fully hierarchical divergence conforming basis functions on tetrahedral cells with applications a new set of hierarchical divergence conforming vector basis functions on curvilinear tetrahedrons is presented the basis can model both mixed and full order polynomial spaces to arbitrary order as defined by raviart and thomas and nedelec solenoidal and non solenoidal components are separately represented on the element except in the case of the mixed first order space for which a decomposition procedure on the global mesh wide level is presented therefore the hierarchical aspect of the basis can be made to extend down to zero polynomial order the basis can be used to model divergence conforming quantities such as electromagnetic flux and current density fluid velocity etc within numerical methods such as the finite element method fem or integral equation based methods the basis is ideally suited to p adaptive analysis the paper concludes with two example applications the first is the fem based solution of the linearized acoustic vector wave equation where it is shown how the decomposition into solenoidal components and their complements can be used to stabilize the method at low frequencies the second is the solution of the electric field volume integral equation for electromagnetic scattering analysis where the benefits of the decomposition are again demonstrated

metrics and isospectral partners for the most generic cubic pt symmetric non hermitian hamiltonian we investigate properties of the most general symmetric non hermitian hamiltonian of cubic order in the annihilation and creation operators as a ten parameter family for various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and moyal products we elaborate on the subtleties of this approach for special choices of the ten parameters the hamiltonian reduces to various models previously studied such as to the complex cubic potential the so called swanson hamiltonian or the transformed version of the from below unbounded quartic x4 potential in addition it also reduces to various models not considered in the present context namely the single site lattice reggeon model and a transformed version of the massive sextic x6 potential which plays an important role as a toy model to identify theories with vanishing cosmological constant

charge iteration a procedure for the finite element computation of unbounded electrical fields an iterative procedure is presented for the finite element computation of unbounded electrical fields created by voltaged conductors the procedure is based on successive evaluations of the potential on a fictitious boundary enclosing all the conductors according to the charge lying on their surface the convergence of the procedure to the solution of the unbounded field problem is demonstrated indications for the optimal placement of the fictitious boundary are provided also taking into account the accuracy of the solution the way in which computational efficiency can be reached is also discussed the main advantage of this procedure lies in its simplicity of implementation in the context of a standard fe code for bounded problems because a very limited amount of additional software is required moreover 2 d and axisymmetric versions can be implemented with minor changes from a suitable 3 d one examples of application are given in order to illustrate the practical use of the procedure and to validate it by comparisons with available solutions

curvilinear vector finite elements using a set of hierarchical basis functions curvilinear finite elements are developed employing a second order geometrical mapping and implemented for a set of hierarchical vector basis functions of up to full second order the development covers the computation of the mass and stiffness matrices and the geometrical transformation from the real curvilinear element to the rectilinear reference element with the basis functions and their curls tabulated in reference coordinates up to full second order the postprocessing of results for curvilinear elements is also discussed the use of these elements is illustrated by applying them to scattering and eigenanalysis problems involving conducting spheres the improvement in the accuracy of the solution is demonstrated by comparing the error in the results obtained using curvilinear elements to those obtained using rectilinear elements

maximally orthogonalized higher order bases over generalized wires quadrilaterals and hexahedra this paper presents a general theory of maximally orthogonalized div and curl conforming higher order basis functions hobfs over generalized wires quadrilaterals and hexahedra in particular all elements of such bases necessary for fast and easy implementation are listed up to order n 8 numerical results given for div conforming bases applied in an iterative method of moments solution of integral equations show that the condition number and the number of iterations are a much lower than in the case of other hobfs of polynomial type and b practically not dependent on the applied expansion order

a higher order parallelized multilevel fast multipole algorithm for 3 d scattering a higher order multilevel fast multipole algorithm mlfma is presented for solving integral equations of electromagnetic wave scattering by three dimensional 3 d conducting objects this method employs higher order parametric elements to provide accurate modeling of the scatterer s geometry and higher order interpolatory vector basis functions for an accurate representation of the electric current density on the scatterer s surface this higher order scheme leads to a significant reduction in the mesh density thus the number of unknowns without compromising the accuracy of geometry modeling it is applied to the electric field integral equation efie the magnetic field integral equation mfie and the combined field integral equation cfie using galerkin s testing approach the resultant numerical system of equations is then solved using the mlfma appropriate preconditioning techniques are employed to speedup the mlfma solution the proposed method is further implemented on distributed memory parallel computers to harness the maximum power from presently available machines numerical examples are given to demonstrate the accuracy and efficiency of the method as well as the convergence of the higher order scheme

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

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

solving the volume integral equations of electromagnetic scattering time harmonic electromagnetic scattering by inhomogeneous three dimensional structures within a free space environment can be described by electric and magnetic field volume integral equations involving the free space green function a comprehensive set of galerkin projection formulations also known as moment methods for the numerical solution of these equations is presented together with comparative numerical results such formulations are widely used for particle scattering analysis optical near field calculation etc results are obtained with higher order divergence and curl conforming basis functions on iso parametric tetrahedral meshes the results demonstrate that all formulations converge with similar accuracy in the case of an analytically solvable test problem when modeling flux densities as solution variables it is argued that solenoidal function spaces should be used rather than the standard divergence conforming function spaces this assertion is supported by the results some of the formulations involve solving for curl conforming fields such fields can be discretized with fewer unknowns than divergence conforming ones implying lower computational costs additionally some formulations yield system matrices which are approximately halfway sparse meaning that computational costs will be down by a factor of 2 when iterative solvers are employed which is the case for the widely used fast methods

