the arlequin method as a flexible engineering design tool by superposing and gluing models the arlequin method offers an extended modelling framework for the design of engineering structures this paper aims at developing the numerical aspects of the approach and at showing how it can be used with great flexibility and in a consistent manner to change locally a global mechanical model the capabilities of the arlequin method and the effectiveness of the implemented numerical tools are exemplified by 1 d 2 d and 3 d numerical applications

computational plasticity for composite structures based on mathematical homogenization theory and practice abstract this paper generalizes the classical mathematical homogenization theory for heterogeneous medium to account for eigenstrains starting from the double scale asymptotic expansion for the displacement and eigenstrain fields we derive a close form expression relating arbitrary eigenstrains to the mechanical fields in the phases the overall structural response is computed using an averaging scheme by which phase concentration factors are computed in the average sense for each micro constituent and history data is updated at two points reinforcement and matrix in the microstructure one for each phase macroscopic history data is stored in the database and then subjected in the post processing stage onto the unit cell in the critical locations for numerical examples considered the cpu time obtained by means of the two point averaging scheme with variational micro history recovery with 30 seconds on sparc 10 51 as opposed to 7 hours using classical mathematical homogenization theory at the same time the maximum error in the microstress field in the critical unit cell was only 3 5 in comparison with the classical mathematical homogenization theory

proper orthogonal decomposition and modal analysis for acceleration of transient fem thermal analysis a method of reducing the number of degrees of freedom and the overall computing times in finite element method fem has been devised the technique is valid for linear problems and arbitrary temporal variation of boundary conditions at the first stage of the method standard fem time stepping procedure is invoked the temperature fields obtained for the first few time steps undergo statistical analysis yielding an optimal set of globally defined trial and weighting functions for the galerkin solution of the problem at hand simple matrix manipulations applied to the original fem system produce a set of ordinary differential equations of a dimensionality greatly reduced when compared with the original fem formulation using the concept of modal analysis the set is then solved analytically treatment of non homogeneous initial conditions time dependent boundary conditions and controlling the error introduced by the reduction of the degrees of freedom are discussed several numerical examples are included for validation of the approach copyright 2004 john wiley sons ltd

model order reduction in hyperelasticity a proper generalized decomposition approach this paper deals with the extension of proper generalized decomposition methods to non linear problems in particular to hyperelasticity among the different approaches that can be considered for the linearization of the doubly weak form of the problem we have implemented a new one that uses asymptotic numerical methods in conjunction with proper generalized decomposition to avoid complex consistent linearization schemes necessary in newton strategies this approach results in an approximation of the problem solution in the form of a series expansion each term of the series is expressed as a finite sum of separated functions the advantage of this approach is the presence of only one tangent operator identical for every term in the series the resulting approach has proved to render very accurate results that can be stored in the form of a meta model in a very compact format this opens the possibility to use these results in real time reaching khz feedback rates or to be used in deployed handheld devices such as smartphones and tablets

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

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

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

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

on the control of the load increments for a proper description of multiple delamination in a domain decomposition framework in quasi static non linear time dependent analysis the choice of the time discretization is a complex issue the most basic strategy consists in determining a value of the load increment that ensures the convergence of the solution with respect to time on the base of preliminary simulations in more advanced applications the load increments can be controlled for instance by prescribing the number of iterations of the non linear resolution procedure or by using an arc length algorithm these techniques usually introduce a parameter whose correct value is not easy to obtain in this paper an alternative procedure is proposed it is based on the continuous control of the residual of the reference problem over time whose measure is easy to interpret this idea is applied in the framework of a multiscale domain decomposition strategy in order to perform 3d delamination analysis copyright 2010 john wiley sons ltd

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

an hp proper orthogonal decomposition moving least squares approach for molecular dynamics simulation abstract the application of model order reduction for molecular dynamic systems exhibits intrinsic complexities due to the highly nonlinear and nonlocal properties of such systems the most costly computational part of such work is the calculation of potential energy and inter atomic force fields we introduce the so called hp proper orthogonal decomposition moving least squares hp pod mls method not only to tackle this force computation but also to improve the accuracy of reduced displacement approximations in an adaptive and controllable manner the approach combines the proper orthogonal decomposition pod method with the moving least squares mls one to interpolate the inter atomic force field at an arbitrary time instance this approximated force field is then fed into the reduced molecular dynamic md governing equation to compute the reduced order displacement field in fact this pod mls approach is already able to handle efficiently the force computation in the online computational stage i e the online force computational cost is independent of the dimension of the full md space however its corresponding approximated displacement error will be very large if the considered total time interval is long the proposed hp pod mls algorithm will divide adaptively and automatically the global temporal interval into many smaller local subintervals h refinement and then build the sets of local reduced basis functions over each of these subintervals p refinement the goal is to reduce the corresponding displacement errors to a desired tolerance over these subintervals the effectiveness of the algorithm is demonstrated by performing reduced order simulations of several molecular dynamic systems

