multi element least square hdmr methods and their applications for stochastic multiscale model reduction stochastic multiscale modeling has become a necessary approach to quantify uncertainty and characterize multiscale phenomena for many practical problems such as flows in stochastic porous media the numerical treatment of the stochastic multiscale models can be very challengeable as the existence of complex uncertainty and multiple physical scales in the models to efficiently take care of the difficulty we construct a computational reduced model to this end we propose a multi element least square high dimensional model representation hdmr method through which the random domain is adaptively decomposed into a few subdomains and a local least square hdmr is constructed in each subdomain these local hdmrs are represented by a finite number of orthogonal basis functions defined in low dimensional random spaces the coefficients in the local hdmrs are determined using least square methods we paste all the local hdmr approximations together to form a global hdmr approximation to further reduce computational cost we present a multi element reduced least square hdmr which improves both efficiency and approximation accuracy in certain conditions to effectively treat heterogeneity properties and multiscale features in the models we integrate multiscale finite element methods with multi element least square hdmr for stochastic multiscale model reduction this approach significantly reduces the original model s complexity in both the resolution of the physical space and the high dimensional stochastic space we analyze the proposed approach and provide a set of numerical experiments to demonstrate the performance of the presented model reduction techniques multi element least square hdmr is proposed to treat stochastic models random domain is adaptively decomposed into some subdomains to obtain adaptive multi element hdmr least square reduced hdmr is proposed to enhance computation efficiency and approximation accuracy in certain conditions integrating msfem and multi element least square hdmr can significantly reduce computation complexity

numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

elastic response of a free half space to random force excitations applied on the boundary abstract a response of an elastic half space to random forces applied normally to the free boundary is studied this paper is the second part of the study we presented in i a shalimova k k sabelfeld the response of an elastic three dimensional half space to random correlated displacement perturbations on the boundary physica a 389 21 2010 4436 4449 where the case of random displacements on the boundary was considered we analyze the white noise excitations in detail and derive explicitly the mean of the elastic energy the strain and displacement correlation tensors simulation algorithms are constructed both for displacement and strain random fields which enables us to calculate any desired statistical characteristics

uncertainty propagation in reduced order models based on crystal plasticity abstract in this work an uncertainty propagation study is performed based on simulated ensembles of statistical volume elements sve used to inform a reduced order internal state variable model focusing on model form and uncertainties in numerical simulations a rate dependent polycrystal plasticity finite element model cp fem of cartridge brass is calibrated to room temperature uniaxial tension testing data of an annealed sample model forms are considered with and without the inclusion of back stress in the cp model three sizes of sve are explored for simulation uncertainty effects effects are explored with regard to the parameters of a bammann chiesa johnson bcj macroscopic viscoplasticity model calibrated element wise to sves from an ensemble modeled using cp fem simulations exploring quasi static uniaxial tension variability in stress strain response at multiple defined length scales is measured the selection of model form is discussed the numerical simulation uncertainty of the model reduction process is quantified

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

on periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure abstract due to high computational costs associated with stochastic computational homogenization a highly complex random material microstructure is often replaced by simplified parametric ergodic and sometimes periodic models this replacement is often criticized in the literature due to unclear error resulting from the periodicity and ergodicity assumptions in the current contribution we perform a validation of both assumptions through various numerical examples to this end we compare large scale non simplified and non ergodic models with simplified ergodic and periodic solutions in addition we analyze the hill mandel condition for stochastic homogenization problems and demonstrate that for a stochastic problem there are more than three classical types of boundary conditions as an example we propose two novel stochastic periodic boundary conditions which possess a clear physical meaning the effect of these novel periodic boundary conditions is also analyzed by comparing with non ergodic simulation results

