the discontinuous enrichment method for medium frequency helmholtz problems with a spatially variable wavenumber numerical dispersion or what is often referred to as the pollution effect presents a challenge to an efficient finite element discretization of the helmholtz equation in the medium frequency regime to alleviate this effect and improve the unsatisfactory pre asymptotic convergence of the classical galerkin finite element method based on piecewise polynomial basis functions several discretization methods based on plane wave bases have been proposed among them is the discontinuous enrichment method that has been shown to offer superior performance to the classical galerkin finite element method for a number of constant wavenumber helmholtz problems and has also outperformed two representative methods that use plane waves the partition of unity and the ultra weak variation formulation methods in this paper the discontinuous enrichment method is extended to the variable wavenumber helmholtz equation to this effect the concept of enrichment functions based on free space solutions of the homogeneous form of the governing differential equation is enlarged to include free space solutions of approximations of this equation obtained in this case by successive taylor series expansions of the wavenumber around a reference point this leads to plane wave enrichment functions based on the piece wise constant approximation of the wavenumber and to airy wave enrichment functions several elements based on these enrichment functions are constructed and evaluated on benchmark problems modeling sound hard scattering by a disk submerged in an acoustic fluid where the speed of sound varies in space all these elements are shown to outperform by a substantial margin their continuous polynomial counterparts

a partition of unity finite element method for three dimensional transient diffusion problems with sharp gradients abstract an efficient partition of unity finite element method for three dimensional transient diffusion problems is presented a class of multiple exponential functions independent of time variable is proposed to enrich the finite element approximations as a consequence of this procedure the associated matrix for the linear system is evaluated once at the first time step and the solution is obtained at subsequent time step by only updating the right hand side of the linear system this results in an efficient numerical solver for transient diffusion equations in three space dimensions compared to the conventional finite element methods with h refinement the proposed approach is simple more efficient and more accurate the performance of the proposed method is assessed using several test examples for transient diffusion in three space dimensions we present numerical results for a transient diffusion equation with known analytical solution to quantify errors for the new method we also solve time dependent diffusion problems in complex geometries we compare the results obtained using the partition of unity finite element method to those obtained using the standard finite element method it is shown that the proposed method strongly reduces the necessary number of degrees of freedom to achieve a prescribed accuracy

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

performances of the partition of unity finite element method for the analysis of two dimensional interior sound fields with absorbing materials the paper deals with the numerical simulation of the acoustic field in two dimensional cavities in which absorbing materials are present though finite element method fem could be employed for this purpose the discretization level required for achieving reasonable accuracy renders the method impractical in the mid frequency range to alleviate this limitation the partition of unity finite element method pufem using plane wave functions has been shown to be very effective for solving short wave helmholtz problems in the present work the method is extended to the computation of the pressure wave field within the absorbing media which is modeled as a bulk reacting material characterized by a complex valued and frequency dependent mean density and dynamic compressibility lagrange multipliers are used to enforce the transmission conditions at the air material interface performances of the pufem are compared with a standard fem in several examples of practical interests it is shown that the technique is a good candidate for solving noise control problems at high frequency

finite element approximation of a displacement formulation for time domain elastoacoustic vibrations this paper deals with the numerical solution of a system of second order in time partial differential equations modeling the vibrations of a coupling between an elastic solid and an inviscid compressible fluid both media are described in terms of their respective displacement fields the problem is solved in the time domain by combining a newmark s scheme for time discretization with a finite element method for space discretization the latter combines standard lagrangian elements in the solid with lowest order raviart thomas elements in the fluid stability is proved and numerical results showing the good behavior of the method are reported

transient analysis of sharp thermal gradients using coarse finite element meshes abstract this paper investigates the application of the generalized finite element method with global local enrichments gfemgl to problems of transient heat transfer involving localized features the gfemgl is utilized in order to numerically construct general specially tailored shape functions yielding high levels of accuracy on coarse fem meshes the use of time dependent shape functions requires that the system of equations be discretized temporally first and then spatially in order to properly account for the time dependency the standard method is used for the time integration scheme the transient three dimensional gfemgl is then applied to a laser heating example in order to demonstrate its ability to resolve localized transient features on a fixed coarse mesh convergence analysis of the proposed method as well as applications to heterogeneous materials and moving heat sources are also provided