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

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

transient analysis of lumped circuit networks loaded thin wires by dgtd method with the purpose of avoiding very fine mesh cells in the proximity of a thin wire the modified telegrapher s equations mtes are employed to describe the thin wire voltage and current distributions which consequently results in reduced number of unknowns and augmented courant friedrichs lewy cfl number as hyperbolic systems both the mtes and the maxwell s equations are solved by the discontinuous galerkin time domain dgtd method in realistic situations the thin wires could be either driven or loaded by circuit networks the thin wire circuit interface performs as a boundary condition for the thin wire solver where the thin wire voltage and current used for the incoming flux evaluation involved in the dgtd analyzed mtes are not available to obtain this voltage and current an auxiliary current flowing through the thin wire circuit interface is introduced at each interface corresponding auxiliary equations derived from the invariable property of characteristic variable for hyperbolic systems are developed and solved together with the circuit equations established by the modified nodal analysis mna modality furthermore in order to characterize the field and thin wire interactions a weighted electric field and a volume current density are added into the mtes and maxwell ampere s law equation respectively to validate the proposed algorithm three representative examples are presented

a structured grid finite element method using computed basis functions structured e g rectangular meshes in finite element fe analysis offer several advantages but are not often used because of the error introduced in fitting material interfaces the nonconforming voxel fe method nvfem is a hierarchical structured method that systematically reduces the geometric error in an adaptive way but it requires large numbers of elements and therefore is expensive by computing basis functions that take into account the interface cutting through an element the number of elements needed is greatly reduced the resulting method cbf nvfem is applied to the computation of the radar cross section of free space scatterers results indicate that cbf nvfem is much more efficient than nvfem and is competitive with fem using unstructured meshes of tetrahedra

an evaluation of mixed order versus full order vector finite elements recently proposed vector finite elements are compared to previously published elements and independently implemented and validated in the context of rectangular waveguide analysis both mixed order and full order elements up to order two are considered the use of quadrature cubature to evaluate the finite element system matrices is discussed details of some analytical work required before quadrature is applied are presented for several types of basis functions for many rf applications mixed order elements offer optimal accuracy for a given number of degrees of freedom and an empty through waveguide solution verifies this with similar convergence rates for mixed and full order elements of the same order however a capacitive iris in the rectangular waveguide with a solution dominated by the quasi static electric field motivates the use of full order elements in such cases results for the electric fields in the vicinity of the iris also demonstrate the enhanced accuracy of the full order elements here this is further illustrated using a dielectric load in a waveguide where complete order elements produce a better approximation of the normal jump discontinuity although this paper addresses guided wave problems the results are also applicable to radiation and scattering problems

symmetric versus hermitian formulations of quantum mechanics a non hermitian hamiltonian that has an unbroken symmetry can be converted by means of a similarity transformation to a physically equivalent hermitian hamiltonian this raises the following question in which form of the quantum theory the non hermitian or the hermitian one is it easier to perform calculations this paper compares both forms of a non hermitian ix3 quantum mechanical hamiltonian and demonstrates that it is much harder to perform calculations in the hermitian theory because the perturbation series for the hermitian hamiltonian is constructed from divergent feynman graphs for the hermitian version of the theory dimensional continuation is used to regulate the divergent graphs that contribute to the ground state energy and the one point green s function the results that are obtained are identical to those found much more simply and without divergences in the non hermitian symmetric hamiltonian the contribution to the ground state energy of the hermitian version of the theory involves graphs with overlapping divergences and these graphs are extremely difficult to regulate in contrast the graphs for the non hermitian version of the theory are finite to all orders and they are very easy to evaluate

general aspects of pt symmetric and p self adjoint quantum theory in a krein space in our previous work we proposed a mathematical framework for symmetric quantum theory and in particular constructed a krein space in which symmetric operators would naturally act in this work we explore and discuss various general consequences and aspects of the theory defined in the krein space not only spectral properties and symmetry breaking but also several issues crucial for the theory to be physically acceptable such as time evolution of state vectors probability interpretation uncertainty relation classical quantum correspondence completeness existence of a basis and so on in particular we show that for a given real classical system we can always construct the corresponding symmetric quantum system which indicates that symmetric theory in the krein space is another quantization scheme rather than a generalization of the traditional hermitian one in the hilbert space we propose a postulate for an operator to be a physical observable in this framework

gradient singular hierarchical finite elements for vector electromagnetics new tetrahedral finite elements are described for the analysis of vector electromagnetic fields they are singular elements that are more accurate than conventional polynomial based elements near sharp edges and corners they are hierarchical meaning that elements of different orders accuracies are available and may be placed together in the same mesh without violating continuity of the field the elements are formed from a series of scalar singular elements by taking the gradient and are therefore called gradient singular results using the new elements in p adaptive analysis orders 1 to 3 demonstrate that when they are used in place of conventional elements the scattering parameters are as much as 10 times more accurate for the same number of unknowns

