rotational degeneracy of hyperbolic systems of conservation laws the purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy in this situation hyperbolicity is necessarily non strict so that the elementary waves have interesting patterns the discussion is centered around a theorem on existence and uniqueness of solutions of the riemann problem we give a unified presentation of examples from continuum mechanics

multiparametric response surface construction by means of proper generalized decomposition an extension of the parafac procedure abstract multidimensional problems are found in almost every scientific field in particular this is standard in parametric design inverse analysis in optimization and in metamodeling analysis whereby statistical or deterministic approximations of multiparametric solutions are built from the results of experimental campaigns or computer simulations multidimensional fitting or approximation of response functions exponentially increase their complexity and computational cost with the number of dimensions responding to the well known curse of dimensionality to reduce the order of complexity and make the solution of many parameter problems affordable we propose to combine the model reduction technique known as proper generalized decomposition pgd and the response surface rsm methodology as a proof of concept we have used a simple fitting procedure as it is least squares although other more complex fitting procedures may be easily included the combined algorithm is presented and its capabilities discussed in a set of multidimensional examples the data samples to be fit in each of these examples are obtained by means of appropriate discretizing the interval of interest for each design factor and then generating the output values exact or stochastically modified by means of virtual experiments to have an idea of the number of discretization points needed along each direction the taguchi s design of experiments is used the obtained results show evident improvements in computer time and accuracy when compared to other traditional multiparametric approximation techniques based on polynomial functions and the standard levenberg marquardt algorithm especially in problems with non linear behavior and with high number of design parameters further comparison was done with the parafac als algorithm the combination of pgd and rsm seems to be an appealing tool for accurately modelling multiparametric problems in almost real time if a sufficient set of previous off line results is available despite the intrinsic complexity of the problem and of the number of parameters involved

the geostatistical characteristics of the borden aquifer a complete reexamination of sudicky s 1986 field experiment for the geostatistical characterization of hydraulic conductivity at the borden aquifer in ontario canada is performed the sampled data reveal that a number of outliers low ln k values are present in the data base these low values cause difficulties in both variogram estimation and determining population statistics the analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate the classical cressie hawkins and squared median of the absolute deviations smad estimators are used to compute experimental variograms none of these estimators provides completely satisfactory variograms for the borden data with the exception of the classical estimator with outliers removed from the data set theoretical exponential variogram parameters are determined from nonlinear nl estimation differences are obtained between nl fits and those of sudicky 1986 for the classical screened estimated variogram nl fits produce an ln k variance of 0 24 nugget of 0 07 and integral scales of 5 1 m horizontal and 0 21 m vertical along a a for b b these values are 0 37 0 11 8 3 and 0 34 the fitted parameter set for b b data horizontal and vertical is statistically different than the parameter set determined for a a we also evaluate a probabilistic form of dagan s 1982 1987 equations relating geostatistical parameters to a tracer cloud s spreading moments the equations are evaluated using the parameter estimates and covariances determined from line a a as input with a velocity equal to 9 0 cm day the results are compared with actual values determined from the field test but evaluated by both freyberg 1986 and rajaram and gelhar 1988 the geostatistical parameters developed from this study produce an excellent fit to both sets of calculated plume moments when combined with dagan s stochastic theory for predicting the spread of a tracer cloud

on the self averaging of dispersion for transport in quasi periodic random media in this study we present a numerical analysis for the self averaging of the longitudinal dispersion coefficient for transport in heterogeneous media this is done by investigating the mean square sample to sample fluctuations of the dispersion for finite times and finite numbers of modes for a random field using analytical arguments as well as numerical simulations we consider transport of point like injections in a quasi periodic random field with a gaussian correlation function in particular we focus on the asymptotic and pre asymptotic behaviour of the fluctuations with the aid of a probability density function for the dispersion and we verify the logarithmic growth of the sample to sample fluctuations as earlier reported in eberhard 2004 j phys a math gen 37 2549 71 we also comment on the choice of the relevant parameters to generate quasi periodic realizations with respect to the self averaging of transport in statistically homogeneous gaussian velocity fields

