generalized parametric solutions in stokes flow abstract design optimization and uncertainty quantification among other applications of industrial interest require fast or multiple queries of some parametric model the proper generalized decomposition pgd provides a separable solution a computational vademecum explicitly dependent on the parameters efficiently computed with a greedy algorithm combined with an alternated directions scheme and compactly stored this strategy has been successfully employed in many problems in computational mechanics the application to problems with saddle point structure raises some difficulties requiring further attention this article proposes a pgd formulation of the stokes problem various possibilities of the separated forms of the pgd solutions are discussed and analyzed selecting the more viable option the efficacy of the proposed methodology is demonstrated in numerical examples for both stokes and brinkman models

convolutional perfectly matched layer for elastic second order wave equation in this work a method is presented to extend the convolutional perfectly matched layer c pml to simulate acoustic wave propagation in elastic media with a second order equation formulation this non physical layer is used at the computational edge of a finite element method algorithm in frequency domain and a pseudo spectral algorithm in time domain as an absorbing boundary condition abc to truncate unbounded media numerical results show that the c pml abc attenuates the outgoing surface waves more effectively than classical pml abc as proposed by berenger j comput phys 114 195 200 1994 for electromagnetic waves in the case of oblique incidence where the pml method suffers from large spurious reflections moreover a modification of the proposed c pml formulation is also discussed in order to stabilize the absorbing layer in anisotropic solids where numerical instabilities can appear

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

a comparison of high order polynomial and wave based methods for helmholtz problems the application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation two common strategies to address this issue are to use high order polynomial shape functions e g hp fem or to use physics based or trefftz methods where the shape functions are local solutions of the problem typically plane waves both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite element methods but they have yet to be compared in this paper a high order polynomial method p fem with lobatto polynomials and the wave based discontinuous galerkin method are compared for two dimensional helmholtz problems a number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties performance and conditioning it is generally assumed that a wave based method naturally provides better accuracy compared to polynomial methods since the plane waves or bessel functions used in these methods are exact solutions of the helmholtz equation results indicate that this expectation does not necessarily translate into a clear benefit and that the differences in performance accuracy and conditioning are more nuanced than generally assumed the high order polynomial method can in fact deliver comparable and in some cases superior performance compared to the wave based dgm in addition to benchmarking the intrinsic computational performance of these methods a number of practical issues associated with realistic applications are also discussed

certified reduced basis method for electromagnetic scattering and radar cross section estimation we study nontrivial applications of the reduced basis method rbm for electromagnetic applications with an emphasis on scattering and the estimation of radar cross section rcs the method and several extensions are explained with two examples with different characteristics parameters that are allowed to vary within the model include frequency incident angle and measurement angle as well as the geometry of the scatterers with appropriate applications of the empirical interpolation method eim transformation of the domain configuration of perfectly matched layer exponential convergence of the reduced basis solution over the entire parameter domain is achieved moreover we demonstrate that this approach allows for the effective capture of the critical behavior in this case through shapes that minimize scattering this further highlights the robustness and quality of the greedy approximation and the reduced basis method approach

analytical approach to the stochastic fitzhugh nagumo system and coherence resonance we consider the fitzhugh nagumo system under the influence of white gaussian noise in the excitable regime we present an analytical approximation in the limit of fast activator time scale marginal probability densities of a reduced system and dynamical quantities such as the pulse rate are found and the mean interspike interval and its relative standard deviation are investigated the latter quantities allow a quantitative description of the phenomenon of coherence resonance as comparisons with simulations show

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

finite number fluctuations in a three state model with growing population abstract we study the role of the finite number of units in a three state model with a growing population the model presents two attractors namely a fixed point and a limit cycle in the mean field approximation it has been shown that there is a bistability region for which both attractors are stable we observe that contrary to constant population models the bistability is preserved in the presence of finite number fluctuations

