Discretization in Multilayered Media with High Contrasts: Is It All About the Boundaries?

TL;DR

This work addresses the computational challenge of high-contrast, multilayered wave propagation by adopting a boundary-integral formulation that restricts discretization to interfaces. It demonstrates that simple rules based on the maximum wavenumber are ineffective for multilayer configurations and shows, through analytic circular-layer solutions, that the exterior wavenumber $k_0$ often governs discretization needs. The authors develop an adaptive, anisotropic boundary-mesh refinement strategy driven by interpolation-error metrics (via a recovered Hessian) and compare two adaptive schemes (ADAPT-ANA and ADAPT-BIE), achieving uniform accuracy across multiple interfaces with substantial efficiency gains. The findings have practical implications for efficiently simulating metamaterials and cloaking designs in 2D, with clear pathways to extension to 3D and improved treatment of nearly singular boundary integrals.

Abstract

Wave propagation in multilayered media with high material contrasts poses significant numerical challenges, as large variations in wavenumbers lead to strong reflections and complex transmission of the incoming wave field. To address these difficulties, we employ a boundary integral formulation thereby avoiding volumetric discretization. In this framework, the accuracy of the numerical solution depends strongly on how the material interfaces are discretized. In this work, we demonstrate that standard meshing strategies based on resolving the maximum wavenumber across the domain become computationally inefficient in multilayered configurations, where high wavenumbers are confined to localized subdomains. Through a systematic study of multilayer transmission problems, we show that no simple discretization rule based on the maximum wavenumber or material contrasts emerges. Instead, the wavenumber of the background (exterior) medium plays a dominant role in determining the optimal boundary resolution. Building on these insights, we propose an adaptive approach that achieves uniform accuracy and efficient computation across multiple layers. Numerical experiments for a range of multilayer configurations demonstrate the scalability and robustness of the proposed approach.

Figures (18)

• Figure 1: Schematic of the concentric layered transmission problem. Each interface $\Gamma_j$ separates regions with wavenumbers $k_j$ and $k_{j+1}$. The exterior field $u_0$ includes an incoming incident wave $u_0^{in}$.
• Figure 2: Schematics of the scattering domain with radius $R_0 = 4$, showing two types of boundary conditions: (a) a sound-hard obstacle and (b) a penetrable interface with transmission conditions.
• Figure 3: Real part of the total field analytic solutions for two contrast cases with $R_0 = 4$. The left plot on each sub-figure shows the solution for the sound-hard boundary condition (Neumann), while the right plot of each sub-figure shows the solution for the transmission boundary condition.
• Figure 4: Real part of the total field analytic solutions for a higher contrast case with sharper contrasts $\beta_0$ as given.
• Figure 5: Schematic of the samplings for $||u_j - u^{ana}_j||_{L^2(r)}$. The boundary solution is computed on the black ring at radius $R_0$, and the error between the off-boundary PTR and analytic solutions is evaluated at each sampling radii (red rings) separately.