modeling radio transmission loss in curved branched and rough walled tunnels with the adi pe method we discuss the use of the parabolic equation pe along with the alternate direction implicit adi method in predicting the loss for three specialized tunnel cases curved tunnels branched tunnels and rough walled tunnels this paper builds on previous work which discusses the use of the adi pe in modeling transmission loss in smooth straight tunnels for each specialized tunnel case the adi pe formulation is presented along with necessary boundary conditions and tunnel geometry limitations to complete the study examples are presented where the adi pe numerical results for the curved and rough walled tunnel are compared to known analytical models and experimental data and the branched tunnel data is compared to the numerical solutions produced by hfss

a time domain volume integral equation and its marching on in degree solution for analysis of dispersive dielectric objects a marching on in degree mod based scheme for analyzing transient electromagnetic scattering from three dimensional dispersive dielectric objects is proposed a time domain volume integral equation tdvie for the electric flux density throughout the object is first formulated and then solved using the mod scheme with the use of weighted laguerre polynomials as entire domain temporal basis functions the convolution of the electric flux density and the medium susceptibility and its derivatives can be handled analytically by employing the galerkin temporal testing procedure the time variable is eliminated in the resultant recursive matrix equation so that the proposed algorithm overcomes the late time instability problem that may occur in the conventional marching on in time mot approach some complex dispersive media such as the debye lorentz and drude media are simulated to illustrate the validity of the tdvie mod algorithm

application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide angle wave equation abstracta classical split perfectly matched layer pml method has recently been applied to the scalar arbitrarily wide angle wave equation awwe in terms of displacement however the classical split pml obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence our goal was to improve the computational efficiency of awwe and to enhance the suppression of edge reflections by applying a convolutional pml cpml we reformulated the original awwe as a first order formulation and incorporated the cpml with a general complex frequency shifted stretching operator into the renewed formulation a staggered grid finite difference fd method was adopted to discretize the first order equation system for wavefield depth continuation the first order awwe with the cpml saved memory compared with the original second order awwe with the conventional split pml with the help of numerical examples we verified the correctness of the staggered

effective local absorbing boundary conditions for a finite difference implementation of the parabolic equation domain truncation by transparent boundary conditions for open problems where parabolic equation is utilized to govern wave propagation are in general computationally costly we utilize two approximations to a convolution in space type discrete boundary condition to reduce the cost while maintaining accuracy in far range solutions perfectly matched layer adapted to the crank nicolson finite difference scheme is also verified for a 2 d model problem where implemented results and stability analyses for different approaches are compared

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

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

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

a novel parallel parabolic equation method for electromagnetic scatterings the alternating group explicit age scheme together with the alternating direction implicit adi scheme is used for the parabolic equation pe method to analyze the electromagnetic scattering from electrically large conducting targets the calculation can be taken plane by plane by using the finite difference scheme along the paraxial direction furthermore each transverse plane is divided into two regions which are a small region around the target and an outer region combined with the air layer and a perfect matching layer the adi scheme is applied to the small region whereas the age scheme is introduced to the outer region in this manner only a few matrix equations are needed to be solved for the small region and the unknowns of the outer region can be directly obtained without solving any matrix equations therefore the cpu time can be reduced significantly by the proposed method and it can be easily adapted to the parallel technology the numerical results verify the accuracy and efficiency of the proposed method compared with the crank nicolson based pe method and the adi based pe method moreover the parallel efficiency is given as the last numerical example

restricted range approximation by splines and variational inequalities error estimates are obtained for the approximation of nonnegative functions from below by nonnegative splines the maximum norm results are of the same order as the jackson type estimates for unconstrained approximation by splines the l 2 results are used to estimate the rate of convergence of the ritz method for certain variational inequalities the more general problem of approximating functions f in the set w g in c a b l x leqq g x leqq u x x in a b by splines in w is also discussed

necessary condition on the dissipative term for the asymptotically free property of dissipative kirchhoff equations abstract it is known that the solutions of linear abstract dissipative wave equation with integrable coefficient of the dissipation term are asymptotically free namely the solutions have the same asymptotic behavior as that of the solutions of an abstract free wave equation in this paper we conclude that the integrability is not enough for asymptotic free property of kirchhoff equation by showing the following for the existence of a non trivial asymptotically free solution of abstract kirchhoff equation with initial data in some class an essentially stronger decay condition on the coefficient of dissipation term is necessary