model order reduction and domain decomposition strategies for the solution of the dynamic elastic plastic structural problem abstract a new strategy for the efficient solution of highly nonlinear structural problems is proposed based on the combined use of domain decomposition dd and snapshots version of the proper orthogonal decomposition pod techniques the formulation here presented is tailored for applications in elastic plastic structural dynamics in this context the pod is applied to domains that remain elastic and a double strategy to update the reduced basis is adopted first the singular value decomposition svd proposed allows to update the reduced basis as soon as a new snapshot is stored secondly an online adaptation technique of the reduced space is performed through a plastic check during the reduced analysis the applications show that the computation time necessary for solving elastic plastic problems can be reduced of approximately 50 while keeping accuracy comparable to that obtained for the full model with a classical monolithic method

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

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

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

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

rectifiable oscillations in second order linear differential equations abstract we study the linear differential equation p y x f x y x 0 on i 0 1 where the coefficient f x is strictly positive and continuous on i and satisfies the hartman wintner condition at x 0 the four main results of the paper are i a criterion for rectifiable oscillations of p characterized by the integrability of f x 4 on i ii a stability result for rectifiable and unrectifiable oscillations of p in terms of a perturbation on f x iii the s dimensional fractal oscillations for which we assume also f x c x when x 0 2 and s max 1 3 2 2 and iv the co existence of rectifiable and unrectifiable oscillations in the absence of the hartman wintner condition on f x explicit examples related to the above results are given

the reference point method a hyperreduction technique application to pgd based nonlinear model reduction a new approximation technique called reference point method rpm is proposed in order to reduce the computational complexity of algebraic operations for constructing reduced order models in the case of time dependent and or parametrized nonlinear partial differential equations even though model reduction techniques enableone to decrease the dimension of the initial problem in the sense that far fewer degrees of freedom are needed to represent the solution the complexity of evaluating the nonlinear terms and assembling the low dimensional operator associated with the reduced order model still scales with the size of the original high dimensional model this point can be critical especially when the reduced order basis changes throughout the solution strategy as it is the case for model reduction techniques based on proper generalized decomposition pgd based on the concept of spatial parameter time reference points and influence patches the rpm defines a compressed version of the data from which an approximate low rank separated representation by patch of the operators can be constructed by explicit formulas at low cost without resorting to svd based techniques an application of the rpm to pgd based model reduction for a nonlinear parametrized elliptic pde previously studied by other authors with reduced basis method and eim is proposed it is shown that computational complexity to construct the reduced order model can be divided in practice by one order of magnitude compared with the classical pgd approach

low rank approximate inverse for preconditioning tensor structured linear systems in this paper we propose an algorithm for the construction of low rank approximations of the inverse of an operator given in low rank tensor format the construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator it provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators these subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank one corrections in order to handle high order tensors approximate projections are computed in low rank hierarchical tucker subsets of the successive subspaces of operators some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step where an optimal rank one correction is searched as the tensor product of operators with the desired properties numerical examples illustrate the ability of this a

a multiscale large time increment fas algorithm with time space model reduction for frictional contact problems a multiscale strategy using model reduction for frictional contact computation is presented this new approach aims to improve computation time of finite element simulations involving frictional contact between linear and elastic bodies this strategy is based on a combination between the latin large time increment method and the fas multigrid solver the latin method is an iterative solver operating on the whole time space domain applying an a posteriori analysis on solutions of different frictional contact problems shows a great potential as far as reducibility for frictional contact problems is concerned time space vectors forming the so called reduced basis depict particular scales of the problem it becomes easy to make analogies with multigrid method to take full advantage of multiscale information

on the verification of model reduction methods based on the proper generalized decomposition in this work we introduce a consistent error estimator for numerical simulations performed by means of the proper generalized decomposition pgd approximation this estimator which is based on the constitutive relation error enables to capture all error sources i e those coming from space and time numerical discretizations from the truncation of the pgd decomposition etc and leads to guaranteed bounds on the exact error the specificity of the associated method is a double approach i e a kinematic approach and a unusual static approach for solving the parameterized problem by means of pgd this last approach makes straightforward the computation of a statically admissible solution which is necessary for robust error estimation an attractive feature of the error estimator we set up is that it is obtained by means of classical procedures available in finite element codes it thus represents a practical and relevant tool for driving algorithms carried out in pgd being possibly used as a stopping criterion or as an adaptation indicator numerical experiments on transient thermal problems illustrate the performances of the proposed method for global error estimation

proper generalized decomposition for nonlinear convex problems in tensor banach spaces tensor based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces a family of methods called proper generalized decompositions pgd methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems in this paper we give a mathematical analysis of a family of progressive and updated pgds for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor banach space

