a multilevel computational strategy for handling microscopic and macroscopic instabilities this paper presents a numerical technique to deal with instability phenomena in the context of heterogeneous materials where buckling may occur at both macroscopic and or microscopic scales we limit ourselves to elastic materials but geometrical nonlinearity is taken into account at both scales the proposed approach combines the multilevel finite element analysis fe2 and the asymptotic method anm in that framework the unknown nonlinear constitutive relationship at the macroscale is found by solving a local finite element problem at the microscale in contrast with fe2 the use of the asymptotic development allows to transform the nonlinear microscopic problems into a sequence of linear problems thus a direct analogy with classical linear homogenization can be made to construct a localisation tensor at each step of the asymptotic development and an explicit macroscopic constitutive relationship can be constructed at each step furthermore the salient features of the anm allow treating instabilities and limit points in a very simple way at both scales the method is tested and illustrated through numerical examples involving local instabilities which have ignificant influence on the macroscopic behavior

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

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

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem

elastic plastic analysis of arbitrary heterogeneous materials with the voronoi cell finite element method abstract in this paper a voronoi cell finite element method is developed to solve small deformation elastic plasticity problems for arbitrary heterogenous materials dirichlet tessellation of microstructural representative materials results in a network of arbitrary sided polygons called voronoi cells each voronoi cell encompasses one second phase heterogeneity at most these are natural elements for the microstructure representing the basic structural elements of the material in this paper formulations are developed for directly considering voronoi cells as elements in a finite element model without any further dissection furthermore a composite voronoi cell finite element method is developed to account for the presence of the second phase within each polygonal element various numerical elastic plastic examples are executed for validating the effectiveness of this formulation finally studies are conducted to understand the effect of size shape and distribution of second phase on the averaged and true local responses of representative material elements

multigrid method for periodic heterogeneous media part 1 convergence studies for one dimensional case abstract a multi grid method for a periodic heterogeneous medium in 1 d is presented based on the homogenization theory special integrid transfer operators have been developed to simulate a low frequency response of the differential equations with oscillatory coefficients the proposed multi grid method has been proved to have a fast rate of convergence governed by the ratio q 4 q where 0

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials the homogenization of nonlinear heterogeneous materials is much more difficult than the homogenization of linear ones this is mainly due to the fact that the general form of the homogenized behavior of nonlinear heterogeneous materials is unknown at the same time the prevailing numerical methods such as concurrent methods require extensive computational efforts a simple numerical approach is proposed to compute the effective behavior of nonlinearly elastic heterogeneous ma terials at small strains the proposed numerical approach comprises three steps at the rst step a representative volume element rve for a given nonlinear heterogeneous material is de ned and a loading space consisting of all the boundary conditions to be imposed on the rve is discretized into a sufficiently large number of points called nodes at the second step the boundary condition corresponding each node is prescribed on the surface of the rve and the resulting nonlinear boundary value problem is solved by the nite element method fem so as to determine the effective response of the heterogeneous material to the loading as sociated to each node of the loading space at the third step the nodal effective responses are interpolated via appropriate interpolation functions so that the effective strain energy stress strain relation and tangent stiffness tensor of the nonlinear heterogeneous material are provided in a numerically explicit way this leads to a non concurrent nonlinear multiscale approach to the computation of structures made of nonlinearly heterogeneous materials the rst version of the proposed approach uses mutidimensional cubic splines to interpolate effective nodal responses while the second version of the proposed approach takes advantage of an outer product decomposition of multidimensional data into rank one tensors to interpolate effective nodal responses and avoid high rank data these two versions of the proposed approach are applied to a few examples where nonlinear composites whose phases are characterized by the power law model are involved the numerical results given by our approach are compared with available analytical estimates exact results and full fem or concurrent multilevel fem solutions

a short and versatile finite element multiscale code for homogenization problems we describe a multiscale finite element fe solver for elliptic or parabolic problems with highly oscillating coefficients based on recent developments of the so called heterogeneous multiscale method hmm the algorithm relies on coupled macro and microsolvers the framework of the hmm allows to design a code whose structure follows the classical finite elements implementation at the macro level to account for the fine scales of the problem elementwise numerical integration is replaced by micro fe methods on sampling domains we discuss a short and flexible fe implementation of the multiscale algorithm which can accommodate simplicial or quadrilateral fe and various coupling conditions for the constrained micro simulations extensive numerical examples including three dimensional and time dependent problems are presented illustrating the efficiency and the versatility of the computational strategy

