Domain decomposition in the rigorous electromagnetic analysis of large structures.

Abstract

A domain decomposition method is introduced to facilitate the efficient and rigorous computation of electromagnetic phenomena in structures that are electrically large in one dimension. Large two-dimensional structures are decomposed into many smaller regions by placing partitions throughout the structure. Sets of independent numerical solutions are generated within each unique block due to excitation by properly chosen expansion functions defined on the partitions as boundary conditions. The finite element method is explained for providing numerical solutions to the modified Helmholtz equation in the smaller blocks. Field continuity conditions are applied at the partitions to superpose the numerical solutions in the blocks and complete the domain decomposition solution for the entire problem. A finite periodic structure is modeled to show the practicality of domain decomposition for similar repetitious structures, and its filter behavior is compared with its infinite periodic analog. A matching problem between two mismatched waveguides demonstrates the building block nature of domain decomposition. The parallelism of domain decomposition and finite elements is demonstrated. The method is re-coded for operation on the Connection Machine CM-2, and various timings are made to illustrate how the method becomes more efficient as the problem becomes more parallel. The second part of the dissertation concentrates on the development of a three-dimensional nodal based finite element code, with applications in domain decomposition. The formulation for domain decomposition is extended in a straightforward fashion to three dimensions. The issues surrounding the more difficult extension of the finite elements for vector, three-dimensional problems are explained. Nodal based finite element techniques are contrasted with edge based finite elements. Boundary conditions are given special consideration to accommodate metallic and dielectric surfaces and edge singularities. A dielectric filter example proves the validity of the dielectric interface condition, and serves as a good filter comparison with a geometry for which the analytic behavior is known. A metallic corrugated filter example is analyzed using a transmission line approximation, and the dispersion curves of the finite and infinite filter are compared.