a novel marching on in degree solver of time domain parabolic equation for transient em scattering analysis a novel marching on in degree mod solver of 3 d time domain parabolic equation is proposed to solve the transient electromagnetic scattering from electrically large perfect electric conductor pec targets the finite difference fd scheme is applied to the spatial discretization while the weighted laguerre polynomials are used as the temporal basis functions in this way a large number of computational resources can be saved by using the fd scheme along the paraxial direction and the late time stability can be guaranteed by the mod method both the fd schemes of crank nicolson and the alternating direction implicit types are discussed in this communication numerical results are given to demonstrate the accuracy and efficiency of the proposed method

piecewise linear recursive convolution for dispersive media using fdtd electromagnetic propagation through linear dispersive media can be analyzed using the finite difference time domain fdtd method by employing the recursive convolution rc approach to evaluate the discrete time convolution of the electric field and the dielectric susceptibility function the rc approach results in a fast and computationally efficient algorithm however the accuracy achieved is not generally as good as that obtained with other methods a new piecewise linear recursive convolution plrc method is described here that has greatly improved accuracy over the original rc approach but retains its speed and efficiency advantages

perfectly matched layers for coupled nonlinear schrodinger equations with mixed derivatives this paper constructs perfectly matched layers pml for a system of 2d coupled nonlinear schrodinger equations with mixed derivatives which arises in the modeling of gap solitons in nonlinear periodic structures with a non separable linear part the pml construction is performed in laplace fourier space via a modal analysis and can be viewed as a complex change of variables the mixed derivatives cause the presence of waves with opposite phase and group velocities which has previously been shown to cause instability of layer equations in certain types of hyperbolic problems nevertheless here the pml is stable if the absorption function s lies below a specified threshold the pml construction and analysis are carried out for the linear part of the system numerical tests are then performed in both the linear and nonlinear regimes checking convergence of the error with respect to the layer width and showing that the pml performs well even in many nonlinear simulations

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

electromagnetic scattering from 3 d targets in a random medium using finite difference frequency domain in this paper a three dimensional 3 d numerical technique based on the finite difference frequency domain fdfd is implemented to calculate the scattering from arbitrary shaped objects embedded in a continuous random medium the total field scattered field tf sf algorithm is integrated with the fdfd to minimize the memory consumption and speed up the calculations for validation purposes the radar cross section of a 2 d conducting cylinder in random medium is calculated using the fdfd technique and compared to previously published data based on the current generator method an upgrade to a 3 d solver was then inspired once the idea of using the multi grid technique was introduced to accelerate the convergence rate of the bicgstab iterative solver thus allowing the fdfd technique to become a robust method to solve the scattering problem from large targets embedded in random medium therefore using the introduced simulating scheme one can easily elucidate any scattering information out of real life targets surrounded by random environmental effects

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

analysis of transient electromagnetic scattering phenomena using a two level plane wave time domain algorithm a fast algorithm is presented for solving electric magnetic and combined field time domain integral equations pertinent to the analysis of surface scattering phenomena the proposed two level plane wave time domain pwtd algorithm permits a numerically rigorous reconstruction of transient near fields from their far field expansion and augments classical marching on in time mot based solvers the computational cost of analyzing surface scattering phenomena using pwtd enhanced mot schemes scales as o n sub i n sub s sup 2 3 log n sub s as opposed to spl oscr n sub t n sub s sup 2 for classical mot methods where n sub t and n sub s are the numbers of temporal and spatial basis functions discretizing the scatterer current numerical results that demonstrate the efficacy of the proposed solver in analyzing transient scattering from electrically large structures and that confirm the above complexity estimate are presented

perfectly matched layer truncation for parabolic wave equation models solutions of the parabolic wave equation are extended to complex values of the height variable the fast decay of the solutions in certain directions of the complex plane is used to improve parabolic wave equation solvers the solution is marched on a suitably chosen contour in the complex plane rather than on the real line this interpretation of berenger s perfectly matched layer allows extremely efficient truncation of the computational domain with a split step fourier algorithm the theory is developed for both the wide angle and narrow angle parabolic wave equations and numerical results are presented

impedance matched absorbers for finite difference parabolic equation algorithms in this paper a perfectly matched layer pml absorber recently introduced into the electromagnetic propagation literature by berenger j comput phys 114 185 200 1994 is adapted for use with both paraxial and wide angle acoustic parabolic equations pes our procedure incorporates an imaginary component into the transverse coordinate that mimics the introduction of a fictitious absorber on the edge of the computational grid use of such an impedance matched layer can significantly reduce spurious reflections compared to physical absorbing layer methods and thus allows a smaller number of boundary points to be employed in pe calculations numerical results obtained with several higher order propagator approximations confirm that such impedance matched absorbers efficiently eliminate reflections