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

restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems the restrictively preconditioned conjugate gradient rpcg method for solving large sparse system of linear equations of a symmetric positive definite and block two by two coefficient matrix is further studied in fact this rpcg method is essentially the classical preconditioned conjugate gradient pcg method with a specially structured preconditioner within this setting we present algorithmic descriptions of two restrictive preconditioners that respectively employ the block jacobi and the block symmetric gauss seidel matrix splitting matrices as approximations to certain matrices involved in them and give convergence analyses of the correspondingly induced two pcg methods numerical results show that these restrictive preconditioners can lead to practical and effective pcg methods for solving large sparse systems of linear equations of symmetric positive definite and block two by two coefficient matrices

two scale kinematics and linearization for simultaneous two scale analysis of periodic heterogeneous solids at finite strain abstract we introduce the notion of two scale kinematics and the procedure of two scale linearization which are indispensable to the simultaneous two scale analysis method for the mechanical behavior of periodic heterogeneous solids at finite strain these are accomplished by formulating the two scale boundary value problem in both material and spatial descriptions with reference to the two scale modeling strategy developed in comput methods appl mech engrg 190 40 41 2001 5427 that utilized the convergence results of mathematical homogenization the formulation brings the intimate relationship between micro and macro scale kinematics in describing the micro macro coupling behavior inherent in heterogeneous media it is also shown that the two scale linearization necessitates the strict consistency with the micro scale equilibrated state and naturally invites the tangential homogenization process for both material and spatial descriptions several numerical examples of simultaneous two scale computations are presented to illustrate the two scale nature of the deformation of a heterogeneous solid at finite strain

multiple scale eigendeformation based reduced order homogenization multiple scale eigendeformation based reduced order homogenization method which provides considerable computational cost saving in comparison to the direct homogenization method has been developed verified against the direct homogenization method and validated against experimental data the salient feature of the method is in the formulation of the unit cell problem in terms of residual free stresses and therefore eliminating the need for costly equilibrium calculations required by the direct homogenization method this is accomplished by introducing discretized eigendeformation fields at different scales which are collectively referred to as eigenstrains and eigenseparations and then precomputing a sequence of elasticity solutions of the unit cell problem subjected to the eigendeformation fields prior to nonlinear analysis

hierarchical model reduction at multiple scales a hierarchical model reduction approach aimed at reducing computational complexity of non linear homogenization at multiple scales is developed the method consists of the following salient features 1 formulation of non linear unit cell problems at multiple scales in terms of eigendeformation modes that a priori satisfy equilibrium equations at multiple scales and thus eliminating the need for costly solution of discretized non linear equilibrium 2 the ability to control the discretization of the eigendeformation modes at multiple scales to maintain desired accuracy and 3 hierarchical solution strategy that requires sequential solution of single scale problems a two scale formulation is verified against an one dimensional model problem for which an analytical solution can be obtained and a three scale formulation is validated against tube crash experiments copyright 2009 john wiley sons ltd

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

a spectral order method for inverting sectorial laplace transforms laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered a spectral order parallelizable method for their numerical inversion is proposed the method takes into account the available information about the errors arising in the evaluations several numerical illustrations are provided

adaptive finite element heterogeneous multiscale method for homogenization problems in this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method unlike standard finite element methods our discretization scheme relies on macro and microfinite elements the desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data as the macroscopic data such as the macroscopic diffusion tensor are not available beforehand appropriate error indicators have to be defined for designing adaptive methods we show that such indicators based only on the available macro and microsolutions used to compute the actual macrosolution can be defined allowing for a macroscopic mesh refinement strategy which is both reliable and efficient the corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm in the case of a uniformly oscillating tensor we recover the standard residual based a posteriori error estimate for the finite element method applied to the homogenized problem numerical experiments confirm the efficiency and reliability of the adaptive multiscale method

