continuum modeling of the mechanical and thermal behavior of discrete large structures in the present paper we introduce a rather straightforward construction procedure in order to derive continuum equivalence of discrete truss like repetitive structures once the actual structure is specified the construction procedure can be outlined by the following three steps a all sets of parallel members are identified b unidirectional effective continuum properties are derived for each of these sets and c orthogonal transformations are finally used to determine the contribution of each set to the overall effective continuum properties of the structure here the properties includes mechanical stiffnesses thermal coefficients of thermal expansions and material densities once expanded descriptions of the steps b and c are done the construction procedure will be applied to a wide variety of discrete structures and the results will be compared with those of other existing methods

transformation optics based local mesh refinement for solving maxwell s equations in this paper a novel local mesh refinement algorithm based on transformation optics to has been developed for solving the maxwell s equations of electrodynamics the new algorithm applies transformation optics to enlarge a small region so that it can be resolved by larger grid cells the transformed anisotropic maxwell s equations can be stably solved by an anisotropic fdtd method while other subgridding or adaptive mesh refinement fdtd methods require time space field interpolations and often suffer from the late time instability problem to avoid small time steps introduced by the transformation optics approach an additional application of the mapping of the material matrix to a dispersive material model is employed numerical examples on scattering problems of dielectric and dispersive objects illustrate the performance and the efficiency of the transformation optics based fdtd method

a block pseudospectral method for maxwell s equations a block pseudospectral bps method is proposed as a new way to couple pseudospectral discretizations across interfaces in computations for a linear hyperbolic system the coupling is achieved via discretized derivative matching conditions obtained from the system compared to the standard technique of imposing compatibility conditions based on characteristics of the system the bps method offers better stability and accuracy especially in the case where equation coefficients are discontinuous computational examples for maxwell s equations in nonhomogeneous media demonstrate that bps retains high accuracy over times that are orders of magnitude larger than those for not only low order methods such as yee s but also high order methods such as characteristic based spectral elements

short note distinguishing correct from incorrect pml proposals and a corrected unsplit pml for anisotropic dispersive media we show that some previous proposals for perfectly matched layer pml absorbers in anisotropic media or for waveguides at oblique incidence are not in fact true pmls in previous work we similarly showed a failure of several pml proposals for periodic media photonic crystals we therefore argue that a more careful validation scheme is required for pml proposals in contrast to past authors who have typically checked only that reflections are small for a fixed resolution and suggest a simple validation scheme that can be readily applied to any pml proposal regardless of derivation or implementation we demonstrate this test for a corrected unsplit field pml valid for anisotropic dispersive media implemented in both planewave expansion and finite difference time domain fdtd methods

new code upper bounds from the terwilliger algebra and semidefinite programming we give a new upper bound on the maximum size a n d of a binary code of word length n and minimum distance at least d it is based on block diagonalizing the terwilliger algebra of the hamming cube the bound strengthens the delsarte bound and can be calculated with semidefinite programming in time bounded by a polynomial in n we show that it improves a number of known upper bounds for concrete values of n and d from this we also derive a new upper bound on the maximum size a n d w of a binary code of word length n minimum distance at least d and constant weight w again strengthening the delsarte bound and yielding several improved upper bounds for concrete values of n d and w

adiabatic theorem and continuous coupled mode theory for efficient taper transitions in photonic crystals we prove that an adiabatic theorem generally holds for slow tapers in photonic crystals and other strongly grated waveguides with arbitrary index modulation exactly as in conventional waveguides this provides a guaranteed pathway to efficient and broad bandwidth couplers with e g uniform waveguides we show that adiabatic transmission can only occur however if the operating mode is propagating nonevanescent and guided at every point in the taper moreover we demonstrate how straightforward taper designs in photonic crystals can violate these conditions but that adiabaticity is restored by simple design principles involving only the independent band structures of the intermediate gratings for these and other analyses we develop a generalization of the standard coupled mode theory to handle arbitrary nonuniform gratings via an instantaneous bloch mode basis yielding a continuous set of differential equations for the basis coefficients we show how one can thereby compute semianalytical reflection and transmission through crystal tapers of almost any length using only a single pair of modes in the unit cells of uniform gratings unlike other numerical methods our technique becomes more accurate as the taper becomes more gradual with no significant increase in the computation time or memory we also include numerical examples comparing to a well established scattering matrix method in two dimensions

