quantitative homogenization in nonlinear elasticity for small loads we study quantitative periodic homogenization of integral functionals in the context of nonlinear elasticity under suitable assumptions on the energy densities in particular frame indifference minimality non degeneracy and smoothness at the identity p geqq d growth from below and regularity of the microstructure we show that in a neighborhood of the set of rotations the multi cell homogenization formula of non convex homogenization reduces to a single cell formula the latter can be expressed with the help of correctors we prove that the homogenized integrand admits a quadratic taylor expansion in an open neighborhood of the rotations a result that can be interpreted as the fact that homogenization and linearization commute close to the rotations moreover for small applied loads we provide an estimate on the homogenization error in terms of a quantitative two scale expansion

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

iterative methods for a fourth order boundary value problem abstract we consider a fourth order nonlinear ordinary differential equation together with two point boundary conditions and provide a priori error estimates on the length of the interval b a so that the picard s iterative method the approximate picard s iterative method and the quasilinear iterative method convergence to the solution of the problem

multi scale second order computational homogenization of multi phase materials a nested finite element solution strategy this paper presents the detailed implementation and computational aspects of a novel second order computational homogenization procedure which is suitable for a multi scale modelling of macroscopic localization and size effects the second order scheme is an extension of the classical first order computational homogenization framework and is based on a proper incorporation of the gradient of the macroscopic deformation gradient tensor into the kinematical macro micro scale transition from the microstructural analysis the macroscopic stress and higher order stress tensors are obtained thus delivering a microstructurally based constitutive response of the macroscopic second gradient continuum the higher order macroscopic constitutive tangents are derived through static condensation of the microscopic global tangent matrix for the solution of the second gradient equilibrium problem on the macrolevel a mixed finite element formulation is developed as an example the second order computational homogenization approach is applied for the multi scale analysis of simple shear of a constrained heterogeneous strip where a pronounced boundary size effect appears

an adaptive method for homogenization in orthotropic nonlinear elasticity a database methodology is developed in order to characterize the constitutive behavior of macroscopically orthotropic nonlinearly elastic composites this methodology discretizes the material test space in terms of the eigenvalues of the macroscopic strain tensor and the orientation of its eigenvectors with respect to the axes of orthotropy the macroscopic properties of the composite are extracted at points of this test space by subjecting a statistically representative sample of the composite to appropriate boundary conditions and are subsequently employed in an interpolation procedure which creates the so called material map that characterizes the macroscopic behavior the macroscopic tangent required in the finite element calculations are derived either by using the tangent induced by the map for the macroscopic stress or by explicitly computing a map for the tangent the range of deformations over which a material map is applicable to a problem is enhanced by an adaptive scheme where a multi level finite element method is employed at highly deformed regions of a finite element mesh where the map limits are exceeded

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

relaxation and optimization for linear growth convex integral functionals under pde constraints abstract we give necessary and sufficient conditions for the minimality of generalized minimizers of linear growth integral functionals of the form f u f x u x d x u r d r n where f r n r is a convex integrand and u is an integrable function satisfying a general pde constraint our analysis is based on two ideas a relaxation argument into a subspace of the space of bounded vector valued radon measures m r n and the introduction of a set valued pairing on m r n l r n by these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle point conditions in particular our results can be applied to relaxation and minimization problems in bv bd and divergence free spaces

a model order reduction method for computational homogenization at finite strains on regular grids using hyperelastic laminates to approximate interfaces abstract fft based homogenization methods operate on regular voxel grids and cannot resolve interfaces exactly in general in this article we study hyperelastic laminates and associate their effective properties to voxels containing interfaces extending previous ideas successfully applied in the framework of linear elasticity more precisely we show that for strictly rank one convex energy densities of the constituents there is precisely one phase wise affine minimizer of the laminate problem a result of independent interest we demonstrate that furnishing interface voxels with appropriately rotated effective hyperelastic properties of a two phase laminate significantly enhances both the local solution quality and the accuracy of the computed effective elastic properties with only a small computational overhead compared to using classical fft based homogenization in particular for equal accuracy problems with a lower number of degrees of freedom suffice

iterative methods for solving right focal point boundary value problems abstract we provide necessary and sufficient conditions for the existence of a solution of n th order differential equation satisfying right focal point boundary conditions for the linear problems we propose forward forward and backward backward one pass practical shooting methods which convert the boundary value problems to its equivalent initial value problems for the nonlinear problems iterative methods are discussed and a priori conditions for the convergence are obtained several examples are also illustrated