impact of sampling volume on the probability density function of steady state concentration 1 in recent years statistical theory has been used to compute the ensemble mean and variance of solute concentration in aquifer formations with second order stationary velocity fields the merit of accurately estimating the mean and variance of concentration however remains unclear without knowing the shape of the probability density function pdf in a setup where a conservative solute is continuously injected into a domain the concentration is bounded between zero and the concentration value in the injected solution at small travel distances close to the fringe of the plume an observation point may fall into the plume or outside so that the statistical concentration distribution clusters at the two limiting values obviously this results in non gaussian pdf s of concentration with increasing travel distance the lateral plume boundaries are smoothed resulting in increased probability of intermediate concentrations likewise averaging the concentration in a larger sampling volume as typically done in field measurements leads to higher probabilities of intermediate concentrations we present semianalytical results of concentration pdf s for measurements with point like or larger support volumes based on stochastic theory applied to stationary media to this end we employ a reversed auxiliary transport problem in which we use analytical expressions for first and second central spatial lateral moments with an assumed gaussian pdf for the uncertainty of the first lateral moment and gauss like shapes in individual cross sections the resulting concentration pdf can be reasonably fitted by beta distributions the results are compared to monte carlo simulations of flow and steady state transport in 3 d heterogeneous domains in both methods the shape of the concentration pdf changes with distance to the contaminant source near the source the distribution is multimodal whereas it becomes a unimodal beta distribution far away from the contaminant source the semianalytical and empirical pdf s differ slightly which we contribute to the numerical artifacts in the monte carlo simulations but also to hard assumptions made in the semianalytical approach our results imply that geostatistical techniques for interpolation and other statistical inferences based on gaussian distributions such as kriging and cokriging may be feasible only far away from the contaminant source for calculations near the source the beta like distribution of concentration should be accounted for

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

non intrusive low rank separated approximation of high dimensional stochastic models this work proposes a sampling based non intrusive approach within the context of low rank separated representations to tackle the issue of curse of dimensionality associated with the solution of models e g pdes odes with high dimensional random inputs under some conditions discussed in details the number of random realizations of the solution required for a successful approximation grows linearly with respect to the number of random inputs the construction of the separated representation is achieved via a regularized alternating least squares regression together with an error indicator to estimate model parameters the computational complexity of such a construction is quadratic in the number of random inputs the performance of the method is investigated through its application to three numerical examples including two ode problems with high dimensional random inputs

on a multiscale computational strategy with time and space homogenization for structural mechanics a new multiscale computational strategy was recently proposed for the analysis of structures described both on a fine space scale and a fine time scale this strategy which involves homogenization in space as well as in time could replace in several domains of application the standard homogenization techniques which are generally limited to the space domain it is an iterative strategy which calls for the resolution of problems on both a micro fine scale and a macro homogenized scale in this paper we review the bases of this approach and present improved approximation techniques to solve the micro and macro problems

a computational strategy for thermo poroelastic structures with a time space interface coupling this paper deals with a computational strategy suitable for the simulation of multiphysics problems based on the large time increment latin method the simulation of such problems must encounter the possible different time and space scales which usually arise with the different physics herein we focus on using different time and space discretizations for each physics by introducing an interface with its own discretization the feasibility of both a time and space couplings is exemplified on a nonlinear 3 physics strongly coupled problem the thermal fluid structure interaction in a thermo poroelastic structure

numerical investigations on ergodicity of solute transport in heterogeneous aquifers 1 darcy velocities for lognormal hydraulic conductivity with small variance and finite correlation length were approximated by periodic random fields accurate simulations of two dimensional advection dispersion processes were achieved with the global random walk algorithm using 1010 particles in every transport realization reliable statistical estimations were obtained by averaging over 256 realizations the main result is a numerical evidence for the mean square convergence of the actual concentrations to the macrodispersion process predicted by a known limit theorem for small initial plumes the ergodic behavior can be expected after thousands of advection timescales when the deviation from the theoretical prediction of the cross section space averaged concentration monotonously decays and falls under 20 the increase of the transverse dimension of the initial plume slows down the approach to the quasi ergodic state and has a nonlinear effect on the variability of the actual concentrations and dispersivities

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

stratified disordered media exact solutions for transport parameters and their self averaging properties we investigate the transport of a passive tracer in a two dimensional stratified random medium with flow parallel and perpendicular to the strata assuming a gaussian random flow with a gaussian correlation function it is not only possible to derive exact expressions for the temporal behaviour of the dispersion coefficients characterizing the tracer transport but also to investigate the self averaging properties of these quantities we give explicit results for the dispersion coefficient and its mean square fluctuations as a function of time as it turns out the sample to sample fluctuations which are encoded in the latter quantity remain finite even in the asymptotic limit of infinite times which implies that the given two dimensional model is not self averaging