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

spatial finite difference approximations for wave type equations the simplest finite difference approximations for spatial derivatives are centered explicit and applied to regular equispaced grids well established generalizations include the use of implicit compact approximations and staggered grids we find here that the combination of these two concepts together with high formal order of accuracy is very effective for approximating the first derivatives in space that occur in many wave type pdes

difference approximations for the second order wave equation difference approximations are derived for the second order wave equation in one and two space dimensions without first writing it as a first order system both the dirichlet and the neumann problems are treated for the one dimensional case relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary general two dimensional domains are considered for the dirichlet problem where the domain is embedded in a cartesian grid and the boundary conditions are approximated by interpolation a stable conservative scheme is derived where the time step is determined only by the interior discretization formula discretization cells cut by the boundary are treated implicitly but the resulting scheme becomes explicit because the implicit dependence only is pointwise numerical examples are provided to verify the stability and accuracy of the proposed method

validation of a finite element error estimator the smoothed solution against which the finite element stresses are compared is based on a reformulation of the finite difference method that permits irregular meshes and complex boundary conditions to be analyzed

a vector finite element time domain method for solving maxwell s equations on unstructured hexahedral grids in this paper the vector finite element time domain vfetd method is derived analyzed and validated the vfetd method uses edge vector finite elements as a basis for the electric field and face vector finite elements as a basis for the magnetic flux density the galerkin method is used to convert maxwell s equations to a coupled system of ordinary differential equations the leapfrog method is used to advance the fields in time the method is shown to be stable and to conserve energy and charge for arbitrary hexahedral grids a numerical dispersion analysis shows the method to be second order accurate on distorted hexahedral grids several computational experiments are performed to determine the accuracy and efficiency of the method

choice of the perfectly matched layer boundary condition for frequency domain maxwell s equations solvers we show that the performance of frequency domain solvers of maxwell s equations is greatly affected by the kind of the perfectly matched layer pml used in particular we demonstrate that using the stretched coordinate pml sc pml results in significantly faster convergence speed than using the uniaxial pml upml such a difference in convergence behavior is explained by an analysis of the condition number of the coefficient matrices additionally we develop a diagonal preconditioning scheme that significantly improves solver performance when upml is used

finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains a finite difference time dependent numerical method for the wave equation supported by recently derived novel elliptic grids is analyzed the method is successfully applied to single and multiple two dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries the new grids have nearly uniform cell area j grids and nearly uniform grid line spacing email protected numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state the restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing as a consequence moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost also numerical solutions for wave problems inside annular regions of complex shapes are obtained the use of the new grids results in late time stability in contrast with other classical finite difference time dependent methods

a high order accurate parallel solver for maxwell s equations on overlapping grids a scheme for the solution of the time dependent maxwell s equations on composite overlapping grids is described the method uses high order accurate approximations in space and time for maxwell s equations written as a second order vector wave equation high order accurate symmetric difference approximations to the generalized laplace operator are constructed for curvilinear component grids the modified equation approach is used to develop high order accurate approximations that use only three time levels and have the same time stepping restriction as the second order scheme discrete boundary conditions for perfect electrical conductors and for material interfaces are developed and analyzed the implementation is optimized for component grids that are cartesian resulting in a fast and efficient method the solver runs on parallel machines with each component grid distributed across one or more processors numerical results in two and three dimensions are presented for the fourth order accurate version of the method these results demonstrate the accuracy and efficiency of the approach

perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods we derive a correct first order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted most previous perturbative methods give incorrect results for this case even to lowest order because of the complicated discontinuous boundary conditions on the electric field at such an interface our final expression is simply a surface integral over the material interface of the continuous field components from the unperturbed structure the derivation is based on a localized coordinate transformation technique which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking the limit in which the coordinate perturbation is infinitesimally localized around the boundary not only is our result potentially useful in evaluating boundary perturbations e g from fabrication imperfections in highly anisotropic media such as many metamaterials but it also has a direct application in numerical electromagnetism in particular we show how it leads to a subpixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes

a novel 3 d cpfdtd scheme for modeling grid nonconformally aligned transmitter structures in this paper a novel three dimensional 3 d subcell algorithm for modeling metallic objects which are nonconformally oriented within the finite difference time domain fdtd grid is proposed the generally applicable algorithm is based on the contour path cp technique and enables in combination with an enhanced nonconformally aligned source model a more accurate spatial representation of antenna structures than the common staircase approach to evaluate the performance of the new algorithm the simulation results are compared with the classical staircase approach for dipole antennas as well as for a generic phone significant improvements were achieved for all antenna parameters from the near and far field i e feedpoint impedance radiation pattern and field distributions

time dependent numerical method with boundary conforming curvilinear coordinates applied to wave interactions with prototypical antennas a new unstaggered finite difference time dependent technique to accurately model scattring from prototypical 2d antenna structures is devised the unbounded boundary value problems defining these phenomena are redefined over bounded domains using appropriate radiation operators over finite artificial boundaries generalized curvilinear coordinates are generated such that physical boundaries correspond to coordinate lines a numerical procedure to generate almost orthogonal boundary conforming fine grids over these bounded regions is developed once the governing equations are written in terms of the new curvilinear coordinates a time dependent numerical method is applied to obtain time harmonic steady state solutions to these problems the electric field wave amplitude as well as the wave pattern inside the waveguide and the scattered field from the prototypical antennas are obtained accuracy and computational cost are compared when almost orthogonal and nonorthogonal grids are in use an optical theorem for a flanged waveguide antenna with perfect electrical conductor walls is derived it is verified that the approximate solutions obtained by application of the time dependent numerical method with boundary conforming curvilinear coordinates satisfy the optical theorem

