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

single and multiple recurrence along non polynomial sequences abstract we establish new recurrence and multiple recurrence results for a rather large family f of non polynomial functions which contains tempered functions and non polynomial functions from a hardy field with polynomial growth in particular we show that somewhat surprisingly and in the contrast to the multiple recurrence along polynomials the sets of return times along functions from f are thick i e contain arbitrarily long intervals a major component of our paper is a new result about equidistribution of sparse sequences on nilmanifolds whose proof borrows ideas from the work of green and tao 26 among other things we show that for any f f any invertible probability measure preserving system x b t any a b with a 0 and any e 0 the sets of returns n n a t f n a 2 a e n n a t f n a t f n 1 a t f n k a 0 are thick our recurrence theorems imply via furstenberg s correspondence principle some new combinatorial results for example we show that given a set e n with positive upper density for every k n there are a n n such that a a f n a f n k e when f n n c with c 0 non integer this result provides a positive answer to a question posed by frantzikinakis 16 problem 23

note on surface radiation conditions the predictions of the surface radiation condition are compared with the exact analytical answer for scattering by a soft elliptical cylinder the agreement is found to be highly satisfactory for a range of parameters which can be handled analytically numerical results are presented that substantiate the analysis the surface radiation condition is generalized to be applicable when the environment is inhomogeneous study of the particular case of a circular cylinder surrounded by an inhomogeneous medium reveals that surface boundary condition gives good results for some values of the parameters where analytic manipulation is feasible

a wide angle on surface radiation condition applied to scattering by spheroids a rational approximation of the pseudo differential dirichlet to neumann operator for a circular cylinder is made using zero placement for an acoustically hard circular cylinder this zero corresponds to the dominant creeping wave singularity in the residue series for the time harmonic free space surface green s function based on this operator a wide angle on surface radiation condition osrc is developed first for two dimensional slender convex bodies for the object scattering problem this osrc is accurate for wide scattering angles relative to the surface normal the notion of creeping waves is then used to develop a wide angle three dimensional osrc applicable to smooth convex bodies application to both hard and soft prolate and oblate spheroids is presented and excellent agreement is found in comparison with a more costly technique

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

non uniform grid accelerated local global boundary condition ng lgbc for acoustic scattering a global absorbing boundary condition utilizing accelerated integration scheme based on the non uniform grid ng approach is hybridized with a local boundary condition producing an accurate and efficient mesh truncation scheme for finite method analysis of open region acoustic scattering problems the method affords arbitrarily shaped in particular non convex boundaries that conform to essentially concave obstacles of dimensions large compared with the wavelength

ergodic averages of commuting transformations with distinct degree polynomial iterates we prove mean convergence as n to infty for the multiple ergodic averages frac 1 n sum n 1 n f 1 t 1 p 1 n x cdot ldots cdot f ell t ell p ell n x where p 1 ldots p ell are integer polynomials with distinct degrees and t 1 ldots t ell are invertible commuting measure preserving transformations acting on the same probability space this establishes several cases of a conjecture of bergelson and leibman that complement the case of linear polynomials recently established by tao furthermore we show that unlike the case of linear polynomials for polynomials of distinct degrees the corresponding characteristic factors are mixtures of inverse limits of nilsystems we use this particular structure together with some equidistribution results on nilmanifolds to give an application to multiple recurrence and a corresponding one to combinatorics

on the approximation solvability of a class of strongly nonlinear elliptic problems on unbounded domains a class of strongly nonlinear boundary value problems posed on unbounded regions is considered a nonlocal coupling of the linearized far field exterior to an auxiliary boundary allows for approximations to be defined on domains of finite extent constructive existence results for bounded domains are then extended by employing an exhausting sequence of approximating domains in particular well posedness is seen to be equivalent to unique approximation solvability with the rate of convergence dependent upon the radius of the auxiliary boundary application to a model of proteins immersed in an electrolyte solution is made

estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the helmholtz equation an estimator for the error in the wave number is presented in the context of finite element approximations of the helmholtz equation the proposed estimate is an extension of the ideas introduced in steffens and d iez comput methods appl mech engng 2009 198 1389 1400 in the previous work the error assessment technique was developed for standard galerkin approximations here the methodology is extended to deal also with stabilized approximations of the helmholtz equation thus the accuracy of the stabilized solutions is analyzed including also their sensitivity to the stabilization parameters depending on the mesh topology the procedure builds up an inexpensive approximation of the exact solution using post processing techniques standard in error estimation analysis from which the estimate of the error in the wave number is computed using a simple closed expression the recovery technique used in steffens and diez comput methods appl mech engng 2009 198 1389 1400 is based in a polynomial least squares fitting here a new recovery strategy is introduced using exponential in a complex setup trigonometric local approximations respecting the nature of the solution of the wave problem copyright 2011 john wiley sons ltd

calculation of electromagnetic scattering by a perfect conductor abstract a new method is presented for calculating the scattering of an arbitrary electromagnetic wave by a bounded perfectly conducting body of general shape the strategy is to replace the corresponding exterior boundary value problem by an approximate problem on the boundary of the scattering body this involves the introduction of a certain bilinear form and non local boundary operator together with the use of a special class of known solutions of the reduced maxwell s equations satisfying the sommerfeld radiation boundary conditions at infinity two computer programs implementing this method are described and numerical results showing the successful application of this method to some model problems are presented

