Wave Propagation In Metamaterials Research Articles

Finite-amplitude nonlinear waves in inertial amplification metamaterials: Theoretical and numerical analyses

The motivation of this study is to understand the unique characteristics of finite-amplitude elastic wave propagation in an inertial amplification (IA) system coupled with a Duffing resonator and leverage the unique opportunities offered by the IA mechanism and nonlinearities to achieve the modulation of elastic waves. Based on the assumption of the perturbation method, the effective range of the wave amplitude was investigated, which is usually ignored. Meanwhile, the obvious phenomenon of bandgap transition (between the single resonance bandgap and hybrid bandgap) is observed with varying IA levels. In particular, we demonstrated a new phenomenon of negative wave propagation and an obvious standing-wave mode in this system. Here, an analytical investigation of the amplitude-dependent and IA-dependent nonlinear dispersion is conducted based on the perturbation approach. Numerical simulations using the time-domain finite element method and Gaussian pulse excitation were performed to validate the nonlinear dispersion and negative motion of the wave packet. The research results can be used to tune wave propagation in metamaterials and provide ideas for the design of metamaterials, which can be used in practical circumstances for different energy transfer needs.

Bandgap merging and backward wave propagation in inertial amplification metamaterials

The aim of this study is to investigate the wave propagation characteristics of one-dimensional inertial amplification (IA) metamaterials with double resonators. The IA Angle is introduced by the IA mechanism as a critical parameter to modulate elastic waves. Double attenuation peaks in the bandgaps caused by coupling the IA mechanism and local resonance can be created in different ranges by adjusting the IA Angle. Moreover, a novel phenomenon of negative relative group velocity is observed in the region of double-negative effective properties, denoting negative energy flow. Numerical simulations using Gaussian pulse excitation are performed to validate the existence of a backward wave corresponding to the negative relative group velocity in the time domain. Furthermore, bandgap merging occurs in the region of double-zero effective properties, resulting in broadband wave attenuation. In particular, instead of using traditional metamaterials, two local resonator bandgaps can be merged to expand the bandgap range by at least 30% without changing the unit cell mass. The research results can be used to tune wave propagation in metamaterials, providing ideas for metamaterial design.

A Dispersive Formulation of Time-Domain Meshless Method With PML Absorbing Boundary Condition for Analysis of Left-Handed Materials

We have proposed a dispersive formulation of scalar-based meshless method for time-domain analysis of electromagnetic wave propagation through left-handed materials (LHMs). Moreover, we have incorporated Berenger’s perfectly matched layer (PML) absorbing boundary condition (ABC) into the dispersive formulation to truncate open-domain structures. In general, the dispersive formulations of conventional numerical methods are proper tools for analysis of left-handed (LH) media, whenever the LH media can be characterized by considering spatial or frequency dispersion effect for the constitutive parameters. In comparison to the grid-based numerical methods, it is proven that meshless methods not only are strong tools for accurate approximation of derivatives in Maxwell’s equations but also can provide more flexibility in modeling the spatial domain of problems. However, we have not seen any reports on using dispersive forms of meshless methods for simulation of wave propagation in metamaterials and applying any PML ABCs to dispersive formulation of meshless method. The proposed formulation enables us to take advantage of meshless methods in analysis of LH media with frequency-dependent parameters. For modeling the frequency behavior of the medium, we have used auxiliary differential equation (ADE) method based on the relations between electromagnetic fields intensities and current densities. Effectiveness of the proposed formulation is verified by analysis of 2-D and 3-D numerical examples; also, some basic factors which affect the accuracy, stability, and computational cost of the simulations are studied.

Controllable elastic wave propagation in metamaterials

Controllable elastic wave propagation in metamaterials

Developing and Analyzing New Unconditionally Stable Finite Element Schemes for Maxwell’s Equations in Complex Media

In this paper we propose and analyze an unconditionally stable leapfrog method for Maxwell’s equations that removes the time step constraint for stability, which makes the proposed scheme more efficient in computation and easier in algorithm implementation compared to the same order Crank–Nicolson scheme. We also prove the unconditional stability and the optimal error estimate of the proposed scheme. To show the generality of our technique, we further develop similar unconditionally stable leapfrog schemes for other complicated Maxwell’s equations. Numerical results are presented to justify our theoretical analysis and demonstrate the practical applications in simulating wave propagation in metamaterials.