perturbation theory for maxwell s equations with shifting material boundaries perturbation theory permits the analytic study of small changes on known solutions and is especially useful in electromagnetism for understanding weak interactions and imperfections standard perturbation theory techniques however have difficulties when applied to maxwell s equations for small shifts in dielectric interfaces especially in high index contrast three dimensional systems due to the discontinous field boundary conditions in fact the usual methods fail even to predict the lowest order behavior by considering a sharp boundary as a limit of anisotropically smoothed systems we are able to derive a correct first order perturbation theory and mode coupling constants involving only surface integrals of the unperturbed fields over the perturbed interface in addition we discuss further considerations that arise for higher order perturbative methods in electromagnetism

a high order mixed vector finite element method for solving the time dependent maxwell equations on unstructured grids we present a mixed vector finite element method for solving the time dependent coupled ampere and faraday laws of maxwell s equations on unstructured hexahedral grids that employs high order discretization in both space and time the method is of arbitrary order accuracy in space and up to 4th order accurate in time making it well suited for electrically large problems where grid anisotropy and numerical dispersion have plagued other methods in addition the method correctly models both the jump discontinuities and the divergence free properties of the electric and magnetic fields is charge and energy conserving conditionally stable and free of spurious modes several computational experiments are performed to demonstrate the accuracy efficiency and benefits of the method

on lee association schemes over z4 and their terwilliger algebra abstract let f 0 1 2 3 and define the set k k 0 k 1 k 2 of relations on f such that x y k i if and only if x y i mod 4 let n be a positive integer we consider the lee association scheme l n over z 4 which is the extension of length n of the initial scheme f k let t denote the terwilliger algebra of l n with respect to the zero codeword of length n we show that t is generated by a homomorphic image of the universal enveloping algebra of the lie algebra sl 3 c and the center z t furthermore we determine the irreducible modules for t using the schur weyl duality

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

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

second order procedure for modeling dielectric material interfaces in structured nonorthogonal finite difference discretizations of maxwell s equations structured nonorthogonal finite difference discretizations of maxwell s equations are often employed to incorporate curved dielectric material interfaces as part of this process it is necessary to employ an adequate procedure to enforce the tangential field continuity conditions at the locations of the interfaces where the components of the permittivity tensor are discontinuous this contribution proposes a second order domain splitting procedure that can accomplish this task without assuming that the permittivity tensor components are piecewise constant or diagonal as explained this feature is needed to exploit the full geometrical flexibility of structured grid generation the proposed domain splitting procedure consistently yields a global error that is second order accurate even in extreme cases where the components of the permittivity tensor have a change of sign or a sharp jump

time domain method having a naturally diagonal mass matrix independent of element shape for general electromagnetic analysis 2 d formulations in this paper we present a new time domain method that has a naturally diagonal mass matrix and thereby a strict linear computational complexity per time step regardless of whether the discretization is a structured grid or an unstructured mesh this property is obtained independent of the element shape used for discretization no interpolations projections and mass lumping are required the accuracy and stability of the proposed method are both theoretically guaranteed in addition no dual mesh is needed and the tangential continuity of the fields is satisfied across the element interface the flexible framework of the proposed method also allows for a straightforward extension to higher order accuracy in both electric and magnetic fields numerical experiments have been conducted on a variety of unstructured triangular element meshes correlations with analytical solutions and the time domain finite element method have validated the accuracy and generality of the proposed new time domain method

an embedded boundary method for the wave equation with discontinuous coefficients a second order accurate embedded boundary method for the two dimensional wave equation with discontinuous wave propagation speed is described the wave equation is discretized on a cartesian grid with constant grid size and the interface across which the wave speed is discontinuous is allowed to intersect the mesh in an arbitrary fashion by using ghost points on either side of the interface previous embedded boundary techniques for the neumann and dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy the resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points resulting in a fully explicit time integration method we prove that the one dimensional restriction of the method is stable without damping for arbitrary locations of the interface relative to the grid for the two dimensional case the previously developed fourth order a t a dissipation is generalized to handle jump conditions we demonstrate that this operator provides sufficient stabilization to enable long time simulations while being weak enough to preserve the accuracy of the solution numerical examples are given where the method is used to study electromagnetic scattering of a plane wave by a dielectric cylinder the numerical solutions are evaluated against the analytical solution due to mie and pointwise second order accuracy is confirmed

