homogenization of two phase elasto plastic composite materials and structures study of tangent operators cyclic plasticity and numerical algorithms we develop homogenization schemes and numerical algorithms for two phase elasto plastic composite materials and structures a hill type incremental formulation enables the simulation of unloading and cyclic loadings it also allows to handle any rate independent model for each phase we study the crucial issue of tangent operators elasto plastic or continuum versus algorithmic or consistent and anisotropic versus isotropic we apply two methods of extraction of isotropic tangent moduli we compare mathematically the stiffnesses of various tangent operators all rate equations are discretized in time using implicit integration we implemented two homogenization schemes mori tanaka and a double inclusion model and two plasticity models classical j 2 plasticity and chaboche s model with non linear kinematic and isotropic hardenings we consider composites with different properties and present several discriminating numerical simulations in many cases the results are validated against finite element fe or experimental data we integrated our homogenization code into the fe program abaqus using a user material interface umat a two scale procedure allows to compute realistic structures made of non linear composite materials within reasonable cpu time and memory usage examples are shown c 2002 elsevier science ltd all rights reserved

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

quantization of filter bank frame expansions through moving horizon optimization this paper describes a novel approach to quantization in oversampled filter banks the new technique is based on moving horizon optimization does not rely on an additive white noise quantization model and allows stability to be explicitly enforced in the associated nonlinear feedback loop moreover the quantization structure proposed here includes sigmadelta and linear predictive subband quantizers as a special case and in general outperforms them

effective properties of composite materials with periodic microstructure a computational approach abstract this study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents the theoretical background of the method is recalled first two different families of numerical methods are considered to solve the problem the first is based on the finite element method the concept of macroscopic degrees of freedom is presented the implementation of periodicity conditions is discussed a general framework permitting either a strain or stress control is proposed the second numerical method is based on fast fourier transforms it considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain the solution of this problem is based on the explicit form of the periodic green s function of the reference medium the relative merits of the two methods are compared and several examples are discussed both methods give very comparable results on test examples and their domains of applications appear to be complementary

determination of rve size for random composites with local volume fraction variation abstract in this paper a novel computational procedure is proposed for the determination of the representative volume element rve size for random composites which is the basis of homogenization methods the proposed approach takes into account the local volume fraction variation by processing computer simulated images of composites with randomly scattered inclusions a variable number of microstructure models are derived directly from the images using the moving window technique each microstructure model contains different amount of inclusions measured by image analysis tools the proposed computational procedure is based on the solution of multiple boundary value problems under kinematic and static uniform boundary conditions using the extended finite element method xfem coupled with monte carlo simulation mcs in this way the statistical characteristics of the upper and lower bounds of the apparent material properties are computed the rve is attained within a prescribed tolerance by examining the convergence of these two bounds with respect to the mesoscale size the numerical results clearly demonstrate the significant effect of matrix inclusion stiffness ratio and volume fraction on the convergence rate the proposed computational procedure leads to rves which can give more realistic results when used in homogenization problems mesoscale random fields describing the spatial variation of the components of the apparent elasticity tensor are also determined using the proposed approach

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

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

micromechanics of inelastic composites with misaligned inclusions numerical treatment of orientation this paper is concerned with the micromechanics of multi phase fiber or inclusion reinforced inelastic composites with a special emphasis on the numerical treatment of misaligned orientation mean field homogenization of these composites involves volume and orientation averaging the latter usually needs average orientation measures known as the second and fourth rank orientation tensors a and a respectively usually the only orientation data available is a and a first issue is to reconstruct a using closure approximations we propose a new weighted regularization method which deals with orientations which are strictly neither 1d 2d nor 3d and ensures a smooth transition between the 3 cases a second issue is that there are situations where the inclusions orientation distribution function odf is needed for homogenization but is unavailable the odf is recovered using a and a and numerically computed at a number of discrete orientations a third issue is the actual computation of orientation averaging integrals an efficient algorithm approximates the integrals with sums over a finite number of orientations orientation and volume averaging concepts are applied to the mean field homogenization of two classes of multi phase composites linear thermo elastic and rate independent inelastic in the latter case an incremental formulation is proposed which enables the simulation of unloading cyclic and otherwise non proportional loadings implicit time discretization and consistent tangent operators are employed the procedures and the corresponding algorithms were implemented in the digimat version 1 4 2004 linear and nonlinear multi scale material modeling software e xstream engineering sa http www e xstream com louvain la neuve belgium software and several numerical simulations show their accuracy and efficiency c 2005 elsevier b v all rights reserved

