A boundary integral equation method for the 3-D Helmholtz equation in multilayered media with many quasi-periodic layers is presented. Compared with conventional quasi-periodic Green's function method, the new method is robust at all scattering parameters. A periodizing scheme is used to decompose the solution into near- and far-field contributions. The near-field contribution uses the free-space Green's function in an integral equation on the interface in the unit cell and its immediate eight neighbors; the far-field contribution uses proxy point sources that enclose the unit cell. A specialized high-order quadrature is developed to discretize the underlying surface integral operators to keep the number of unknowns per layer small. We achieve overall linear computational complexity in the number of layers by reducing the linear system into block tridiagonal form and then solving the system directly via block LU decomposition. The new solver is capable of handling a 100-interface structure with 961.3k unknowns to $10^{-5}$ accuracy in less than 2 hours on a desktop workstation.
翻译:提供了多层介质中多半周期性层3-D Helmholtz 方程式的边界整体方程方法。 与常规的半定期Green函数法相比,新法在所有散射参数上都很稳健。 使用一个周期性办法将解决方案分解成近地和远地贡献物。 近地贡献方在单元单元格及其近邻8个界面的界面上使用自由空间Green函数的一体化方程; 远地贡献方块使用嵌入单元单元格的替代点源。 正在开发一个专门的高阶二次曲线二次曲线,使底表面整体操作器分离,使每个层的未知数保持在小处。 我们通过将线性系统缩小成三角方形块,然后通过LU decomposition街区直接解析系统,实现层数的总体线性计算复杂度。 新求解器能够在台式工作站上用不到2小时的时间处理一个100个界面结构,961.3k 至10 ⁇ -5}的准确度为 。