contribution of leaky modes in the modal analysis of unbounded problems with perfectly matched layers the modal analysis of wave problems of unbounded type involves a continuous sum of radiation modes this continuum is difficult to handle mathematically and physically it can be approximated by a discrete set of leaky modes corresponding to improper modes growing to infinity perfectly matched layers pmls have been widely applied in numerical methods to efficiently simulate infinite media most often without considering a modal approach this letter aims to bring insight into the modal basis computed with pmls pmls actually enable to reveal of the contribution of leaky modes by redefining the continua two for elastodynamics discretized after pml truncation

a krylov stability corrected coordinate stretching method to simulate wave propagation in unbounded domains the krylov subspace projection approach is a well established tool for the reducedorder modeling of dynamical systems in the time domain in this paper we address the main issues obstructing the application of this powerful approach to the time domain solution of exterior wave problems we use frequency independent perfectly matched layers to simulate the extension to infinity pure imaginary stretching functions based on zolotarev s optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest a new krylov based solution method via stabilitycorrected operator exponents is presented which allows us to construct reduced order models roms that respect the delicate spectral properties of the original scattering problem the roms are unconditionally stable and are based on a renormalized bi lanczos algorithm we give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate two dimensional electromagnetic wave propagation in unbounded domains including a photonic waveguide example the new algorithm outperforms the conventional finitedifference time domain method for problems on large time intervals

coherent families of weight modules of lie superalgebras and an explicit description of the simple admissible sl n 1 1 modules abstract let g be a classical lie superalgebra of type i we introduce coherent families of weight g modules with bounded weight multiplicities and establish a correspondence between cuspidal and highest weight submodules of these families by extending mathieu s work ann inst fourier 50 2000 537 this enables us to reduce the description of the g 0 module structure of arbitrary simple weight g modules with bounded weight multiplicities to the g 0 module structure of highest weight modules with bounded weight multiplicities we then construct tensor coherent families for g sl n 1 1 which yield an explicit description of the g 0 structure of an arbitrary simple weight module with bounded weight multiplicities in particular we show that for g sl n 1 1 the maximal length of an indecomposable g 0 submodule of a simple weight module with bounded multiplicities equals 5

an efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures an efficient multilevel fast multipole algorithm mlfma formalism to model radiation and scattering by from large planar microwave structures is presented the technique relies on an electric field integral equation efie formulation and a series expansion for the green dyadic based on the use of perfectly matched layers pml in this way a new pml mlfma is developed to efficiently evaluate matrix vector multiplications arising in the iterative solution of the scattering problem the computational complexity of the new algorithm scales down to o n for electrically large structures the theory is validated by means of several illustrative numerical examples

discretization of continuous spectra based on perfectly matched layers as a tool of analysis in physics wavefields are often expanded in a set of eigensolutions obtained from a sturm liouville problem for singular sturm liouville problems subject to radiation boundary conditions i e problems defined on an infinite domain this set of eigensolutions has continuous parts in this paper we will show that it is possible to approximate this continuous set of eigensolutions by a discrete set of eigensolutions of the same sturm liouville operator but subject to dirichlet boundary conditions in complex space the idea of dirichlet boundary conditions in complex space stems from the perfectly matched layer pml absorbing boundary condition the pml was introduced in 1994 j p berenger j comput phys 114 1994 pp 185 200 as an absorbing termination of a finite difference time domain grid these complex space dirichlet boundary conditions have been used recently to close open electromagnetic waveguide structures in the present paper we aim at developing a mathemati

series representation of green dyadics for layered media using pmls the green dyadics for closed layered media i e layered media bounded by a perfectly conducting plate at the bottom and top of the structure can be expanded in a discrete surface wave series for open layered media with semi infinite layers at the top and or bottom of the structure the discrete series needs to be complemented by a branch cut integral of space waves in this paper we present a technique to circumvent this branch cut integral by truncating the semi infinite layers with a perfectly matched layer pml that is backed by a perfect electric conductor pec it is demonstrated that in this way it is possible to obtain an accurate series or closed form representation for the green dyadic of the open layered medium the series allows a very efficient calculation and storage of the green dyadic if it is needed for multiple observation and or excitation points very close to the source the series loses efficiency it is shown that the determination of the surface waves in the pml truncated layered medium has the same complexity as the determination of the surface waves in a pec truncated layered medium without a pml