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

fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains abstract a tensor based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains it provides an accurate solution which is explicit with respect to parameters defining the shape of the domain thus allowing efficient a posteriori probabilistic or parametric analyses in the proposed method a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain yielding a weak formulation in a tensor product space product of space functions and parametric functions the paper is limited to the case of neumann conditions on uncertain parts of the boundary the proper generalized decomposition method is then introduced for the construction of a tensor product approximation separated representation of the solution it can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution this tensor based method is made computationally tractable by introducing separated representations of variational forms resulting from separated representations of the parameterized indicator function of the uncertain domain for this purpose a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well posedness of problems associated with approximate indicator functions moreover a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations

latin a new view and an extension to wave propagation in nonlinear media summary the latin acronym of large time increment method was originally devised as a non incremental procedure for the solution of quasi static problems in continuum mechanics with material nonlinearity in contrast to standard incremental methods like newton and modified newton latin is an iterative procedure applied to the entire loading path in each latin iteration two problems are solved a local problem which is nonlinear but algebraic and miniature and a global problem which involves the entire loading process but is linear the convergence of these iterations which has been shown to occur for a large class of nonlinear problems provides an approximate solution to the original problem in this paper the latin method is presented from a different viewpoint taking advantage of the causality principle in this new view latin is an incremental method and the latin iterations are performed within each load step similarly to the way that newton iterations are performed the advantages of the new approach are discussed in addition latin is extended for the solution of time dependent wave problems as a relatively simple model for illustrating the new formulation lateral wave propagation in a flat membrane made of a nonlinear material is considered numerical examples demonstrate the performance of the scheme in conjunction with finite element discretization in space and the newmark trapezoidal algorithm in time copyright 2017 john wiley sons ltd

a priori space time separated representation for the reduced order modeling of low reynolds number flows abstract reduced order models roms in fluid dynamics are nowadays mostly developed by performing a projection of the navier stokes equations onto a low dimensional space basis this basis is usually obtained through proper orthogonal decomposition pod which remains one the most efficient techniques to compress precomputed data the main drawback of a posteriori pod based roms is however their lacks of reliability as their parameters are varied preventing their direct uses within optimization algorithms the goal of the present article is to obtain an a priori low dimensional space time separated representation of the fluid fields without precomputed data the approach is based on the use of space time proper generalized decomposition pgd definitions which are successfully applied in several fields but whose uses in fluid dynamics remain scarce their applications to the navier stokes equations are indeed not straightforward due to the pressure velocity coupling the divergence free constraint and the non linear convective term the roms are built here from a space time weak formulation of the chorin temam prediction correction scheme more particularly a priori space time separated representations are obtained by applying the galerkin based progressive pgd definition the minimax pgd definition is moreover experimented on a linear stokes simplified case the related algorithms are explicitly given and illustrated on a transient lid driven cavity flow it is shown that the a priori space time separated representations converge to the full model solution as the decomposition order increases the ability of the resulting rom to learn iteratively from its own error is highlighted the progressive pgd algorithm can be used to effectively enrich incomplete pod and pgd precomputed space bases such as those obtained with different parameter values this may allow to ensure the accuracy of a rom as the parameter is varied which is of crucial interest for optimization problems

