asymptotics of work distributions in nonequilibrium systems the asymptotic behavior of the work distribution in driven nonequilibrium systems is determined using the method of optimal fluctuations for systems described by langevin dynamics the corresponding euler lagrange equation together with the appropriate boundary conditions and an equation for the leading pre exponential factor are derived the method is applied to three representative examples and the results are used to improve the accuracy of free energy estimates based on the application of the jarzynski equation

fft based homogenization for microstructures discretized by linear hexahedral elements summary the fft based homogenization method of moulinec suquet has recently emerged as a powerful tool for computing the macroscopic response of complex microstructures for elastic and inelastic problems in this work we generalize the method to problems discretized by trilinear hexahedral elements on cartesian grids and physically nonlinear elasticity problems we present an implementation of the basic scheme that reduces the memory requirements by a factor of four and of the conjugate gradient scheme that reduces the storage necessary by a factor of nine compared with a naive implementation for benchmark problems in linear elasticity the solver exhibits mesh and contrast independent convergence behavior and enables the computational homogenization of complex structures for instance arising from computed tomography computed tomography ct imaging techniques there exist 3d microstructures involving pores and defects for which the original fft based homogenization scheme does not converge in contrast for the proposed scheme convergence is ensured also the solution fields are devoid of the spurious oscillations and checkerboarding artifacts associated to conventional schemes we demonstrate the power of the approach by computing the elasto plastic response of a long fiber reinforced thermoplastic material with 172 106 displacement degrees of freedom copyright 2016 john wiley sons ltd

reconstructing displacements from the solution to the periodic lippmann schwinger equation discretized on a uniform grid uniform grid solvers of the periodic lippmann schwinger equation have been introduced by moulinec and suquet for the numerical homogenization of heterogeneous materials based on the fast fourier transform these methods use the strain as main unknown and usually do not produce displacement fields while this is generally not perceived as a restriction for homogenization purposes some tasks might require kinematically admissible displacement fields in this paper we show how the numerical solution to the periodic lippmann schwinger equation can be post processed to reconstruct a displacement field our procedure is general it applies to any variant of the moulinec suquet solver the reconstruction is formulated as an auxiliary elastic equilibrium problem of a homogeneous material which is solved with displacement based finite elements taking advantage of the periodicity the uniformity of the grid and the homogeneity of the material the resulting linear system is formulated and solved efficiently in fourier space the cost of our procedure is lower than that of one iteration of the lippmann schwinger solver an application of this post processing procedure is proposed in which the reconstructed displacement field is used to compute a rigorous upper bound on the effective shear modulus of some model microstructure this is the pre peer reviewed version of the following article reconstructing displacements from the solution to the periodic lippmann schwinger equation discretized on a uniform grid which has been published in final form at 10 1002 nme 5263 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving sections 1 introduction and 5 numerical results have been significantly reworked in the published final version of this article the remainder of this article including sections 3 and 4 where the method itself is presented is essentially unchanged

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

trigonometric collocation methods with product integration for boundary integral equations on closed curves considering boundary integral equations on smooth closed curves the product integration methods have turned out to be very efficient to define fully discrete schemes with a high convergence rate here we give a general approach to product integration methods in connection with the trigonometric collocation the discretization scheme has arbitrarily high convergence rate even exponential if the solution and the curve are analytic applications and numerical experiments confirming our theoretical results are discussed

an accelerated fft algorithm for thermoelastic and non linear composites a fast numerical algorithm to compute the local and overall responses of non linear composite materials is developed this alternative formulation allows us to improve the convergence of the existing method of moulinec and suquet e g comput meth appl mech eng 1998 157 1 2 69 94 in the present method a non linear elastic or conducting material is replaced by infinitely many locally linear thermoelastic materials with moduli that depend on the values of the local fields this makes it possible to use the advantages of an algorithm developed by eyre and milton eur phys j appl phys 1999 6 1 41 47 which has faster convergence the method is applied to compute the local fields as well as the effective response of non linear conducting and elastic periodic composites copyright 2008 john wiley sons ltd

filtering material properties to improve fft based methods for numerical homogenization fft based solvers introduced in the 1990s for the numerical homogenization of heterogeneous elastic materials have been extended to a wide range of physical properties in parallel alternative algorithms and modified discrete green operators have been proposed to accelerate the method and or improve the description of the local fields in this short note filtering material properties is proposed as a third complementary way to improve fft based methods it is evidenced from numerical experiments that the grid refinement and consequently the computation time and or the spurious oscillations observed on local fields can be significantly reduced in addition if the two filters based on voigt or reuss homogenization rules can improve or deteriorate the method depending on the microstructure a stiff inclusion within a compliant matrix or the reverse the proposed 2 layer filter is efficient in both situations the study is proposed in the context of linear elasticity but similar results are expected in a different physical context thermal electrical

