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Scattering from a random thin coating of nanoparticles: the Dirichlet case

Amandine Boucart, Sonia Fliss, Laure Giovangigli

TL;DR

This work addresses time-harmonic scattering by a planar object coated with a thin, randomly distributed layer of Dirichlet nanoparticles. It develops a two-scale asymptotic framework that replaces the nanoparticle layer with an effective boundary condition, combining far-field Helmholtz behavior and near-field Laplace-type problems on a random half-space. The authors prove well-posedness in the random setting, construct a formal two-scale expansion, and derive quantitative error estimates, including improved rates under a quantitative mixing hypothesis. They also provide numerical experiments validating the theory and illustrating the accuracy of first- and second-order effective models. The results significantly reduce computational cost while delivering provable error control for stochastic thin-layer scattering problems in acoustics and related settings.

Abstract

We study the time-harmonic scattering by a heterogeneous object covered with a thin layer of randomly distributed sound-soft nanoparticles. The size of the particles, their distance between each other and the layer's thickness are all of the same order but small compared to the wavelength of the incident wave. Solving the Helmholtz equation in this context can be very costly and the simulation depends on the given distribution of particles. To circumvent this, we propose, via a multi-scale asymptotic expansion of the solution, an effective model where the layer of particles is replaced by an equivalent boundary condition. The coefficients that appear in this equivalent boundary condition depend on the solutions to corrector problems of Laplace type defined on unbounded random domains. Under the assumption that the particles are distributed given a stationary and mixing random point process, we prove that those problems admit a unique solution in the proper space. We then establish quantitative error estimates for the effec tive model and present numerical simulations that illustrate our theoretical results.

Scattering from a random thin coating of nanoparticles: the Dirichlet case

TL;DR

This work addresses time-harmonic scattering by a planar object coated with a thin, randomly distributed layer of Dirichlet nanoparticles. It develops a two-scale asymptotic framework that replaces the nanoparticle layer with an effective boundary condition, combining far-field Helmholtz behavior and near-field Laplace-type problems on a random half-space. The authors prove well-posedness in the random setting, construct a formal two-scale expansion, and derive quantitative error estimates, including improved rates under a quantitative mixing hypothesis. They also provide numerical experiments validating the theory and illustrating the accuracy of first- and second-order effective models. The results significantly reduce computational cost while delivering provable error control for stochastic thin-layer scattering problems in acoustics and related settings.

Abstract

We study the time-harmonic scattering by a heterogeneous object covered with a thin layer of randomly distributed sound-soft nanoparticles. The size of the particles, their distance between each other and the layer's thickness are all of the same order but small compared to the wavelength of the incident wave. Solving the Helmholtz equation in this context can be very costly and the simulation depends on the given distribution of particles. To circumvent this, we propose, via a multi-scale asymptotic expansion of the solution, an effective model where the layer of particles is replaced by an equivalent boundary condition. The coefficients that appear in this equivalent boundary condition depend on the solutions to corrector problems of Laplace type defined on unbounded random domains. Under the assumption that the particles are distributed given a stationary and mixing random point process, we prove that those problems admit a unique solution in the proper space. We then establish quantitative error estimates for the effec tive model and present numerical simulations that illustrate our theoretical results.

Paper Structure

This paper contains 30 sections, 34 theorems, 292 equations, 8 figures.

Table of Contents

  1. Presentation of the model
  2. Domain of propagation
  3. Problem formulation
  4. Outgoing wave condition based on a Dirichlet-to-Neumann operator
  5. Well-posedness of the scattering problem
  6. Random setting
  7. Well-posedness of the scattering problem in the random setting
  8. A formal asymptotic expansion
  9. Well-posedness of the near-field problems
  10. Functional inequalities in $\mathcal{W}_0(D)$
  11. Proof of proposition \ref{['prop:Utnwellposed']}
  12. Application to the first near-field terms
  13. Integral representation and behavior at infinity of the near-fields
  14. Effective model
  15. Construction of $U^{NF}_1$
  16. ...and 15 more sections

Key Result

Proposition 1

The operator $\Lambda^k:H^{\frac{1}{2}}(\Sigma_{L})\to H^{-\frac{1}{2}}(\Sigma_{L})$ is a continuous operator such that

Figures (8)

  • Figure 1: Illustration of the geometry of the model
  • Figure 2: Illustration of the different domains where the errors are estimated in section \ref{['sec:errest']}
  • Figure 3: Notations used in Section \ref{['subsubsec:osc']}
  • Figure 4: Real part of the total field $u_\varepsilon$ for $\rho=0.4$, frequency$=2$GHz, $\theta=\pi/4$, $k_2\varepsilon=10^{-1}$m, $\gamma=1+i$
  • Figure 5: Profile function $W_1$ for $\rho=0.1$ (left) and $\rho=0.4$ (right)
  • ...and 3 more figures

Theorems & Definitions (69)

  • Definition 1
  • Proposition 1
  • Theorem 2
  • Definition 2: Stationarity
  • Definition 3: Ergodicity
  • Theorem 3: Birkhoff ergodic Theorem
  • Corollary 4
  • Remark 1
  • Proposition 5
  • Remark 2
  • ...and 59 more