an analysis of finite volume finite element and finite difference methods using some concepts from algebraic topology in this paper we apply the ideas of algebraic topology to the analysis of the finite volume and finite element methods illuminating the similarity between the discretization strategies adopted by the two methods in the light of a geometric interpretation proposed for the role played by the weighting functions in finite elements we discuss the intrinsic discrete nature of some of the factors appearing in the field equations underlining the exception represented by the constitutive term the discretization of which is maintained as the key issue for numerical methods devoted to field problems we propose a systematic technique to perform this task present a rationale for the adoption of two dual discretization grids and point out some optimization opportunities in the combined selection of interpolation functions and cell geometry for the finite volume method finally we suggest an explanation for the intrinsic limitations of the classical finite difference method in the construction of accurate high order formulas for field problems

on the terwilliger algebra of distance biregular graphs abstract let denote a distance biregular graph with vertex set x fix x x and let t t x denote the terwilliger algebra of with respect to x in this paper we consider irreducible t modules with endpoint 1 we show that there are no such modules if and only if is the complete bipartite graph k 1 n n 1 and x is a vertex of with valency 1 if the valency of x is at least 2 then we show that up to isomorphism there is a unique irreducible t module of endpoint 1 and this module is thin

high order finite differences and the pseudospectral method on staggered grids finite difference approximations for the first derivative valid halfway between equidistant gridpoints are in general much more accurate than the corresponding approximations which are valid at gridpoints the pseudospectral fourier method can be viewed as the limit of finite difference approximations when the order of accuracy tends to infinity a fundamentally different and more accurate pseudospectral method is obtained from the halfway approximations this study derives these and related methods and discusses their accuracies

modeling of sloped interfaces on a yee grid to represent material boundaries in the finite difference time domain or frequency domain method effective cell permittivity eeff can be introduced for each grid cell crossed by material interface in this paper we revisit the derivation of tensorial eeff for a sloped interface and describe possible interpolation schemes for coupling of different effective electric field and induction components near the interface we put the resulting non symmetric and symmetrized effective permittivity matrices to numerical tests in the frequency domain for very high contrast interfaces the symmetrized schemes perform worse than simple staircasing while non symmetrized interpolation retains the second order convergence

accuracy of staircase approximations in finite difference methods for wave propagation while a number of increasingly sophisticated numerical methods have been developed for time dependent problems in electromagnetics the yee scheme is still widely used in the applied fields mainly due to its simplicity and computational efficiency a fundamental drawback of the method is the use of staircase boundary approximations giving inconsistent results usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result in this paper we derive exact discrete solutions to the yee scheme close to the staircase approximated boundary enabling a detailed theoretical study of the amplitude phase and frequency errors created furthermore we show how evanescent waves of amplitude o 1 o 1 occur along the boundary these characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why in practice the yee scheme works as well as it does the analysis is supported by detailed proofs and numerical examples

equivalent continuum representation of structures composed of repeated elements a procedure for determining the equivalent continuum properties of a structure composed of repeated patterns of discrete elements with both displacement and rotational coordinates is presented these nodal coordinates are transformed to rigid body and strain gradient variables using a polynomial approximation the maximum number of independent variables that may be retained is determined by applying a ranking procedure to the resulting transformation matrix the possibility of introducing errors by requiring the analyst to supply the strain gradient terms directly is reduced by identifying the appropriate variables through the use of the polynomial expansion and the ranking procedure additional constraints may be imposed in this analysis the equivalent continuum parameters result when a further transformation to the appropriate kinematic variables is applied and the strain energy expression is reduced to these variables three dimensional beamand plate like structures are treated the results correspond to findings using other approaches

a new approach to boundary modelling for finite difference applications in solid mechanics procedures for extending the useful scope of the finite difference method in solid mechanics applications are presented the improvements centre around the introduction of the physical nature of the deformations into the equations used to formulate the approximate solution this is accomplished by evaluating the coefficients of taylor series expansions for the displacement approximations in terms of rigid body motions strains and derivatives of strains this procedure is demonstrated with plane stress applications the ability to interpret the derivative approximations physically allows the fictitious nodes typical of the finite difference method to be rationally incorporated into the model as a means of enforcing traction boundary conditions an example problem is solved using both regular and irregular meshes the displacements and stresses compare well with finite element solutions

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

a more accurate stable fdtd algorithm for electromagnetics in anisotropic dielectrics a more accurate stable finite difference time domain fdtd algorithm is developed for simulating maxwell s equations with isotropic or anisotropic dielectric materials this algorithm is in many cases more accurate than previous algorithms g r werner et al 2007 5 a f oskooi et al 2009 7 and it remedies a defect that causes instability with high dielectric contrast usually for e 10 with either isotropic or anisotropic dielectrics ultimately this algorithm has first order error in the grid cell size when the dielectric boundaries are sharp due to field discontinuities at the dielectric interface accurate treatment of the discontinuities in the limit of infinite wavelength leads to an asymmetric unstable update c a bauer et al 2011 6 but the symmetrized version of the latter is stable and more accurate than other fdtd methods the convergence of field values supports the hypothesis that global first order error can be achieved by second order error in bulk material with zero order error on the surface this latter point is extremely important for any applications measuring surface fields