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

sound wave propagation from a point source over a homogeneous surface and over a surface with an impedance discontinuity the sound field from a point source above an infinite plane with an impedance discontinuity is studied by considering the helmholtz equation with boundary conditions the helmholtz equation is transformed into a singular integral equation and an exact solution formula is given the formula is evaluated for the case when the source and the receiver are close to the boundary many wavelengths from the discontinuity the evaluation gives among other things information on the dependence of the surface wave structure on the impedances and in the case of no surface waves information on the farfield solution if one of the impedances gives a surface wave in the case of a homogeneous boundary then a surface wave appears also in the nonhomogeneous case and is modified depending on the other impedance in the case of no surface waves the rapid decay of the sound field in the homogeneous case does not appear in the nonhomogeneous case since the discontinuity acts as a weak source of slowly decaying waves a numb

on the structure of weight modules given any simple lie superalgebra g we investigate the structure of an arbitrary simple weight 0 module we introduce two invariants of simple weight modules the shadow and the small weyl group generalizing results of fernando and futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a levi subsuperalgebra then we classify the cuspidal levi subsuperalgebras of all simple classical lie superalgebras and of the lie superalgebra w n most of them are simply levi subalgebras of go in which case the classification of all finite cuspidal representations has recently been carried out by one of us mathieu our results reduce the classification of the finite simple weight modules over all classical simple lie superalgebras to classifying the finite cuspidal modules over certain lie superalgebras which we list explicitly

note on a result of morse and bolt abstract a result given without derivation by morse and bolt 1 pertaining to the reflection of a spherical sound wave from an absorbent surface is investigated it is shown that the result as given is not quite accurate

error control in the perfectly matched layer based multilevel fast multipole algorithm the development of efficient algorithms to analyze complex electromagnetic structures is of topical interest application of these algorithms in commercial solvers requires rigorous error controllability in this paper we focus on the perfectly matched layer based multilevel fast multipole algorithm pml mlfma a dedicated technique constructed to efficiently analyze large planar structures more specifically the crux of the algorithm viz the pertinent layered medium green functions is under investigation therefore particular attention is paid to the plane wave decomposition for 2 d homogeneous space green functions in very lossy media as needed in the pml mlfma the result of the investigations is twofold first upper bounds expressing the required number of samples in the plane wave decomposition as a function of a preset accuracy are rigorously derived these formulas can be used in 2 d homogeneous lossy media mlfmas second a more heuristic approach to control the error of the pml mlfma s green functions is presented the theory is verified by means of several numerical experiments

sound propagation above an inhomogeneous plane boundary integral equation methods abstract a boundary integral equation method is used to compute the sound pressure emitted by a harmonic source above an inhomogeneous plane first the theoretical aspects of the problem behaviour of the pressure around the discontinuities are studied then a comparison between theoretical levels and experimental levels obtained in an anechoic room is presented it shows that the boundary integral equation bie method is quite convenient for solving this kind of problem two interesting results are pointed out i if only a prediction of maximum sound levels is needed the attenuation is the same for a cylindrical source a spherical source and n spherical sources and so it is possible to transform some three dimensional problems into two dimensional ones ii a numerical method of computation of the sound field above an inhomogeneous plane does not provide a correct prediction if each part of the plane is not accurately described by the boundary condition chosen

a note on reflection of spherical waves in 1909 sommerfeld gave an exact solution for the reflection of a spherical wave from a plane surface in terms of an oscillatory integral and also presented an asymptotic solution for the case where both source and receiver are at the boundary weyl 1919 presented an alternative solution and also an asymptotic solution for the case where the source is at the boundary it is known that the general case is solved if a general solution for the case where the source is at the boundary is known here it is demonstrated that it is sufficient to have the general solution for the case where both source and receiver are at the boundary this is mainly of theoretical interest but may have practical applications as an example it is demonstrated that sommerfeld s approximate solution gives ingard s 1951 approximate solution which is valid for arbitrary source and receiver heights

on the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination an explicit proof of the completeness of the eigenmodes of a grounded parallel plate waveguide with a perfectly matched layer pml termination is given the proof is based on a general theorem governing the completeness of sets of complex exponentials the green s function of the structure is then obtained by means of a biorthogonalization procedure a comparison with the green s function of a grounded halfspace indicates that a pml termination simulates the open halfspace efficiently index terms completeness eigenmodes green s function parallel plate waveguide perfectly matched layer pml

scattering and radiation from by 1 d periodic metallizations residing in layered media an efficient technique is proposed to compute the scattering or radiation from by 1 d periodic structures residing in a layered background medium the technique is based on a mixed potential integral equation mpie combined with the method of moments mom solving for the unknown current density flowing within a unit cell of the periodic structure the formalism requires the knowledge of the pertinent layered medium green s functions with 1 d periodicity here these green s functions are derived in closed form by invoking the perfectly matched layer pml paradigm the stationary phase method is applied to determine the far field of the infinite periodic structure leading to a series of floquet modes in addition approximating this series leads to an efficient technique to estimate the scattering or radiation from by large but finite periodic structures with a 1 d periodic character the theory is illustrated and validated by means of various examples stemming from scattering and radiation applications from by antenna arrays residing on microstrip substrates the efficiency of the method is also demonstrated

sound propagation above an inhomogeneous impedance plane abstract boundary integral equation bie methods are described for the prediction of sound propagation in particular from a line source over a flat plane of inhomogeneous impedance approximate methods which satisfy reciprocity for the calculation of the wave field over a two impedance plane are proposed these approximations applied to propagation from a line source give results agreeing well with those of the bie method when they are applied to propagation from a point source agreement with experiment is shown