solving stochastic systems with low rank tensor compression abstract for parametrised equations which arise for example in equations dependent on random parameters the solution naturally lives in a tensor product space the application which we have in mind is a stochastic linear elliptic partial differential equation spde usual spatial discretisation leads to a potentially large linear system for each point in the parameter space approximating the parametric dependence by a galerkin ansatz the already large number of unknowns for a fixed parameter value is multiplied by the dimension of the galerkin subspace for the parametric dependence and thus can be very large therefore we try to solve the total system approximately on a smaller submanifold which is found adaptively through compression during the solution process by alternating iteration and compression we show that for general linearly converging stationary iterative schemes and general adaptation processes which can be seen as a modification or perturbation of the iteration the interlaced procedure still converges our proposed modification can be used for most stationary solvers for systems on tensor products we demonstrate this on an example of a discretised spde with a simple preconditioned iteration we choose a truncated singular value decomposition svd for the compression and give details of the implementation finishing with some examples

fuzzy stochastic fem based homogenization framework for materials with polymorphic uncertainties in the microstructure uncertainties in the macroscopic response of heterogeneous materials result from two sources the natural variability in the microstructure s geometry and the lack of sufficient knowledge regarding the microstructure the first type of uncertainty is denoted aleatoric uncertainty and may be characterized by a known probability density function the second type of uncertainty is denoted epistemic uncertainty this kind of uncertainty cannot be described using probabilistic methods models considering both sources of uncertainties are called polymorphic in the case of polymorphic uncertainties some combination of stochastic methods and fuzzy arithmetic should be used thus in the current work we examine a fuzzy stochastic finite element method based homogenization framework for materials with random inclusion sizes we analyze an experimental radii distribution of inclusions and develop a stochastic representative volume element the stochastic finite element method is used to obtain the material response in the case of random inclusion radii due to unavoidable noise in experimental data an insufficient number of samples and limited accuracy of the fitting procedure the radii distribution density cannot be obtained exactly thus it is described in terms of fuzzy location and scale parameters the influence of fuzzy input on the homogenized stress measures is analyzed

wavelet based numerical homogenization a numerical homogenization procedure for elliptic differential equations is presented it is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions if the operator is in divergence form this is preserved by the homogenization procedure for periodic problems results similar to classical effective coefficient theory are proved the procedure can be applied to problems that are not cell periodic

singular value decomposition via gradient and self equivalent flows the task of finding the singular value decomposition svd of a finite dimensional complex linear operator is here addressed via gradient flows evolving on groups of complex unitary matrices and associated self equivalent flows the work constitutes a generalization of that of brockett on the diagonalization of real symmetric matrices via gradient flows on orthogonal matrices and associated isospectral flows it complements results of symes chu and other authors on continuous analogs of the classical qr algorithm as well as earlier work by the authors on svd via gradient flows on positive definite matrices

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

a multiscale stochastic finite element method on elliptic problems involving uncertainties in this study a multiscale stochastic finite element method mssfem is developed to resolve scale coupling stochastic elliptic problems based on formulations of a stochastic variational approach and scale bridging multiscale shape functions by employing polynomial chaos expansions and lagrange polynomials stochastic galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods i e variational multiscale method and multiscale finite element method the resulting spectral and pseudo spectral stochastic finite element methods are incorporated into the fine and coarse mesh computation of the mssfem a benchmark multiscale model and the numerical experiment based on the formulated mssfem are provided for illustration it is expected that the proposed mssfem can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

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 reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

cross correlation neural network models in this paper we provide theoretical foundations for a new neural model for singular value decomposition based on an extension of the hebbian learning rule called the cross coupled hebbian rule the model is extracting the svd of the cross correlation matrix of two stochastic signals and is an extension on previous work on neural network related principal component analysis pca we prove the asymptotic convergence of the network to the principal normalized singular vectors of the cross correlation and we provide simulation results which suggest that the convergence is exponential the new model may have useful applications in the problems of filtering for signal processing and signal detection

recursive updating the eigenvalue decomposition of a covariance matrix the author addresses the problem of computing the eigensystem of the modified hermitian matrix given the prior knowledge of the eigensystem of the original hermitian matrix specifically an additive rank k modification corresponding to adding and deleting blocks of data to and from the covariance matrix is considered an efficient and parallel algorithm which makes use of a generalized spectrum slicing theorem is derived for computing the eigenvalues the eigenvector can be computed explicitly in terms of the solution of a much reduced k k homogeneous hermitian system the overall computational complexity is shown to be improved by an order of magnitude from o n sup 3 to o n sup 2 k where n n is the size of the covariance matrix it is pointed out that these ideas can be applied to adaptive signal processing applications such as eigen based techniques for frequency or angle of arrival estimation and tracking specifically adaptive versions of the principal eigenvector method and the total least squares method are derived

