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

two scale analysis of heterogeneous elastic plastic materials with asymptotic homogenization and voronoi cell finite element model in this paper a multiple scale finite element model vcfem homo has been developed for elastic plastic analysis of heterogeneous porous and composite materials by combining asymptotic homogenization theory with the voronoi cell finite element model vcfem vcfem for microstructural modeling originates from dirichlet tessellation of representative material elements at sampling points in the structure structural modeling is done by the general purpose finite element code abaqus and interfacing with the microscale vcfem analysis is done through the user subroutine in abaqus for material constitutive relation umat asymptotic homogenization in umat generates macroscopic material parameters for abaqus following the macroscopic analysis a local vcfem analysis is invoked to depict the true evolution of microstructural state variables various numerical examples are executed for validating the effectiveness of vcfem homo and the effect of size shape and distribution of heterogeneities on local and global response is examined

a random field based method to estimate convergence of apparent properties in computational homogenization abstract this work addresses the challenge of efficiently determining the representative volume size in computational homogenization by exploiting the requirement that mechanical response quantities are statistically homogeneous and ergodic random fields the proposed computational homogenization approach focuses on empirically determining the autocorrelation functions of output response quantities through post processing finite element analyses once the autocorrelation functions are known only trivial computations are needed to determine the variance of apparent properties as a function of domain size without any additional finite element computation by utilizing a simple formula relating the autocorrelation function to the variance of spatially averaged quantities this approach improves upon the current established approach by circumventing the need to analyze numerous successively larger domains in order to determine convergence of apparent properties which can be computationally prohibitive furthermore similar previous analytical expressions for the variance of apparent properties are asymptotic with respect to domain size while the expression in the proposed approach is exact after presenting the formulation the method is first demonstrated for a one dimensional bar model where analytical expressions can be derived then the approach is demonstrated on two numerical examples 1 a plane strain linear elastic domain with a lognormal random field describing its compliance and 2 a stochastic polycrystalline microstructure of a nickel super alloy having a crystal plasticity model for its constitutive law

voronoi cell finite element model based on micropolar theory of thermoelasticity for heterogeneous materials in this paper a new voronoi cell finite element model is developed for solving steady state heat conduction and micropolar thermoelastic stress analysis problems in arbitrary heterogeneous materials the method is based on the natural discretization of a multiple phase domain into basic structural elements by dirichlet tessellation tessellation process results in a network of polygons called voronoi polygons in this paper formulations are developed for treating these polygons as elements in a finite element mesh furthermore a composite voronoi cell finite element model is developed to account for the presence of a second phase inclusion within a polygonal element various numerical examples are executed for validating the effectiveness of this model in the analysis of the temperature and stress fields for micropolar elastic materials effective material properties are derived for microstructures containing different distributions of second phase

a multi level computational model for multi scale damage analysis in composite and porous materials abstract an adaptive multi level methodology is developed in this paper to create a hierarchy of computational sub domains with varying resolution for multiple scale problems it is intended to concurrently predict evolution of variables at the structural and microstructural scales as well as to track the incidence and propagation of microstructural damage in composite and porous materials the microstructural analysis is conducted with the voronoi cell finite element model vcfem while a conventional displacement based fem code executes the macroscopic analysis the model introduces three levels in the computational domain which include macro macro micro and microscopic analysis it differentiates between non critical and critical regions and ranges from macroscopic computations using continuum constitutive relations to zooming in at hotspots for pure microscopic simulations coupling between the scales in regions of periodic microstructure is accomplished through asymptotic homogenization an adaptive process significantly increases the efficiency while retaining appropriate level of accuracy for each region numerical examples are conducted for composite and porous materials with a variety of microscopic architectures to demonstrate the potential of the model