pseudo hermitian versus hermitian position dependent mass hamiltonians in a perturbative framework we formulate a systematic algorithm for constructing a whole class of hermitian position dependent mass hamiltonians which to lowest order of perturbation theory allow a description in terms of symmetric hamiltonians the method is applied to the hermitian analogue of the symmetric cubic anharmonic oscillator a new example is provided by a hamiltonian approximately equivalent to a symmetric extension of the one parameter trigonometric poschl teller potential

numerical characterization of divergence conforming constrained basis functions for surface integral equations this paper presents an algebraic method of generating arbitrary order basis functions suitable for use in moment methods the basis functions are constructed by enforcing a set of suitable constraints on a more general set of functions leading to a matrix representation of the constraint condition the desired bases are then found through use of the singular value decomposition divergence conforming constrained bases are presented for quadrilateral meshes for use in electric and magnetic field integral equations appropriate constraints for enforcing normal continuity at cell boundaries are discussed a variety of problems are analyzed using the constrained bases and results show that the constrained bases exhibit exponential convergence for smooth structures and produce systems whose condition numbers grow relatively slowly with increasing basis order

analysis of a plane wave excited subwavelength circular aperture in a planar conducting screen illuminating a multilayer uniaxial sample a spectral domain analysis is presented of a plane wave excited subwavelength circular aperture in a planar perfectly conducting screen in close proximity to a multilayer stack which may comprise uniaxially anisotropic slices the formulation employs the bethe bouwkamp aperture model for the general case of oblique incidence and it combines the hankel transform method with a transmission line network analogue of the medium which facilitates the inclusion of any number of material layers the electromagnetic fields are expressed in terms of one dimensional 1 d hankel transform integrals which are efficiently evaluated by a numerical quadrature along a suitably selected path with convergence acceleration by extrapolation sample numerical results are presented to demonstrate the validity of the method and its utility in the modeling of multilayer near field lenses and aperture type probes of near field optical microscopy

maximally orthogonalized higher order basis functions in large domain finite element modeling in electromagnetics curl conforming max ortho basis functions mobfs are coupled with higher order large domain curved finite elements fes the performance of the functions is compared with that of the classical and near ortho basis functions through numerical experiments it is shown that max ortho fes yield highly orthogonal mass matrices for practically arbitrarily high orders of polynomial field approximations this facilitates the usage of iterative solvers and it significantly increases their efficiency accurate and fast computation of mobfs of arbitrarily high orders is enabled by the proposed two term recurrent formula

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

a hybrid fem mom method for 3 d electromagnetic scattering in layered medium accurate and efficient prediction of electromagnetic scattering from inhomogeneous objects in layered medium is one of the most challenging issues in engineering applications this paper presents the first 3 d higher order hybrid finite element method fem and method of moments mom for the accurate modeling of inhomogeneous dielectric objects in multilayered medium the main challenges of this paper include 1 the integration of these algorithms for layered medium and 2 the higher order computational approach involved in layered medium for high efficiency in the proposed method the mom with the layered medium dyadic green s function is used as the exact radiation boundary condition in an inhomogeneous background and the fem is applied to model the inhomogeneous objects furthermore the higher order maximally orthogonal basis functions with curl conforming and divergence conforming properties are used in the fem and mom respectively to improve the modeling capability of this algorithm for 3 d inhomogeneous objects scattering in multilayered medium this new method requires a much more tightly truncated simulation domain than the traditional fem and provides much higher flexibility than the pure surface integral equation method finally some numerical results are provided to validate the accuracy efficiency and flexibility of this method

construction of nearly orthogonal nedelec bases for rapid convergence with multilevel preconditioned solvers this paper presents a systematic approach to constructing high order tangential vector basis functions for the multilevel finite element solution of electromagnetic wave problems the new bases allow easy computation of a preconditioner to eliminate or at least weaken the indefiniteness of the system matrix and thus reduce the condition number of the system matrix when these bases are used in multilevel solutions where the multilevels correspond to the order of the basis functions the resulting p multilevel ilu preconditioned conjugate gradient method mpcg provides an optimal rate of convergence we first derive an admissible set of vectors of order p and decompose this set into two subspaces rotational and irrotational gradient we then reduce the number of vectors by making them orthogonal to all previously constructed lower order bases the remaining vectors are made mutually orthogonal in both the vector space and in the range space of the curl operator the resulting vector basis functions provide maximum orthogonality while maintaining tangential continuity of the field the zeroth order space is further decomposed using a scalar vector formulation to eliminate convergence problems at extremely low frequencies numerical experiments show that number of iterations needed for the solution by mpcg is basically constant regardless of the order of the basis or of the matrix size computational speed is improved by several orders of magnitude due to the fast matrix solution of mpcg and to the high accuracy of the higher order bases

the implicit element residual method for a posteriori error estimation in fe bi analysis hybrid finite element boundary integral fe bi formulations are widely used for electromagnetic analysis in such analysis estimation of the mesh wide error distribution and global solution quality may be used to drive adaptive refinements in this paper an a posteriori error estimator based on the implicit element residual method is constructed for the cavity fe bi formulation with dominant mode coaxial port excitation it is based on solving elemental variational boundary value problems of the error using approximate localized boundary conditions results show that the scheme can be applied to practical problems yielding significant reductions in computational cost