stokes flows under random boundary velocity excitations a viscous stokes flow over a disc under random fluctuations of the velocity on the boundary is studied we give exact karhunen loeve k l expansions for the velocity components pressure stress and vorticity and the series representations for the corresponding correlation tensors both the white noise fluctuations and general homogeneous random excitations of the velocities prescribed on the boundary are studied we analyze the decay of correlation functions in angular and radial directions both for exterior and interior stokes problems numerical experiments show the fast convergence of the k l expansions the results indicate that ignoring the stochastic fluctuations in boundary conditions dramatically underestimates the variance of the velocity and pressure in the interior exterior of the domain

dynamical systems that perform the singular value decomposition abstract a dynamical system is presented whose solutions are shown to converge exponentially to the singular value decomposition of a given matrix provided that its singular values are distinct this system has potential application to certain problems in adaptive control and adaptive filtering it is one of several techniques that employ dynamical systems to perform tasks done conventionally with the techniques of numerical linear algebra

discrete to continuum bridging based on multigrid principles abstract the paper presents a new concurrent multiscale approach based on multigrid principles with intent of solving large molecular statics and molecular dynamics systems we show that the coarse model effective stiffness matrix obtained by variational restriction of the atomistic model effective stiffness matrix coincides with the effective stiffness matrix of the equivalent continuum scale model the equivalent continuum model is defined for both molecular statics and molecular dynamics cases and theoretical estimates of the rate of convergence for the proposed concurrent multiscale approach are obtained

integral formulas for chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions we evaluate explicitly the integrals 1 1 n t r t dt r 1 with the n being any one of the four chebyshev polynomials of degree n these integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with chebyshev abscissae when the function to be integrated is analytic in a domain containing 1 1 in its interior

stochastic aspects of localised failure material and boundary imperfections this paper presents the application of the finite element reliability method to the evaluation of the statistical properties of localisation phenomena material and boundary constraint imperfections are considered the latter are imposed through lagrange multipliers the evaluation of the sensitivity to the stochastic basic variables which is needed in the reliability method is outlined numerical simulations of mode i and mode ii localisation phenomena are presented

dynamic system methods for solving mixed linear matrix inequalities and linear vector inequalities and equalities a novel idea is proposed for solving a system of mixed linear matrix inequalities and linear vector inequalities and equalities first the problem is converted into an unconstrained minimization problem with a continuously differentiable convex objective function then a continuous time dynamic system and a discrete time dynamic system are proposed for solving it under some mild conditions the proposed dynamic systems are shown to be globally convergent to a solution of the problem the merits of the methods refer to their simple numerical implementations and capability for handling nonstrict lmis easily in addition the methods are promising in neural circuits realization and therefore have potential applications in many online control problems several numerical examples are presented to illustrate the performance of the methods and substantiate the theoretical results

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

strong control of the familywise error rate in observational studies that discover effect modification by exploratory methods an effect modifier is a pretreatment covariate that affects the magnitude of the treatment effect or its stability when there is effect modification an overall test that ignores an effect modifier may be more sensitive to unmeasured bias than a test that combines results from subgroups defined by the effect modifier if there is effect modification one would like to identify specific subgroups for which there is evidence of effect that is insensitive to small or moderate biases in this paper we propose an exploratory method for discovering effect modification and combine it with a confirmatory method of simultaneous inference that strongly controls the familywise error rate in a sensitivity analysis despite the fact that the groups being compared are defined empirically a new form of matching strength k matching permits a search through more than k covariates for effect modifiers in such a way that no pairs are lost provided that at most k covariates are selected to group the pairs in a strength k match each set of k covariates is exactly balanced although a set of more than k covariates may exhibit imbalance we apply the proposed method to study the effects of the earthquake that struck chile in 2010

