Topological and geometrical considerations for Maxwell's equations on unstructured meshes

Persistent Link:
http://hdl.handle.net/10150/282472
Title:
Topological and geometrical considerations for Maxwell's equations on unstructured meshes
Author:
Kaus, Cynthia Christine, 1965-
Issue Date:
1997
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
A discrete differential form approach to solving Maxwell's equations numerically on unstructured meshes is presented. A differential form representation of Maxwell's equations provides a natural and coordinate-free means of studying these equations and their solutions in the presence of curved objects. We begin by reviewing basic properties of differential forms and the operators associated with them for their use in describing electromagnetic fields and sources. Because we are interested in numerically solving Maxwell's equations on unstructured meshes, we introduce discrete representations of these differential forms and the underlying manifolds. This allows a discrete representation of Maxwell's equations in terms of chains and cochains on an arbitrary polyhedral cell complex. The discrete boundary operator, coboundary operator and star operator on cochains are constructed and shown to maintain divergence-free regions. The constructions of the dual of a polyhedral cell complex and the star operator, which give a one-to-one correspondence between the primary and the dual cell complexes, are introduced. This star operation gives the relationship between the magnetic field 1-cochain on the dual cell complex and the magnetic flux 2-cochain on the primary cell complex, and the relationship between the electric field 1-cochain on the primary cell and the electric flux 2-cochain on the dual cell complex. With the construction of these operators, the dual cell complex, and the associated cochains, we have determined the corresponding numerical update equations for the electromagnetic fields on unstructured meshes. The numerical update equations provided by the discrete differential form approach are determined explicitly for cubical, parallelepiped, tetrahedral, and trapezoid cell complexes. For the special cases of an orthogonal complex and a parallelepiped complex, these discrete differential form update equations recover those provided by both Yee's algorithm and the discrete surface integral (DSI) algorithm. It is demonstrated that the discrete differential form update equations differ from those obtained with the DSI approach on the more irregular trapezoid cell complex and, hence, may overcome the known late-time instabilities associated with the DSI approach applied to such highly unstructured meshes.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.; Engineering, Electronics and Electrical.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Ziolkowski, Richard

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleTopological and geometrical considerations for Maxwell's equations on unstructured meshesen_US
dc.creatorKaus, Cynthia Christine, 1965-en_US
dc.contributor.authorKaus, Cynthia Christine, 1965-en_US
dc.date.issued1997en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractA discrete differential form approach to solving Maxwell's equations numerically on unstructured meshes is presented. A differential form representation of Maxwell's equations provides a natural and coordinate-free means of studying these equations and their solutions in the presence of curved objects. We begin by reviewing basic properties of differential forms and the operators associated with them for their use in describing electromagnetic fields and sources. Because we are interested in numerically solving Maxwell's equations on unstructured meshes, we introduce discrete representations of these differential forms and the underlying manifolds. This allows a discrete representation of Maxwell's equations in terms of chains and cochains on an arbitrary polyhedral cell complex. The discrete boundary operator, coboundary operator and star operator on cochains are constructed and shown to maintain divergence-free regions. The constructions of the dual of a polyhedral cell complex and the star operator, which give a one-to-one correspondence between the primary and the dual cell complexes, are introduced. This star operation gives the relationship between the magnetic field 1-cochain on the dual cell complex and the magnetic flux 2-cochain on the primary cell complex, and the relationship between the electric field 1-cochain on the primary cell and the electric flux 2-cochain on the dual cell complex. With the construction of these operators, the dual cell complex, and the associated cochains, we have determined the corresponding numerical update equations for the electromagnetic fields on unstructured meshes. The numerical update equations provided by the discrete differential form approach are determined explicitly for cubical, parallelepiped, tetrahedral, and trapezoid cell complexes. For the special cases of an orthogonal complex and a parallelepiped complex, these discrete differential form update equations recover those provided by both Yee's algorithm and the discrete surface integral (DSI) algorithm. It is demonstrated that the discrete differential form update equations differ from those obtained with the DSI approach on the more irregular trapezoid cell complex and, hence, may overcome the known late-time instabilities associated with the DSI approach applied to such highly unstructured meshes.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
dc.subjectEngineering, Electronics and Electrical.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorZiolkowski, Richarden_US
dc.identifier.proquest9806846en_US
dc.identifier.bibrecord.b37563713en_US
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