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

the latin multiscale computational method and the proper generalized decomposition this paper deals with the synergy between the latin multiscale method and what is called the proper generalized decomposition pgd which is the key of its performances

stability and accuracy of finite element methods for flow acoustics i general theory and application to one dimensional propagation the dispersion properties of finite element models for aeroacoustic propagation based on the convected scalar helmholtz equation and on the galbrun equation are examined the current study focusses on the effect of the mean flow on the dispersion and amplitude errors present in the discrete numerical solutions a general two dimensional dispersion analysis is presented for the discrete problem on a regular unbounded mesh and results are presented for the particular case of one dimensional acoustic propagation in which the wave direction is aligned with the mean flow the magnitude and sign of the mean flow is shown to have a significant effect on the accuracy of the numerical schemes quadratic helmholtz elements in particular are shown to be much less effective for downstream as opposed to upstream propagation even when the effect of wave shortening or elongation due to the mean flow is taken into account these trends are also observed in solutions obtained for simple test problems on finite meshes a similar analysis of two dimensional propagation is presented in an accompanying article

discontinuous galerkin methods with plane waves for time harmonic problems a general framework for discontinuous galerkin methods in the frequency domain with numerical flux is presented the main feature of the method is the use of plane waves instead of polynomials to approximate the solution in each element the method is formulated for a general system of linear hyperbolic equations and is applied to problems of aeroacoustic propagation by solving the two dimensional linearized euler equations it is found that the method requires only a small number of elements per wavelength to obtain accurate solutions and that it is more efficient than high order drp schemes in addition the conditioning of the method is found to be high but not critical in practice it is shown that the ultra weak variational formulation is in fact a subset of the present discontinuous galerkin method a special extension of the method is devised in order to deal with singular solutions generated by point sources like monopoles or dipoles aeroacoustic problems with non uniform flows are also considered and results are presented for the sound radiated from a two dimensional jet

from h to p efficiently implementing finite and spectral hp element methods to achieve optimal performance for low and high order discretisations the spectral hp element method can be considered as bridging the gap between the traditionally low order finite element method on one side and spectral methods on the other side consequently a major challenge which arises in implementing the spectral hp element methods is to design algorithms that perform efficiently for both low and high order spectral hp discretisations as well as discretisations in the intermediate regime in this paper we explain how the judicious use of different implementation strategies can be employed to achieve high efficiency across a wide range of polynomial orders furthermore based upon this efficient implementation we analyse which spectral hp discretisation which specific combination of mesh size h and polynomial order p minimises the computational cost to solve an elliptic problem up to a predefined level of accuracy we investigate this question for a set of both smooth and non smooth problems

proper generalized decomposition of time multiscale models summary models encountered in computational mechanics could involve many time scales when these time scales cannot be separated one must solve the evolution model in the entire time interval by using the finest time step that the model implies in some cases the solution procedure becomes cumbersome because of the extremely large number of time steps needed for integrating the evolution model in the whole time interval in this paper we considered an alternative approach that lies in separating the time axis one dimensional in nature in a multidimensional time space then for circumventing the resulting curse of dimensionality the proper generalized decomposition was applied allowing a fast solution with significant computing time savings with respect to a standard incremental integration copyright 2011 john wiley sons ltd

schwarz pick estimates for holomorphic mappings from the polydisk to the unit ball abstract in this paper schwarz pick estimates of arbitrary order partial derivatives for holomorphic mappings from the polydisk to the unit ball are presented we generalize the early work on schwarz pick estimates of higher order derivatives for bounded holomorphic functions on the unit disk and the polydisk

parametric solutions involving geometry a step towards efficient shape optimization abstract optimization of manufacturing processes or structures involves the optimal choice of many parameters process parameters material parameters or geometrical parameters usual strategies proceed by defining a trial choice of those parameters and then solving the resulting model then an appropriate cost function is evaluated and its optimality checked while the optimum is not reached the process parameters should be updated by using an appropriate optimization procedure and then the model must be solved again for the updated process parameters thus a direct numerical solution is needed for each choice of the process parameters with the subsequent impact on the computing time in this work we focus on shape optimization that involves the appropriate choice of some parameters defining the problem geometry the main objective of this work is to describe an original approach for computing an off line parametric solution that is a solution able to include information for different parameter values and also allowing to compute readily the sensitivities the curse of dimensionality is circumvented by invoking the proper generalized decomposition pgd introduced in former works which is applied here to compute geometrically parametrized solutions

