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Virginia Tech

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

Degree: PhD, Mathematics, 2012, Virginia Tech

This dissertation is a uniï¬ cation of an analysis-based approach and the traditional topological-based approach to Poincaré duality. We examine the role of the discrete Hodge star operator in proving and in realizing the Poincaré 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 Poincaré 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 Poincaré 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; Poincaré Duality; Discrete Hodge Star

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APA (6th Edition):

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

Chicago Manual of Style (16th Edition):

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

MLA Handbook (7th Edition):

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

Vancouver:

Arnold RF. The Discrete Hodge Star Operator and Poincaré 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 Poincaré 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

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

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

APA (6th 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 (16th 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 (7th 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

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 19, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299.

MLA Handbook (7th 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

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