maximum entropy approach for modeling random uncertainties in transient elastodynamics a new approach is presented for analyzing random uncertainties in dynamical systems this approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics the information used does not require the description of the local parameters of the mechanical model the probability model is deduced from the use of the entropy optimization principle whose available information is constituted of the algebraic properties related to the generalized mass damping and stiffness matrices which have to be positive definite symmetric matrices and the knowledge of these matrices for the mean reduced matrix model an explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to monte carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads the fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed finally an example is presented

non gaussian positive definite matrix valued random fields for elliptic stochastic partial differential operators this paper deals with the construction of a class of non gaussian positive definite matrix valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled the properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given such a matrix valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters the class of random fields which can be approximated is presented and their experimental identification is analyzed an example is given in three dimensional linear elasticity for which the fourth order elasticity tensor valued random field is constructed for a random non homogeneous anisotropic elastic material

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

de rham cohomology and hodge decomposition for quantum groups istvan heckenberger and axel schuler abstra t let z be one of the n dimensional bicovariant first order differential calculi for the quantum groups glq n slq n soq n or spq n where q is a transcendental complex number and z is a regular parameter it is shown that the de rham cohomology of woronowicz external algebra coincides with the de rham cohomologies of its left coinvariant its right coinvariant and its twosided coinvariant subcomplexes in the cases glq n and slq n the cohomology ring is isomorphic to the coinvariant external algebra inv and to the vector space of harmonic forms we prove a hodge decomposition theorem in these cases the main technical tool is the spectral decomposition of the quantum laplace beltrami operator

strain based transient gradient damage model for failure analyses a transient gradient enhanced damage model has been developed for the numerical modelling of the damage and fracture process within a continuum damage mechanics framework some deficiencies of existing gradient enhanced damage formulations for the simulation of macroscopic crack propagation are pointed out the transient gradient approach assumes a direct coupling between the material length parameter and the local strain state of the material which leads to a transient behaviour of the nonlocal effect details of the method are presented and fully elaborated in an incremental iterative solution scheme mesh objectivity and physical relevance of the method are analysed by one dimensional and two dimensional numerical examples

on locking free shear deformable beam finite elements abstract this is an elementary exposition of beam finite element models an overview of existing and new displacement finite element models of the euler bernoulli and shear deformation beam theories is presented a locking free finite element model using the form of the exact solution of the timoshenko beam theory is developed the element contains the euler bernoulli beam element as a special case and yields exact nodal values for the generalized displacements for constant material and geometric properties of beams the procedure is extended to develop a locking free finite element model of the simplified third order theory of beams the element contains the euler bernoulli and the locking free timoshenko beam finite elements as special cases the element stiffness matrix of the unified beam bending element is 4 4 and yet has the superconvergence character the advantage of this element is that it can be used to obtain exact nodal displacements and forces from the equilibrium equations of the element in a frame structural analysis using one finite element per a structural member this cannot be achieved with the traditional equal interpolation reduced integration element because the element exhibits shear locking for thin members unless two or more elements per a structural member are used

a finite strain incremental secant homogenization model for elasto plastic composites abstract this paper presents a finite strain extension of the incremental secant mean field homogenization mfh formulation for two phase elasto plastic composites the formulation of the local finite strain elasto plastic constitutive equations of each phase is based on a multiplicative decomposition of the deformation gradient as suggested by simo in computer methods in applied mechanics and engineering 99 1 61 112 1992 the latter has proposed algorithms which preserve the classical return mapping schemes of the infinitesimal theory by using principal kirchhoff stresses and logarithmic eigenvalues of the left elastic cauchy green strain relying on this property we show that by considering a quadratic logarithmic free energy and j2 flow theory at the local level infinitesimal strain incremental secant mfh is readily extended to finite strains the proposed formulation and corresponding numerical algorithms are then presented finally the predictions are illustrated with several numerical simulations which are verified against full field finite element simulations of composite cells demonstrating that the micro mechanically based approach is able to predict the influence of the micro structure and of its evolution on the macroscopic properties in a very cost effective manner

multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous galerkin formulation abstract when considering problems of dimensions close to the characteristic length of the material the size effects can not be neglected and the classical so called first order multiscale computational homogenization scheme fmch looses accuracy motivating the use of a second order multiscale computational homogenization smch scheme this second order scheme uses the classical continuum at the micro scale while considering a second order continuum at the macro scale although the theoretical background of the second order continuum is increasing the implementation into a finite element code is not straightforward because of the lack of high order continuity of the shape functions in this work we propose a smch scheme relying on the discontinuous galerkin dg method at the macro scale which simplifies the implementation of the method indeed the dg method is a generalization of weak formulations allowing for inter element discontinuities either at the c 0 level or at the c 1 level and it can thus be used to constrain weakly the c 1 continuity at the macro scale the c 0 continuity can be either weakly constrained by using the dg method or strongly constrained by using usual c 0 displacement based finite elements therefore two formulations can be used at the macro scale i the full discontinuous galerkin formulation fdg with weak c 0 and c 1 continuity enforcements and ii the enriched discontinuous galerkin formulation edg with high order term enrichment into the conventional c 0 finite element framework the micro problem is formulated in terms of standard equilibrium and periodic boundary conditions a parallel implementation in three dimensions for non linear finite deformation problems is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way

numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage this paper deals with a class of rate independent metal plasticity models which exhibit non linear isotropic hardening non linear kinematic hardening chaboche marquis model and ductile damage lemaitre chaboche model the backward euler scheme is used to integrate the rate constitutive relations the non linear equations obtained are solved by the newton method the consistent tangent operator is obtained by exact linearization of the algorithm despite the complexity of the constitutive equations closed form expressions are derived without any approximations analytical numerical and experimental results are presented acid discussed

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

statistical volume element method for predicting microstructure constitutive property relations abstract there is an inevitable need to establish multilength scale statistical microstructure constitutive property relations in materials design in this paper we have developed a statistical volume element method to analyze quantify and calibrate such microstructure constitutive property relations by statistical means statistical volume element simulations are adopted to predict material constitutive properties corresponding to various realizations of random microstructure configurations a computing framework that links random configuration generators and finite element analysis has been developed a statistical cause effect analysis approach is proposed to study the influence of random material microstructure on material constitutive properties within the proposed approach statistically significant microstructure parameters are first identified based on their linear impacts on material constitutive properties global sensitivity analysis is then employed to provide a more comprehensive importance ranking of these critical microstructure parameters considering both main and interaction effects the uncertainties in material constitutive properties due to random microstructure configurations are quantified in terms of distributions statistical moments and correlations the obtained probabilistic constitutive relations are used to calibrate the model parameters in a constitutive relation model following a statistical calibration process the proposed approach is applied to examine a porous steel alloy material for demonstrative purposes

near perfect reconstruction pseudo qmf banks a novel approach to the design of m channel pseudo quadrature mirror filter qmf banks is presented in this approach the prototype filter is constrained to be a linear phase spectral factor of a 2mth band filter as a result the overall transfer function of the analysis synthesis system is a delay moreover the aliasing cancellation ac constraint is derived such that all the significant aliasing terms are canceled consequently the aliasing level at the output is comparable to the stopband attenuation of the prototype filter in other words the only error at the output of the analysis synthesis system is the aliasing error which is at the level of stopband attenuation using this approach it is possible to design a pseudo qmf bank where the stopband attenuation of the analysis and thus synthesis filters is on the order of spl minus 100 db moreover the resulting reconstruction error is also on the order of spl minus 100 db several examples are included

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

a new variational principle and its application to nonlinear heterogeneous systems a new variational principle is proposed for estimating the effective properties of nonlinear heterogeneous systems the variational principle expresses the effective energy density function of a given nonlinear heterogeneous dielectric in terms of an infinite dimensional optimization problem involving the effective energy density functions of the class of linear comparison materials with variable dielectric cofficients the variational principle can alternatively be used in an approximate fashion by optimizing over finite dimensional subspaces of the original space of comparison dielectric coefficients to obtain bounds on the effective dielectric properties of nonlinear composite materials with isotropic constituent phases in given proportions the application of the procedure is illustrated by computing an exact estimate for the effective energy density function of a nonlinear laminated dielectric material which has overall anisotropic effective behavior it is also demonstrated that the talbot willis

