In many applications, electromagnetic scattering from inhomogeneous objects embedded in multiple layers needs to be simulated numerically. The straightforward solution by the method of moments (MoM) for the volume integral equation method is computationally expensive. Due to the shift-invariance and correlation properties of the layered-medium Green’s functions, the stabilized-biconjugate gradient fast Fourier transform (BCGS-FFT) has been developed to greatly reduce the computational complexity of the MoM, but so far this method is limited to objects located in a homogeneous background or in the same layer of a layered medium background. For those problems with objects located in different layers, FFT cannot be applied directly in the direction normal to the layer interfaces, thus the MoM solution requires huge computer memory and CPU time. To overcome these difficulties, the BCGS-FFT method combined with the domain decomposition method (DDM) is proposed in this article. With the BCGS-FFT-DDM, the objects or different parts of an object are first treated separately in several subdomains, each of which satisfies the 3-D shift-invariance and correlation properties; the couplings among the different objects/parts are then taken into account, where the coupling matrices can be built to satisfy the 2-D shift-invariance property if the objects/subdomains have the same mesh size on the $xy$ plane. Hence, 3-D FFT and 2-D FFT can respectively be applied to accelerate the self- and mutual-coupling matrix-vector multiplications. By doing so, the impedance matrix is explicitly formed as one including both the self- and mutual-coupling parts, and the solver converges well for problems with considerable conductivity contrasts. The computational complexity in memory and CPU time for self-coupling matrix-vector multiplication are $O(N_{{z}}^{q} N_{x} N_{y})$ and $O(N_{{z}} N_{x} N_{y} \log (N_{{z}} N_{x} N_{y}))$ respectively, and for mutual-coupling matrix-vector multiplication are $O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y})$ and $O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y} \log (N_{x} N_{y}))$ , respectively, for the proposed method, where $N_{x}$ and $N_{y}$ are the cell numbers of all the subdomains in the $x$ - and $y$ -directions, and $N_{{z}}^{p}$ and $N_{{z}}^{q}$ the cell numbers of different subdomains in the ${z}$ -direction. Several results of different subsurface sensing scenarios are presented to show the capabilities of this method.

中文翻译：

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复杂非均匀物体穿过多层的电磁散射的快速3D体积积分方程域分解方法

在许多应用中，来自嵌入多层的非均匀物体的电磁散射需要进行数值模拟。体积积分方程方法的矩量法 (MoM) 的直接求解在计算上是昂贵的。由于分层介质格林函数的平移不变性和相关性，已经开发了稳定双共轭梯度快速傅立叶变换（BCGS-FFT）以大大降低 MoM 的计算复杂度，但到目前为止这种方法是有限的位于同质背景或分层介质背景的同一层中的对象。对于那些对象位于不同层的问题，FFT 不能直接应用于层接口的法线方向，因此 MoM 解决方案需要大量的计算机内存和 CPU 时间。为了克服这些困难，本文提出了结合域分解方法（DDM）的BCGS-FFT方法。使用BCGS-FFT-DDM，对象或对象的不同部分首先在几个子域中单独处理，每个子域都满足3-D平移不变性和相关性；然后考虑不同对象/部分之间的耦合，如果对象/子域在对象/子域上具有相同的网格大小，则可以构建耦合矩阵以满足二维平移不变性。每个都满足 3-D 平移不变性和相关性；然后考虑不同对象/部分之间的耦合，如果对象/子域在对象/子域上具有相同的网格大小，则可以构建耦合矩阵以满足二维平移不变性。每个都满足 3-D 平移不变性和相关性；然后考虑不同对象/部分之间的耦合，如果对象/子域在对象/子域上具有相同的网格大小，则可以构建耦合矩阵以满足二维平移不变性。 $xy$ 飞机。因此，可以分别应用 3-D FFT 和 2-D FFT 来加速自耦合和互耦合矩阵向量乘法。通过这样做，阻抗矩阵被明确地形成为包括自耦合和互耦合部分的阻抗矩阵，并且求解器可以很好地收敛于具有相当大的电导率对比的问题。自耦合矩阵向量乘法的内存计算复杂度和 CPU 时间为 $O(N_{{z}}^{q} N_{x} N_{y})$ 和 $O(N_{{z}} N_{x} N_{y} \log (N_{{z}} N_{x} N_{y}))$ 分别，对于互耦矩阵向量乘法是 $O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y})$ 和 $O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y} \log (N_{x} N_{y}))$ ，分别对于所提出的方法，其中 $N_{x}$ 和 $N_{y}$ 是所有子域的单元格编号 $x$ - 和 $y$ - 方向，和 $N_{{z}}^{p}$ 和 $N_{{z}}^{q}$ 不同子域的细胞数 ${z}$ -方向。展示了不同地下传感场景的几个结果，以展示该方法的能力。