interior penalty discontinuous galerkin time domain method based on wave equation for 3 d electromagnetic modeling built on the vector wave equation an interior penalty discontinuous galerkin time domain ipdg we method is presented for 3 d real life electromagnetic modeling in this paper we develop the interior penalty fluxes corresponding to different types of boundary conditions which are commonly used in electromagnetic modeling to construct the ipdg we system in local elements in addition to model the incident wave excitation the total field scattered field approach is incorporated into the proposed method the stability of the ipdg we method is studied analytically and demonstrated numerically we also compare the performance between wave equation based finite element time domain method the maxwell s equation based discontinuous galerkin time domain method and the proposed method both in cpu time and memory usage which demonstrates that the ipdg we method is very competitive among time domain solvers besides the presented method shows good characteristics including the energy conservation and the optimal convergence rate of o h p 1 numerical examples are presented to illustrate the validity versatility and potential of the proposed method in 3 d electromagnetic problems modeling

four easy pieces this paper is a mini summary of four new advances in the theory of pt symmetric quantum mechanics it describes some new calculations that were completed in 2005 the first advance concerns the classical coordinate space trajectories in some pt symmetric theories depending on the initial conditions one can find arbitrarily long periodic pt symmetric classical trajectories the longer trajectories originate from smaller regions of initial conditions there is an interesting resemblance to the so called period doubling route to chaos the second advance concerns the perturbative construction of the c operator for a pt symmetric square well the result has notable similarities to and differences from that for the pt symmetric cubic oscillator the third advance is a detailed comparison of a pt symmetric hamiltonian and the corresponding hermitian hamiltonian it is argued that a perturbative construction of the hermitian hamiltonian leads to an almost intractable theory the fourth advance concerns reflectionless potentials and pt symmetry and a possible connection with problems in cosmology

hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements new vector finite elements are proposed for electromagnetics the new elements are triangular or tetrahedral edge elements tangential vector elements of arbitrary polynomial order they are hierarchal so that different orders can be used together in the same mesh and p adaption is possible they provide separate representation of the gradient and rotational parts of the vector field explicit formulas are presented for generating the basis functions to arbitrary order the basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning matrix assembly for the frequency domain curl curl equation is conveniently carried out by means of universal matrices application of the new elements to the solution of a parallel plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient for tetrahedral elements to order 4 in particular the full order elements have only the same asymptotic convergence rate as elements with a reduced gradient space such as the whitney element however further tests reveal that the optimum balance of the gradient and rotational components is problem dependent

curl conforming hierarchical vector bases for triangles and tetrahedra a new family of hierarchical vector bases is proposed for triangles and tetrahedra these functions span the curl conforming reduced gradient spaces of nedelec the bases are constructed from orthogonal scalar polynomials to enhance their linear independence which is a simpler process than an orthogonalization applied to the final vector functions specific functions are tabulated to order 6 5 preliminary results confirm that the new bases produce reasonably well conditioned matrices

analysis of periodic structures via a time domain finite element formulation with a floquet abc an accurate absorbing boundary condition abc is presented for three dimensional time domain finite element fem td analysis of infinitely periodic structures the proposed abc serves to truncate the computational domain in the nonperiodic directions and it is highly effective to absorb both the fundamental and the higher order floquet modes the formulation is applicable both for scattering and radiation analysis validation results are presented that demonstrate the ability of the fem td to obtain highly accurate results over a broad frequency band that includes frequencies where a higher order floquet mode is propagating nonevanescent

a 3 d interface enriched generalized fem for electromagnetic problems with nonconformal discretizations an interface enriched generalized finite element method igfem is introduced for efficient 3 d electromagnetic analysis of heterogeneous materials without using meshes that conform to the material microstructures which greatly lessens the burden of mesh generation the method assigns generalized degrees of freedom dofs at material interfaces to capture the discontinuities of the field and its derivatives the generalized dofs are supported by enriched vector basis functions vbfs which are constructed through a linear combination of the vbfs from the subelements several verification examples are provided to show that the igfem is not sensitive to the quality of the subelements and maintains the same level of solution accuracy and computational complexity as the standard finite element method fem based on conformal meshes the potential of the proposed igfem is demonstrated by simulating some engineering problems with complex periodic internal structures including composite materials with randomly distributed spherical particles and ellipsoidal inclusions and microvascular channels

an auxiliary differential equation formulation for the complex frequency shifted pml an efficient auxiliary differential equation ade form of the complex frequency shifted perfectly matched layer cpml absorbing media derived from a stretched coordinate pml formulation is presented it is shown that a unit step response of the ade cpml equations leads to a discrete form that is identical to roden s convolutional pml method for fdtd implementations the derivation of discrete difference operators for the ade cpml equations for fdtd is also presented the ade cpml method is also extended in a compact form to a multiple pole pml formulation the advantage of the ade cpml method is that it provides a more flexible representation that can be extended to higher order methods in this paper it is applied to the discontinuous galerkin finite element time domain dgfetd method it is demonstrated that the ade cpml maintains the exponential convergence of the dgfetd method