Simultaneous energy harvesting and vibration attenuation in piezo-embedded negative stiffness metamaterial

Mechanical metamaterials are uniquely engineered form of periodically arranged unit cells that exhibit interesting frequency-dependent physical properties like negative effective mass, Young’s modulus and Poisson’s ratio. These extreme engineering properties are beyond the natural properties of a material, which can modulate the propagation of wave. In this article, a mechanical realization of one of these uncommon properties called negative stiffness is emulated through analytical simulation. Wave propagation in metamaterials is contingent on frequency, which in turn results in transmission and attenuation bands. Simultaneous vibration control and energy harvesting can be executed by embedding energy harvesting smart material within the resonating units of the metamaterial. However, this needs careful design studies to outline the range of parameters. In this work, first, the band structure of a piezo-embedded negative stiffness metamaterial is studied using generalized Bloch’s theorem. Subsequently, harvested power along with the transmissibility is computed for a chain of finite number of metamaterial units by using backward substitution method. The results of the parametric studies elucidate that piezo-embedded negative stiffness metamaterial can enhance the performance in terms of vibration attenuation and harvested energy.

Mathematical analysis of Ziolkowski’s PML model with application for wave propagation in metamaterials

In this paper we investigate one Perfectly Matched Layer (PML) model proposed by Ziolkowski in 1999. Various schemes for solving this PML model have been developed and have been shown to be effective in absorbing outgoing waves when the wave propagation in unbounded domain problem is reduced to a bounded domain problem. However, a rigorous analysis of this model is lacking. In this paper we establish the stability of this PML model and propose a fully-discrete finite element scheme to solve this model with edge elements. Discrete stability and optimal error estimate of this scheme are proved. Numerical results justifying the analysis and demonstrating the effectiveness of this PML model are presented.

Analysis and application of an equivalent Berenger’s PML model

Abstract The perfectly matched layer (PML) is a technique initially proposed by Berenger for solving unbounded electromagnetic problems with the finite-difference time-domain method. In this work, we first formulate an equivalent PML model from the original Berenger PML model in the corner region, and then establish its stability. We further develop a discontinuous Galerkin method to solve this PML model, and discrete stability similar to the continuous case is proved. To demonstrate the absorbing property of this PML model, we apply it to simulate wave propagation in metamaterials.

Long-range elastic metamaterials

The problem of wave propagation control in one-dimensional systems, including electrical charges and dipole magnetic moments is investigated. The waveguide is characterized by long-range and nonlinear interaction forces of Coulomb and Lorentz nature. Wave propagation properties are derived by a method based on an equivalent partial differential equation that replaces the discrete equation of motion of the chain. The paper shows how the waves propagating in these special systems have characteristics, such as phase and group velocity, that are function of the electrical and magnetic property distribution along the chain. Of interest are also possible wave-stopping phenomena. The paper presents an outline of some basic principles developed by some of the authors in recent theoretical papers and shows also numerical experiments illustrating wave propagation in metamaterials characterized by long-range elastic-electromagnetic interactions.

Optimal non-dissipative discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials

Simulation of electromagnetic wave propagation in metamaterials leads to more complicated time domain Maxwell’s equations than the standard Maxwell’s equations in free space. In this paper, we develop and analyze a non-dissipative discontinuous Galerkin (DG) method for solving the Maxwell’s equations in Drude metamaterials. Previous discontinuous Galerkin methods in the literature for electromagnetic wave propagation in metamaterials were either non-dissipative but sub-optimal, or dissipative and optimal. Our method uses a different and simple choice of numerical fluxes, achieving provable non-dissipative stability and optimal error estimates simultaneously. We prove the stability and optimal error estimates for both semi- and fully discrete DG schemes, with the leap-frog time discretization for the fully discrete case. Numerical results are given to demonstrate that the DG method can solve metamaterial Maxwell’s equations effectively.

Nonconforming Finite Element Methods for Wave Propagation in Metamaterials

AbstractIn this paper, nonconforming mixed finite element method is proposed to simulate the wave propagation in metamaterials. The error estimate of the semi-discrete scheme is given by convergence order O(h2), which is less than 40 percent of the computational costs comparing with the same effect by using Nédélec-Raviart element. A Crank-Nicolson full discrete scheme is also presented with O(τ2 + h2) by traditional discrete formula without using penalty method. Numerical examples of 2D TE, TM cases and a famous re-focusing phenomena are shown to verify our theories.

Metamaterials: supra-classical dynamic homogenization**Dedicated with gratitude to the memory of Prof Yves C Angel.

Metamaterials are artificial composite structures designed for controlling waves or fields, and exhibit interaction phenomena that are unexpected on the basis of their chemical constituents. These phenomena are encoded in effective material parameters that can be electronic, magnetic, acoustic, or elastic, and must adequately represent the wave interaction behavior in the composite within desired frequency ranges. In some cases—for example, the low frequency regime—there exist various efficient ways by which effective material parameters for wave propagation in metamaterials may be found. However, the general problem of predicting frequency-dependent dynamic effective constants has remained unsolved. Here, we obtain novel mathematical expressions for the effective parameters of two-dimensional metamaterial systems valid at higher frequencies and wavelengths than previously possible. By way of an example, random configurations of cylindrical scatterers are considered, in various physical contexts: sound waves in a compressible fluid, anti-plane elastic waves, and electromagnetic waves. Our results point towards a paradigm shift in our understanding of these effective properties, and metamaterial designs with functionalities beyond the low-frequency regime are now open for innovation.

