Rigorous Coupled Wave Analysis (RCWA) (used for electromagnetic (EM) analysis of planar diffraction gratings) has been applied to solve EM scattering and diffraction problems for spatially inhomogeneous, cylindrical, elliptical systems. The RCWA algorithm and an appropriate method for matching EM boundary conditions in the elliptical system are described herein. Comparisons of the eigenfunctions determined by RCWA (found in spatially homogeneous elliptical regions) and Mathieu functions are presented and shown to agree closely with one another. Numerical results of scattering from a uniform elliptical shell system (excited by an electrical surface current) obtained by using both a Mathieu function expansion method and by using the RCWA algorithm are presented and also shown to agree closely with one another. The RCWA algorithm was used to study EM scattering and diffraction from an elliptical, azimuthally inhomogeneous dielectric permittivity, step profile system. EM field matching and power conservation were shown to hold for this step profile example. A comparison of the EM fields of the step profile elliptical shell example and that of a uniform profile elliptical shell having the same excitation and bulk material parameters (permittivity and permeability) was made and significant differences of the EM fields of the two systems were observed.

We present a method giving the field scattered by a plane surface with a cylindrical local perturbation illuminated by a plane wave. The theory is based on Maxwell's equations in covariant form written in a nonorthogonal coordinate system fitted to the surface profile. The covariant components of electric and magnetic vectors are solutions of a differential eigenvalue system. A Method of Moments (PPMoM) with Pulses for basis and weighting functions is applied for solving this system in the spectral domain. The scattered field is expanded as a linear combination of eigensolutions satisfying the outgoing wave condition. Their amplitudes are found by solving the boundary conditions. Above a given deformation, the Rayleigh integral is valid and becomes identified with one of covariant components of the scattered field. Applying the PPMo Method to this equality, we obtain the asymptotic field and the scattering pattern. The method is numerically investigated in the far-field zone, by means of convergence tests on the spectral amplitudes and on the power balance criterion. The theory is verified by comparison with results obtained by a rigorous method.

The diffraction of E-polarized plane waves by a tandem impedance slit waveguide is investigated rigorously by using the Fourier transform technique in conjunction with the Mode Matching method. This mixed method of formulation gives rise to a scalar Wiener-Hopf equation of the second kind,the solution of which contains infinitely many constants satisfying an infinite system of linear algebraic equations. A numerical solution of this system is obtained for various values of the surface impedances,slit width and the distance between the slits,through which the effect of these parameters on the diffraction phenomenon are studied.

The field scattered by a perfectly conducting plane surface with a perturbation illuminated by an E//-polarized plane wave is determined by means of a Rayleigh method. This cylindrical surface is described by a local function. The scattered field is supposed to be represented everywhere in space by a superposition of a continuous spectrum of outgoing plane waves. A "triangle/Dirac" method of moments applied to the Dirichlet boundary condition in the spectral domain allows the wave amplitudes to be obtained. For a half cosine arch,the proposed Rayleigh method is numerically investigated in the far-field zone,b y means of convergence tests on the spectral amplitudes and on the power balance criterion. We show that the Rayleigh integral can be used for perturbations,the amplitudes of which are close to half the wavelength.

Accurate determination of the generalized scattering matrix of the crossed waveguides junction, T-junction and right-angle bend containing a cylindrical post is presented. The generalized scattering matrix of the four port structure is obtained from those of the two ports right angle bend with different combinations of perfect electric and/or magnetic walls. The generalized scattering matrix of the right angle bend (quarter of the structure) is obtained by the mode matching method where the electromagnetic fields in rectangular waveguides are matched to those in a junction section formed by asectoral region.

Luebbers' and Maliuzhinets' solutions for diffraction by alossy wedge are compared in order to model the urban propagationchannel. Derivation, validity criteria and accuracy of impedanceboundary conditions (IBCs) --- as a main approximation embedded inthe Maliuzhinets' solution—are discussed. A modified (a slightlyimproved) Luebbers diffraction coefficient is proposed. The UniformTheory of Diffraction (UTD) Maliuzhinets diffraction coefficient isgiven in a structured form that might facilitate its use. Detailednumerical comparisons of the above-mentioned solutions with method-of-moments solutions using either exact boundary conditions or IBCsare done for canonical configurations.