hierarchic finite element bases on unstructured tetrahedral meshes the problem of constructing hierarchic bases for finite element discretization of the spaces h1 h curl h div and l2 on tetrahedral elements is addressed a simple and efficient approach to ensuring conformity of the approximations across element interfaces is described hierarchic bases of arbitrary polynomial order are presented it is shown how these may be used to construct finite element approximations of arbitrary non uniform local order approximation on unstructured meshes of curvilinear tetrahedral elements

evidence of exponential convergence in the computation of maxwell eigenvalues abstract this paper describes a finite element approach for the approximation of maxwell eigenvalues for three dimensional cavity problems hierarchical h 1 and h curl conforming discretisations are employed in conjunction with a technique that reduces the size of the problem to be solved numerical results are given on both polyhedral and curvilinear domains these results indicate that exponential rates of convergence can be obtained with the appropriate combination of h and p refinements

efficient surface integral equation using hierarchical vector bases for complex em scattering problems a set of hierarchical divergence conforming vector basis functions based on curved triangular patches is presented for method of moment mom solutions of surface integral equations in this paper the higher order method of combined field integral equation cfie solves perfect electric conductor electromagnetic scattering problems the higher order electric magnetic current combined field integral equation jmcfie formulations solves dielectric objects and their combination is able to efficiently analyzes the scattering of hybrid pec dielectric objects the expressions of the divergence conforming hierarchical basis functions up to order 3 5 are reported in this paper the multilevel fast multipole algorithm mlfma is then employed to reduce the memory requirements and computational complexity numerical experiments indicate that the proposed hierarchical vector basis functions can provide well conditioned linear system for iterative solution

hierarchal triangular elements using orthogonal polynomials hierarchal elements are finite elements which have the useful property that elements with different polynomial orders can be used together in the same mesh without causing discontinuities this paper introduces a new hierarchal triangular element in which the basis functions are constructed from orthogonal polynomials jacobi polynomials the resulting element is shown to be better conditioned than the earlier hierarchal element of rossow and katz 1 recursive formulas allow the complete set of basis functions for an element to be efficiently evaluated at a given point in addition the formulas can be used to generate pre computed universal matrices examples are given of universal matrices up to order 4 for the generalized helmholtz equation an electromagnetic problem involving a length of transmission line is used to show the usefulness of the new elements

evaluation of a spherical pml for vector fem applications the implementation and evaluation of a spherical perfectly matched layer pml within a cartesian finite element method context using standard curl conforming elements is presented in this paper results are compared to the long standing 1st order absorbing boundary condition abc and a new rigorous implementation of a 2nd order abc for curl conforming elements the 4 and 8 layer spherical pmls are shown to offer very attractive levels of absorption with reflections on the order of 60 to 70db demonstrated numerical tests show that the guidelines for cartesian pml absorbers in terms of maximum conductivity also carry over to the spherical pml the 2nd order abc is also shown to offer very good performance finally coding issues for both the spherical pml and the analytical abcs are briefly addressed

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

high order divergence conforming constrained bases for triangular cells this paper presents an algebraic technique for generating arbitrary order divergence conforming bases for curvilinear triangular cells the bases are constructed by enforcing appropriate constraints on a linear combination of general functions and then extracting the desired bases using singular value decompositions koornwinder dubiner polynomials are chosen as the general function set basic constraints are presented to obtain divergence conforming bases and additional constraints are presented to further enforce the bases to be nedelec results from a variety of problems are given to show that the bases exhibit high order convergence and also produce systems that are relatively well conditioned compared to other basis sets

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

explicit time marching schemes for solving the magnetic field volume integral equation a method for constructing explicit marching on in time mot schemes to solve the time domain magnetic field volume integral equation td mfvie on inhomogeneous dielectric scatterers is proposed the td mfvie is cast in the form of an ordinary differential equation ode and the unknown magnetic field is expanded using curl conforming spatial basis functions inserting this expansion into the td mfvie and spatially testing the resulting equation yield an ode system with a gram matrix this system is integrated in time for the unknown time dependent expansion coefficients using a linear multistep method the gram matrix is sparse and well conditioned for galerkin testing and consists of only four diagonal blocks for point testing the resulting explicit mot schemes which call for the solution of this matrix system at every time step are more efficient than their implicit counterparts which call for inversion of a fuller matrix system at lower frequencies numerical results compare the efficiency accuracy and stability of the explicit mot schemes and their implicit counterparts for low frequency excitations the results show that the explicit mot scheme with point testing is significantly faster than the other three solvers without sacrificing from accuracy

exact pt symmetry is equivalent to hermiticity we show that a quantum system possessing an exact antilinear symmetry in particular pt symmetry is equivalent to a quantum system having a hermitian hamiltonian we construct the unitary operator relating an arbitrary non hermitian hamiltonian with exact pt symmetry to a hermitian hamiltonian we apply our general results to pt symmetry in finite dimensions and give the explicit form of the above mentioned unitary operator and hermitian hamiltonian in two dimensions our findings lead to the conjecture that non hermitian cpt symmetric field theories are equivalent to certain nonlocal hermitian field theories