perfectly matched absorbing layers for the paraxial equations a new absorbing boundary technique for the paraxial wave equations is proposed and analyzed numerical results show the efficiency of the method

solution of time domain electric field integral equation using the laguerre polynomials in this paper we propose a numerical method to obtain a solution for the time domain electric field integral equation td efie for arbitrary shaped conducting structures this method does not utilize the customary marching on in time mot solution method often used to solve a hyperbolic partial differential equation instead we solve the wave equation by expressing the transient behaviors in terms of laguerre polynomials by using these causal orthonormal basis functions for the temporal variation the time derivatives can be handled analytically in order to solve the wave equation we introduce two separate testing procedures a spatial and temporal testing by introducing first the galerkin temporal testing procedure the mot procedure is replaced by a recursive relation between the different orders of the weighted laguerre polynomials the other novelty of this approach is that through the use of the entire domain laguerre polynomials for the expansion of the temporal variation of the current the spatial and the temporal variables can be separated and the temporal variables can be integrated out for convenience we use the hertz vector as the unknown variable instead of the electric current density to verify our method we compare the results of a td efie and inverse fourier transform of a frequency domain efie

a high order finite element discretization with local absorbing boundary conditions of the linear schrodinger equation the goal of this paper is to obtain a high order full discretization of the initial value problem for the linear schrodinger equation in a finite computational domain for this we use a high order finite element discretization in space together with an adaptive implementation of local absorbing boundary conditions specifically obtained for linear finite elements and a high order symplectic time integrator the numerical results show that it is possible to obtain simultaneously a very good absorption at the boundary and a very small error in the interior of the computational domain

an analysis of new mixed finite elements for the approximation of wave propagation problems we construct and analyze a new family of rectangular two dimensional or cubic three dimensional mixed finite elements for the approximation of the acoustic wave equations the main advantage of this element is that it permits us to obtain through mass lumping an explicit scheme even in an anisotropic medium nonclassical error estimates are given for this new element

local soltutions for a nonlinear degenerate hyperbolic equation on etudie l existence et l unicite des solutions locales pour le probleme mixte associe a l equation hyperbolique degenere du type 2 u 2 m u x t 2 dx u 0 ou est un ouvert borne de r n a frontiere lisse et x est un vecteur de r n

wide angle shift map pe for a piecewise linear terrain a finite difference approach a wide angle parabolic wave equation solution is presented using shift map and finite difference techniques the corresponding split step fourier solution is well known the solution using finite difference technique where the standard parabolic wave equation is modified into the so called claerbout equation allowing propagation angles up to 45deg from the paraxial direction is also well known here we present an extension to that solution in which the shift map technique is incorporated into the finite difference scheme allowing a varying terrain to be considered the result is a solution that corresponds to the well known split step solution which is believed to perform well for terrain slopes up to 10deg 15deg and discontinuous slope changes on the order of 15deg 20deg this solution is a first order one with respect to the terrain slope however when using the finite difference technique it is also possible to find a second order solution with respect to the terrain slope this new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg which is an improvement

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

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

global solutions for dissipative kirchhoff strings with m r rp p 1 abstract we investigate the evolution problem u u m a 1 2 u 2 h au 0 u 0 u 0 u 0 u 1 where h is a hilbert space a is a self adjoint non negative operator on h with domain d a 0 is a parameter and m r r p with p u 0 u 1 d a i 2 d a i 1 2 satisfy a suitable smallness assumption and the non degeneracy condition m a 1 2 u 0 2 h 0 where p 2 i and i 2 i 1 moreover we prove for this solution decay with a polynomial rate as t these results apply to degenerate hyperbolic pdes with non local non linearities

an adi pe approach for modeling radio transmission loss in tunnels alternate direction implicit adi method is used to study radio wave propagation in tunnels using the parabolic equation pe we formulate the adi technique for use in tunnels with rectangular circular and arched cross sections and with lossy walls the electrical parameters of the lossy walls are characterized by an equivalent surface impedance a vector pe is also formulated for use in tunnels with lossy walls it is shown that the adi is more computationally efficient than the crank nicolson method however boundary conditions become more difficult to model the boundary conditions of the adi intermediate planes are given the same boundary conditions as the physical plane and the overall accuracy is reduced also when implementing the adi in tunnels with circular cross sections the order at which the line by line decomposition occurs becomes important to validate the adi pe we show simulation results for tunnel test cases with known analytical solutions furthermore the adi pe is used to simulate real tunnels in order to compare with experimental data it is shown that the pe models the electric fields most accurately in real tunnels at large distances where the lower order modes dominate

