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H-matrix based fast direct finite-element methods for large-scale electromagnetic analysis.

机译：基于H矩阵的快速直接有限元方法，用于大规模电磁分析。

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摘要

In this work, we introduce a general mathematical framework called the "hierarchical ( H ) matrix" framework to reduce the computational complexity of finite-element-based analysis of electromagnetic problems. In the mathematical literature, the existence of an H -matrix approximation was only proved for elliptic partial differential equations that govern static phenomena. We provide the first proof for electrodynamic analysis. Based on the proof, we develop an H -matrix-based direct finite-element solver of O( kNlogN) memory complexity and O( k2Nlog2N) time complexity for solving electromagnetic problems, where k is a small parameter that is adaptively determined based on accuracy requirements. Both inverse-based and LU-based direct solutions are developed. The LU-based solution is further accelerated by nested dissection. A comparison with the state-of-the-art direct finite element solver that employs the most advanced sparse matrix solution has shown clear advantages of the proposed direct solver. In addition, the proposed solver is applicable to arbitrarily-shaped three-dimensional structures and arbitrary inhomogeneity.;To further reduce the computational complexity of the proposed methods, we develop a layered H -inverse and a layered H -LU to fully utilize the large zero blocks present in a finite-element-based system matrix. As a result, the storage complexity of the proposed methods is reduced from O(NlogN) to O(M logM), and the time complexity is reduced from O( Nlog2N) to O(Nlog 2M), where M is the number of unknowns in a single layer, which is in general orders of magnitude smaller than N. For periodic structures, we develop a layered H-matrix based reduction and cascading algorithm, which further reduces the time complexity to log 2(p)O(Mlog 2M), where p is the number of periods.;To reduce the complexity down to linear, we take advantage of the layered and periodic properties along both transverse and layer growth directions, and develop an H -matrix parametric cascading (HPC) based fast direct solver to solve a large-scale integrated circuit problem. The storage complexity is reduced to a single period storage of O(NsNx), where Ns is the number of surface unknowns along the transverse direction in one period and Nx is the number of surfaces in one period. Both Ns and Nx are constant numbers, despite the number of periods in both directions, therefore our proposed solver is able to solve arbitrarily large structures on a single machine with constant memory.;We also provide the first theoretical study on the rank of the inverse finite-element matrix for 1-D, 2-D, and 3-D electrodynamic problems. We find that the rank of the inverse finite-element matrix is a constant, irrespective of electric size for 1-D electrodynamic problems. For 2-D electrodynamic problems, the rank grows very slowly with electric size as the square root of the logarithm of the electric size of the problem. For 3-D electrodynamic problems, the rank scales linearly with the electric size. The findings of this work are both theoretically proved and numerically verified. They are applicable to problems with inhomogeneous materials, arbitrarily shaped structures, and truncated with any kind of absorbing boundary condition. The findings on the rank of the inverse finite element matrix also lead to a theoretical proof on the fact that the rank of the interaction between two separated geometry blocks in a surface integral-equation based system scales linearly with the electric size of the block diameter instead of electric size square. (Abstract shortened by UMI.).
机译：在这项工作中，我们引入了一个通用的数学框架，称为“分层（H）矩阵”框架，以减少基于有限元素的电磁问题分析的计算复杂性。在数学文献中，仅对于控制静态现象的椭圆型偏微分方程，证明了H矩阵逼近的存在。我们为电动分析提供了第一个证据。基于该证明，我们开发了一种基于H矩阵的直接有限元求解器，用于求解电磁问题的O（kNlogN）和O（k2Nlog2N）时间复杂度，其中k是根据精度自适应确定的小参数要求。开发了基于逆的和基于LU的直接解决方案。基于LU的解决方案通过嵌套解剖进一步加速。与采用最先进的稀疏矩阵解决方案的最新直接有限元求解器进行比较，显示了所提出的直接求解器的明显优势。此外，所提出的求解器适用于任意形状的三维结构和任意不均匀性。为了进一步降低所提出方法的计算复杂度，我们开发了分层的H-逆和分层的H-LU以充分利用大基于有限元的系统矩阵中存在零个块。结果，所提出的方法的存储复杂度从O（NlogN）降低到O（M logM），时间复杂度从O（Nlog2N）降低到O（Nlog 2M），其中M是未知数在单层中，通常比N小几个数量级。对于周期结构，我们开发了一种基于分层H矩阵的约简和级联算法，该算法进一步降低了log 2（p）O（Mlog 2M）的时间复杂度为了将复杂度降低到线性，我们利用沿横向和分层生长方向的分层和周期性特性，并开发了基于H矩阵参数级联（HPC）的快速直接求解器解决大规模集成电路问题。存储复杂性降低为O（NsNx）的单周期存储，其中Ns是一个周期内沿横向方向的表面未知数，而Nx是一个周期内的表面数。 Ns和Nx都是常数，尽管在两个方向上都有一定数量的周期，因此我们提出的求解器能够在具有恒定内存的单机上求解任意大的结构。一维，二维和3-D电动力学问题的有限元矩阵。我们发现，有限元逆矩阵的秩是一个常数，而与一维电动力学问题的电大小无关。对于二维电动力学问题，等级随电尺寸的增长非常缓慢，而电尺寸是问题电尺寸的对数的平方根。对于3D电动力学问题，等级与电尺寸成线性比例关系。这项工作的发现在理论上和数值上都得到了证明。它们适用于不均匀材料，任意形状的结构以及由于任何类型的吸收边界条件而被截断的问题。关于逆有限元矩阵的秩的发现还导致以下事实的理论证明：基于表面积分方程的系统中两个分离的几何块之间的相互作用的秩与块直径的电尺寸成线性比例电尺寸的平方。 （摘要由UMI缩短。）。

著录项

  • 作者

    Liu, Haixin.;

  • 作者单位

    Purdue University.;

  • 授予单位 Purdue University.;
  • 学科 Engineering Electronics and Electrical.;Physics Electricity and Magnetism.
  • 学位 Ph.D.
  • 年度 2012
  • 页码 148 p.
  • 总页数 148
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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