numerical identification of a viscoelastic substitute model for heterogeneous poroelastic media by a reduced order homogenization approach the paper deals with the computational homogenization of pressure diffusion processes in a poroelastic medium the underlying physical phenomena are of interest for the interpretation of seismic data with applications in hydrocarbon production and geothermal energy pressure diffusion is assumed to take place on a length scale much smaller than the observer scale thus the macroscopic observer is not able to measure the properties of the poroelastic medium directly but notices an intrinsic viscous attenuation under these circumstances the macro scale can be interpreted as a single phase solid with apparent viscoelastic properties in this paper we establish a numerical upscaling procedure based on a volume averaging concept this enables us to identify the material properties of the viscoelastic substitute model in a numerically efficient manner for this purpose the poroelastic medium on the small scale is modelled in terms of the momentum balance of the biphasic mixture and a coupled diffusion equation we approximate the poroelastic pressure field on the small scale by a linear combination of pressure modes forming a reduced orthogonal basis and being identified by a proper orthogonal decomposition pod technique from the superposition principle the evaluation of the poroelastic continuity equation results in a proper identification of the evolution equations defining the apparent viscoelastic model in comparison to the nested fe2 solution schemes the reduced order approach only requires a small amount of off line precomputations and therefore causes very low numerical costs the proposed method is validated for the simple setup of a layered porous rock with alternating water and gas saturated zones

a berndtsson andersson operator solving overline partial equation with wα estimates on convex domains of finite type the carleson measures were first introduced by carleson in order to solve the corona problem for the disc in mathbb c the notion of carleson measure can be generalized to any homogeneous space and were also used in the context of the corona problem in mathbb c n for example by varopoulos amar and andersson and carlsson one of the steps to solve the hp corona problem in a pseudoconvex domain is to solve the overline partial equation for a form satisfying a carleson condition and get norm estimates of the solution in term of the carleson norm of the main goal of this paper is to consider this question in the case of convex domains of finite type and to get estimates linked to the multitype of the domain

a reduced multiscale model for nonlinear structural topology optimization abstract this paper presents a reduced multiscale model for macroscopic structural design considering microscopic material nonlinear microstructures this work introduces reduced order model rom to alleviate the heavy computational demand of nonlinear nested multiscale procedures particularly in an optimization framework which requires multiple loops involving similar computations the surrogate model constructed using proper orthogonal decomposition pod and diffuse approximation reduces the computational effort for solving the microscopic boundary value problems multiscale analysis model fe 2 is applied to link structure and microstructures in the two scales maximum stiffness design of the macroscopic structure is realized using a discrete level set topology optimization model it is shown by means of numerical tests that the reduced multiscale model provides reasonable designs as compared to those obtained by the unreduced model while with a significantly reduced computational effort

a non intrusive reduced basis method for elastoplasticity problems in geotechnics abstract this work aims at investigating the use of reduced basis rb methods to diminish the cost of numerical simulation of elastoplasticity problems arising from geotechnics modeling and involving parameter dependent partial differential equations pdes computation times for large three dimensional analysis commonly take tens of hours making optimization procedures or sensitivity analysis relying on repeated simulations hardly feasible in many cases the geotechnical analysis requires very specific features such as highly non linear constitutive laws making the necessary modification of the fe calculation code for a standard rb method impossible an approach making it possible to use the reduced basis framework without having to modify the code gives the so called non intrusive reduced basis method a versatility of great practical interest our approach involves the computation of less expensive but less accurate fe approximation during the online stage and improvement of those solutions using a rb based rectification method

a multiscale strategy for structural optimization this paper presents a multiscale strategy dedicated to structural optimization the applications concern the study of geometric details such as holes surface profiles etc within the structures with frictional contacts the first characteristic of the method is that it uses a micro macro approach this approach is based on a domain decomposition into substructures and interfaces which involves the resolution of independent micro problems in each substructure and transfers macro information only through the interfaces the second characteristic is the use of a multiresolution strategy in order to reduce the computation cost for problems with evolving design parameters the last characteristic is the capability to model the geometry of details without remeshing thanks to two features the use of a local enrichment method and the use of level set functions to easily modify the boundary of the detail during the optimization process