the generalized finite element method for helmholtz equation part ii effect of choice of handbook functions error due to absorbing boundary conditions and its assessment abstract in part i t strouboulis i babuska r hidajat the generalized finite element method for helmholtz equation theory computation and open problems comput methods appl mech engrg 195 2006 4711 4731 we introduced the q version of the generalized finite element method gfem for the helmholtz equation and we addressed its a pollution error due to the wave number b exponential q convergence c robustness to perturbations of the mesh the roundoff and numerical quadrature errors and d a posteriori error estimation here we continue the development of the gfem for helmholtz and we address the effects of a alternative handbook functions and mesh types b the error due to the artificial truncation boundary conditions and its assessment the conclusions are 1 the employment of plane wave wave band and vekua handbook functions lead to equivalent results and 2 for high q the most significant component of error may be the one due to the artificial truncation boundary conditions a rather straightforward approach for assessing this error is proposed

finite element vibration analysis of fluid solid systems without spurious modes this paper deals with the finite element approximation of the vibration modes of a problem with fluid structure interaction displacement variables are used for both the fluid and the solid to avoid the typical spurious modes of this formulation we introduce a nonconforming discretization error estimates for the approximation of eigenvalues and eigenvectors are given

time independent hybrid enrichment for finite element solution of transient conduction radiation in diffusive grey media we investigate the effectiveness of the partition of unity finite element method for transient conduction radiation problems in diffusive grey media the governing equations consist of a semi linear transient heat equation for the temperature field and a stationary diffusion approximation to the radiation in grey media the coupled equations are integrated in time using a semi implicit method in the finite element framework we show that for the considered problems a combination of hyperbolic and exponential enrichment functions based on an approximation of the boundary layer leads to improved accuracy compared to the conventional finite element method it is illustrated that this approach can be more efficient than using h adaptivity to increase the accuracy of the finite element method near the boundary walls the performance of the proposed partition of unity method is analyzed on several test examples for transient conduction radiation problems in two space dimensions

mixed enrichment for the finite element method in heterogeneous media summary problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media these problems are usually very demanding in terms of computations with the conventional finite element method on the other hand different enriched finite element methods such as the partition of unity which proved to be very successful in treating similar problems are developed and studied for homogeneous media in this work we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials the idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method although presented within the partition of unity context the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials copyright 2014 john wiley sons ltd

wave interpolation finite elements for helmholtz problems with jumps in the wave speed finite elements for short wave scattering problems have recently been developed by various authors these have almost exclusively dealt with the helmholtz equation the elements have been very successful in terms of drastic reductions of the number of degrees of freedom in the numerical model however most of them are not directly applicable to problems in which the wave speed is not constant but varies with position many important wave problems fall into this latter category this is because short waves are often present in materials whose properties vary in space the present paper demonstrates how the method may be extended so as to deal with problems in which the wave speed is piecewise constant in various regions of the problem domain lagrange multipliers are used to enforce the necessary conditions of compatibility between the different regions the paper gives numerical results for problems for which the analytical solution is known this shows how these methods may be extended in a relatively simple fashion to solve a much larger class of wave problems of great practical interest

finitely repeated games etude systematique des equilibres parfaits de sous jeux pour des jeux repetes de facon finie avec information complete

