Development of Surface Integral Equation Methods for Electromagnetic Problems

In the field of computational electromagnetism, Integral Equation Methods (IEM) represent a valid alternative to Finite Element Methods (FEM), implemented in most commercial software. The peculiarity of IEM lies in the fact that discretization of electromagnetically inactive domains, such as the volume of air in which the studied device can be immersed, is not required, thus reducing the number of degrees of freedom of the problem (DOFs). This characteristic becomes particularly interesting for analyzing open boundary problems, for example involving antennas, where mesh generation would be particularly demanding. IEM are subdivided into Volume Integral Methods (VIE) and Surface Integral Methods (SIE). In particular, the second type results advantageous in physical cases where currents are located in surficial regions, thus requiring the discretization only of the surfaces of the device. This situation applies, for example, to the analysis of high-frequency interconnections in which current density naturally distributes on the surfaces of the conductors due to a marked skin effect. A limitation for integral methods compared to FEM, however, is the generation of dense matrices, coming from Galerkin formulation, rather than sparse ones for solving the problem. In this way, given the quadratic growth of the matrix dimensions with the DOFs, the computational load for memorization and inversion of the matrices could become onerous. Therefore, compression techniques based on the geometry of the problem are necessary, where elements distant from each other in space and experiencing weak mutual interactions are stored within a convenient way. An interesting technique which implements this condition is based on Hierarchical Matrix method (H-matrices). In this context, the thesis work focuses on the development of integral methods for the analysis of electromagnetic problems with specific attention to SIE. In particular, the state of art of the techniques presented in the literature is analyzed from both a theoretical and experimental point of view through the development of codes in MATLAB. Furthermore, the aforementioned compression techniques (H-matrices) are applied to large-scale problems to demonstrate their validity.