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

asymmetric non linear forced vibrations of free edge circular plates part 1 theory in this article a detailed study of the forced asymmetric non linear vibrations of circular plates with a free edge is presented the dynamic analogue of the von karman equations is used to establish the governing equations the plate displacement at a given point is expanded on the linear natural modes the forcing is harmonic with a frequency close to the natural frequency omega n of one asymmetric mode of the plate thus the vibration is governed by the two degenerated modes corresponding to omega n which are one to one internally resonant an approximate analytical solution using the method of multiple scales is presented attention is focused on the case where one configuration which is not directly excited by the load gets energy through non linear coupling with the other configuration slight imperfections of the plate are taken into account experimental validations are presented in the second part of this paper

applying the method of normal forms to second order nonlinear vibration problems vibration problems are naturally formulated with second order equations of motion when the vibration problem is nonlinear in nature using normal form analysis currently requires that the second order equations of motion be put into first order form in this paper we demonstrate that normal form analysis can be carried out on the second order equations of motion in addition for forced damped nonlinear vibration problems we show that the invariance properties of the first and second order transforms differ as a result using the second order approach leads to a simplified formulation for forced damped nonlinear vibration problems

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

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

a computational scheme for linear and non linear composites with arbitrary phase contrast a numerical method making use of fast fourier transforms has been proposed in moulinec and suquet 1994 1998 to investigate the effective properties of linear and non linear composites this method is based on an iterative scheme the rate of convergence of which is proportional to the contrast between the phases composites with high contrast typically above 104 or infinite contrast those containing voids or rigid inclusions or highly non linear materials cannot be handled by the method this paper presents two modified schemes the first one is an accelerated scheme for composites with high contrast which extends to elasticity a scheme initially proposed in eyre and milton 1999 its rate of convergence varies as the square root of the contrast the second scheme adequate for composites with infinite contrast is based on an augmented lagrangian method the resulting saddle point problem involves three steps the first step consists of solving a linear elastic problem using the fast fourier transform method in the second step a non linear problem is solved at each individual point in the volume element the third step consists of updating the lagrange multiplier applications of this scheme to rigidly reinforced and to voided composites are shown copyright 2001 john wiley sons ltd

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

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

analysis of non linear dynamical systems by the normal form theory abstract a method is proposed for calculating the periodic solutions of non linear mechanical systems with analytical non linearities the jordan normalization procedure for the case of non linear autonomous systems is described and generalized to dampened harmonically excited oscillators non linear modes for hamiltonian systems are introduced normal forms simplify the analysis of bifurcation it is shown how to extend the modal synthesis procedure the proposed non linear modes obtained from free vibrations are used to construct a superposition technique to describe the forced response of harmonically excited systems these results are tested for one and two degrees of freedom systems with cubic non linearities the results are compared with expressions obtained by classical analytical methods averaging or multiple scales methods or runge kutta numerical methods

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

the two machine total completion time flow shop problem in this paper we study the np hard scheduling problem of minimizing total completion time in a two machine flow shop five known lower bounds are discussed and two new ones are presented a new dominance criterion is also proposed several versions of a branch and bound method are derived by applying both individually and combined these lower bounds a heuristic procedure is also presented that uses a constructive o n2 time method which computes a good starting solution together with a neighborhood search based on pairwise interchanges computational results show that the exact method can handle problems of up to 30 jobs in size within a reasonable amount of time and that the heuristic procedure has an average error of less than 0 5 from the optimal value and less than 2 7 from the lower bound

modal substructuring of geometrically nonlinear finite element models with interface reduction substructuring methods have been widely used in structural dynamics to divide large complicated finite element models into smaller substructures for linear systems many methods have been develop

dynamics and energy exchanges between a linear oscillator and a nonlinear absorber with local and global potentials abstract the dynamical behavior of a two degree of freedom system made up of a linear oscillator and a coupled nonlinear energy sink with nonlinear global and local potentials is studied the nonlinear global potential of the energy sink performs direct interactions with the linear oscillator while its local potential depends only on its own behavior during vibratory energy exchanges between two oscillators a time multiple scale method around 1 1 1 resonance is used to detect slow invariant manifold of the system its equilibrium and singular points detected equilibrium points permit us to predict periodic regime s while singular points can lead the system to strongly modulated responses characterized by persistent bifurcations several possible scenarios occurring during these strongly modulated regimes are highlighted all analytical predictions are compared with those which are obtained by direct numerical integration of system equations

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

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

identifying the significance of nonlinear normal modes nonlinear normal modes nnms are widely used as a tool for understanding the forced responses of nonlinear systems however the contemporary definition of an nnm also encompasses a large number of dynamic behaviours which are not observed when a system is forced and damped as such only a few nnms are required to understand the forced dynamics this paper firstly demonstrates the complexity that may arise from the nnms of a simple nonlinear system highlighting the need for a method for identifying the significance of nnms an analytical investigation is used alongside energy arguments to develop an understanding of the mechanisms that relate the nnms to the forced responses this provides insight into which nnms are pertinent to understanding the forced dynamics and which may be disregarded the nnms are compared with simulated forced responses to verify these findings

interpreting the forced responses of a two degree of freedom nonlinear oscillator using backbone curves in this paper the backbone curves of a two degree of freedom nonlinear oscillator are used to interpret its behaviour when subjected to external forcing the backbone curves describe the loci of dynamic responses of a system when unforced and undamped and are represented in the frequency amplitude projection in this study we provide an analytical method for relating the backbone curves found using the second order normal form technique to the forced responses this is achieved using an energy based analysis to predict the resonant crossing points between the forced responses and the backbone curves this approach is applied to an example system subjected to two different forcing cases one in which the forcing is applied directly to an underlying linear mode and the other subjected to forcing in both linear modes additionally a method for assessing the accuracy of the prediction of the resonant crossing points is then introduced and these predictions are then compared to responses found using numerical continuation

robust identification of backbone curves using control based continuation abstract control based continuation is a recently developed approach for testing nonlinear dynamic systems in a controlled manner and exploring their dynamic features as system parameters are varied in this paper control based continuation is adapted to follow the locus where system response and excitation are in quadrature extracting the backbone curve of the underlying conservative system the method is applied to a single degree of freedom oscillator under base excitation and the results are compared with the standard resonant decay method

nonlinear normal modes for damped geometrically nonlinear systems application to reduced order modelling of harmonically forced structures in order to build efficient reduced order models roms for geometrically nonlinear vibrations of thin structures a normal form procedure is computed for a general class of nonlinear oscillators with quadratic and cubic nonlinearities the linear perturbation brought by considering a modal viscous damping term is especially addressed in the formulation a special attention is focused on how all the linear modal damping terms are gathered together in order to define a precise decay of energy onto the invariant manifolds also defined as nonlinear normal modes nnms then this time independent formulation is used to reduce the dynamics governing the oscillations of a structure excited by an external harmonic force the validity of the proposed roms is systematically discussed and compared with other available methods in particular it is shown that large values of the modal damping of the slave modes may change the type of nonlinearity hardening softening behaviour of the directly excited master mode two examples are used to illustrate the main features of the method a two degrees of freedom dof system allows presentation of the main results through a simple example then a water filled circular cylindrical shell with external resonant forcing is considered in order to show the ability of the method to substantially reduce the dynamics of a continuous structure 2006 elsevier ltd all rights reserved