Non-periodic Media Research Articles

Dynamic homogenization of heterogeneous piezoelectric media: A polarization approach using infinite-body Green’s function

Dynamic homogenization theories are powerful tools for describing and understanding the behavior of heterogeneous media such as composites and metamaterials. However, a major challenge in the homogenization theory is determining Green’s function of these media, which makes it difficult to predict their effective constitutive relations, particularly for the finite-size and/or non-periodic media in real-world applications. In this paper, we present a formulation for finding the elastodynamic effective constitutive relations for general heterogeneous media, including finite-size and non-periodic ones, via a polarization approach based on the Hashin–Shtrikman principle along with Green’s identities. Our proposed formulation relies on the infinite-body Green’s function of a homogeneous reference medium, making it free from the difficulty of determining Green’s function even for the homogenization of finite-size and/or non-periodic media. Additionally, we demonstrate the universal applicability of this formulation for both random and deterministic heterogeneous media. This work contributes to a better understanding of the homogenization theory and the design of next-generation metamaterials that require the accurate prediction of effective material characteristics for dynamic wave manipulation under desired operating environments.

Dissipation driven degeneracies

Dissipation alone can produce counterintuitive topological wave transport that is otherwise absent in a non-dissipative system. This work demonstrates the influence of dissipation on degeneracies that arise in the context of elastic wave transport. The conditions on the parameters necessary to observe non-Hermitian degeneracies in the Bloch spectrum are precisely derived. It will be shown—contrary to the expectation from singularity theory of a linear eigenvalue problem—that a proportionally damped system with commutative damping does not exhibit non-Hermitian degeneracies. The necessity of a non-commutative and non-proportional dissipation model to observe non-Hermitian degeneracies (or exceptional points) is emphasized. Non-proportional dissipation is used to induce a non-Hermitian degeneracy in a local resonance sub-Bragg bandgap of a linear chain, without using negative damping. While Bloch waves are chosen to illustrate the influence of dissipation, the results readily extend to waves in non-periodic media as well as other wave and vibration transport problems.

Deformation activated negative group velocity state in soft laminates

We report the observation of deformation-induced negative group velocity (NGV) state in the non-periodic media. The phenomenon is illustrated by examples of the shear waves traveling along the non-periodic direction of pre-strained soft laminates. We show that the NGV state can be induced in absolutely stable laminates without invoking buckling. These stable composites with activated NGV states are achieved through tailored stiffening behavior of their non-Gaussian soft phases. The deformation range – for which the NGV state is activated – is multiple times higher in the proposed absolutely-stable composites than those prone to buckling. Finally, we analyze how this unusual NGV state can be induced and further tuned by variations in the material and geometric parameter spaces.

Numerical homogenization of non-linear parabolic problems on adaptive meshes

We propose an efficient numerical strategy for solving non-linear parabolic problems defined in a heterogeneous porous medium. This scheme is based on the classical homogenization theory and uses a locally mass-conservative formulation at different scales. In addition, we discuss some properties of the proposed non-linear solvers and use an error indicator to perform a local mesh refinement. The main idea is to compute the effective parameters in such a way that the computational complexity is reduced but preserving the accuracy. We illustrate the behavior of the homogenization scheme and of the non-linear solvers by performing two numerical tests. We consider both a quasi-periodic example and a problem involving strong heterogeneities in a non-periodic medium.

Intrinsic transfer matrix method and split quaternion formalism for multilayer media

The intrinsic transfer matrix method for 1D longitudinal elastic wave propagation through multilayer media is used to obtain an equivalent split quaternion formalism. Periodic media are analysed in this framework and the presence of a defect is considered. A simple one layer defect and an inversion defect are analysed. A commutative type of split quaternion is identified, which corresponds to defects of the periodic structure that can be placed in any position, the overall acoustic properties of the medium being conserved. Also, a nonperiodic medium composed of such commutative elements has a behaviour independent of the order of elements. Several possible applications in sound wave measurement and processing are outlined. The proposed split quaternion formalism is compact and can make analytical and numerical computations easier.

