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Virginia Tech

1. Arnold, Rachel Florence. The Discrete Hodge Star Operator and PoincarÃ© Duality.

Degree: PhD, Mathematics, 2012, Virginia Tech

URL: http://hdl.handle.net/10919/27485

This dissertation is a uniï¬ cation of an analysis-based approach and the traditional topological-based approach to PoincarÃ© duality. We examine the role of the discrete Hodge star operator in proving and in realizing the PoincarÃ© duality isomorphism (between cohomology and ho-
mology in complementary degrees) in a cellular setting without reference to a dual cell complex. More speciï¬ cally, we provide a proof of this version of PoincarÃ© duality over R via the simplicial discrete Hodge star deï¬ ned by Scott Wilson in [19] without referencing a dual cell complex. We also express the PoincarÃ© duality isomorphism over both R and Z in terms of this discrete operator. Much of this work is dedicated to extending these results to a cubical setting, via the introduction of a cubical version of Whitney forms. A cubical setting provides a place for Robin Formanâ s complex of nontraditional differential forms, deï¬ ned in [7], in the uniï¬ cation of analytic and topological perspectives discussed in this dissertation. In particular, we establish a ring isomorphism (on the cohomology level) between Formanâ s complex of differential forms with his exterior derivative and product and a complex of cubical cochains with the discrete coboundary operator and the standard cubical cup product.
*Advisors/Committee Members: Haskell, Peter E. (committeechair), Thomson, James E. (committee member), Rossi, John F. (committee member), Floyd, William J. (committee member).*

Subjects/Keywords: Cell Complex; Cubical Whitney Forms; PoincarÃ© Duality; Discrete Hodge Star

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

APA (6^{th} Edition):

Arnold, R. F. (2012). The Discrete Hodge Star Operator and PoincarÃ© Duality. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/27485

Chicago Manual of Style (16^{th} Edition):

Arnold, Rachel Florence. “The Discrete Hodge Star Operator and PoincarÃ© Duality.” 2012. Doctoral Dissertation, Virginia Tech. Accessed January 19, 2020. http://hdl.handle.net/10919/27485.

MLA Handbook (7^{th} Edition):

Arnold, Rachel Florence. “The Discrete Hodge Star Operator and PoincarÃ© Duality.” 2012. Web. 19 Jan 2020.

Vancouver:

Arnold RF. The Discrete Hodge Star Operator and PoincarÃ© Duality. [Internet] [Doctoral dissertation]. Virginia Tech; 2012. [cited 2020 Jan 19]. Available from: http://hdl.handle.net/10919/27485.

Council of Science Editors:

Arnold RF. The Discrete Hodge Star Operator and PoincarÃ© Duality. [Doctoral Dissertation]. Virginia Tech; 2012. Available from: http://hdl.handle.net/10919/27485

The Ohio State University

2. Na, Dong-Yeop, NA. Electromagnetic Particle-in-Cell Algorithms on Unstructured Meshes for Kinetic Plasma Simulations.

Degree: PhD, Electrical and Computer Engineering, 2018, The Ohio State University

URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1543398838970791

Plasma is a significantly ionized gas composed of a
large number of charged particles such as electrons and ions. A
distinct feature of plasmas is the collective interaction among
charged particles. In general, the optimal approach used for
modeling a plasma system depends on its characteristic (temporal
and spatial) scales. Among various kinds of plasmas, collisionless
plasmas correspond to those where the collisional frequency is much
smaller than the frequency of interests (e.g. plasma frequency) and
the mean free path is much longer than the characteristic length
scales (e.g. Debye length). Collisionless plasmas consisting of
kinetic space charge particles interacting with electromagnetic
fields are well-described by Maxwell-Vlasov equations.
Electromagnetic particle-in-cell (EM-PIC) algorithms solve
Maxwell-Vlasov systems on a computational mesh by employing
coarse-grained superparticle. The concept of superparticle, which
may represent millions of physical charged particles
(coarse-graining of the phase space), facilitates the realization
of computer simulations for underscaled kinetic plasma systems
mimicking the physics of real kinetic plasma systems. In this
dissertation, we present an EM-PIC algorithm on general (irregular)
meshes based on discrete exterior calculus (DEC) and Whitney forms.
DEC and Whitney forms are utilized for consistent discretization of
Maxwell’s equation on general irregular meshes. The proposed EM-PIC
algorithm employs a mixed finite-element time-domain (FETD) field
solver which yields a symplectic integrator satisfying energy
conservation. Importantly, we employ Whitney-forms-based gather and
scatter schemes to obtain exact charge conservation from first
principles, which had been a long-standing challenge for PIC
algorithms on irregular meshes. Several further contributions are
made in this dissertation: (i) We develop a local and explicit
EM-PIC on unstructured grids using sparse approximate inverse
(SPAI) strategy and study macro- and microscopic residual errors in
motions of charged particles affected by the approximate inverse
errors. (ii) We extend the present EM-PIC algorithm to the
relativistic regime with several relativistic particle-pushers and
compare their performance. (iii) We implement a secondary electron
emission (SEE) processor based on probabilistic Furman-Pivi model
and numerically investigate multipactor effects that are resonant
electron discharges from conducting surfaces by external RF fields.
(iv) We diagnose numerical Cherenkov radiation, which is a
detrimental effect frequently found in EM-PIC simulations involving
relativistic plasma beams, for the present EM-PIC algorithm on
general meshes. (v) We extend the FETD field solver for the
solution of Maxwell's equations in circularly symmetric or
body-of-revolution (BOR) geometries. (vi) Lastly, we combine the
EM-PIC algorithm with the BOR-FETD field solver for the efficient
analysis of vacuum electronic devices (VED).
*Advisors/Committee Members: Teixeira, Fernando (Advisor).*

Subjects/Keywords: Electrical Engineering; Electromagnetics; Plasma Physics; particle-in-cell, unstructured grids, discrete exterior calculus, Whitney forms, finite-element time-domain, charge conservation, sparse approximate inverse, multipactor, numerical Cherenkov radiation, body-of-revolution, transformation optics, BWO

Record Details Similar Records

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

APA (6^{th} Edition):

Na, Dong-Yeop, N. (2018). Electromagnetic Particle-in-Cell Algorithms on Unstructured Meshes for Kinetic Plasma Simulations. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1543398838970791

Chicago Manual of Style (16^{th} Edition):

Na, Dong-Yeop, NA. “Electromagnetic Particle-in-Cell Algorithms on Unstructured Meshes for Kinetic Plasma Simulations.” 2018. Doctoral Dissertation, The Ohio State University. Accessed January 19, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1543398838970791.

MLA Handbook (7^{th} Edition):

Na, Dong-Yeop, NA. “Electromagnetic Particle-in-Cell Algorithms on Unstructured Meshes for Kinetic Plasma Simulations.” 2018. Web. 19 Jan 2020.

Vancouver:

Na, Dong-Yeop N. Electromagnetic Particle-in-Cell Algorithms on Unstructured Meshes for Kinetic Plasma Simulations. [Internet] [Doctoral dissertation]. The Ohio State University; 2018. [cited 2020 Jan 19]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1543398838970791.

Council of Science Editors:

Na, Dong-Yeop N. Electromagnetic Particle-in-Cell Algorithms on Unstructured Meshes for Kinetic Plasma Simulations. [Doctoral Dissertation]. The Ohio State University; 2018. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1543398838970791

The Ohio State University

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

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

URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299

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

Record Details Similar Records

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

APA (6^{th} 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 (16^{th} Edition):

He, Bo. “Compatible discretizations for Maxwell equations.” 2006. Doctoral Dissertation, The Ohio State University. Accessed January 19, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299.

MLA Handbook (7^{th} Edition):

He, Bo. “Compatible discretizations for Maxwell equations.” 2006. Web. 19 Jan 2020.

Vancouver:

He B. Compatible discretizations for Maxwell equations. [Internet] [Doctoral dissertation]. The Ohio State University; 2006. [cited 2020 Jan 19]. 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