the finite element method with nonuniform mesh sizes for unbounded domains the finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains consider the following model problem du f in w sup c u g on partialw partialu partialr 1 r u o 1 r as r vertical barxvertical bar infinity where w sup c is the complement in r sup 3 three dimensional eucidean space of a bounded set w with smooth boundary partialw f and g are smooth functions and f has bounded support this problem is approximately solved by introducing an artifical bound g sub r near infinity e g a sphere of sufficiently large radius r the intersection of this sphere with w sup c is denoted by w sub r sup c and the given problem is replaced by du sub r f in w sup c sub r u sub r g on partialw partialu sub r partialr 1 r u sub r 0 on g sub r this problem is then solved approximately by the finite element method resulting in an approximate solution u sub r sup h for each h 0 in order to obtain a reasonably small error for u u sub r sup h u u sub r u sub r u sub r sup h it is necessary to make r large this necessitates the solution of amore large number of linear equations so that this method is often not very good when a uniform mesh size h is employed it is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by ch sup 3 with c independent of h and r less

a note on the radiation boundary conditions for the helmholtz equation the radiation boundary conditions in two dimensions are derived in the body fitted coordinate system using the method of successive approximations for the wave envelope function results are valid for a convex scatterer with continuous radius of curvature on the surface of the scatterer in the special case when the computational boundary is circular the boundary operators derived are identical to the bayliss operators the on surface radiation condition is examined it is shown that for a large conducting circular cylinder a boundary operator of infinite order should be used due to the breakdown of the asymptotic expansions of the boundary operators on the surface of the scatterer the leading order result based on the boundary operator of infinite order applied on the surface of the cylinder is the same as the result obtained by the method of physical optics

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

flow shops with a dominant machine abstract flow shop problems with a dominant machine are introduced we show that for this kind of problems we can restrict our search for an optimal schedule to permutation schedules if the optimality criterion is regular furthermore we give an expression for the completion times with respect to a semi active permutation schedule and we give fast algorithms for the weighted completion times and the maximal lateness criterion

a multidimensional szemeredi theorem for hardy sequences of different growth abstract we prove a variant of the multidimensional polynomial szemeredi theorem of bergelson and leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some hardy field and satisfy certain growth conditions we do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula a consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences another consequence is that every syndetic set of integers contains certain nonshift invariant patterns and every finite coloring of n with each color class a syndetic set contains certain polychromatic patterns results very particular to our non polynomial setup

error analysis and preconditioning for an enhanced dtn fe algorithm for exterior scattering problems in this paper we present an error analysis for a high order accurate combined dirichlet to neumann dtn map finite element fe algorithm for solving two dimensional exterior scattering problems we advocate the use of an exact dtn or steklov poincare map at an artificial boundary exterior to the scatterer to truncate the unbounded computational region the advantage of using an exact dtn map is that it provides a transparent condition which does not reflect scattered waves unphysically our algorithm allows for the specification of quite general artificial boundaries which are perturbations of a circle to compute the dtn map on such a geometry we utilize a boundary perturbation method based upon recent theoretical work concerning the analyticity of the dtn map we also present some preliminary work concerning the preconditioning of the resulting system of linear equations including numerical experiments

a comparison of nrbcs for pufem in 2d helmholtz problems at high wave numbers in this work exact and approximate non reflecting boundary conditions nrbcs are implemented with the partition of unity finite element method pufem to solve short wave scattering problems governed by the helmholtz equation in two dimensions by short wave problems we mean situations in which the wavelength is a small fraction of the characteristic dimension of the scatterer various nrbcs are implemented and a comparison of their performance is carried out based on the accuracy of the results ease of implementation and computational cost the aim is to accurately model such problems in a reduced computational domain around the scatterer with fewer elements and without refining the mesh at each wave number

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

efficient iterative solution of the three dimensional helmholtz equation we examine preconditioners for the discrete indefinite helmholtz equation on a three dimensional box shaped domain with sommerfeld like boundary conditions the preconditioners are of two types the first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions the second is derived by a block toeplitz approximation to the descretized problem the resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete helmholtz operator by an operator of low rank we present experimental results demonstrating that when these methods are combined with krylov subspace iteration convergence rates depend only mildly on both the wave number and discretization mesh size in addition the methods display high efficiencies in an implementation on an ibm sp 2 parallel computer

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

absorbing boundary conditions on arbitrary boundaries for the scalar and vector wave equations the solution of open region scattering problems involving inhomogeneous arbitrarily shaped objects may be performed through the use of partial differential equation techniques which require enclosing the scatterer by an outer boundary on which an absorbing boundary condition abc is applied in order to minimize the size of the domain to be meshed and consequently the number of unknowns if may be advisable to implement abc s devised for outer boundaries of arbitrary shapes such abc s are obtained for the 3d scalar and vector wave equations they incorporate most of existing boundary conditions when used in conjunction with a finite element technique the numerical results derived by using a simplified form of these abc s compare favourably to those obtained by using a rigorous hybrid finite element integral equation formulation these boundary conditions have been obtained in the frequency domain framework they may however be used in time domain calculations

finite element methods applied to nearly one way wave propagation abstract a numerical method is developed for approximately solving stationary wave propagation problems for which most of the energy propagates in an angle band of less than 90 about a fixed direction this property often occurs in realistic physical models the numerical method is based on factoring out the principal propagation direction in the solution and then applying the finite element method the method is applied to wave propagation in a duct or waveguide and to the exterior scattering problem it is shown both theoretically and by means of numerical examples that this method requires considerably less grid points and computational cost than standard discretization methods

on surface radiation conditions for an ellipse we compare several on surface radiation boundary conditions in two dimensions for solving the helmholtz equation exterior to an ellipse we also introduce a new boundary condition for an ellipse based on a modal expansion in mathieu functions we compare the osrc to a finite difference method

