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.

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。该方法利用平面波的叠加在有限元网格上逐单元表示解。我们重点介绍并行UWVF实现——ParMax，特别强调在具有分段光滑边界的散射体环境中求解高阶解。本文阐述了弯曲曲面三角形在UWVF中的集成方法，这需要针对系统矩阵组装进行数值积分。我们还展示了如何实现总场与散射场分解策略，并结合跨界面传输条件处理电阻薄片。值得注意的是，该方法支持多种单元形状，单元尺寸可远大于辐射波长，且低内存版本易于实现（尽管计算成本较高）。通过数值算例验证了弯曲单元对UWVF计算效率的提升效果。该方法能够精确处理电阻屏、理想导体及可穿透散射体。采用大型弯曲单元与低内存方案，我们成功模拟了X波段频率下飞行器的电磁散射特性。这些创新证明了UWVF在工业应用中的实用性。