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

perfectly matched transmission problem with absorbing layers application to anisotropic acoustics in convex polygonal domains this paper presents an original approach to design perfectly matched layers pml for anisotropic scalar wave equations this approach is based first on the introduction of a modified wave equation and second on the formulation of general perfectly matched transmission conditions for this equation the stability of the transmission problem is discussed by way of the adaptation of a high frequency stability necessary condition and we apply our approach to construct pml suited for any convex domain with straight boundaries a new variational formulation of the problem including a lagrange multiplier at the interface between the physical and the absorbing domains is then set and numerical results are presented in 2d and 3d these results show the efficiency of our approach when using constant damping coefficients combined with high order elements

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

globally coupled stochastic two state oscillators synchronization of infinite and finite arrays we consider arrays of the simplest two state on off stochastic units the units are markovian that is the transitions between the two states occur at a given rate we construct arrays of n globally coupled binary units and observe a remarkable richness of behavior as the control parameter that measures the coupling strength is increased in the mean field limit as we consider the four simplest polynomial forms of coupling that lead to bifurcations and characterize the associated phase transitions of the arrays when n is finite there are fluctuations about the well defined steady states of the infinite arrays we study the nature of these fluctuations and their effects on the bifurcations in all cases by constructing the appropriate langevin equations and the associated fokker planck equations

the partition of unity finite element method for short wave acoustic propagation on non uniform potential flows a novel numerical method is proposed for modelling time harmonic acoustic propagation of short wavelength disturbances on non uniform potential flows the method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space the basis functions are local solutions of the governing equations they are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non uniform many wavelengths can be included within a single element leading to ultra sparse meshes results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid based schemes numerical results for lined uniform two dimensional ducts and for non uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved numerical studies indicate that the pollution effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated

non overlapping domain decomposition method for a nodal finite element method a new approach is proposed for constructing nonoverlapping domain decomposition procedures for solving a linear system related to a nodal finite element method it applies to problems involving either positive semi definite or complex indefinite local matrices the main feature of the method is to preserve the continuity requirements on the unknowns and the finite element equations at the nodes shared by more than two subdomains and to suitably augment the local matrices we prove that the corresponding algorithm can be seen as a converging iterative method for solving the finite element system and that it cannot break down each iteration is obtained by solving uncoupled local finite element systems posed in each subdomain and in contrast to a strict domain decomposition method is completed by solving a linear system whose unknowns are the degrees of freedom attached to the above special nodes

synchronization of weakly perturbed markov chain oscillators rate processes are simple and analytically tractable models for many dynamical systems that switch stochastically between a discrete set of quasistationary states however they may also approximate continuous processes by coarse grained symbolic dynamics in contrast to limit cycle oscillators that are weakly perturbed by noise in such systems stochasticity may be strong and topologies more complicated than a circle can be considered here we apply a second order time dependent perturbation theory to derive expressions for the mean frequency and phase diffusion constant of discrete state oscillators coupled or driven through weakly time dependent transition rates we also describe a method of global control to optimize the response of the mean frequency in complex transition networks

special short wave elements for flow acoustics a numerical formulation based on the partition of unity method pum is proposed for modelling the propagation of short acoustic waves on irrotational mean flows the method seeks to reduce the pollution error which exists in conventional fe schemes at high frequencies by using a local basis which is enriched by plane wave solutions of the convected helmholtz equation initially the method is demonstrated with reference to a one dimensional model consisting of a variable area converging diverging duct with mean flow next a simple two dimensional model of a straight duct with uniform flow is considered in both cases the accuracy and the conditioning of the numerical solution is investigated for ranges of frequency and mach number characteristic of aero engine bypass ducts

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

boundary schwarz lemma for pluriharmonic mappings between unit balls abstract in this paper we get a schwarz lemma for pluriharmonic mappings between the unit balls of any dimensions which generalizes classical schwarz lemma for bounded harmonic functions to higher dimensions as an application of the schwarz lemma obtained a boundary schwarz lemma is established for pluriharmonic mappings between unit balls with any dimensions