fft based homogenization of hypoelastic plasticity at finite strains abstract fast fourier transform fft based methods are becoming increasingly popular for modeling the texture evolution and local mechanical response of polycrystalline materials within the representative volume element rve originally the fft based method was formulated through the lippman schwinger l s equation in terms of a homogeneous elastic reference medium more recently the fourier galerkin method was proposed to discretize the rve weak form independently of the reference medium using trigonometric polynomials the fourier galerkin method albeit efficient and robust has not been extended to solve crystal plasticity cp problems with spatial heterogeneity and fine resolution also its algorithmic homogenization has not yet been investigated which is essential for concurrent multiscale modeling in this paper a general framework is proposed that connects the fft based method and objective rate constitutive models in particular a generalized implementation of crystal plasticity consistent linearization is achieved by pulling back the tangent stiffness from unrotated configuration to reference configuration which improves the convergence behavior of the newton krylov solver applying the inexact newton method further improves the numerical efficiency also the algorithmic homogenized tangent stiffness for the fourier galerkin method is derived to enable mixed boundary conditions and concurrent multiscale modeling lastly the fourier galerkin method s accuracy and efficiency for solving cp problems are studied in a parallelized environment and significant speedup is observed versus the finite element method and l s based fft method

combining galerkin approximation techniques with the principle of hashin and shtrikman to derive a new fft based numerical method for the homogenization of composites we report on the mathematical analysis of two different fft based numerical schemes for the homogenization of composite media within the framework of linear elasticity the basic scheme of moulinec and suquet 1994 1998 9 and 10 and the energy based scheme of brisard and dormieux 2010 13 casting these two schemes as galerkin approximations of the same variational problem allows us to assert their well posedness and convergence more importantly we extend in this work their domains of application by relieving some stringent conditions on the reference material which were previously thought necessary the origins of the flaws of each scheme are identified and a third scheme is proposed which seems to combine the strengths of the basic and energy based schemes while leaving out their weaknesses finally a rule is proposed for handling heterogeneous pixels voxels a situation frequently met when images of real materials are used as input to these schemes

short note accelerating a fft based solver for numerical homogenization of periodic media by conjugate gradients in this short note we present a new technique to accelerate the convergence of a fft based solver for numerical homogenization of complex periodic media proposed by moulinec and suquet 1 the approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to vainikko 2 to give a linear system which can be efficiently solved by conjugate gradient methods computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high contrast coefficients at a low overhead per iteration

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

field dislocation mechanics for heterogeneous elastic materials a numerical spectral approach spectral methods using fast fourier transform fft algorithms have recently seen a surge in interest in the mechanics of materials community the present contribution addresses the critical question of determining accurate local mechanical fields using fft methods without artificial fluctuations arising from materials and defects induced discontinuities precisely the present work introduces a numerical approach based on intrinsic discrete fourier transforms for the simultaneous treatment of material discontinuities arising from the presence of dislocations and from elastic stiffness heterogeneities to this end the elasto static equations of the field dislocation mechanics theory for periodic heterogeneous materials are numerically solved with fft in the case of dislocations in proximity of inclusions of varying stiffness an optimal intrinsic discrete fourier transform method is sought based on two distinct schemes a centered finite difference scheme for differential rules are used for numerically solving the poisson type equation in the fourier space while centered finite differences on a rotated grid is chosen for the computation of the modified fourier green s operator associated with the lippmann schwinger type equation by comparing different methods with analytical solutions for an edge dislocation in a composite material it is found that the present spectral method is accurate devoid of any numerical oscillation and efficient even for an infinite phase elastic contrast like a hole embedded in a matrix containing a dislocation the present fft method is then used to simulate physical cases such as the elastic fields of dislocation dipoles located near the matrix inclusion interface in a 2d composite material and the ones due to dislocation loop distributions surrounding cubic inclusions in 3d composite material in these configurations the spectral method allows investigating accurately the elastic interactions and image stresses due to dislocation fields in the presence of elastic inhomogeneities