ito sde based generator for a class of non gaussian vector valued random fields in uncertainty quantification this paper is concerned with the derivation of a generic sampling technique for a class of non gaussian vector valued random fields such an issue typically arises in uncertainty quantification for complex systems where the input coefficients associated with the elliptic operators must be identified by solving statistical inverse problems specifically we consider the case of non gaussian random fields with values in some arbitrary bounded or semi bounded subsets of r n the approach involves two main features the first one is the construction of a family of random fields converging at a user controlled rate towards the target random field each of these auxialiary random fields can be subsequently simulated by solving a family of ito stochastic differential equations the second ingredient is the definition of an adaptive discretization algorithm the latter allows refining the integration step on the fly and prevents the scheme from diverging the proposed strategy is finally exemplified on three examples each of which serving as a benchmark either for the adaptivity procedure or for the convergence of the diffusions

homogenization of first order equations with u varepsilon periodic hamiltonians part i local equations in this paper we present a result of homogenization of first order hamilton jacobi equations with u varepsilon periodic hamiltonians on the one hand under a coercivity assumption on the hamiltonian and some natural regularity assumptions we prove an ergodicity property of this equation and the existence of non periodic approximate correctors on the other hand the proof of the convergence of the solution usually based on the introduction of a perturbed test function in the spirit of evans work uses here a twisted perturbed test function for a higher dimensional problem

existence uniqueness and iterative methods for third order boundary value problems abstract in this paper we shall provide necessary and sufficient conditions for the existence and uniqueness of solutions of third order nonlinear differential equations satisfying three point boundary conditions for the linear case we propose a constructive method which is a variation of the method of chasing for the nonlinear problems sufficient conditions are provided to ensure the convergence of a general class of iterative methods several examples are also included

on the commutability of homogenization and linearization in finite elasticity we consider a family of non convex integral functionals frac 1 h 2 int omega w x varepsilon rm id h nabla g x rm d x quad g in w 1 p omega mathbb r n where w is a caratheodory function periodic in its first variable and non degenerate in its second we prove under suitable conditions that the limits corresponding to linearization h 0 and homogenization varepsilon rightarrow 0 commute provided w is minimal at the identity and admits a quadratic taylor expansion at the identity moreover we show that the homogenized integrand which is determined by a multi cell homogenization formula has a quadratic taylor expansion with a quadratic term that is given by the homogenization of the second variation of w

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

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

theoretical aspects of gray level morphology after a brief discussion of the extension of mathematical morphology to complete lattices the space of gray level functions is considered and the concept of a threshold set is introduced it is shown how one can use binary morphological operators and thresholding techniques to build a large class of gray level morphological operators particular attention is given to the class of so called flat operators i e operators which commute with thresholding it is also shown how to define dilations and erosions with nonflat structuring elements if the gray level set is finite it is reported that mere truncation yields wrong results

uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials abstract this paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials at the microscale level the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used the geometry of the microstructure is random and characterized by a high number of random parameters the method is based on a deterministic non concurrent multiscale approach devoted to micro macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale cauchy green strain states to overcome the curse of dimensionality due to the high number of involved random variables the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion several strategies based on novel algorithms dedicated to high stochastic dimension are used and adapted for the class of multi modal random variables which may characterize the potential numerical examples at both small and large scales allow analyzing the efficiency of the approach through comparisons with classical methods

m idempotent and self dual morphological filters in this paper we present a comprehensive analysis of self dual and m idempotent operators we refer to an operator as m idempotent if it converges after m iterations we focus on an important special case of the general theory of lattice morphology spatially variant morphology which captures the geometrical interpretation of spatially variant structuring elements we demonstrate that every increasing self dual morphological operator can be viewed as a morphological center necessary and sufficient conditions for the idempotence of morphological operators are characterized in terms of their kernel representation we further extend our results to the representation of the kernel of m idempotent morphological operators we then rely on the conditions on the kernel representation derived and establish methods for the construction of m idempotent and self dual morphological operators finally we illustrate the importance of the self duality and m idempotence properties by an application to speckle noise removal in radar images