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

numerical material representation using proper orthogonal decomposition and diffuse approximation from numerical point of view analysis and optimization in computational material engineering require efficient approaches for microstructure representation this paper develops an approach to establish an image based interpolation model in order to efficiently parameterize microstructures of a representative volume element rve based on proper orthogonal decomposition pod reduction of density maps snapshots when the parameters of the rve snapshot are known a priori the geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients in terms of these parameters otherwise a set of pseudo parameters corresponding to the detected dimensionality of the data set are taken from learning the manifolds of the projection coefficients we showcase the approach and its potential applications by considering a set of two phase composite snapshots the choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization

multidimensional a priori hyper reduction of mechanical models involving internal variables this paper concerns the adaptation of reduced order models during simulations of series of elastoviscoplastic problems in continuation with previous works this paper aimed at extending the a priori hyper reduction method aphr method for nonlinear thermal problems to nonlinear mechanical problems involving internal variables this method is an a priori approach because full incremental responses of detailed models are not forecasted in order to build reduced order models the recent extension of the hyper reduction method to reduction of mechanical models involving internal variables makes possible the reduction of degrees of freedom and the reduction of integration points a multi level formulation is introduced to focus on the capability of the method to perform efficient parallel computations to adapt reduced order models

diffusion over a saddle with a langevin equation the diffusion problem over a saddle is studied using a multidimensional langevin equation an analytical solution is derived for a quadratic potential and the probability to pass over the barrier deduced a very simple solution is given for the one dimensional problem and a general scheme is shown for higher dimensions

multiple parameter reduced basis technique for bifurcation and post buckling analyses of composite plates a multiple parameter reduced basis technique and a problem adaptive computational algorithm are presented for the bifurcation and post buckling analyses of composite plates subjected to combined loadings the computational algorithm can be conveniently divided into three distinct stages the first stage is that of determining the stability boundary the plate is discretized by using displacement finite element models and the analysis region is reduced by exploiting the special symmetries exhibited by the response of the plate the vector of unknown nodal displacements is expressed as a linear combination of a small number of path derivatives derivatives of the nodal displacements with respect to path parameters and a rayleigh ritz technique is used to approximate the finite element equations by a small system of algebraic equations the reduced equations are used to determine the stability boundary of the plate in the second stage a nonllnear solution in the vicinity of the stability boundary is obtained by using a bifurcation buckling mode as a predictor and a set of reduced equations is generated in the third stage the reduced equations are used to trace post buckling paths corresponding to various combinations of the load parameters the potential of the proposed approach is discussed and its effectiveness is demonstrated by means of a numerical example of laminated composite plate subjected to combined compressive and shear loadings

a best points interpolation method for efficient approximation of parametrized functions we present an interpolation method for efficient approximation of parametrized functions the method recognizes and exploits the low dimensional manifold structure of the parametrized functions to provide good approximation basic ingredients include a specific problem dependent basis set defining a low dimensional representation of the parametrized functions and a set of best interpolation points capturing the spatial parameter variation of the parametrized functions the best interpolation points are defined as solution of a least squares minimization problem which can be solved efficiently using standard optimization algorithms the approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure in addition an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost numerical results are presented to demonstrate the accuracy and efficiency of the method copyright 2007 john wiley sons ltd

efficient non linear model reduction via a least squares petrov galerkin projection and compressive tensor approximations the first author acknowledges the partial support by a national science foundation graduate fellowship and the partial support by a national defense science and engineering graduate fellowship the second and third authors acknowledge the partial support by the motor sports division of the toyota motor corporation under agreement number 48737 and the partial support by a research grant from the academic excellence alliance program between king abdullah university of science and technology kaust and stanford university all authors also acknowledge the constructive comments received during the review process

anomalous diffusion exact solution of the generalized langevin equation for harmonically bounded particle we study the effect of a disordered or fractal environment in the irreversible dynamics of a harmonic oscillator starting from a generalized langevin equation and using laplace analysis we derive exact expressions for the mean values variances and velocity autocorrelation function of the particle in terms of generalized mittag leffler functions the long time behaviors of these quantities are obtained and the presence of a whip back effect is analyzed

