Date of Award

9-26-2000

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics & Astrophysics

First Advisor

William A. Schwalm

Abstract

The analysis of physical problems on complex structures often involves the use of a lattice model of the underlying physical space. The lattice sites provide a convenient basis for the representation of physical quantities. Analogous to the use of differential forms in continuous space, a discrete formalism is developed in which geometrical simplices can be used as a basis for vector as well as scalar quantities. The space of p-forms at a point in the theory of differential forms is replaced by the lattice vertex and the p-dimensional geometrical simplices which contain it. These p-simplices serve as a basis for scalar, polar vector, axial vector and pseudoscalar fields or, in general, p-fields. The boundary ∂ and coboundary ∂† maps of algebraic topology define vector difference maps loosely analogous to the differential vector operators divergence, gradient and curl. Defining scalar and vector products in a natural way on the set of p-simplices results in a difference formalism in which most standard vector identities hold exactly. The definition of an inner product together with the vector difference maps leads to difference analogs of vector integral theorems including a generalized Stoke's theorem. Metric properties are added to the formalism through the introduction of quadrature weights. Models of electromagnetism and fluid dynamics are defined on a variety of regular and irregular lattices. In particular, the formalism will be used to study the electromagnetic modes inside hollow cavities with conducting boundaries and fluid flow through various structures.

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