transient acoustic radiation from impulsively accelerated bodies by the finite element method bodies under impulsive motion immersed in an infinite acoustic fluid severely put to test any numerical method for the transient exterior acoustic problem such problems in the context of the finite element method fem are not well studied fe modeling of such problems requires truncation of the infinite fluid domain at a certain distance from the structure the volume of computation depends upon the extent of this domain as well as the mesh density the modeling of the fluid truncation boundary is crucial to the economy and accuracy of solution and various boundary dampers have been proposed in the literature for this purpose the second order damper leads to unsymmetric boundary matrices and this necessitates the use of an unsymmetric equation solver for large problems the present paper demonstrates the use of fem with zeroth first and second order boundary dampers in conjunction with an unsymmetric out of core banded equation solver for impulsive motion problems of rigid bodies in an acoustic fluid the results compare well with those obtained from analytical methods

on controllability methods for the helmholtz equation abstract when the helmholtz equation is discretized by standard finite difference or finite element methods the resulting linear system is highly indefinite and thus notoriously difficult to solve in fact increasingly so at higher frequency the exact controllability approach bristeau et al 1994 instead reformulates the problem in the time domain and seeks the time harmonic solution of the corresponding wave equation by iteratively reducing the mismatch between the solution at initial time and after one period the controllability method greatly speeds up the convergence to the time harmonic asymptotic limit moreover each conjugate gradient iteration solely relies on standard numerical algorithms which are inherently parallel and robust against higher frequencies the original energy functional used to penalize the departure from periodicity is strictly convex only for sound soft scattering problems to extend the controllability approach to general boundary value problems governed by the helmholtz equation new penalty functionals are proposed which are numerically efficient numerical experiments for wave scattering from sound soft and sound hard obstacles inclusions but also for wave propagation in closed wave guides illustrate the usefulness of the resulting controllability methods

the quantitative behaviour of polynomial orbits on nilmanifolds a theorem of leibman asserts that a polynomial orbit g n g n z on a nilmanifold g g is always equidistributed in a union of closed sub nilmanifolds of g g in this paper we give a quantitative version of leibman s result describing the uniform distribution properties of a finite polynomial orbit g n g n n in a nilmanifold more specifically we show that there is a factorisation g eg where e n is smooth n g n z is periodic and rational and g n g n p is uniformly distributed up to a specified error d inside some subnilmanifold g g of g g for all sufficiently dense arithmetic progressions p n our bounds are uniform in n and are polynomial in the error tolerance d in a companion paper we shall use this theorem to establish the mobius and nilsequences conjecture from an earlier paper of ours

intersective polynomials and the polynomial szemeredi theorem abstract let p p 1 p r q n 1 n m be a family of polynomials such that p i z m z i 1 r we say that the family p has the psz property if for any set e z with d e lim sup n m e m n 1 n m 0 there exist infinitely many n z m such that e contains a polynomial progression of the form a a p 1 n a p r n we prove that a polynomial family p p 1 p r has the psz property if and only if the polynomials p 1 p r are jointly intersective meaning that for any k n there exists n z m such that the integers p 1 n p r n are all divisible by k to obtain this result we give a new ergodic proof of the polynomial szemeredi theorem based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds we also obtain as a corollary the following generalization of the polynomial van der waerden theorem if p 1 p r q n are jointly intersective integral polynomials then for any finite partition z i 1 k e i of z there exist i 1 k and a n e i such that a a p 1 n a p r n e i

numerical solution of the three dimensional ginzburg landau models using artificial boundary for three dimensional simulations of the vortex phenomena in superconductors based on the ginzburg landau gl theory the physical variables must be solved in the whole space in general exact boundary conditions on an artificial boundary are discussed to reformulate the problem in a finite domain approximations of these boundary conditions with high accuracy are given and their convergence properties are examined error estimates are derived for the finite element approximations with the approximate boundary conditions

transient spheroidal elements for unbounded wave problems a wave envelope element numerical scheme is applied to the solution of unbounded wave problems the scheme is based on a fourier transformation of a discrete model formulated in the frequency domain this yields a discrete system of ordinary differential equations in time which are local in space oblate and prolate spheroidal element geometries are used the accuracy of the scheme is demonstrated by a comparison of computed and analytic solutions for axisymmetric test cases time harmonic and transient solution are presented an indirect solution procedure is also presented which permits the sparsity of the transient equations to be utilised more effectively

three dimensional finite element calculations in acoustic scattering using arbitrarily shaped convex artificial boundaries we report on a generalization of the bayliss gunzburger turkel non reflecting boundary conditions to arbitrarily shaped convex artificial boundaries for elongated scatterers such as submarines we show that this generalization can improve significantly the computational efficiency of finite element methods applied to the solution of three dimensional acoustic scattering problems copyright 2001 john wiley sons ltd

higher order impedance and absorbing boundary conditions traditionally generalized impedance boundary conditions gibcs have been used to model dielectrics and coated surfaces and absorbing boundary conditions abcs have been used to simulate nonreflecting surfaces the two types have the same mathematical form and in most instances a higher order condition involving higher order field derivatives has a better accuracy we demonstrate that there is a close connection between the two and this enables us to use a systematic method which is available for generating gibcs of any desired order to derive new two and three dimensional abcs the method is applicable to curvilinear doubly curved surfaces and examples are given finally curves are presented that quantify the accuracy of two dimensional abcs up to the fourth order and show how higher order abcs can improve the efficiency of large scale partial differential equation pde solutions