fdtd algorithm in curvilinear coordinates em scattering the finite difference time domain fdtd algorithm for the solution of electromagnetic scattering problems is formulated in generalized coordinates in two dimensions and implemented in a code with the lowest order bayliss turkel radiation boundary condition expressed in cylindrical coordinates it is shown that for a perfect conductor such a formulation leads to a stable well posed algorithm and that in regions where the curvature of the coordinate lines is not great the dispersion and anisotropy effects are negligible such effects become more pronounced in regions of high curvature leading to unphysical phase shifts the magnitude of such shifts and the amount of wavefront distortion is studied via numerical experiments using a cylindrical mesh near field results are given for two canonical shapes for each polarization the circular cylinder and cylinders of square and rectangular cross sections these results are compared with those obtained by exact eigenfunction expansion techniques with method of moments mm solutions and with solutions obtained from an alternate fdtd approach in each case agreement is excellent the propagation of a plane wave through a polar space in the absence of a scatterer is also examined and it is shown that the fdtd algorithm is capable of tracking the incident wave closely

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

numerical stability of nonorthogonal fdtd methods in this paper a sufficient test for the numerical stability of generalized grid finite difference time domain fdtd schemes is presented it is shown that the projection operators of such schemes must be symmetric positive definite without this property such schemes can exhibit late time instabilities the origin and the characteristics of these late time instabilities are also uncovered based on this study nonorthogonal grid fdtd schemes nfdtd and the generalized yee gy methods are proposed that are numerically stable in the late time for quadrilateral prism elements allowing these methods to be extended to problems requiring very long time simulations the study of numerical stability that is presented is very general and can be applied to most solutions of maxwell s equations based on explicit time domain schemes

the correct formulation of intermediate boundary conditions for runge kutta time integration of initial boundary value problems a disturbing feature of applying explicit high order runge kutta rk time integrators to initial boundary value problems solved by the method of lines is the loss of accuracy that results from wrong specifications of intermediate stage boundary conditions the solution proposed here is to prescribe analytically those values that would result from applying the rk solver at the boundaries and hence maintain the order of the rk method without a reduction in the step size limit the procedure is described in detail for nonlinear hyperbolic equations and both scalar and vector examples are presented

a second order 3d electromagnetics algorithm for curved interfaces between anisotropic dielectrics on a yee mesh a new frequency domain electromagnetics algorithm is developed for simulating curved interfaces between anisotropic dielectrics embedded in a yee mesh with second order error in resonant frequencies the algorithm is systematically derived using the finite integration formulation of maxwell s equations on the yee mesh second order convergence of the error in resonant frequencies is achieved by guaranteeing first order error on dielectric boundaries and second order error in bulk possibly anisotropic regions convergence studies conducted for an analytically solvable problem and for a photonic crystal of ellipsoids with anisotropic dielectric constant both show second order convergence of frequency error the convergence is sufficiently smooth that richardson extrapolation yields roughly third order convergence the convergence of electric fields near the dielectric interface for the analytic problem is also presented

a finite volume method for the approximation of maxwell s equations in two space dimensions on arbitrary meshes a new finite volume method is presented for discretizing the two dimensional maxwell equations this method may be seen as an extension of the covolume type methods to arbitrary possibly non conforming or even non convex n sided polygonal meshes thanks to an appropriate choice of degrees of freedom an equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles the main properties of the scheme are its energy conservation its stability under a cfl like condition and the fact that it preserves gauss law and divergence free magnetic fields second order convergence is demonstrated numerically on non conforming and distorted meshes

the ode formulation of hyperbolic pdes discretized by the spectral collocation method we formulate systems of hyperbolic partial differential equations pdes with characteristic boundary conditions as pure initial value problems of ordinary differential equations odes this enables a straightforward application of any explicit or implicit method for the time integration of the spatially discretized system we also show how this method relates to the correctional method of treating the boundary points to exemplify we work out the ode formulation for the one dimensional compressible euler equations and a simple two dimensional linear system and conclude with some numerical calculations on these using both explicit and implicit runge kutta methods

a pseudospectral fictitious point method for high order initial boundary value problems when pseudospectral approximations are used for space derivatives one often encounters spurious eigenvalues these can lead to severe time stepping difficulties for pdes this is especially the case for equations with high order derivatives in space requiring multiple conditions at one or both boundaries we note here that a very simple to implement fictitious point approach circumvents most of these difficulties the new approach is tested on the kuramoto sivashinsky equation and on a dispersive linear pde featuring a time space corner singularity

dynamic continuum modeling of beamlike space structures using finite element matrices subscripts a rational and straight forward method is introduced for developing equivalent continuum models of large beam like periodic lattice structures based on energy equivalence extended timoshenko bean model is chosen to take account of the effects due to couplings between extension transverse shear and bending deformations the procedure for developing continuum models involves utilizing well defined existing finite element matrices directly in caluclating strain and kinetic energies from which equivalent continuum structural and dynamic properties are induced the numerical results of free vibration analysis show that the method developed in this paper gives very reliable dynamic characteristics compared to other methods nomenclature

three dimensional computer modeling of electromagnetic fields a global lookback lattice truncation scheme abstract a new lattice truncation scheme for the finite difference time domain approach to the solution of maxwell s equations has been developed the problem space is truncated near the sources and the field components on its boundary are generated from those field values known at retarded times on an interior surface one cell from it with an integral representation of the electromagnetic field the numerical implementation of this global lookback scheme is discussed examples which have been used to determine its characteristics and its validity are given

