On the Computation of Power in Volume Integral Equation Formulations

We present simple and stable formulas for computing power (including absorbed/radiated, scattered and extinction power) in current-based volume integral equation formulations. The proposed formulas are given in terms of vector-matrix-vector products of quantities found solely in the associated linear system. In addition to their efficiency, the derived expressions can guarantee the positivity of the computed power. We also discuss the application of Poynting's theorem for the case of sources immersed in dissipative materials. The formulas are validated against results obtained both with analytical and numerical methods for scattering and radiation benchmark cases

The Orthogonal Expansion in Time-Domain Method for Solving Maxwell Equations Using Paralleling-in-Order Scheme

The orthogonal expansion in time-domain method is a new kind of unconditionally stable finite-difference time-domain (FDTD) method for solving the Maxwell equation efficiently. Generally, it can be implemented by two schemes: marching-on-in-order and paralleling-in-order, which, respectively, use weighted Laguerre polynomials and associated Hermite functions as temporal expansions and testing functions. This chapter summarized paralleling-in-order-based FDTD method using associated Hermite functions and Legendre polynomials. And a comparison from theoretical analysis to numerical examples is shown. The LD integral transfer matrix can be considered as a “dual” transformation for AH differential matrix, which gives a possible way to find more potential orthogonal basis function to implement a paralleling-in-order scheme. In addition, the differences with these two orthogonal functions are also analyzed. From the numerical results, we can see their agreements in some general cases while differing in some cases such as shielding analysis with the long-time response requirement

A new hybrid implicit-explicit FDTD method for local subgridding in multiscale 2-D TE scattering problems

The conventional finite-difference time-domain (FDTD) method with staggered Yee scheme does not easily allow including thin material layers, especially so if these layers are highly conductive. This paper proposes a novel subgridding technique for 2-D problems, based on a hybrid implicit-explicit scheme, which efficiently copes with this problem. In the subgrid, the new method collocates field components such that the thin layer boundaries are defined unambiguously. Moreover, aspect ratios of more than a million do not impair the stability of the method and allow for very accurate predictions of the skin effect. The new method retains the Courant limit of the coarse Yee grid and is easily incorporated into existing FDTD codes. A number of illustrative examples, including scattering by a metal grating, demonstrate the accuracy and stability of the new method

Numerical methods for computing Casimir interactions

We review several different approaches for computing Casimir forces and related fluctuation-induced interactions between bodies of arbitrary shapes and materials. The relationships between this problem and well known computational techniques from classical electromagnetism are emphasized. We also review the basic principles of standard computational methods, categorizing them according to three criteria---choice of problem, basis, and solution technique---that can be used to classify proposals for the Casimir problem as well. In this way, mature classical methods can be exploited to model Casimir physics, with a few important modifications.Comment: 46 pages, 142 references, 5 figures. To appear in upcoming Lecture Notes in Physics book on Casimir Physic

Theoretical Developments and Simulation Tools for Discrete Geometric Computational Electromagnetics in the Time Domain