three dimensional stochastic finite element method for elasto plastic bodies a new stochastic finite element method sfem is formulated for three dimensional softening elasto plastic bodies with random material properties the method is based on the karhunen loeve and polynomial chaos expansions and able to efficiently estimate complete probabilistic characteristics of the response such as moments or pdfs to reduce the computational complexity in the three dimensional setting two alterations are made with respect to the two dimensional sfem proposed earlier by the authors first a variability preserving modification of the karhunen loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties second an efficient algorithm for parallel processing is developed with time consumption being the same order as for an ordinary fem rendering the proposed sfem an effective alternative to monte carlo simulation the applicability of the proposed method to stochastic analysis of strain localization is examined using monte carlo simulation then it is applied to a fault formation problem which is a recent concern of earthquake engineering ground surface layers are modelled by a softening elasto plastic body and the evolution of probabilistic characteristics of the rupture process is analysed in detail some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint copyright 2001 john wiley sons ltd

cohesive zone models higher order continuum theories and reliability methods for computational failure analysis a concise overview is given of various numerical methods that can be used to analyse localization and failure in engineering materials the importance of the cohesive zone approach is emphasized and various ways to incorporate the cohesive zone methodology in discretization methods are discussed numerical representations of cohesive zone models suffer from a certain mesh bias for discrete representations this is caused by the initial mesh design while for smeared representations it is rooted in the ill posedness of the rate boundary value problem that arises upon the introduction of decohesion a proper representation of the discrete character of cohesive zone formulations which avoids any mesh bias can be obtained elegantly when exploiting the partition of unity property of finite element shape functions the effectiveness of the approach is demonstrated for some examples at different scales moreover examples are shown how this concept can be used to obtain a proper transition from a plastifying or damaging continuum to a shear band with gross sliding or to a fully open crack true discontinuum when adhering to a continuum description of failure higher order continuum models must be used meshless methods are ideally suited to assess the importance of the higher order gradient terms as will be shown finally regularized strain softening models are used in finite element reliability analyses to quantify the probability of the emergence of various possible failure modes

a neural network learning for adaptively extracting cross correlation features between two high dimensional data streams this paper proposes a novel cross correlation neural network cnn model for finding the principal singular subspace of a cross correlation matrix between two high dimensional data streams we introduce a novel nonquadratic criterion nqc for searching the optimum weights of two linear neural networks lnn the nqc exhibits a single global minimum attained if and only if the weight matrices of the left and right neural networks span the left and right principal singular subspace of a cross correlation matrix respectively the other stationary points of the nqc are unstable saddle points we develop an adaptive algorithm based on the nqc for tracking the principal singular subspace of a cross correlation matrix between two high dimensional vector sequences the nqc algorithm provides a fast online learning of the optimum weights for two lnn the global asymptotic stability of the nqc algorithm is analyzed the nqc algorithm has several key advantages such as faster convergence which is illustrated through simulations

a cross associative neural network for svd of non squared data matrix in signal processing this paper proposes a cross associative neural network cann for singular value decomposition svd of a non squared data matrix in signal processing in order to improve the convergence speed and avoid the potential instability of the deterministic networks associated with the cross correlation neural network models we study the global asymptotic stability of the network for tracking all the singular components and show that the selection of its learning rate in the iterative algorithm is independent of the singular value distribution of a non squared matrix the performances of cann are shown via simulations

sampling free linear bayesian update of polynomial chaos representations we present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes described by possibly non gaussian distributions the description of the introduced random fields is given in a white noise framework which enables us to solve the stochastic forward problem through galerkin projection onto polynomial chaos with the help of such a representation the probabilistic identification problem is cast in a polynomial chaos expansion setting and the baye s linear form of updating by introducing the hermite algebra this becomes a direct purely algebraic way of computing the posterior which is comparatively inexpensive to evaluate in addition we show that the well known kalman filter is the low order part of this update the proposed method is here tested on a stationary diffusion equation with prescribed source terms characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity

stochastic collocation and mixed finite elements for flow in porous media the aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments the governing equations are based on darcy s law with stochastic permeability starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product hermite polynomials is used in the stochastic dimensions error analysis is performed and experimentally verified with numerical simulations computational results include incompressible and slightly compressible single and two phase flow