adaptive multiresolution antenna modeling using hierarchical mixed order tangential vector finite elements hierarchical mixed order tangential vector finite elements tvfes for tetrahedra are attractive for accurate and efficient finite element or hybrid finite element boundary integral fe bi simulation of complicated electromagnetic problems they provide versatility in the geometrical modeling of physical structures guarantee solutions free of spurious modes and allow a local increase of resolution by combination of mixed order tvfes of different orders within a computational domain regions with higher order mixed order tvfes can be selected a priori or be found adaptively via methods for a posteriori error estimation or indication this paper demonstrates the merits of various explicit residual methods for a posteriori error indication via adaptive refinement of fe bi solutions of electromagnetic radiation problems using hierarchical mixed order tvfes for tetrahedra

pt symmetric cubic anharmonic oscillator as a physical model there is a factor of 2 error in equation 61 of this paper correcting this error leads to minor changes in equations 62 and 63 please see pdf for details

small ϵ behavior of the non hermitian symmetric hamiltonian h p2 x2 ix ϵ the energy eigenvalues of the class of non hermitian pt symmetric hamiltonians h p 2 x 2 ix 0 are real positive and discrete the behavior of these eigenvalues has been studied perturbatively for small however until now no other features of h have been examined perturbatively in this paper the small expansion of the c operator and the equivalent isospectral dirac hermitian hamiltonian h are derived

optimal modeling parameters for higher order mom sie and fem mom electromagnetic simulations general guidelines and quantitative recipes for adoptions of optimal higher order parameters for computational electromagnetics cem modeling using the method of moments and the finite element method are established and validated based on an exhaustive series of numerical experiments and comprehensive case studies on higher order hierarchical cem models of metallic and dielectric scatterers the modeling parameters considered are electrical dimensions of elements h refinement polynomial orders of basis functions p refinement orders of gauss legendre integration formulas integration accuracy and geometrical curvature orders of elements in the model the goal of the study which is the first such study of higher order parameters in cem is to reduce the dilemmas and uncertainties associated with the great modeling flexibility of higher order elements basis and testing functions and integration procedures this flexibility is the principal advantage but also the greatest shortcoming of the higher order cem and to ease and facilitate the decisions to be made on how to actually use them by both cem developers and practitioners

incorporation of a feed network into the time domain finite element modeling of antenna arrays an accurate and efficient numerical scheme is presented for incorporating a feed network into the time domain finite element modeling of antennas and antenna arrays the feed network is represented in terms of a rational function form of its scattering matrix in the frequency domain that enables its interfacing with the time domain finite element modeling of the array through a fast recursive time convolution algorithm the exchange of information between the antenna elements and the feed network occurs through the incident and reflected modal voltages currents at properly defined port interfaces the proposed numerical scheme significantly simplifies the simulation of an antenna system that comprises both antennas and a feed network and allows one to fully utilize the power of an antenna simulation technique to deal with large and complex antennas several examples are presented to demonstrate the accuracy and efficiency of the method

higher order brick tetrahedron hybrid method for maxwell s equations in time domain we present a higher order brick tetrahedron hybrid method for maxwell s equations in time domain brick shaped elements are used for large homogeneous parts of the computational domain where we exploit mass lumping and explicit time stepping in regions with complex geometry we use an unstructured mesh of tetrahedrons that share an interface with the brick shaped elements and at the interface tangential continuity of the electric field is imposed in the weak sense by means of nitsche s method implicit time stepping is used for the tetrahedrons together with the interface for cavity resonators the hybrid method reproduces the lowest non zero eigenvalues with correct multiplicity and for geometries without field singularities from sharp corners or edges the numerical eigenvalues converge towards the analytical result with an error that is approximately proportional to h 2 p where h is the cell size and p is the polynomial order of the elements for a rectangular waveguide a layer of tetrahedrons embedded in a grid of brick shaped elements yields a low reflection coefficient that scales approximately as h 2 p finally we demonstrate hybrid time stepping for a lossless closed cavity resonator where the time domain response is computed for 300 000 time steps without any signs of instabilities new hybrid fem interface between tetrahedrons and bricks for maxwell s equations nitsche s method is applied to higher order curl conforming basis functions allows for both incomplete and complete order basis functions on tetrahedrons stable hybrid explicit implicit time stepping scheme computational results up to fourth order curl conforming basis functions

nedelec spaces in affine coordinates in this note we provide a conveniently implementable basis for simplicial nedelec spaces of any order in any space dimension the main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex

spectral design for matrix hamiltonians different methods of constructing of a matrix intertwining operator we study relations describing the intertwining of n n matrix in general non hermitian one dimensional hamiltonians by matrix linear differential operators with nondegenerate coefficients of in the highest degree some methods for constructing matrix intertwining operators of the first order of general form are proposed and their interrelations are examined we construct as an example a 2 2 matrix hamiltonian of general form intertwined by an operator of the first order with the hamiltonian with zero matrix potential it is shown that one can add for the final 2 2 matrix hamiltonian with respect to the initial matrix hamiltonian with the help of an intertwining operator of first order up to two bound states for different energy values or up to two bound states described by vector eigenfunctions for the same energy value or up to two bound states described by a vector eigenfunction and an associated vector function for the same energy value

