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 by element on a finite element mesh. We discuss the use of our parallel UWVF implementation called ParMax, and concentrate on methods for obtaining high order solutions in the presence of scatterers with piecewise smooth boundaries. In particular, we show how curved surface triangles can be incorporated in the UWVF. This requires quadrature to assemble the system matrices. 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 matrix free version is easy to implement (although computationally costly). Our contributions are illustrated by several numerical examples showing that curved elements can improve the efficiency of the UWVF, and that the method accurately handles resistive screens as well as PEC and penetrable scatterers. Using large curved elements and the matrix free approach, we are able to simulate scattering from an aircraft at X-band frequencies. The innovations here demonstrate the applicability of the UWVF for industrial examples.

翻译：超弱变分公式(UWVF)是一种特殊的Trefftz间断伽辽金方法，本文将其应用于时谐麦克斯韦方程组。该方法通过平面波叠加在有限元网格上逐单元表示解。我们讨论了并行UWVF实现程序ParMax的使用，重点研究了在具有分段光滑边界的散射体存在时获取高阶解的方法。特别地，我们展示了如何将曲面三角形融入UWVF中，这需要利用数值积分组装系统矩阵。同时说明了如何实现总场与散射场方法，以及处理电阻薄层时跨界面传输条件的设置。我们还注意到，该方法可兼容多种单元形状，单元尺寸可远大于辐射波长，且无矩阵版本易于实现（尽管计算成本较高）。通过多个数值算例验证了我们的贡献：曲面单元能提升UWVF效率，且该方法能精确处理电阻屏、理想导体及可穿透散射体。采用大型曲面单元与无矩阵方法，我们实现了X波段飞机散射仿真。这些创新证明了UWVF在工业案例中的适用性。