a class of general algorithms for multi scale analyses of heterogeneous media abstract a class of computational algorithms for multi scale analyses is developed in this paper the two scale modeling scheme for the analysis of heterogeneous media with fine periodic microstructures is generalized by using relevant variational statements instead of the method of two scale asymptotic expansion the mathematical results on the generalized convergence are utilized in the two scale variational descriptions accordingly the global local type computational schemes can be unified in association with the homogenization procedure for general nonlinear problems after formulating the problem in linear elastostatics that with local contact condition and the elastoplastic problem we present representative numerical examples along with the computational algorithm consistent with our two scale modeling strategy as well as some direct approaches

multiple scale computational model for damage in composite materials abstract in this paper a hierarchical multiple scale computational model is proposed to concurrently predict evolution of variables at the structural and microstructural scales as well as to track the incidence and propagation of microstructural damage the microscopic analysis is conducted with the voronoi cell finite element model vcfem while a conventional displacement based fem code executes the macroscopic analysis adaptive schemes and mesh refinement strategies are developed to create a hierarchy of computational sub domains with varying resolution such hierarchy allow for differentiation between non critical and critical regions and help in increasing the efficiency of computations through preferential zoom in regions coupling between the scales for regions with periodic microstructure is accomplished through asymptotic homogenization whereas regions of nonuniformity and non periodicity are modeled by true microstructural analysis with vcfem various numerical examples are executed for validating the effectiveness of the model for different microstructural configurations

the caratheodory fejer method for real rational approximation a caratheodory fejer method is presented for near best real rational approximation on intervals based on the eigenvalue or singular value analysis of a hankel matrix of chebyshev coefficients in approximation of a smooth function f the cf approximant r cf frequently differs from the best approximation r by only one part in millions or billions to account for this we show here under weak assumptions that if f is approximated on varepsilon varepsilon then as varepsilon to 0 f r o varepsilon m n 1 while r cf r o varepsilon 3m 2n 3 in contrast the latter figure would be o varepsilon m n 2 for the chebyshev economization approximant of maehly or the chebyshev pade approximant of gragg it follows that as varepsilon to 0 best approximation error curves approach the real parts of m n 1 winding rational functions of constant modulus to within o varepsilon 3m 2n 3 numerical examples are given

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

on multiscale fe analyses of heterogeneous structures from homogenization to multigrid solvers heterogeneous structures like composites often need a fine scale resolution of micro effects which influence the macroscopic overall response this is of particular relevance in the fully non linear range of large strains and inelastic material response of the constituents suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales for big differences of micro and macro scales the argument of scale separation induces the application of homogenization methods such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine scale micro structure as well as the macroscopic boundary value problem in contrast small scale differences require full resolution of the heterogeneous structure effective solution methods for the resulting large scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques which may be considered as numerical multiscale approaches in both scenarios a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales the paper outlines new mesh bridging techniques in a deformation driven context for fully non linear response which exploit in a non trivial manner weak constraints on the average deformation in typical finite element patches the framework is based on an incremental variational structure of finite inelasticity the proposed new formulations provide variational based homogenization algorithms for physical multiscale scenarios and problem dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials copyright 2007 john wiley sons ltd

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

an offline online homogenization strategy to solve quasilinear two scale problems at the cost of one scale problems inspired by recent analyses of the finite element heterogeneous multiscale method and the reduced basis technique for nonlinear problems we present a simple and concise finite element algorithm for the reliable and efficient resolution of elliptic or parabolic multiscale problems of nonmonotone type solutions of appropriate cell problems on sampling domains are selected by a greedy algorithm in an offline stage and assembled in a reduced basis rb this rb is then used in an online stage to solve two scale problems at a computational cost comparable to the single scale case both the offline and the online cost are independent of the smallest scale in the physical problem the performance and accuracy of the algorithm are illustrated on 2d and 3d stationary and evolutionary nonlinear multiscale problems