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

synchronization and fluctuations coupling a finite number of stochastic units it is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit infinite number of elements in fact since the early work of kuramoto mean field theory has been used to analyze this transition in contrast work that directly deals with finite arrays is relatively scarce in the context of synchronization and yet it is worth noting that finite number effects should be seriously taken into account since in general the limits n where n is the number of units and t where t is time do not commute mean field theory implements the particular choice first n and then t here we analyze an ensemble of three state coupled stochastic units which has been widely studied in the thermodynamic limit we formally address the finite n problem by deducing a fokker planck equation that describes the system we compute the steady state solution of this fokker planck equation that is finite n but t we use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions

continued fraction absorbing boundary conditions for convex polygonal domains continued fraction absorbing boundary conditions cfabcs are highly effective boundary conditions for modelling wave absorption into unbounded domains they are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries in this paper cfabcs are extended to the more general case of polygonal computational domains the key to the current development is the surprising link found between the cfabcs and the complex co ordinate stretching of perfectly matched layers pmls this link facilitates the extension of cfabcs to oblique corners and thus to polygonal domains it is shown that the proposed cfabcs are easy to implement expected to perform better than pmls and are effective for general polygonal computational domains in addition to the derivation of cfabcs a novel explicit time stepping scheme is developed for efficient numerical implementation numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain although this paper focuses on wave propagation its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients copyright 2005 john wiley sons ltd

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

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

non markovian approach to globally coupled excitable systems we consider stochastic excitable units with three discrete states each state is characterized by a waiting time density function this approach allows for a non markovian description of the dynamics of separate excitable units and of ensembles of such units we discuss the emergence of oscillations in a globally coupled ensemble with excitatory coupling in the limit of a large ensemble we derive the non markovian mean field equations nonlinear integral equations for the populations of the three states we analyze the stability of their steady solutions collective oscillations are shown to persist in a large parameter region beyond supercritical and subcritical hopf bifurcations we compare the results with simulations of discrete units as well as of coupled fitzhugh nagumo systems

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

multiscale model reduction for fluid infiltration simulation through dual continuum porous media with localized uncertainties abstract here we present some reduced basis rb methods for fluid infiltration problems through certain porous media modeled as dual continuum with localized uncertainties we apply dimension reduction techniques to construct a reduced order model in the rb methods to perform the offline online computation decomposition the model inputs need to be affinely dependent on the uncertainties we develop a proper orthogonal decomposition and greedy pod greedy rb method for stochastic dual continuum models in the pod greedy rb framework for heterogeneous porous media we need to solve the stochastic dual continuum models many times using very fine grid to construct a set of snapshots for building optimal reduced basis this offline computation may be very expensive to improve the offline computational efficiency we further develop a local global rb method which integrates the coupled multiscale and multicontinuum approach using generalized multiscale finite element method gmsfem to the pod greedy rb method to illustrate the efficiency of the proposed methods we present two numerical examples for stochastic dual continuum models our numerical results show that both the pod greedy rb method and the local global rb method greatly improve the computation efficiency with high approximation accuracy

growth of domains under global constraints abstract we develop growth rates for domains in bistable systems far from equilibrium the governing equation is a bistable reaction diffusion equation including a nonlocal inhibiting term the latter one will be responsible for the existence of stable localized domains the growth of the size of different domains shows quite surprising similarities with coalescence phenomena of droplets in van der waals gases eventually we discuss the possibility of breathing of the domains induced by a time delayed nonlocal interaction

a few new facts about infinite elements we discuss the conjugated bubnov galerkin and petrov galerkin infinite element ie discretizations to helmholtz equation including the use of elements of locally variable order optimal choice of ie shape functions calculation of echo area ea and automatic hp adaptivity the discussion is illustrated with 2d numerical experiments

recent advances on the use of separated representations separated representations based on finite sum decompositions constitute an appealing strategy for reducing the computer resources and the calculation costs by reducing drastically the number of degrees of freedom that the functional approximations involve the number of degrees of freedom scale linearly with the dimension of the space in which the model is defined instead of the exponential growing characteristic of mesh based discretization strategies in our knowledge the use of separated representations is the only possibility for circumventing the terrific curse of dimensionality related to some highly multidimensional models involving hundreds of dimensions as we proved in some of our former works its application is not restricted to multidimensional models obviously separated representation can also be applied in standard 2d or 3d models allowing for high resolution computations because its early life numerous issues persist many of them attracting the curiosity of many research groups within the computational mechanics community in this paper we are focusing in two issues never until now addressed i the imposition of non homogenous essential boundary conditions and ii the consideration of complex geometries copyright 2009 john wiley sons ltd

