Ph.D. Theses

Diakoptics Solution Techniques for Electromagnetic Fields in Time and Frequency Domain

Mario Righi, University of Victoria, December 1995.


The solution of complex electromagnetic fields problems cannot usually be found by applying a single technique to the entire solution domain. It is necessary to divide the problem space into manageable subdomains, solve the sub-problems with the most appropriate techniques and then recombine these solutions to obtain the answer to the overall problem. The sub-solutions could be obtained either in the time or in the frequency domain. The main challenge resides in the proper combination or reconnection of the various sub-solutions.

The objective of this thesis is to develop, implement and test a number of diakoptics procedures suitable for the analysis of complex microwave and millimeter wave structures.

The theory of the Transmission Line Matrix (TLM) method is reviewed and two novel hybridization methods based on diakoptics are proposed. In the first approach TLM and the time domain mode matching are combined. This link allows the model to take advantage of an analytical description for uniform sub-volumes while using the tra ditional TLM method in highly complex regions. Applications include: efficient analysis of packaged components, TLM computation of the generalized S-matrix, generation of high quality wideband dispersive multi-modal Absorbing Boundary Conditions (ABCs) for homogenous waveguides. Secondly, the systematic extraction of the lumped element equivalent circuit of distributed components is described. This provides a link with net work based solvers, thus allowing the use of TLM results in connection with SPICE-type simulators and the creation of a library of elementary components.

By taking advantage of these techniques it is possible to incorporate more information of the structure under study into the TLM solution, thus creating an accurate, efficient and flexible CAD tool.

Development and Characterization of Generalized TLM Networks for Electromagnetic Modeling

Qi Zhang, University of Victoria, March 1996.


A set of two- and three-dimensional TLM node structures with cells of arbitrary aspect ratio for the time domain analysis of electromagnetic field problems have been developed. These node structures increase the flexibility of space discretization and allow the modeling of media with arbitrary constitutive parameters without the need for reactive stubs. They provide substantial improvement in accuracy when modeling microwave and millimeter-wave structures. The algorithms were implemented in a numerically efficient manner and validated extensively by applying them to solve field problems.

It is shown that the anisotropic rectangular or cuboid TLM network can be built in such a way that the propagation vector remains independent of the direction of propagation in the infinitesimal approximation. The dispersion error related to the modeling of the TLM method is studied. A full dispersion analysis of the rectangular and cuboid meshes is performed for the general case, and results are compared to those of the traditional square and cubic cells.

In order to verify the conformity of the rectangular TLM algorithm with Maxwell's equations, and to place it on a sound field-theoretical basis, the properties of the rectangular TLM were derived using the Method of Moments (MOM). This derivation extends the work by Krumpholz and Russer from the square to the more general rectangular case. Hilbert space representation was also extended to the general rectangular case; it represents the most elegant and compact formulation of the new TLM scheme. This approach leads to more efficient analysis of structures sustaining waves with different wave numbers in two coordinate directions.

Design of Complex Electromagnetic Structures with TLM - Theory and Practice

Poman So, University of Victoria, June 1996.


To design high frequency complex electromagnetic structures with TLM is not trivial in spite of the simplicity of the fundamental TLM algorithm. This is because TLM is a time and space discretization method in which the entire computational domain must be filled with TLM cells. In three-dimensional cases, the computational effort to solve realistic problems soon becomes intractable unless special techniques are used. To be attractive to the design engineer, field simulation tools must include these advanced techniques automatically in the models. New TLM features and computational techniques are developed in this thesis to overcome the above mentioned problems.

An experimental multi-purpose electromagnetic field simulation tool has been created to demonstrate the features and techniques developed in this thesis can be easily integrated into a well designed tool.

Modeling of General Medium Constitutive Relationships in the Transmission Line Matrix Method (TLM)

Leonardo de Menezes, University of Victoria, September 1996.


This thesis presents the modeling of general medium constitutive relationships in the Transmission Line Matrix (TLM) method. The technique is shown for two- and three dimensional cases. The procedure consists of decoupling the impulse scattering at the nodes from equations describing the medium. This is achieved by using nodal sources connected to the TLM node. The nodal sources are implemented with the state-variable description of the constitutive relationships. The technique requires only few modifications to the TLM algorithm. The procedure is validated for frequency-dependent, nonlinear, anisotropic and gyromagnetic media.

This thesis also presents a dispersion analysis of TLM with frequency-dependent dielectrics. This study is performed in two- and three-dimensions by solving the dispersion relationship of TLM with nodal sources. The sources are used to model the frequency dependent dielectric. The study shows that the nodal source and stub-loaded models are equivalent for frequency independent dielectrics. The accuracy bounds of the TLM frequency-dependent dielectric model are presented.

This thesis also investigates the physical origin of the coarseness and dispersion errors influencing two-dimensional TLM solutions of Maxwell's equations. The study is performed by solving the difference equations of the numerical method analytically. The results confirm a reduction of the accuracy of the discrete solution near field singularities. The solution of a partially filled waveguide is also investigated. The results show that TLM can have positive or negative frequency shifts, depending on the dielectric filling, excited mode and geometry. These results are also valid for the finite difference time domain method (FDTD).