continuous approximation of material distribution for topology optimization in this paper we propose a checkerboard free topology optimization method without introducing any additional constraint parameter this aim is accomplished by the introduction of finite element approximation for continuous material distribution in a fixed design domain that is the continuous distribution of microstructures or equivalently design variables is realized in the whole design domain in the context of the homogenization design method hdm by the discretization with finite element interpolations by virtue of this continuous fe approximation of design variables discontinuous distribution like checkerboard patterns disappear without any filtering schemes we call this proposed method the method of continuous approximation of material distribution camd to emphasize the continuity imposed on the material field two representative numerical examples are presented to demonstrate the capability and the efficiency of the proposed approach against some classes of numerical instabilities copyright 2004 john wiley sons ltd

consistent micro macro transitions at large objective strains in curvilinear convective coordinates this paper presents a computational homogenization scheme that is of particular interest for problems formulated in curvilinear coordinates the main goal of this contribution is to generalize the computational homogenization scheme to a formulation of micro macro transitions in curvilinear convective coordinates where different physical spaces are considered at the homogenized macro continuum and at the locally attached representative micro structures the deformation and the coordinate system of the micro structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro continuum for the consistent formulation of micro macro transitions the operations scale up and scale down are introduced considering the rotated representation of tensor variables at the different physical reference frames of micro and macro structure the second goal of this paper is to use objective strain measures like the green lagrange strain tensor for the solution of boundary value problems on the micro and macro scale by providing the required transformations for the work conjugate stress strain and tangent tensors into variables admissible for the considered micro macro transitions and satisfying the averaging theorem copyright 2007 john wiley sons ltd

a nested dynamic multi scale approach for 3d problems accounting for micro scale multi physics in this paper a first order multi scale approach for three dimensional structural problems is proposed a nested solution technique enables the mutual transfer of information between global and local scales which are solved simultaneously in a finite element analysis therefore apart from improving the coarse scale solution by considering fine scale features a detailed simulation of the micro level also enables the determination of local fine scale quantities of interest to account for transient effects a dynamic simulation on the macro scale is locally coupled to a quasi static simulation of the discretized micro level for time integration the generalized ff scheme is employed on both scales the presented algorithm is applicable for large deformations rotations and arbitrary micro structural behavior on the macro level no constitutive assumption is required since the stress strain relationship is directly computed from the micro scale using established computational homogenization procedures first simulations validate the novel approach and demonstrate its suitability for lung parenchyma considering complex surfactant film dynamics as an example of fine scale multi physics

a fast null space method for the unsteady stokes equations abstract this work is a first taste of using null space technique to deal with a basic model of unsteady partial differential equations pdes i e the unsteady stokes equations any inf sup stable semi discretization of the unsteady stokes equations yields a system of differential algebraic equations daes i e the unsteady discrete stokes equations the exact solution to the unsteady discrete stokes equations is explicitly constructed with the help of the null space of the discrete divergence operator and a matrix exponential in practical implementation the explicit use of the null space of the discrete divergence operator is avoided therefore the main workload of solving the unsteady discrete stokes equations is the matrix exponential vector product the matrix exponential vector product is written in terms of an integral relation which is approximated by a linear combination of values of the integrand evaluated at a select number of complex numbers each evaluation of the integrand needs to solve a linear system in numerical experiments the null space method outperforms frequently used time stepping methods in terms of accuracy and computing time

fully implicit formulation of elastoplastic homogenization problem for two scale analysis abstract in this study using a virtual work equation a micro macro kinematic relation and a linearized constitutive relation a boundary value problem is fully implicitly formulated to determine perturbed displacement increment fields in elastoplastic unit cells for two scale analysis it is shown that this implicit homogenization problem can be iteratively solved with quadratic convergences by successively updating strain increment fields in unit cells and that the boundary value problem formulated provides a computational algorithm which is versatile for initial setting of strain increment fields the computational algorithm developed is then examined by performing a two scale analysis of a holed plate with an elastoplastic micro structure subjected to tensile loading this demonstrates that the convergence in iteratively solving the implicit homogenization problem strongly depends on the initial setting of strain increment fields in unit cells