quantum group gauge theory on quantum spaces we construct quantum group valued canonical connections on quantum homogeneous spaces including aq deformed dirac monopole on the quantum sphere of podles with quantum differential structure coming from the 3d calculus of woronowicz onsu q 2 the construction is presented within the setting of a general theory of quantum principal bundles with quantum group hopf algebra fibre associated quantum vector bundles and connection one forms both the base space spacetime and the total space are non commutative algebras quantum spaces

optimal signal reconstruction in noisy filter bank systems multirate kalman synthesis filtering approach a multirate kalman synthesis filter is proposed in this paper to replace the conventional synthesis filters in a noisy filter bank system to achieve optimal reconstruction of the input signal based on an equivalent block representation of subband signals a state space model is introduced for an m band filter bank system with subband noises the composite effect of the input signal analysis filter bank decimators and interpolators is represented by a multirate state space model the input signal is embedded in the state vector and the corrupting noises in subband paths are generally considered as additive noises hence the signal reconstruction problem in the m band filter bank systems with subband noises becomes a state estimation procedure in the resultant multirate state space model the multirate kalman filtering algorithm is then derived according to the multirate state space model to achieve optimal signal reconstruction in noisy filter bank systems based on the optimal state estimation theory the proposed multirate kalman synthesis filter provides the minimum variance reconstruction of the input signal two numerical examples are also included the simulation results indicate that the performance improvement of signal reconstruction in noisy filter bank systems is remarkable

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

pre torsors and equivalences abstract properties of most general non commutative torsors or a b torsors are analysed starting with pre torsors it is shown that they are equivalent to a certain class of galois extensions of algebras by corings it is shown that a class of faithfully flat pre torsors induces equivalences between categories of comodules of associated corings it is then proven that a b torsors correspond to monoidal functors and under some additional conditions equivalences between categories of comodules of bialgebroids

identification of polynomial chaos representations in high dimension from a set of realizations this paper deals with the identification in high dimensions of a polynomial chaos expansion of random vectors from a set of realizations due to numerical and memory constraints the usual polynomial chaos identification methods are based on a series of truncations that induce a numerical bias this bias becomes very detrimental to the convergence analysis of polynomial chaos identification in high dimensions this paper therefore proposes a new formulation of the usual polynomial chaos identification algorithms to avoid this numerical bias after a review of the polynomial chaos identification method the influence of the numerical bias on the identification accuracy is quantified the new formulation is then described in detail and illustrated using two examples

surrogate modeling of multiscale models using kernel methods this work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods the key element is to learn the interface between the different scales using a fast surrogate for the microscale model which is given by multivariate kernel expansions the expansions are computed using statistically representative samples of in and output of the microscale model as learning methods we apply both support vector machines and a vectorial kernel greedy algorithm we demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines first we consider a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and second a model for simulation of saturation overshoots in porous media involving nonclassical shock waves

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

noncommutative riemannian and spin geometry of the standard q sphere we study the quantum sphere open image in new window as a quantum riemannian manifold in the quantum frame bundle approach we exhibit its 2 dimensional cotangent bundle as a direct sum 0 1 1 0 in a double complex we find the natural metric volume form hodge operator laplace and maxwell operators and projective module structure we show that the q monopole as spin connection induces a natural levi civita type connection and find its ricci curvature and q dirac operator open image in new window we find the possibility of an antisymmetric volume form quantum correction to the ricci curvature and lichnerowicz type formulae for open image in new window we also remark on the geometric q borel weil bott construction

the reduced model multiscale method r3m for the non linear homogenization of hyperelastic media at finite strains this paper presents a new multi scale method for the homogenization analysis of hyperelastic solids undergoing finite strains the key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method in order to alleviate the complexity of multiscale procedures which usually involve a large number of nonlinear nested problems to be solved the problem associated with the representative volume element rve is solved via a model reduction method proper orthogonal decomposition the reduced basis is obtained through pre computations on the rve the technique coined as reduced model multiscale method r3m allows reducing significantly the computation times as no large matrix needs to be inverted and as the convergence of both macro and micro problems is enhanced furthermore the r3m drastically reduces the size of the data base describing the history of the micro problems in order to validate the technique in the context of porous elastomers at finite strains a comparison between a full and a reduced multiscale analysis is performed through numerical examples involving different micro and macro structures as well as different nonlinear models neo hookean mooney rivlin it is shown that the r3m gives good agreement with the full simulations at lower computational and data storage requirements