nonlinear multi scale homogenization with different structural models at different scales summary we present an extension of the computational homogenization theory to cases where different structural models are used at different scales and no energy potential can be defined at the small scale we observe that volumetric averaging which is not applicable in such cases unless similarities exist in the macro scale and micro scale models is not a necessary prerequisite to carry out computational homogenization at each material point of the macro model we replace the conventional representative volume element with a representative domain element rde to link the large scale and small scale problems we then introduce a linear operator mapping the smooth part of the small scale displacement field of each rde to the large scale strain field and a trace operator to impose boundary conditions in the rde the latter is defined on the basis of engineering judgement analogously to the conventional theory a generalized hill s condition rather than being invoked is derived from duality principles and is used to recover the stress measures at the large scale for the implementation in a nonlinear finite element analysis control nodes and constraint equations are used the effectiveness of the procedure is demonstrated for three beam to truss example problems for which multi scale convergence is numerically analysed copyright 2013 john wiley sons ltd

identification of elastoplastic microscopic material parameters within a homogenization scheme summary a two scale parameter identification approach is investigated the microscopic material parameters of a two scale model are identified by comparing macroscopic simulation data with macroscopic full field measurements of the micro structured specimen gradient based solution strategies are employed for the optimization problem of the two scale parameter identification in particular two approaches for the gradient calculation are investigated the finite difference method is compared with a newly introduced semi analytical scheme the focus lies on the identification of microscopic elastoplastic material parameters the presented identification example with artificial data confirms a reduced computational effort and advantageous convergence for the semi analytical approach within the two scale parameter identification a drawback is the increase in memory requirement copyright 2015 john wiley sons ltd

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

probabilistic effective characteristics of cables for superconducting coils the article is devoted to computational estimation of the probabilistic effective properties of the superconducting coil cable 4 component composite for this purpose the well known deterministic effective modules method has been used and reformulated for the case where the composite constituents elastic characteristics in the form of the young moduli and poisson coefficients have random elastic character the variational formulation being a modified virtual work principle has been implemented into the mcceff system which is a probabilistic homogenization oriented finite element program the probabilistic approach implemented in the program is the monte carlo simulation mcs technique based on the random sampling and statistical estimation method by using the program the first four probabilistic moments of the effective elasticity tensor have been computed for the superconductors for coils with periodic cross section and compared against the statistical estimators of the upper and lower bounds of the tensor

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

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

novel formulations of microscopic boundary value problems in continuous multiscale finite element methods abstract this article explores the use of a homogenization method inspired by the classical irving kirkwood procedure in the continuous multiscale modeling of elastic solids within the context of the finite element method this homogenization method gives rise to a broad range of allowable boundary conditions for the rve which in turn yield a rich spectrum of estimates of the macroscopic stress response for two representative problems involving heterogeneous elastic bodies

an adaptive sub incremental strategy for the solution of homogenization based multi scale problems abstract in this contribution several schemes for the solution of homogenization based multi scale constitutive problems undergoing finite strains with inelastic material behavior are investigated these schemes are aimed at improving the robustness and efficiency of the newton raphson method in the multilevel finite element ml fem framework an adaptive sub incremental strategy is proposed for the discrete representative volume element rve boundary value problem the procedure is able to ensure the convergence of the solution algorithm in the presence of several sources of non linearity and obtains improved initial guesses for the newton raphson scheme in the ml fem framework the enlargement of the convergence bowl of the newton raphson procedure at the micro scale allows larger macroscopic deformation gradients to be prescribed and significantly reduces the overall computational cost of ml fem analyses the proposed strategy preserves the quadratic rates of asymptotic convergence that characterize the newton raphson scheme at the macroscopic level numerical examples of both micro scale and two scale finite element simulations are presented to demonstrate the improved robustness and efficiency of the solution procedures proposed

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

reliability of recurrence quantification analysis of postural sway data a comparison of two methods to determine recurrence thresholds ageing affects balance control resulting in a greater amplitude of sway and alterations in structure of the sway time series recurrence quantification analysis rqa has been used to determine the structure of center of pressure cop a measure that reflects standing postural control data as a means to quantify how cop repeats itself recurs below a certain threshold this study aimed to determine how the method of threshold determination below which a recurrence is defined affects the within session reliability of rqa in an elderly population within session reliability of rqa of cop motion in the anterior posterior and mediolateral directions was assessed in 267 individuals 65 years old when standing on firm or foam surface with eyes open or closed for each of two recurrence threshold methods one threshold method sets the recurrence threshold level such that the recurrence rate is fixed to 5 the other method sets the recurrence threshold based on 27 of the mean distance between all points from which recurrences are quantified reliability across four 30 s balance trials within each of four balance conditions firm vs foam eyes open vs closed was determined using intra class correlation standard error of measurement and minimal detectable change iccs were better the standard error of measurement and minimal detectable change were smaller when the recurrence threshold was set to 5 using the fixed recurrence threshold fixing recurrence rate improves the within session reliability of rqa and could increase sensitivity to identify fall risk

