We present algorithms for solving high-frequency acoustic scattering problems in complex domains. The eikonal and transport partial differential equations from the WKB/geometric optic approximation of the Helmholtz equation are solved recursively to generate boundary conditions for a tree of eikonal/transport equation pairs, describing the phase and amplitude of a geometric optic wave propagating in a complicated domain, including reflection and diffraction. Edge diffraction is modeled using the uniform theory of diffraction. For simplicity, we limit our attention to domains with piecewise linear boundaries and a constant speed of sound. The domain is discretized into a conforming tetrahedron mesh. For the eikonal equation, we extend the jet marching method to tetrahedron meshes. Hermite interpolation enables second order accuracy for the eikonal and its gradient and first order accuracy for its Hessian, computed using cell averaging. To march the eikonal on an unstructured mesh, we introduce a new method of rejecting unphysical updates by considering Lagrange multipliers and local visibility. To handle accuracy degradation near caustics, we introduce several fast Lagrangian initialization algorithms. We store the dynamic programming plan uncovered by the marcher in order to propagate auxiliary quantities along characteristics. We introduce an approximate origin function which is computed using the dynamic programming plan, and whose 1/2-level set approximates the geometric optic shadow and reflection boundaries. We also use it to propagate geometric spreading factors and unit tangent vector fields needed to compute the amplitude and evaluate the high-frequency edge diffraction coefficient. We conduct numerical tests on a semi-infinite planar wedge to evaluate the accuracy of our method. We also show an example with a more realistic building model with challenging architectural features.

翻译：本文提出了求解复杂域中高频声散射问题的算法。通过递归求解亥姆霍兹方程的WKB/几何光学近似中的程函和输运偏微分方程组，生成描述几何光学波在复杂域（含反射和衍射）中传播的相位与振幅的程函/输运方程对树结构。边缘衍射采用一致性绕射理论进行建模。为简化问题，本文限定研究具有分段线性边界和恒定声速的域。将计算域离散为一致四面体网格。针对程函方程，将喷射推进法扩展到四面体网格。采用埃尔米特插值实现程函及其梯度二阶精度、黑塞矩阵一阶精度（通过单元平均计算）。为在非结构网格上推进程函，引入基于拉格朗日乘子和局部可见性的非物理解剔除新方法。针对焦散面附近精度退化问题，提出多种快速拉格朗日初始化算法。存储推进算法揭示的动态规划路径，以便沿特征线传播辅助量。引入基于动态规划路径计算的近似原点函数，其1/2水平集近似几何光学阴影边界和反射边界。进而利用该函数传播计算振幅所需的几何扩散因子和单位切向量场，并评估高频边缘衍射系数。在半无限平面楔形体上进行数值测试以评估方法精度，并以具有复杂建筑特征的真实建筑模型为例进行验证。