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

a high performance upgrade of the perfectly matched layer multilevel fast multipole algorithm for large planar microwave structures an improvement of the perfectly matched layer multilevel fast multipole algorithm pml mlfma for simulating large planar microwave structures is presented by exploiting the low rank property of the pml mlfma multimodal aggregation and disaggregation matrices considerable reductions in memory usage and computation time are obtained the method has been extensively validated demonstrating complete error controllability when simulating large planar microwave structures reductions in memory requirements and cpu time of more than 60 and 40 have been achieved

invariant supersymmetric multilinear forms and the casimir elements of p type lie superalgebras the casimir elements of the p type lie superalgebras are investigated depending on the class of algebras under consideration either there do not exist any nontrivial casimir elements at all or else the casimir elements are highly degenerate basic to the investigation is a lemma about invariant supersymmetric multilinear forms on a finite dimensional module over a lie superalgebra some comments on the cartan subalgebras of a lie superalgebra are also included an appendix provides some information on multilinear algebra with e commuting scalars

long distance sound propagation over an impedance discontinuity abstract a calculation method is presented for sound propagation over an impedance discontinuity which is an improvement over previous methods being particularly efficient for long distances a homogeneous still atmosphere and flat ground are assumed the starting point is to make an approximation in a two dimensional boundary integral equation formulation of the problem this leads to an expression of the solution as the solution for homogeneous ground plus an integral over part of the boundary this integral is recognized as a generalized fourier integral to which standard techniques of asymptotic analysis can be applied an asymptotic approximation is found which can replace the part of the integral which is most expensive to evaluate by numerical quadrature the remaining integral is evaluated by numerical quadrature monofrequency excess attenuation results for propagation from a point source above rigid ground to a receiver above absorbing ground are discussed when the source and receiver are low enough very orderly trends are observed the results are applied with air attenuation and a weighting to a notional jet engine noise source simple trends are noted

extrapolation methods for sommerfeld integral tails a review is presented of the extrapolation methods for accelerating the convergence of sommerfeld type integrals i e semi infinite range integrals with bessel function kernels which arise in problems involving antennas or scatterers embedded in planar multilayered media attention is limited to partition extrapolation procedures in which the sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real axis tail segment spl alpha spl infin of the integration path where spl alpha 0 is selected to ensure that the integrand is well behaved an analytical form of the asymptotic truncation error or the remainder which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods is derived several extrapolation algorithms deemed to be the most suitable for the sommerfeld integrals are described and their performance is compared it is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points spl rho and by the choice of the subinterval break points furthermore it is found that some well known extrapolation techniques may fail for a number of values of spl rho and ways to remedy this are suggested finally the most effective extrapolation methods for accelerating sommerfeld integral tails are recommended

ground effect analysis surface wave and layer potential representations abstract this paper presents two kinds of analytical exact expressions of the sound field reflected by a plane boundary as obtained by using either surface wave or layer potentials representations both solutions are first expressed as a sum of integrals which have a form suitable for numerical computation then these integrals are expanded into convergent series which provide analytical approximations of the solution numerical techniques are proposed for computing a the surface wave representation and b the approximation deduced from the layer potentials representation this approximation and the classical one sum of the surface wave and the first two terms of the asymptotic series are compared with the exact solution several examples show that the approximate formulas established here are valid on a range much wider than the validity domain of the classical ones

an exact laplace transform formulation for a point source above a ground surface an exact analysis is given for a point source in air above a ground surface by representing the plane wave reflection coefficient as the laplace transform of an image source distribution a well behaved image integral instead of the usual sommerfeld integral is obtained the approach is valid for both locally and extended reacting surfaces for a locally reacting ground surface the image integral is an especially simple rapidly convergent integral the integral for local reaction is investigated analytically for a number of limiting cases the resulting analytic solutions are compared with analytic solutions obtained from more standard approaches finally the image integral for local reaction is analyzed numerically and an upper limit on the numerical integration is given it is shown that with realistic values of ground impedance the prescribed integration limit allows the image integral to be easily and accurately computed numerically

a reduction method for atypical representations of classical lie superalgebras abstract we introduce a reduction method for studying representations of classical lie superalgebras with atypical central character we show that the atypical quotient of universal enveloping algebra has a non trivial jacobson radical the factor by this radical has a new center which is calculated for sl 1 n and psl 2 2 using this center we obtain new character formulae generalization of borel weil bott and beilinson bernstein localization theorems

field decomposition and factorization of the helmholtz determinant operator for bianisotropic media the paper generalizes the class of media for which the electromagnetic em fields can be decomposed in two independent components each of the two components see the medium as a simpler so called equivalent medium this generalization was possible because it was realized that the plane wave fields satisfy e spl middot d h spl middot b 0 this discovery allows us to come to a consistent structure for the class of decomposable and factorizable bianisotropic media encompassing all previously studied media as special cases the class of media of the present paper is shown to be closed with respect to duality transformations we dare to say that the present paper puts the crown on a long quest for always more general classes of media that allow decomposition and factorization the equivalent media yield a new class of media that allow a closed form representation of the green dyadics