a posteriori error estimates for maxwell equations maxwell equations are posed as variational boundary value problems in the function space h curl and are discretized by nedelec finite elements in beck et al 2000 a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain in the present paper we prove the reliability of that error estimator on lipschitz domains the key is to establish new error estimates for the commuting quasi interpolation operators recently introduced in j schoberl commuting quasi interpolation operators for mixed finite elements similar estimates are required for additive schwarz preconditioning to incorporate boundary conditions we establish a new extension result

second order nedelec curl conforming prismatic element for computational electromagnetics a systematic approach to obtaining mixed order curl conforming basis functions for a triangular prism is presented focus is made on the second order case space of functions for the prism is given basis functions are obtained as the dual basis with respect to suitably discretized nedelec degrees of freedom functionals acting on elements of the space thus the linear independence of the basis functions is assured while the belonging of the basis to the a priori given space of functions is guaranteed different strategies for the finite element assembly of the basis are discussed numerical results showing the verification procedure of the correctness of the implemented basis functions are given numerical results about sensibility of the condition number of the basis obtained concerning the quality of the elements of the mesh are also shown comparison with other representative sets of basis functions for prisms is included

goal oriented error estimation and h adaptivity for maxwell s equations goal oriented error estimates based on dual solutions are derived for the s parameters of a waveguide cavity resonator and a computationally cheap method is proposed to compute these numerical results show that the error estimators are relatively accurate usually well within a factor 2 from the exact errors adaptive mesh refinement based on these estimates recovers optimal convergence rates for complete and incomplete first and second order curl conforming finite elements of nedelec type despite singularities at reentrant corners it is also shown numerically that the convergence rate is independent of the element order when singularities are present and the mesh is refined uniformly

an explicit a posteriori error indicator for electromagnetic finite element boundary integral analysis within the finite element method fem there is always an error associated with the solution quantifying this error is a very important aspect of fe analysis since it can increase confidence in a solution and can be used to drive adaptive refinements of the discretization optimizing the quality of the solution relative to the number of degrees of freedom and in turn reducing computational cost this paper presents an a posteriori error indicator for time harmonic electromagnetic analysis in three dimensions when using a galerkin fe approach with curl conforming elements and including a nonlocal boundary integral boundary condition as well as a dominant mode port boundary condition the error indicator is residual based of an explicit nature and constitutes a bound on an error energy measure the error indicator is evaluated by presenting results relating to common microwave engineering applications which shows that it can be used successfully to estimate the relative error distribution and that it can successfully serve within an adaptive algorithm

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

a stable and efficient direct time integration of the vector wave equation in the finite element time domain method for dispersive media the finite element time domain fetd solution of the vector wave equation vwe is directly extended to doubly dispersive media using the bilinear transform approach the proposed formulation is quite general and flexible in the sense of material dispersion model in contrast to limitations of existing convolutional based approaches in addition it can be implemented in a very straightforward and efficient manner a stability analysis is performed that demonstrates the unconditional stability of the original formulation is preserved in the dispersive case however in order to avoid the well known late time instabilities that arise in this formulation an alternative formulation is considered and extended to the dispersive case with the same approach several numerical examples are simulated to verify the validity and accuracy of the proposed formulations the late time stability of the alternative formulation is tested through a numerical example with 20 million time steps the solution is completely free from late time growth

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

on the low order partial fraction fitting of dielectric functions at optical wavelengths low order partial fraction pf fitting of optical dielectric functions from tabulated frequency data is investigated two rational function identification techniques are examined viz the sanathanan koerner iteration and vector fitting and it is found that these methods generate pf models of similar quality and both provide excellent starting solutions for further improvement by nonlinear optimizers high quality wide band pf models are obtained for several metals frequently encountered in plasmonic applications these models are stable and well suited for the finite difference time domain fdtd simulation of light interaction with nanostructures an efficient fdtd updating scheme for pf media is also demonstrated

parametric identification of transfer functions in the frequency domain a survey this paper gives a survey of frequency domain identification methods for rational transfer functions in the laplace s or z domain the interrelations between the different approaches are highlighted through a study of the equivalent cost functions the properties of the various estimators are discussed and illustrated by several examples

analysis of volume integral equation formulations for scattering by high contrast penetrable objects the volume integral equation method is applied in electromagnetic scattering from arbitrarily shaped three dimensional inhomogeneous objects the properties of the volume electric and magnetic field integral equations vefie and vmfie are investigated numerical experiments show that if the galerkin s method with the lowest mixed order basis functions is used to discretize the equations the accuracy of the vmfie can be significantly poorer than the accuracy of the vefie in particular for high contrast objects at high frequencies the accuracy of the vmfie can be essentially improved with full first order linear basis functions the linear basis functions are found to be useful also when a single volume integral equation is used to model a general scatterer where both permittivity and permeability differ from the background

