Global Energy-conserved Splitting FDTD Schemes For Maxwell’s Equations In Metamaterials: Methods,Theories And Applications

Early in 1968, the concept of "metamaterial" has been first proposed by Veselago ([67]) which possesses negative permittivity and negative permeabil-ity. Till 2000, inspired by the work of Pendry et al. in [55], the first negative index metamaterial was successfully constructed by Shelby and Smith, which was a composite "medium" in the microwave regime by arranging periodic arrays of small metallic wires and split-ring resonators (SRRs). It lead-s to a significant revolution in the development of electromagnetics (EMs) and materials sciences ([58] [60]). Due to the unusual physical properties of these special materials that could not generally been found in nature ma-terials, they are called "metamaterials" (Metamaterials, MTMs), and also referred as double negative (DNG) media, negative-index materials (NIM), left-handed media (LHM) et al. ([8]). So far, some kinds of metamaterials have been artificially composed and studied such as double negative media (DNG), zero-index media (ZIM), etc. ([2][8] [47] [55] [57] [81]). These meta-materials are engineered to have exotic properties that may not be found in conventional materials at the frequencies of interest. For example, the double negative (DNG) medium has a negative refractive index which makes the wave propagate backward or bend at an arbitrary angle in it ([17] [18] [35] [54] [68] [79]); the zero-index medium (ZIM) could not only control the propagation of wave completely but also provide a potential method to shel-ter objects ([81]). Applications of metamaterials are diverse and include 3-D display, remote aerospace applications, air defence radar, optical nano-lithography, medical imaging devices, lenses for high-gain antennas, cloaking devices, shielding structures from earthquakes, etc ([5][19] [20] [35] [47]).Modeling and analysis of phenomena within metamaterials play an im-portant role both in the studies and applications of metamaterials inspired by the recent advancements and exciting potential applications of metama-terials. Modeling of electromagnetic wave propagation in metamaterials is becoming urgent and a hot topic which attracts many researches ([14] [15] [30] [68] [80] [81] [82]). The corresponding wave propagates with the per-mittivity and permeability of the media dependent of the frequency since the metamaterial is classified as a kind of dispersive medium. Considering that the electromagnetic parameters of metamaterials vary with frequency, some methods were proposed to deal with it such as the Recursive Convolu-tion (R.C) ([40] [46] [59]); the Auxiliary Differential Equation (ADE) methods ([16][25] [52] [65]); and Z-transform methods ([56][61]) etc. Auxiliary Differ-ential Equation(ADE) methods transfer the frequency dependent Maxwell’s equations in frequency domains to be a new set of equations including four electromagnetic vectors in time and space domains by auxiliary differential equations and auxiliary electromagnetic variables.Much work and achievements have been obtained on the study of the finite difference time domain methods for classic Maxwell’s equations in free space since Yee’s FDTD method was initially proposed in 1966 ([74]). As an explicit scheme based on the staggered grids, Yee’s FDTD scheme is easy to implement and widely adopted ([48] [63] [64]). However, its condition-al stability makes Yee’s FDTD scheme disadvantaged, since it will increase the computational cost dramatically especially in the computing of prob-lems with large domains and high dimensions. To overcome the shortcom-ing of Yee’s scheme and further improve the accuracy, many uncondition-ally methods were developed including the Alternating Direction Implicit (ADI) method ([24] [28] [50] [72] [77] [78]); the Local One Dimension (LOD) method ([1] [45]):the Splitting FDTD (S-FDTD)([27]); the Symplcctic FDT-D method ([62] [71]); the Energy-Conserved Splitting FDTD (EC-S-FDTD) method ([9] [10] [11] [26]) and so on.With respect to the numerical methods for the Maxwell’s equations in metamaterials, [69] simulated the interaction between Gaussian beam and in-definite anisotropic metamaterial (AMM) slabs with ADE method and FDT-D scheme. However, although it is easy to implement, the conventional FDT-D formulation seriously bears the conditional stability. [51] assessed the per-formance of unconditionally stable FDTD methods containing ADI-FDTD scheme and LOD-FDTD schemes for the modeling of 2D doubly dispersive metamaterials. On the other hand, in recent years, some efforts in developing and analyzing finite element methods for solving the time-domain Maxwell’s equations with Drudc model have been made ([7][37] [38] [41] [42] [43]).[43] proposed the standard finite element method for the Maxwell’s equations in metamaterials and strictly analyzed the error estimate. [37] studied the 3D Maxwell’s equations in metamaterials, constructed a fully-discrete leap-frog type mixed clement method and proved the superconvergence of the scheme. In addition, [42] developed the discontinuous Galcrkin method for Maxwell’s equations in metamaterials.The electromagnetic energy of classic Maxwell’s equations in lossless medium without source is preserved. For long-time electromagnetic simula-tions, it is of significance to construct numerical schemes which can preserve the electromagnetic energy. [9] and [10] creatively developed the energy-conserved splitting FDTD methods for 2D standard Maxwell’s equations in free space, and strictly proved that the methods preserve the discrete elec-tromagnetic energy and are of second order accuracy both in time and