optimal control of the cylinder wake in the laminar regime by trust region methods and pod reduced order models in this paper optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime re 200 the control parameters are the amplitude and the frequency of the time harmonic cylinder rotation in order to reduce the size of the discretized optimality system a proper orthogonal decomposition pod reduced order model rom is derived to be used as state equation we then propose to employ the trust region proper orthogonal decomposition trpod approach originally introduced by fahl m fahl trust region methods for flow control based on reduced order modeling ph d thesis trier university 2000 to update the reduced order models during the optimization process a lot of computational work is saved because the optimization process is now based only on low fidelity models a particular care was taken to derive a pod rom for the pressure and velocity fields with an appropriate balance between model accuracy and robustness the key enablers are the extension of the pod basis functions to the pressure data the use of calibration methods for the pod rom and the addition in the pod expansion of several non equilibrium modes to describe various operating conditions when the trpod algorithm is applied to the wake flow configuration this approach converges to the minimum predicted by an open loop control approach and leads to a relative mean drag reduction of 30 at reduced cost

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

the reduced basis method for initial value problems this paper deals with the application of the reduced basis method to the initial value problem of ordinary differential equations a global approximation of the true solution curve is defined in terms of a sequence of reduced basis curves error bounds are established and these lead to order estimates for specific choices of reduced basis subspaces the effects of numerical integration are then incorporated into the estimates a numerical study is included

estimation of the error in the reduced basis method solution of nonlinear equations the reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension in this paper the reduced basis error is shown to be dominated by an approximation error this in turn leads to error estimates for projection onto specific subspaces for example subspaces related to taylor lagrange and discrete least squares approximation

bifurcation and post buckling analysis of laminated composite plates via reduced basis technique abstract a reduced basis technique and a problem adaptive computational algorithm are presented for the bifurcation and post buckling analysis of laminated anisotropic plates the computational algorithm can be conveniently divided into three distinct stages the first stage is that of determining the bifurcation point the plate is discretized by using displacement finite element or finite difference models the special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region the vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors and a rayleigh ritz technique is used to approximate the finite element equations by a small system of algebraic equations the reduced equations are used to determine the bifurcation point and the associated eigen mode of the panel in the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new updated sets of basis vectors and reduced equations are generated in the third stage the reduced equations are used to trace the post buckling paths the effectiveness of the proposed technique for predicting the bifurcation and post buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements

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

diffusion over a fluctuating barrier in underdamped dynamics we apply a langevin model by imposing additive and multiplicative noises to study thermally activated diffusion over a fluctuating barrier in underdamped dynamics the barrier fluctuation is characterized by gaussian colored noise with exponential correlation we present the exact solutions for the first and second moments furthermore we use direct simulations to calculate the asymptotic probability for a brownian particle passing over the fluctuating barrier the results indicate that the correlation of the fluctuating barrier is crucial for barrier crossing dynamics

reduction methods for nonlinear steady state thermal analysis hybrid analysis techniques based on the combined use of finite elements and the classical bubnov galerkin approximation are presented for predicting nonlinear steady state temperature distributions in structures and solids in these hybrid techniques the modelling versatility of the finite element method is preserved and a substantial reduction in the number of degrees of freedom is achieved by expressing the vector of nodal temperatures as a linear combination of a small number of global temperature modes or basis vectors the bubnov galerkin technique is then used to compute the coefficients of the linear combination i e the amplitudes of the global temperature modes the basis vectors chosen are the path derivatives commonly used in perturbation techniques namely the derivatives of the nodal temperature vector with respect to a preselected control or path parameter s the vectors are generated by using the finite element model of the initial discretization also the performance of alternate sets of basis vectors is investigated in the alternate sets only a few path derivatives are generated and they are augmented by a constant vector representing a uniform temperature rise or drop and by reciprocal vectors with nonzero components equal to the reciprocals of the nonzero components of the path derivatives a problem adaptive computational algorithm is presented for efficient evaluation of global approximation vectors and generation of the reduced system of equations and for monitoring the accuracy of the reduced system of equations the potential of the proposed reduction methods for the solution of large scale nonlinear steady state thermal problems is also discussed the effectiveness of these methods is demonstrated by means of four numerical examples including conduction convection and radiation modes of heat transfer this study shows that the use of the uniform temperature mode and the path derivatives as global approximation vectors significantly increases the accuracy of the solutions obtained by reduction methods thereby enhancing the effectiveness of these methods for the solution of large scale nonlinear thermal problems