plane wave enriched partition of unity isogeometric analysis puiga for 2d helmholtz problems abstract in general numerical concerns in simulation of time harmonic acoustic waves are mainly due to the dispersion errors and the demands in modeling the problems with higher number of degrees of freedom per wavelength to obtain acceptable results as a remedy plane wave enriched partition of unity finite element method pufem was earlier proposed by melenk and babuska 1996 even though pufem gives accurate results with lesser degrees of freedom and shows exponential convergence it suffers from the approximate representation of geometry and leads to the boundary representation induced errors in this regard plane wave enriched pufem under isogeometric framework named partition of unity isogeometric analysis puiga is proposed for accurate geometry representation and smooth solution approximation which can lead to a better solution several benchmark problems are considered in the present study to illustrate the computational effectiveness of puiga

h and p adaptivity driven by recovery and residual based error estimators for pht splines applied to time harmonic acoustics abstract in this work we demonstrate the application of pht splines for time harmonic acoustic problems modeled by the helmholtz equation solutions of the helmholtz equation have two features global oscillations associated with the wave number and local gradients caused by geometrical irregularities we show that after a sufficient number of degrees of freedom is used to approximate global oscillations adaptive refinement can capture local features of the solution we compare residual based and recovery based error estimators and investigate the performance of p refinement the simulations are done in the context of recently introduced geometry independent field approximation gift where pht splines are only used to approximate the solution while the computational domain is parameterized with nurbs this approach builds on the natural adaptation ability of pht splines and avoids the re parameterization of the nurbs geometry during the solution refinement process

a survey of finite element methods for time harmonic acoustics many of the current issues and methodologies related to finite element methods for time harmonic acoustics are reviewed the effective treatment of unbounded domains is a major challenge most prominent among the approaches that have been developed for this purpose are absorbing boundary conditions infinite elements and absorbing layers standard computational methods are unable to cope with wave phenomena at short wave lengths due to resolutions required to control dispersion and pollution errors leading to prohibitive computational demands since computation naturally separates the scales of a problem according to the mesh size multiscale considerations provide a useful framework for viewing these difficulties and developing methods to counter them other issues addressed are related to the efficient solution of systems of specialized algebraic equations and inverse problems of acoustics the tremendous progress that has been made in all of the above areas in recent years will surely continue leading to many more exciting developments

construction of local boundary conditions for an eigenvalue problem using micro local analysis application to optical waveguide problems we present a general procedure based on the pseudo differential calculus for deriving artificial boundary conditions for an eigenvalue problem that characterizes the propagation of guided modes in optical waveguides this new approach allows the construction of local conditions that a are independent of the frequency regime b preserve the sparsity pattern of the finite element discretization and c are applicable to arbitrarily shaped convex artificial boundaries the last feature has the potential for reducing the size of the computational domain numerical results are presented to highlight the potential of conditions of order 1 2 and 1 for improving significantly the computational efficiency of finite element methods for the solution of optical waveguide problems

computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements in this paper we analyze a galerkin procedure based on a combination of finite and spectral elements for approximating a time harmonic acoustic wave scattered by a bounded inhomogeneity the finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition which consists in a linear integral equation this integral equation is discretized by a spectral galerkin approximation method we provide error estimates for the galerkin method propose fully discrete schemes based on elementary quadrature formulas and show that the perturbation due to this numerical integration gives rise to a quasi optimal rate of convergence we also suggest a method for implementing the algorithm using the preconditioned gmres method and provide some numerical results

a higher order on surface radiation condition derived from an analytic representation of a dirichlet to neumann map on surface radiation conditions are useful for obtaining approximate solutions to scattering problems involving compact obstacles an analytic representation of the dirichlet to neumann map for a circle is derived and used to construct a higher order on surface radiation condition for a generally convex perfectly conducting body in two dimensions this approach is based on a hankel function in which a tangential operator appears in the index in the high frequency limit this analytic representation approaches the square root of a differential operator which commonly arises in the application of parabolic equation techniques to propagation problems treating the scattered field propagation angle relative to the surface normal and the surface curvature as independent parameters the representation is fit to a rational function to provide an accurate and efficient on surface radiation condition that is tested for various examples

numerical simulation for the problem of infinite elastic foundation in this paper we consider the numerical simulation for the problem of infinite elastic foundation a half circle artificial boundary is introduced and discrete artificial boundary condition on it is designed by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain furthermore finite element approximation of this boundary value problem defined in the bounded domain is considered the numerical results show that the artificial boundary condition given in this paper is very effective and more accurate than the neumann boundary condition which is often used in engineering literatures

higher order extensions of a discontinuous galerkin method for mid frequency helmholtz problems recently a discontinuous galerkin method with plane wave basis functions and lagrange multiplier degrees of freedom was proposed for the efficient solution of helmholtz problems in the mid frequency regime in this paper this method is extended to higher order elements performance results obtained for various two dimensional problems highlight the advantages of these elements over classical higher order galerkin elements such as q2 and q4 for the discretization of interior and exterior helmholtz problems copyright 2004 john wiley sons ltd

comparison of local radiation boundary conditions for the scalar helmholtz equation with general boundary shapes the relative accuracy of several local radiation boundary conditions based on the second order bayliss turkel 1980 condition are evaluated these boundary conditions permit the approximate solution of the scalar helmholtz equation in an infinite domain using traditional finite element and finite difference techniques unlike the standard bayliss turkel condition the generalizations considered here are applicable to noncircular solution domains the accuracy of these conditions are investigated for elliptical and linear circular boundaries