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

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

towards optimal lower bounds for clique and chromatic number it was previously known that neither max clique nor min chromatic number can be approximated in polynomial time within n1 e for any constant e 0 unless np zpp in this paper we extend the reductions used to prove these results and combine the extended reductions with a recent result of samorodnitsky and trevisan to show that unless np zptime 2o log n log log n 3 2 neither max clique nor min chromatic number can be approximated in polynomial time within n1 e n where e o log log n 1 2 since there exists polynomial time algorithms approximating both problems within n1 e n where e n log log n log n our result shows that the best possible ratio we can hope for is of the form n1 o 1 for some yet unknown value of o 1 between o log log n 1 2 and log log n log n

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

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

a material based finite element analysis of heterogeneous media involving dirichlet tessellations abstract in this paper a new finite element formulation has been developed for analysis of heterogeneous media in which the second phase is randomly dispersed within the matrix it introduces a tessellation based mesh generation technique to account for the arbitrariness in location shape and size of the second phase an assumed stress hybrid formulation has been implemented to utilize the resulting voronoi polygons as elements in a finite element model the effect of the second phase has been accounted for by introducing a transformation strain based constitutive formulation numerical examples have been conducted to validate the model

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

prediction of the mechanical behavior of nonlinear heterogeneous systems by multi level finite element modeling an accurate homogenization method that accounts for large deformations and viscoelastic material behavior on microscopic and macroscopic levels is presented this method is based on the classical homogenization theory assuming local spatial periodicity of the microstructure consequently the microstructure is identified by a representative volume element rve with conformity of opposite boundaries at any stage of the deformation process the local macroscopic stress is obtained by applying the local macroscopic deformation represented by the deformation tensor on a unique rve through imposing appropriate boundary conditions and averaging the resulting rve stress field if the assumption of local periodicity of the morphology is valid this homogenization procedure supplies a consistent objective relationship between the local macroscopic deformation and the microstructural deformation the homogenization method was implemented in a multi level finite element program with meshes on macroscopic level mesh of entire structure and microscopic level meshes of rves the performance was successfully verified by the comparison of the deformation of a perforated macroscopic sheet to the response of a homogenized sheet

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

adaptivity and convergence in the voronoi cell finite element model for analyzing heterogeneous materials in this paper an adaptive voronoi cell finite element model is presented for analyzing micromechanical response of composites and porous materials both elastic and elastic plastic materials are considered two error measures viz a traction reciprocity error and an error in the kinematic relation are formulated as indicators of the quality of vcfem solutions based on a posteriori evaluation of these error measures element adaptation is executed in two consecutive stages in the first stage displacement function adaptations on the element boundaries and matrix inclusion void interfaces are carried out to minimize the corresponding traction reciprocity errors this is accomplished through a sequence of h refinement and spectral p enrichment strategy in an optimal displacement direction following this an enrichment of matrix and inclusion stress functions or enrp adaptation is conducted to reduce the error in kinematic relations the complete process improves convergence characteristics of the vcfem solution numerical analysis is conducted to examine the potential of the resulting vcfem code in analyzing microstructures with different distributions sizes and shapes of heterogeneities the method is seen to perform very well for the wide variety of problems solved

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

bridging multiple structural scales with a generalized finite element method summary this paper presents a generalized fem based on the solution of interdependent coarse scale global and fine scale local problems in order to resolve multiscale effects due to fine scale heterogeneities overall structural behavior is captured by the global problem while local problems focus on the resolution of fine scale solution features in regions where material heterogeneities may govern the structural response fine scale problems are accurately solved in parallel and to address the intrinsic coupling of scales these solutions are embedded into the global solution space using a partition of unity approach this method is demonstrated on representative heat transfer examples in order to examine its accuracy efficiency and flexibility copyright 2014 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