consistent tangent matrices for substepping schemes a very simple and general expression of the consistent tangent matrix for substepping time integration schemes is presented if needed the derivatives required for the computation of the consistent tangent moduli can be obtained via numerical differentiation these two strategies substepping and numerical differentiation lead to quadratic convergence in complex nonlinear inelasticity problems

coreness of cooperative games with truncated submodular profit functions coreness represents solution concepts related to core in cooperative games which captures the stability of players motivated by the scale effect in social networks economics and other scenario we study the coreness of cooperative game with truncated submodular profit functions specifically the profit function f cdot is defined by a truncation of a submodular function sigma cdot f cdot sigma cdot if sigma cdot ge eta and f cdot 0 otherwise where eta is a given threshold in this paper we study the core and three core related concepts of truncated submodular profit cooperative game we first prove that whether core is empty can be decided in polynomial time and an allocation in core also can be found in polynomial time when core is not empty when core is empty we show hardness results and approximation algorithms for computing other core related concepts including relative least core value absolute least core value and least average dissatisfaction value

exact integration of constitutive equations in elasto plasticity a unified approach is presented for establishing exact integration of the constitutive equations in elastoplasticity assuming the total strain rate direction to be constant this unified approach includes all previous exact integration procedures as special cases and in addition some new closed form solutions are derived for combined kinematic and isotropic hardening special emphasis is laid on combined kinematic and isotropic hardening for von mises material and on isotropic hardening for mohr coulomb and tresca materials

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

on the implementation of inelastic constitutive equations with special reference to large deformation problems abstract methods for finite element analysis of inelastic problems are reviewed problems with currently popular methods based on rate principles of virtual work are pointed out a formulation is presented based on the exact incremental virtual work equation the procedure for integration of the constitutive rate equations to obtain incremental constitutive equations is based on the assumption of a constant strain rate or straight strain path during the increment full newton raphson iteration is used to solve the resulting set of nonlinear equations the treatment of the constitutive equations in this newton raphson scheme leads to a linearized form which is substantially different from the classical rate equations the procedure is implemented in a finite element program and its effectiveness is shown in a number of examples

the numerical evaluation of very oscillatory infinite integrals by extrapolation recently the author has given two modifications of a nonlinear extrapolation method due to levin and sidi which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity in this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity the new method is applied to a number of complicated integrals among them the solution to a problem in viscoelasticity some convergence results for this method are presented

two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series two new classes of nonlinear transformations the d transformation to accelerate the convergence of infinite integrals and the d transformation to accelerate the convergence of infinite series are presented in the course of the development of these transformations two interesting asymptotic expansions one for infinite integrals and the other for infinite series are derived the transformations d and d can easily be applied to infinite integrals 0f t dt whose integrands f t satisfy linear differential equations of the form f t s m k 1p k t f k t and to infinite series s r 1f r w terms f r satisfy a linear difference equation of the form f r s m k 1p k r d kf r such that in both cases the p k have asymptotic expansions in inverse powers of their arguments in order to be able to apply these transformations successfully one need not know explicitly the differential equation that the integrand satisfies or the difference equation that the terms of the series satisfy mere knowledge of the existence of such a differential or difference equation and its order m is enough this broadens the areas to which these methods can be applied the connection between the d and d transformations with some known transformations in shown the use and the remarkable efficiency of the d and d transformations are demonstrated through several numerical examples the computational aspects of these transformations are described in detail

iterative lcp solvers for non local loading unloading conditions this paper deals with the finite element analysis of a certain class of non local dissipative constitutive models where the canonical pointwise backward euler scheme cannot be employed for satisfying the loading unloading conditions in the presence of a non local dissipation the admissibility conditions in a point depend on the inelastic strain increment of the surrounding points and can be cast as a linear complementarity problem lcp involving all gauss points of the process zone in order to actually solve the lcp the use of iterative algorithms that can be easily embodied into existing fe codes is discussed the performance of the proposed algorithms is tested in 1d and 2d examples for both elastoplastic and damaging materials copyright 2003 john wiley sons ltd