local absorbing boundary conditions for elliptical shaped boundaries we compare several local absorbing boundary conditions for solving the helmholtz equation by a finite difference or finite element method exterior to a general scatterer these boundary conditions are imposed on an artificial elliptical or prolate spheroid outer surface in order to compare the computational solution with an analytical solution we consider as an example scattering about an ellipse we solve the helmholtz equation with both finite differences and finite elements we also introduce a new boundary condition for an ellipse based on a modal expansion

analytical and numerical investigation of the performance of the bgt2 condition for low frequency acoustic scattering problems a mathematical and numerical analysis is performed to assess the performance of the second order bayliss gunzburger turkel bgt2 condition when applied to solving low frequency acoustic scattering problems in the case of elongated scatterers this investigation suggests that bgt2 retains an acceptable level of accuracy for relatively low wavenumber a damping effect is incorporated to the bgt2 condition in order to extend the range of satisfactory performance this damping procedure consists in adding only a constant imaginary part to the wavenumber the numerical results indicate that the modified version of bgt2 extends the range of satisfactory performance by improving the level of accuracy by up to two orders of magnitude guidelines on the appropriate choice of the damping coefficient are provided

makespan minimization in machine dominated flowshop abstract in a flowshop scheduling problem a set of jobs is processed by a set of machines the jobs follow the same sequence in all machines we study the flowshop scheduling problem under a new case of machine dominance that is often found in the manufacturing of computers and electronic devices we provide a formula for makespan value for a given sequence show that the makespan value depends only on certain jobs in the sequence and present an algorithm that finds a sequence with minimum makespan numerical examples of the solution approaches are provided

fast frequency sweep computations using a multi point pade based reconstruction method and an efficient iterative solver problems of the form z u f where z is a given matrix f is a given vector and is a circular frequency or circular frequency related parameter arise in many applications including computational structural and fluid dynamics and computational acoustics and electromagnetics the straightforward solution of such problems for fine increments of is computationally prohibitive particularly when z is a large scale matrix this paper discusses an alternative solution approach based on the efficient computation of u and its successive derivatives with respect to at a few sample values of this parameter and the reconstruction of the solution u in the frequency band of interest using multi point pade approximants this computational methodology is illustrated with applications from structural dynamics and underwater acoustic scattering in each case it is shown to reduce the cpu time required by the straightforward approach to frequency sweep computations by two orders of magnitude copyright 2006 john wiley sons ltd

determination of the complex acoustic scattering matrix of a right angled duct a method for determining the complete higher order scattering matrix of an acoustic discontinuity is developed the method is demonstrated for a right angled waveguide bend and the magnitude and phase of the reflection and transmission coefficients are extracted precisely the procedure is straightforward and based on the solutions to the helmholtz equation by the finite element method fem the consistency of the scattering coefficients found by this method is verified by their properties of symmetry and their accuracy is established by the conservation of energy the reliability of the new technique is further proved by means of an arbitrary sound source and by comparing the direct fem response to the reflection matrix calculation some features of the scattering matrix as a function of frequency are surprising such as the steps and reversion of the phase evolution or the complete loss of transmission of the incoming wave the methodology detailed in this paper can be extended to other multiport junctions such as t junctions or size discontinuities in ducts

on the coupling of mixed fem and bem for an exterior helmholtz problem in the plane this paper deals with an elliptic boundary value problem posed in the plane with variable coefficients but whose restriction to the exterior of a bounded domain reduces to a helmholtz equation we consider a mixed variational formulation in a bounded domain that contains the heterogeneous medium coupled with a boundary integral method applied to the helmholtz equation in we utilize suitable auxiliary problems duality arguments and fredholm alternative to show that the resulting formulation of the problem is well posed then we define a corresponding galerkin scheme by using rotated raviart thomas subspaces and spectral elements on the interface we show that the discrete problem is uniquely solvable and convergent and prove optimal error estimates finally we illustrate our analysis with some results from computational experiments

flow shop scheduling problems with deteriorating jobs on no idle dominant machines abstract in this paper we study the permutation flow shop scheduling problems with deteriorating jobs on no idle dominant machines the objective is to minimize one of the two regular performance criteria namely makespan and total completion time for each objective the following dominant machines constraint idm ddm idm ddm and ddm idm are considered we present a polynomial time solution algorithm for each of the four cases

high frequency numerical analytic approximations for the separable acoustic wave equation abstract an algorithm which employs a combination of asymptotic and numerical methods to solve a sturm liouville problem arising from a separable acoustic wave equation is presented error criteria for switching between methods are derived implementation requires a decomposition of the index of refraction profile into separable subprofiles computing times and accuracies are compared with conventional methods

efficient enforcement of far field boundary conditions in the transformed field expansions method the method of transformed field expansions tfe has been demonstrated to be a robust and highly accurate numerical scheme for simulating solutions of boundary value and free boundary problems from the sciences and engineering as a boundary perturbation method it builds highly accurate solutions based upon exact solutions in a simple canonical geometry and corrects these via taylor series to fit the actual geometry at hand the tfe method has significantly enhanced stability properties when compared with other boundary perturbation approaches however this comes at the cost of requiring a full volumetric discretization as opposed the surface formulation that other methods can realize in this paper we outline two techniques for ameliorating this shortcoming first by employing a legendre spectral element method to implement efficient graded meshes and second by utilizing an artificial boundary with a transparent boundary condition placed quite close to the boundary of the domain in this contribution we focus on the specific problem of simulating the dirichlet neumann operator associated to laplace s equation on a periodic cell which arises in the water wave problem while the details of our results are specific to this problem the general conclusions are valid for the wider class of problems to which the tfe method can be applied for each technique we discuss implementation details and display numerical results which support the conclusion that each of these techniques can greatly reduce the computational cost of using the tfe method

