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The Ohio State University

1. He, Bo. Compatible discretizations for Maxwell equations.

Degree: PhD, Electrical Engineering, 2006, The Ohio State University

The main focus of this dissertation is the study and development of numerical techniques to solve Maxwell equations on irregular lattices. This is achieved by means of compatible discretizations that rely on some tools of algebraic topology and a discrete analog of differential forms on a lattice. Using discrete Hodge decomposition and Euler’s formula for a network of polyhedra, we show that the number of dynamic degrees of freedom (DoFs) of the electric field equals the number of dynamic DoFs of the magnetic field on an arbitrary lattice (cell complex). This identity reflects an essential property of discrete Maxwell equations (Hamiltonian structure) that any compatible discretization scheme should observe. We unveil a new duality called Galerkin duality, a transformation between two (discrete) systems, primal system and dual system. If the discrete Hodge operators are realized by Galerkin Hodges, we show that the primal system recovers the conventional edge-element FEM and suggests a geometric foundation for it. On the other hand, the dual system suggests a new (dual) type of FEM. We find that inverse Hodge matrices have strong localization properties. Hence we propose two thresholding techniques, viz., algebraic thresholding and topological thresholding, to sparsify inverse Hodge matrices. Based on topological thresholding, we propose a sparse and fully explicit time-domain FEM for Maxwell equations. From a finite-difference viewpoint, topological thresholding provides a general and systematic way to derive stable local finite-difference stencils in irregular grids. We also propose and implement an E-B mixed FEM scheme to discretize first order Maxwell equations in frequency domain directly. This scheme results in sparse matrices. In order to tackle low-frequency instabilities in frequency domain FEM and spurious linear growth of time domain FEM solutions, we propose some gauging techniques to regularize the null space of a curl operator. Advisors/Committee Members: Teixeira, Fernando (Advisor).

Subjects/Keywords: differential forms; chains and cochains; Whitney forms; de Rham diagram; gauging; compatible discretization; Hodge operator; Hodge decomposition; Euler's formula; FDTD; FEM; Galerkin duality; primal and dual; pure Neumann boundary condition; mixed FEM

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

He, B. (2006). Compatible discretizations for Maxwell equations. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299

Chicago Manual of Style (16th Edition):

He, Bo. “Compatible discretizations for Maxwell equations.” 2006. Doctoral Dissertation, The Ohio State University. Accessed January 25, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299.

MLA Handbook (7th Edition):

He, Bo. “Compatible discretizations for Maxwell equations.” 2006. Web. 25 Jan 2020.

Vancouver:

He B. Compatible discretizations for Maxwell equations. [Internet] [Doctoral dissertation]. The Ohio State University; 2006. [cited 2020 Jan 25]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299.

Council of Science Editors:

He B. Compatible discretizations for Maxwell equations. [Doctoral Dissertation]. The Ohio State University; 2006. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299

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