a brick tetrahedron finite element interface with stable hybrid explicit implicit time stepping for maxwell s equations a new brick tetrahedron finite element interface with stable hybrid explicit implicit time stepping for maxwell s equations is described and tested the tetrahedrons are connected directly to the bricks as opposed to previous curl conforming formulations that use an intermediate layer of pyramids the electric field is expanded in linear edge elements which yields a discontinuous tangential electric field at the brick tetrahedron interface and this discontinuity is treated by nitsche s method in addition tangential continuity for an arbitrary constant electric field is imposed in the strong sense at the interface which makes it possible to avoid penalization that perturbs the frequency spectrum this hybridization preserves the null space of the curl curl operator and is free from non physical spurious modes which is confirmed by numerical tests the implicit newmark time stepping scheme is employed for the tetrahedrons which allows for local mesh refinement without reduced time step for the brick elements spatial lumping and explicit time stepping is employed which yields the standard finite difference time domain scheme furthermore we prove that the explicit implicit time stepping employed at the hybrid interface is stable for time steps up to the courant limit of the explicitly time stepped bricks second order of convergence is achieved for field solutions without singular behavior the reflection from the brick tetrahedron interface is small and scattering from a thin layer of tetrahedrons indicates levels at approximately 49db for a resolution of 14 cells per wavelength

investigation of nonplanar perfectly matched absorbers for finite element mesh truncation we present a detailed theoretical and numerical investigation of the perfectly matched layer pml concept as applied to the problem of mesh truncation in the finite element method fem we show that it is possible to extend the cartesian pml concepts involving half spaces to cylindrical and spherical geometries appropriate for closed boundaries in two and three dimensions by defining lossy anisotropic layers in the relevant coordinate systems by using the method of separation of variables it is possible to solve the boundary value problems in these geometries the analytical solutions demonstrate that under certain conditions outgoing waves are absorbed with negligible reflection and the transmitted wave is attenuated within the pml to reduce the white space in radiation or scattering problems conformal pmls are constructed via parametric mappings it is also verified that the pml concept which was originally introduced for problems governed by maxwell s equations can be extended to cases governed by the scalar helmholtz equation finally numerical results are presented to demonstrate the use of the pml in fem mesh truncation

finite element analysis of axisymmetric radomes a two step technique for the analysis of axisymmetric radomes is presented initially an axisymmetric finite element approach is employed together with an absorbing boundary condition for mesh truncation to determine the near fields scattered by an empty radome illuminated by a distant source next the reciprocity theorem is invoked to determine the far field pattern of an antenna encased by the radome by computing the interaction between the current distribution on the antenna and the near field data determined in the first step the details of the formulation are presented along with numerical results for two different arrays enclosed by radomes of varying permittivities

on convergence rates of simultaneously triangularizable switched linear systems in this note we introduce the concepts of best case and worst case convergence rates for switched linear systems their connections to the problems of stability and stabilization are briefly discussed for simultaneously triangularizable switched systems explicit formulas are presented for the convergence rates in terms of the system eigenvalues

high spatial order finite element method to solve maxwell s equations in time domain this paper presents a finite element method with high spatial order for solving the maxwell equations in the time domain in the first part we provide the mathematical background of the method then we discuss the advantages of the new scheme compared to a classical finite difference time domain fdtd method several examples show the advantages of using the new method for different kinds of problems comparisons in terms of accuracy and cpu time between this method the fdtd and the finite volume time domain methods are given as well

a novel td vie based on mot scheme for analysis of dispersive objects a novel time domain volume integral equation td vie is proposed to analyze the transient electromagnetic scattering from inhomogeneous dispersive objects in this td vie method the higher order nystrom method is applied in spatial domain namely the curvilinear tetrahedron elements are utilized to discretize dielectric objects and resulting unknowns are volume electric current density the lagrange interpolation polynomials are used to represent spatial basis functions in temporal domain the marching on in time mot scheme is applied and the shifted lagrange interpolation functions are used to represent temporal basis functions moreover the point matching method is applied both in spatial and temporal domain compared with the traditional spatial schaubert wilton glisson basis based on td vie this novel td vie method is more efficient several numerical examples have demonstrated that the proposed method is accurate stable and also more efficient than the traditional td vie

time domain finite difference and finite element methods for maxwell equations in complex media extensions of finite difference time domain fdtd and finite element time domain fetd algorithms are reviewed for solving transient maxwell equations in complex media also provided are a few representative examples to illustrate the modeling capabilities of fdtd and fetd for complex media the term complex media refers here to media with dispersive bi anisotropic inhomogeneous and or nonlinear properties present in the constitutive tensors

some remarks on the stability of time domain electromagnetic simulations aspects of the stability of time domain update schemes for electromagnetic problems are examined based on the consideration of system amplification matrices arising from the discretization of maxwell s equations in the sequence space we discuss time domain stability for a given discretization scale and under three different assumptions of decreasing generality general updates general leapfrog updates and reciprocal leapfrog updates

a new explicit time domain finite element method based on element level decomposition a new explicit time domain finite element method tdfem which is fundamentally different from traditional explicit tdfem formulations for solving maxwell s equations is introduced this new explicit tdfem is derived from a recently developed tdfem domain decomposition algorithm by extending domain decomposition to the element level this method solves dual field second order wave equations and computes the electric and magnetic fields in a leapfrog fashion with the element level decomposition no global system matrix has to be assembled and solved as required in the implicit tdfem and each element is related to its neighboring elements in an explicit manner hence the computational complexity of the method is reduced to only o n for both computer memory and cpu time a hybrid explicit implicit scheme is also proposed to alleviate a major disadvantage of the proposed explicit tdfem formulation in addition the convergence behavior of the corresponding higher order algorithms and the stability issues are discussed