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

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

a stitching method for the generation of unstructured meshes for use with co volume solution techniques abstract the successful implementation of co volume time domain solution techniques requires the use of high quality smooth dual meshes the generation of delaunay voronoi diagrams with these properties for two dimensional domains of complicated geometrical shape is considered in the adopted approach near boundary regions are discretised according to prescribed criteria and the remainder of the computational domain is discretised using an ideal mesh the two meshes are stitched together to provide a consistent mesh for the complete domain after smoothing the resulting mesh is found to be of a quality which exceeds that of meshes produced by standard automatic unstructured mesh generation methods examples involving electromagnetic wave scattering are included to demonstrate the computational performance that can be achieved with a co volume time domain solution algorithm on these meshes

krawtchouk polynomials the lie algebra sl2 and leonard pairs abstract a leonard pair is a pair of diagonalizable linear transformations of a finite dimensional vector space each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one in the present paper we give an elementary but comprehensive account of how the following are related i krawtchouk polynomials ii finite dimensional irreducible modules for the lie algebra sl 2 iii a class of leonard pairs said to have krawtchouk type along the way we obtain elementary proofs of some well known facts about krawtchouk polynomials such as the three term recurrence the orthogonality the difference equation and the generating function the paper is a tutorial meant for a graduate student or a researcher unfamiliar with the above topics

the finite difference time domain fdtd and the finite volume time domain fvtd methods in solving maxwell s equations the finite difference time domain fdtd and its current generalizations have been demonstrated to be useful and powerful tools for the calculation of the radar cross section rcs of complicated objects the radiation of antennas in the presence of other structures and other applications the mathematical techniques for conformal fdtd have matured the primary impediments to its implementation are the complex geometries and material properties associated with the problem even under these circumstances fdtd is more flexible to implement because it is based on first principles instead of a clever mathematical trick this paper gives an account of some new results on conformal fdtd obtained by the authors and their associates at lockheed martin space company since 1988 the emphasis is on nonsmooth boundary condition simulation

validation of finite difference boundary condition models for solid mechanics applications five basic finite difference boundary condition models are validated using a process that embeds the boundary conditions in a larger problem with a known solution the boundary conditions validated are 1 fixed boundaries 2 straight boundaries 3 90 deg re entrant concave corners 4 curved boundaries

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

a stable and consistent generalization of the fdtd technique to nonorthogonal unstructured grids a new way to extend the finite difference time domain fdtd technique to nonorthogonal grids is described in this paper the method whose theoretical foundations are based on the finite integration technique and on recently extended affine theories can be proven to be both stable and consistent in order to show its benefits the nonorthogonal fdtd is applied to the analysis of a cylindrical cavity and to the characterization of biconical antennas for ultrawideband applications

a domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change a domain decomposition technique is proposed for the computation of the acoustic wave equation in which the bulk modulus and density fields are allowed to be discontinuous at the interfaces inside each subdomain the method presented coincides with the second order finite difference schemes traditionally used in geophysical modeling however the possibility of assigning to each subdomain its own space step makes numerical simulations much less expensive another interest of the method lies in the fact that its hybrid variational formulation naturally leads to exact equations for gridpoints on the interfaces transposing babuska brezzi s formalism on mixed and hybrid finite elements provides a suitable functional framework for this domain decomposition formulation and shows that the inf sup condition remains the basic requirement for convergence to occur

a three dimensional fdtd algorithm in curvilinear coordinates em scattering the finite difference time domain fdtd algorithm for the solution of electromagnetic scattering problems is formulated in numerically defined generalized coordinates in three dimensions and implemented in a code with the lowest order bayliss turkel radiation boundary condition expressed in spherical coordinates it is shown that the algorithm is capable of accurately tracking the progress of a pulse of electromagnetic radiation through the curvilinear mesh generated by a body of revolution the only problems occurring in the vicinity of the rotation axis which represents a coordinate singularity a simple method to deal with this singular line is presented and discussed and its is shown that at least for the test problem this approximation is sufficient the algorithm discussed is useful for the solution of the exterior problem in the presence of conductors and dielectrics with complicated shapes and electrical compositions and for near field problems such as cavity penetration problems the far fields are obtained by replacing the scatterer with a virtual surface enclosing all sources