integration of tresca and mohr coulomb constitutive relations in plane strain elastoplasticity summary this paper considers the problem of integrating the constitutive relations for tresca and mohr coulomb materials under conditions of plane strain in the case of a tresca material we show that the constitutive law may be integrated exactly by assuming the strain rates de dt to be constant we also derive a semi analytic method for integrating both types of constitutive law which assumes that the quantities d dl are constant this approach is motivated by the fact that the exact variation of the strains during a time interval is unknown and leads to a single non linear equation in a which can be solved efficiently to yield the unknown stresses finally we compare the results from the analytic and semi analytic methods with those from a variety of numerical integration schemes

a sub stepping scheme for multi scale analysis of solids abstract this paper proposes a sub stepping procedure aimed at saving overall computing time in the finite element solution of non linear solid mechanics problems where homogenization based multi scale constitutive models are used to describe the material response the procedure obtains better initial guesses for the newton raphson scheme adopted in the solution of the incremental equilibrium problem of the representative volume element rve associated with each gauss quadrature point of the discretised macroscopic continuum the resulting gain in robustness of the rve newton raphson iteration allows larger time load steps to be prescribed at the macroscopic scale leading to substantial savings in computing time the proposed sub stepping preserves the quadratic rate of asymptotic convergence of the macroscopic scale newton raphson scheme without compromising the accuracy of the solution numerical examples demonstrate that the achieved improvement in efficiency is particularly noticeable under the assumption of finite stains where overall speed up factors of up to four are observed

an automatic load stepping algorithm with error control summary this paper presents an algorithm for controlling the error in non linear finite element analysis which is caused by the use of finite load steps in contrast to most recent schemes the proposed technique is non iterative and treats the governing load deflection relations as a system of ordinary differential equations this permits the governing equations to be integrated adaptively where the step size is controlled by monitoring the local truncation error the latter is measured by computing the difference between two estimates of the displacement increments for each load step with the initial estimate being found from the first order euler scheme and the improved estimate being found from the second order modified euler scheme if the local truncation error exceeds a specified tolerance then the load step is abandoned and the integration is repeated with a smaller load step whose size is found by local extrapolation local extrapolation is also used to predict the size of the next load step following a successful update in order to control not only the local load path error but also the global load path error the proposed scheme incorporates a correction for the unbalanced forces overall the cost of the automatic error control is modest since it requires only one additional equation solution for each successful load step because the solution scheme is non iterative and founded on successful techniques for integrating systems of ordinary differential equations it is particularly robust to illustrate the ability of the scheme to constrain the load path error to lie near a desired tolerance detailed results are presented for a variety of elastoplastic boundary value problems

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

limit analysis with the dual affine scaling algorithm the collapse state of a rigid plastic material with the linearized mises yield condition is computed we use an infeasible point variant of the dual affine scaling algorithm for linear programming which is extremely efficient for this large sparse and ill conditioned problem for a classical test problem we obtain better results than previously both for the limit load and for the collapse fields thus setting a new standard for computations in limit analysis

stress update algorithm for elastoplastic models with nonconvex yield surfaces a stress update algorithm for elastoplastic models with nonconvex yield surfaces is presented an explicit algorithm based on the runge kutta embedded method of second order accuracy is developed the crossing of the yield surface is properly taken in to account by means of a robust intersection finding algorithm this algorithm is based on a simple multiple root finding technique which requires the yield function evaluated along a given secant stress path to be continuously differentiable to the second order the accuracy of the intersection finding algorithm is illustrated through examples using simple yield functions similar to the ones adopted in models for unsaturated soils and modified cam clay model yield surface with nonconvex argyris et al failure criterion isoerror surfaces and finite element simulations are used to investigate the accuracy and efficiency of the stress update algorithm it is observed that although the algorithm for nonconvex surfaces is slower than its equivalent for convex surfaces the accuracy can be controlled locally by means of specified tolerances copyright c 2008 john wiley sons ltd

