Existence of ${\cal H}$-Matrix Representations of the Inverse Finite-Element Matrix of Electrodynamic Problems and ${\cal H}$-Based Fast Direct Finite-Element Solvers

Liu, Haixin; Jiao, Dan

doi:10.1109/tmtt.2010.2086532

Cited by 18 publications

(37 citation statements)

References 19 publications

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“…An FEM-based solution to the above boundary value problem results in a linear system of equation [14]: (4) where is the right hand side, is sparse and can be written as (5) in which (6) where denotes the computational domain, and is the vector basis used to expand unknown . In (6), is known to be a mass matrix, and is a stiffness matrix.…”

Section: A Formulate An Fem System Matrixmentioning

confidence: 99%

“…When using -matrix based mathematics for electrodynamic analysis, the rank of the inverse finite element matrix is electric size, and hence dependent in 2-and 3-D analysis [10], [11]. However, is still less than , thus being low rank [5], [10], [11].…”

Section: Generate An -Matrix Representation Of the Fem System Matrixmentioning

confidence: 99%

“…In other words, all the admissible blocks in the original FEM matrix have a zero rank [5]. Thus, an FEM matrix can be represented exactly by an -matrix without approximation, and the assembly of an FEM matrix has optimal complexity .…”

Section: Generate An -Matrix Representation Of the Fem System Matrixmentioning

confidence: 99%

“…When or is large, an iterative solver becomes inefficient. Manuscript In [3]- [5], an -matrix-based mathematical framework [6]- [9] was introduced to accelerate the direct solution of the FEM-based linear system of equations for large-scale electromagnetic analysis. It is proved in [5] that the sparse matrix resulting from a finite-element-based analysis of electrodynamic problems can be represented by an matrix without any approximation, and the inverse of this sparse matrix has a data-sparse -matrix approximation with error well controlled.…”

Section: Introductionmentioning

confidence: 99%

“…Manuscript In [3]- [5], an -matrix-based mathematical framework [6]- [9] was introduced to accelerate the direct solution of the FEM-based linear system of equations for large-scale electromagnetic analysis. It is proved in [5] that the sparse matrix resulting from a finite-element-based analysis of electrodynamic problems can be represented by an matrix without any approximation, and the inverse of this sparse matrix has a data-sparse -matrix approximation with error well controlled. Based on this proof, an -matrix-based direct finite-element solver having storage cost and operation counts is developed for solving electrodynamic problems, where rank is adaptively determined based on a prescribed accuracy for each low-rank block in the inverse or LU factors of the finite-element matrix.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Layered ${\cal H}$-Matrix Based Inverse and LU Algorithms for Fast Direct Finite-Element-Based Computation of Electromagnetic Problems

Liu

Jiao

2013

IEEE Trans. Antennas Propagat.

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A layered -matrix based inverse algorithm and a layered -matrix based LU factorization and solution algorithm are developed to accelerate the direct solution of the finite element matrix resulting from electromagnetics-based analysis of general 3-D problems. In these algorithms, the direct solution of the original 3-D system matrix is transformed to the direct solution of multiple 2-D problems. The size of the matrix to be computed is thus reduced from a 3-D size to a 2-D size. Moreover, the growth rate of the rank of the inverse finite element matrix with electric size is reduced from a 3-D based growth rate to a 2-D based growth rate. Numerical experiments have demonstrated the clear advantages of the proposed direct solvers over a state-of-the-art multifrontal based direct sparse solver as well as existing fast -matrix based direct solvers, in both CPU time and memory consumption, for finite-element analysis of general 3-D electromagnetic problems.Index Terms--matrix, direct matrix solution, finite element methods, electromagnetic analysis.

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Section: A Formulate An Fem System Matrixmentioning

confidence: 99%

Section: Generate An -Matrix Representation Of the Fem System Matrixmentioning

confidence: 99%

Section: Generate An -Matrix Representation Of the Fem System Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Layered ${\cal H}$-Matrix Based Inverse and LU Algorithms for Fast Direct Finite-Element-Based Computation of Electromagnetic Problems

Liu

Jiao

2013

IEEE Trans. Antennas Propagat.

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Data‐sparse numerical models for SNOM tips

Hafner

Hiptmair

Souzangar

2016

Int J Numerical Modelling

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Summary We propose a compressed non‐local surface‐impedance‐type boundary condition for the efficient numerical modeling of large geometrically persistent parts in multi‐scale electromagnetic simulations. The underlying compressed model is an approximate Schur complement of finite element Galerkin matrix. Our approach relies on local low‐rank representation in the framework of the scriptℋ‐matrix storage format. We discuss two ways to build scriptℋ‐matrix approximations of Schur‐complement matrices: adaptive cross approximation and scriptℋ‐arithmetics. Profound numerical tests and studies of accuracy are carried out for an axisymmetric setting, employing the open source library AHMED by M. Bebendorf. To demonstrate the use of our method, we build the compressed model for an axisymmetric scanning near‐field optical microscopy tip. The model can be embedded in three‐dimensional tip‐sample simulations. Copyright © 2016 John Wiley & Sons, Ltd.

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Multilevel compressed block decomposition–based finite‐element domain decomposition method for the fast analysis of finite periodic structures

Wan

Jiang

2016

Int J Numerical Modelling

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An efficient finite‐element domain decomposition method based on the multilevel compressed block decomposition (MLCBD) algorithm is presented. The dual‐primal finite element tearing and interconnecting (FETI‐DP) method is first introduced to generate a nonoverlapping domain decomposition method with good convergence properties. A second‐order transmission condition is adopted to further accelerate the convergence rate of the FETI‐DP. With the MLCBD algorithm, the computational complexity for the direct solution of FETI‐DP subdomain equations can be reduced to be almost logarithmic linear. The accuracy of the MLCBD algorithm is highly adjustable according to practical requirements. Exploiting the repetitiveness of finite periodic structures, the proposed method can significantly reduce the computational costs. Numerical examples demonstrate the effectiveness of the proposed method for the electromagnetic simulation of finite periodic structures.

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Existence of ${\cal H}$-Matrix Representations of the Inverse Finite-Element Matrix of Electrodynamic Problems and ${\cal H}$-Based Fast Direct Finite-Element Solvers

Cited by 18 publications

References 19 publications

Layered ${\cal H}$-Matrix Based Inverse and LU Algorithms for Fast Direct Finite-Element-Based Computation of Electromagnetic Problems

Layered ${\cal H}$-Matrix Based Inverse and LU Algorithms for Fast Direct Finite-Element-Based Computation of Electromagnetic Problems

Data‐sparse numerical models for SNOM tips

Multilevel compressed block decomposition–based finite‐element domain decomposition method for the fast analysis of finite periodic structures

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