accelerating iterative solution methods using reduced order models as solution predictors we propose the use of reduced order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large scale numerical simulation the acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps the algorithm basically consists of two projection steps 1 projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions and 2 solving the governing equations in this reduced space and projecting the solution back on the original high dimensional one we applied the algorithm to numerical models for simulation of two phase flow through heterogeneous porous media in particular we considered implicit pressure explicit saturation impes schemes and investigated the scope to accelerate the iterative solution of the pressure equation which is by far the most time consuming part of any impes scheme we achieved a substantial reduction in the number of iterations and an associated acceleration of the solution our largest test problem involved 93 500 variables in which case we obtained a maximum reduction in computing time of 67 the method is particularly attractive for problems with time varying parameters or source terms copyright 2006 john wiley sons ltd

noise induced anomalous diffusion over a periodically modulated saddle we study analytically and numerically the anomalous diffusion across periodically modulated parabolic potential within langevin and fokker planck descriptions we find that the probability of particles passing over the saddle is affected strikingly by the periodical modulation with average zero bias particularly the initial phase plays an important role in the modulation effect the effect of the correlation time of external ornstein uhlenbeck noise on dynamical process is also discussed a reduction in overpassing probability is observed due to finite correlation time

a method for interpolating on manifolds structural dynamics reduced order models a rigorous method for interpolating a set of parameterized linear structural dynamics reduced order models roms is presented by design this method does not operate on the underlying set of parameterized full order models hence it is amenable to an online real time implementation it is based on mapping appropriately the rom data onto a tangent space to the manifold of symmetric positive definite matrices interpolating the mapped data in this space and mapping back the result to the aforementioned manifold algorithms for computing the forward and backward mappings are offered for the case where the roms are derived from a general galerkin projection method and the case where they are constructed from modal reduction the proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input in all cases good accuracy is demonstrated at real time processing speeds copyright 2009 john wiley sons ltd

a new multi solution approach suitable for structural identification problems abstract in this paper a multi solution strategy aimed at greatly decreasing the cost of structural identification problems is proposed it is based on the large time increment method and on the possibility offered by this approach to obtain a family of admissible displacement and stress fields in order to solve a nonlinear time dependent problem this technique relies on a time and space decomposition of the solution it has been extended to solve at a low cost the inverse problem of identification for various techniques in particular it has been adapted to the minimization of the gradient of the gap between test and simulation in this first attempt it is applied to the simple case of bending of an elastic viscoplastic beam the issue of the numerical efficiency of this type of strategy is studied

rb reduced basis for rb rayleigh benard abstract the reduced basis approximation is a discretization method that can be implemented for solving parameter dependent problems in cases of many queries in this work it is applied to a two dimensional rayleigh benard problem that depends on the rayleigh number which measures buoyancy for each fixed aspect ratio multiple steady solutions can be found for different rayleigh numbers and stable solutions coexist at the same values of external physical parameters the reduced basis method permits to speed up the computations of these solutions at any value of the rayleigh number chosen in a fixed interval associated with a single bifurcation branch while maintaining accuracy

reduced basis approximation and a posteriori error estimation for stress intensity factors we develop reduced basis approximations rigorous a posteriori error bounds and offline online computational procedures for the accurate fast and reliable prediction of stress intensity factors or strain energy release rates for mode i linear elastic crack problems relevant to fracture mechanics we demonstrate the efficiency and rigor of our numerical method in several examples