perfectly matched layer in three dimensions for the time domain finite element method applied to radiation problems a new perfectly matched layer pml formulation for the time domain finite element method is described and tested for problems in three dimensional space an integral part of this pml algorithm is a novel time integration scheme based on galerkin s method with a piecewise linear temporal expansion of the electric field our time integration procedure is constructed by forming a linear combination of exact and trapezoidal integration applied to the temporal weak form which reduces to the well known newmark scheme in the case where the pml is not present this technique allows for a consistent time discretization of the electric field vector wave equation augmented with the pml which includes terms that involve the electric field itself time derivatives thereof and exponential functions temporally convolved with the electric field the auxiliary variables that are associated with such convolutions are efficiently updated in an explicit manner by matrix vector products the new pml formulation is successfully tested on canonical problems the pml can very efficiently absorb propagating electromagnetic waves which is demonstrated by reflections coefficients as low as 100 db the pml formulation is capable of giving highly accurate results for very broad frequency bands which is confirmed by tests that show an error smaller than 2 over more than six octaves in frequency finally we apply the pml formulation to a bow tie antenna and comparisons with measurements show good agreement

marching on in degree solver of time domain finite element boundary integral method for transient electromagnetic analysis a marching on in degree solver of time domain finite element boundary integral method is proposed for the analysis of transient electromagnetic scattering from inhomogeneous objects weighted laguerre polynomials are employed as temporal basis functions and testing functions in the proposed algorithm which leads to a recursive matrix equation without time variable and it can be solved degree by degree then the time domain response can be obtained by post processing furthermore a hybrid algorithm using both direct method and iterative method is introduced to accelerate the convergence in the solving of matrix equation numerical results are presented to demonstrate the accuracy stability and effectiveness of the proposed method

source and boundary implementation in vector and scalar dgtd we summarize the boundary and source implementation for the several formulations of the discontinuous galerkin time domain method dgtd since dgtd with zeroth order scalar basis functions using the upwind flux coincides with the finite volume time domain fvtd many of the concepts developed for fvtd can be ported to dgtd in any of its different formulations scalar vector basis upwind centered flux numerical examples illustrate the different alternatives

continuously inhomogeneous higher order finite elements for 3 d electromagnetic analysis a novel higher order entire domain finite element technique is presented for accurate and efficient full wave three dimensional 3d analysis of electromagnetic structures with continuously inhomogeneous material regions using large generalized curved hierarchical curl conforming hexahedral vector finite elements that allow continuous change of medium parameters throughout their volumes this is the first general 3d implementation and numerical demonstration of the inherent theoretical ability of the finite element method fem to directly treat arbitrarily continuously inhomogeneous materials the results demonstrate considerable reductions in both number of unknowns and computation time of the entire domain fem modeling of continuously inhomogeneous materials over piecewise homogeneous models they indicate that in addition to theoretical relevance and interest large curved higher order continuous fem elements also have great potential for practical applications that include structures with pronounced material inhomogeneities and complexities

perfectly matched layer for the time domain finite element method a new perfectly matched layer pml formulation for the time domain finite element method is described and tested for maxwell s equations in particular we focus on the time integration scheme which is based on galerkin s method with a temporally piecewise linear expansion of the electric field the time stepping scheme is constructed by forming a linear combination of exact and trapezoidal integration applied to the temporal weak form which reduces to the well known newmark scheme in the case without pml extensive numerical tests on scattering from infinitely long metal cylinders in two dimensions show good accuracy and no signs of instabilities for a circular cylinder the proposed scheme indicates the expected second order convergence toward the analytic solution and gives less than 2 root mean square error in the bistatic radar cross section rcs for resolutions with more than 10 points per wavelength an ogival cylinder which has sharp corners supporting field singularities shows similar accuracy in the monostatic rcs

parallel and explicit finite element time domain method for maxwell s equations we construct a parallel and explicit finite element time domain fetd algorithm for maxwell s equations in simplicial meshes based on a mixed e b discretization and a sparse approximation for the inverse mass matrix the sparsity pattern of the approximate inverse is obtained from edge adjacency information which is naturally encoded by the sparsity pattern of successive powers of the mass matrix each column of the approximate inverse is computed independently allowing for different processors to be used with no communication costs and hence linear ideal speedup in parallel processors the convergence of the approximate inverse matrix to the actual inverse full matrix is investigated numerically and shown to exhibit exponential convergence versus the density of the approximate inverse matrix the resulting fetd time stepping is explicit is the sense that it does not require a linear solve at every time step akin to the finite difference time domain fdtd method

irreducible numerical semigroups with arbitrary multiplicity and embedding dimension we give families of irreducible numerical semigroups with even conductor and with arbitrary multiplicity and embedding dimension we also study minimal presentations for these families of numerical semigroups

differential forms galerkin duality and sparse inverse approximations in finite element solutions of maxwell equations we identify primal and dual formulations in the finite element method fem solution of the vector wave equation using a geometric discretization based on differential forms these two formulations entail a mathematical duality denoted as galerkin duality galerkin dual fem formulations yield identical nonzero dynamical eigenvalues up to machine precision but have static zero eigenvalue solution spaces of different dimensions algebraic relationships among the degrees of freedom of primal and dual formulations are explained using a deep rooted connection between the hodge helmholtz decomposition of differential forms and descartes euler polyhedral formula and verified numerically in order to tackle the fullness of dual formulation algebraic and topological thresholdings are proposed to approximate inverse mass matrices by sparse matrices

scattering from complex bodies of revolution using a high order mixed finite element method and locally conformal perfectly matched layer an efficient finite element method fem is presented to analyze the scattering from complex bodies of revolution bor made of perfectly electric conductors and or dielectrics in the proposed method high order edge based vector basis functions are used to expand the transverse field components and high order node based scalar basis functions are used to expand the angular component the fem mesh is truncated using a locally conformal perfectly matched layer pml by the complex coordinate stretching such a kind of pml is very easy to implement in the numerical process and is able to enclose an arbitrarily shaped convex object in the spatial domain numerical examples are presented to demonstrate the accuracy and efficiency of the presented method

