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Discretization of a cube into tetrahedra. The first mesh is obtained by dividing all cubes in which the domain has been divided in this way.
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The aim of this work is to analyze the role of the impressed sources in determining the well or ill-posedness of time harmonic electromagnetic boundary value problems involving isotropic effective media. It is shown, in particular, that, even if all interfaces are regular, the class of ill-posed problems can be very large in the presence of general...
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... the first simulation we consider a mesh obtained by dividing all cubes into six tetrahedra, as partially shown in Figure 2. ...
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... order to show the difficulties one can encounter in analyzing these kinds of well posed problems with finite element simulators we carried out a second simulation. In this case, the mesh is obtained by dividing the domain into macrocubes, as shown in Figure 12. It is important to observe that the set of nodes is exactly the same as before, that we have simply changed the way in which the tetrahedral elements are generated to obtain an "isotropic" mesh and, finally, that users of finite element simulators are not at all used to considering this details, as a consequence of the robustness of this type of simulators in more traditional conditions. ...
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... Figure 19) and, together with the analytical solution, on the line y = 0.005001, z = −0.000001 ( Figure 20). From Figures 18, 19 and 20 we can now conclude that, on the one hand, the errors near the interface are small and, on the other hand, these errors are limited to a neighbourhood of the interface of thickness lower than 1 mm, a very small fraction of the wavelength of the TE 10 mode at the considered frequency. ...
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... using the anisotropic mesh we have exploited in our first simulations, which was the most critical one, we have obtained the results shown in Figures 21 and 22. ...
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... the contrary, for n ≥ 2 the approximation is extremely good even if the anisotropic mesh is used. In the presence of the additional fictitious layer the results obtained using the isotropic mesh are essentially unchanged with respect to those shown in Figures 21 and 22. ...
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... practice, finite precision arithmetic is necessarily used, of course, and for high value of n it is possible to incur in prob- lems of accuracy because we expect that a layer with negligible losses is not able to contrast in a significant way the formation of fictitious sources at the dangerous interface. The beginning of this effect is marginally present, but very difficult to see, in Figures 21 and 22. It can be easily recognized, on the contrary, in Figure 23, where |E x | is shown. ...
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... beginning of this effect is marginally present, but very difficult to see, in Figures 21 and 22. It can be easily recognized, on the contrary, in Figure 23, where |E x | is shown. The analytical solution is given by |E x | = 0 and it is possible to notice that with n = 3 the error is not negligible with respect to the corresponding value of |E y |, shown in Figure 21. ...
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... can be easily recognized, on the contrary, in Figure 23, where |E x | is shown. The analytical solution is given by |E x | = 0 and it is possible to notice that with n = 3 the error is not negligible with respect to the corresponding value of |E y |, shown in Figure 21. The error is much smaller when a layer with n = 2 is present. ...
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... these settings the mesh obtained in the case the fifth layer is not present is shown in Figure 24. ...
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... this mesh we have 23302 degrees of freedom. In Figure 25 we show the magnitude of E on the plane y = 0.005 m. ...
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... amplitude of the errors does not seem to be very significant, however. In order to provide more precise indications on these errors, in Figure 26, 27 and 28 we report the transversal behaviour of the amplitude of the electric field, respectively on the planes z = −0.01 m, z = −0.001 ...
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Citations
In this chapter, we introduce several discontinuous Galerkin (DG) methods for solving time-dependent Maxwell’s equations in dispersive media and metamaterials. We first present a succint review of DG methods in Sect. 4.1. Then we present some DG methods for the cold plasma model in Sect. 4.2. Here the DG methods are developed for a second-order integro-differential vector wave equation. We then consider DG methods for the Drude model written in a system of first-order differential equations in Sect. 4.3. Finally, we extend the nodal DG methods developed by Hesthaven and Warburton [141] to metamaterial Maxwell’s equations in Sect. 4.4.
A set of sufficient conditions for the well-posedness and the convergence of the finite element approximation of two-dimensional time-harmonic electromagnetic boundary value problems involving non-conducting moving objects with stationary boundaries is provided for the first time to the best of authors's knowledge. The set splits into two parts. The first of these is made up of traditional conditions, which are not restrictive for practical applications and define the usual requirements for the domain, its boundary, its subdomains and their boundaries, the boundary conditions and the constitutive parameters. The second part consists of conditions which are specific for the problems at hand. In particular, these conditions are expressed in terms of the constitutive parameters of the media involved and of the velocity field. It is shown that it is not difficult to check the validity of these conditions and that they hold true for broad classes of practically important problems which involve axially moving media.