The original Finite Difference Time Domain (FDTD) method, devised by Yee in 1966, inspired a conspicuous amount of research in the field of numerical schemes for solving Maxwell's equations in the time domain, thanks to its simplicity and computational efficiency. The original algorithm, which computes the values of electric and magnetic fields on the points of two interlocked Cartesian orthogonal grids, has also been rewritten as a Finite Integration Technique (FIT) algorithm, where the computed quantities are the integrals of the field over geometric elements of the grids. Both formulations suffer from the so-called staircase approximation problem: when an interface between regions with discontinuous material properties is not flat, the expected convergence properties of the numerical solution are not guaranteed if an exaggeratedly fine grid is not used. In this regard, even recent improved techniques based on combined arithmetic and harmonic averaging techniques cannot achieve second order accuracy in time in the neighborhood of the interface. This problem is inherent to the Cartesian orthogonal discretization of the domain, as unstructured grids (tetrahedral or polyhedral) mesh generators avoid it with grids conformal to the discontinuities in material properties. Approaches that have had some degree of success in adapting the FDTD algorithm to unstructured grids include schemes based on the Finite Element method (FEM), on the Cell Method and, more recently, formulations based on the Discontinuous Galerkin (DG) approach. Yet, consistency issues of discontinuous methods question their accuracy, since these methods do not explicitly force tangential continuity of the fields across mesh element interfaces, weakening the local fulfillment of physical conservation laws (charge conservation in particular). On the other hand, classical FEM formulations, which do not share this drawback, trade their geometric flexibility with an implicit time-stepping scheme, i.e. the computation includes solving a linear system of algebraic equations at each time-step. This severely limits the scalability of the algorithm. Recently, a technique has been introduced by Codecasa et al., based on a Discrete Geometric Approach (DGA) which instead yields an explicit, consistent and conditionally stable algorithm on tetrahedral grids. Due to the promising features of this approach, a thorough analysis of its performance and accuracy is in order, since neither have been widely tested yet. This work addresses the issue and shows that the latter approach compares favorably with equal order FEM approaches on unstructured grids. An important drawback of the DGA approach is that it was originally formulated for strictly dielectric materials. The way to overcome this limitation is unfortunately not obvious. The present work addresses this issue and solves it without sacrificing any property of the original algorithm. Furthermore, although the properties of the material operators in the original formulation show that the resulting scheme is conditionally stable, a Courant--Friedrich--Lewy (CFL) condition equivalent to the one of the original FDTD algorithm is not given. This is also dealt with in the bulk of this thesis and a sufficient condition for the stability of this algorithm is given with proof. Finally a practical toolbox for time domain electromagnetic simulations, tentatively named TetFIT and resulting from the coding efforts of the author is presented, with preliminary results on its performance when running on Graphical Processing Units (GPUs).The original Finite Difference Time Domain (FDTD) method, devised by Yee in 1966, inspired a conspicuous amount of research in the field of numerical schemes for solving Maxwell's equations in the time domain, thanks to its simplicity and computational efficiency. The original algorithm, which computes the values of electric and magnetic fields on the points of two interlocked Cartesian orthogonal grids, has also been rewritten as a Finite Integration Technique (FIT) algorithm, where the computed quantities are the integrals of the field over geometric elements of the grids. Both formulations suffer from the so-called staircase approximation problem: when an interface between regions with discontinuous material properties is not flat, the expected convergence properties of the numerical solution are not guaranteed if an exaggeratedly fine grid is not used. In this regard, even recent improved techniques based on combined arithmetic and harmonic averaging techniques cannot achieve second order accuracy in time in the neighborhood of the interface. This problem is inherent to the Cartesian orthogonal discretization of the domain, as unstructured grids (tetrahedral or polyhedral) mesh generators avoid it with grids conformal to the discontinuities in material properties. Approaches that have had some degree of success in adapting the FDTD algorithm to unstructured grids include schemes based on the Finite Element method (FEM), on the Cell Method and, more recently, formulations based on the Discontinuous Galerkin (DG) approach. Yet, consistency issues of discontinuous methods question their accuracy, since these methods do not explicitly force tangential continuity of the fields across mesh element interfaces, weakening the local fulfillment of physical conservation laws (charge conservation in particular). On the other hand, classical FEM formulations, which do not share this drawback, trade their geometric flexibility with an implicit time-stepping scheme, i.e. the computation includes solving a linear system of algebraic equations at each time-step. This severely limits the scalability of the algorithm. Recently, a technique has been introduced by Codecasa et al., based on a Discrete Geometric Approach (DGA) which instead yields an explicit, consistent and conditionally stable algorithm on tetrahedral grids. Due to the promising features of this approach, a thorough analysis of its performance and accuracy is in order, since neither have been widely tested yet. This work addresses the issue and shows that the latter approach compares favorably with equal order FEM approaches on unstructured grids. An important drawback of the DGA approach is that it was originally formulated for strictly dielectric materials. The way to overcome this limitation is unfortunately not obvious. The present work addresses this issue and solves it without sacrificing any property of the original algorithm. Furthermore, although the properties of the material operators in the original formulation show that the resulting scheme is conditionally stable, a Courant--Friedrich--Lewy (CFL) condition equivalent to the one of the original FDTD algorithm is not given. This is also dealt with in the bulk of this thesis and a sufficient condition for the stability of this algorithm is given with proof. Finally a practical toolbox for time domain electromagnetic simulations, tentatively named TetFIT and resulting from the coding efforts of the author is presented, with preliminary results on its performance when running on Graphical Processing Units (GPUs)

Mathematical modeling of metamaterials

Metamaterials are artificially structured nano materials with negative refraction index. The successful construction of such metamaterials in 2000 triggered a great interest in study of metamaterials by researchers from different areas. The discovery of metamaterials opened a wide potential for applications in diverse areas such as cloaking, sub-wavelength imaging, solar cell design and antennas. In this thesis, we investigate the most popular Drude metamaterial model. More specifically, we first present a brief overview of metamaterials and their potential applications, then we discuss the well-posedness of this model, and develop several numerical schemes to solve it. We implement our schemes using MATLAB, and demonstrate their effectiveness through numerical simulations of the negative refraction and cloaking phenomena