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Efficient time-domain scattering synthesis via frequency-domain singularity subtraction

Oscar P. Bruno, Manuel A. Santana

TL;DR

This work tackles the challenge of time-domain scattering by trapping obstacles, where near-real complex resonances hinder inverse Fourier transforms. It introduces a frequency-domain singularity-subtraction (SS) technique together with an incidence-excitation (IE) adaptive rational-approximation framework to identify resonances from real-frequency data, enabling regularization of the inverse transform and efficient large-time evaluation. The approach combines boundary-integral formulations for 2D exterior problems, an adaptive IE algorithm to obtain resonance poles and residues, and a large-time asymptotic expansion for the subtracted polar contributions, yielding the FTH-SS method. Numerical results across multiple trapping and non-trapping geometries demonstrate dispersion-free, accurate, and efficient long-time simulations, confirming the practical impact of the method for challenging scattering problems.

Abstract

Fourier transform-based methods enable accurate, dispersion-free simulations of time-domain scattering problems by evaluating solutions to the Helmholtz equation at a discrete set of frequencies sufficient to approximate the inverse Fourier transform. However, in the case of scattering by trapping obstacles, the Helmholtz solution exhibits nearly-real complex resonances -- which significantly slows the convergence of numerical inverse transform. To address this difficulty this paper introduces a frequency-domain singularity subtraction technique that regularizes the integrand of the inverse transform and efficiently computes the singularity contribution via a combination of a straightforward and inexpensive numerical technique together with a large-time asymptotic expansion. Crucially, all relevant complex resonances and their residues are determined via rational approximation of integral equation solutions at real frequencies. An adaptive algorithm is employed to ensure that all relevant complex resonances are properly identified.

Efficient time-domain scattering synthesis via frequency-domain singularity subtraction

TL;DR

This work tackles the challenge of time-domain scattering by trapping obstacles, where near-real complex resonances hinder inverse Fourier transforms. It introduces a frequency-domain singularity-subtraction (SS) technique together with an incidence-excitation (IE) adaptive rational-approximation framework to identify resonances from real-frequency data, enabling regularization of the inverse transform and efficient large-time evaluation. The approach combines boundary-integral formulations for 2D exterior problems, an adaptive IE algorithm to obtain resonance poles and residues, and a large-time asymptotic expansion for the subtracted polar contributions, yielding the FTH-SS method. Numerical results across multiple trapping and non-trapping geometries demonstrate dispersion-free, accurate, and efficient long-time simulations, confirming the practical impact of the method for challenging scattering problems.

Abstract

Fourier transform-based methods enable accurate, dispersion-free simulations of time-domain scattering problems by evaluating solutions to the Helmholtz equation at a discrete set of frequencies sufficient to approximate the inverse Fourier transform. However, in the case of scattering by trapping obstacles, the Helmholtz solution exhibits nearly-real complex resonances -- which significantly slows the convergence of numerical inverse transform. To address this difficulty this paper introduces a frequency-domain singularity subtraction technique that regularizes the integrand of the inverse transform and efficiently computes the singularity contribution via a combination of a straightforward and inexpensive numerical technique together with a large-time asymptotic expansion. Crucially, all relevant complex resonances and their residues are determined via rational approximation of integral equation solutions at real frequencies. An adaptive algorithm is employed to ensure that all relevant complex resonances are properly identified.
Paper Structure (33 sections, 3 theorems, 139 equations, 15 figures, 2 algorithms)

This paper contains 33 sections, 3 theorems, 139 equations, 15 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frequency-time hybrid method
  4. Frequency-domain integral equation solutions
  5. $O(1)$-cost Fourier transform at large times
  6. Complex resonances
  7. Incidence-excited resonances from real-frequency data
  8. Scalar-valued and random-sketching vector-valued rational approximation
  9. Adaptive random-excitation (RE) resonance evaluation
  10. Incidence-excitation (IE) resonance evaluation for time-domain problems
  11. Adaptive IE resonance evaluation
  12. Computation of the residues $d_{\mathbf{p},n}$ \ref{['eqn:res']}
  13. Frequency-domain singularity subtraction
  14. Singularity subtraction and the integrals $\mathcal{I}_{1}(\mathbf{r},t)$ and $\mathcal{I}_{2}(\mathbf{r},t)$
  15. Numerical evaluation of $\mathcal{I}_{1}(\mathbf{r},t)$ and numerical/asymptotic evaluation of $\mathcal{I}_{2}(\mathbf{r},t)$
  16. ...and 18 more sections

Key Result

Theorem 1

Let $\mu(W_1,W_2)$, $\varepsilon(\mu(W_1,W_2))$ and $N_h^I$, be defined as in mu, eqn:eps_m_def and eq:nih, respectively. Further, assume that the relation holds, and that there exists a constant $D > 0$ such that, for any $W_1 < 0$ and $W_2 > 0$, the residues $c_{\mathbf{p},n}(\mathbf{r})$eqn:res of the poles contained in the set $\mathcal{M}_h^I$eq:box associated with the interval $I = I(W_1,W_

Figures (15)

  • Figure 1: Scatterers used in some of the examples presented in this section. From left to right: large-aperture circle ($1.25$-radian aperture), small-aperture circle ($0.125$-radian aperture), open rocket-shaped cavity, closed-curve cavity.
  • Figure 2: FTH and FTH-SS solution errors for the closed-circle scatterer, as a function of the number of integral equation inverses used. Due to its non-trapping nature, this scatterer does not generate complex resonances near the real axis. Consequently, the FTH-SS method performed no actual singularity subtraction, and its results coincide with those of the standard FTH method in this case.
  • Figure 3: FTH and FTH-SS solution errors for the open circle scatterer depicted on the leftmost panel in Figure \ref{['fig:scatterers']}, as a function of the number of integral equation inverses used. Since a number $J < 200$ of inverses was used for these test cases, the FTH-SS method did not trigger the IE-algorithm's adaptivity.
  • Figure 4: Same as Figure \ref{['fig:FTH_comp_no_adapt']} but using a different range of numbers of integral equation inverses---for which the IE-algorithm's adaptivity was triggered. The triangles mark the errors corresponding to three different numbers of inverses actually used by FTH-SS method---which are determined by each one of the three adaptivity levels triggered in the adaptive IE method.
  • Figure 5: Temporal evolution of scattering from the large-aperture circle, shown at an increasing sequence of times from left to right and top to bottom. Each panel shows the absolute value of the real part of the total field.
  • ...and 10 more figures

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Theorem 1
  • ...and 7 more