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

an error estimator for separated representations of highly multidimensional models fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics most of these models consist of a partial differential equation defined in a highly multidimensional domain e g schrodinger equation fokker planck equations among many others the main challenge related to these models is their associated curse of dimensionality we proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations also known as finite sums decomposition this technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates the issue related to error estimation has never been addressed this paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques

perfectly matched layers for time harmonic elastodynamics of unbounded domains theory and finite element implementation abstract one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies this paper develops the pml concept for time harmonic elastodynamics in cartesian coordinates utilising insights obtained with electromagnetics pmls and presents a novel displacement based symmetric finite element implementation of the pml for time harmonic plane strain or three dimensional motion the pml concept is illustrated through the example of a one dimensional rod on elastic foundation and through the anti plane motion of a two dimensional continuum the concept is explored in detail through analytical and numerical results from a pml model of the semi infinite rod on elastic foundation and through numerical results for the anti plane motion of a semi infinite layer on a rigid base numerical results are presented for the classical soil structure interaction problems of a rigid strip footing on a i half plane ii layer on a half plane and iii layer on a rigid base the analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by pml models even with small bounded domains

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

assessment of finite and spectral element shape functions for efficient iterative simulations of interior acoustics several shape function families currently used in the spectral element and the finite element method fem are reviewed widely known shape functions such as the conventional lagrange functions various p fem shapes and recent developments in the area of spectral methods are compared especially shape functions derived from bernstein polynomials are discussed due to their ease of formulation for almost any geometric entity their applicability and efficiency for acoustic simulations are assessed in particular the efficiency is measured in terms of performance of current krylov solvers

a spectral finite element approach to modeling soft solids excited with high frequency harmonic loads abstract an approach for efficient and accurate finite element analysis of harmonically excited soft solids using high order spectral finite elements is presented and evaluated the helmholtz type equations used to model such systems suffer from additional numerical error known as pollution when excitation frequency becomes high relative to stiffness i e high wave number which is the case for example for soft tissues subject to ultrasound excitations the use of high order polynomial elements allows for a reduction in this pollution error but requires additional consideration to counteract runge s phenomenon and or poor linear system conditioning which has led to the use of spectral element approaches this work examines in detail the computational benefits and practical applicability of high order spectral elements for such problems the spectral elements examined are tensor product elements i e quad or brick elements of high order lagrangian polynomials with non uniformly distributed gauss lobatto legendre nodal points a shear plane wave example is presented to show the dependence of the accuracy and computational expense of high order elements on wave number then a convergence study for a viscoelastic acoustic structure interaction finite element model of an actual ultrasound driven vibroacoustic experiment is shown the number of degrees of freedom required for a given accuracy level was found to consistently decrease with increasing element order however the computationally optimal element order was found to strongly depend on the wave number

on wave based weak trefftz discontinuous galerkin approach for medium frequency heterogeneous helmholtz problem abstract medium frequency heterogeneous helmholtz problem gives rise to severe numerical challenge to finite element method fem an extended variational theory of complex rays vtcr is developed for solving this problem since vtcr is kind of trefftz method analytic solutions of governing equation need to be known a priori but this requirement limits the application of vtcr to some specific forms of governing equations recently a new computational method named weak trefftz discontinuous galerkin method wtdg is proposed to enable hybrid finite element method fem variational theory of complex rays vtcr strategies to be developed easily in acoustics and vibrations this hybrid method combines the advantages of fem and vtcr to solve efficiently low frequency and medium frequency problems this paper proposes a wave based wtdg for heterogeneous helmholtz problem without knowing analytic solution of governing equation wtdg method provides a simple and general resolution strategy for medium frequency heterogeneous helmholtz problem

proper generalized decomposition for parameterized helmholtz problems in heterogeneous and unbounded domains application to harbor agitation abstract solving the helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge this is due to the particular nature of the helmholtz operator and the sensibility of the solution to small variations of the data here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency moreover this is applied to a real engineering problem water agitation inside real harbors for low to mid high frequencies the proper generalized decomposition pgd model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency here its applicability to such a problem is discussed and demonstrated more precisely the contributions of the paper include the pgd implementation into a perfectly matched layer framework to model the unbounded domain and the separability of the operator which is addressed here using an efficient higher order projection scheme then the performance of the pgd in this framework is discussed and improved using the higher order projection and a petrov galerkin approach to construct the separated basis moreover the efficiency of the higher order projection scheme is demonstrated and compared with the higher order singular value decomposition