the dtn nonreflecting boundary condition for multiple scattering problems in the half plane abstract the multiple dirichlet to neumann multiple dtn non reflecting boundary condition is adapted to acoustic scattering from obstacles embedded in the half plane the multiple dtn map is coupled with the method of images as an alternative model for multiple acoustic scattering in the presence of acoustically soft and hard plane boundaries as opposed to the current practice of enclosing all obstacles with a large semicircular artificial boundary that contains portion of the plane boundary the proposed technique uses small artificial circular boundaries that only enclose the immediate vicinity of each obstacle in the half plane the adapted multiple dtn condition is simultaneously imposed in each of the artificial circular boundaries as a result the computational effort is significantly reduced a computationally advantageous boundary value problem is numerically solved with a finite difference method supported on boundary fitted grids approximate solutions to problems involving two scatterers of arbitrary geometry are presented the proposed numerical method is validated by comparing the approximate and exact far field patterns for the scattering from a single and from two circular obstacles in the half plane

composite boundary valued solution of the 2 5 d green s function for arbitrary acoustic media theoretically the green s function can be used to calculate the wavefield response of a specified source and the frechet derivative with respect to the model parameters for crosshole seismic full waveform inversion in this paper we apply the finite element method to numerically compute the 2 5 d green s function for an arbitrary acoustic medium by solving a composite boundary valued problem in the wavenumber frequency domain the composite boundary condition consists of a 2 5 d absorbing boundary condition for the propagating wave field and a mixed boundary condition for the evanescent field in inhomogeneous media modeling a numerical experiment performed for a uniform earth having a known exact solution shows the accuracy of the computation in the frequency and time domain an inhomogeneous medium test involving an embedded low velocity layer demonstrates that the permissible range of ky at each frequency can be determined rationally from the critical wavenumber value of the medium around the sou

a hardy field extension of szemeredi s theorem abstract in 1975 szemeredi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions bergelson and leibman showed in 1996 that the common difference of the arithmetic progression can be a square a cube or more generally of the form p n where p n is any integer polynomial with zero constant term we produce a variety of new results of this type related to sequences that are not polynomial we show that the common difference of the progression in szemeredi s theorem can be of the form n where is any positive real number and x denotes the integer part of x more generally the common difference can be of the form a n where a x is any function that is a member of a hardy field and satisfies a x x k and a x x k 1 0 for some non negative integer k the proof combines a new structural result for hardy sequences techniques from ergodic theory and some recent equidistribution results of sequences on nilmanifolds

high order numerical simulation of the transmission and scattering of waves using the method of difference potentials the method of difference potentials generalizes the method of calderon s operators from pdes to arbitrary difference equations and systems it offers several key advantages such as the capability of handling boundaries interfaces that are not aligned with the discretization grid variable coefficients and nonstandard boundary conditions in doing so the complexity of the algorithm remains comparable to that of an ordinary finite difference scheme on a regular structured grid previously we have applied the method of difference potentials to solving several variable coefficient interior helmholtz problems with fourth and sixth order accuracy we have employed compact finite difference schemes as a core discretization methodology those schemes enable high order accuracy on narrow stencils and hence require only as many boundary conditions as needed for the underlying differential equation itself numerical experiments corroborate the high order accuracy of our method for variable coefficients regular grids and non conforming boundaries in the current paper we extend the previously developed methodology to exterior problems we present a complete theoretical analysis of the algorithm as well as the results of a series of numerical simulations specifically we study the scattering of time harmonic waves about smooth shapes subject to various boundary conditions we also solve the transmission scattering problems in which not only do the waves scatter off a given shape but also propagate through the interface and travel across the heterogeneous medium inside in all the cases our methodology guarantees high order accuracy for regular grids and non conforming boundaries and interfaces

characterization of woodwind instrument toneholes with the finite element method a method is proposed to determine the transfer matrix parameters of a discontinuity in a waveguide with the finite element method fem this is used to characterize open and closed woodwind instrument toneholes and develop expressions for the shunt and series equivalent lengths two types of toneholes are characterized unflanged toneholes made of thin material such as found on saxophones and concert flutes and toneholes drilled through a thick material such as found on most instruments made of wood the results are compared with previous tonehole models from the literature in general the proposed expressions provide a better fit across a wide range of frequencies and tonehole sizes than previous results for tall toneholes the results are in general agreement with previous models for shorter tonehole heights some discrepancies from previous results are found that are most important for larger diameter toneholes finally the impact of a main bore taper conicity on the characterization of toneholes was investigated and found to be negligible for taper angles common in musical instruments

a new mapped infinite wave element for general wave diffraction problems and its validation on the ellipse diffraction problem the theory for an improved infinite element for wave diffraction is presented the improvement is on the original wave element presented by bettess zienkiewicz and others the modification involves the introduction of one extra term in the original shape function derivatives and a change in the numerical integration sequence the modification enables the element to be used with elliptical and other non circular meshes similar mappings have already been used with wave envelope infinite elements by astley and others as a test of the element accurate results are demonstrated for a range of elliptical cylinder diffraction problems the element has not yet been tested for other non circular problems

a fast method for solving acoustic scattering problems in frequency bands abstract we present a methodology for computing efficiently scattered fields in a frequency band the main feature of this methodology is the construction of a series of discrete problems that differ only by their right hand sides rather than by both their left and right hand sides its key steps are a the reformulation of the acoustic scattering problem in a bounded domain using any preferred absorbing boundary condition b the characterization of the repeated derivatives with respect to the frequency of the scattered field as solutions of scattering type problems with different source terms and boundary conditions and c the reconstruction of a scattered field by either the pade approximants or wynn s algorithm we report on several multifrequency acoustic scattering examples that illustrate the accuracy and computational efficiency of the proposed solution methodology

