power law distributions in some random boolean networks the kauffman net is a dynamical system of logical variables receiving two random inputs and each randomly assigned a boolean function we show that the attractor and transient lengths exhibit scaleless behavior with power law distributions over up to 10 orders of magnitude in probability our results provide evidence for the existence of the edge of chaos as a distinct regime between the ordered and chaotic phases analogous to a critical point in statistical mechanics the power law distributions are robust to the changes in the composition of the transition rules and network dynamics

relevant elements magnetization and dynamical properties in kauffman networks a numerical study abstract this is the first of two papers about the structure of kauffman networks in this paper we define the relevant elements of random networks of automata following previous work by flyvbjerg j phys a 21 1988 l955 l960 and flyvbjerg and kjaer j phys a 21 1988 1695 1718 and we study numerically their probability distributions in the chaotic phase and on the critical line of the model a simple approximate argument predicts that their number scales as n on the critical line while it is linear with n in the chaotic phase and independent on system size in the frozen phase this argument is confirmed by numerical results the study of the relevant elements gives useful information about the properties of the attractors in critical networks where the pictures coming from either approximate computation methods or from simulations are not very clear

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

modeling and computation of shakedown problems for nonlinear hardening materials abstract the paper is concerned with shakedown analysis of perfectly plastic nonlinear kinematic hardening linear unlimited kinematical hardening and cyclic kinematical hardening materials a 3 d overlay model being able to describe nonlinear kinematic hardening material behavior is reviewed and a static shakedown theorem is formulated which is shown to be an extension of the melan s static shakedown theorems for both perfectly plastic and linear unlimited kinematic hardening materials different numerical algorithms are presented which efficiently solve the nonlinear discretized optimization problems using the finite element method the influence of kinematic hardening on shakedown limits is discussed it is shown how the exactness of the numerical shakedown limits can efficiently be improved by using the adaptive mesh refinement a possibility for analytical determination of shakedown limits in case of alternating plasticity is demonstrated by a square plate with a central circular hole under biaxial uniform loading the shakedown behaviour of materials with cyclic kinematical hardening has been investigated by an incremental computation method the theoretical and numerical aspects discussed are illustrated by the examples given in the paper

an energy conserving ultra weak discontinuous galerkin method for the generalized korteweg de vries equation abstract we propose an energy conserving ultra weak discontinuous galerkin dg method for the generalized korteweg de vries kdv equation in one dimension optimal a priori error estimate of order k 1 is obtained for the semi discrete scheme for the kdv equation without the convection term on general nonuniform meshes when polynomials of degree k 2 are used we also numerically observe optimal convergence of the method for the kdv equation with linear or nonlinear convection terms it is numerically observed for the new method to have a superior performance for long time simulations over existing dg methods

scale free network with boolean dynamics as a function of connectivity in this work we analyze scale free networks with different power law spectra n k ensuremath sim k ensuremath ensuremath gamma under a boolean dynamic where the boolean rule that each node obeys is a function of its connectivity k this is done by using only two logical functions and and xor which are controlled by a parameter q using a damage spreading technique we show that the hamming distance and the number of 1 s exhibit power law behavior as a function of q the exponents appearing in the power laws depend on the value of ensuremath gamma

nested canalyzing unate cascade and polynomial functions this paper focuses on the study of certain classes of boolean functions that have appeared in several different contexts nested canalyzing functions have been studied recently in the context of boolean network models of gene regulatory networks in the same context polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks finally unate cascade functions have been studied in the design of logic circuits and binary decision diagrams this paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions furthermore it provides a description of nested canalyzing functions as a certain type of boolean polynomial function using the polynomial framework one can show that the class of nested canalyzing functions or equivalently the class of unate cascade functions forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra as a corollary of the functional equivalence derived here a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions

lyapunov exponents in random boolean networks a new order parameter approximation to random boolean networks rbn is introduced based on the concept of boolean derivative a statistical argument involving an annealed approximation is used allowing to measure the order parameter in terms of the statistical properties of a random matrix using the same formalism a lyapunov exponent is calculated allowing to provide the onset of damage spreading through the network and how sensitive it is to minimal perturbations finally the lyapunov exponents are obtained by means of dierent approximations through distance method and a discrete variant of the wolf s method for continuous systems c 2000 elsevier science b v all rights reserved

