Scattered wavefield in the stochastic homogenization regime

TL;DR

This paper develops a mathematical framework using stochastic homogenization to analyze ultrasound wave scattering in random multiscale media, providing asymptotic expansions and error estimates supported by numerical validation.

Contribution

It introduces a novel stochastic homogenization approach with boundary layer correctors for wave scattering in random media, including rigorous error estimates.

Findings

Asymptotic expansions of scattered fields are derived.

Quantitative error estimates are established.

Numerical experiments support theoretical results.

Abstract

In the context of providing a mathematical framework for the propagation of ultrasound waves in a random multiscale medium, we consider the scattering of classical waves (modeled by a divergence form scalar Helmholtz equation) by a bounded object with a random composite micro-structure embedded in an unbounded homogeneous background medium. Using quantitative stochastic homogenization techniques, we provide asymptotic expansions of the scattered field in the background medium with respect to a scaling parameter describing the spatial random oscillations of the micro-structure. Introducing a boundary layer corrector to compensate the breakdown of stationarity assumptions at the boundary of the scattering medium, we prove quantitative $L^2$- and $H^1$- error estimates for the asymptotic first-order expansion. The theoretical results are supported by numerical experiments.