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

finding hypernetworks in directed hypergraphs abstract the term hypernetwork more precisely s hypernetwork and s d hypernetwork has been recently adopted to denote some logical structures contained in a directed hypergraph a hypernetwork identifies the core of a hypergraph model obtained by filtering off redundant components therefore finding hypernetworks has a notable relevance both from a theoretical and from a computational point of view in this paper we provide a simple and fast algorithm for finding s hypernetworks which substantially improves on a method previously proposed in the literature we also point out two linearly solvable particular cases finding an s d hypernetwork is known to be a hard problem and only one polynomially solvable class has been found so far here we point out that this particular case is solvable in linear time

objective evaluation of linearization procedures in nonlinear homogenization a methodology and some implications on the accuracy of micromechanical schemes a systematic methodology for an accurate evaluation of various existing linearization procedures sustaining mean fields theories for nonlinear composites is proposed and applied to recent homogenization methods it relies on the analysis of a periodic composite for which an exact resolution of both the original nonlinear homogenization problem and the linear homogenization problems associated with the chosen linear comparison composite lcc with an identical microstructure is possible the effects of the sole linearization scheme can then be evaluated without ambiguity this methodology is applied to three different two phase materials in which the constitutive behavior of at least one constituent is nonlinear elastic or viscoplastic a reinforced composite a material in which both phases are nonlinear and a porous material comparisons performed on these three materials between the considered homogenization schemes and the reference solution bear out the relevance and the performances of the modified second order procedure introduced by ponte castaneda in terms of prediction of the effective responses however under the assumption that the field statistics first and second moments are given by the local fields in the lcc all the recent nonlinear homogenization procedures still fail to provide an accurate enough estimate of the strain statistics especially for composites with high contrast

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

new method based on the hpm and rkhsm for solving forced duffing equations with integral boundary conditions this paper investigates the forced duffing equation with integral boundary conditions its approximate solution is developed by combining the homotopy perturbation method hpm and the reproducing kernel hilbert space method rkhsm hpm is based on the use of the traditional perturbation method and the homotopy technique the hpm can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases rkhsm is also an analytical technique which can solve powerfully linear boundary value problems therefore the forced duffing equation with integral boundary conditions can be solved using advantages of these two methods two numerical examples are presented to illustrate the strength of the method

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

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

flows on hypergraphs we consider the capacitated minimum cost flow problem on directed hypergraphs we define spanning hypertrees so generalizing the spanning tree of a standard graph and show that like in the standard and in the generalized minimum cost flow problems a correspondence exists between bases and spanning hypertrees then we show that like for the network simplex algorithms for the standard and for the generalized minimum cost flow problems most of the computations performed at each pivot operation have direct hypergraph interpretations

hypernetworks in a directed hypergraph abstract recently novel distributed approaches to the formation of collaborative networked organizations have been proposed some of them make use of logical formal structures i e hypernetworks and hyperpaths based on concepts related to directed hypergraphs in order to formally represent competitive inter cluster and collaborative intra cluster relationships in this paper we introduce the concepts of the s hypernetwork the structure consisting of all hyperpaths from a node s to any node in a directed hypergraph the s d hypernetwork the structure consisting of all hyperpaths from a node s to a node d in a directed hypergraph moreover we present characterization theorems for both structures a polynomial algorithm to determine the s hypernetwork and some conditions under which the problem of finding the s d hypernetwork is solvable in polynomial time

numerical solutions of reaction diffusion equations with nonlocal boundary conditions abstract the purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction diffusion equations with nonlocal boundary conditions using the finite difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time dependent and the steady state finite difference systems each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence comparison theorem for the corresponding finite difference system the existence comparison theorems are used to investigate the asymptotic behavior of the discrete time dependent solution in relation to the discrete maximal and minimal solutions of the steady state problem numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes

finding optimal hyperpaths in large transit networks with realistic headway distributions abstract this paper implements and tests a label setting algorithm for finding optimal hyperpaths in large transit networks with realistic headway distributions it has been commonly assumed in the literature that headway is exponentially distributed to validate this assumption the empirical headway data archived by chicago transit agency are fitted into various probabilistic distributions the results suggest that the headway data fit much better with loglogistic gamma and erlang distributions than with the exponential distribution accordingly we propose to model headway using the erlang distribution in the proposed algorithm because it best balances realism and tractability when headway is not exponentially distributed finding optimal hyperpaths may require enumerating all possible line combinations at each transfer stop which is tractable only for a small number of alternative lines to overcome this difficulty a greedy method is implemented as a heuristic and compared to the brute force enumeration method the proposed algorithm is tested on a large scale cta bus network that has over 10 000 stops the results show that 1 the assumption of exponentially distributed headway may lead to sub optimal route choices and 2 the heuristic greedy method provides near optimal solutions in all tested cases

max horn sat and the minimum cut problem in directed hypergraphs in this paper we consider the maximum horn satisfiability problem which is reduced to the problem of finding a minimum cardinality cut on a directed hypergraph for the latter problem we propose different ip formulations related to three different definitions of hyperpath weight we investigate the properties of their linear relaxations showing that they define a hierarchy the weakest relaxation is shown to be equivalent to the relaxation of a well known ip formulation of max horn sat and to a max flow problem on hypergraphs the tightest relaxation which is a disjunctive programming problem is shown to have integer optimum the intermediate relaxation consists in a set covering problem with a possible exponential number of constraints this latter relaxation provides an approximation of the convex hull of the integer solutions which as proven by the experimental results given is much tighter than the one known in the literature 1998 the mathematical programming society inc published by elsevier science b v

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

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

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

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

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a finite difference method for the singularly perturbed problem with nonlocal boundary condition the purpose of this paper is to present a finite difference method for numerical solutions of singularly perturbed boundary value problem for second order ordinary differential equation with nonlocal boundary condition by the method of integral identities with the use exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form an exponentially fitted difference scheme on an uniform mesh is developed which is shown to be original e uniformly first order accurate in the discrete maximum norm for original problem numerical results are presented which illustrate the theoretical results

gainfree leontief substitution flow problems leontief substitution systems have been studied by economists and operations researchers for many years we show how such linear systems are naturally viewed asleontief substitution flow problems on directed hypergraphs and that important solution properties follow from structural characteristics of the hypergraphs we give a strongly polynomial non simplex algorithm for leontief substitution flow problems that satisfy againfree property leading to acyclic extreme solutions integrality conditions follow easily from this algorithm another structural property support disjoint reachability leads to necessary and sufficient conditions for extreme solutions to be binary in a survey of applications we show how the leontief flow paradigm links polyhedral combinatorics expert systems mixed integer model formulation and some problems in graph optimization

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

constitutive modeling of anisotropic plasticity with application to fiber reinforced composites abstract this article proposes two constitutive models to describe the nonlinear elastic plastic behavior of unidirectional fiber reinforced composites at the meso level key ingredients are the formulation of two anisotropic yield functions with the aid of representation theorems invariant formulation and the set up of governing equations for plastic variables with respect to associative and non associative flow response the governing equations are solved using an elastic predictor plastic corrector algorithm which imposes the constraint posed by the yield condition both the models are evaluated qualitatively and quantitatively by comparison to micromechanics simulations as well as to experimental data

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