FFT-acceleration and stabilization of the 3D Marching-on-in-Time Contrast Current Density Volume Integral Equation for scattering from high contrast dielectrics

TL;DR

This work develops an FFT-accelerated and stabilization-enabled MOT-JVIE for 3D scattering from high-permittivity dielectrics by combining spatial and temporal FFT techniques with hierarchical Toeplitz division. It analyzes stability through a positive definite stability framework and introduces FIR-based regularization to maintain stability with minimal accuracy loss, validated on large voxel counts. Numerical results on cubic and spherical targets demonstrate good agreement with CST and Mie-series benchmarks, confirming both computational efficiency (scaling ~ $\mathcal{O}(M\log^2 M)$) and practical accuracy. The methods enable large-scale time-domain simulations of complex dielectrics with robust stability and high fidelity, suitable for wideband and resonant scenarios.

Abstract

An implicit causal space-time Galerkin scheme applied to the contrast current density volume integral equation gives rise to a marching-on-in-time scheme known as the MOT-JVIE, which is accelerated and stabilized via a fully embedded FIR filter to compute the electromagnetic scattering from high permittivity dielectric objects discretized with over a million voxels. A review of two different acceleration approaches previously developed for two-dimensional time-domain surface integral equations based on fast Fourier transforms (FFTs), leads to an understanding why these schemes obtain the same order of acceleration and the extension of this FFT-acceleration to the three-dimensional MOT-JVIE. The positive definite stability analysis (PDSA) for the MOT-JVIE shows that the number of voxels for a stable MOT-JVIE discretization is restricted by the finite precision of the matrix elements. The application of the PDSA provides the insight that stability can be enforced through regularization, at the cost of accuracy. To minimize the impact in accuracy, FIR-regularization is introduced, which is based on low group-delay linear-phase high-pass FIR-filters. We demonstrate the capabilities of the FFT-accelerated FIR-regularized MOT-JVIE for a number of numerical experiments with high permittivity dielectric scatterers.

Figures (16)

• Figure 1: The scattering setup consists of a sphere with a constant dielectric contrast value represented by the color red in a background medium. The sphere is enclosed by a box illustrated by the black dashed lines.
• Figure 2: The box around the scatter in Figure \ref{['fig:Sphere']} is divided evenly along each Cartesian direction, i.e. $U$ times in the $\hat{\mathbf{x}}$-direction, $V$ times in the $\hat{\mathbf{y}}$-direction, $W$ times the in $\hat{\mathbf{z}}$-direction. So, the scattering setup is discretized using $M = U \times V \times W$ voxels, each of a dimension $\Delta x \times \Delta y \times \Delta z$. We have visualized the voxels in Green at $[1,1,1]$ and $[U,V,W]$. The dielectric contrast of each voxel is equal to the dielectric contrast of the scatterer setup shown in Figure \ref{['fig:Sphere']} sampled at $\mathbf{r}_{\mathbf{u}}$, i.e. the center of each voxel.
• Figure 3: The matrix elements shown here illustrate a part of the block-lower triangular matrix in Equation \ref{['eq:PnMatrixVector']}. We have computed the values of $\mathbf{J}_n$ up to the red-dashed line $\mathbin{{\stackinset{c}{0pt}{c}{0pt}{1}{\bigcirc}}}$, thus we can perform any on the left of the line $\mathbin{{\stackinset{c}{0pt}{c}{0pt}{1}{\bigcirc}}}$. Because we have computed up to $\mathbf{J}_n$, according to Equation \ref{['eq:MOTInversion']}, we have already solved all matrix vector products above the line $\mathbin{{\stackinset{c}{0pt}{c}{0pt}{2}{\bigcirc}}}$ to find $\mathbf{P}_{n-1}$. The optimally largest Toeplitz matrix $\mathbf{U}_0$ is the one enclosed by the horizontal and vertical red lines and the horizontal and vertical black dashed lines marked as $\mathbin{{\stackinset{c}{0pt}{c}{0pt}{3}{\bigcirc}}}$. The strictly lower triangular Toeplitz matrix $\mathbf{L}_0$ is the matrix enclosed by the horizontal and vertical red-dashed lines and the horizontal and vertical black-dashed lines marked as $\mathbin{{\stackinset{c}{0pt}{c}{0pt}{4}{\bigcirc}}}$
• Figure 4: The division of the strictly lower triangular Toeplitz matrix $\mathbf{L}_0$\ref{['eq:LowerTriangularToeplitzL']} into strictly lower triangular Toeplitz matrices $\mathbf{L}_1$ and Toeplitz matrix $\mathbf{U}_1$ to which we can apply FFT-acceleration as it does not include elements above the diagonal in Equation \ref{['eq:PnMatrixVector']} here marked by the red triangle. We indicate the approximate sizes of the matrices.
• Figure 5: The definition of the sets $\mathcal{M}_k$ is used to divided the interaction matrices $\mathbf{Z}_{1}$ through $\mathbf{Z}_\ell$ into $K$ sets. The $k$-th set is used to construct the four-level Toeplitz matrix $\mathbf{U}_k$.