a generalized space time mathematical homogenization theory for bridging atomistic and continuum scales a generalized space time mathematical homogenization theory which constructs an equivalent continuum description directly from molecular dynamics md equations is developed the noteworthy theoretical findings of this work are i the coarse scale continuum equations pdes obtained from the homogenization of md equations are identical to those obtained from the classical homogenization of the fine scale continuum ii the lower order effective continuum properties such as the fourth order elasticity tensor have similar characteristics to those resulting from the homogenization of fine scale continuum iii the higher order effective properties such as polarization and dispersion tensors substantially differ from those of continuum due to the discreteness effect both the heterogeneous continuum and discrete media possess the size effect but only atomistic medium has the discreteness effect some demonstrative numerical examples are presented to verify the formulation copyright 2005 john wiley sons ltd

a generalized hill s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials abstract based on the hill s lemma for classical cauchy continuum a generalized hill s lemma for micro macro homogenization modeling of gradient enhanced cosserat continuum is presented in the frame of the average field theory in this context not only the strain and stress tensors defined in classical cosserat continuum but also their gradients are attributed to assigned micro structural representative volume element rve that leads to a higher order macroscopic cosserat continuum modeling and enables to incorporate the micro structural size effects the enhanced hill mandel condition for gradient enhanced cosserat continuum is extracted as a corollary of the presented generalized hill s lemma the derived admissible boundary conditions for the modeling are deduced to direct the proper presentation of boundary conditions to be prescribed on the rve in order to ensure the satisfaction of the hill mandel energy condition with the link between the discrete particle assembly and its effective cosserat continuum in an individual rve the boundary conditions prescribed on the rve modeled as cosserat continuum are transformed into those prescribed to the peripheral particles of the rve modeled as the discrete particle assembly the micromechanically based macroscopic constitutive model and corresponding rate forms of the macroscopic stress strain relations taking into account the local microstructure and its evolution are formulated with neither need of specifying the macroscopic constitutive relation nor need of providing macroscopic material parameters

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

finite deformation plasticity for composite structures computational models and adaptive strategies abstract we develop computational models and adaptive modeling strategies for obtaining an approximate solution to a boundary value problem describing the finite deformation plasticity of heterogeneous structures a nearly optimal mathematical model consists of an averaging scheme based on approximating eigenstrains and elastic concentration factors in each micro phase by a constant in the portion of the macro domain where modeling errors are small whereas elsewhere a more detailed mathematical model based on a piecewise constant approximation of eigenstrains and elastic concentration factors is utilized the methodology is developed within the framework of statistically homogeneous composite material and local periodicity assumptions

computational damage mechanics for composite materials based on mathematical homogenization this paper is aimed at developing a non local theory for obtaining numerical approximation to a boundary value problem describing damage phenomena in a brittle composite material the mathematical homogenization method based on double scale asymptotic expansion is generalized to account for damage effects in heterogeneous media a closed form expression relating local fields to the overall strain and damage is derived non local damage theory is developed by introducing the concept of non local phase fields stress strain free energy density damage release rate etc in a manner analogous to that currently practiced in concrete 1 2 with the only exception being that the weight functions are taken to be c0 continuous over a single phase and zero elsewhere numerical results of our model were found to be in good agreement with experimental data of 4 point bend test conducted on composite beam made of blackglas nextel 5 harness satin weave copyright 1999 john wiley sons ltd

a geometric schur weyl duality for quotients of affine hecke algebras abstract after establishing a geometric schur weyl duality in a general setting we recall this duality in type a in the finite and affine case we extend the duality in the affine case to positive parts of the affine algebras the positive parts have nice ideals coming from geometry allowing duality for quotients some of the quotients of the positive affine hecke algebra are then identified to some cyclotomic hecke algebras and the geometric setting allows the construction of canonical bases

