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Asymptotic models for time-domain scattering by small particles of arbitrary shapes

Maryna Kachanovska, Adrian Savchuk

TL;DR

This work develops high-order time-domain models for scattering by many small particles of arbitrary shapes, formulating the problem via boundary integral equations and an asymptotic Galerkin discretization. It introduces three models: a Galerkin-Foldy-Lax (GFL) model, a simplified version, and a Born-type approximation, with stability and convergence results established in the frequency domain and transferred to the time domain. The first two models achieve $O(\varepsilon^{3})$-level accuracy, while the Born model attains $O(\varepsilon^{1})$-level accuracy, with proofs supported by detailed operator-norm bounds and density-decomposition arguments. Computational efficiency is addressed through the simplified model, which relies on capacitances $c_k^{\varepsilon}$ and first moments $\mathbf{p}_k^{\varepsilon}$, enabling scalable simulations for larger $N$. Numerical experiments validate the theoretical convergence rates and demonstrate stability under periodic perturbations, establishing the practical viability of the proposed asymptotic framework for time-domain scattering by many small particles.

Abstract

In this work, we investigate time-dependent wave scattering by multiple small particles of arbitrary shape. To approximate the solution of the associated boundary-value problem, we derive an asymptotic model that is valid in the limit as the particle size tends to zero. Our method relies on a boundary integral formulation, semi-discretized in space using a Galerkin approach with appropriately chosen basis functions, s.t. convergence is achieved as the particle size vanishes rather than by increasing the number of basis functions. Since the computation of the Galerkin matrix involves double integration over particles, the method can become computationally demanding when the number of obstacles is large. To address this, we also derive a simplified model and consider the Born approximation to improve computational efficiency. For the high-order models, we provide an error analysis, supported and validated by numerical experiments.

Asymptotic models for time-domain scattering by small particles of arbitrary shapes

TL;DR

This work develops high-order time-domain models for scattering by many small particles of arbitrary shapes, formulating the problem via boundary integral equations and an asymptotic Galerkin discretization. It introduces three models: a Galerkin-Foldy-Lax (GFL) model, a simplified version, and a Born-type approximation, with stability and convergence results established in the frequency domain and transferred to the time domain. The first two models achieve -level accuracy, while the Born model attains -level accuracy, with proofs supported by detailed operator-norm bounds and density-decomposition arguments. Computational efficiency is addressed through the simplified model, which relies on capacitances and first moments , enabling scalable simulations for larger . Numerical experiments validate the theoretical convergence rates and demonstrate stability under periodic perturbations, establishing the practical viability of the proposed asymptotic framework for time-domain scattering by many small particles.

Abstract

In this work, we investigate time-dependent wave scattering by multiple small particles of arbitrary shape. To approximate the solution of the associated boundary-value problem, we derive an asymptotic model that is valid in the limit as the particle size tends to zero. Our method relies on a boundary integral formulation, semi-discretized in space using a Galerkin approach with appropriately chosen basis functions, s.t. convergence is achieved as the particle size vanishes rather than by increasing the number of basis functions. Since the computation of the Galerkin matrix involves double integration over particles, the method can become computationally demanding when the number of obstacles is large. To address this, we also derive a simplified model and consider the Born approximation to improve computational efficiency. For the high-order models, we provide an error analysis, supported and validated by numerical experiments.

Paper Structure

This paper contains 59 sections, 53 theorems, 332 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem setting and main result
  3. Problem setting
  4. Preliminary notation and definitions
  5. Functions on $\Gamma^{\varepsilon}_k$
  6. $H^{1/2}(\Gamma^{\varepsilon})$, $H^{-1/2}(\Gamma^{\varepsilon})$ spaces
  7. Fourier-Laplace transform
  8. Auxiliary notaiton
  9. Principal results of the paper
  10. An asymptotic model 1
  11. An asymptotic model 2
  12. An asymptotic model 3 (Born approximation)
  13. Well-posedness, stability and the error estimates
  14. Derivation of the asymptotic model (\ref{['eq:tdsys']})
  15. An abstract asymptotic Galerkin method
  16. ...and 44 more sections

Key Result

Theorem 2.1

The models eq:tdsys, eq:tdsys2 are well-posed; together with the model eq:tdsysborn, they are uniformly stable. The models eq:tdsys and eq:tdsys2 are convergent of order $p=2$. The model eq:tdsysborn is convergent of order $p=1$.

Figures (5)

  • Figure 1: An example of domains $\Omega_k$ and $\Omega_{\ell}$, where $d_{k \ell} := \operatorname{dist}(B_k, B_{\ell})$.
  • Figure 2: Geometric configuration of the numerical experiments for $\varepsilon=0.5$. The obstacles are centred at $\boldsymbol{c}_1 = (0,0,0), ~ \boldsymbol{c}_2 = (-1,-1,1), ~\boldsymbol{c}_3 = (-1,1,1), ~ \boldsymbol{c}_4 = (1,-1,-1),~ \boldsymbol{c}_5 = (1,1,1)$.
  • Figure 3: (Left) Time evolution of the scattered field for $\varepsilon = 0.1$. (Right) Absolute error of the scattered field computed using the three models described in Section \ref{['sec:principal_result']}: the Galerkin-Foldy-Lax (GFL), Simplified Galerkin-Foldy-Lax (SGFL), and Born models.
  • Figure 4: Dependence of the scattered field on time for $\varepsilon = 0.05$.
  • Figure 5: (Left) Geometric configuration from Section \ref{['sec:many_non_convex_particles']} with $\varepsilon = 0.25$; (Top right) Time evolution of the scattered field at $\boldsymbol{x}_0 = (-1.5, -1.5, -1.5)$ for $\varepsilon = 0.05$. (Bottom right) An absolute error between the scattered field obtained from the simplified and Born models, measured in the $L^{\infty}(0,T)$ norm.

Theorems & Definitions (115)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 4
  • Theorem 2.1
  • proof
  • Remark 5
  • ...and 105 more