Contributions to the Art of Finite Element Analysis in Electromagnetics

The finite element method is a powerful method for the approximate solution of boundary value problems governed by partial differential equations. A really first application to structural engineering problems, dating 1943, is attributed to R. Courant. Since then, there has been a lot of successful tentatives to apply the method to other fields. In particular, Silvester showed in 1969 that waveguide modes could be easily computed with the method. His work started a long path for finite elements in electromagnetics, with multiple assessments of the method with real-world problems and gradually improving the efficiency of the algorithms. Nowadays, finite elements in computational electromagnetics has become an invaluable part in radio frequency and microwave application designs, and many packages are widely available to perform these tasks. However, there remain a lot of problems to be solved. In this dissertation, we have inquired in two of these. The first, the efficient solution of large problems which may not be solvable on a single modern computer. Domain decomposition methods have been thus investigated, these allowing to solve smaller parts of a large problem and to achieve the whole solution upon proper interconnection. Two types of domain decomposition methods have been analyzed, leading to the construction of algorithms for solving large electromagnetic problems at a nearly linear complexity. The other, the accurate solution of electromagnetic problems in which some materials behave nonlinearly, that is their properties vary depending on the intensity of the fields they imbue. Almost all materials behave nonlinearly and their effect is just a matter of fields intensities and accuracy requirements. In many microwave applications, the nonlinear effects, necessary for information processing and control, are still limited to lumped devices for their highly developed models. Accurate modeling of bulk or films of nonlinear materials may open the way to a new variety of controllable materials in flexible, reconfigurable, electromagnetic devices. A finite element package has been implemented to perform several tests here documented.