The Ohio State University
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
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
to Zotero / EndNote / Reference
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 February 20, 2020.
MLA Handbook (7th Edition):
He, Bo. “Compatible discretizations for Maxwell equations.” 2006. Web. 20 Feb 2020.
He B. Compatible discretizations for Maxwell equations. [Internet] [Doctoral dissertation]. The Ohio State University; 2006. [cited 2020 Feb 20].
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