implicit integration procedure for viscoplastic gurson materials abstract constitutive relations and numerical integration algorithms for elastic viscoplastic ductile materials are investigated the gurson yield function is considered together with a coalescence criterion from thomason the constitutive equations are reformulated using the assumption of constant strain rate within an increment a consistency condition is incorporated into the traditional viscoplastic rate format and the tangential material operator expression is derived the algorithm is used to investigate the influence of triaxiality in the critical coalescence values of porosity

remarks on the interpretation of current non linear finite element analyses as differential algebraic equations for the numerical solution of materially non linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point wise defined evolution equations characterizing the material behaviour the interpretation of such systems as differential algebraic equations dae allows modern day integration algorithms from numerical mathematics to be efficiently applied especially the application of diagonally implicit runge kutta methods dirk together with a multilevel newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward euler schemes for the local time integration moreover the notion of the consistent tangent operator becomes more obvious in this context the quadratical order of convergence of the multilevel newton algorithm is usually validated by numerical studies however an analytical proof of this second order convergence has already been given by authors in the field of non linear electrical networks we show that this proof can be applied in the current context based on the dae interpretation mentioned above we finally compare the proposed procedure to several well known stress algorithms and show that the inclusion of a step size control based on local error estimations merely requires a small extra time investment copyright 2001 john wiley sons ltd

upper bound limit analysis using discontinuous velocity fields abstract a new method for computing rigorous upper bounds under plane strain conditions is described it is based on a linear three noded triangular element which has six unknown nodal velocities and a fixed number of unknown plastic multiplier rates and uses the kinematic theorem to define a kinematically admissible velocity field as the solution of a linear programming problem unlike existing formulations which permit only a limited number of velocity discontinuities whose directions of shearing must be specified a priori the new formulation permits velocity discontinuities at all edges shared by adjacent triangles and the directions of shearing are found automatically the variation of the velocity jump along each discontinuity is described by an additional set of four unknowns all of the unknowns are subject to the constraints imposed by an associated flow rule and the velocity boundary conditions the objective function corresponds to the dissipated power or some related load parameter of interest and is minimised to yield the desired upper bound since plastic deformation may occur not only in the discontinuities but also throughout the triangular elements as well the method is capable of modelling complex velocity fields accurately and typically produces tight upper bounds on the true limit load the formulation is applicable to materials whose strength is cohesive frictional purely cohesive and uniform or purely cohesive and linearly varying and thus quite general the new procedure is very efficient and always requires fewer elements than existing methods to obtain useful upper bound solutions moreover because of the extra degrees of freedom introduced by the discontinuities the linear elements no longer need to be arranged in a special pattern to model incompressible behaviour accurately

numerical technique in plasticity including solution advancement control numerical techniques applied to the consistent formulation of plasticity which is based on convex analysis are investigated for each time step the stress is found as the projection in complementary energy of the elastic stress onto the set of plastically admissible stresses while the velocity field is the extremal of a non quadratic functional explicit formulas for von mises yield criterion with mixed hardening are developed and the nonlinear equations arising from finite element discretization are solved for comparison by a number of newton type iteration procedures with line search and are length control a few numerical examples with proportional and non proportional loading are analyzed

submodularity of influence in social networks from local to global social networks are often represented as directed graphs where the nodes are individuals and the edges indicate a form of social relationship a simple way to model the diffusion of ideas innovative behavior or word of mouth effects on such a graph is to consider an increasing process of infected or active nodes each node becomes infected once an activation function of the set of its infected neighbors crosses a certain threshold value such a model was introduced by kempe kleinberg and tardos kkt in maximizing the spread of influence through a social network in proceedings of the 9th acm sigkdd international conference on knowledge discovery and data mining 2003 pp 137 146 and influential nodes in a diffusion model for social networks in proceedings of the 32nd international colloquium on automata languages and programming icalp 2005 where the authors also impose several natural assumptions the threshold values are random and the activation functions are monotone and submodular the monotonicity condition indicates that a node is more likely to become active if more of its neighbors are active while the submodularity condition indicates that the marginal effect of each neighbor is decreasing when the set of active neighbors increases for an initial set of active nodes s let sigma s denote the expected number of active nodes at termination here we prove a conjecture of kkt we show that the function sigma s is submodular under the assumptions above we prove the same result for the expected value of any monotone submodular function of the set of active nodes at termination roughly our results demonstrate that local submodularity is preserved globally under this diffusion process this is of natural computational interest as many optimization problems have good approximation algorithms for submodular functions