comparison between the modal identification method and the pod galerkin method for model reduction in nonlinear diffusive systems this paper presents a comparison between the modal identification method mim and the proper orthogonal decomposition galerkin pod g method for model reduction an example of application on a nonlinear diffusive system is used to illustrate the study the study shows that in both methods the state formulation of the nonlinear diffusive equation may be similar however the ideas behind both methods are completely different the considered example shows that for both methods reducing the order up to 99 5 gives enough accuracy to simulate the dynamic of the original system it is also seen in this example that the reduced model given through the mim are slightly faster and more accurate than the ones given through the pod g method copyright 2006 john wiley sons ltd

an optimal projection method for the reduced order modeling of incompressible flows a new method to build reduced order models roms for incompressible flows is addressed in this paper proper orthogonal decompositions pods are used to efficiently represent both the fluid velocity and pressure fields with a small number of spatial modes a minimum residual projection is then developed for building the reduced models in which the velocity and pressure temporal coefficients are fully coupled comparisons of numerical calculations based on the proposed approach with results arising from the classical galerkin projection are exposed on three fluid flows a transient ventilated cavity a periodic lid driven cavity and a transient mixed convection flow it is shown that both stability and accuracy of the roms are strongly enhance when the minimum residual projection is used

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

a posteriori error estimation and basis adaptivity for reduced basis approximation of nonaffine parametrized linear elliptic partial differential equations in this paper we extend the earlier work m barrault y maday n c nguyen a t patera an empirical interpolation method application to efficient reduced basis discretization of partial differential equations c r acad sci paris serie i 339 2004 667 672 m a grepl y maday n c nguyen a t patera efficient reduced basis treatment of nonaffine and nonlinear partial differential equations m2an math model numer anal 41 3 2007 575 605 to provide a posteriori error estimation and basis adaptivity for reduced basis approximation of linear elliptic partial differential equations with nonaffine parameter dependence the essential components are i rapidly convergent reduced basis approximations galerkin projection onto a space w n u spanned by n global hierarchical basis functions which are constructed from solutions of the governing partial differential equation at judiciously selected points in parameter space ii stable and inexpensive interpolation procedures methods which allow us to replace nonaffine parameter functions with a coefficient function expansion as a sum of m products of parameter dependent coefficients and parameter independent functions iii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp error bounds for the error in the outputs of interest iv optimal basis construction processes which make use of the error bounds as an inexpensive surrogate for the expensive true error to explore the parameter space in the quest for an optimal sampling set and v offline online computational procedures methods which decouple the generation and projection stages of the approximation process the operation count for the online stage in which given a new parameter value we calculate the output of interest and associated error bounds depends only on n m and the affine parametric complexity of the problem the method is thus ideally suited for repeated and reliable evaluation of input output relationships in the many query or real time contexts

modified kramers formulas for the decay rate in agreement with dynamical modeling abstract accuracy of the kramers approximate formulas for the thermal decay rate of the metastable state is studied for the anharmonic shapes of the potential pocket and the barrier this is done by the comparison with the quasistationary rate resulting from the dynamical modeling disagreement between the kramers rate and the dynamical one is shown to reach 15 in the cases when much better agreement is expected corrections to the kramers formulas accounting for the higher derivatives of the potential are obtained the small parameters are the ratios of the thermal energy to the stiffnesses at the extremes of the potential the distance between the potential barrier and the absorptive border is accounted for as well this corrected kramers rate is demonstrated to agree with the dynamical rate typically within 2 probably the most interesting result is that despite the corrections are derived in the case of the overdamped brownian motion the above 2 agreement holds even in the case of medium friction

non markovian diffusion over a saddle with a generalized langevin equation the diffusion over a simple parabolic barrier is exactly solved with a non markovian generalized langevin equation for a short relaxation time the problem is shown to be similar to a markovian one with a smaller effective friction but for longer relaxation time the average trajectory starts to oscillate and the system can have a very fast first passage over the barrier for very long relaxation times the solution tends to a zero friction limit

a priori reduction method for solving the two dimensional burgers equations abstract the two dimensional burgers equations are solved here using the a priori reduction method this method is based on an iterative procedure which consists in building a basis for the solution where at each iteration the basis is improved the method is called a priori because it does not need any prior knowledge of the solution which is not the case if the standard karhunen loeve decomposition is used the accuracy of the apr method is compared with the standard newton raphson scheme and with results from the literature the apr basis is also compared with the karhunen loeve basis