the finite element method with non uniform mesh sizes applied to the exterior helmholtz problem the finite element method with non uniform mesh sizes is employed to approximately solve helmholtz type equations in unbounded domains the given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary this approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases this results in a great reduction in the number of equations to be solved it is proved that optimal error estimates hold inl 2 h 1 andl provided that certain relationships hold between the frequency mesh size and outer radius

application of pade via lanczos approximations for efficient multifrequency solution of helmholtz problems this paper addresses the efficient solution of acoustic problems in which the primary interest is obtaining the solution only on restricted portions of the domain but over a wide range of frequencies the exterior acoustics boundary value problem is approximated using the finite element method in combination with the dirichlet to neumann dtn map the restriction domain problem is formally posed in transfer function form based on the finite element solution in order to obtain the solution over a range of frequencies a matrix valued pade approximation of the transfer function is employed using a two sided block lanczos algorithm this approach provides a stable and efficient representation of the pade approximation in order to apply the algorithm it is necessary to reformulate the transfer function due to the frequency dependency in the nonreflecting boundary condition this is illustrated for the case of the dtn boundary condition but there is no restriction on the approach which can also be applied to other radiation boundary conditions numerical tests confirm that the approach offers significant computational speed up

finite difference analysis of borehole flows involving domain contractions around three dimensional anomalies a finite difference analysis in cylindrical coordinates of borehole flows governed by poisson s equation yields an infinite cylindrical electrical grid as a model the customary technique of truncating the medium in order to obtain a finite grid can be avoided at least to some extent by using the recently developed theory of infinite electrical networks when the medium around the borehole is either uniform or cylindrically layered except for a three dimensional anomalous region near the borehole infinite electrical network theory allows one to represent the medium outside any cylinder s containing the anomaly and centered on the borehole by an infinite continued fraction of circulant laurent operators the continued fraction in turn can be represented by scalar conductances connected to the grid s nodes on s by choosing s just large enough to contain the anomaly one can replace the large system of equations resulting from the usual medium truncations far removed from the anomaly by a much smaller systems of equations for the grid within and on s an operational calculus for the circulant laurent operators and the fast fourier transform expedites the computations and moreover allows a rapidly computed determination of the solution outside s

high order surface radiation conditions for time harmonic waves in exterior domains abstract we formulate a new family of high order on surface radiation conditions to approximate the outgoing solution to the helmholtz equation in exterior domains motivated by the pseudo differential expansion of the dirichlet to neumann operator developed by antoine et al 1999 we design a systematic procedure to apply pseudo differential symbols of arbitrarily high order numerical results are presented to illustrate the performance of the proposed method for solving both the dirichlet and the neumann boundary value problems possible improvements and extensions are also discussed

a fast finite difference method for propagation predictions over irregular inhomogeneous terrain propagation of electromagnetic waves over irregular inhomogeneous terrain is solved by a finite difference scheme the method is fast and requires considerably less memory than the integral equation methods the method requires a storage space of order o n and an execution time of order o n sup 2 fields generated by a te sub 2 line source are represented in an integral form in terms of the field over a flat constant impedance plane the incident field and the field scattered by the terrain irregularities and inhomogeneities accurate expressions are provided for the incident field and the green s function whose evaluation is otherwise accomplished by the rather time consuming sommerfeld s integrals measured equation of invariance is used to terminate the computational domain the sparse matrix generated by the method is inverted by the ricatti transform numerical results are presented for the ground wave as well as for the sky wave comparison is made for known geometries to establish the validity and limitations of the method

permutation flow shop scheduling with dominant machines to minimize discounted total weighted completion time this paper deals with some special cases of general no wait and no idle permutation flow shop scheduling problems respectively special cases means that the machines form an increasing series of dominant machines a decreasing series of dominant machines an increasing decreasing series of dominant machines and a decreasing increasing series of dominant machines efficient polynomial time algorithms for finding the optimal schedules to minimize discounted total weighted completion time are developed

numerical accuracy of a pade type non reflecting boundary condition for the finite element solution of acoustic scattering problems at high frequency the present text deals with the numerical solution of two dimensional high frequency acoustic scattering problems using a new high order and asymptotic pade type artificial boundary condition the pade type condition is easy to implement in a galerkin least squares iterative finite element solver for arbitrarily convex shaped boundaries the method accuracy is investigated for different model problems and for the scattering problem by a submarine shaped scatterer as a result relatively small computational domains optimized according to the shape of the scatterer can be considered while yielding accurate computations for high frequencies copyright 2005 john wiley sons ltd

absorbing interface conditions for domain decomposition methods a general presentation the continuity conditions and the transmission conditions involved in domain decomposition methods are of major importance for the fast and robust convergence of these algorithms in this review paper we study step by step the methodology to derive various continuity conditions and compatible transmission conditions for two different types of problems i e the laplace equation and the helmholtz equation an original homogeneous formulation is also presented both in the continuous and in the discrete analysis numerical experiments show the relative efficiency of these continuity conditions and of these transmission conditions on academic problems