a comparative study on low memory iterative solvers for fft based homogenization of periodic media in this paper we assess the performance of four iterative algorithms for solving non symmetric rank deficient linear systems arising in the fft based homogenization of heterogeneous materials defined by digital images our framework is based on the fourier galerkin method with exact and approximate integrations that has recently been shown to generalize the lippmann schwinger setting of the original work by moulinec and suquet from 1994 it follows from this variational format that the ensuing system of linear equations can be solved by general purpose iterative algorithms for symmetric positive definite systems such as the richardson the conjugate gradient and the chebyshev algorithms that are compared here to the eyre milton scheme the most efficient specialized method currently available our numerical experiments carried out for two dimensional elliptic problems reveal that the conjugate gradient algorithm is the most efficient option while the eyre milton method performs comparably to the chebyshev semi iteration the richardson algorithm equivalent to the still widely used original moulinec suquet solver exhibits the slowest convergence besides this we hope that our study highlights the potential of the well established techniques of numerical linear algebra to further increase the efficiency of fft based homogenization methods

fluctuation properties of an effective nonlinear system subject to poisson noise we study the work fluctuations of a particle confined to a moving harmonic potential under the influence of friction and external asymmetric poissonian shot noise this type of noise generalizes the usual gaussian noise and induces an effective interaction between the noise and the potential leading to an effectively nonlinear system with singular features on the basis of an analytic solution we investigate the roles of time scales symmetries and singularities in the context of nonequilibrium fluctuations our results highlight the nonuniversality of the steady state fluctuation theorem in stochastic systems

the thermostatistical aspect of werner type states and quantum entanglement a general werner type state is studied from two viewpoints i an application of the dynamical interaction of the objective system with its environment represented by a unital positive operator valued measure povm which ensures an increase of entropy making the system evolve from an initial pure state to a mixed state of the werner type and ii the space of the objective system is constrained to have a given value of the fidelity then the maximum entropy principle is shown to yield the werner type state as a canonical ensemble with a projector hamiltonian this novel observation is illustrated by examples of bipartite systems the separability criteria of which are given in terms of the values of temperature the present viewpoint may cast light on the relevance of thermostatistics to the physics of quantum entanglement in addition the povm scheme presented here offers a way of experimentally generating the werner type thermal states

guaranteed upper lower bounds on homogenized properties by fft based galerkin method abstract guaranteed upper lower bounds on homogenized coefficients arising from the periodic cell problem are calculated in a scalar elliptic setting our approach builds on the recent variational reformulation of the moulinec suquet 1994 fast fourier transform fft homogenization scheme by vond ejc et al 2014 which is based on the conforming galerkin approximation with trigonometric polynomials upper lower bounds are obtained by adjusting the primal dual finite element framework developed independently by dvo ak 1993 and wi ckowski 1995 to the fft based galerkin setting we show that the discretization procedure differs for odd and non odd number of grid points thanks to the helmholtz decomposition inherited from the continuous formulation the duality structure is fully preserved for the odd discretizations in the latter case a more complex primal dual structure is observed due to presence of the trigonometric polynomials associated with the nyquist frequencies these theoretical findings are confirmed with numerical examples to conclude the main advantage of the fft based approach over conventional finite element schemes is that the primal and the dual problems are treated on the same basis and this property can be extended beyond the scalar elliptic setting

combining fft methods and standard variational principles to compute bounds and estimates for the properties of elastic composites abstract in this paper we develop a computational approach based on variational principles combined with fft technique to determine the effective elastic properties of composite materials the unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches based on the strain and the stress potential and the hashin shtrikman variational principles the problem is discretized with fourier series and the stationary point is computed numerically by means of an iterative scheme these algorithms use a representation of the local elastic tensor on the double grid twice the size of the discretization which is introduced by the integral formulation it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients in the second part of the paper we establish a hierarchy between the different fft solutions the strain the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the hashin shtrikman principle which depends on the choice of the elastic moduli of the reference material it is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the hashin shtrikman variational principle when the same discretization is used alternatively when the elastic moduli of the reference material is negative the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound

a finite element perspective on non linear fft based micromechanical simulations fourier solvers have become efficient tools to establish structure property relations in heterogeneous materials introduced as an alternative to the finite element fe method they are based on fixed point solutions of the lippmann schwinger type integral equation their computational efficiency results from handling the kernel of this equation by the fast fourier transform fft however the kernel is derived from an auxiliary homogeneous linear problem which renders the extension of fft based schemes to nonlinear problems conceptually difficult this paper aims to establish a link between fe based and fft based methods in order to develop a solver applicable to general history dependent and time dependent material models for this purpose we follow the standard steps of the fe method starting from the weak form proceeding to the galerkin discretization and the numerical quadrature up to the solution of nonlinear equilibrium equations by an iterative newton krylov solver no auxiliary linear problem is thus needed by analyzing a two phase laminate with nonlinear elastic elastoplastic and viscoplastic phases and by elastoplastic simulations of a dual phase steel microstructure we demonstrate that the solver exhibits robust convergence these results are achieved by re using the nonlinear fe technology with the potential of further extensions beyond small strain inelasticity considered in this paper

a closed form expression for the sharma mittal entropy of exponential families the sharma mittal entropies generalize the celebrated shannon renyi and tsallis entropies we report a closed form formula for the sharma mittal entropies and relative entropies for arbitrary exponential family distributions we explicitly instantiate the formula for the case of the multivariate gaussian distributions and discuss its estimation

fft based homogenisation accelerated by low rank tensor approximations abstract fast fourier transform fft based methods have turned out to be an effective computational approach for numerical homogenisation in particular fourier galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials their computational effectiveness benefits from efficient fft based algorithms as well as a favourable condition number here these kinds of methods are accelerated by low rank tensor approximation techniques for a solution field using canonical polyadic tucker and tensor train formats this reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem it significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation the advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients this approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material

energy based comparison between the fourier galerkin method and the finite element method abstract the fourier galerkin method in short ffth has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom because the method incorporates the fast fourier transform fft in the linear solver it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method fem here we systematically compare these two methods using the energetic norm of local fields which has the clear physical interpretation as being the error in the homogenised properties this enables the comparison of memory and computational requirements at the same level of approximation accuracy we show that the methods effectiveness relies on the smoothness regularity of the solution and thus on the material coefficients thanks to its approximation properties fem outperforms ffth for problems with jumps in material coefficients while ambivalent results are observed for the case that the material coefficients vary continuously in space ffth profits from a good conditioning of the linear system independent of the number of degrees of freedom but generally needs more degrees of freedom to reach the same approximation accuracy more studies are needed for other fft based schemes non linear problems and dual problems which require special treatment in fem but not in ffth

power fluctuation theorem for a brownian oscillator in a thermal bath in this paper we prove the validity of the total power fluctuation theorem for a brownian particle in a harmonic trap when it is dragged out of equilibrium taking into account the inertia term in the langevin equation we first consider the theorem for an ordinary brownian harmonic oscillator embedded in a thermal bath as a second case we assume the presence of an electromagnetic field acting on the charged brownian oscillator in both cases the theorem is proved for two examples in the trap motion linear and oscillatory drag on the potential minimum

fourier based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields a modified green operator is proposed as an improvement of fou rier based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media contrary to other methods the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite furtherm ore it is shown that the method produces much more accurate local field s inside highly conducting and quasi insulating phases as well as in the vicinity of the phases interfaces these good propert ies stem from the discretization of green s function which is consistent with the pixel grid while retaining the local nature of the op erator that acts on the polarization field finally a fast im plementation of the direct scheme of moulinec et al 1994 that allows for parcimonious memory use is proposed

anomalous fluctuation properties we complement and extend our work on fluctuations arising in nonequilibrium systems in steady states driven by l evy noise h touchette and e g d cohen phys rev e 76 020101 r 2007 as a concrete example we consider a particle subjected to a drag force and a l evy white noise with tail index ensuremath mu 0 2 and calculate the probability distribution of the work done on the particle by the drag force as well as the probability distribution of the work dissipated by the dragged particle in a nonequilibrium steady state for 0l ensuremath mu l2 both distributions satisfy what we call an anomalous fluctuation property characterized by positive and negative fluctuations that asymptotically have the same probability for ensuremath mu 2 by contrast the work and dissipated work distributions satisfy the known conventional and extended fluctuation relations respectively which are both characterized by positive fluctuations that are exponentially more probable than negative fluctuations the difference between these different fluctuation behaviors is discussed in the context of large deviation theory experiments that could probe or reveal an anomalous fluctuation property are also discussed

use of composite voxels in fft based homogenization abstract fft based homogenization methods operate on regular voxel grids in general such grids cannot resolve interfaces exactly in this article we assign voxels containing an interface a stiffness different from the constituent materials in a systematic fashion more precisely we characterize the class of these so called composite voxels leading to convergence of the discretizations considering the interface in the composite voxel as linear we furnish the voxel with the corresponding laminate stiffness these laminate voxels are shown to increase both the accuracy of the calculated effective properties and the local quality of the strain and stress fields dramatically