stochastic homogenization of nonconvex unbounded integral functionals with convex growth we consider the well trodden ground of the problem of the homogenization of random integral functionals when the integrand has standard growth conditions the qualitative theory is well understood when it comes to unbounded functionals that is when the domain of the integrand is not the whole space and may depend on the space variable there is no satisfactory theory in this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth we first prove that if the integrand is convex and has p growth from below with p d the dimension then it admits homogenization regardless of growth conditions from above this result that crucially relies on the existence and sublinearity at infinity of correctors is also new in the periodic case in the case of nonconvex integrands we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two sided estimate by a convex integrand the domain of which may depend on the space variable that itself admits homogenization this result is of interest to the rigorous derivation of rubber elasticity from polymer physics which involves the stochastic homogenization of such unbounded functionals

on three point boundary value problems associated with third order differential equations introduction in this paper we shall be concerned with the existence and uniqueness of solutions to three point boundary value problems associated with the differential equation y f x y y s y 1 1 where f x y z w is assumed to be continuous on a subset of r4 initial value problems associated with 1 1 exist and are unique in the interval x1 x a monotonicity restriction on f x y z w ensures that the following boundary value problems y j x y y y y xj y y xj y y x m i 1 2 12i y j x y y y y x y y x m2 2 2 9 y x y i 421 le3i

existence and uniqueness of solutions for nth order nonlinear two point boundary value problems in this paper we present some general results of existence and uniqueness of solutions for nth order nonlinear two point boundary value problems by using shooting method together with maximum principle and kneser hukahara continuum theorem as applications we give certain concrete sufficient conditions for the existence and uniqueness of solutions of nonlinear two point boundary value problems of nth order lipschitz equations

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

a quasistatic evolution model for the interaction between fracture and damage this paper is devoted to the investigation of a quasistatic evolution model for a continuum which undergoes damage and possibly fracture in both cases the model appears to be ill posed so that it is necessary to introduce a relaxed variational evolution preserving the irreversibility of the process the minimality at each time and the energy balance from a mechanical point of view it turns out that the material prefers to form microstructures through the creation of fine mixtures between the damaged and healthy parts of the medium the brutal character of the damage process is then replaced by a progressive one where the original damage internal variable that is the characteristic function of the damaged part is replaced by the local volume fraction the analysis rests on a locality property for mixtures which enables us to use an alternative formula for the lower semicontinuous envelope of the elastic energy in terms of the g closure set

homogenization of multiparameter integrals where is a bounded domain in rn u rm 0 and w rn rmn 0 is supposed to be 0 1 periodic with respect to the 7rst variable x rn the distinguishing feature of 1 here is that the integrand is permitted to depend on a vector of parameters 1 k r with k 1 we are interested in the asymptotic behavior of 1 as 0 and 0 integral functionals of this type appear in the mathematical modeling of cellular composite materials when n m 3 1 can be interpreted as the stored strain energy of an elastic and heterogeneous material u being a deformation or displacement 7eld when m 1 u may be a di erence of potential in a condenser in any case the medium under consideration is composed of several materials which are periodically distributed at the microscopic scale given by in the applications we often deal with

adaptive reduced basis finite element heterogeneous multiscale method an adaptive reduced basis finite element heterogeneous multiscale method rb fe hmm is proposed for elliptic problems with multiple scales the multiscale method is based on the rb fe hmm introduced in a abdulle y bai reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems j comput phys 231 21 2012 7014 7036 it couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains unlike classical numerical homogenization methods the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions the reduced basis obtained by a greedy algorithm in an offline stage in this paper we present a residual based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest for both type of adaptive strategies rigorous a posteriori error estimates are derived and corresponding error estimators are proposed in contrast to the adaptive finite element heterogeneous multiscale method fe hmm there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement up to an offline preliminary stage the rb fe hmm has the same computational complexity as a standard adaptive fem for the effective problem two and three dimensional numerical experiments confirm the efficiency of the rb fe hmm and illustrate the improvements compared to the adaptive fe hmm c 2013 elsevier b v all rights reserved

quasilinearization and approximate quasilinearization for multipoint boundary value problems quasilinearization technique has been applied to a general nonlinear lidstone boundary value problem for the construction of a sequence of its approximate solutions xn t sufficient conditions