optimizing the perfectly matched layer the berenger perfectly matched layer or pml 4 3 is used in computational electromagnetism as a sponge layer to terminate finite element approximations of scattering problems an infinite perfectly matched layer creates no reflection for incident waves of any frequency or any incident direction and the waves decay exponentially in magnitude into the layer this property is lost when the layer is truncated since the truncation boundary generates a reflected wave discretization of the differential equations further changes the reflection coefficient of the layer in this talk we shall describe the pml in the simple case of a two dimensional electromagnetic problem the first part of this abstract is expository we shall show how to derive the layer using a change of variable approach 5 9 6 after this derivation we describe some new analysis to show the effects of truncating and discretizing the pml 9 we end by showing how to compute an optimal layer and make some remarks about pml for the full maxwell system

the variational theory of complex rays a predictive tool for medium frequency vibrations abstract today the main numerical modeling techniques for the analysis of medium frequency vibrations are all based on finite element or boundary element approaches in order to represent small wavelength phenomena in complex structures such as car chassis satellites or ships these techniques require a huge number of degrees of freedom at least seven elements per wavelength are required to represent oscillating solutions in addition the solution obtained is highly sensitive to material properties and boundary conditions the use of high frequency approaches such as the statistical energy analysis sea or any of its improvements does not appear to be suitable for medium frequency vibrations the vibrational behavior becomes too smooth and in general the coupling loss factor cannot be calculated in a predictive way in this paper a new and multiscale approach to the calculation of the vibrations of elastic structures in the medium frequency range called the variational theory of complex rays is presented the feasibility and effectiveness of the method are demonstrated for structures made of plate assemblies several 3d examples are analyzed and compared to the results from a classical finite element approach and from a sea code this comparison shows that our method is able to predict the effective quantities at a very low numerical cost

a perfectly matched layer for the helmholtz equation in a semi infinite strip the perfectly matched layer pml has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains we develop a discretization to solve the helmholtz equation in an infinite two dimensional strip we solve the interior equation using high order finite differences schemes the combined helmholtz pml problem is then analyzed for the parameters that give the best performance we show that the use of local high order methods in the physical domain coupled with a specific second order approximation in the pml yields global high order accuracy in the physical domain we discuss the impact of the parameters on the effectiveness of the pml numerical results are presented to support the analysis

proper generalized decomposition applied to linear acoustic a new tool for broad band calculation the calculation of the acoustic response of systems in frequency bands is becoming increasingly important in simulation based engineering design this is particularly true in medium frequency bands where the response is very sensitive to the frequency some standard techniques for addressing these problems present a frequency dependent formulation and may involve fixed frequency calculations at many different frequencies in this paper we propose a new technique which combines the variational theory of complex rays vtcr with proper generalized decomposition pgd and does not require the resolution of acoustic problems at many frequencies in this approach the vtcr is used to find an approximate solution of a medium frequency acoustic problem using only a few degrees of freedom dofs then pgd is used to find a representation of the approximate solution which is separated between two variables the wave propagation direction and the frequency a relevant numerical example is used to present the strategy and illustrate its applicability for frequency band calculations

a quasi optimal petrov galerkin method for helmholtz problem a petrov galerkin finite element formulation is introduced for helmholtz problem in two dimensions using polynomial weighting functions at each node of the mesh a global basis function for the weighting space is obtained adding to the bilinear c 0 lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node quasi optimal weighting functions with the same support of the corresponding global test functions are obtained after computing the coefficients of these linear combinations attending to optimality criteria this is done numerically through a simple preprocessing technique naturally applied to non uniform and unstructured meshes in particular for uniform mesh a quasi optimal interior stencil of the same order of the quasi stabilized finite element method stencil derived by babuska et al comput methods appl mech engrg 1995 128 325 359 is obtained numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non uniform meshes