a framework for fft based homogenization on anisotropic lattices abstract in order to take structural anisotropies of a given composite and different shapes of its unit cell into account we generalize the basic scheme in homogenization by moulinec and suquet to arbitrary sampling lattices and tilings of the d dimensional euclidean space we employ a fourier transform for these lattices by introducing the corresponding set of sample points the so called pattern and its frequency set the generating set the pattern and the generating set represent the anisotropy of both the shape of the unit cell and the chosen preferences in certain sampling directions in several cases this fourier transform is of lower dimension than the space itself for the so called rank 1 lattices it reduces to a one dimensional fourier transform having the same leading coefficient as the fastest fourier transform implementation available we illustrate the results using the generalized basic scheme on an anisotropic laminate and on a generalized ellipsoidal hashin structure for both we give an analytical solution to the elasticity problem in two and three dimensions respectively we then illustrate the possibilities of choosing a pattern compared to classical grids this introduces both a reduction of computation time and a reduced error of the numerical method it also allows for anisotropic subsampling i e choosing a sub lattice of a pixel or voxel grid based on anisotropy information of the material at hand

an fft based galerkin method for homogenization of periodic media in 1994 moulinec and suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media the scheme is based on a fixed point iterative solution to an integral equation of the lippmann schwinger type with action of its kernel efficiently evaluated by the fast fourier transform techniques the aim of this work is to demonstrate that the moulinec suquet setting is actually equivalent to a galerkin discretization of the cell problem based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme for the latter framework and scalar elliptic problems we prove convergence of the approximate solution to the weak solution including a priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration moreover we also show that the variational structure implies that the resulting non symmetric system of linear equations can be solved by the conjugate gradient method apart from providing a theoretical support to fast fourier transform based methods for numerical homogenization these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature

effect of correlated dichotomous noises on stochastic resonance in a linear system a linear system driven by correlated asymmetric dichotomous noises and periodic signal was investigated in the overdamped case the exact expressions of output signal amplitude and signal to noise ratio snr of the system were derived by means of numerical calculations we found that i at some fixed multiplicative noise intensities the output signal amplitude with frequency exhibits the structure of a weak peak even no peak as the dichotomous noise is asymmetric ii in the case of asymmetric dichotomous noise the signal frequency can cause non monotonous behavior of the output signal amplitude with respect to multiplicative noise intensity iii the curve of snr with frequency has a weak peak and a trough in the case of symmetric dichotomous noise but no peak with asymmetric iv whether the multiplicative noise is symmetric or asymmetric the noise can enhance response of the system v the snr increases with the correlation strength between the two noises decreasing in addition the plane of multiplicative noise intensity versus noise symmetric parameter was plotted

convergence of iterative methods based on neumann series for composite materials theory and practice iterative fast fourier transform methods are useful for calculating the fields in composite materials and their macroscopic response by iterating back and forth until convergence the differential constraints are satisfied in fourier space and the constitutive law in real space the methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli it is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative fast fourier transform methods we look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity obnosov theoretically some of the methods converge when the inclusions have zero or even negative conductivity however the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations there is little point in iterating beyond this accuracy improves when the grid size is reduced showing that the discrepancy is linked to the discretization finally it is shown that none of the three iterative schemes investigated over performs the others for all possible microstructures and all contrasts

a note on exponential families of distributions we show that an arbitrary probability distribution can be represented in an exponential form in physical contexts this implies that the equilibrium distribution of any classical or quantum dynamical system is expressible in a grand canonical form

computational homogenization of elasticity on a staggered grid summary in this article we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers our method shares some properties with the fft based homogenization technique of moulinec and suquet which has received widespread attention recently because of its robustness and computational speed these similarities include the use of fft and the resulting performing solvers the staggered grid discretization however offers three crucial improvements firstly solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by moulinec suquet s discretization secondly the iteration numbers of our solvers are bounded independently of the grid size and the contrast in particular our solvers converge for three dimensional porous structures which cannot be handled by moulinec suquet s method thirdly the finite difference discretization allows for algorithmic variants with lower memory consumption more precisely it is possible to reduce the memory consumption of the moulinec suquet algorithms by 50 we underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale copyright 2015 john wiley sons ltd