the folk theorem for repeated games a neu condition we are concerned here with perfect folk theorems for infinitely repeated games with complete information folk theorems assert that any feasible and individually rational payoff vector of the stage game is a subgame perfect equilibrium payoff in the associated infinitely repeated game with little or no discounting where payoff streams are evaluated as average discounted or average values respectively it is obvious that feasibility and individual rationality are necessary conditions for a payoff vector to be an equilibrium payoff the surprising content of the folk theorems is that these conditions are also almost sufficient perhaps the first folk theorem type result is due to friedman 1971 who showed that any feasible payoff which pareto dominates a nash equilibrium payoff of the stage game will be an equilibrium payoff in the associated repeated game with sufficiently patient players this kind of result is sometimes termed a nash threats folk theorem a reference to its method of proof for the more permissive kinds of folk theorems considered here the seminal results are those of aumann and shapley 1976 and rubinstein 1977 1979 these authors assume that payoff streams are undiscounted 2 fudenberg and maskin 1986 establish an analogous result for discounted repeated games as the discount factor goes to 1 their result uses techniques of proof rather different from those used by aumann shapley and rubinstein respectively see their paper for an insightful discussion of this point and quite generally for more by way of background it is a key reference for subsequent work in this area including our own for the two player case the result of fudenberg and maskin 1986 is a complete if and only if characterization modulo the requirement of strict rather than weak individual rationality which we retain in this note and does not employ additional conditions for three or more players fudenberg and maskin introduced a full dimensionality condition the convex hull f of the set of feasible payoff vectors of the stage game must have dimension n where n is the number of players or equivalently a nonempty interior this condition has been widely adopted in proving folk theorems for related environments such as finitely repeated games benoit and krishna 1985 and overlapping generations games kandori 1992 smith 1992 full dimensionality is a sufficient condition fudenberg and maskin present an example of a three player stage game in which the conclusion of the folk theorem is false in this example all players receive the same payoffs in all contingencies the convex hull of the set of feasible payoffs is one dimensional this example violates full dimensionality in a rather extreme way less extreme violations may also lead to

modelling of short wave diffraction problems using approximating systems of plane waves this paper describes a finite element model for the solution of helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element the approach is based on partition of unity isoparametric elements at each finite element node the potential is expanded in a discrete series of planar waves each propagating at a specified angle these angles can be uniformly distributed or may be carefully chosen they can also be the same for all nodes of the studied mesh or may vary from one node to another the implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres the wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced issues related to the integration and the conditioning are also discussed

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

pollution studies for high order isogeometric analysis and finite element for acoustic problems abstract it is well known that galerkin finite element methods suffer from pollution error when solving wave problems to reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave like functions so that the exact wavenumber is incorporated into the finite element approximation space solving wave problems with isogeometric analysis was also investigated in the literature where the superior behaviour of isogeometric analysis due to higher continuity in the underlying basis has been studied recently a plane wave enriched isogeometric analysis was introduced for acoustic problems however it remains unquantified the impact of these different approaches on the pollution or how they perform compared to each other in this work we show that isogeometric analysis outperforms finite element method in dealing with pollution we observe similar behaviour when both the methods are enriched with plane waves using higher order polynomials with fewer enrichment functions seems to improve the pollution compared to lower order polynomials with more functions however the latter still leads to smaller errors using similar number of degrees of freedom in conclusion we propose that partition of unity isogeometric analysis can be an efficient tool for wave problems as enrichment eliminates the need for domain re meshing at higher frequencies and also due to its ability to capture the exact geometry even on coarse meshes as well as its improved pollution behaviour

generalized finite element analysis of three dimensional heat transfer problems exhibiting sharp thermal gradients abstract in this paper heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods the effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated convergence studies show that low order linear and quadratic elements require strongly refined meshes for acceptable accuracy we propose a generalized fem with global local enrichments for the class of problems investigated in the paper in this procedure a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized fem with solutions of local boundary value problems the local problems are defined using the same procedure as in the global local fem where boundary conditions are provided by a coarse scale global solution coarse uniform global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence the use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem the two way information transfer provided by the proposed generalized fem is appealing to several classes of problems especially those involving multiple spatial scales the proposed methodology brings the benefits of generalized fem to problems where limited or no information about the solution is known a priori