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

accelerating numerical simulations of strain adaptive bone remodeling predictions abstract bone modeling and remodeling have been extensively studied in many fields of research the process of adaptive bone remodeling can be described mathematically and simulated in a computer model integrated with the finite element method fem the fem can determine the long term behavior of bone and the impact on bone biomechanics produced by prosthetic models as whole bone remodeling simulations are time consuming the aim of this paper is to combine fe analyses with numerical extrapolation techniques to accelerate bone remodeling simulations two vector extrapolation methods reduce ranked extrapolation rre and minimal polynomial extrapolation mpe are used to reduce the simulation time these extrapolation techniques are illustrated by numerical examples of 2 d and 3 d models based on a strain adaptive bone remodeling theory

temporal behavior of a solute cloud in a heterogeneous porous medium 3 numerical simulations 1 the article presents systematic numerical simulations of the temporal behavior of a passive solute in a saturated three dimensional heterogeneous medium the groundwater flow is derived from the linearized solution of the darcy equation with gauss distributed log hydraulic conductivity the transport of a passive solute is studied by a random walk method which allows for a systematic study of the temporal behavior of the effective and ensemble dispersion coefficients the numerical results are compared to the second order perturbation theory expressions given in two companion papers dentz et al 2000a 2000b and to nonperturbative results which follow from corrsin s conjecture the low order perturbation theory is intrinsically based on the assumption of small heterogeneity while corrsin s conjecture does not take into account certain contributions due to higher order terms of the perturbation series the simulations yield for the first time systematic quantitative information on the validity and the limitations of these analytic approximations for increasing heterogeneities considerable deviations from the theoretically predicted transport behavior are observed

a pgd based homogenization technique for the resolution of nonlinear multiscale problems abstract this paper deals with the offline resolution of nonlinear multiscale problems leading to a pgd reduced model from which one can derive microinformation which is suitable for design here we focus on an improvement to the proper generalized decomposition pgd technique which is used for the calculation of the homogenized operator and which plays a central role in our multiscale computational strategy this homogenized behavior is calculated offline using the latin multiscale strategy and pgd leads to a drastic reduction in cpu cost indeed the multiscale aspect of the latin method is based on splitting the unknowns between a macroscale and a microcomplement this macroscale is used to capture the homogenized behavior of the structure and then to accelerate the convergence of the iterative algorithm the novelty of this work lies in the reduction of the computation cost of building the homogenized operator in order to do that the problems defined within each subdomain are solved using a new pgd representation of the unknowns in which the separation of the unknowns is performed not only in time and in space but also in terms of the interface macrodisplacements the numerical example presented shows that the convergence rate of the approach is not affected by this new representation and that a significant reduction in computation cost can be achieved particularly in the case of nonlinear behavior the technique used herein is then particularly important to enhance the performances of the latin strategy but is not limited to this method the more general path which is followed in this paper is to solve the microproblems arising in the homogenization for all the configurations that can be encountered in the calculation an additional and novel benefit of this approach is that it includes the calculation of the homogenized tangent operator which gives over the whole time interval and for any cell the relation between the perturbation in terms of macroforces and the perturbation in terms of macrodisplacements in linear viscoelasticity this operator represents the homogenized behavior of the structure exactly

a priori hyperreduction method an adaptive approach model reduction methods are usually based on preliminary computations to build the shape function of the reduced order model rom before the computation of the reduced state variables they are a posteriori approaches most of the time these preliminary computations are as complex as the simulation which we want to simplify by the rom the reduction method we propose avoids such preliminary computations it is an a priori approach based on the analysis of some state evolutions such that all the state evolutions needed to perform the model reduction are described by an approximate rom the rom and the state evolution are simultaneously improved by the method thanks to an adaptive strategy obviously an initial set of known shape functions can be used to define the rom to adapt but it is not necessary the adaptive procedure includes extensions of the subspace spanned by the shape functions of the rom and selections of the most relevant shape functions in order to represent the state evolution the hyperreduction is achieved by selecting a part of the integration points of the finite element model to forecast the evolution of the reduced state variables hence both the number of degrees of freedom and the number of integration points are reduced to perform the adaptive procedure different computational strategies can be developed in this paper we propose an incremental algorithm involving adaptive periods during these adaptive periods the incremental computation is restarted until a quality criterion is satisfied this approach is compatible with classical formulations of the equations

asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations we determine the asymptotic dispersion coefficients in 2d exponentially correlated lognormally distributed permeability fields by using parallel computing fluid flow is computed by solving the flow equation discretized on a regular grid and transport triggered by advection and diffusion is simulated by a particle tracker to obtain a well defined asymptotic regime under ergodic conditions initial plume size much larger than the correlation length of the permeability field the characteristic dimension of the simulated computational domains was of the order of 103 correlation lengths with a resolution of ten cells by correlation length we determine numerically the asymptotic effective longitudinal and transverse dispersion coefficients over 100 simulations for a broad range of heterogeneities s 2 0 9 where s 2 is the lognormal permeability variance for purely advective transport the asymptotic longitudinal dispersion coefficient depends linearly on s 2 for s 2 1 and the asymptotic transverse dispersion coefficient is zero addition of homogeneous isotropic diffusion induces an increase of transverse dispersion and a decrease of longitudinal dispersion

certification of projection based reduced order modelling in computational homogenisation by the constitutive relation error in this paper we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites the upper bound relies on the construction of a reduced model for the stress field upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense the desired bounding property is obtained the lower bound is obtained by defining a hierarchical enriched reduced model for the displacement we show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model

evaluation of the first order approximations for transport in heterogeneous media 1 longitudinal dispersion coefficients in given realizations of the transport computed by two currently used approximations of the first order in velocity variance are compared with accurate global random walk simulations the comparisons are performed for the same ensemble of realizations of the darcy velocity field approximated by a quasiperiodic random field for lognormal hydraulic conductivity with small variance and finite correlation lengths the results show that at finite times of about one dispersion timescale the mean coefficient is underestimated by 20 and the fluctuations are overestimated by 80 at larger times the errors decrease monotonously and the first order approximations yield fairly good predictions for the mean and the fluctuations of the dispersion coefficient

an exact nonlinear correction to transverse macrodispersivity for transport in heterogeneous formations the random two dimensional and steady in time eulerian velocity field is assumed to be given it pertains to flow through heterogeneous porous media of spatially variable permeability advective transport of a passive scalar is modeled by the lagrangian approach by seeking the statistical moments of the trajectory of a tagged particle the trajectories variances and the associated dispersion coefficients are evaluated by a perturbation expansion in the velocity variances and by corrsin s conjecture first order results obtained in the past show that for flow through porous formations of log permeability two point covariance of finite integral scale the transverse dispersion coefficient tends asymptotically to zero in contrast corrsin s conjecture leads to a finite dispersion coefficient quadratic in the velocity variance to investigate this effect an exact nonlinear correction to the transverse dispersion coefficient is derived for a normal velocity field it is shown that this term tends to zero asymptotically in contrast with the result based on corrsin s conjecture furthermore an illustrative computation shows the transport related nonlinear correction to be small at any time

a manifold learning based reduced order model for springback shape characterization and optimization in sheet metal forming abstract the parameters of a stamping process include the geometry of the tools the shape of the initial sheet blank the material constitutive law and the process parameters when designing the overall process one has to also take into account the springback effect that appears when the tools are removed and additional surfaces are cut off the goal then is to obtain a final shape as close as possible to the desired shape while satisfying the admissibility constraints on the variable parameters as well as the feasibility constraints frequently expressed in the form of forming limit diagrams in the present paper we represent the post springback shape by a level set function then rather than rely on arbitrarily selected case dependent measurement locations as in the numisheet benchmark problems we build a reduced order shape space where this level set evolves by extending our recent shape manifold approach to the problem of springback assessment for 3d shapes next we propose an optimization algorithm designed to minimize the gap between the post springback and the desired final shapes the required level set functions are generated from a corresponding set of springback shapes predicted by finite element simulations using our approach we determine the minimal number of parameters needed in order to uniquely characterize the final formed shape regardless of complexity finally we demonstrate the approach using an industrial test case springback assessment of the deep drawing operation of an automotive strut tower

