Numerical solution of many-body wave scattering problem for small particles

A numerical approach to the problem of wave scattering by many small particles is developed under the assumptions k«1, d»a, where a is the size of the particles and d is the distance between the neighboring particles. On the wavelength one may have many small particles. An impedance boundary conditions are assumed on the boundaries of small particles. The results of numerical simulation show good agreement with the theory. They open a way to numerical simulation of the method for creating materials with a desired refraction coefficient.

💡 Research Summary

The paper develops a numerical framework for solving the many‑body wave scattering problem when a large number of small particles are embedded in a host medium. The authors assume that the particle radius a is much smaller than the wavelength (k a ≪ 1) and that the distance d between neighboring particles satisfies d ≫ a, so that many particles can fit within a single wavelength. Each particle is a ball with an impedance boundary condition ∂u/∂n = ζₘ u on its surface, where ζₘ may vary from particle to particle.

Starting from the Helmholtz equation with the usual radiation condition, the total field u(x) is expressed as the incident field u₀(x) plus the sum of single‑layer potentials over all particle surfaces. By introducing the “effective field” eₘ acting on the m‑th particle, the authors obtain an exact integral equation for the unknown surface densities σₘ(y). In the limit a → 0, asymptotic formulas for the total charge Qₘ and the surface density σₘ are derived: Qₘ ≈ −4π a / (1 + i k a/ζₘ), σₘ ≈ −4π ζₘ / (1 + i k a/ζₘ).
These lead to an explicit approximation for the effective field (equation 15), which involves a sum over all particles of the Green’s function G(xₘ,xⱼ) multiplied by the asymptotic charges.

Two linear algebraic systems (LAS) are then formulated. The first system (16) directly solves for the charges Qₘ using the exact particle positions; its matrix is almost diagonal and is provably invertible for sufficiently small a. The second system (17) results from a collocation method applied to the limiting integral equation (9). The domain D is partitioned into P small cubes; the unknowns are the average fields in each cube. Because P ≪ M, solving (17) is computationally much cheaper while still capturing the collective scattering effect.

Numerical experiments are performed with k = 1, κ = 0.9, and a constant particle density. Various values of a, d, and the number of collocation points P are tested. The relative error of the solution to (9) decreases as P increases: with P = 125 the error is about 1 % (real part) and 0.02 % (imaginary part); with P = 512 the errors drop to 0.29 % and 0.005 %; and with P ≈ 4096 the error falls below 0.1 % for both components. Comparisons between the solutions of LAS (16) and LAS (17) show that the discrepancy diminishes when the particles are smaller and more widely spaced; for a = 0.001 and d = 15 a the difference is less than 0.08 %.

The study demonstrates that the asymptotic formulas derived in earlier theoretical work are accurate for realistic configurations involving many small scatterers, and that the collocation‑based LAS provides an efficient computational tool for large‑scale problems. The authors conclude that the presented methodology opens the way to numerical design of materials with prescribed refraction coefficients, such as metamaterials, acoustic cloaks, or engineered photonic structures.