an inverse micro mechanical analysis toward the stochastic homogenization of nonlinear random composites abstract an inverse mean field homogenization mfh process is developed to improve the computational efficiency of non linear stochastic multiscale analyzes by relying on a micro mechanics model first full field simulations of composite stochastic volume element sve realizations are performed to characterize the homogenized stochastic behavior the uncertainties observed in the non linear homogenized response which result from the uncertainties of their micro structures are then translated to an incremental secant mfh formulation by defining the mfh input parameters as random effective properties these effective input parameters which correspond to the micro structure geometrical information and to the material phases model parameters are identified by conducting an inverse analysis from the full field homogenized responses compared to the direct finite element analyzes on sves the resulting stochastic mfh process reduces not only the computational cost but also the order of uncertain parameters in the composite micro structures leading to a stochastic mean field reduced order model mf rom a data driven stochastic model is then built in order to generate the random effective properties under the form of a random field used as entry for the stochastic mf rom embedded in a stochastic finite element method sfem the two cases of elastic unidirectional ud fibers embedded in an elasto plastic matrix and of elastic ud fibers embedded in a damage enhanced elasto plastic matrix are successively considered in order to illustrate the capabilities of the method the stochastic response of a ply is studied under transverse loading condition

a stochastic computational multiscale approach application to mems resonators abstract the aim of this work is to develop a stochastic multiscale model for polycrystalline materials which accounts for the uncertainties in the micro structure at the finest scale we model the micro structure using a random voronoi tessellation each grain being assigned a random orientation then we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso scale a random field of the meso scale elasticity tensor can thus be generated based on the information obtained from the sve simulations finally using a stochastic finite element method these meso scale uncertainties are propagated to the coarser scale as an illustration we study the resonance frequencies of mems micro beams made of poly silicon materials and we show that the stochastic multiscale approach predicts results in agreement with a monte carlo analysis applied directly on the fine finite element model i e with an explicit discretization of the grains

minimax deconvolution design of multirate systems with channel noises a unified approach the conventional signal reconstruction problem of multirate systems with channel noises can be cast as a robust multirate deconvolution design problem we investigate a unified minimax approach for the robust deconvolution design of multirate systems we discuss two typical multirate systems the multirate filter bank system and the transmultiplexer system we consider transmission noises resulting from quantization coding errors or external noises the deconvolution filters for these systems that we derive are all iir filters the keypoint is converting the original robust deconvolution design problem to an equivalent minimax matching problem via polyphase decomposition and noble identities then in spite of the presence of input signals and channel noises we can solve this minimax matching problem by an optimization technique the proposed method can be interpreted as designing an optimal multirate deconvolution filter such that the worst case multirate system reconstruction error is minimized over all possible inputs and noises from the energy perspective therefore our proposed design method is more robust than the conventional design method for multirate systems in the presence of uncertain input signals and channel noises we present several numerical examples that show the good performance of our design method

multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method abstract this paper deals with the development of a multiple scale finite element method by combining the asymptotic homogenization theory with voronoi cell vcfem for microstructural modeling the voronoi cell finite element model originates from dirichlet tessellation of a representative material element or a base cell in the microstructure homogenized material coefficients for a global displacement finite element model are generated by vcfem analysis using periodic boundary conditions on the base cell following the macroscopic analysis the local vcfem analysis is implemented to depict the true evolution of microstructural stresses and strains various numerical examples are executed for validating the effectiveness of vcfem macro micro modeling of elastic materials the effect of size shape orientation and distribution of heterogeneities on the local and global response are examined

a probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures in this paper we address the construction of a prior stochastic model for non gaussian deterministically bounded positive definite matrix valued random fields in the context of mesoscale modeling of heterogeneous elastic microstructures we first introduce the micromechanical framework and recall in particular huet s partition theorem based on the latter we discuss the nature of hierarchical bounds and define under some given assumptions deterministic bounds for the apparent elasticity tensor having recourse to the maximum entropy principle under the constraints defined by the available information we then introduce two random matrix models it is shown that an alternative formulation of the boundedness constraints further allows constructing a probabilistic model for deterministically bounded positive definite matrix valued random fields such a construction is presented and relies on a class of random fields previously defined we finally exemplify the overall methodology considering an experimental database obtained from ebsd measurements and provide a simple numerical application