a novel reduced basis method with upper and lower bounds for real time computation of linear elasticity problems abstract this paper presents a novel reduced basis method for analyzing problems of linear elasticity in a systematical rapid and reliable fashion for solutions with both upper and lower bounds to the exact solution in the form of energy norm or compliance output the lower bound of the solution output is obtained form the well known reduced basis method based on the galerkin projection used in the finite element method which is termed as gp rbm for the upper bound a new reduced basis approach is developed by the combination of the reduced basis method and a smoothed galerkin projection used in the linearly conforming point interpolation method and it is thus termed as sgp rbm to examine the present sgp rbm we first conduct a theoretical study on the very important upper bound property reduced basis models for both gp rbm and sgp rbm are constructed with the aid of an asymptotic error estimation and greedy adaptive procedure the gp rbm and the newly proposed sgp rbm are applied to analyze a cantilever beam with an oblique crack to verify the proposed rbm technique in terms of accuracy convergence bound properties and computational savings both theoretical analysis and numerical results have demonstrated that the present method is a very efficient method for real time solutions providing exact output bounds

reduced basis method for linear elasticity problems with many parameters he reduced basis rb methods are proposed here for the solution of parametrized equations in linear elasticity problems the fundamental idea underlying rb methods is to decouple the generation and projection stages offline online computational procedures of the approximation process in order to solve parametrized equations in a rapid inexpensive and reliable way the method allows important computational savings with respect to the classical galerkin finite element method ill suited to a repetitive environment like the parametrized contexts of optimization many queries and sensitivity analysis we consider different parametrization for the systems either physical quantities to model the materials and loads and geometrical parameters to model different geometrical configurations then we describe three different applications of the method in problems with isotropic and orthotropic materials working in plane stress and plane strain approximation and subject to harmonic loads

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

a new estimate for the vector valued corona problem abstract we improve an estimate of fuhrmann and vasyunin in the vector valued corona theorem

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

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

an h2 corona theorem on the bidisk for infinitely many functions we extend the h 2 corona theorem on the bidisk to infinitely many functions the main tool is a linear algebra result which simplifies the bookkeeping for estimates of solutions

model order reduction of nonlinear homogenization problems using a hashin shtrikman type finite element method abstract this work presents a computational nonlinear homogenization approach the starting point of which is a model order reduction method based on data clustering to this end the micromechanical data from numerical experiments snapshots is analyzed in order to identify characteristic microstructural deformation patterns these describe how the macroscopic strain typically localizes within the microstructure the outcome of the procedure is a subdivision of the microstructure into a set of clusters of material points within each cluster the strain is then approximated as being constant the mechanical problem is formulated in terms of a three field hashin shtrikman type variational formulation which is based on the introduction of a linear elastic reference medium after discretization most of the global unknowns can be eliminated via static condensation leaving the piecewise constant cluster strains as the primary unknowns the resulting homogenization scheme includes as special cases the finite element method as well as hashin shtrikman and talbot willis type homogenization approaches with phase wise constant trial fields as well as related bounds the limit case finite element method allows to transfer knowledge from finite element technology and thus provides new strategies for the choice of the stiffness of the reference material the method is applied to several nonlinear microstructures with different inclusion volume fractions and varying degree of anisotropy the results are shown to be in good agreement with full field fe simulations furthermore the method is used to compute a refined upper bound of the talbot willis type compared to phase wise constant trial fields which converges to the finite element solution with increasingly refined discretization

reduced basis hybrid computational homogenization based on a mixed incremental formulation a new reduced basis method for the computer assisted homogenization of microheterogeneous materials is proposed key features of the nonuniform transformation field analysis ntfa 1 2 are taken as a point of departure a short coming of the ntfa method is its limitation to simple constitutive models on the microscale a second disadvantage is the possible loss of accuracy when the reduced basis approximating the plastic strains has increasing dimension both issues are related to the simple evolution law for the mode activity coefficients which are the new macroscopic internal variables of the homogenized material in the present contribution a generalization of the ntfa is proposed in which the evolution of the new internal variables is derived from a mixed incremental variational formulation the derivation is based on purely micro mechanical considerations the modification allows for arbitrary generalized standard materials 3 on the microscopic scale including e g crystal visco plasticity additionally the fidelity of the homogenized response is now directly linked to the reduced basis approximating the plastic strain field numerical examples for nonlinear viscous materials and single crystal plasticity outline the accuracy of the proposed multi scale method