a computation of fermi dirac integrals by asymptotics and the hermite corrector formula we present a computation of fermi dirac integrals based on asymptotic expansions and the hermite corrector formula

slope stability analysis based on elasto plastic finite element method summary the paper deals with two essential and related closely processes involved in the finite element slope stability analysis in two dimensional problems i e computation of the factors of safety fos and location of the critical slide surfaces a so called inequality sin 1 2 is proved for any elasto plastic material satisfying mohr coulomb s yield criterion in order to obtain an fos of high precision with less calculation and a proper distribution of plastic zones in the critical equilibrium state it is stated that the poisson s ratio should be adjusted according to the principle that the inequality always holds as reducing the strength parameters c and while locating the critical slide surface represented by the critical slide line csl under the plane strain condition an initial value problem of a system of ordinary differential equations defining the csl is formulated a robust numerical solution for the initial value problem based on the predictor corrector method is given in conjunction with the necessary and sufficient condition ensuring the convergence of solution a simple example the kinematic solution of which is available and a challenging example from a hydraulic project in construction are analysed to demonstrate the effectiveness of the proposed procedures copyright 2005 john wiley sons ltd

ring fragmentation predictions using the gurson model with material stability conditions as failure criteria the gurson constitutive model used in combination with failure criteria based on material stability and bifurcation has been used in a finite element model to predict fracture and fragmentation in a dynamic expanding ring experiment integration of the constitutive relation is through a two level iteration scheme operating only on scalar equations it provides a robust solution for the large strain increments expected in shock problems numerically efficient failure criteria based on a bifurcation analysis and drucker s condition for material stability are evaluated at each integration point and at every time step the models were used to predict the fracture strain and number of fragments for thin electromagnetically loaded expanding rings the results show good predictions of time to failure predictions of the number of fragments are generally good with the model based on drucker s hypothesis being better at the higher strain rates

on steplength algorithms for a class of continuation methods the continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form fx b f d subset r n 1 to r n for given b in r n while these methods are similar in structure to those used for ode solvers their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step this suggests the use of a posteriors estimates of the convergence radii of the corrector it is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about f however it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors this is used for the design of various step algorithms for continuation processes two of them are based on a newton corrector while the third o

complexity of equilibrium in competitive diffusion games on social networks in this paper we consider the competitive diffusion game and study the existence of its pure strategy nash equilibrium when defined over general undirected networks we first determine the set of pure strategy nash equilibria for two special but well known classes of networks namely the lattice and the hypercube characterizing the utility of the players in terms of graphical distances of their initial seed placements to other nodes in the network we show that in general networks the decision process on the existence of pure strategy nash equilibrium is an np hard problem following this we provide some necessary conditions for a given profile to be a nash equilibrium furthermore we study players utilities in the competitive diffusion game over erdos renyi random graphs and show that as the size of the network grows the utilities of the players are highly concentrated around their expectation and are bounded below by some threshold based on the parameters of the network finally we obtain a lower bound for the maximum social welfare of the game with two players and study sub modularity of the players utilities

maximizing stochastic monotone submodular functions we study the problem of maximizing a stochastic monotone submodular function with respect to a matroid constraint because of the presence of diminishing marginal values in real world problems our model can capture the effect of stochasticity in a wide range of applications we show that the adaptivity gap the ratio between the values of optimal adaptive and optimal nonadaptive policies is bounded and is equal to e e 1 we propose a polynomial time nonadaptive policy that achieves this bound we also present an adaptive myopic policy that obtains at least half of the optimal value furthermore when the matroid is uniform the myopic policy achieves the optimal approximation ratio of 1 1 e this paper was accepted by dimitris bertsimas and yinyu ye optimization