fdtd simulations of tem horns and the implications for staircased representations two dimensional 2 d tem horns are modeled using the finite difference time domain fdtd method the boundary walls are perfect electric conductors and one wall which does not align with the cartesian grid is approximated using a staircased representation by carefully comparing the fdtd results to those of the analytic solution one can make conclusions about the coarseness with which a boundary can be represented it is found that staircasing errors are small when the staircase diagonal the hypotenuse of the right triangle created by the stairstep is smaller than half a wavelength at the highest significant frequency in the excitation this rule of thumb is put forward as a necessary condition for the discretization of general problems results are also provided for some simple fdtd schemes that are designed to reduce staircasing errors by using large aspect ratio cells a grid can be constructed that satisfies the rule of thumb given above while this approach eliminates general staircasing errors some errors persist owing to the presence of step discontinuities immediately adjacent to the horn feed these errors can be further reduced by using a cell splitting approach it is shown that the contour path fdtd technique can be used to eliminate nearly all staircasing errors while some additional improvement is shown to be provided by using a stabilized contour path fdtd approach finally a recently proposed conformal technique that permits simple implementation is shown to provide results comparable with those of the stabilized contour path approach

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

convergence analysis of a covolume scheme for maxwell s equations in three dimensions this paper contains error estimates for covolume discretizations of maxwell s equations in three space dimensions several estimates are proved first an estimate for a semi discrete scheme is given second the estimate is extended to cover the classical interlaced time marching technique third some of our unstructured mesh results are specialized to rectangular meshes both uniform and nonuniform by means of some additional analysis it is shown that the spatial convergence rate is one order higher than for the unstructured case

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

block pseudospectral methods for maxwell s equations ii two dimensional discontinuous coefficient case block pseudospectral bps methods are examined for maxwell s equations in two dimensional inhomogeneous media for the case of a rectangular strip with a straight line interface blocks may be coupled via fictitious points or a generalization of characteristic outflow conditions the bps methods generalize to curvilinear strips described by a change of variables such strips can conform to interfaces while overlapping with a high order free space grid to form a composite grid method numerical experiments on strips with dielectrics lossy materials perfect conductors and absorbing layers indicate that the two coupling methods are comparable and accurate with just 3 4 points per wavelength full composite examples are included to demonstrate high accuracy and geometric flexibility

a note on mass lumping in the finite element time domain method mass lumping has been used to produce explicit time integration schemes for the finite element time domain method in this paper we show that a specific choice in the lumping procedure can produce a scheme that is equivalent to the finite difference time domain method if lumping is done in the time integration process one can show that it can be equivalent to the newmark beta method

divergence preserving discrete surface integral methods for maxwell s curl equations using non orthogonal unstructured grids abstract several new discrete surface integral methods for solving maxwell s equations in the time domain are presented these methods which allow the use of general non orthogonal mixed polyhedral unstructured grids are direct generalizations of the canonical staggered grid finite difference method these methods are conservative in that they locally preserve divergence or charge employing mixed polyhedral cells hexahedral tetrahedral etc these methods allow more accurate modeling of non rectangular structures and objects because the traditional stair stepped boundary approximations associated with the orthogonal grid based finite difference methods can be avoided numerical results demonstrating the accuracy of these new methods are presented

convergence analysis of fully discrete finite volume methods for maxwell s equations in nonhomogeneous media we will consider both explicit and implicit fully discrete finite volume schemes for solving three dimensional maxwell s equations with discontinuous physical coefficients on general polyhedral domains stability and convergence for both schemes are analyzed we prove that the schemes are second order accurate in time both schemes are proved to be first order accurate in space for the voronoi delaunay grids and second order accurate for nonuniform rectangular grids we also derive explicit expressions for the dependence on the physical parameters in all estimates

on the role of polynomials in rbf fd approximations ii numerical solution of elliptic pdes rbf generated finite differences rbf fd have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of pdes we find in the present study that combining polyharmonic splines phs with multivariate polynomials offers an outstanding combination of simplicity accuracy and geometric flexibility when solving elliptic equations in irregular or regular regions in particular the drawbacks on accuracy and stability due to runge s phenomenon are overcome once the rbf stencils exceed a certain size due to an underlying minimization property test problems include the classical 2 d driven cavity and also a 3 d global electric circuit problem with the earth s irregular topography as its bottom boundary the results we find are fully consistent with previous results for data interpolation

numerical approximation of the maxwell equations in inhomogeneous media by ap1conforming finite element method the aim of this paper is to present an extension of thep1conforming finite element method for the time dependent maxwell equations that has been previously exposed we shall consider inhomogeneous media with the piecewise constant dielectric and magnetic parameters and precisely the interface of two such media by analogy with the idea developed we propose a method based on the dualization and then the approximation of the interface conditions in a way consistent with the one derived for the fields and the lagrange multipliers of the divergence constraints inside the domain

fdtd method on a lebedev grid for anisotropic materials the finite difference time domain method is derived on a lebedev grid for lossy anisotropic media the lebedev grid uses collocated field components and supports spurious solutions but an intuitive method for removing the extra solutions is presented update equations at material discontinuities and metal planes are derived and shown to take the same form as updates in bulk media additionally a dispersion relation and stability criteria are presented a numerical comparison with the yee grid shows that the lebedev grid suffers from greater numerical dispersion but better represents material discontinuities when compared using equal memory requirements furthermore the small stencil on the lebedev grid decreases the required number of calls to memory and simplifies the programming structure

high order finite difference methods for wave propagation in discontinuous media high order finite difference approximations are derived for the second order wave equation with discontinuous coefficients on rectangular geometries the discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion each sub domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method this guarantees a conservative strictly stable and high order accurate scheme the analysis is verified by numerical simulations in one and two spatial dimensions