computational homogenization for heterogeneous thin sheets in this paper a computational homogenization technique for thin structured sheets is proposed based on the computational homogenization concepts for first and second order continua the actual three dimensional 3d heterogeneous sheet is represented by a homogenized shell continuum for which the constitutive response is obtained from the nested analysis of a microstructural representative volume element rve incorporating the full thickness of the sheet and an in plane representative cell of the macroscopic structure at an in plane integration point of the macroscopic shell the generalized strains i e the membrane deformation and the curvature are used to formulate the boundary conditions for the microscale rve problem at the rve scale all microstructural constituents are modeled as an ordinary 3d continuum described by the standard equilibrium and the constitutive equations upon proper averaging of the rve response the macroscopic generalized stress and the moment resultants are obtained in this way an in plane homogenization is directly combined with a through thickness stress integration from a macroscopic point of view a numerical generalized stress strain constitutive response at every macroscopic in plane integration point is obtained additionally the simultaneously resolved microscale rve local deformation and stress fields provide valuable information for assessing the reliability of a particular microstructural design

a coupled two scale shell model with applications to layered structures summary in this paper a coupled two scale shell model is presented a variational formulation and associated linearization for the coupled global local boundary value problem is derived for small strain problems various numerical solutions are computed within the so called fe 2 method the discretization of the shell is performed with quadrilaterals whereas the local boundary value problems at the integration points of the shell are discretized using 8 noded or 27 noded brick elements or so called solid shell elements at the bottom and top surface of the representative volume element stress boundary conditions are applied whereas at the lateral surfaces the in plane displacements are prescribed for the out of plane displacements link conditions are applied the coupled nonlinear boundary value problems are simultaneously solved within a newton iteration scheme with an important test the correct material matrix for the stress resultants assuming linear elasticity and a homogeneous continuum is verified copyright 2013 john wiley sons ltd

a micro macro homogenization approach for discrete particle assembly cosserat continuum modeling of granular materials based on hill s lemma for classical cauchy continuum a version of hill s lemma for micro macro homogenization modeling of cosserat continuum is derived the derived admissible boundary conditions for the modeling are used to direct the proper prescription of boundary conditions on the representative volume element rve in order to ensure the satisfaction of the hill mandel energy condition a micro macro computational homogenization procedure for modeling the mechanical response of granular material modeled as heterogeneous cosserat continuum is presented taking into account both physical and geometrical non linear evolution of microstructures of the material with the link between the discrete particle assembly and its cosserat continuum equivalent in an individual rve the boundary conditions prescribed on the rve modeled as cosserat continuum are transformed into those prescribed to the peripheral particles of the rve modeled as the discrete particle assembly the average stresses and strains and their rates over the rve defined for the cosserat continuum equivalent are then determined by the physical and geometrical quantities of the discrete particle assembly the consistent macroscopic modulus tensor and the macroscopic constitutive relation defined at the integration point are formulated in terms of the averaged behavior of the associated microstructures since the proposed micro macro computational homogenization procedure is used within the finite element framework there is no need to specify the macroscopic constitutive relation at the macroscopic integration points

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

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

nonlocal constitutive models with gradients of internal variables derived from a micro macro homogenization procedure finite element simulations involving strain softening materials or finite strains are faced with problems of loss of uniqueness and strain localization if localization occurs the size of the region where the strains concentrate is determined by the mesh size and no convergence can be obtained with mesh refinement a non local approach has been proposed recently by andrieux et al 15 16 in order to solve the problem with this approach spatial gradients of internal variables are incorporated in the macroscopic constitutive equations via a homogenization procedure in this paper this new formulation is investigated and numerical algorithms are presented for the one dimensional case the results of the numerical simulations demonstrate that the size of the strain localization zone cannot be arbitrarily small anymore c 1999 elsevier science s a all rights reserved