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

the reduced basis element method application to a thermal fin problem the reduced basis element method is a new approach for the approximation of partial differential equations that takes its roots in the domain decomposition method and in the reduced basis discretization the basic idea is to decompose the domain of computation into a series of subdomains the elements that are similar to a few reference domains these reference domains are actually filled with reduced basis functional spaces that are mapped to each subdomain together with the geometry the discrete approximation space is then composed of functions with the property that a function restricted to a subdomain belongs to the mapped reduced spaces finally a mortar type method is applied to glue the various local functions in this paper we focus on the definition of the reference shapes and together with theoretical and numerical justifications of the method we provide a posteriori error analysis tools that allow us to certify the computational results

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

proper general decomposition pgd for the resolution of navier stokes equations in this work the pgd method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions in the first stage the equations of stokes and burgers will be solved then we will solve the navier stokes problem in the case of the lid driven cavity for different reynolds numbers re 100 1000 and 10 000 finally the pgd method will be compared to the standard resolution technique both in terms of cpu time and accuracy

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

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

time parallel multiscale multiphysics framework we introduce the time parallel compound wavelet matrix method tpcwm for modeling the temporal evolution of multiscale and multiphysics systems the method couples time parallel tp and cwm methods operating at different spatial and temporal scales we demonstrate the efficiency of our approach on two examples a chemical reaction kinetic system and a non linear predator prey system our results indicate that the tpcwm technique is capable of accelerating time to solution by 2 3 orders of magnitude and is amenable to efficient parallel implementation

fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition abstract in recent years a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials where the complexity of solving numerically all microscale details is not feasible in this paper we present a multiscale framework based on the decomposition of the displacement field into coarse macro and fine micro scales earlier proposed in the variational multiscale approach or the hp d method the novelty of this work lies in solving the microscale step where a multidimensional parametrized model of a generic rve is solved by means of the multidimensional model reduction technique named as proper generalized decomposition pgd as result of this previous and off line step the displacement field over the rve is obtained by simple algebraic operations for any combination of parameters boundary condition material properties loads etc allowing a significant reduction in computational cost when solving the macro scale problem for problems where the detailed structure of the localized displacement strain and stress fields are of interest recovery of the fine scale components can be performed immediately the basic concepts and several 1 d and 2 d linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving

a pgd based multiscale formulation for non linear solid mechanics under small deformations abstract model reduction techniques have became an attractive and a promising field to be applied in multiscale methods the main objective of this work is to formulate a multiscale procedure for non linear problems based on parametrized microscale models the novelty of this work relies in the implementation of the model reduction technique known as proper generalized decomposition for solving the high dimensional parametrized problem resulting from the microscale model the multiscale framework here proposed is formulated to non linear problems specifically to material non linearities where material response is governed by a strain dependent evolution law two strategies to deal with this kind of problem under small deformations are detailed in this work both strategies based on parametrized microscale models solved by pgd have been applied to a problem with a rate dependent isotropic damage model first a procedure where the problem is solved by uncoupling the equilibrium equation to the state variable expression has been explored in order to alleviate the parametrized microscale problem a second strategy for problems with material non linearity has been proposed incorporating a parametrized microscale problem at each macroscale increment fe pgd the basis of those procedures are described and compared highlighting the solution accuracy and computer time consumption in comparison to a traditional finite element analysis

a numerical solver for high dimensional transient fokker planck equation in modeling polymeric fluids in this paper a tensor decomposition approach combined with chebyshev spectral differentiation is presented to solve the high dimensional transient fokker planck equations fpe arising in the simulation of polymeric fluids via multi bead spring mbs model generalizing the authors previous work on the stationary fpe the transient solution is obtained in a single candecomp parafac decomposition cpd form for all times via the alternating least squares algorithm this is accomplished by treating the temporal dimension in the same manner as all other spatial dimensions thereby decoupling it from them as a result the transient solution is obtained without resorting to expensive time stepping schemes a new relaxed approach for imposing the vanishing boundary conditions is proposed improving the quality of the approximation the asymptotic behavior of the temporal basis functions is studied the proposed solver scales very well with the dimensionality of the mbs model numerical results for systems up to 14 dimensional state space are successfully obtained on a regular personal computer and compared with the corresponding matrix riccati differential equation for linear models or monte carlo simulations for nonlinear models

