排序和三种列排序GPU压缩存贮格式，通过性能比较表明提出的存贮格式能够快速有效地提高计算性能。在研究中，首先分析了GPU的内存模型，并基于不同内存模型的特性提出了相应的内存访问准则；进而，基于该准则研究了GPU计算采用的压缩存贮格式，将所有格式分为行排序和列排序两大类；随后，提出更高效率的MCTO行存贮格式，该格式能够合并同时访问同一内存的地址，从而满足内存访问准则；最后，提出sliced EET, sliced EEV-T and sliced EEV-F三种列存贮格式，三种格式均满足内存访问准则。性能比较表明提出的格式能够具有快速有效的计算性能。
4.针对本文提出的四种压缩存贮格式，提出了相应的GPU并行计算策略。基于对行排序并行策略的研究，提出了适合MCTO格式的并行计算方法，并分别给出该算法在加法、点乘和稀疏矩阵矢量乘中的具体实施策略；基于对列排序并行策略的研究，提出了适合sliced EET、sliced EEV-T以及sliced EEV-F格式的并行计算方法。基于两种压缩存贮格式的并行算法性能评估的结果表明，并行策略能得到更好的计算性能。
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.
引文
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