finite element analysis of complex antennas and arrays various technical issues are addressed in the frequency and time domain finite element analysis of complex antennas and arrays that may consist of fine structures and composite materials the paper starts with the finite element formulation of a generic antenna problem formulation for the modeling of general electrically and magnetically dispersive lossy and anisotropic materials in the time domain finite element analysis is presented for the first time this is followed by discussion of three typical techniques for truncation of the finite element computational domain via the use of absorbing boundary conditions perfectly matched layers and boundary integral equations it then describes the modeling of antenna feeds using a current probe including an improved model a voltage gap and a waveguide port boundary condition after that numerical approaches to modeling infinite phased arrays large finite arrays and antenna platform interaction are presented including two novel domain decomposition algorithms in both the frequency and time domains numerical examples are given to demonstrate the capability and application of the finite element analysis of a variety of complex antennas and arrays

conservative and provably stable fdtd subgridding the finite difference time domain fdtd method is a common robust simulation technique for transient electromagnetic interactions with complicated structures however the standard fdtd method is limited to cartesian grids everywhere in the computational grid many practitioners have extended fdtd to handle multiresolution problems by using finer grids near structures with small geometrical features abutted to coarse grids in regions of empty space unfortunately subgridding implementations based on interpolation or extrapolation of neighboring field values can exhibit late time instabilities herein a subgridding method based on multigrid finite element principles will be developed and its stability proven numerical results will assess its performance in 2 d and 3 d

technical communique existence and stability of periodic trajectories in switched server systems the paper analyzes an example of the discrete control of a continuous variable system we prove that the switched server system with one server and an arbitrary number of buffers has a set of periodic trajectories that attract all other trajectories of the system

a perfectly matched layer for fluid solid problems application to ocean acoustics simulations with solid ocean bottoms a time domain legendre spectral element method is described for full wave simulation of ocean acoustics models i e coupled fluid solid problems in unbounded or semi infinite domains taking into account shear wave propagation in the ocean bottom the technique can accommodate range dependent and depth dependent wave speed and density as well as steep ocean floor topography for truncation of the infinite domain to efficiently absorb outgoing waves a fluid solid complex frequency shifted unsplit perfectly matched layer is introduced based on the complex coordinate stretching technique the complex stretching is rigorously taken into account in the derivation of the fluid solid matching condition inside the absorbing layer which has never been done before in the time domain two implementations are designed a convolutional formulation and an auxiliary differential equation formulation because the latter allows for implementation of high order time schemes leading to reduced numerical dispersion and dissipation a topic of importance in particular in long range ocean acoustics simulations the method is validated for a two dimensional fluid solid pekeris waveguide and for a three dimensional seamount model which shows that the technique is accurate and numerically long time stable compared with widely used paraxial absorbing boundary conditions the perfectly matched layer is significantly more efficient at absorbing both body waves and interface waves

hybrid time domain solvers for the maxwell equations in 2d numerical simulation is an important tool in understanding the electromagnetic field and how it interacts with the environment different topics for time domain finite difference fdtd and finite element fetd methods for maxwell s equations are treated in this thesis subcell models are of vital importance for the efficient modeling of small objects that are not resolved by the grid a novel model for thin sheets using shell elements is proposed this approach has the advantage of taking into account discontinuities in the normal component of the electric field unlike previous models based on impedance boundary conditions ibcs several results are presented to illustrate the capabilities of the shell element approach waveguides are of fundamental importance in many microwave applications for example in antenna feeds the key issues of excitation and truncation of waveguides are addressed a complex frequency shifted form of the uniaxial perfectly matched layer upml absorbing boundary condition abc in fetd is developed prism elements are used to promote automatic grid generation and enhance the performance results are presented where reflection errors below 70db are obtained for different types of waveguides including inhomogeneous cases excitation and analysis via the scattering parameters are achieved using waveguide modes computed by a general frequency domain mode solver for the vector helmholtz equation huygens surfaces are used in both fdtd and fetd for excitation in waveguide ports inverse problems have received an increased interest due to the availability of powerful computers an important application is non destructive evaluation of material a time domain minimization approach is presented where exact gradients are computed using the adjoint problem the approach is applied to a general form of maxwell s equations including dispersive media and upml successful reconstruction examples are presented both using synthetic and experimental measurement data parameter reduction of complex geometries using simplified models is an interesting topic that leads to an inverse problem gradients for subcell parameters are derived and a successful reconstruction example is presented for a combined dielectric sheet and slot geometry

alternative fdtd updating schemes for anisotropic materials a novel way of looking at the anisotropic finite difference time domain fdtd method is proposed based on the finite element point of view which allows for a systematic approach of deriving anisotropic fdtd stencils without resorting to interpolation a new stencil for anisotropic material media that properly accounts for material interfaces and perfect electric conductor boundaries is derived this new stencil is tested and compared to analytical solutions and numerical solutions based on implicit finite element methods

a fast higher order time domain finite element boundary integral method for 3 d electromagnetic scattering analysis a novel hybrid time domain finite element boundary integral method for analyzing three dimensional 3 d electromagnetic scattering phenomena is presented the method couples finite element and boundary integral field representations in a way that results in a sparse system matrix and solutions that are devoid of spurious modes to accurately represent the unknown fields the scheme employs higher order vector basis functions defined on curvilinear tetrahedral elements to handle problems involving electrically large objects the multilevel plane wave time domain algorithm is used to accelerate the evaluation of the boundary integrals numerical results demonstrate the accuracy and versatility of the proposed scheme

numerical evaluation of high order finite element time domain formulations in electromagnetics this paper compares three full wave finite element time domain fetd formulations the first is based on the vector wave equation the others on maxwell s equations viz the ebhd formulation that discretizes e rarr b rarr h rarr and drarr and the eb formulation that discretizes only e rarr and b rarr the latter two formulations use a combination of 1 and 2 form discretization to avoid an auxiliary mesh a novel method for making the ebhd formulation operational is presented conditions for finite difference time domain fdtd like explicit operation are discussed the formulations are compared numerically by solving a three dimensional cavity and a rectangular waveguide using high order field representations up to mixed fourth order the error balance between time integration and field representation is investigated difficulties in making the ebhd formulation operational which have not previously been addressed in the literature are discussed and worked around novel numerical results show that the ebhd formulation has serious performance limitations

