Date of Award

Summer 2004

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Jonathan H.C. Luke

Second Advisor

Demetrius T. Papageorgiou

Third Advisor

Peter G. Petropoulos

Fourth Advisor

David C. Stickler

Fifth Advisor

Gerald Martin Whitman

Abstract

This dissertation extends the work done by Hue and Kriegsmann in 1998 on microwave heating of a ceramic sample in a single-mode waveguide cavity. In that work, they devised a method combining asymptotic and numerical techniques to speed up the computation of electromagnetic fields inside a high-Q cavity in the presence of low-loss target. In our problem, the dependence of the electrical conductivity on temperature increases the complexity of the problem. Because the electrical conductivity depends on temperature, the electromagnetic fields must be recomputed as the temperature varies. We then solve the coupled heat equation and Maxwell's equations to determine the history and distribution of the temperature in the ceramic sample. This complication increases the overall computational effort required by several orders of magnitude.

In their work, Hile and Kriegsmann used the established technique of solving the time-dependent Maxwell's equations with the finite-difference time domain method (FDTD) until a time-harmonic steady state is obtained. Here we replace this technique with a more direct solution of a finite-difference approximation of the Helmholtz equation. The system of equations produced by this finite-difference approximation has a matrix that is large and non-Hermitian. However, we find that it may be splitted into the sum of a real symmetric matrix and a relatively low-rank matrix. The symmetric system represents the discretization of Helmholtz equation inside an empty and truncated waveguide; this system can be solved efficiently with the conjugate gradient method or fast Fourier transform. The low-rank matrix carries the information at the truncated boundaries of the waveguide and the properties of the sample. The rank of this matrix is approximately the sum of twice the number of grid spacings across waveguide and the number of grid points in the target. As a result of the splitting, we can handle this part of the problem by solving a system having as many unknowns as the rank of this matrix.

With the above algorithmic innovations, substantial computational efficiencies have been obtained. We demonstrate the heating of a target having a temperature dependent electrical conductivity. Comparison with computations for constant electrical conductivity demonstrate significant difference in the heating histories. The computational complexity of our approach in comparison with that of using the FDTD solver favors the FDTD method when ultra-fine grids are used. However, in cases where grids are refined simply to reduce asymptotic truncation error, our method can retain its advantages by reducing truncation error through higher-order discretization of the Helmholtz operator.

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Mathematics Commons

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