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

proper generalized decomposition solution of the parameterized helmholtz problem application to inverse geophysical problems this is the peer reviewed version of the following article signorini m zlotnik s and diez p 2017 proper generalized decomposition solution of the parameterized helmholtz problem application to inverse geophysical problems int j numer meth engng 109 1085 1102 doi 10 1002 nme 5313 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 5313 full this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

dispersion and pollution of the fem solution for the helmholtz equation in one two and three dimensions for high wave numbers the helmholtz equation suffers the so called pollution effect this effect is directly related to the dispersion a method to measure the dispersion on any numerical method related to the classical galerkin fem is presented this method does not require to compute the numerical solution of the problem and is extremely fast numerical results on the classical galerkin fem p method is compared to modified methods presented in the literature a study of the influence of the topology triangles is also carried out the efficiency of the different methods is compared the numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well the most effective modified method is the qsfem 1 2 but it is also very complicated in the general setting the residual free bubble 3 4 is effective in one dimension but not in higher dimensions the least square method 1 5 approach lowers the dispersion but relatively little the results for triangular meshes show that the best topology is the criss cross pattern copyright 1999 john wiley sons ltd

increase of coherence in excitable systems by delayed feedback the control of coherence and spectral properties of noise induced oscillations by time delayed feedback is studied in a fitzhugh nagumo system which serves as a paradigmatic model of excitable systems a semianalytical approach based on a discrete model with waiting time densities is developed which allows one to predict quantitatively the increase of coherence measured by the correlation time and the modulation of the main frequencies of the stochastic dynamics in dependence on the delay time the analytical mean field approximation is in good agreement with numerical results for the full nonlinear model

phase velocity and phase diffusion in periodically driven discrete state systems we develop a theory to calculate the effective phase diffusion coefficient and the mean phase velocity in periodically driven stochastic models with two discrete states this theory is applied to a dichotomically driven markovian two state system explicit expressions for the mean phase velocity the effective phase diffusion coefficient and the pe clet number are analytically calculated the latter indicates as a measure of phase coherence forced synchronization of the stochastic system with respect to the periodic driving and exhibits a bona fide resonance in a second step the theory is applied to a non markovian two state system modeling excitable systems the results prove again stochastic synchronization to the periodic driving and are in good agreement with simulations of a stochastic fitzhugh nagumo system

nurbs enhanced finite element method nefem the development of nurbs enhanced finite element method nefem is revisited this technique allows a seamless integration of the cad boundary representation of the domain and the finite element method fem the importance of the geometrical model in finite element simulations is addressed and the benefits and potential of nefem are discussed and compared with respect to other curved finite element techniques

analysis of high order finite elements for convected wave propagation in this paper we examine the performance of high order finite element methods fem for aeroacoustic propagation based on the convected helmholtz equation a methodology is presented to measure the dispersion and amplitude errors of the p fem including non interpolating shape functions such as bubble shape functions a series of simple test cases are also presented to support the results of the dispersion analysis the main conclusion is that the properties of p fem that make its strength for standard acoustics e g exponential p convergence low dispersion error remain present for flow acoustics as well however the flow has a noticeable effect on the accuracy of the numerical solution even when the change in wavelength due to the mean flow is accounted for and an approximation of the dispersion error is proposed to describe the influence of the mean flow also discussed is the so called aliasing effect which can reduce the accuracy of the solution in the case of downstream propagation this can be avoided by an appropriate choice of mesh resolution

a schwarz lemma at the boundary using the julia wolff caratheodory type condition on finite dimensional irreducible bounded symmetric domains abstract in this paper we prove a schwarz lemma at the boundary for holomorphic self mappings f of finite dimensional irreducible bounded symmetric domains without assuming the boundary regularity of f our result generalizes the previous results obtained for holomorphic self mappings f of the euclidean unit ball or of the classical cartan domains of type i and of type ii which are smooth up to the boundary

alleviating mesh constraints model reduction parallel time integration and high resolution homogenization industrial processes involving composite materials need for efficient numerical simulations in order to optimize the process parameters even if the thermo mechanical models are nowadays well established efficient simulations need for further developments in this work we are addressing some of these issues in particular the one related to fast solutions combining model reduction and parallel time integration a separated representation will be also proposed in the context of material homogenization allowing to alleviate the usual mesh constraints

coupled three state oscillators abstract we investigate globally coupled stochastic three state oscillators which we consider as general models of stochastic excitable systems we compare two situations in the first case the transitions between the three states of each unit 1 2 3 1 are determined by poissonian waiting time distributions in the second case only transition 1 2 is poissonian whereas the others are deterministic with a fixed delay when coupled the second system shows coherent oscillations whereas the first remains in a stable stationary state we show that the coherent oscillations are due to a hopf bifurcation in the dynamics of the occupation probabilities of the discrete states and discuss the bifurcation diagram