application of stable fem fdtd hybrid to scattering problems a recently developed stable finite element method fem finite difference time domain fdtd hybrid that eliminates the staircase approximation of complex geometries is tested by convergence studies for radar cross sections for a conducting sphere 1 db accuracy in all directions is obtained with nine cells per wavelength whereas the nasa almond requires a higher resolution of about 15 cells per wavelength for scatterers with a smooth boundary the results converge quadratically with the mesh size but for a horizontally polarized wave incident on the nasa almond the order of convergence is lower because of singular fields at the tip

a new 3 d nonspurious discontinuous galerkin spectral element time domain dg setd method for maxwell s equations a new discontinuous galerkin spectral element time domain dg setd method for maxwell s equations based on the field variables mathbf e and mathbf b is proposed to analyze three dimensional 3 d transient electromagnetic phenomena compared to the previous setd method based on the field variables mathbf e and mathbf h the mathbf eh scheme in which different orders of interpolation polynomials for electric and magnetic field intensities are required the newly proposed method can eliminate spurious modes using basis functions with the same order interpolation for electric field intensity and magnetic flux density the mathbf eb scheme consequently it can reduce the number of unknowns and computation load domain decomposition for the mathbf eb scheme setd method is completed via the dg method in addition the mathbf eb scheme setd method is extended to the well posed time domain perfectly matched layer pml to truncate the computation domain when solving open region problems the effectiveness and advantages of the new dg setd method are validated by eigenvalue analysis and numerical results

integral equation analysis of radiating structures of revolution a methods for solving radiation problems involving structures of revolution is presented that is based on the direct solutions of one dimensional integral equations for the currents the kernels of these equations are derived explicitly and their singularities are studied efficient methods for their computation are presented the numerical solution is obtained by a method of moments technique using sinusoidal basis functions applications include the determination of the current densities reflection coefficient and far field patterns for several waveguide fed structures numerical checks and comparisons with experiment demonstrate the accuracy of the method

full wave analysis and optimization of a tara like shield assisted paraboloidal reflector antenna using a nystrom type method using the recently developed method based on the rigorous theory of singular and hypersingular integral equations ies and nystrom type discretization a paraboloidal reflector assisted with a conical shield is investigated to decrease the computation time we propose a new method of calculation of the modal green s functions in the quasi optical range the far and near fields of the studied reflector are analyzed in the transmission and reception cases and reveal fine features that escape asymptotic analysis elementary numerical optimization of the shield assisted paraboloidal antenna is performed

a hybrid time domain technique that combines the finite element finite difference and method of moment techniques to solve complex electromagnetic problems this paper describes a hybrid technique directly operating in time domain that combines the finite element time domain fetd the finite difference time domain fdtd and the integral equation based method of moments in the time domain momtd techniques to analyze complex electromagnetic problems involving thin wire antennas radiating in the presence of inhomogeneous dielectric bodies whose shape can be arbitrary the method brings together the ability of the fdtd scheme to deal with arbitrary material properties the versatility of the fetd to accurately model curved geometries and that of the mom to analyze thin wire structures working in the time domain provides wide band information from a single execution of the marching on in time procedure and simplifies the interfacing of the fe and mom methods with the fdtd an approach specifically designed for time domain analysis numerical results that validate the hybrid method and show its capabilities are presented in the paper

a hybrid finite element method for axisymmetric waveguide fed horns a new method for finding radiation patterns and the reflection coefficients associated with an axisymmetric waveguide fed horn is presented the approach is based on a hybrid finite element method fem wherein the electromagnetic fields in the fem region are coupled to the fields outside by two surface integral equations because of the local nature of the fem this formalism allows for the presence of inhomogeneities to be included in the problem domain the matrix equation which results from the application of this method is shown to be complex symmetric comparisons of calculated and measured data for two different horns show good agreement

envelope tracking adaptive integral method for band pass transient scattering analysis an envelope tracking et algorithm is presented for efficiently solving integral equations pertinent to the analysis of transient scattering from perfect electrically conducting surfaces the algorithm solves space time samples of the complex envelope of the surface current density by implicit time marching thus the time step size is inversely proportional to the bandwidth not the maximum frequency content of the fields of interest at each time step anterpolation propagation interpolation and near zone correction stages are performed to compute the right hand side of a matrix equation which involves 4 d space time ffts and to accelerate its iterative solution which involves 3 d space ffts these ffts dominate the computational costs of the method they require o n mathrm c log 2 n mathrm g bar n mathrm i log n mathrm c operations per time step here n mathrm c is the number of nodes on an auxiliary regular grid n mathrm g is determined by the number of time steps that fields take to travel the maximum distance between two points on the scattering surface and bar n mathrm i is the average number of iterations needed for the iterative solution to converge at each time step moreover the algorithm is systematically compared to its time and frequency domain counterparts to this end radar profiles of benchmark targets are computed a time domain db error norm is proposed the algorithms parameters are optimized subject to at most 2 error in the radar profile according to this norm and the computational costs are measured as target size shape and excitation pulse are varied the results show that the algorithm outperforms its time domain counterpart and is competitive with its frequency domain counterpart

electromagnetic scattering by an inhomogeneous gyroelectric sphere using volume integral equation and orthogonal dini type basis functions the electromagnetic scattering by an inhomogeneous gyroelectric sphere is studied in this work the solution is constructed using the well known electric field volume integral equation efvie the kernel that incorporates the green s function is expanded in spherical vector wave functions and the unknown electric field and electric displacement is expanded in spherical vector wave functions using fully orthogonal dini type basis functions the permittivity tensor of the spherical object supports a continuously varying radial inhomogeneity while its permeability is that of free space this approach allows a fully analytical computation of the unknown matrix coefficients in special cases of homogeneous permittivity tensors the validation of our method is performed by comparisons with published results in the literature new numerical results are given for various continuously varying permittivity tensors

time domain integration of the maxwell equations on finite elements an explicit time domain integration scheme is developed for finite element spatial discretization of the maxwell equations a generalized wave equation is used in weak form which is free of vector parasites when discretized on single c sup 0 elements the use of integral lumping renders the mass matrix diagonal with no necessary degradation in accuracy the explicit method which results combines the economy of established finite difference techniques with the geometric flexibility of finite elements test problems representative of practical hyperthermia applications show agreement with analytic and frequency domain finite element methods and confirm scaling arguments on run time and memory