convergence behavior and n roots of stack filters the convergence behavior of two types of stack filters is investigated both types are shown to possess the convergence property and to exhibit nontrivial behavior the first type of stack filter has the erosive property it erodes any input signal to a root after a sufficient number of passes the second type of stack filter has the dilative property it dilates any input signal to a root after a sufficient number of passes for each type of stack filter an algorithm is presented which can determine a filter that has any specific signal or set of signals as roots these two algorithms are efficient in that their execution time is a linear function of the length of the input signal the width of the filter window and the number of signals to be preserved since some stack filters have the phenomenon of oscillations when they filter some input signals successively a partial ordering is defined over the set of stack filters which makes it possible to determine upper and lower bounds for these oscillations

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

numerical shakedown analysis of damaged structures a numerical method is proposed to predict the failure of mechanical structures it is based on the generalization of the classical static shakedown theorem to damaged inelastic structures ductile plastic damage is taken into account by using the models of lemaitre and shichun hua the obtained results are compared in the special case of limit analysis to those obtained by incremental methods

lower bound limit analysis by using the efg method and non linear programming intended to avoid the complicated computations of elasto plastic incremental analysis limit analysis is an appealing direct method for determining the load carrying capacity of structures on the basis of the static limit analysis theorem a solution procedure for lower bound limit analysis is presented firstly making use of the element free galerkin efg method rather than traditional numerical methods such as the finite element method and boundary element method the numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration the reduced basis technique is adopted to solve the mathematical programming iteratively in a sequence of reduced self equilibrium stress subspaces with very low dimensions the self equilibrium stress field is expressed by a linear combination of several self equilibrium stress basis vectors with parameters to be determined these self equilibrium stress basis vectors are generated by performing an equilibrium iteration procedure during elasto plastic incremental analysis the complex method is used to solve these non linear programming sub problems and determine the maximal load amplifier numerical examples show that it is feasible and effective to solve the problems of limit analysis by using the efg method and non linear programming copyright 2007 john wiley sons ltd

order or chaos in boolean gene networks depends on the mean fraction of canalizing functions we explore the connection between order chaos in boolean networks and the naturally occurring fraction of canalizing functions in such systems this fraction turns out to give a very clear indicati

optimal energy conserving discontinuous galerkin methods for linear symmetric hyperbolic systems abstract we propose energy conserving discontinuous galerkin dg methods for symmetric linear hyperbolic systems on general unstructured meshes optimal a priori error estimates of order k 1 are obtained for the semi discrete scheme in one dimension and in multi dimensions on cartesian meshes when tensor product polynomials of degree k are used a high order energy conserving lax wendroff time discretization is also presented extensive numerical results in one dimension and two dimensions on both rectangular and triangular meshes are presented to support the theoretical findings and to assess the new methods one particular method with the doubling of unknowns is found to be optimally convergent on triangular meshes for all the examples considered in this paper the method is also compared with the classical dissipative upwinding dg method and conservative dg method with a central flux it is numerically observed for the new method to have a superior performance for long time simulations

the classification and associative memory capability of stack filters the notion of the on set of a positive boolean function is used to classify stack filters into three different types called decreasing increasing and mixed the associative memory capability of each of these three types of stack filters is then investigated the associative memory of a stack filter is defined to be the set of root signals of that filter in a class of stack filters in which each filter s root set contains a desired set of patterns those filters whose root sets have the smallest cardinality are said to be minimal among all filters in that class for that set of patterns some learning schemes are proposed to find minimal decreasing and increasing stack filters it is also shown that for any specified set of patterns there is always a mixed stack filter which is minimal when one considers all stack filters which preserve those patterns in this sense mixed stack filters are always at least as good as decreasing or increasing stack filters