a simple computational homogenization method for structures made of linear heterogeneous viscoelastic materials in this paper a numerical multiscale method is proposed for computing the response of structures made of linearly non aging viscoelastic and highly heterogeneous materials in contrast with most of the approaches reported in the literature the present one operates directly in the time domain and avoids both defining macroscopic internal variables and concurrent computations at micro and macro scales the macroscopic constitutive law takes the form of a convolution integral containing an effective relaxation tensor to numerically identify this tensor a representative volume element rve for the microstructure is first chosen relaxation tests are then numerically performed on the rve correspondingly the components of the effective relaxation tensor are determined and stored for different snapshots in time at the macroscopic scale a continuous representation of the effective relaxation tensor is obtained in the time domain by interpolating the data with the help of spline functions the convolution integral characterizing the time dependent macroscopic stress strain relation is evaluated numerically arbitrary local linear viscoelastic laws and microstructure morphologies can be dealt with implicit algorithms are provided to compute the time dependent response of a structure at the macroscopic scale by the finite element method accuracy and efficiency of the proposed approach are demonstrated through 2d and 3d numerical examples and applied to estimate the creep of structures made of concrete

on the mathematical foundations of the self consistent clustering analysis for non linear materials at small strains abstract we investigate both mathematically and numerically the self consistent clustering analysis recently introduced by liu bessa liu and independently by wulfinghoff cavaliere reese we establish in the small strain setting and non softening material behavior existence and uniqueness of the solution to the discretized equations for fixed possibly anisotropic reference material by a constructive method furthermore we establish convergence of the solution to the discretized equation to the continuous solution upon refinement of the clusters thus we generalize the work of tang zhang liu to spatial dimensions larger than 1 we explore the specific structure of the lippmann schwinger equation governing clustering analysis proving strict equivalence to a problem of eyre milton type for the latter formulation existence and uniqueness are easily established based on recent progress in the understanding of polarization schemes in fft based computational homogenization methods last but not least for elasto viscoplastic constituents we clarify the relationship of the self consistent clustering analysis to the transformation field analysis of dvorak benveniste our theoretical considerations are confirmed by pertinent numerical experiments

functional approximation and projection of stored energy functions in computational homogenization of hyperelastic materials a probabilistic perspective abstract this work is concerned with the construction of a surrogate model for the homogenized stored energy functions defining the effective behavior of nonlinear elastic microstructures here a probabilistic standpoint is adopted and allows for the definition of a nonlinear mapping between the macroscopic deformations and the homogenized potential this functional approximation is specifically obtained by means of a polynomial chaos expansion the coefficients of which are computed through a gauss legendre quadrature rule by invoking well known results related to projections in hilbert spaces closest approximations of arbitrary potentials to standard e g ogden type models are subsequently defined and characterized by appropriate residuals numerical illustrations on various microstructures are finally provided in order to discuss the relevance of the proposed framework in particular it is shown that the surrogate model compares very well with reference results at both the macroscale and the structural scale even for moderate order expansions and that the aforementioned closest approximations constitute very accurate approximation provided that the subset onto which the homogenized response is projected is properly chosen this result readily allows for nonconcurrent coupling using standard constitutive models available in commercial codes and therefore makes the approach very attractive for engineering applications

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

on full two scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher order homogenised variational problems we consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations which may be a model of e g nonlinear conducting dielectric or deforming in a restricted way hardening elastic plastic composites with outer periodicity conditions of a fixed large period under some natural growth assumptions on the stored energy function we construct for uniformly elliptic problems a full two scale asymptotic expansion which has a precise double series structure separating the slow and the fast variables in all orders so that its slowly varying part solves asymptotically an infinite order homogenised equation cf bakhvalov n s panasenko g p homogenisation averaging processes in periodic media nauka moscow 1984 in russian english translation kluwer 1989 and whose higher order terms depend on the higher gradients of the slowly varying part we prove the error bound i e that the truncated asymptotic expansion is higher order close to the actual solution in appropriate norms the approach is extended to a non uniformly elliptic case for two dimensional power law potentials we prove the non degeneracy using topological index methods examples and explicit formulae for the higher order terms are given in particular we prove that the first term in the higher order homogenised equations is related to the first order corrector to the mean flux and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest second derivative being a linear combination of the second minors of the hessian with coefficients depending on the first gradient and in dimension two is of monge ampere type we show that this term is present at least for some examples three phase power law laminates

