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.

Figures (20)

• Figure 1: Geometry and notation for the resistive sheet calculation. The normal $\nu$ is outward to $\Omega_-$.
• Figure 2: Sketch of mapping from the reference face to the face of an element in the volume mesh. Here we sketch a quadratic map requiring that the midpoint of each edge in the curvilinear face be given.
• Figure 3: Cross-sections of the computational grids used to approximate scattering from a PEC sphere. The colorbar shows the number of plane wave directions on each element. The major difference is the grid density on the surface of the sphere. This figure shows how wedge and hexahedral elements can be usefully employed in the outer PML layer. In all figures showing grids, the colorbar shows the number of plane wave direction, see Eq. (\ref{['eq:Nell']}), for each element.
• Figure 4: Snapshots of the scattered electric field component $|E_y^s|$ for the meshes considered here. There is good agreement between mesh 1 and mesh 2 with curved faces. Mesh 2 with flat faces produces unacceptable error due to the coarse boundary approximation.
• Figure 5: Bistatic RCS at 2 GHz for the PEC sphere. We show the RCS computed using meshes 1 and 2 compared to the Mie series. The bottom panel shows the difference between the numerical solution and the Mie series. Note that the 'Mesh 2, flat - Mie' curve is omitted from the bottom panel due to its significantly larger amplitude compared to the other two curves.