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

the bird s eye view on finite element method for structures with large stochastic variations abstract extensive motivation to do an additional work in the finite element method in stochastic problems femsp is discussed the qualitative comparison of femsp with the state of the art of the deterministic fem is given these observations and thoughts are then realized in several manners we first present the exact inverse femsp which however cannot serve as a general tool for stochastic analysis of complex structures the new variational principles for stochastic beams for the mean response function as well as response s auto correlation function are formulated and the fem based on the variational principles is presented finally a general non perturbative fem for stochastic problems with large variations is developed based on the element level flexibility it is concluded that much work needs to be done in order femsp to be at the level compared to that of the deterministic fem the advantage of the main methods presented here over the conventional ones lies in their non perturbational nature numerical examples are presented

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

bubble functions prompt unusual stabilized finite element methods a second order linear scalar differential equation including a zero th order term is approximated using first the standard galerkin method enriched with bubble functions static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations the method is generalized to allow for a convection operator in the equation this work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena flows with chemical reactions and other important problems

estimation of local modeling error and goal oriented adaptive modeling of heterogeneous materials i error estimates and adaptive algorithms abstract a theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented these quantities may for example represent averaged stresses on the surface of inclusions or mollifications of pointwise stresses or displacements or in general local features of the fine scale solution characterized by continuous linear functionals these estimators are used to construct goal oriented adaptive procedures in which models of the microstructure are adapted to deliver local features to a preset level of accuracy algorithms for implementing these procedures are discussed and preliminary numerical results are given the analysis is restricted to linear static heterogeneous elastic materials

elastostatics of a half plane under random boundary excitations a stochastic analysis of an elastostatics problem for a half plane under random white noise excitations of the displacement vector prescribed on the boundary is given solutions of the problem are inhomogeneous random fields governed by the lame equation with random boundary conditions this is used to model the displacements strain tensor vorticity and the deformation energy and to give exact representations for their correlation tensors as well as the corresponding karhunen loeve k l expansions numerical calculations illustrating the rate of convergence of the spectral and k l expansions are also given an interesting behaviour of the strain correlation tensor for the increasing value of the elasticity constant is found theoretically and confirmed by calculations the paper presents the second part of our study the first being published recently in sabelfeld and shalimova j stat phys 132 6 1071 1095 2008 where only the displacement correlation tensor was derived and analyzed

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

stochastic computation based on orthogonal expansion of random fields abstract in solving stochastic differential equations recently a random variable based polynomial chaos rv pc method has been developed as a major numerical solver for many realistic random media problems the rv pc method however confronts a critical challenge of curse of dimensionality since a random field is represented by random variables the use of various optimal sampling techniques and conventional high dimensional methods still faces the curse of dimensionality distinguished from all the random variable based methods in this study a novel random field based orthogonal expansion rf oe method is proposed in aim to circumvent the curse of dimensionality for many physical systems whereas the input information is represented as random fields or stochastic processes e g seismic ocean wave wind load shock wave and geophysical media multiscale modeling of random media problems is selected as the benchmark problem to test the rf oe method especially the rf oe method provides a perfect matching with the higher order mehler s formula by replacing high dimensional random variable representations with a series of orthogonal expansion terms about an underlying random field process the rf oe method reduces the number of dimensions of a stochastic differential equation exponentially in the first example the rf oe method is verified with monte carlo simulation on a lognormal random media flow transport problem in the second example the rf oe method is applied to a time domain problem involving orthogonal expansion of random excitations in the conclusion the items for further development of the rf oe method are identified

perturbation based stochastic multi scale computational homogenization method for the determination of the effective properties of composite materials with random properties quantifying uncertainty in the overall elastic properties of composite materials arising from randomness in the material properties and geometry of composites at microscopic level is crucial in the stochastic analysis of composites in this paper a stochastic multi scale finite element method which couples the multi scale computational homogenization method with the second order perturbation technique is proposed to calculate the statistics of the overall elasticity properties of composite materials in terms of the mean value and standard deviation the uncertainties associated with the material properties of the constituents are considered performance of the proposed method is evaluated by comparing mean values and coefficients of variation for components of the effective elastic tensor against corresponding values calculated using monte carlo simulation for three numerical examples results demonstrate that the proposed method has sufficient accuracy to capture the variability in effective elastic properties of the composite induced by randomness in the constituent material properties