a stable fdtd subgridding method based on finite element formulation with hanging variables the finite element time domain fetd method based on the use of hanging variables to generate nested grids is used as an interface between the coarse and fine grids in the fdtd subgridding method since the formulation for treating hanging variables is based on a galerkin type intergrid boundary operator the resulting fdtd subgridding algorithm is guaranteed to be stable numerical examples such as the computation of resonant modes of a 3 d rectangular resonant cavity and 2 d tez scattering by a pec cylinder and a naca airfoil are presented to verify the stability and accuracy of the proposed subgridding algorithm

numerical simulation of bor scattering and radiation using a higher order fem numerical simulations of body of revolution geometries for scattering and radiation problems are presented the formulation consists of a finite element boundary integral fe bi method which is based on a finite element method that uses higher order nodal based scalar basis functions for the azimuthal field component and higher order edge based vector basis functions for the transverse field this formulation when combined with a symmetric fe bi hybridization scheme yields a final system of equations that is more accurate than earlier first order formulations numerical examples are given to demonstrate the accuracy and capabilities of the higher order solution

an efficient algorithm for implementing the crank nicolson scheme in the mixed finite element time domain method in this paper an efficient algorithm for implementing crank nicolson scheme in the finite element time domain fetd method is presented based on a direct discretization of the first order coupled maxwell curl equations this algorithm employs edge elements whitney 1 form to expand the electric field and face elements whitney 2 form for the magnetic field since the curl of an edge element is the linear combination of those face elements whose faces contain the given edge only the maxwell ampere equation composes a sparse linear matrix equation for the electric field update the maxwell faraday equation is explicit the crank nicolson scheme is implemented leading to an unconditionally stable vector fetd method and the matrix inverse is not required to be computed explicitly therefore only one matrix equation is required to be solved at each time step numerical results demonstrate that the proposed method is efficient when compared with the conventional leap frog mixed fetd method and the crank nicolson fdtd method

symplectic discretization for spectral element solution of maxwell s equations applying the spectral element method sem based on the gauss lobatto legendre gll polynomial to discretize maxwell s equations we obtain a poisson system or a poisson system with at most a perturbation for the system we prove that any symplectic partitioned runge kutta prk method preserves the poisson structure and its implied symplectic structure numerical examples show the high accuracy of sem and the benefit of conserving energy due to the use of symplectic methods

stability of explicit implicit hybrid time stepping schemes for maxwell s equations an improved version of the stable fem fdtd hybrid method t rylander and a bondeson comput phys commun 125 75 2000 for maxwell s equations is presented the new formulation has a modified time stepping scheme and is rigorously proven to be stable for time steps up to the stability limit for the fdtd the new scheme gives less reflection at the boundary between the structured and unstructured grids than the original formulation the hybrid method is compared to the fdtd with staircasing for scattering from a conducting sphere the discretization errors of the hybrid show quadratic dependence on mesh size while the scaling is less clear for the fdtd the fdtd gives errors that are 5 60 times higher than that of the hybrid depending on resolution and staircasing strategy

3d time domain simulation of electromagnetic diffusion phenomena a finite element electric field approach we present a finite element time domain fetd approach for the simulation of 3d electromagnetic em diffusion phenomena the finite element algorithm efficiently simulates transient electric fields and the time derivatives of magneticfields in general anisotropic earth media excited by multiple arbitrarily configured electric dipoles with various signal waveforms to compute transient electromagnetic fields theelectricfielddiffusionequationistransformedinto asystemofdifferentialequationsviagalerkin smethodwith homogeneous dirichlet boundary conditions to ensure numerical stability and an efficient time step the system of the differential equations is discretized in time using an implicit backward euler scheme the resultant fetd matrix vector equation is solved using a sparse direct solver along with a fill inreducedorderingtechnique whenadvancingthesolution in time the fetd algorithm adjusts the time step by examining whether or not the current step size can be doubled without unacceptably affecting the accuracy of the solution to simulate a step off source waveform the 3d fetd algorithm also incorporates a 3d finite element direct current fedcalgorithmthatsolvespoisson sequationusingasecondarypotentialmethodforageneralanisotropicearthmodel examples of controlled source fetd simulations are compared with analytic and or 3d finite difference time domain solutions and are used to confirm the accuracy and efficiencyofthe3dfetdalgorithm

on symmetric numerical semigroups a numerical semigroup is a subset s of n such that is closed under sums contains the zero and generates z as a group from this definition w x one obtains see for example 2 that s has a conductor c i e the maximum among all the natural numbers not belonging to s a numerical semigroup s is called symmetric if for every integer z f s c y z g s the study of the subsemigroups of n is a classical problem which is equivalent to study of the sets of natural solutions of linear equations with w x coefficients in n from 1970 and after the works 5 6 and others the study of the subsemigroups of n is also motivated by its applications to algebraic geometry an example arises when we consider an analytically irreducible one dimensional noetherian local domain r the integral closure v of r in the quotient field of r is a discrete valuation ring if we assume that r and v have the same residue class field and we denote by the valuation attached to v then s s r is a numerical semigroup r then properties on the semigroup s imply properties of the ring r for r w x w x example 6 proves that r is gorenstein see 1 if and only if s is r symmetric the close relation between numerical semigroups and monomial curves has produced the usage of algebraic geometry terminology in the field of numerical semigroups for example the smallest integer m greater than zero in s is usually called the multiplicity of s the cardinal e of a minimal system of generators for s is called the embedding dimension of s and a finite presentation for s is called a relation for s

analysis of a time domain finite element method for 3 d maxwell s equations in dispersive media abstract we consider the time dependent maxwell s equations in dispersive media on a bounded domain in three dimensional space a fully discrete finite element scheme is developed to approximate the electric field equation derived from the maxwell s equations optimal energy norm error estimates are proved for nedelec curl conforming edge elements this is the first finite element error analysis for maxwell s equations in dispersive media