Some computational problems in electromagnetic wave simulations with metamaterials

Metamaterial is an artificially made nano electromagnetic material, whose permittivity and permeability can be both negative for some frequencies. Hence the reflective index of metamaterial is negative. The study of metamaterial started from the beginning of 2000, and has been one of the hottest research topics in the past decade. We focus on introducing those modeling equations appeared in simulating wave propagation in metamaterials and developing effective finite element methods for solving them. This paper is a short survey of the works carried out in the computational mathematics community in the past ten years. In the end, we mention some open problems that worth further study.

Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials

In this paper, we develop a nodal discontinuous Galerkin method for solving the time-dependent Maxwellʼs equations when metamaterials are involved. Both semi- and fully-discrete schemes are constructed. Numerical stability and error estimate are proved for both schemes. Numerical results are presented to demonstrate that the method is not only efficient but also very effective in solving metamaterial Maxwellʼs equations.

Mathematical analysis of a PML model obtained with stretched coordinates and its application to backward wave propagation in metamaterials

We investigate the well‐posedness of a perfectly matched layer model developed by Cohen and Monk [Comput Methods Appl Mech Eng 169 (1999), 197–217]. A new time‐domain finite element method is proposed to solve the model. A novel feature of the scheme is that the number of unknowns is reduced to two from four in the original form. The scheme's effectiveness is demonstrated through numerical experiments. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1558–1574, 2014

Modeling Backward Wave Propagation in Metamaterials by the Finite Element Time-Domain Method

In this paper we develop a leap-frog-type finite element method for modeling the electromagnetic wave propagation in metamaterials. The metamaterial model equations are represented by integrodifferential Maxwell's equations, which are quite challenging to analyze and solve in that we have to solve a coupled problem with different partial differential equations given in different material regions. Our method is based on a mixed finite element method using edge elements, which can easily handle the tangential continuity of the electric field. Stability analysis and optimal error estimate are carried out for the proposed scheme. The scheme is implemented and confirmed to obey the proved optimal convergence rate by using a smooth analytical solution. Then the scheme is extended to model wave propagation in heterogeneous media composed of metamaterials and free space, and extensive numerical results (using a rectangular edge element, a triangular edge element, and mixed edge elements) demonstrate the effectiveness of our algorithm for modeling the exotic backward wave propagation phenomenon in metamaterials. To the best of our knowledge, this is the first paper with an exhaustive simulation of backward wave propagation in metamaterials using time-domain finite element methods.

Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials

AbstractIn this article, we consider the time‐dependent Maxwell's equations modeling wave propagation in metamaterials. One‐order higher global superclose results in the L2 norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011

Padé approximate solution for wave propagation in graded-index metamaterials

A new approximate solution for wave propagation in graded-index media is developed wherebythe exact wave propagation equation is replaced by any one of a sequence of higher-order(m, n) Padé approximant operators. The resulting formulations may be discretized bycommon numerical schemes and solvable by existing numerical techniques. Theaccuracy of this approximate calculation of the wave propagator is demonstratedin comparison with the exact result. We then employ the resulting method toinvestigate wave propagation in metamaterials with graded-index profiles whichchange according to a hyperbolic tangent function along the propagation direction.

Backward self-induced transparency in metamaterials

Dispersion and dissipation are inherent to electromagnetic wave propagation in metamaterials. We show that these inherent limitations can be overcome by a self-induced transparency pulse that stably propagates through a metamaterial sparsely doped with resonantly absorbing dopants. This pulse has the form of a hitherto unexplored backward soliton. We discuss the observable signature of this soliton.

Anomalous phenomena of electromagnetic wave propagation in metamaterials for arbitrary orientation of optical axis

When there is an angle between the anisotropic metamaterials' optical axis and the electromagneticwave propagation axis, anomalous omnidirectional total reflection and anomalous negative refraction are discovered at the interface. The functional relation between incident angle and refractive angle of energy flow is obtained. According to this relation, we can restructure the refractive angle as negative or positive readily by selecting suitable permittivity and permeability tensors. And under these conditions, an anomalous transmittance emerges，such that when incident angle is less than a value, especially in the vicinity of vertical incident, the transmittance is minimum or even zero. According to these characteristics, a new slab spatial filter is put forward. Compared with the traditional space filter system, this new spatial filter's volume is smaller. And these characteristics can also be used to build a reflector of low spatial frequency.