a partition of unity fem for time dependent diffusion problems using multiple enrichment functions an enriched partition of unity fem is developed to solve time dependent diffusion problems in the present formulation multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space the resulting enrichment is in the form of a local asymptotic expansion unlike previous works in this area where the enrichment must be updated at each time step here the temporal decay in the solution is embedded in the asymptotic expansion thus the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right hand side of the linear system of equations the advantage is a significant saving in the computational effort where previously the linear system must be reevaluated and resolved at every time step in comparison with the traditional finite element analysis with p version refinements the present approach is much simpler more efficient and yields more accurate solutions for a prescribed number of dofs numerical results are presented for a transient diffusion equation with known analytical solution the performance of the method is analyzed on two applications the transient heat equation with a single source and the transient heat equation with multiple sources the aim of such a method compared with the classical fem is to solve time dependent diffusion applications efficiently and with an appropriate level of accuracy

elasto acoustic and acoustic acoustic coupling on non matching grids flexible discretization techniques for the approximative solution of coupled wave propagation problems are investigated focussing on aero acoustic and elasto acoustic coupling in particular the advantages of using non matching grids are presented when one subregion has to be resolved by a substantially finer grid than the other subregion for the elasto acoustic coupling the problem formulation remains essentially the same as for the matching situation while for the aero acoustic coupling the formulation is enhanced with lagrange multipliers within the framework of mortar finite element methods several numerical examples are presented to demonstrate the flexibility and applicability of the approach copyright 2006 john wiley sons ltd

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

a discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid solid problems abstract an evanescent wave is produced when a propagating incident wave impinges on an interface between two media or materials at a sub critical angle the sub scale nature of such a wave makes it difficult to be captured computationally in this paper the discontinuous enrichment method dem developed in c farhat i harari l p franca the discontinuous enrichment method comput methods appl mech engrg 190 2001 6455 6479 c farhat i harari u hetmaniuk a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime comput methods appl mech engrg 192 2003 1389 1419 c farhat i harari u hetmaniuk the discontinuous enrichment method for multiscale analysis comput methods appl mech engrg 192 2003 3195 3210 c farhat r tezaur p weidemann goiran higher order extensions of a discontinuous galerkin method for mid frequency helmholtz problems int j numer methods engrg 61 2004 1938 1956 c farhat p wiedemann goiran r tezaur a discontinuous galerkin method with plane waves and lagrange multipliers for the solution of short wave exterior helmholtz problems on unstructured meshes new computational methods for wave propagation wave motion 39 4 2004 307 317 l zhang r tezaur c farhat the discontinuous enrichment method for elastic wave propagation in the medium frequency regime int j numer methods engrg 66 2006 2086 2114 r tezaur c farhat three dimensional discontinuous galerkin elements with plane waves and lagrange multipliers for the solution of mid frequency helmholtz problems int j numer methods engrg 66 2006 796 815 is extended to a class of evanescent wave problems new dem elements are constructed by enriching polynomial approximations with free space evanescent solutions in order to achieve high accuracy for problems with fluid fluid or fluid solid interfaces on which evanescent waves may occur the new dem elements are applied to the solution of various two dimensional model problems with evanescent waves their performance is observed to be better than that of the basic helmholtz type dem elements and superior to that of the classical higher order polynomial finite element method

time dependent shape functions for modeling highly transient geothermal systems this paper presents new time dependent finite element shape functions suitable for modeling high gradient transient conductive heat flow in geothermal systems the shape functions are made adaptive by enhancing the approximation functions with time dependent variables which may vary according to the transient process without adding extra degrees of freedom or applying mesh adaptation two different approaches are presented first an iterative method is proposed in which an exponential approximation function which is optimized continually during the transient process is incorporated in the shape function second an analytical method is suggested in which an analytical solution of a simplified process is incorporated in the shape function enabling an explicit update of the shape functions in each time step a methodology for modeling the variation of temperature in one and two dimensions is introduced the ability of the method to capture high gradient temperature profiles using relatively large elements is illustrated with numerical examples of cases in which equally large standard finite elements fail copyright 2008 john wiley sons ltd