a hybrid variationally consistent homogenization approach based on ritz s method summary multiscale approaches based on homogenization theory provide a suitable framework to incorporate information associated with a small scale microscale problem into the considered large scale macroscopic problem in this connection the present paper proposes a novel computationally efficient hybrid homogenization method its backbone is a variationally consistent fe2 approach in which every aspect is governed by energy minimization in particular scale bridging is realized by the canonical principle of energy equivalence as a direct implementation of the aforementioned variationally consistent fe2 approach is numerically extensive an efficient approximation based on ritz s method is advocated by doing so the material parameters defining an effective macroscopic material model capturing the underlying microstructure can be efficiently computed furthermore the variational scale bridging principle provides some guidance to choose a suitable family of macroscopic material models comparisons between the results predicted by the novel hybrid homogenization method and full field finite element simulations show that the novel method is indeed very promising for multiscale analyses copyright 2013 john wiley sons ltd

reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems a new finite element method for the efficient discretization of elliptic homogenization problems is proposed these problems characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed recently the finite element heterogeneous multiscale method fe hmm has been proposed for such problems based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh departing from the approach used in the fe hmm we show that interpolation techniques based on the reduced basis methodology an offline online strategy allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions this new method called the reduced basis finite element heterogeneous multiscale method rb fe hmm is significantly more efficient than the fe hmm for high order macroscopic discretizations and for three dimensional problems when the repeated computation of micro problems over the whole computational domain is expensive a priori error estimates of the rb fe hmm are derived numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method

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

multiscale structural topology optimization with an approximate constitutive model for local material microstructure abstract this paper builds on our recent work xia and breitkopf 2014 on multiscale structural topology optimization where at the microscopic scale local materials are optimized concurrently according to current loading status the former design framework requires intensive computational cost due to large number of repetitive local material optimizations to circumvent this limitation in the present work we construct a reduced database model viewing the local material optimization process as a generalized constitutive behavior using separated representations in this model the database is built from a set of numerical experiments of local material optimizations in the macroscopic strain tensor space each value in the database corresponds to the strain energy density evaluated on a material microstructure optimized according to the imposed macroscopic strain by tensor decomposition a continuous representation of the strain energy density is built as a sum of products of one dimensional interpolation functions as a result of this a priori off line step the effective strain energy and stress strain relations required for macroscopic structural evaluation and optimization are provided in a numerically explicit manner the results given by the reduced database model are compared with full scale results it is also shown that this explicit constitutive behavior representation can well serve multiscale structural design at a significantly reduced computational cost

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

heterogeneous thin films of martensitic materials we study the effective behavior of heterogeneous thin films with three competing length scales the film thickness and the length scales of heterogeneity and material microstructure we start with three dimensional nonhomogeneous nonlinear elasticity enhanced with an interfacial energy of the van der waals type and derive the effective energy density as all length scales tend to zero with given limiting ratios we do not require any a priori selection of asymptotic expansion or ansatz in deriving our results depending on the dominating length scale the effective energy density can be identified by three procedures averaging homogenization and thin film limit we apply our theory to martensitic materials with multi well energy density and use a model example to show that the shape memory behavior can crucially depend on the ratios of these length scales we comment on the effective conductivity of linear composites and also on multilayers made of shape memory and elastic materials

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

packing hyperspheres in high dimensional euclidean spaces we present a study of disordered jammed hard sphere packings in four five and six dimensional euclidean spaces using a collision driven packing generation algorithm we obtain the first estimates for the packing fractions of the maximally random jammed mrj states for space dimensions d 4 5 and 6t o be mrj0 46 0 31 and 0 20 respectively to a good approximation the mrj density obeys the scaling form mrj c1 2 d c2d 2 d where c1 2 72 and c2 2 56 which appears to be consistent with the highdimensional asymptotic limit albeit with different coefficients calculations of the pair correlation function g2r and structure factor sk for these states show that short range ordering appreciably decreases with increasing dimension consistent with a recently proposed decorrelation principle which among other things states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d as in three dimensions where mrj0 64 the packings show no signs of crystallization are isostatic and have a power law divergence in g2r at contact with power law exponent 0 4 across dimensions the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2r has its first minimum additionally we obtain estimates for the freezing and melting packing fractions for the equilibrium hard sphere fluid solid transition f0 32 and m0 39 respectively for d 4 and f0 20 and m0 25 respectively for d 5 although our results indicate the stable phase at high density is a crystalline solid nucleation appears to be strongly suppressed with increasing dimension

computational homogenization of nonlinear elastic materials using neural networks in this work a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks nn in this method the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters the discrete values of the effective potential are computed by fem through random sampling in the parameter space and nn are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components we show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10 allowing to consider 3d representative volume elements rve and an explicit dependence of the effective behavior on microstructural parameters like volume fraction we present several applications of this technique to the homogenization of nonlinear elastic composites involving a two scale example of heterogeneous structure with graded nonlinear properties

existence and uniqueness of solutions for third order nonlinear boundary value problems in this paper we present some general results of existence and uniqueness of solutions of nonlinear two point boundary value problems for third order nonlinear differential equations by using the shooting method as applications we give certain concrete sufficient conditions for the existence and uniqueness