stochastic analysis of an elastic 3d half space respond to random boundary displacements exact results and karhunen loeve expansion a stochastic response of an elastic 3d half space to random displacement excitations on the boundary plane is studied we derive exact results for the case of white noise excitations which are then used to give convolution representations for the case of general finite correlation length fluctuations of displacements prescribed on the boundary solutions to these elasticity problem are random fields which appear to be horizontally homogeneous but inhomogeneous in the vertical direction this enables us to construct explicitly the karhunen loeve k l series expansion by solving the eigen value problem for the correlation operator simulation results are presented and compared with the exact representations derived for the displacement correlation tensor this paper is a complete 3d generalization of the 2d case study we presented in sabelfeld and shalimova j stat phys 132 6 1071 1095 2008

a stochastic computational method for evaluation of global and local behavior of random elastic media a stochastic computational method is developed to evaluate global effective properties and local probabilistic behavior of random elastic media the elasticity solution for random media is addressed with a method that combines a fast iterative numerical method with stochastic decomposition techniques stochastic homogenization problems are mathematically formulated with an ensemble average approximation scheme based on a concept of stochastic representative volume element srve probabilistic descriptors of local strain are generated based on convolution equations derived from the equilibrium equations of elasticity with the application of the galerkin formulation in probability space and the fourier transform of the convolution equations an efficient numerical scheme is implemented using an iterative algorithm that takes advantage of fast fourier transform two algorithms are provided for gaussian and non gaussian random media respectively with corresponding sufficient conditions for convergence derived finally numerical experiments are conducted for a gaussian and a non gaussian random medium respectively with verifications obtained from monte carlo simulation

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

on stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix abstract in applications of signal processing and pattern recognition eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed iterative methods are suggested and analyzed in which no sample moments are used convergence is shown by stochastic approximation theory

an adaptive and powerful randomization test reference stap article 1992 005view record in web of science record created on 2006 04 04 modified on 2017 05 12

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

the response of an elastic three dimensional half space to random correlated displacement perturbations on the boundary abstract the response of an elastic half space to random excitations of displacements on the boundary under the condition of no shearing forces is studied we analyze the white noise excitations and general random fluctuations of displacements prescribed on the boundary we consider the case of partially ordered defects on the boundary whose positions are governed by an exponential cosine type correlation function the analysis is based on a poisson type integral formula which we derive here for the case of zero shearing forces on the boundary we obtain exact representations for the displacement correlation tensor and the karhunen loeve expansion for the solution of the lame equation itself and analyze some features of the correlation structure of the displacements the monte carlo technique developed can be applied to a wide class of differential equations with random boundary conditions

identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data this paper is devoted to the identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to partial experimental data made up of an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem has to be solved to carry out the identification of the random field a complete methodology is proposed to solve this challenging problem and consists in introducing a family of prior probability models in identifying an optimal prior model in the constructed family using the experimental data in constructing a statistical reduced order optimal prior model in constructing the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal prior model and finally in constructing the probability distribution of random coefficients of the polynomial chaos expansion and in identifying the parameters using experimental data an application is presented for which several millions of random coefficients are identified solving an inverse stochastic problem c 2010 elsevier b v all rights reserved

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

stochastic finite elements as a bridge between random material microstructure and global response this paper presents a finite element method aimed at the introduction of microstructural material randomness below the level of a single finite element a consideration of dependence of effective moduli on scale and on the defining boundary conditions leads to an identification of a finite element as a mesoscale window or a mesoscale finite element in a stochastic finite element method sfe an estimation of the global response can be obtained through bounds stemming from minimum potential energy and complementary energy principles which involve dirichlet and neumann boundary conditions on all the mesoscale finite elements respectively while in the classical case of a homogeneous material these two bounds converge to each other as the finite element mesh becomes sufficiently fine an optimal mesoscale with respect to the difference between both bounds may exist in the case of a heterogeneous material the proposed sfe method is illustrated with a numerical example of a sample two phase voronoi composite with some 26 000 grains where a reference solution taking into account the entire microstructure without any smearing out is shown to fall between both energy bounds a generalization to an ensemble response is straightforward