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach

a conservative staggered grid chebyshev multidomain method for compressible flows we present a new multidomain spectral collocation method that uses a staggered grid for the solution of compressible flow problems the solution unknowns are defined at the nodes of a gauss quadrature rule the fluxes are evaluated at the nodes of a gauss lobatto rule the method is conservative free stream preserving and exponentially accurate a significant advantage of the method is that subdomain corners are not included in the approximation making solutions in complex geometries easier to compute

residue codes of extremal type ii mathbb z _4 codes and the moonshine vertex operator algebra in this paper we study the residue codes of extremal type ii mathbb z 4 codes of length 24 and their relations to the famous moonshine vertex operator algebra voa the main result is a complete classification of all residue codes of extremal type ii mathbb z 4 codes of length 24 some corresponding results associated to the moonshine voa are also discussed

a 3 d tensor fdtd formulation for treatment of sloped interfaces in electrically inhomogeneous media a general integral finite difference time domain fdtd formulation on cubical grids for the modeling of electrically inhomogeneous media of arbitrary shape is derived the simple cubical structure is maintained and no modifications of the integral paths are necessary instead a three dimensional tensor relationship between the average electric flux density and electric field is determined beforehand and used during the simulation to account for discontinuities in the neighborhood of sloped interfaces the accuracy of the method is confirmed by comparisons with analytical solutions

a meshless local petrov galerkin method for the time dependent maxwell equations abstract in this paper the meshless local petrov galerkin mlpg method is employed to solve the 2 d time dependent maxwell equations the mlpg method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces in the current work the moving least square reproducing kernel mlsrk scheme is chosen to be the trial function the method is applied for the unsteady maxwell equations in different media in the local weak form by employing the difference operator for evolution in time and simultaneously in time and space the semi discrete and fully discrete schemes are obtained respectively the error estimation is discussed for both the semi discrete and fully discrete numerical schemes for modelling the time dependent maxwell equations we show that provided that the time step size is sufficiently small the proposed scheme yields an error of o 2 m 1 2 in the l 2 norm for the square of error the new scheme is implemented and the numerical results are provided to justify our theoretical analysis

improved finite difference method for equilibrium problems based on differentiation of the partial differential equations and the boundary conditions a numerical algorithm for producing high order solutions for equilibrium problems is presented the approximated solutions are improved by differentiating both the governing partial differential equations and their boundary conditions the advantages of the proposed method over standard finite difference methods are the possibility of using arbitrary meshes the possibility of using simultaneously approximations with different distinct orders of accuracy at different locations in the problem domain an improvement in approximating the boundary conditions the elimination of the need for fictitious or external nodal points in treating the boundary conditions furthermore the proposed method is capable of reaching approximate solutions which are more accurate than other finite difference methods when the same number of nodal points participate in the local scheme a computer program was written for solving two dimensional problems in elasticity the solutions of a few examples clearly illustrate these advantages

numerical dispersion and stability characteristics of time domain methods on nonorthogonal meshes the familiar finite difference time domain method for discretizing maxwell s curl equations on orthogonal grids has been extended to nonorthogonal grids by a number of researchers while it is difficult to determine the dispersion and stability characteristics of these methods when applied on arbitrary grids analysis of the idealized but representative case of a uniform skewed mesh proves to be quite tractable in 2 d this analysis demonstrates that numerical dispersion errors are small for well resolved spatial wavelengths and that these methods converge to the continuous space solution in the limit as the cell and time step sizes vanish grid anisotropy variations in wave propagation speed as a function of the propagation angle relative to the mesh coordinates increases as the mesh is skewed in spite of this there exist some angles where waves propagate through the skewed mesh with virtually no dispersion this analysis also provides a stability limit for the time step size in terms of geometrical mesh quantities

an explicit fourth order staggered finite difference time domain method for maxwell s equations a new explicit fourth order accurate staggered finite difference time domain fdtd scheme is proposed and applied to electromagnetic wave problems it is fourth order accurate in both space and time conditionally stable and highly efficient with respect to yee s scheme and still retains much of the original simplicity of yee s scheme both extension to perfectly matched layers and modification to deal with dielectric interfaces and perfectly conducting boundaries of the scheme have also been presented numerical examples are shown to illustrate the efficiency of the method