In this chapter, we first give a quick review of superconvergence analysis in Sect. 5.1. Then we carry out the superclose analysis for 3-D metamaterial Maxwell’s equations represented by the Drude model. The analysis for a semi-discrete scheme is presented in Sect. 5.2, which is followed by the analysis for two fully-discrete schemes in Sect. 5.3. In Sect. 5.4, a superconvergence result in the discrete l 2 norm is proved. Finally, the superconvergence analysis is extended to the 2-D case in Sect. 5.5.
The finite element method (FEM) is arguably one of the most robust and popular numerical methods used for solving various partial differential equations (PDEs). Due to the diligent work of many researchers over the past several decades, the fundamental theory and implementation of FEM have been well established as evidenced by many excellent books published in this area (e.g., [4, 20, 21, 39, 51, 54, 65, 78, 158, 163, 243]).
In this chapter, we present some basic techniques for developing a posteriori error estimation for solving Maxwell’s equations. It is known that the a posteriori error estimation plays a very important role in adaptive finite element method. In Sect. 6.1, we provide a brief overview of a posteriori error estimation. Then in Sect. 6.2, through time-harmonic Maxwell’s equations, we demonstrate the fundamental ideas on how to obtain the upper and lower posteriori error estimates. In Sect. 6.3, we present a posteriori error estimator obtained for a cold plasma model described by integro-differential Maxwell’s equations.
In this chapter, we present several fully discrete mixed finite element methods for solving Maxwell’s equations in metamaterials described by the Drude model and the Lorentz model. In Sects. 3.1 and 3.2, we respectively discuss the constructions of divergence and curl conforming finite elements, and the corresponding interpolation error estimates. These two sections are quite important, since we will use both the divergence and curl conforming finite elements for solving Maxwell’s equations in the rest of the book. The material for Sects. 3.1 and 3.2 is quite classic, and we mainly follow the book by Monk (Finite element methods for Maxwell’s equations. Oxford Science Publications, New York, 2003). After introducing the basic theory of divergence and curl conforming finite elements, we focus our discussion on developing some finite element methods for solving the time-dependent Maxwell’s equations when metamaterials are involved. More specifically, in Sect. 3.3, we discuss the well posedness of the Drude model. Then in Sects. 3.4 and 3.5, we present detailed stability and error analysis for the Crank-Nicolson scheme and the leap-frog scheme, respectively. Finally, we extend our discussion on the well posedness, scheme development and analysis to the Lorentz model and the Drude-Lorentz model in Sects. 3.6 and 3.7, respectively.
In this chapter, we present some interesting simulations of wave propagation in metamaterials. We start in Sect. 9.1 with a perfectly matched layer model, which allows us to reduce the simulation on an infinite domain to be realized on a bounded domain. Here we present a simulation demonstrating the negative refraction index phenomenon for metamaterials. In Sects. 9.2 and 9.3, we present invisibility cloak simulations using metamaterials in frequency domain and time domain, respectively. In Sect. 9.4, we present an interesting application of metamaterials for solar cell design. In Sect. 9.5, we end this chapter by presenting some open mathematical problems related to metamaterials.
One common problem in computational electromagnetics is how to simulate wave propagation on an unbounded domain accurately and efficiently. One typical technique is to use the absorbing boundary conditions (ABCs) to truncate the unbounded domain to a bounded domain. The solution computed with an ABC on a bounded domain should be a good approximation to the solution originally given on the unbounded domain. Hence constructing a good ABC is quite challenging.
In this chapter, we demonstrate the practical implementation of a mixed finite element method (FEM) for a 2-D Drude metamaterial model (5.1)–(5.4).
In this chapter, we start with a brief discussion on the origins of metamaterials, and their basic electromagnetic and optical properties. We then present some metamaterial structures and potential applications in areas such as sub-wavelength imaging, antenna design, invisibility cloak, and biosensing. After all these, we then move to the related mathematical problems by introducing the governing equations used to model the wave propagation in metamaterials. Finally, a brief overview of some popular computational methods for solving Maxwell’s equations is provided.
