Computational Methods for Complex Electromagnetic Geometries and Media

Additive manufacturing has enabled electromagnetic devices with increasingly complex geometries, but existing numerical tools remain limited in their ability to model and design such structures. This dissertation presents two major advancements that address these restrictions in the finite-difference frequency-domain method (FDFD) and the spatially-variant lattice algorithm (SVLA). These two numerical methods provide a foundation for future exploration in the simulation, optimization, and realization of next-generation electromagnetic devices. First, a general bianisotropic FDFD formulation based on the vector wave equation is presented that enables practical modeling of metamaterials using effective medium homogenized parameters rather than explicitly resolving subwavelength metamaterial features. The formulation fully supports bianisotropic media, includes total-field/scattered-field (TF/SF) and scattered-field (SF) source injection techniques, and reduces to the standard anisotropic FDFD with the appropriate material tensors. This method is validated through radar cross section (RCS) simulations against analytical solutions and other numerical methods. Second, the dissertation introduces a finite element method (FEM) adaptation of the SVLA. Traditional SVLA tools operate on rectangular finite-difference grids, limiting geometric flexibility. The FEM-based formulation enables the creation of spatially variant lattices on flat surfaces, curved surfaces, and volumetric structures, greatly improving geometric accuracy. Detailed formulation of basis functions, weak-form derivations, and solver strategies are provided, along with design demonstrations for 2D flat and curved surfaces and 3D volumetric structures.