bmo estimates for the h bn corona problem we study the h bn corona problem j 1nfjgj h and show it is always possible to find solutions f that belong to bmoa bn for any n 1 including infinitely many generators n this theorem improves upon both a 2000 result of andersson and carlsson and the classical 1977 result of varopoulos the former result obtains solutions for strictly pseudoconvex domains in the larger space h bmoa with n while the latter result obtains bmoa bn solutions for just n 2 generators with h 1 our method of proof is to solve problems and to exploit the connection between bmo functions and carleson measures for h2 bn key to this is the exact structure of the kernels that solve the equation for 0 q forms as well as new estimates for iterates of these operators a generalization to multiplier algebras of besov sobolev spaces is also given

enablers for robust pod models this paper focuses on improving the stability as well as the approximation properties of reduced order models rom based on proper orthogonal decomposition pod the rom is obtained by seeking a solution belonging to the pod subspace and that at the same time minimizes the navier stokes residuals we propose a modified rom that directly incorporates the pressure term in the model the rom is then stabilized making use of a method based on the fine scale equations an improvement of the pod solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations the methods proposed are tested on the two dimensional confined square cylinder wake flow in laminar regime

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

hyper reduction of mechanical models involving internal variables we propose to improve the efficiency of the computation of the reduced state variables related to a given reduced basis this basis is supposed to be built by using the snapshot proper orthogonal decomposition pod model reduction method in the framework of non linear mechanical problems involving internal variables the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model this drawback is due to the fact that using a galerkin formulation the size of the reduced basis has no effect on the complexity of the constitutive equations in this paper we show how a reduced basis approximation and a petrov galerkin formulation enable to reduce the computational effort related to the internal variables the key concept is a reduced integration domain where the integration of the constitutive equations is performed the local computations being not made over the entire domain we extrapolate the computed internal variable over the full domain using pod vectors related to the internal variables this paper shows the improvement of the computational saving obtained by the hyper reduction of the elasto plastic model of a simple structure

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

non markovian diffusion over a parabolic potential barrier influence of the friction memory function the over passing probability across an inverted parabolic potential barrier is investigated according to the classical and quantal generalized langevin equations it is shown that in the classical case the asymptotic value of the over passing probability is determined by a single dominant root of the characteristic function and it is given by a simple expression the expression for the over passing probability is quite general and details of dissipation mechanism and memory effects enter into the expression only through the dominant root of the characteristic equation

the matrix valued hp corona problem in the disk and polydisk abstract in this paper we consider the matrix valued h p corona problem in the disk and polydisk the result for the disk is rather well known and is usually obtained from the classical carleson corona theorem by linear algebra our proof provides a streamlined way of obtaining this result and allows one to get a better estimate on the norm of the solution in particular we were able to improve the estimate found in the recent work of trent in j funct anal 189 2002 267 282 note that the solution of the h matrix corona problem in the disk can be easily obtained from the h 2 corona problem either by factorization or by the commutant lifting theorem the h p corona problem in the polydisk was originally solved by lin in bull sci math 110 2 1986 69 84 trans amer math soc 341 1994 371 375 the solution used koszul complexes and was rather complicated because one had to consider higher order equations our proof is more transparent and it improves upon lin s result in several ways first we deal with the more general matrix corona problem second we were able to show that the norm of the solution is independent of the number of generators additionally we illustrate that the norm of the solution of the h 2 corona problem in the polydisk d n grows at most proportionally to n our approach is based on one that was originated by andersson in math z 201 1989 121 130 in the disk it essentially depends on green s theorem and duality to obtain the estimate in the polydisk we use riesz projections to reduce the problem to the disk case