performance of a causal anisotropic absorber for the truncation of finite element meshes in time domain electromagnetic unbounded problems the application of a new causal anisotropic absorber for the truncation of time domain finite element meshes is presented numerical results are illustrated in order to show the performance of the absorber both homogeneous and stratified absorbers are considered and their performances compared the computational efficiency of the finite element code implementing this absorber is also proven by comparing its computational cost with that one of a conventional time domain finite element code with boundary conditions given by perfect conductors and no absorber

an investigation of numerical dispersion in the vector finite element method using quadrilateral elements the discretization inherent in the vector finite element method results in the numerical dispersion of a propagating wave the numerical dispersion of a time harmonic plane wave propagating through an infinite two dimensional vector finite element mesh composed of uniform quadrilateral elements is investigated the effects on the numerical dispersion of the propagation direction of the wave the order of the polynomials used for the basis functions and the electrical size of the elements are investigated simple formulas are presented which are excellent approximations to the exact numerical dispersion the numerical dispersion is validated by a numerical example

em field induced in inhomogeneous dielectric spheres by external sources the electromagnetic field induced in the interior of inhomogeneous dielectric bodies by external sources can be evaluated by solving the well known electric field integrodifferential equation efide for spheres with constant magnetic permeability but variable dielectric constant a direct mainly analytical solution can be used even in case when the inhomogeneity in renders separation of variables inapplicable this approach constitutes a generalization of the hybrid analytical numerical scalar method developed by the authors in two recent papers for the corresponding acoustic scalar field induced in spheres with variable density and or compressibility this extension by no means trivial owing to the vector and integrodifferential nature of the equation is based on field vector expansions using the set of three harmonic surface vectors orthogonal and complete over the surface of the sphere for their angular dependence and dini s expansions of a general type for their radial functions the use of the latter has been shown to be superior to other possible sets of orthogonal expansions and as far as its convergence is concerned it may further be improved by properly choosing a crucial parameter in their eigenvalue equation the restriction to the spherical shape is imposed here to allow use of the well known expansion of green s dyadic in spherical eigenvectors of the vector wave equation

anisotropic medium pml for vector fetd with modified basis functions a novel anisotropic medium perfectly matched layer pml implementation is presented for boundary truncation in the three dimensional vector finite element time domain fetd method this approach utilizes a complex set of vector test and basis functions in the pml region that recover the usual basis functions in the non pml region numerical examples show that the proposed pml fetd approach has small reflection errors and no observed late time instabilities for simulations involving up to 10 sup 6 time steps

time domain finite element methods various time domain finite element methods for the simulation of transient electromagnetic wave phenomena are discussed detailed descriptions of test trial spaces explicit and implicit formulations nodal and edge facet element basis functions are given along with the numerical stability properties of the different methods the advantages and disadvantages of mass lumping are examined finally the various formulations are compared on the basis of their numerical dispersion performance

periodicity and chaos from switched flow systems contrasting examples of discretely controlled continuous systems two examples of the discrete control of a continuous variable system are analyzed these examples exhibit what may be regarded as the two extremes of complexity of the closed loop behavior one is eventually periodic the other is chaotic the examples are derived from sampled deterministic flow models these are of interest in their own right but have also been used as models for certain aspects of manufacturing systems in each case the authors give a precise characterization of the closed loop behavior

the gautschi time stepping scheme for edge finite element discretizations of the maxwell equations for the time integration of edge finite element discretizations of the three dimensional maxwell equations we consider the gautschi cosine scheme where the action of the matrix function is approximated by a krylov subspace method first for the space discretized edge finite element maxwell equations the dispersion error of this scheme is analyzed in detail and compared to that of two conventional schemes second we show that the scheme can be implemented in such a way that a higher accuracy can be achieved within less computational time as compared to other implicit schemes we also analyzed the error made in the krylov subspace matrix function evaluations although the new scheme is unconditionally stable it is explicit in structure as an explicit scheme it requires only the solution of linear systems with the mass matrix

electric and magnetic losses modeled by a stable hybrid with explicit implicit time stepping for maxwell s equations a stable hybridization of the finite element method fem and the finite difference time domain fdtd scheme for maxwell s equations with electric and magnetic losses is presented for two dimensional problems the hybrid method combines the flexibility of the fem with the efficiency of the fdtd scheme and it is based directly on ampere s and faraday s law the electric and magnetic losses can be treated implicitly by the fem on an unstructured mesh which allows for local mesh refinement in order to resolve rapid variations in the material parameters and or the electromagnetic field it is also feasible to handle larger homogeneous regions with losses by the explicit fdtd scheme connected to an implicitly time stepped and lossy fem region the hybrid method shows second order convergence for smooth scatterers the bistatic radar cross section rcs for a circular metal cylinder with a lossy coating converges to the analytical solution and an accuracy of 2 is achieved for about 20 points per wavelength the monostatic rcs for an airfoil that features sharp corners yields a lower order of convergence and it is found to agree well with what can be expected for singular fields at the sharp corners a careful convergence study with resolutions from 20 to 140 points per wavelength provides accurate extrapolated results for this non trivial test case which makes it possible to use as a reference problem for scattering codes that model both electric and magnetic losses

a mixed method for approximating maxwell s equations a semidiscrete mixed finite element approximation to the time dependent maxwell s equations on a bounded smooth domain is analyzed a variational problem for the electric and magnetic fields in which the boundary conditions are enforced naturally is derived then a general convergence result for mixed methods is proven and it is shown how this result may be used to prove various error estimates when nedelec s curl conforming finite elements are used jz3 with connected boundary f and unit outward normal n let e x and 1 x denote respectively the dielectric constant and magnetic permeability of the medium occupying q let a x denote the conductivity of the medium then if e x t and h x t denote respectively the electric and magnetic fields maxwell s equations 10 state that