some simple explicit bounds for the overall behaviour of nonlinear composites abstract variational expressions developed over the last few years provide bounds for the overall energy functions of a range of nonlinear composite materials the evaluation of a bound requires the solution of a system of nonlinear algebraic equations and this generally involves a computation there are however certain simple composites comprising a nonlinear matrix containing either rigid inclusions or cavities for which very simple explicit formulae can be given these formulae are displayed here at a level of generality greater than in any previous presentation the energy density function of the matrix is arbitrary and the microgeometry of the composite appears through an expression which bounds the energy of a linear composite with the same geometry to the extent that such linear bounds can be developed making allowance for any amount of statistical information on the composite the new nonlinear bounds collect this new results at the level of employing bounds of hashin shtrikman type for the linear problem are given for an incompressible matrix reinforced by aligned rigid platelets or weakened by aligned cracks in the course of the work a recently derived formula more general than any available previously is presented and developed explicitly for any two phase composite

design of multirate filter banks by spl hscr sub spl infin optimization a procedure is developed to design iir synthesis filters in a multirate filter bank the filters minimize the l sub 2 induced norm of the error system between the multirate filter bank and a desired pure time delay system this criterion is reduced to one of spl hscr sub spl infin optimization for which there is ready made software

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

fully implicit integration and consistent tangent modulus in elasto plasticity this paper deals with the numerical integration of a class of rate independent elasto plastic models the backward euler scheme is used to integrate the rate constitutive relations the non linear equations obtained are solved by the newton method the consistent tangent modulus is obtained by exact linearization of the algorithm in the case of j 2 elastoplasticity with non linear isotropic hardening and non linear kinematic hardening chaboche marquis model explicit formulas are derived without any approximations

a critical comparison of nonlocal and gradient enhanced softening continua continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients these gradients may occur for instance in the propagation of waves with high wave numbers and at stress concentrators adding nonlocal or gradient terms to the constitutive modelling may enhance the ability of the models to describe such situations the effect of adding nonlocal or gradient terms and the relation between these enhancements are examined in a continuum damage setting a nonlocal damage model and two different gradient damage models are considered in one of the gradient models higher order deformation gradients enter the equilibrium equations explicitly while in the other model the gradient influence follows in a more implicit way from an additional partial differential equation the latter implicit gradient formulation can be rewritten in the integral format of the nonlocal model and can therefore be regarded as truly nonlocal this is not true for the explicit formulation in which the nonlocality is limited to an infinitesimal volume this fundamental difference between the formulations results in quite different behaviour in wave propagation localisation and at crack tips this is shown for the propagation of waves in the models their localisation properties and the behaviour at a crack tip the responses of the nonlocal model and the implicit gradient model agree remarkably well in these situations while the explicit gradient formulation shows an entirely different and sometimes nonphysical response

stochastic finite element analysis of composite structures based on mesoscale random fields of material properties abstract the linking of microstructure uncertainty with the random variation of material properties at higher scales is of paramount importance in the framework of the stochastic finite element method sfem an efficient computational scheme has been recently proposed by the authors for the determination of mesoscale random fields for the apparent properties and of the rve size of particle reinforced composites based on computer simulated images of their microstructure the proposed approach exploits the excellent synergy of the extended finite element method xfem and monte carlo simulation in order to analyze the microstructure models and obtain statistical information probability distribution correlation structure for the apparent properties of the composite in each mesoscale size in this paper sample functions of the statistically homogeneous mesoscale random fields are generated using the spectral representation method in conjunction with translation field theory moreover micro mechanically consistent lower and upper bounds on the macroscopic response of composite structures are computed in the framework of sfem useful conclusions are derived regarding the effect of particle matrix stiffness ratio on the probabilistic characteristics of the mesoscale random fields as well as the influence of mesoscale size on the response variability