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

tensor based methods for numerical homogenization from high resolution images we present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images we first introduce specific numerical treatments for the translation of image based homogenization problems into a tensor framework it includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images we then introduce some variants of proper generalized decomposition pgd methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems a new definition of pgd is introduced which allows the progressive construction of a tucker decomposition of the solution this tensor format is well adapted to the present application and improves convergence properties of tensor decompositions finally we use a dual based error estimator on quantities of interest which was recently introduced in the context of pgd we exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material we also provide a complete goal oriented adaptive strategy for the progressive construction of tensor decompositions of primal and dual solutions yielding to predictions of homogenized quantities with a prescribed accuracy

effects of epidemic threshold definition on disease spread statistics we study the statistical properties of sir epidemics in random networks when an epidemic is defined as only those sir propagations that reach or exceed a minimum size sc using percolation theory to calculate the average fractional size view the mathml source of an epidemic we find that the strength of the spanning link percolation cluster p is an upper bound to view the mathml source for small values of sc p is no longer a good approximation and the average fractional size has to be computed directly we find that the choice of sc is generally but not always guided by the network structure and the value of t of the disease in question if the goal is to always obtain p as the average epidemic size one should choose sc to be the typical size of the largest percolation cluster at the critical percolation threshold for the transmissibility we also study q the probability that an sir propagation reaches the epidemic mass sc and find that it is well characterized by percolation theory we apply our results to real networks dimes and tracerouter to measure the consequences of the choice sc on predictions of average outcome sizes of computer failure epidemics

on a mixed and multiscale domain decomposition method this paper presents a reexamination of a multiscale computational strategy with homogenization in space and time for the resolution of highly heterogeneous structural problems focusing on its suitability for parallel computing spatially this strategy can be viewed as a mixed multilevel domain decomposition method or more accurately as a structure decomposition method regarding time a parallel property is also described we also draw bridges between this and other current approaches

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

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

proper generalized decomposition method for incompressible navier stokes equations with a spectral discretization proper generalized decomposition pgd is a method which consists in looking for the solution to a problem in a separate form this approach has been increasingly used over the last few years to solve mathematical problems the originality of this work consists in the association of pgd with a spectral collocation method to solve transfer equations as well as navier stokes equations in the first stage the pgd method and its association with spectral discretization is detailed this approach was tested for several problems the poisson equation the darcy problem navier stokes equations the taylor green problem and the lid driven cavity in the navier stokes problems the coupling between velocity and pressure was performed using a fractional step scheme and a p n p n 2 discretization for all problems considered the results from pgd simulations were compared with those obtained by a standard solver and or with the results found in the literature the simulations performed showed that pgd is as accurate as standard solvers pgd preserves the spectral behavior of the errors in velocity and pressure when the time step or the space step decreases moreover for a given number of discretization nodes pgd is faster than the standard solvers

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

a stable and optimally convergent latin cutfem algorithm for multiple unilateral contact problems in this paper we propose a novel unfitted finite element method for the simulation of multiple body contact the computational mesh is generated independently of the geometry of the interacting solids which can be arbitrarily complex the key novelty of the approach is the combination of elements of the cutfem technology namely the enrichment of the solution field via the definition of overlapping fictitious domains with a dedicated penalty type regularisation of discrete operators and the latin hybrid mixed formulation of complex interface conditions furthermore the novel p1 p1 discretisation scheme that we propose for the unfitted latin solver is shown to be stable robust and optimally convergent with mesh refinement finally the paper introduces a high performance 3d level set cutfem framework for the versatile and robust solution of contact problems involving multiple bodies of complex geometries with more than two bodies interacting at a single point

a posteriori error estimation and adaptive strategy for pgd model reduction applied to parametrized linear parabolic problems abstract we define an a posteriori verification procedure that enables to control and certify pgd based model reduction techniques applied to parametrized linear elliptic or parabolic problems using the concept of constitutive relation error it provides guaranteed and fully computable global goal oriented error estimates taking both discretization and pgd truncation errors into account splitting the error sources it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of pgd approximations the focus of the paper is on two technical points i construction of equilibrated fields required to compute guaranteed error bounds ii error splitting and adaptive process when performing pgd based model reduction performances of the proposed verification and adaptation tools are shown on several multi parameter mechanical problems

toward an optimal a priori reduced basis strategy for frictional contact problems with latin solver abstract in this paper an efficient a priori model reduction strategy for frictional contact problems is presented we propose to solve this problem by using the finite element method and the non linear latin solver basically this non linear solver assumes a space time separated representation presaging nowadays pgd strategies we extend this family of solvers to frictional engineering applications with reduced subspaces and no prior knowledge about the solution contrary to a posteriori model reduction techniques hereinafter a hybrid a priori a posteriori latin pgd formulation for frictional contact problems is proposed indeed the suggested algorithm may or may not start with an initial guess of the reduced basis and is able to enrich the basis in order to reach a given level of accuracy moreover it provides progressively the solution of the considered problem into a quasi optimal space time separated form compared to the singular value decomposition svd some examples are provided in order to illustrate the efficiency and quasi optimality of the proposed a priori reduced basis latin solver

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