dispersion analysis of finite difference and discontinuous galerkin schemes for maxwell s equations in linear lorentz media abstract in this paper we consider maxwell s equations in linear dispersive media described by a single pole lorentz model for electronic polarization we study two classes of commonly used spatial discretizations finite difference methods fd with arbitrary even order accuracy in space and high spatial order discontinuous galerkin dg finite element methods both types of spatial discretizations are coupled with second order semi implicit leap frog and implicit trapezoidal temporal schemes by performing detailed dispersion analysis for the semi discrete and fully discrete schemes we obtain rigorous quantification of the dispersion error for lorentz dispersive dielectrics in particular comparisons of dispersion error can be made taking into account the model parameters and mesh sizes in the design of the two types of schemes this work is a continuation of our previous research on energy stable numerical schemes for nonlinear dispersive optical media 6 7 the results for the numerical dispersion analysis of the reduced linear model considered in the present paper can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models the numerical dispersion analysis of the fully discrete fd and dg schemes for the dispersive maxwell model considered in this paper clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations their order of accuracy mesh discretization parameters and model parameters the results obtained here cannot be arrived at by considering discretizations of maxwell s equations in free space in particular our results contrast the advantages and disadvantages of using high order fd or dg schemes and leap frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors finally we highlight the limitations of the second order accurate temporal discretizations considered

application of tree cotree splitting to the time domain finite element analysis of electromagnetic problems the tree cotree splitting algorithm is applied to the time domain finite element analysis of electromagnetic problems to accelerate the time marching process based on the splitting of finite element edges into tree and cotree edges two sets of basis functions are formed to expand the electric field and construct the time marching system inclusion of the time domain waveguide port boundary condition and lumped port interfaces is also discussed it is shown that application of the tree cotree splitting to the time domain finite element analysis maintains the solution accuracy while significantly reducing the iteration count per time step for a preconditioned iterative solution when the time step size becomes relatively large as a result it allows adoption of a larger time step size within the requirement of the temporal sampling rate to achieve a faster time marching process with a marginal additional cost the tree cotree splitting algorithm also suppresses the late time linear drift associated with the conventional time domain finite element method

the extension of time domain integral equation analysis to scattering from imperfectly conducting bodies an extension of integral equation time domain analysis to treat electromagnetic scattering from imperfect conductors embedded in a dielectric is presented it employs isoparametric curvilinear representations and an implicit time stepping approach is used to achieve stability the principal difficulties arise from the diffusive wake term which the finite conductivity introduces into the green s function the consequent computational complexity and costs both significantly greater than for perfectly conducting scatterers are discussed examples of the application of the methods are presented

a study of discretization error in the finite element approximation of wave solutions a dispersion analysis is used to study the errors caused by the spatial discretization of the finite element method for the two dimensional scalar helmholtz equation it is shown that the error can be determined analytically for a uniform mesh of infinite extent numerical results are presented to show the effects of several parameters on the error these parameters are the nodal density the electrical size of the mesh the direction of propagation of the incident wave the type of element and the type of boundary condition

implementation of material interface conditions in the radial point interpolation meshless method we propose a systematic approach to accurate imposition of material interface conditions for the meshless radial point interpolation method rpim a new set of equations for updating fields at interfaces is derived a piecewise polynomial and a modified radial basis function are proposed and applied to account for field discontinuities near the interfaces in contrast to the previous work the proposed approach incorporates material properties and reduces geometrical dependence of the interpolation function numerical results show that the proposed approach significantly improves the simulation accuracy of the rpim at a cost of a small overhead in computational time

the advantages of triangular and tetrahedral edge elements for electromagnetic modeling with the finite element method in this paper we examine the numerical dispersion properties of the tetrahedral edge element in comparison to other edge and nodal elements for all nodal elements as well as hexahedral edge elements the phase error is always either positive or negative for waves propagating at any incidence angle which means that the error accumulates as the wave propagates from element to element this effect can produce large errors for electrically large geometries on the other hand the tetrahedral edge elements can produce either a negative or positive phase error depending on the direction of propagation through the element for an unstructured mesh phase cancellation occurs since the orientation of the tetrahedral elements are arbitrary because of the cancellation the numerical dispersion error is very low for meshes composed of well shaped tetrahedral edge elements numerical results are presented to demonstrate this point

a topological analysis of a family of dynamical systems with non standard chaotic and periodic behaviour we examine a family of maps which are motivated by a specific flow model for a manufacturing system that act on polygons inscribed within an equilateral triangle the family of maps is interesting because it provides an example of chaotic behaviour in one extreme and periodic behaviour in the other with a non standard transition between these extremes we characterize the behaviour of this family of maps and describe the interesting phenomena which occur between the two extremes of chaos and periodicity we also use symbolic dynamics to relate one of the maps to a particular subshift of finite type

the lobatto cell robust explicit higher order fdtd that handles inhomogeneous media to improve numerical dispersion and second order convergence behavior of the traditional yee finite difference time domain fdtd algorithm various higher order finite difference implementations have been proposed many have difficulty handling interfaces between distinct dielectric media or metallic sheets because the high order taylor expansions on which these codes are based are invalidated by field or derivative discontinuities one sided extensions extrapolations are a common approach to remedy interface discontinuities here another approach based upon whitney form mass lumped finite elements is pursued to develop the ldquolobatto cell rdquo a high order replacement for the yee cell with appropriate continuity behavior at contrasting media interfaces the lobatto cell retains many robust features of the yee method discrete conservation laws a guarantee of conditional stability and straightforward explicit updating but provides higher order interpolation and dispersion error convergence a notable drawback of the method is larger modeling granularity staircasing which is managed here by combining the lobatto cell method with preexisting robust subgridding techniques

analysis of exponential time differencing for fdtd in lossy dielectrics we determine the stability condition and analyze the accuracy of the exponential and centered time differencing schemes for finite difference time domain fdtd in an isotropic homogeneous lossy dielectric with electric and magnetic conductivities spl sigma and spl sigma respectively we show that these schemes are equivalent and determine that for accuracy both schemes must be used with a time step that finely resolves the electric and magnetic conduction current relaxation time scales the implications of these results for perfectly matched layer pml type absorbing boundary conditions are discussed

