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

TL;DR

This paper develops an effective boundary model for scattering by a thin layer of randomly distributed nanoparticles, simplifying complex simulations and providing error estimates with numerical validation.

Contribution

It introduces a multi-scale asymptotic expansion approach to replace a random nanoparticle layer with an effective boundary condition, including rigorous error analysis.

Findings

Effective boundary condition accurately approximates scattering

Unique solutions exist for the corrector problems in random domains

Numerical simulations confirm theoretical error estimates

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.