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Full text release has been delayed at the author's request until May 10, 2024
ETD Abstract Container
Abstract Header
Acceleration Methods of Discontinuous Galerkin Integral Equation for Maxwell's Equations
Author Info
Lee, Chung Hyun
ORCID® Identifier
http://orcid.org/0000-0001-8180-1122
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu1649937976475563
Abstract Details
Year and Degree
2022, Doctor of Philosophy, Ohio State University, Electrical and Computer Engineering.
Abstract
Recently, the discontinuous Galerkin integral equation (IEDG) method has been introduced in the computational electromagnetic community. This method can be helpful for mesh preparation because of the guaranteed flexibility of the geometry and the basis functions. However, this method has the intrinsic drawback that additional integral terms are required during integration by parts, hence potentially increasing the computational cost. In addition, because this method is relatively new, it is not yet an active focus of research, despite its obvious advantages. Therefore, in this article, three different methods for accelerating the IEDG method are proposed using the unique characteristics of the method itself. Similar to other partial differential equation methods, the IEDG method can be divided into three basic stages: computation of the impedance matrix using the IEDG formulation, preparation of the right-hand side (RHS) matrix for multiple angular responses, and solution of the matrix equations with iterative solvers. Accordingly, a different method is proposed for each stage of the IEDG process. For the first stage, a new procedure for efficient numerical integration is presented based on the fact that each integration term can be divided into three submatrices that are universally used for entire elements, can be stored in advance for each patch, and are related to two source and field elements. Because those matrices can also be shared between integration terms, one can take advantage of this to achieve the acceleration of numerical integration. In addition, approximation by the impedance boundary condition (IBC) in the IEDG framework can also be easily formulated with the proposed numerical integration method. Second, based on rigorous error analysis, an efficient and error-controllable angular sweep algorithm is proposed. With this algorithm, the solution process can be accelerated based on two factors: only a minimum set of singular vectors can be stored for the solution process, and the tolerance of the iterative solver for each singular vector can be mitigated. It is proven that two types of bounded numerical errors arise: the solution error computed based on the vector-induced matrix norm and the root mean square (RMS) of the solution errors for individual RHS vectors. In addition, a block version of the generalized conjugate residual (GCR) is proposed to accelerate the solution process with a limited increase in memory. For various targets of different sizes and materials, the above errors and accelerations are numerically validated. Finally, a robust preconditioner for accelerating the Krylov subspace method is presented, which is constructed by stacking two preconditioners to alleviate different physical problems. The first is the near-field preconditioner, which is the inverse of a sparse matrix consisting only of interactions within a predetermined distance. Nested dissection reordering is used to conserve memory and time for factorization. In addition, a global preconditioner is stacked on top of the near-field preconditioner to enhance the robustness. The global preconditioner is based on the approximate inverse of the summation matrix of the identity matrix and off-diagonal terms. Principal component analysis (PCA) and the Sherman-Morrison-Woodbury formula are used to obtain the inverse. Additionally, the transpose of the multilevel fast multipole method is described, along with numerical validations, since left vector-matrix multiplication is required to implement the PCA process. Finally, the deflation technique is used for the further acceleration of iterative solvers.
Committee
Jin-Fa Lee (Advisor)
Robert Lee (Committee Member)
Kubilay Sertel (Committee Member)
Subject Headings
Electromagnetics
Keywords
computational electromagnetics
;
discontinuous Galerkin method
;
nested dissection
;
IEDG
;
multiple RHS
;
robust preconditioner
;
numerical integration
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RIS
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Citations
Lee, C. H. (2022).
Acceleration Methods of Discontinuous Galerkin Integral Equation for Maxwell's Equations
[Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1649937976475563
APA Style (7th edition)
Lee, Chung Hyun.
Acceleration Methods of Discontinuous Galerkin Integral Equation for Maxwell's Equations.
2022. Ohio State University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=osu1649937976475563.
MLA Style (8th edition)
Lee, Chung Hyun. "Acceleration Methods of Discontinuous Galerkin Integral Equation for Maxwell's Equations." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1649937976475563
Chicago Manual of Style (17th edition)
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Document number:
osu1649937976475563
Copyright Info
© 2022, some rights reserved.
Acceleration Methods of Discontinuous Galerkin Integral Equation for Maxwell's Equations by Chung Hyun Lee is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by The Ohio State University and OhioLINK.