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

a general perfectly matched layer model for hyperbolic parabolic systems this paper describes a very general absorbing layer model for hyperbolic parabolic systems of partial differential equations for linear systems with constant coefficients it is shown that the model possesses the perfect matching property i e it is a perfectly matched layer pml the model is applied to two linear systems a linear wave equation with a viscous damping term and the linearized navier stokes equations the resulting perfectly matched layer for the viscous wave equation is proved to be stable the paper also presents how the model can be used to construct an absorbing layer for the full compressible navier stokes equations for all three applications numerical experiments are presented especially for the linear problems the results are very promising in one experiment where the performance of a hyperbolic pml and the new hyperbolic parabolic pml is compared for a hyperbolic parabolic system an improvement of six orders of magnitude is observed for the compressible navier stokes equations results obtained with the presented layer are competitive with existing methods

on the non equivalence of perfectly matched layers and exterior complex scaling abstract the perfectly matched layers pml and exterior complex scaling ecs methods for absorbing boundary conditions are analyzed using spectral decomposition both methods are derived through analytical continuations from unitary to contractive transformations we find that the methods are mathematically and numerically distinct ecs is complex stretching that rotates the operator s spectrum into the complex plane whereas pml is a complex gauge transform which shifts the spectrum consequently the schemes differ in their time stability numerical examples are given

global solutions for dissipative kirchhoff strings with non lipschitz nonlinear term we investigate the evolution problem u u m a 1 2 u 2 au 0 u 0 u0 u 0 u1 where h is a hilbert space a is a self adjoint nonnegative operator on h with domain d a 0i s a parameter and m r is a nonnegative function such that m 0 0a ndm is nonnecessarily lipschitz continuous in a neighborhood of 0 we prove that this problem has a unique global solution for positive times provided that the initial data u0 u 1 d a d a 1 2 satisfy a suitable smallness assumption and the nondegeneracy condition m a 1 2 u0 2 0 moreover we study the decay of the solution as t these results apply to degenerate hyperbolic pdes with nonlocal nonlinearities

dispersive waves and pml performance for large stencil differencing in the parabolic equation the discrete reflection absorption performance of perfectly matched layer pml boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the pml region most previous work analyzing pml reflection performance has assumed use of low order second or fourth small stencil three or five point spatial discretizations whose dispersion is poor outside a narrow low wavenumber band large stencil differencing schemes regardless of order are easily created with physically realistic wide band dispersive characteristics but the effect their adoption has on pml performance has not been analyzed previously in the context of a semi discrete approximation to the parabolic paraxial equation this work derives the analytic form of the discrete pml reflection coefficient for an arbitrary order arbitrary stencil interior differencing scheme coupled to an arbitrary order arbitrary stencil pml region this result is then used to show that the absorption performance of the pml regions is largely insensitive to changes in interior differencing as a consequence the benefits of superior dispersion characteristics provided by use of large stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the pml

transient analysis of dispersive periodic structures for oblique plane wave incidence using laguerre marching on in degree mod transient analysis of dispersive periodic structures in the case of obliquely incident plane wave is presented in this work the laguerre marching on in degree mod is used for the development of an unconditionally stable algorithm due to the unconditional stability courant freidrich levy cfl temporal discretization limit can be circumvented resulting in less simulation time compared to the conventional dispersive finite difference time domain fdtd this is particularly advantageous for multiscale dispersive structures the uniaxial perfectly matched layer upml is implemented in the laguerre domain by virtue of the laguerre decomposition the presented method can simulate general dispersive media compared to conventional dispersive fdtd which can simulate only standard dispersion models such as debye lorentz and drude the formulations are validated by estimating the transmission spectra of an infinite double pole debye slab and comparing the results with analytical results further confirmation is obtained by estimating the transmission spectra of zeroth order floquet mode of a 1d periodic array of dispersive square cylinders and comparing the results with periodic method of moments solution

target scattering calculations with the parabolic equation method the parabolic equation technique is used to solve the helmholtz equation in the presence of scatterers of arbitrary shape in two and three dimensions the scattered field is computed directly using non homogeneous boundary conditions on the scattering object to represent the incident field effectively this decouples the pe paraxial direction from the direction of incidence for convex objects the whole range of scattering angles can be covered with a small number of narrow angle calculations finite difference implementations involve tridiagonal matrices in two dimensions and more general sparse matrices in three dimensions the resulting codes can be used to solve scattering problems for objects ranging in size from a few wavelengths to hundreds of wavelengths the method has been tested against analytical solutions for soft and rigid circular cylinders in 2d and soft and rigid spheres in 3d showing good agreement at all scattering angles