Constitutive relations for isotropic material media are formulated in a manifestly covariant manner. Clifford's geometric algebra is used throughout. Polarisable,c hiral and Tellegen medium are investigated. The investigation leads to the discovery of an underlying algebraic structure that completely classifies isotropic media. Variational properties are reviewed,sp ecial attention is paid to the imposed constraints on material parameters. Covariant reciprocity condition is given. Finally,dualit y transformations and their relevance to constitutive relations are investigated. Duality is shown to characterise ‘well-behavedness' of medium which has an interesting metric tensor related implication.

This paper concerns the implementation of the perfectly matched layer (PML) absorbing boundary condition in the framework of a mimetic differencing scheme for Maxwell's Equations. We use mimetic versions of the discrete curl operator on irregular logically rectangular grids to implement anisotropic tensor formulation of the PML. The form of the tensor we use is fixed with respect to the grid and is known to be perfectly matched in the continuous limit for orthogonal coordinate systems in which the metric is constant, i.e. Cartesian coordinates, and quasi-perfectly matched (quasi-PML) for curvilinear coordinates. Examples illustrating the effectiveness and long-term stability of the methods are presented. These examples demonstrate that the grid-based coordinate implementation of the PML is effective on Cartesian grids, but generates systematic reflections on grids which are orthogonal but non-Cartesian (quasi-PML). On non-orthogonal grids progressively worse performance of the PML is demonstrated. The paper begins with a summary derivation of the anisotropic formulation of the perfectly matched layer and mimetic differencing schemes for irregular logically rectangular grids.

The focus of this paper is on the solution of Maxwell's equations for time-harmonic fields on triangular, possibly nonorthogonal meshes. The method is based on the well-known Finite Integration Technique (FIT) [33, 35] which is a proven consistent discretization method for the computation of electromagnetic fields. FIT on triangular grids was first introduced in [29, 31] for eigenvalue problems arising in the design of accelerator components and dielectric loaded waveguides. For many technical applications the 2D simulation on a triangular grid combines the advantages of FIT, as e.g. the consistency of the method or the numerical advantage of banded system matrices, with the geometrical flexibility of non-coordinate grids. The FIT-discretization on non-orthogonal 2D grids has close relations [26] to the N´ed´elec elements [14, 15] or edge elements in the Finite Element Method.

A desire to unify the mathematical description of many physical theories, such as electrodynamics, mechanics, thermal conduction, has led us to understand that global (=integral) physical variables of eachth eory can be associated to spatial geometrical elements suchas points, lines, surfaces, volumes and temporal geometrical elements suchas instants and intervals. This association has led us to build a space-time classification diagram of variables and equations for each theory. Moreover, the possibility to express physical laws directly in a finite rather than differential form has led to the development of a computational methodology called Cell Method [12]. This paper discusses some practical aspects of this method and how it may overcome some of the main limitations of FDTD method in electrodynamics. Moreover, we will provide some numerical examples to compare the two methodologies.

We report some properties of the Finite Integration Technique (FIT), which are related to the definition of a discrete energy quantity. Starting with the well-known identities for the operator matrices of the FIT, not only the conservation of discrete energy in time and frequency domain simulations is derived, but also some important orthogonality properties for eigenmodes in cavities and waveguides. Algebraic proofs are presented, which follow the vectoranalytical proofs of the related theorems of the classical (continuous) theory. Thus, the discretization approach of the FIT can be considered as the framework for a consistent discrete electromagnetic field theory.

The calculus of differential forms can be used to devise a unified description of discrete differential forms of any order and polynomial degree on simplicial meshes in any spatial dimension. A general formula for suitable degrees of freedom is also available. Fundamental properties of nodal interpolation can be established easily. It turns out that higher order spaces, including variants with locally varying polynomial order, emerge from the usual Whitneyforms by local augmentation. This paves the way for an adaptive pversion approach to discrete differential forms.

Discrete differential forms should be used to deal with the discretization of boundary value problems that can be stated in the calculus of differential forms. This approach preserves the topological features of the equations. Yet, the discrete counterparts of the metricdependent constitutive laws remain elusive. I introduce a few purely algebraic constraints that matrices associated with discrete material laws have to satisfy. It turns out that most finite element and finite volume schemes comply with these requirements. Thus convergence analysis can be conducted in a unified setting. This discloses basic sufficient conditions that discrete material laws have to meet in order to ensure convergence in the relevant energy norms.

This paper outlines a generic algorithm to generate cuts for magnetic scalar potentials in 3-dimensional multiply-connected finite element meshes. The algorithm is based on the algebraic structures of (co)homology theory with differential forms and developed in the context of the finite element method and finite element data structures. The paper also studies the computational complexity of the algorithm and examines how the topology of the region can create an obstruction to finding cuts in O(m20) time and O(m0) storage, where m0 is the number of vertices in the finite element mesh. We argue that in a problem where there is no a priori data about the topology, the algorithm complexity is O(m20) in time and O(m4/30 ) in storage. We indicate how this complexity can be achieved in implementation and optimized in the context of adaptive mesh refinement.