penalty factor threshold and time step bound estimations for discontinuous galerkin time domain method based on helmholtz equation in this article penalty factor threshold and time step bound in discontinuous galerkin time method based on vector wave equation dgtd we method are well estimated based on the semidiscrete form of the dgtd we method properties of the system matrices are studied and the stability condition related to the penalty factor is derived by decomposing the global system matrices of the dgtd we method into the local ones and developing an efficient iteration procedure the lower bound threshold of the penalty factor is well estimated to guarantee the positive semidefinite property of the global system matrices with the calculated penalty factor the maximum time step is analytically determined by approximating spectral radius of the local system matrix both the penalty factor bound and the maximal time step are computed element wise instead of a global system matrix operation and thus the proposed method can be efficiently applied into the large scale meshes with different types of the basis functions and boundary conditions numerical examples are presented to demonstrate the validity and good performance of the proposed methods

a p mus preconditioner for the efie this paper considers the solution of the electric field integral equation efie in electromagnetics as with associated finite element methods their solution relies upon the construction of conforming bases while lowest order rwg spaces are near ubiquitous their extension to higher order offers potentially a number of benefits in terms of accuracy and efficiency which has been well documented in both finite elements and integral equation formulations a further evolution of higher order conforming bases is the hierarchical basis these have demonstrated considerable gains in efficiency in finite element applications such bases allow for the development of effective acceleration schemes for instance the multilevel schwarz type preconditioner p mus an obvious question arises as to the applicability of such hierarchical bases and their associated acceleration schemes to integral equations it is seen that the conclusions as to their efficacy depend strongly on the scattering regime in particular high frequency problems those where the wavelength is the principal determinant of mesh size are shown to benefit little from hierarchical functions on the other hand for low frequency problems where geometry is the main determinant of mesh size there are significant improvements in performance over corresponding interpolatory schemes

geometric decompositions and local bases for spaces of finite element differential forms abstract we study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions these spaces are generalizations of the classical finite element spaces for vector fields frequently referred to as raviart thomas brezzi douglas marini and nedelec spaces in the present paper we derive geometric decompositions of these spaces which lead directly to explicit local bases for them generalizing the bernstein basis for ordinary lagrange finite elements the approach applies to both families of finite element spaces for arbitrary polynomial degree arbitrary order of the differential forms and an arbitrary simplicial triangulation in any number of space dimensions a prominent role in the construction is played by the notion of a consistent family of extension operators which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

effective and practical h p version adaptive analysis procedures for the finite element method two practical and effective h p type finite element adaptive procedures are presented the procedures allow not only the final global energy norm error to be well estimated using hierarchic p refinement but in addition give a nearly optimal mesh the design of this is guided by the local information computed on the previous mesh the desired accuracy can always be obtained within one or at most two h p refinements the rate of convergence of the adaptive h p version analysis procedures has been tested for some examples and found to be very strong the presented procedures can easily be incorporated into existing p or h type code structures

computation of resonant modes for axisymmetric maxwell cavities using hp version edge finite elements the computation of the resonant frequencies for closed cavities is not a trivial task multi materials and sharp corners all give rise to highly singular eigenfunctions however an approach using hp finite elements is well suited to such problems and with the correct combination of h and p refinements it yields the theoretically predicated exponential rates of convergence in this paper we present a novel approach to the solution of axisymmetric cavity problems which uses a hierarchic h1 and h curl conforming finite element basis a selection of numerical examples is included and these demonstrate that the exponential rates of convergence are achieved in practice copyright 2005 john wiley sons ltd

universal finite element matrices for tetrahedra methods are described for forming finite element matrices for a wide variety of operators on tetrahedral finite elements in a manner similar to that previously employed for line segments and triangles this technique models the differentiation and product embedding operators as rectangular matrices and produces finite element matrices by replacing all required analytic operations by their finite matrix analogues the method is illustrated by deriving the conventional matrix representation for laplace s equation brief computer programs are given which generate universal finite element matrices for use in various applications

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

a differential quadrature hierarchical finite element method using fekete points for triangles and tetrahedrons and its applications to structural vibration abstract a differential quadrature hierarchical finite element method dqhfem using fekete points was formulated for triangles and tetrahedrons and applied to structural vibration analyses first orthogonal polynomials on triangles and tetrahedrons that can be used as bases of the hierarchical finite element method hfem were derived and simple formulas of transforming one dimensional non uniform nodes to simplexes were presented then the non uniform nodes were used as initial guesses to solve the fekete points on simplexes through newton raphson s method together with the orthogonal polynomials new differential quadrature dq rules on simplexes were formulated using the hfem bases and the fekete points the numbers of nodes or bases on different edges and faces and inside the body of the new dq elements do not relate with each other like the hfem that can freely assign different numbers of bases on different edges and faces and inside the body so the new dq method was named as a differential quadrature hierarchical dqh method that uses either interpolation functions or orthogonal polynomials as bases inside the element its weak form was named as the dqhfem besides the dqh method and its weak form a simple method of generating high quality linear and high order triangular and tetrahedral meshes from a single nurbs patch was presented numerical tests of the dqhfem through structural vibration analyses showed that high accuracy results can be obtained using only a few nodes even on curvilinear domains using the dqh bases on both physical and geometric fields it was concluded that wide applications of the dqh method and the dqhfem to science and engineering are possible and commercial codes based on them are deserved to be developed