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

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

highly accurate absorbing boundary conditions for wide angle wave equations we develop a new class of absorbing boundary conditions abcs to prevent unwanted artifacts and wraparounds associated with aperture truncation in migration modeling using high order one way wave equations the fundamental approach behind the proposed development is the efficient discretization of the half space beyond the boundary of interest using midpoint integrated imaginary finite elements an idea recently utilized in the development of effective one way wave equations the proposed absorbing boundary conditions essentially add absorbing layers at the aperture truncation points we derive the absorbing boundary conditions analyze their properties and develop a stable explicit finite difference scheme to solve the downward continuation problem modified by these boundary conditions with the help of numerical examples we conclude that with as few as three absorbing layers i e two additional gridpoints the waves can be absorbed completely thus preventing associated artifacts

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

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

the degree of copositive approximation and a computer algorithm the main results are as follows 1 let f in bf c 0 1 change its sign a finite number of times then the degree of copositive approximation of f by splines with n equally spaced knots is bounded by c omega 3 f 1 n for n large enough this rate is the best in the sense that omega 3 cannot be replaced by omega 4 2 an algorithm is developed basedon the proof 3 the first result above holds for a copositive polynomial approximation of f 4 if f in c 1 0 1 then the degree of approximation by copositive splines of order r is bounded by cn 1 omega r 1 f 1 n the results on f in bf c 0 1 fill a gap left by s p zhou israel j math 78 1992 pp 75 83 and y k hu d leviatan and x m yu j anal 1 1993 pp 85 90 j approx theory 80 1995 pp 204 218

a vector meshless parabolic equation method for three dimensional electromagnetic scatterings the three dimensional 3 d vector parabolic equation pe method was used for electromagnetic scattering with reasonable accuracy the algorithm was marched from plane to plane along some preferred directions so that the computational load could be reduced significantly however the traditional 3 d pe method was discretized with the finite difference fd scheme and was not flexible for complex geometries in this paper the radial point interpolation meshless rpim method is used for its flexibility to model complex geometries in each transverse plane when the pe method is applied to analyze the electromagnetic scattering from electrically large objects numerical results demonstrate that the proposed technique can efficiently give reasonably accurate results for the complex objects when compared with both the multilevel fast multipole method mlfmm and the traditional pe method

the wlp fdtd method for periodic structures with oblique incident wave a new unconditionally stable finite difference time domain fdtd method for periodic structures is presented which is based on the field transformation method and the weighted laguerre polynomials fdtd wlp fdtd the proposed method uses a field transformation to remove the time gradient across the grid and then uses the concept of the wlp fdtd to get the implicit relationship between the transformed field variables it holds the advantages of the wlp fdtd can eliminate the restriction of the courant friedrich levy cfl stability condition compared with other field transform methods the new method needn t to do special treatment for the additional terms it appears to be much more efficient than the other field transformation fdtd method for solving periodic structures with fine structures and large incident angle to verify the accuracy and the efficiency of the proposed method we compare the results of the split field fdtd method with the proposed method

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

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

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

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

positive and copositive spline approximation in lp 0 1 the order of positive and copositive spline approximation in the lp norm 1 p is studied the main results are 1 1 the error of positive approximation by splines is bounded by c 2 f 1n p if f has a nonnegative extension 2 2 the order deteriorates to 1 if f does not have such an extension 3 3 the error of copositive spline approximation is bounded by c f 1n p 4 4 if f is also continuous the error in 3 can be estimated in terms of the third modulus 3 f 1n p all constants in the error bounds are absolute

optimizing the perfectly matched layer the berenger perfectly matched layer or pml 4 3 is used in computational electromagnetism as a sponge layer to terminate finite element approximations of scattering problems an infinite perfectly matched layer creates no reflection for incident waves of any frequency or any incident direction and the waves decay exponentially in magnitude into the layer this property is lost when the layer is truncated since the truncation boundary generates a reflected wave discretization of the differential equations further changes the reflection coefficient of the layer in this talk we shall describe the pml in the simple case of a two dimensional electromagnetic problem the first part of this abstract is expository we shall show how to derive the layer using a change of variable approach 5 9 6 after this derivation we describe some new analysis to show the effects of truncating and discretizing the pml 9 we end by showing how to compute an optimal layer and make some remarks about pml for the full maxwell system

