Time-domain simulation of wave propagation across resonant meta-interfaces

Abstract

Meta-interface models stem from the homogenization, in a low-frequency dynamic regime, of thin heterogeneous layers that are structured to achieve uncommon properties at the macroscopic level. When the layer is composed of a thin periodic array of highly-contrasted inclusions embedded within a homogeneous background medium then the corresponding effective interface model is characterized by jump conditions that, in the harmonic regime, involve some singular frequency-dependent terms. In this context, the article is concerned with the simulation of transient waves across such resonant meta-interfaces and a numerical method is proposed to handle the associated resonant jump conditions. To do so, a set of auxiliary variables is introduced locally along the interface and an augmented system of first-order equations in time accompanied with local-in-time jump conditions is derived. This system is then discretized on a Cartesian grid and solved using a high-order finite-difference scheme while the complexity associated with the geometry of the interface and the jump conditions is handled using an immersed interface method. A set of numerical examples in 1D and 2D is proposed to illustrate and validate the overall numerical approach, and quantitative comparisons with semi-analytical solutions are also provided.

Effective Resonant Model and Simulations in the Time-Domain of Wave Scattering from a Periodic Row of Highly-Contrasted Inclusions

“…Integrating (25) in time and differentiating it with respect to x 2 , the second integral I 2 writes:…”

“…Finally, the solution U h = (V h , Σ h ) of the time-domain effective problem (37) is discretized with a mesh size ∆X and a time step ∆t and we denote by (U h ) n i,j the approximation of U h at point (i∆X, j∆X) and time t n = n∆t. A specific numerical method has been developed in (25) to handle the resonant jump conditions considered, which are non-local in time. The latter relies on the introduction of a set of auxiliary variables, locally along the interface, to derive some equivalent jump conditions that are local in time.…”

“…The term ε 1 is the numerical error associated with the simulation in the microstructured configuration while ε 3 is the one associated with the simulation based on the homogenized model. Both are governed and controlled by the numerical methods employed, see (11) and (25), respectively. In appropriate implementations of the latter, these errors are considered to be negligible compared to ε 2 .…”

“…Figure 8b compares the profiles of velocity for the homogenized model, computed either numerically on a grid of mesh size ∆X = 0.2 m or semi-analytically. The latter computation is performed using the method described in (25) which relies on the calculation of the reflexion and transmission coefficients from (A.4). On the one hand, the numerical solution for the microstructured configuration on the fine grid ∆X = 0.025 m is assumed to have converged, see Fig.…”

“…microstructure-based, simulations. One notes that the time-domain simulations presented here rely on the numerical method introduced in (25) to handle resonant jump conditions on meta-interfaces. Numerical results are discussed to highlight the computational merits of the homogenized model for the configuration of interest compared to full-field simulations.…”