Aspectos computacionais da formulação variacional ultra fraca aplicada ao eletromagnetismo

Abstract

Numerical methods that use plane waves as basis functions to approximate the solution of the Helmholtz equation have received considerable attention in computational electromagnetics. These approaches do not require the usual ten nodal points per wavelength to approximate oscillatory solutions. The main drawback of these methods are: imposing the essential boundary conditions, the domain discontinuities and ill-conditioning of the system. Although Lagrange multipliers and Mortar elements can be used to incorporate boundary conditions and domain discontinuities, they increase the computational cost and the condition number. Alternatively, the Ultra Weak Variational Formulation uses the numerical upwind flux of the Discontinuous Galerkin Method. This scheme ensures the weak incorporation of the essential boundary conditions and the transmission conditions between medias. These features have arisen great interest in several applications, such as to compute the Helmholtz equation. In this thesis, we investigate the Ultra Weak Variational Formulation for the electromagnetic wave propagation, using uniform and non-uniform meshes. For the uniform meshes, we investigate the ill-conditioning of the global system as function of the degrees of freedom, the heterogeneous medium, the basis functions as plane waves and Bessel functions and the truncation of the domain by means of the Perfectly Matched Layers (PML). For non-uniform meshes, which is the main contribution of this research, we propose a new methodology to control the condition number and how to predetermine the maximum number of basis functions based on maximum edge. The proposed methodology is validated through the simulation of electromagnetic scattering by a perfect conductor cylinder coated by a dielectric layer. Also is solved an electrically large problem as an example, the electromagnetic scattering by a perfect conductor airfoil.