Software systems designed to solve Maxwell's equations need abstractions that accurately explain what different kinds of electromagnetic problems really do have in common. Computational electromagnetics calls for higher level abstractions than what is typically needed in ordinary engineering problems. In this paper Maxwell's equations are described by exploiting basic concepts of set theory. Although our approach unavoidably increases the level of abstraction,it also simplifies the overall view making it easier to recognize a topological problem behind all boundary value problems modeling the electromagnetic phenomena. This enables us also to construct an algorithm which tackles the topological problem with basic tools of linear algebra.

Aspects of the geometric discretization of electromagnetic fields on simplicial lattices are considered. First, the convenience of the use of exterior differential forms to represent the field quantities through their natural role as duals (cochains) of the geometric constituents of the lattice (chains = nodes, edges, faces, volumes) is briefly reviewed. Then, the use of the barycentric subdivision to decompose the (ordinary) simplicial primal lattice together with the (twisted) non-simplicial barycentric dual lattice into simplicial elements is considered. Finally, the construction of lattice Hodge operators by using Whitney maps on the first barycentric subdivision is described. The objective is to arrive at a discrete formulation of electromagnetic fields on general lattices which better adheres to the underlying physics.

This paper uses simplicial complexes and simplicial (co)homology theory to expose a foundation for data structures for tetrahedral finite element meshes. Identifying tetrahedral meshes with simplicial complexes leads, by means of Whitney forms, to the connection between simplicial cochains and fields in the region modeled by the mesh. Furthermore, lumped field parameters are tied to matrices associated with simplicial (co)homology groups. The data structures described here are sparse, and the computational complexity of constructing them is O(n) where n is the number of vertices in the finite element mesh. Non-tetrahedral meshes can be handled by an equivalent theory. These considerations lead to a discrete form of Poincar´e duality which is a powerful tool for developing algorithms for topological computations on finite element meshes. This duality emerges naturally in the data structures. We indicate some practical applications of both data structures and underlying theory.

The space-time geometric structure of Maxwell's equations is examined and a subset of them is found to define a pair of exact discrete time-stepping relations. The desirability of adopting an approach to the discretization of electromagnetic problems which exploits this fact is advocated, and the name topological time-stepping for numerical schemes complying with it is suggested. The analysis of the equations leading to this kind of time-stepping reveals that these equations are naturally written in terms of integrated field quantities associated with space-time domains. It is therefore suggested that these quantities be adopted as state variables within numerical methods. A list of supplementary prescriptions for a discretization of electromagnetic problems suiting this philosophy is given, with particular emphasis on the necessity to adopt a space-time approach in each discretization step. It is shown that some existing methods already comply with these tenets, but that this fact is not explicitly recognized and exploited. The role of the constitutive equations in this discretization philosophy is briefly analyzed. The extension of this approach to more general kinds of space-time meshes, to other sets of basic time-stepping equations and to other field theories is finally considered.

We have constructed mimetic finite difference methods for both the TE and TM modes for 2-D Maxwell's curl equations and equations of magnetic diffusion with discontinuous coefficients on nonorthogonal, nonsmooth grids. The discrete operators were derived using the discrete vector and tensor analysis to satisfy discrete analogs of the main theorems of vector analysis. Because the finite difference methods satisfy these theorems, they do not have spurious solutions and the "divergence-free" conditions for Maxwell's equations are automatically satisfied. The tangential components of the electric field and the normal components of magnetic flux used in the FDM are continuous even across discontinuities. This choice guarantees that problems with strongly discontinuous coefficients are treated properly. Furthermore on rectangular grids the method reduces to the analytically correct averaging for discontinuous coefficients. We verify that the convergence rate was between first and second order on the arbitrary quadrilateral grids and demonstrate robustness of the method in numerical examples.

The Finite Integration Technique (FIT) is a consistent discretization scheme for Maxwell's equations in their integral form. The resulting matrix equations of the discretized fields can be used for efficient numerical simulations on modern computers. In addition, the basic algebraic properties of this discrete electromagnetic field theory allow to analytically and algebraically prove conservation properties with respect to energy and charge of the discrete formulation and gives an explanation of the stability properties of numerical formulations in the time domain.