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.

The geometrical approach to Maxwell's equations promotes a way to discretize them that can be dubbed "Generalized Finite Differences", which has been realized independently in several computing codes. The main features of this method are the use of two grids in duality, the "metric-free" formulation of the main equations (Amp`ere and Faraday), and the concentration of metric information in the discrete representation of the Hodge operator. The specific role that finite elements have to play in such an approach is emphasized, and a rationale for Whitney forms is proposed, showing why they are the finite elements of choice.

The objective of this paper is to present an approach to electromagnetic field simulation based on the systematic use of the global (i.e. integral) quantities. In this approach, the equations of electromagnetism are obtained directly in a finite form starting from experimental laws without resortingto the differential formulation. This finite formulation is the natural extension of the network theory to electromagnetic field and it is suitable for computational electromagnetics.

In this paper, we introduce a modified FDTD model in order to investigate the behavior of the induced voltage of a non-uniform transmission line (TL) excited by lumped voltage sources or external electromagnetic wave. The parameters to be taken into account for this specific coupling phenomenon are explicitly analyzed and the affection on the behavior of the induced voltage is discussed in detail. To confirm the validity of this model, results obtained by this model, for two typical transmission line configurations, are compared to results obtained by other models, already published in the literature. Finally, several numerical calculations of the line responses are provided for non-uniform TL's that can be present in practical configurations. These results indicate that a configuration slightly different from the uniform one can cause large discrepancies on the termination voltages of the TL.

This study is concerned with the development of a model to describe microwave emission from terrain covered by wet snow. The model is based on the radiative transfer theory and the strong fluctuation theory. Wet snow is treated in the model as a mixture of dry snow and water inclusions. The shape of the water inclusions is taken into account. The effective permittivity is calculated by using the two-phase strong fluctuation theory model with nonsymmetrical inclusions. The phase matrix and the extinction coefficient of wet snow for an anisotropic correlation function with azimuth symmetric are used. The vector radiative transfer equation for a layer of a random medium was solved by using Gaussian quadrature and eigen analysis. The model behaviour is illustrated by using typical parameters encountered in microwave remote sensing of wet snow. Comparisons with emissivity data at 11, 21 and 35 GHz are made. It is shown that the model predictions fit the experimental data.

The strong fluctuation theory is applied to calculate the effective permittivity of wet snow by a two-phase model with nonsymmetrical inclusions. In the two-phase model, wet snow is assumed to consist of dry snow (host) and liquid water (inclusions). Numerical results for the effective permittivity of wet snow are illustrated for random media with isotropic and anisotropic correlation functions. A three-phase strong fluctuation theory model with symmetrical inclusions is also presented for theoretical comparison. In the three-phase model, wet snow is assumed to consist of air (host), ice (inclusions) and water (inclusions) and the shape of the inclusions is spherical. The results are compared with the Debye-like semi-empirical model and a comparison with experimental data at 6, 18 and 37 GHz is also presented. The results indicate that (a) the shape and the size of inclusions are important, and (b) the two-phase model with non-symmetrical inclusions provides the good results to the effective permittivity of wet snow.

In this paper, we introduce a new classification scheme for dual frequency polarimetric SAR data sets. A (6×6) polarimetric coherency matrix is defined to simultaneously take into account the full polarimetric information from both images. This matrix is composed of the two coherency matrices and their cross-correlation. A decomposition theorem is applied to both images to obtain 64 initial clusters based on their scattering characteristics. The data sets are then classified by an iterative algorithm based on a complex Wishart density function of the 6 by 6 matrix. A class number reduction technique is then applied on the 64 resulting clusters to improve the efficiency of the interpretation and representation of each class characteristics. An alternative technique is also proposed which introduces the polarimetric cross-correlation information to refine the results of classification to a small number of clusters using the conditional probability of the crosscorrelation matrix. The analysis of the resulting clusters is realized by determining the rigorous change in polarimetric properties from one image to the other. The polarimetric variations are parameterized by 8 real coefficients derived from the decomposition of a special unitary operator on the Gell-Mann basis. These classification and analysis schemes are applied to full polarimetric P, L, and C bands SAR images of the Nezer forest acquired by NASA/JPL AIRSAR sensor (1989).