stratification and enumeration of boolean functions by canalizing depth abstract boolean network models have gained popularity in computational systems biology over the last dozen years many of these networks use canalizing boolean functions which has led to increased interest in the study of these functions the canalizing depth of a function describes how many canalizing variables can be recursively picked off until a non canalizing function remains in this paper we show how every boolean function has a unique algebraic form involving extended monomial layers and a well defined core polynomial this generalizes recent work on the algebraic structure of nested canalizing functions and it yields a stratification of all boolean functions by their canalizing depth as a result we obtain closed formulas for the number of n variable boolean functions with depth k which simultaneously generalizes enumeration formulas for canalizing and nested canalizing functions

superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids in this paper we present a superconvergence result for the local discontinuous galerkin ldg method for a model elliptic problem on cartesian grids we identify a special numerical flux for which the l2 norm of the gradient and the l2 norm of the potential are of orders k 1 2 and k 1 respectively when tensor product polynomials of degree at most k are used for arbitrary meshes this special ldg method gives only the orders of convergence of k and k 1 2 respectively we present a series of numerical examples which establish the sharpness of our theoretical results

on spectral techniques in analysis of boolean networks abstract in this work we present results that can be used for analysis of boolean networks the results utilize fourier spectra of the functions in the network an accurate formula is given for derrida plots of networks of finite size n based on a result on boolean functions presented in another context derrida plots are widely used to examine the stability issues of boolean networks for the limit n we give a computationally simple form that can be used as a good approximation for rather small networks as well a formula for derrida plots of random boolean networks rbns presented earlier in the literature is given an alternative derivation it is shown that the information contained in the derrida plot is equal to the average fourier spectrum of the functions in the network in the case of random networks the mean derrida plot can be obtained from the mean spectrum of the functions the method is applied to real data by using the boolean functions found in genetic regulatory networks of eukaryotic cells in an earlier study conventionally derrida plots and stability analysis have been computed with statistical sampling resulting in poorer accuracy

a nonlinear programming approach to kinematic shakedown analysis of frictional materials abstract this paper develops a novel nonlinear numerical method to perform shakedown analysis of structures subjected to variable loads by means of nonlinear programming techniques and the displacement based finite element method the analysis is based on a general yield function which can take the form of most soil yield criteria e g the mohr coulomb or drucker prager criterion using an associated flow rule a general yield criterion can be directly introduced into the kinematic theorem of shakedown analysis without linearization the plastic dissipation power can then be expressed in terms of the kinematically admissible velocity and a nonlinear formulation is obtained by means of nonlinear mathematical programming techniques and the finite element method a numerical model for kinematic shakedown analysis is developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints the objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load can be calculated an effective direct iterative algorithm is then proposed to solve the resulting nonlinear programming problem the calculation is based on the kinematically admissible velocity with one step calculation of the elastic stress field only a small number of equality constraints are introduced and the computational effort is very modest the effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples

the number of multistate nested canalyzing functions abstract identifying features of molecular regulatory networks is an important problem in systems biology it has been shown that the combinatorial logic of such networks can be captured in many cases by special functions called nested canalyzing in the context of discrete dynamic network models it was also shown that the dynamics of networks constructed from such functions has very special properties that are consistent with what is known about molecular networks and that simplify analysis it is important to know how restrictive this class of functions is for instance for the purpose of network reverse engineering this paper contains a formula for the number of such functions and a comparison to the class of all functions in particular it is shown that as the number of variables becomes large the ratio of the number of nested canalyzing functions to the number of all functions converges to zero this shows that the class of nested canalyzing functions is indeed very restrictive the principal tool used for this investigation is a description of these functions as polynomials and a parametrization of the class of all such polynomials in terms of relations on their coefficients this parametrization can also be used for the purpose of network reverse engineering using only nested canalyzing functions

the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function the paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms this study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms available at present the purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail in part 1 the exactl 1 penalty function of han and powell is replaced by a differentiable augmented lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution in part 2 the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems

optimal non dissipative discontinuous galerkin methods for maxwells equations in drude metamaterials simulation of electromagnetic wave propagation in metamaterials leads to more complicated time domain maxwells equations than the standard maxwells equations in free space in this paper we develop and analyze a non dissipative discontinuous galerkin dg method for solving the maxwells equations in drude metamaterials previous discontinuous galerkin methods in the literature for electromagnetic wave propagation in metamaterials were either non dissipative but sub optimal or dissipative and optimal our method uses a different and simple choice of numerical fluxes achieving provable non dissipative stability and optimal error estimates simultaneously we prove the stability and optimal error estimates for both semi and fully discrete dg schemes with the leap frog time discretization for the fully discrete case numerical results are given to demonstrate that the dg method can solve metamaterial maxwells equations effectively

lower bound shakedown analysis by using the element free galerkin method and non linear programming abstract shakedown analysis is a powerful tool for assessing the safety of structures under variable repeated loads by using the element free galerkin efg method and non linear programming a novel numerical solution procedure is developed to perform lower bound shakedown analysis of structures made up of elasto perfectly plastic material the numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration the reduced basis technique is adopted here to solve the mathematical programming iteratively in a sequence of reduced self equilibrium stress subspaces with very low dimensions the self equilibrium stress field is expressed by linear combination of several self equilibrium stress basis vectors with parameters to be determined these self equilibrium stress basis vectors are generated by performing an equilibrium iteration procedure during elasto plastic incremental analysis the complex method is used to solve the non linear programming and determine the lower bound of shakedown load the proposed numerical method is verified by using several numerical examples and the results show good agreement with other available solutions

canalization and the stability of nk kauffman networks abstract boolean variables are such that they take values on z 2 0 1 nk kauffman networks are dynamical deterministic systems of n boolean functions that depend only on k n boolean variables they were proposed by kauffman as a first step to understand cellular behavior kauffman 1969 with great success among the problems that still have not been well understood in kauffman networks is the mechanism that regulates the phase transition of the system from an ordered phase where small changes of the initial state decay to a chaotic one where they grow exponentially i show that this mechanism is regulated through the irreducible decomposition of boolean functions proposed in zertuche 2009 this is in contrast to previous knowledge that attributed it to canalization i also review other statistical properties of kauffman networks that boolean irreducibility explains

discontinuous galerkin methods for maxwell s equations in drude metamaterials on unstructured meshes abstract in this follow up work we extend the discontinuous galerkin dg methods previously developed on rectangular meshes li et al 2017 to triangular meshes the dg schemes in li et al 2017 are both optimally convergent and energy conserving however as we shall see in the numerical results section the dg schemes on triangular meshes only have suboptimal convergence rate we prove the energy conservation and an error estimate for the semi discrete schemes the stability of the fully discrete scheme is proved and its error estimate is stated we present extensive numerical results with convergence consistent of our error estimate and simulations of wave propagation in drude metamaterials to demonstrate the flexibility of triangular meshes

the number and probability of canalizing functions canalizing functions have important applications in physics and biology for example they represent a mechanism capable of stabilizing chaotic behavior in boolean network models of discrete dynamical systems when comparing the class of canalizing functions to other classes of functions with respect to their evolutionary plausibility as emergent control rules in genetic regulatory systems it is informative to know the number of canalizing functions with a given number of input variables this is also important in the context of using the class of canalizing functions as a constraint during the inference of genetic networks from gene expression data to this end we derive an exact formula for the number of canalizing boolean functions of n variables we also derive a formula for the probability that a random boolean function is canalizing for any given bias p of taking the value 1 in addition we consider the number and probability of boolean functions that are canalizing for exactly k variables finally we provide an algorithm for randomly generating canalizing functions with a given bias p and any number of variables which is needed for monte carlo simulations of boolean networks