axial green s function method for multi dimensional elliptic boundary value problems axial green s function method is proposed to solve multi dimensional second order elliptic partial differential equations with the geometrical complexity of solution domains instead of directly dealing with 2d or 3d green s functions the proposed method systematically decomposes a multi dimensional elliptic differential operator into id ones consequently the method only requires id green s function associated with each coordinate axis theoretical formulation and simple axial discretization result in the system of equations of which the solutions represent the numerical solution of the original multi dimensional elliptic problem which is otherwise intractable to obtain by the traditional multi dimensional green s function technique the method employs analytical form of id green s function and integral schemes so that it can potentially be more robust than both the finite difference method and the boundary element method in addition the simple discretization procedure which uses only axially orthogonal straight lines and boundary data demonstrates its convenience over the finite element method of comparable accuracy numerical examples verify these advantages by demonstrating the second order convergence in various discretized models of complex geometries

numerically derived absorbing boundary condition for the solution of open region scattering problems an absorbing boundary condition is developed by means of a numerical approximation of the analytical behavior of the exact boundary condition the boundary operator is more accurate than other analytically derived differential operators having the same order and it can be applied to arbitrarily shaped scatterer geometries that can be handled most efficiently through the use of outer boundaries that conform to the body of the scatterer examples demonstrate the improvement in accuracy and efficiency achieved by the numerical boundary condition the enhancement in accuracy is attributable to the inclusion of the evanescent harmonics behavior in the model

a note on flow shop scheduling problems with a learning effect on no idle dominant machines abstract this paper considers some permutation flow shop scheduling problems with a learning effect on no idle dominant machines the objective is to minimize maximum completion time this objective is considered under following dominant machines constraint idm ddm idm ddm and ddm idm we present a polynomial time solution algorithm for each of the above cases

edged based smoothed point interpolation method for acoustic radiation with perfectly matched layer abstract it is known that for exterior acoustics the domain based finite element method always encounters the accuracy loss for high wavenumbers and the difficulty in dealing with the sommerfeld radiation condition in this work by using the polynomial and radial bases with specific node selection schemes the point interpolation method with the generalized gradient smoothing technique is employed to improve the accuracy for acoustic computations the appropriately softened acoustical stiffness of the discrete model is constructed based on the edge based smoothing domains via the gradient smoothing technique in addition the perfectly matched layer technique enables the smoothed point interpolation method to investigate exterior acoustics the complex coordinate transformation is utilized to determine the proper parameters for the perfectly matched layer through theoretical and numerical analyses several typical acoustic problems are analyzed revealing that better accuracy is obtained by the present method compared with the finite element method especially for high wavenumbers

three dimensional approximate local dtn boundary conditions for prolate spheroid boundaries we propose a new class of approximate local dtn boundary conditions to be applied on prolate spheroidal shaped exterior boundaries when solving problems of acoustic scattering by elongated obstacles these conditions are a exact for the first modes b easy to implement and to parallelize c compatible with the local structure of the computational finite element scheme and d applicable to exterior ellipsoidal shaped boundaries that are more suitable in terms of cost effectiveness for surrounding elongated scatterers we investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions we also compare their performance to the second order absorbing boundary condition bgt2 designed by bayliss gunzburger and turkel when expressed in prolate spheroid coordinates the analysis reveals that in the low frequency regime the new second order dtn condition dtn2 retains a good level of accuracy regardless of the slenderness of the boundary in addition the dtn2 boundary condition outperforms the bgt2 condition such superiority is clearly noticeable for large eccentricity values

sequencing games with controllable processing times in this paper we study a class of cooperative sequencing games that arise from sequencing situations in which the processing times are not fixed we obtain two core elements that depend only on the optimal schedule for the grand coalition furthermore we show that although these games are not convex in general many marginal vectors are core elements we also consider convexity for special instances of the sequencing situation

enriched isogeometric collocation for two dimensional time harmonic acoustics abstract in this work the isogeometric collocation iga c is paired with two types of enrichment plane wave pw and generalized harmonic polynomial ghp functions to solve 2d problems for the helmholtz equation a parametric study is conducted and a detailed assessment of the performance of the method in a number of benchmark problems is provided three different collocation methods are tested for the non enriched formulation namely greville abscissae ga approximated cauchy galerkin acg and superconvergent sc points showing that the acg scheme is the best choice in terms of overall error convergence rate and ease of solving the linear system then the influence of the number of shape functions in the original and enriched basis location and number of collocation points and wave number over both the convergence rate as well as the condition number of the stiffness matrix are studied the numerical results show that 1 there is an improvement over the non enriched formulation 2 the improvement depends on the choice of the number and type of enrichment and 3 the pollution error is not completely alleviated with the enriched formulations

a boundary condition in pade series for frequency domain solution of wave propagation in unbounded domains a boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain in a frequency domain analysis using finite elements this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary a method for determining a pade series of the dynamic stiffness matrix is proposed in this paper this method starts from the scaled boundary finite element equation which is a system of ordinary differential equations obtained by discretizing the boundary only the coefficients of the pade series are obtained directly from the ordinary differential equations which are not actually solved for the dynamic stiffness matrix the high rate of convergence of the pade series with increasing order is demonstrated numerically this technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry it can be combined seamlessly with standard finite elements copyright 2006 john wiley sons ltd

non overlapping domain decomposition algorithm based on modified transmission conditions for the helmholtz equation abstract a square root based transmission conditions domain decomposition method was recently introduced for the helmholtz equation it produces an effective algorithm where the convergence is independent of the wavenumber and the mesh discretization we modify here these conditions in order to guarantee well posedness of local problems and further improve the efficiency of the whole method numerical results in particular in the three dimensional case show significant reduction of the computational time needed in the iterative procedure while preserving the iteration number when compared with the original algorithm

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use