Solutions to electromagnetic scattering at any angle of incidence by a perfectly conducting spheroid and a homogeneous dielectric spheroid coated with a dielectric layer are obtained by solving Maxwell's equations together with boundary conditions. The method used is that of expanding electric and magnetic fields in the spheroidal coordinates in terms of the spheroidal vector wave functions and matching their respective boundary conditions at spheroidal interfaces. In this formulation, the column vector of the series expansion coefficients of the scattered field is obtained from that of the incident field by means of a matrix transformation, which is in turn obtained from a system of equations derived from boundary conditions. The matrix depends only on the scatterer's properties; hence the scattered field at a different direction of incidence is obtained without repeatedly solving a new set of simultaneous equations. Different from the previous work, the present work developed an accurate and efficient Mathematica source code for more accurate solution to the problem. Normalized bistatic and backscattering cross sections are obtained for conducting (for verification purpose), homogeneous dielectric (for verification purpose), and coated dielectric prolate (for some new results) spheroids, with real and complex permittivities. Numerically exact results for the coated dielectric prolate spheroids are newly obtained and are not found in existing literature.

This paper considers expressions of the dyadic Green's function for rectangular enclosures, which are efficient from a computational point of view. The inherent application is to solve numerically electromagnetic scattering problems with an integral equation formulation, using the Method of Moments. The Green's dyadic is derived from an image-spectral approach, which has the flexibility to generate directly the expression with the fastest convergence once the locations of both the observation and the source points are given. When the observation point is at the source region, slowly converging sums arise that are overcome by extending the method of Ewald to the dyadic case. Numerical proofs are reported in tabular form to validate the technique developed.

This paper presents an alternative vector analysis of the electromagnetic (EM) fields radiated from electrically small thin square and rectangular loop antennas of arbitrary length 2a and width 2b . This method employs the dyadic Green's function (DGF) in the derivation of the EM radiated fields and thus makes the analysis general, compact and straightforward. Both near- and far-zones are considered so that the EM radiated fields are expressed in terms of the vector wave eigenfunctions. Not only the exact solution of the EM fields in the near and far zones outside the region (where r > √a2 + b2 ) is derived by use of spherical Hankel functions of the first kind, but also the closed series form of the EM fields radiated in the near zone inside the region 0 ≤ r < b is obtained in series of spherical Bessel functions of the first kind. The two regions between the radial distances b and √a2 + b2 which are defined as intermediate zones are characterized by both the spherical Bessel and Hankel functions of the first kind. Validity of the numerical results is discussed and clarified.

In this paper, the scanning characteristics of an infinite stacked microstrip phased array in a three-layered structure are analysed. In the analysis of the field distribution, the spectral-domain Galerkin Method of Moments together with the planar dyadic Green's function is applied. An attachment mode current is employed to model the singularity of currents nearby the feed point so as to facilitate the fast convergence. The currents on all patches are expanded in terms of the trigonometric basis and weighting functions in the entire domain. The normalized patterns of the infinite microstrip array are computed in this paper and the scanning features of the antenna against the scanning angle and frequency are discussed in both the E- and H-planes.

To see the similarities and differences with eletromagnetics, basic concepts and equations of elastodynamics are formulated in coordinate-free form applying concepts from Polyadic Algebra. Planewave propagation is studied for time-harmonic equations in isotropic and simple uniaxially anisotropic media and the Green dyadic is derived for the isotropic medium. As an extension, the Green dyadic for the anisotropic elastic medium is derived in perturbational approximation. In the Appendix A, some basic properties for tetradics, useful for practical analysis both in elastodynamics as well as in electromagnetics, are derived.

An effective wavelet based multigrid preconditioned conjugate gradient method is developed to solve electromagnetic large matrix problem for millimeter wave scattering application. By using wavelet transformation we restrict the large matrix equation to a relative smaller matrix and which can be solved rapidly. The solution is prolonged as the new improvement for the conjugate gradient (CG) method. Numerical result shows that our developed wavelet based multigrid preconditioned CG method can reach large improvement of computational complexity. Due to the automaticity of wavelet transformation, this method is potential to be a block box solver without physical background.