homogenization based finite element analysis of unidirectional composites by classical and multiresolutional techniques computational studies of unidirectional transient problems in multiresolutional periodic heterogeneous media using homogenization technique and the finite element method are documented below this homogenization method being the extension of the classical two scale asymptotic approach and based on the wavelet representation for composite parameters is used here to calculate effective material parameters of a composite efficiency of this method is compared against asymptotic homogenization technique and simple spatial averaging of material properties using the fem solution for some transient heat transfer and free vibration problems on the other hand a comparison with the solutions obtained for a multiresolutional decomposition of material parameters for composites is also made numerical results show that the homogenized parameters resulting from the new approach are bounded by spatial averages and by asymptotic method results this result is valid also for the eigenvalues of a periodic and simply supported composite beam the multiresolutional homogenization proposed appears to be especially efficient in case of smaller numbers of periodicity cells in the structure while increased number of cells needs application of an asymptotic homogenization method further computational studies are necessary in this area but the application of the multiresolutional technique is natural because of multiscale character of composites moreover this method may be very efficient in common symbolic fem programs implementation

a stochastic multiscale framework for modeling flow through random heterogeneous porous media flow through porous media is ubiquitous occurring from large geological scales down to the microscopic scales several critical engineering phenomena like contaminant spread nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system any realistic modeling of these transport phenomena has to resolve two key issues i the multi length scale variations in permeability that these systems exhibit and ii the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes a stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features a stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process recent developments in linear and non linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model an adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations spdes the framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given

computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis this paper is devoted to the computational nonlinear stochastic homogenization of a hyperelastic heterogeneous microstructure using a non concurrent multiscale approach the geometry of the microstructure is random since the non concurrent multiscale approach is based on the use of a tensorial decomposition which is then submitted to the curse of dimensionality we perform an analysis with respect to the stochastic dimension the technique uses a database describing the strain energy density function potential in both the macroscopic cauchy green strain space and the geometrical random parameters domain each value of the potential is numerically computed by means of the finit element method on an elementary cell an interpolation scheme is finall introduced to obtain a continuous explicit form of the potential which by derivation allows to evaluate the macroscopic stress and elastic tangent tensors during the macroscopic structural computations two numerical examples are presented

elastic half plane under random displacement excitations on the boundary we study in this paper a respond of an elastic half plane to random boundary excitations we treat both the white noise excitations and more generally homogeneous random fluctuations of displacements prescribed on the boundary solutions to these problems are inhomogeneous random fields which are however homogeneous with respect to the longitudinal coordinate this is used to represent the displacements as series expansions involving a complete set of deterministic functions with corresponding random coefficients we construct the karhunen loeve k l series expansion which is based on the eigen decomposition of the correlation operator the k l expansion can be used to calculate the statistical characteristics of other functionals of interest in particular the strain and stress tensors and the elastic energy tensor

multiscale methods for mechanical science of complex materials bridging from quantum to stochastic multiresolution continuum recent development in the multiscale method based on the bridging scale concept is presented with an emphasis on complex material systems the bridging scale method bsm was originally proposed by wagner and liu j comput phys 2003 190 249 274 as an effective way of treating the interface in coupled atomistic continuum simulation since its publication the bsm has become a very useful paradigm that has been applied to solve a host of problems in mechanical sciences of complex material systems in this paper we present a review on the recent developments we first describe the application of bsm to the coupled atomistic continuum simulation of dynamic fracture the important extensions within the framework of space time method and multiscale non equilibrium molecular dynamics are then presented we then focus on the multiresolution continuum theory that inherits the bsm concept in the analysis of heterogeneous material structures recent work of incorporating statistical factors into this model based on the concurrent nested homogenization of randomness of the material structures is highlighted finally we present the use of the bridging scale concept in resolving the electron mechanical coupling mechanism the robustness of the bsm is demonstrated through many benchmark problems and application examples copyright 2010 john wiley sons ltd