A High-Order Ultra-Weak Variational Formulation for Electromagnetic Waves Utilizing Curved Elements

Abstract

The Ultra Weak Variational Formulation (UWVF) is a special Trefftz discontinuous Galerkin method, here applied to the time-harmonic Maxwell's equations. The method uses superpositions of plane waves to represent solutions element-wise on a finite element mesh. We focus on our parallel UWVF implementation, called ParMax, emphasizing high-order solutions in the presence of scatterers with piecewise smooth boundaries. We explain the incorporation of curved surface triangles into the UWVF, necessitating quadrature for system matrix assembly. We also show how to implement a total field and scattered field approach, together with the transmission conditions across an interface to handle resistive sheets. We note also that a wide variety of element shapes can be used, that the elements can be large compared to the wavelength of the radiation, and that a low memory version is easy to implement (although computationally costly). Our contributions are illustrated through numerical examples demonstrating the efficiency enhancement achieved by curved elements in the UWVF. The method accurately handles resistive screens, as well as perfect electric conductor and penetrable scatterers. By employing large curved elements and the low memory approach, we successfully simulated X-band frequency scattering from an aircraft. These innovations demonstrate the practicality of the UWVF for industrial applications.