an enriched finite element model with q refinement for radiative boundary layers in glass cooling radiative cooling in glass manufacturing is simulated using the partition of unity finite element method the governing equations consist of a semi linear transient heat equation for the temperature field and a stationary simplified p 1 approximation for the radiation in non grey semitransparent media to integrate the coupled equations in time we consider a linearly implicit scheme in the finite element framework a class of hyperbolic enrichment functions is proposed to resolve boundary layers near the enclosure walls using an industrial electromagnetic spectrum the proposed method shows an immense reduction in the number of degrees of freedom required to achieve a certain accuracy compared to the conventional h version finite element method furthermore the method shows a stable behaviour in treating the boundary layers which is shown by studying the solution close to the domain boundaries the time integration choice is essential to implement a q refinement procedure introduced in the current study the enrichment is refined with respect to the steepness of the solution gradient near the domain boundary in the first few time steps and is shown to lead to a further significant reduction on top of what is already achieved with the enrichment the performance of the proposed method is analysed for glass annealing in two enclosures where the simplified p 1 approximation solution with the partition of unity method the conventional finite element method and the finite difference method are compared to each other and to the full radiative heat transfer as well as the canonical rosseland model

an interface enriched generalized fem for problems with discontinuous gradient fields summary a new generalized fem is introduced for solving problems with discontinuous gradient fields the method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges the proposed approach has several advantages over conventional generalized fem formulations such as a lower computational cost easier implementation and straightforward handling of dirichlet boundary conditions a detailed convergence study of the proposed method and a comparison with the standard fem are presented for heat transfer problems the method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard fem with conforming meshes various application problems are presented including the conjugate heat transfer problem encountered in microvascular materials copyright 2011 john wiley sons ltd

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

a robust weakly compressible sph method and its comparison with an incompressible sph summary this paper presents a comparative study for the weakly compressible wcsph and incompressible isph smoothed particle hydrodynamics methods by providing numerical solutions for fluid flows over an airfoil and a square obstacle improved wcsph and isph techniques are used to solve these two bluff body flow problems it is shown that both approaches can handle complex geometries using the multiple boundary tangents mbt method and eliminate particle clustering induced instabilities with the implementation of a particle fracture repair procedure as well as the corrected sph discretization scheme wcsph and isph simulation results are compared and validated with those of a finite element method fem the quantitative comparisons of wcsph isph and fem results in terms of strouhal number for the square obstacle test case and the pressure envelope surface traction forces and velocity gradients on the airfoil boundaries as well as the lift and drag values for the airfoil geometry indicate that the wcsph method with the suggested implementation produces numerical results as accurate and reliable as those of the isph and fem methods copyright 2011 john wiley sons ltd

identifying the wavenumber for the inverse helmholtz problem using an enriched finite element formulation abstract we investigate the inverse problem of identifying the wavenumber for the helmholtz equation the problem solution is based on measurements taken at few points from inside the computational domain or on its boundary a novel iterative approach is proposed based on coupling the secant and the descent methods with the partition of unity method starting from an initial guess for the unknown wavenumber the forward problem is solved using the partition of unity method then the secant descent methods are used to improve the initial guess by minimizing a predefined objective function based on the difference between the solution and a set of data points in the next round of iterations the improved wavenumber estimate is used for the forward problem solution and the partition of unity approximation is improved by adding more enrichment functions the iterative process is terminated when the objective function has converged and a set of two predefined tolerances are met to evaluate the estimate accuracy we propose to utilize extra data points to validate the approach and test its efficiency two wave applications with known analytical solutions are studied the results show that the proposed approach can achieve high accuracy for the studied applications even when the considered data is contaminated with noise despite the clear advantages that were previously shown in the literature for solving the forward helmholtz problem this work presents a first attempt to solve the inverse helmholtz problem with an enriched finite element approach

finite element computation of the vibration modes of a fluid solid system abstract in this paper we solve the interior elastoacoustic problem by a finite element method which does not present spurious or circulation modes for nonzero frequencies it consists of classical triangular lagrangian elements for the solid and lowest order triangular raviart thomas elements for the fluid transmission conditions at the fluid solid interface are taken into account in a weak sense numerical results for some test examples are presented which show the good performance of the proposed methodology