reversible boolean networks ii phase transitions oscillations and local structures we continue our consideration of a class of models describing the reversible dynamics of n boolean variables each with k inputs we investigate in detail the behavior of the hamming distance as well as of the distribution of orbit lengths as n and k are varied we present numerical evidence for a phase transition in the behavior of the hamming distance at a critical value kc 1 62 and also an analytic theory that yields the exact bounds 1 5 k c 2 we also discuss the large oscillations that we observe in the hamming distance for k k c as a function of time as well as in the distribution of cycle lengths as a function of cycle length for moderate k both greater than and less than kc we propose that local structures or subsets of spins whose dynamics are not fully coupled to the other spins in the system play a crucial role in generating these oscillations the simplest of these structures are linear chains called linkages and rings called circuits we discuss the properties of the linkages in some detail and sketch the properties of circuits we argue that the observed oscillation phenomena can be largely understood in terms of these local structures 2001 published by elsevier science b v

on forcing functions in kauffman s random boolean networks the phase transition between frozen and chaotic behavior in kauffman s cellular automata on a nearest neighbor square lattice does not agree with the percolation threshold of the forcing functions

optimal energy conserving local discontinuous galerkin methods for second order wave equation in heterogeneous media abstract solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of applications in physics and engineering the design of numerical methods for such general wave propagation problems is challenging because the energy conserving property has to be incorporated in the numerical algorithms in order to minimize the phase or shape errors after long time integration in this paper we focus on multi dimensional wave problems and consider linear second order wave equation in heterogeneous media we develop and analyze an ldg method in which numerical fluxes are carefully designed to maintain the energy conserving property and accuracy compatible high order energy conserving time integrators are also proposed the optimal error estimates and the energy conserving property are proved for the semi discrete methods our numerical experiments demonstrate optimal rates of convergence and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach

the p version of the finite element method in incremental elasto plastic analysis summary whereas the higher order versions of the finite element method p and hp versions are fairly well established as highly efficient methods for monitoring and controlling the discretization error in linear problems little has been done to exploit their benefits in elasto plastic structural analysis in this paper we discuss which aspects of incremental elasto plastic finite element analysis are particularly amenable to improvements by the p version these theoretical considerations are supported by several numerical experiments first we study an example for which an analytical solution is available it is demonstrated that the p version performs very well even in cycles of elasto plastic loading and unloading not only as compared with the traditional h version but also with respect to the exact solution finally an example of considerable practical importancethe analysis of a cold working lug is presented which demonstrates how the modelling tools offered by higher order finite element techniques can contribute to an improved approximation of practical problems

boolean nested canalizing functions a comprehensive analysis abstract boolean network models of molecular regulatory networks have been used successfully in computational systems biology the boolean functions that appear in published models tend to have special properties in particular the property of being nested canalizing a concept inspired by the concept of canalization in evolutionary biology it has been shown that networks comprised of nested canalizing functions have dynamic properties that make them suitable for modeling molecular regulatory networks namely a small number of large attractors as well as relatively short limit cycles this paper contains a detailed analysis of this class of functions based on a novel normal form as polynomial functions over the boolean field the concept of layer is introduced that stratifies variables into different classes depending on their level of dominance using this layer concept a closed form formula is derived for the number of nested canalizing functions with a given number of variables additional metrics considered include hamming weight the activity number of any variable and the average sensitivity of the function it is also shown that the average sensitivity of any nested canalizing function is between 0 and 2 this provides a rationale for why nested canalizing functions are stable since a random boolean function in n variables has average sensitivity n 2 the paper also contains experimental evidence that the layer number is an important factor in network stability

a non linear programming approach to kinematic shakedown analysis of composite materials using a representative volume element rve to represent the microstructure of periodic composite materials this paper develops a non linear numerical technique to calculate the macroscopic shakedown domains of composites subjected to cyclic loads the shakedown analysis is performed using homogenization theory and the displacement based finite element method with the aid of homogenization theory the classical kinematic shakedown theorem is generalized to incorporate the microstructure of composites using an associated flow rule the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity by means of non linear mathematical programming techniques a finite element formulation of kinematic shakedown analysis is then developed leading to a non linear mathematical programming problem subject to only a small number of equality constraints the objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a composite is then obtained an effective direct iterative algorithm is proposed to solve the non linear programming problem the effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples this can serve as a useful numerical tool for developing engineering design methods involving composite materials copyright 2005 john wiley sons ltd