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

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

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

some nonlinear degenerate wave equations on etudie la resolubilite locale du probleme aux valeurs limites et initiales suivant 2 u t 2 m gradu x t 2 dx u 0 dans 0 u 0 sur w 0 u 0 0 u t 0 dans ou m r est une fonction c 1 non negative sur 0 un domaine borne de r n a frontiere lisse w

transient electromagnetic scattering from dielectric objects using the electric field integral equation with laguerre polynomials as temporal basis functions in this paper we propose a time domain electric field integral equation td efie formulation for analyzing the transient electromagnetic response from three dimensional 3 d dielectric bodies the solution method in this paper is based on the galerkin s method that involves separate spatial and temporal testing procedures triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3 d dielectric structures the time domain unknown coefficients of the equivalent electric and magnetic currents are approximated using a set of orthonormal basis function that is derived from the laguerre functions these basis functions are also used as the temporal testing functions use of the laguerre polynomials as expansion functions for the transient portion of response enables one not only to handle the time derivative terms in the integral equation in an analytic fashion but also completely separates the space and the time variables thus the time variable along with the courant condition can be eliminated in a galerkin formulation using this procedure we also propose an alternative formulation using a different expansion of the magnetic current the total computational cost for this new method is similar to that of an implicit marching on in time mot efie scheme even though at each step this procedure requires more computations numerical results involving equivalent currents and far fields computed by the two proposed methods are presented and compared

higher order paraxial wave equation approximations in heterogenous media a new family of paraxial wave equation approximations is derived these approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa

unconditionally stable discretization schemes of non reflecting boundary conditions for the one dimensional schrodinger equation this paper addresses the problem of the construction of stable approximation schemes for the one dimensional linear schrodinger equation set in an unbounded domain after a study of the initial boundary value problem in a bounded domain with a transparent boundary condition some unconditionally stable discretization schemes are developed for this kind of problem the main difficulty is linked to the involvement of a fractional integral operator defining the transparent operator the proposed semi discretization of this operator yields with a very different point of view the one proposed by yevick friese and schmidt j comput phys 168 2001 433 two possible choices of transparent boundary conditions based on the dirichlet neumann dn and neumann dirichlet nd operators are presented to preserve the stability of the fully discrete scheme conform galerkin finite element methods are employed for the spatial discretization finally some numerical tests are performed to study the respective accuracy of the different schemes

a generalized impedance method for application of the parabolic approximation to underwater acoustics the parabolic equation pe has become an important computational tool for approximating underwater acoustic path losses in inhomogeneous environments the numerical solution of this equation is performed on a grid over the propagation medium which includes the water column and has also included a homogeneous false bottom below the actual ocean bottom this situation is viewed as one in which the propagation medium can be divided into a horizontally inhomogeneous upper layer and a horizontally homogeneous lower layer the pressure field in the latter layer is found using standard techniques valid for homogeneous media and in the former layer is found using an implicit finite difference scheme these fields and their normal derivatives are matched along the interface between the layers the result is referred to as a generalized impedance boundary condition gibc which serves to eliminate the need for a false bottom predictions obtained using the standard parabolic equation and a completely homoge

an alternative direction decomposition scheme and error analysis for parabolic equation model alternate direction implicit adi method has been widely used to analyze the tunnel propagation or electromagnetic scattering problem for the finite difference method in this paper a novel alternate direction decomposition add scheme is proposed to simplify the split step fourier transform ssft algorithm in parabolic equation pe model the spatial matrix of electric field is decoupled into two subarrays by add ssft algorithm which is used to decrease the computational burden for 3 d pe problems in a large scale environment after the integration order exchanged also the phase error of this approximation is analyzed to show the algorithm performance along with several numerical results including the tunnel cases with traditional modal theory and they are provided to validate and calibrate the pe model and demonstrate the efficiency and accuracy of the proposed algorithm moreover the scattering computation is investigated and compared with multilevel fast multipole method to show that add ssft is appropriate for solving the scattering problems for the electrical large target