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

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

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

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

simulation of the multi scale convergence in computational homogenization approaches although the asymptotic homogenization is known to explicitly predict the thermo mechanical behaviors of an overall structure as well as the microstructures the current developments in engineering fields introduce some kinds of approximation about the microstructural geometry in order for the homogenization method for periodic media to apply for general heterogeneous ones the problems arising from mathematical modeling are examined in the framework of representative volume element rve analyses here the notion of homogenization convergence allows us to eliminate the geometrical periodicity requirement when the size of rve is sufficiently large then the numerical studies in this paper realize the multi scale nature of the convergence of overall material properties as the unit cell size is increased in addition to such dependency of the macroscopic field variables on the selected size of unit cells the convergence nature of microscopic stress values is also studied quantitatively via the computational homogenization method similar discussions are made for the elastoplastic mechanical responses in both macro and microscopic levels in these multi scale numerical analyses the specific effects of the microstructural morphology are reflected by using the digital image based dib finite element fe modeling technique which enables the construction of accurate microstructural models

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

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

data driven probability concentration and sampling on manifold a new methodology is proposed for generating realizations of a random vector with values in a finite dimensional euclidean space that are statistically consistent with a dataset of observations of this vector the probability distribution of this random vector while a priori not known is presumed to be concentrated on an unknown subset of the euclidean space a random matrix is introduced whose columns are independent copies of the random vector and for which the number of columns is the number of data points in the dataset the approach is based on the use of i the multidimensional kernel density estimation method for estimating the probability distribution of the random matrix ii a mcmc method for generating realizations for the random matrix iii the diffusion maps approach for discovering and characterizing the geometry and the structure of the dataset and iv a reduced order representation of the random matrix which is constructed using the diffusion maps vectors associated with the first eigenvalues of the transition matrix relative to the given dataset the convergence aspects of the proposed methodology are analyzed and a numerical validation is explored through three applications of increasing complexity the proposed method is found to be robust to noise levels and data complexity as well as to the intrinsic dimension of data and the size of experimental datasets both the methodology and the underlying mathematical framework presented in this paper contribute new capabilities and perspectives at the interface of uncertainty quantification statistical data analysis stochastic modeling and associated statistical inverse problems

large strain two scale computational approach using c1 continuity finite element employing a second gradient theory a two scale computational algorithm employing second order computational homogenization under assumption of large strain is presented a micro macro transition has been proposed where the c1 continuity displacement based finite element is used for discretization at the macrolevel while the representative volume element at the microlevel is discretized by c0 continuity finite elements the performed finite element verifications show an advantage in comparison to available finite element formulations in order to improve numerical efficiency of the proposed computational procedures the algorithms are embedded into the finite element program abaqus the performance of the presented multiscale homogenization approach is demonstrated in numerical examples where the elastoplastic deformation responses are considered and compared with the small strain analysis

the serre spectral sequence of a noncommutative fibration for de rham cohomology for differential calculi on noncommutative algebras we construct a twisted de rham cohomology using flat connections on modules this has propertie s similar in some respects to sheaf cohomology on topological spaces we also discuss generalised mapping properties of these theories and relations of these properties to corings using t his we give conditions for the serre spectral sequence to hold for a noncommutative fibration th is might be better read as giving the definition of a fibration in noncommutative differential geo metry we also study the multiplicative structure of such spectral sequences finally we show that some noncommutative homogeneous spaces satisfy the conditions to be such a fibration and in th e process clarify the differential structure on these homogeneous spaces we also give two explicit examples of differential fibrations these are built on the quantum hopf fibration with two differe nt differential structures

high performance model reduction techniques in computational multiscale homogenization a novel model order reduction technique for the solution of the fine scale equilibrium problem appearing in computational homogenization is presented the reduced set of empirical shape functions is obtained using a partitioned version that accounts for the elastic inelastic character of the solution of the proper orthogonal decomposition pod on the other hand it is shown that the standard approach of replacing the nonaffine termby an interpolant constructed using only pod modes leads to ill posed formulations we demonstrate that this ill posedness can be avoided by enriching the approximation space with the span of the gradient of the empirical shape functions furthermore interpolation points are chosen guided not only by accuracy requirements but also by stability considerations the approach is assessed in the homogenization of a highly complex porous metal material computed results show that computational complexity is independent of the size and geometrical complexity of the representative volume element the speedup factor is over three orders of magnitude as compared with finite element analysis whereas the maximum error in stresses is less than 10