Fast 3-D Volume Integral Equation Domain Decomposition Method for Electromagnetic Scattering by Complex Inhomogeneous Objects Traversing Multiple Layers

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 <inline-formula> <tex-math notation="LaTeX">$xy$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$O(N_{{z}}^{q} N_{x} N_{y})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$O(N_{{z}} N_{x} N_{y} \log (N_{{z}} N_{x} N_{y}))$ </tex-math></inline-formula> respectively, and for mutual-coupling matrix-vector multiplication are <inline-formula> <tex-math notation="LaTeX">$O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y} \log (N_{x} N_{y}))$ </tex-math></inline-formula>, respectively, for the proposed method, where <inline-formula> <tex-math notation="LaTeX">$N_{x}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N_{y}$ </tex-math></inline-formula> are the cell numbers of all the subdomains in the <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>- and <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula>-directions, and <inline-formula> <tex-math notation="LaTeX">$N_{{z}}^{p}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N_{{z}}^{q}$ </tex-math></inline-formula> the cell numbers of different subdomains in the <inline-formula> <tex-math notation="LaTeX">${z}$ </tex-math></inline-formula>-direction. Several results of different subsurface sensing scenarios are presented to show the capabilities of this method.