Light scattering calclulations in planar non-homogeneous dielectric media by means of the light diffraction calculation on gratings

The generalized source method previously developed for the light diffraction calculation on periodic structures is applied for the light scattering calculation in non-periodic media. This greatly enlarges the domain of applicability of two-dimensional Fourier-methods in light scattering applications since the generalized source method has much less numerical complexity than widely used Fourier modal method. It is also demonstrated on numerical examples that for pure dielectric structures the use of lossy perfectly matching layers is not necessary for removing the effects of periodicity and taking of a large grating period is sufficient.

Application of simple, periodic homogenization techniques to non-linear heat conduction problems in non-periodic, porous media

Often, detailed simulations of heat conduction in complicated, porous media have large runtimes. Then homogenization is a powerful tool to speed up the calculations by preserving accurate solutions at the same time. Unfortunately real structures are generally non-periodic, which requires unpractical, complicated homogenization techniques. We demonstrate in this paper, that the application of simple, periodic techniques to realistic media, that are just close to periodic, gives accurate, approximative solutions. In order to obtain effective parameters for the homogenized heat equation, we have to solve a so called “cell problem”. In contrast to periodic structures it is not trivial to determine a suitable unit cell, which represents a non-periodic media. To overcome this problem, we give a rule of thumb on how to choose a good cell. Finally we demonstrate the efficiency of our method for virtually generated foams as well as real foams and compare these results to periodic structures.

Observation of a Localization Transition in Quasiperiodic Photonic Lattices

We report the observation of the signature of a localization phase transition for light in one-dimensional quasiperiodic photonic lattices, by directly measuring wave transport inside the lattice. Below the predicted transition point an initially narrow wave packet expands as it propagates, while above the transition expansion is fully suppressed. In addition, we measure the effect of focusing nonlinear interaction on the propagation and find it increases the width of the localized wave packets.

Second order homogenization of the elastic wave equation for non-periodic layered media

In many cases, in the seismic wave propagation modelling context, scales much smaller than the minimum wavelength are present in the earth model in which we wish to compute seismograms. For many numerical methods these small scales are a challenge leading to high numerical cost. The purpose of this paper is to understand and to build the effective medium and equations allowing to average the small scales of the original medium without losing the accuracy of the wavefield computation. In this paper, only the simple layered medium case is studied, leaving the general 3-D medium case for future work. To obtain such an effective medium and equations, we use high order two scale homogenization applied to the wave equation for layered media with rapid variation of elastic properties and density compared to the smallest wavelength of the wavefield. We show that the order 0 homogenization gives the result that was obtained by Backus in 1962. Order 0 homogenized models are transversely isotropic even though the original model is isotropic. It appears that order 0 is not enough to obtain surface waves with correct group and phase velocities and higher order homogenization terms up to two are often required. In many cases, the order one and two simply require to correct the boundary conditions of the wave equation to obtain an accurate solution, even for surface waves. We show how to extend the theory from the periodic case to the non-periodic case. Examples in periodic and non-periodic media are shown. The accuracy of the results obtained by homogenization is checked against the normal mode solution computed in the original medium and shows good agreement.

Effective acoustic equations for a two-phase medium with microstructure

We study acoustic wave propagation in a two-phase medium in which the solid phase is a linear elastic material, and the fluid phase is assumed to be a compressible Newtonian barotropic fluid. Assuming that properties of the medium change rapidly on the small scale ε, we analyze the microscopic nonlinear Navier-Stokes equations and show that they can be linearized when ε tends to zero. Using a variant of Tartar's method of oscillating test functions, we derive effective acoustic equations which turn out to be viscoelastic. In order to treat disordered materials occurring in nature, we develop a new approach to describing geometry of a nonperiodic medium with length scale separation. Our approach is not based on probabilistic considerations. Instead, we postulate that certain inequalities hold uniformly on the microscale.

Propagation of optical radiation through nonperiodic media

Results are presented of a numerical analysis of the propagation of light through a nonperiodic one-dimensional structure made of layers of dielectric material. In particular, for a Fibonacci sequence of quarter-wave layers, we examine the ratio of the optical reflectivity to the transmission [R/(1 − R), which is the analog of the Landauer resistivity for electrons] as a function of the number of layers, the optical reflectance spectrum, and the spatial distribution of the electric field. We compare the results with those for an ordinary periodic quarter-wave stack and a random quarter-wave stack. For the Fibonacci sequence the observations are consistent with analyses showing that the allowed states form a Cantor set with a Lebesgue measure of zero.