finite element analysis of complex axisymmetric radiating structures a finite element method fem is developed for the analysis of complex axisymmetric radiating structures the method is based on the electric field formulation with the transverse field expanded in terms of edge based vector basis functions and the azimuth component expanded using nodal based scalar basis functions this mixed representation of the electric field eliminates spurious solutions and permits an easy treatment of boundary conditions on conducting surfaces as well as across material interfaces the fem mesh is truncated using a previously developed cylindrical perfectly matched layer pml the method has been successfully applied to three radiating structures a corrugated horn antenna a spherical luneburg lens and a half maxwell fish eye numerical results are presented to show the validity accuracy and efficiency of the method

a general approach for the stability analysis of the time domain finite element method for electromagnetic simulations this paper presents a general approach for the stability analysis of the time domain finite element method tdfem for electromagnetic simulations derived from the discrete system analysis the approach determines the stability by analyzing the root locus map of a characteristic equation and evaluating the spectral radius of the finite element system matrix the approach is applicable to the tdfem simulation involving dispersive media and to various temporal discretization schemes such as the central difference forward difference backward difference and newmark methods it is shown that the stability of the tdfem is determined by the material property and by the temporal and spatial discretization schemes the proposed approach is applied to a variety of tdfem schemes which include 1 time domain finite element modeling of dispersive media 2 time domain finite element boundary integral method 3 higher order tdfem and 4 orthogonal tdfem numerical results demonstrate the validity of the proposed approach for stability analysis

application of the symplectic finite difference time domain scheme to electromagnetic simulation an explicit fourth order finite difference time domain fdtd scheme using the symplectic integrator is applied to electromagnetic simulation a feasible numerical implementation of the symplectic fdtd sfdtd scheme is specified in particular new strategies for the air dielectric interface treatment and the near to far field nff transformation are presented by using the sfdtd scheme both the radiation and the scattering of three dimensional objects are computed furthermore the energy conserving characteristic hold for the sfdtd scheme is verified under long term simulation numerical results suggest that the sfdtd scheme is more efficient than the traditional fdtd method and other high order methods and can save computational resources

discretization errors in finite methods issues and possible solutions in this paper we review many of the sources of discretization error when finite methods is applied to high frequency electromagnetic problems a major source of error is the numerical dispersion error this error is probably the most serious error for electrically large geometries and we review past efforts to reduce this error we also present a new edge based finite difference method which offers several improvements to current finite difference this new method has less numerical dispersion error and less errors due to material discontinuities than current finite difference methods we compare its performance to vector finite elements and show its superior accuracy finally we propose a new subgridding scheme to reduce discretization error which maintains second order accuracy

a novel efficient algorithm for scattering from a complex bor using mixed finite elements and cylindrical pml an efficient finite element method fem is developed to compute scattering from a complex body of revolution bor the bor is composed of a perfect conductor and impedance surfaces and arbitrary inhomogeneous materials the method uses edge based vector basis functions to expand the transverse field components and node based scalar basis functions to expand the angular component the use of vector basis functions eliminates the problem of spurious solutions suffered by other three component fem formulations the fem mesh is truncated with a perfectly matched layer pml in cylindrical coordinates the use of pml in cylindrical coordinates avoids the wasted computation which results from a spherical mesh boundary with an elongated scatterer the fem equations are solved by ordering the unknowns with a reverse cuthill mckee algorithm and applying a banded matrix solution algorithm the method is capable of handling large realistic radar targets and good agreement with measured results is achieved for benchmark targets

efficient computation of electromagnetic waves in anisotropic orthogonal plano cylindrically layered media using the improved numerical mode matching nmm method an orthogonal plano cylindrically layered opcl medium consists of materials stratified planarly and layered concentrically in the orthogonal directions the numerical mode matching nmm method has previously been shown to be a fast and robust semianalytical solver to investigate the propagation of electromagnetic em waves in such a complex but isotropic or transversely isotropic medium in this paper several important improvements have been made to extend applications of this efficient solver to the anisotropic ocpl medium the formulas for anisotropic media with three different diagonal elements in the cylindrical coordinate system are deduced to expand its application the perfectly matched layer pml is incorporated along the radial direction as an absorbing boundary condition abc to make the nmm method more accurate and efficient for unbounded low conductivity media and applicable to lossless media we manipulate the weak form of maxwell s equations and impose the correct boundary conditions at the cylindrical axis to solve the singularity problem finally we also offer formulas for computing em fields excited by a magnetic dipole located at any position with an arbitrary orientation numerical results have demonstrated the efficiency and accuracy of this method

on numerical semigroups with high embedding dimension abstract we compute the number of elements of a minimal system of generators for the congruence of a numerical semigroup with embedding dimension minus multiplicity equal to zero one and two for symmetric numerical semigroups we study the cases of embedding dimension minus multiplicity equal to one two and three

a linear nonconforming finite element method for maxwell s equations in two dimensions part i frequency domain we suggest a linear nonconforming triangular element for maxwell s equations and test it in the context of the vector helmholtz equation the element uses discontinuous normal fields and tangential fields with continuity at the midpoint of the element sides an approximation related to the crouzeix raviart element for stokes the element is stabilized using the jump of the tangential fields giving us a free parameter to decide we give dispersion relations for different stability parameters and give some numerical examples where the results converge quadratically with the mesh size for problems with smooth boundaries the proposed element is free from spurious solutions and for cavity eigenvalue problems the eigenfrequencies that correspond to well resolved eigenmodes are reproduced with the correct multiplicity

the discrete origin of fetd newmark late time instability and a correction scheme the finite element time domain fetd method is commonly used for transient simulation of electromagnetic wave phenomena most practitioners consider fetd when time integrated using the newmark beta method to be unconditionally stable when s 0 25 unlike the finite difference time domain fdtd courant criterion fetd newmark has no limiting timestep above which the method exhibits exponential growth however herein the stability properties of fetd newmark will be rigorously investigated by deducing the jordan canonical form of the fetd newmark amplification matrix and it will be demonstrated that the method does exhibit linear growth for certain field configurations these modes are none other than the pure gradient fields associated with late time instability though many practical simulations are of short duration and will never observe a linearly growing gradient solution it can be problematic for simulations which require long time periods to be integrated a correction scheme for eliminating this late time instability shall be suggested and numerical results will verify its performance