Compatible discretizations for maxwell equations

 
 
 
 

Compatible discretizations for maxwell equations

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dc.contributor.advisor Teixeira, Fernando en_US
dc.contributor.author He, Bo en_US
dc.date.accessioned 2008-07-07T18:44:42Z
dc.date.available 2008-07-07T18:44:42Z
dc.date.created 2006 en_US
dc.date.issued 2008-07-07T18:44:42Z
dc.identifier.uri http://rave.ohiolink.edu/etdc/view?acc_num=osu1143171299 en_US
dc.identifier.uri http://hdl.handle.net/2374.OX/4253
dc.description 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. en_US
dc.format application/pdf en_US
dc.rights unrestricted en_US
dc.rights Copyright and permissions information available at the source archive en_US
dc.subject differential forms en_US
dc.subject chains and cochains en_US
dc.subject Whitney forms en_US
dc.subject de Rham diagram en_US
dc.subject gauging en_US
dc.subject compatible discretization en_US
dc.subject Hodge operator en_US
dc.subject Hodge decomposition en_US
dc.subject Euler's formula en_US
dc.subject FDTD en_US
dc.subject FEM en_US
dc.subject Galerkin duality en_US
dc.subject primal and dual en_US
dc.subject pure Neumann boundary condition en_US
dc.subject mixed FEM en_US
dc.subject sp en_US
dc.title Compatible discretizations for maxwell equations en_US
dc.type Electronic Thesis or Dissertation en_US
dc.degree.name PhD en_US
dc.degree.level doctoral en_US
dc.degree.discipline Electrical Engineering en_US
dc.degree.grantor Ohio State University en_US
dc.contributor.publisher Ohio State University / OhioLINK en_US

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