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

nonuniform transformation field analysis the exact description of the overall behavior of composites with nonlinear dissipative phases requires an infinity of internal variables approximate models involving only a finite number of those can be obtained by considering a decomposition of the microscopic anelastic strain field on a finite set of transformation fields the transformation field analysis of dvorak proc r soc lond a 437 1992 311 corresponds to piecewise uniform transformation fields the present theory considers nonuniform transformation fields comparison with numerical simulations shows the accuracy of the proposed model

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

computational aspects for constructing realizations of polynomial chaos in high dimension this paper deals with computational aspects related to the construction of realizations of polynomial chaos expansion in high dimension the method proposed consists of 1 constructing the realizations of the multivariate monomials using a generator of independent realizations of the germs whose probability distribution is the given arbitrary measure and 2 performing an orthogonalization of the realizations of the multivariate monomials with an algorithm different from the gram schmidt orthogonalization algorithm which is not stable in high dimension a brief review of polynomial chaos expansion with arbitrary measure is given the statistically independent realizations of multivariate monomials are introduced the centered statistically independent realizations of orthonormal multivariate polynomials are developed finally a quantification of the errors induced by the usual methods is given

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

a nonlinear manifold based reduced order model for multiscale analysis of heterogeneous hyperelastic materials a new manifold based reduced order model for nonlinear problems in multiscale modeling of heterogeneous hyperelastic materials is presented the model relies on a global geometric framework for nonlinear dimensionality reduction isomap and the macroscopic loading parameters are linked to the reduced space using a neural network the proposed model provides both homogenization and localization of the multiscale solution in the context of computational homogenization to construct the manifold we perform a number of large three dimensional simulations of a statistically representative unit cell using a parallel finite strain finite element solver the manifold based reduced order model is verified using common principles from the machine learning community both homogenization and localization of the multiscale solution are demonstrated on a large three dimensional example and the local microscopic fields as well as the homogenized macroscopic potential are obtained with acceptable engineering accuracy

method of chasing for multipoint boundary value problems the method of chasing developed by gel fand and lokutsiyevskii is completely formulated here for nth order ordinary linear differential equations subject to separated multipoint boundary conditions

annular filters for binary images a binary annular filter removes isolated points in the foreground and the background of an image here the adjective isolated refers to an underlying adjacency relation between pixels which may be different for foreground and background pixels in this paper annular filters are represented in terms of switch pairs a switch pair consists of two operators which govern the removal of points from foreground and background respectively in the case of annular filters switch pairs are completely determined by foreground and background adjacency it is shown that a specific triangular condition in terms of both adjacencies is required to establish idempotence of the resulting annular filter in the case of translation invariant operators an annular filter takes the form x spl rarr x spl oplus a spl cap x spl cup x spl ominus b where a and b are structuring elements satisfying some further conditions when a spl cap b spl cap a spl oplus b spl ne o it is an idempotent morphological filter when a spl cup b spl sub a spl oplus b it is a strong filter and in this case it can be obtained by composing in either order the annular opening x spl rarr x spl oplus a spl cap x and the annular closing x spl rarr spl cup x spl oplus b

second and higher order systems of boundary value problems abstract existence results for systems of kth order boundary value problems are presented our analysis is based on a nonlinear alternative of a granas and use of a priori bounds

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

nonincremental proper generalized decomposition solution of parametric uncoupled models defined in evolving domains this work addresses the recurrent issue related to the existence of reduced bases related to the solution of parametric models defined in evolving domains in this first part of the work we address the case of decoupled kinematics that is models whose solution does not affect the domain in which they are defined the chosen framework considers an updated lagrangian description of the kinematics solved by using natural neighbor galerkin methods within a nonincremental space time framework that can be generalized for addressing parametric models examples showing the performance and potentialities of the proposed methodology are included

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