Equations In Metamaterials Research Articles

An adaptive edge finite element method for the Maxwell's equations in metamaterials

<p style='text-indent:20px;'>In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. <i>A-posteriori</i> error estimators based on the recovery type and residual type are proposed, respectively. Based on our <i>a-posteriori</i> error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed <i>a-posteriori</i> error estimations for the adaptive method.</p>

Mathematical analysis of transmission properties of electromagnetic meta-materials

We study time-harmonic Maxwell's equations in meta-materials that use either perfect conductors or high-contrast materials. Based on known effective equations for perfectly conducting inclusions, we calculate the transmission and reflection coefficients for four different geometries. For high-contrast materials and essentially two-dimensional geometries, we analyze parallel electric and parallel magnetic fields and discuss their potential to exhibit transmission through a sample of meta-material. For a numerical study, one often needs a method that is adapted to heterogeneous media; we consider here a Heterogeneous Multiscale Method for high contrast materials. The qualitative transmission properties, as predicted by the analysis, are confirmed with numerical experiments. The numerical results also underline the applicability of the multiscale method.

Simulating backward wave propagation in metamaterial with radial basis functions

In this paper we revisit the radial basis function (RBF) meshless method and implement it to solve the time-dependent Maxwell's equations in metamaterials. Numerical simulations of the backward wave propagation phenomena are successfully achieved using this RBF method with properly chosen free parameters in the RBFs.

A Fully Fourth Order Accurate Energy Stable Finite Difference Method for Maxwell's Equations in Metamaterials

We present a novel fully fourth order in time and space finite difference method (FDM) for the time domain Maxwell's equations in metamaterials. We consider a Drude metamaterial model for the material response to incident electromagnetic fields. We consider the second-order formulation of the system of partial differential equations that govern the evolution in time of electric and magnetic fields along with the evolution of the polarization and magnetization current densities. Our discretization employs fourth-order staggering in space of different field components and the modified equation approach to obtain fourth-order accuracy in time. Using the energy method, we derive energy relations for the continuous models and design numerical schemes that preserve a discrete analogue of the energy relation. Numerical simulations are provided in one- and two-dimensional settings to illustrate fourth-order convergence as well as to compare with second-order schemes.

Nodal Discontinuous Galerkin Method for Time-Domain Lorentz Model Equations in Meta-Materials

Nodal Discontinuous Galerkin Method for Time-Domain Lorentz Model Equations in Meta-Materials

Superconvergence analysis of Yee scheme for metamaterial Maxwell’s equations on non-uniform rectangular meshes

Since the development of Yee scheme back in 1966, it has become one of the most popular simulation tools for modeling electromagnetic wave propagation in various situations. However, its rigorous error analysis on nonuniform rectangular type grids was carried out until 1994 by Monk and Suli. They showed that the Yee scheme is still second-order convergent on a nonuniform mesh even though the local truncation error is only of first order. In this paper, we extend their results to Maxwell's equations in metamaterials by a simpler proof, and show the second-order superconvergence in space for the true Yee scheme instead of the only semi-discrete form discussed in Monk and Suli's original work. Numerical results supporting our analysis are presented.

Implicit DG Method for Time Domain Maxwell’s Equations Involving Metamaterials

AbstractAn implicit discontinuous Galerkin method is introduced to solve the time-domain Maxwell’s equations in metamaterials. The Maxwell’s equations in metamaterials are represented by integral-differential equations. Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain. The fully discrete numerical scheme is proved to be unconditionally stable. When polynomial of degree at most p is used for spatial approximation, our scheme is verified to converge at a rate of O(τ2+hp+1/2). Numerical results in both 2D and 3D are provided to validate our theoretical prediction.

Averaging for superconvergence: Verification and application of 2D edge elements to Maxwell’s equations in metamaterials

In this paper, a simple averaging technique for edge elements is justified to improve the standard convergence rate (hence achieving superconvergence) when they are used to solve the Maxwell’s equations. For simplicity, here we focus on the lowest-order triangular edge element, which is widely used in practice. Though there exists no natural superconvergence points for the numerical solution obtained on such an edge element, superconvergence can be obtained after a simple average of solutions over the neighboring elements. A comprehensive analysis for the lowest-order triangular edge element is carried out, and one-order higher convergence rate than the standard interpolation error estimate is proved for the averaged solution at midpoints of those interior edges of parallelograms (formed by two triangles), i.e., superconvergence happens at the parallelogram centers. We also provide detailed analysis to explain why several cases do not have superconvergence. Extensive numerical results consistent with our analysis are presented.

An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwell’s equations in metamaterials

Numerical simulation of metamaterials play a very important role in the design of invisibility cloak, and sub-wavelength imaging. In this paper, we propose a leap-frog discontinuous Galerkin method to solve the time-dependent Maxwell’s equations in metamaterials. Conditional stability and error estimates are proved for the scheme. The proposed algorithm is implemented and numerical results supporting the analysis are provided.

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

Superconvergence of mixed finite element approximations to 3-D Maxwell’s equations in metamaterials

Numerical simulation of metamaterials has attracted more and more attention since 2000, after the first metamaterial with negative refraction index was successfully constructed. In this paper we construct a fully-discrete leap-frog type finite element scheme to solve the three-dimensional time-dependent Maxwell’s equations when metamaterials are involved. First, we obtain some superclose results between the interpolations of the analytical solutions and finite element solutions obtained using arbitrary orders of Raviart–Thomas–Nédélec mixed spaces on regular cubic meshes. Then we prove the superconvergence result in the discrete l 2 norm achieved for the lowest-order Raviart–Thomas–Nédélec space. To our best knowledge, such superconvergence results have never been obtained elsewhere. Finally, we implement the leap-frog scheme and present numerical results justifying our theoretical analysis.

Development of discontinuous Galerkin methods for Maxwell’s equations in metamaterials and perfectly matched layers

The discontinuous Galerkin method has proved to be an accurate and efficient way to numerically solve many differential equations. In this paper, we extend this method to solve the time-dependent Maxwell’s equations when metamaterials and perfectly matched layers are involved. Numerical results are presented to demonstrate that our method is not only simple to implement, but also quite effective in solving Maxwell’s equations in complex media.

Numerical convergence and physical fidelity analysis for Maxwell’s equations in metamaterials

In this paper, we develop a leap-frog mixed finite element method for solving Maxwell’s equations resulting from metamaterials. Our scheme is similar to the popular Yee’s FDTD scheme used in electrical engineering community, and is preferable for three dimensional large scale modeling since no storage of the large coefficient matrix is needed. Our scheme is proved to obey the Gauss’s law automatically if the initial fields satisfy that. Furthermore, the conditional stability and optimal error estimate for the proposed scheme are proved. To our best knowledge, we are unaware of any other publications devoted to the convergence analysis of this leap-frog explicit scheme for Maxwell’s equations even in a simple medium, while our results for metamaterials automatically reduce to the standard Maxwell’s equations in vacuum by dropping some terms resulting from the constitutive equations. Numerical results confirming our analysis are presented.