bridging proper orthogonal decomposition methods and augmented newton krylov algorithms an adaptive model order reduction for highly nonlinear mechanical problems this article describes a bridge between pod based model order reduction techniques and the classical newton krylov solvers this bridge is used to derive an efficient algorithm to correct on the fly the reduced order modelling of highly nonlinear problems undergoing strong topological changes damage initiation problems tackled via a corrected hyperreduction method are used as an example it is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm even when strong topological changes are involved

numerical solution of high dimensional stationary fokker planck equations via tensor decomposition and chebyshev spectral differentiation abstract this paper focuses on the curse of dimensionality in the numerical solution of the stationary fokker planck equation for systems with state independent excitation a tensor decomposition approach is combined with chebyshev spectral differentiation to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases following the enforcement of the normality condition via a penalty method the discretized system is solved using alternating least squares algorithm numerical results for a variety of systems including separable non separable systems linear nonlinear systems and systems with without closed form stationary solutions up to ten dimensional state space are presented to illustrate the effectiveness of the proposed method

approximations for transport parameters and self averaging properties for point like injections in heterogeneous media we focus on transport parameters in heterogeneous media with a flow modelled by an ensemble of periodic and gaussian random fields the parameters are determined by ensemble averages we study to what extent these averages represent the behaviour in a single realization we calculate the centre of mass velocity and the dispersion coefficient using approximations based on a perturbative expansion for the transport equation and on the iterative solution of the langevin equation compared with simulations the perturbation theory reproduces the numerical results only poorly whereas the iterative solution yields good results using these approximations we investigate the self averaging properties the ensemble average of the velocity characterizes the behaviour of a realization for large times in both ensembles the dispersion coefficient is not self averaging in the ensemble of periodic fields for the gaussian ensemble the asymptotic dispersion coefficient is self averaging for finite times however the fluctuations are so large that the average does not represent the behaviour in a single realization

dispersion variance for transport in heterogeneous porous media 1 we study dispersion in heterogeneous porous media for solutes evolving from point like and extended source distributions in d 2 and d 3 spatial dimensions the impact of heterogeneity on the dispersion behavior is captured by a stochastic modeling approach that represents the spatially fluctuating flow velocity as a spatial random field we focus here on the sample to sample fluctuations of the dispersion coefficients about their ensemble mean for finite source sizes the definition of dispersion coefficients in single realizations is not unique we consider dispersion measures that describe the extension of the solute distribution as well as dispersion coefficients that quantify the solute spreading relative to injection points of the partial plumes that constitute the solute distribution while the ensemble averages of these dispersion quantities may be identical their fluctuation behavior is found to be different using a perturbation approach in the fluctuations of the random flow field we derive explicit expressions for the temporal evolution of the variances of the dispersion coefficients between realizations their evolution is governed by the typical dispersion time over the characteristic heterogeneity scale and the dimensions of the source distribution we find that the dispersion variance decreases toward zero with time in d 3 spatial dimensions while in d 2 it converges toward a finite long time value that is independent of the source dimensions

a framework for optimal temporal reduced order modeling of nonlinear dynamical systems abstract an optimal temporal reduced order modeling framework is proposed for nonlinear dynamical systems the governing equations are modified for an under resolved simulation with an arbitrary scheme and a coarse temporal grid subgrid scale models are developed to account for the unresolved temporal structure via inclusion of statistical information on a multi point temporal stencil these models are based upon principles of mean square error minimization conditional expectations and stochastic estimation in order to validate the proposed framework we investigate time periodic solutions for a canonical duffing oscillator using a coarse harmonic balance scheme in order to demonstrate application of the proposed framework to a high dimensional nonlinear dynamical system we also investigate a simply supported geometrically nonlinear beam under the influence of time periodic external forcing for both problems the subgrid scale models are shown to significantly improve the accuracy of coarse grained solutions

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques

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

multiscale temporal integration abstract we study the application of the variational multiscale method to the problem of temporal integration with the final goal of designing integration schemes that go beyond the classical notion of upwinding we develop a formulation based on a mixed hybrid finite element method where the fine scale mode problems automatically decouple at the element level without the need to resort to a localization assumption we give general orthogonality conditions for the trial and test spaces that allow to construct hierarchical p methods we test some simple ideas for the modeling of the unresolved scales the resulting algorithms are analyzed using classical analytical measures