nonlinear analyses with a micromorphic computational homogenization framework for composite materials abstract a small deformation micromorphic computational homogenization framework for matrix inclusion composites was recently presented in biswas and poh 2017 where standard continuum models at the micro scale are translated consistently onto the macro scale to recover a micromorphic continuum in this contribution the micromorphic framework is extended to the regime of significant geometrical and material nonlinearities following the small deformation framework an additional degree of freedom is introduced to capture the influence of rapid fluctuations within a unit cell in this contribution we elaborate on the specific choice of decomposition for the kinematic fields where certain higher order modes are deliberately neglected several examples are considered to illustrate the influence of different higher order modes and how the corresponding size effect emerges naturally in the homogenized micromorphic model in the regime of geometrical softening the higher order term provides a regularizing effect to give mesh independent solutions albeit with a stiffer response a detailed discussion on this phenomenon is provided the homogenization framework is implemented using a client server based parallel processing algorithm to reduce the computational time the excellent predictive capability and efficiency of the micromorphic approach are demonstrated with a mixed loading problem on a composite plate with non uniform cross section furthermore the micromorphic solutions are shown to be independent of the choice of a unit cell

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

numerical plate testing for linear two scale analyses of composite plates with in plane periodicity summary a method of numerical plate testing npt for composite plates with in plane periodic heterogeneity is proposed in the two scale boundary value problem a thick plate model is employed at macroscale while three dimensional solids are assumed at microscale the npt which is nothing more or less than the homogenization analysis is in fact a series of microscopic analyses on a unit cell that evaluates the macroscopic plate stiffnesses the specific functional forms of microscopic displacements are originally presented so that the relationship between the macroscopic resultant stresses moments and strains curvatures to be consistent with the microscopic equilibrated state in order to perform npt by using general purpose fem programs we introduce control nodes to facilitate the multiple point constraints for in plane periodicity numerical examples are presented to verify that the proposed method of npt reproduces the plate stiffnesses in classical plate and laminate theories we also perform a series of homogenization macroscopic and localization analyses for an in plane heterogeneous composite plate to demonstrate the performance of the proposed method copyright 2015 john wiley sons ltd

appropriate number of unit cells in a representative volume element for micro structural bifurcation encountered in a multi scale modeling the paper proposes a method to determine the number of unit cells basic structural elements to be employed for a representative volume element rve of the multi scale modeling for a solid with periodic micro structures undergoing bifurcation main difficulties for the multi scale modeling implementing instability are twofold loss of convexity of the total potential energy that should be homogenized and determination of a pertinent rve that contains multiple unit cells in order to resolve these difficulties variational formulation is achieved with the help of convergence theory within the framework of non convex homogenization method while the number of unit cells in an rve is determined by the block diagonalization method of group theoretic bifurcation theory the latter method enables us to identify the most critical bifurcation mode among possible bifurcation patterns for an assembly of arbitrary number of periodic micro structures thus the appropriate number of unit cells to be employed in the rve can be determined in a systematic manner representative numerical examples for a cellular solid show the feasibility of the proposed method and illustrate material instability at a macroscopic point due to geometrical instability in a micro scale

a method for solving contact problems in this paper a further method is presented to solve problems involving contact mechanics the basic idea is related to a special modification of the unconstrained functional to include inequality constraints the modification is constructed in such a way that minimal point of the unconstrained potential can be exactly shifted to the constraint limit moreover the functional remains smooth and the admissible range of the solution is not restricted the solution search process with iterative techniques takes advantage from these features in fact due to a better control of gap status changes a more stable solution path with respect to other methods is usually obtained the characteristics of the method are evidenced and compared to other classical techniques like penalty and barrier methods the finite element discretization of the proposed method is included and some numerical applications are shown 1998 john wiley sons ltd

a multilevel finite element method fe2 to describe the response of highly non linear structures using generalized continua abstract a general method called fe2 has been introduced which consists in describing the behavior of heterogeneous structures using a multiscale finite element model instead of trying to build differential systems to establish a stress strain relation at the macroscale a finite element computation of the representative volume element is carried out simultaneously doing so does not require any constitutive equations to be written at the macroscopic scale all non linearities come directly from the microscale in this paper we describe how this method can be used in the context of generalized continua for such continua constitutive equations are very difficult to write and a new set of material is difficult to fit to experimental data the use of fe2 models bypasses this problem because no analytical equation is needed at the macroscale an academic application is presented to show that generalized continua are necessary when the size of the heterogeneities increases and that fe2 models behave well compared to a reference solution