space. [11] considered the 3D Maxwell’s equations in free space and proposed the EC-S-FDTD schemes, which were proved to conserve the discrete electromag-netic energy. They analyzed the second-order convergence of the schemes. However, there is a few work on the energy-conserved methods for Maxwell’s equations in metamaterials.It is important to develop numerical schemes preserving the discrete en-ergy in metamaterials to model and simulate the practical electromagnetic problems in metamaterials much more efficiently and effectively. We consider the 2D Maxwell’s equations in metamaterials and the 3D Maxwell’s equations with the Drude model in metamaterials and obtain the new energy-conserved properties for the continuous system. Then the new energy-conserved schemes including EC-S-FDTD, S-EC-S-FDTD schemes are proposed for the 2D mod-els and GEC-S-FDTD schemes arc developed for the 3D problems in meta-materials, all of which are strictly proved to be energy conserved in the discrete sense. We also demonstrate the optimal error estimates for these proposed energy-conserved schemes. Numerical tests confirm the theoretical conclusions. Further, numerical simulation results clearly exhibit the unusu-al physical properties of metamaterials such as the negative refractive index and "perfect lens", etc.. In addition, we apply the methods and theory of the splitting schemes to some important electromagnetic applications:the 2D transient electromagnetic (TEM) exploration, electromagnetic propaga-tion in the inhomogeneous ring domain, scattering of the discontinuous media and the perfectly matched layer problems.The dissertation is divided into five chapters. The outline is as follows.In Chapter 1, considering the 2D Maxwell’s equations with Drude model for TM wave, we derive out the time-domain governing equations with the electromagnetic variables E, H, J and K by ADE method and prove the en-ergy preservation property for this model with the perfect electric conductor (PEC) boundary condition. Based on the work on EC-S-FDTD scheme for the standard Maxwell’s equations in [9], we develop a new energy-conserved splitting FDTD scheme for the 2D Maxwell’s equations with Drude model in metamaterials, and prove the discrete energy conservation of the scheme. Moreover, we analyze the convergence of the proposed EC-S-FDTD scheme and prove that it is of first order in time and second order in space.In Chapter 2, we further construct a symmetric EC-S-FDTD scheme for the 2D Maxwell’s equations in metamaterials which contains four step from the 2k-th time step to the 2k+1-th time step. Numerical analysis proves that the S-EC-S-FDTD scheme preserves the energy-conserved property in the discrete sense and improves the accuracy in time by one order, that is, of second order accuracy both in time and space. Numerical tests are done to compare the EC-S-FDTD, S-EC-S-FDTD and ADI-FDTD schemes and confirm that both the EC-S-FDTD scheme and the S-EC-S-FDTD scheme hold the discrete energy conservation. Numerical error orders confirm the theoretical conclusions. By using the perfectly matched layers to the compu-tational domain, we simulate the long-time interactions of continuous wave (CW) Gaussian beams with double negative (DNG) metamaterials in the case of normal incident and oblique incident by the S-EC-S-FDTD scheme. The simulation results demonstrate the exotic physical properties of metama-terials such as backward-wave propagation. The experiments with sin point source are also implemented, whose results present the process of electro-magnetic propagation to indicate the influence of negative refractive index in metamaterials and the "perfect lens" phenomenon.In Chapter 3, we study the important 3D Maxwell’s equations with the Drude model in metamaterials. We first derive out the new global energy-conserved identities for the continuous system. We then propose a kind of new global energy-conserved S-FDTD (GEC-S-FDTDD) scheme for the 3D model in metamaterials. We prove strictly the scheme to satisfy the corre-sponding discrete global energy conservations. Thirdly, the theoretical anal-ysis on the convergence is given to show the second order accuracy both in time and space. We also prove the super-convergence of the GEC-S-FDTD scheme and obtain the second-order convergence in terms of the discrete di-vergence. Numerical examples demonstrate the theoretical results. Some practical simulations are presented to show the propagations of electromag-netic waves from the free space to metamaterials, where unusual phenomena could be clearly observed.In Chapter 4, we focus on the important 2D transient electromagnetic exploration problems with an integral boundary condition. We propose the ADI-FDTD algorithm for the problems and study the theoretical analysis for the algorithm. Numerical examples illustrate that the proposed ADI-FDTD algorithm is more accurate compared with the classic DF-FFT algorithm and is efficient in computing the induced electric field in the earth and further orientating the location of anomy in the earth.In Chapter 5, we study the numerical analysis and simulations of the EC-S-FDTD schemes for some practical electromagnetic problems including the inhomogeneous ring domain problem, the electromagnetic scattering in discontinuous media and the perfectly matched layer problem. The corre- sponding discrete energy-conserved equalities are proved. Numerical exper-iments further confirm the discrete energy conservations and convergence order of numerical schemes. We carry out numerical experiments to present the electromagnetic propagation in ring domain and scattering in discontin-uous media and with the perfectly matched layers.