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

an evaluation of substepping and implicit stress point algorithms abstract elasto plastic finite element analyses of geotechnical boundary value problems involve the use of a non linear solution strategy a popular and efficient strategy is the modified newton raphson approach of which a key element is the integration of the constitutive laws over a finite strain step two algorithms for performing such an integration are the substepping and return methods in this paper the basic assumptions behind these methods are described and the associated errors compared this comparative study is undertaken both at a fundamental level and when applied to a typical boundary value problem the errors associated with both methods are increment size dependent it is shown in the problems considered that for a particular increment size the substepping method is the more accurate

a numerical evaluation of very oscillatory infinite integrals by asymptotics recently levin and sidi have given some nonlinear methods for the accurate evaluation of slowly converging infinite integrals but one needs to solve some very instable linear systems when for example the integrand have an increasingly rapid oscillatory behavior at infinity in this work we present a direct method which avoids the instable linear systems

on the consistency of viscoplastic formulations a viscoplastic consistency condition is incorporated into the traditional viscoplastic rate format with two objectives in mind i to develop a tangential material operator for rate sensitive behavior similarly to rate independent elastoplasticity and ii to make an analytical reference solution available for evaluating the accuracy of new and wellestablished computational strategies to integrate the viscoplastic evolution equations the viscoplastic tangent operator provides the missing link between rate independent plasticity and rate dependent viscoplasticity it also furnishes the acoustic tensor which is required for localization analysis of discontinuous failure besides the formulation of the viscoplastic tangent operator we present an analytical reference solution for perfect j2 viscoplasticity this analytical result is subsequently used to determine the accuracy of simplifying assumptions behind the algebraic evaluation of the viscoplastic multiplier and to quantify the error of the radial return mapping strategy for viscoplastic computations 2000 elsevier science ltd all rights reserved

return mapping for nonsmooth and multiscale elastoplasticity we present a semi implicit return mapping algorithm for integrating generic nonsmooth elastoplastic models the semi implicit nature of the algorithm stems from freezing the plastic internal variables at their previous state followed by implicitly integrating the stresses and plastic multiplier the plastic internal variables are incrementally updated once convergence is achieved a posteriori locally the algorithm behaves as a classic return mapping for perfect plasticity and hence inherits the stability of implicit integrators however it differs from purely implicit integrators by keeping the plastic internal variables locally constant this feature affords the method the ability to integrate nonsmooth c0 evolution laws that may not be integrable using implicit methods as a result we propose and use the algorithm as the backbone of a semi concurrent multiscale framework in which nonsmooth constitutive relationships can be directly extracted from the underlying micromechanical processes and faithfully incorporated into elastoplastic continuum models though accuracy of the proposed algorithm is step size dependent its simplicity and its remarkable ability to handle nonsmooth relations make the method promising and computationally appealing

a rigorous bound on error in backward difference elastoplastic time integration abstract the finite element analysis of elastoplastic structures requires in general a time stepping procedure and in most cases the integration of the constitutive law within each time step has to be carried out by numerical integration the error associated to this numerical integration depends on the degree of non linearity of the structural response and can be used as an indicator for the adaptive definition of the time step size based on martin s and ortiz theorem on minimum total work a simple estimate of the integration error associated to a backward difference scheme for elastoplastic models is derived it is shown that the proposed estimate is a rigorous upper bound on the error in the case of assigned constant strain rate finally a simple strategy for the automatic definition of the time step size is proposed the estimator and the adaptive strategy are validated by application to problems with a perfectly plastic material model

an interior point algorithm for elastoplasticity the problem of small deformation rate independent elastoplasticity is treated using convex programming theory and algorithms a finite step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the lagrangian associated with a convex mathematical program next an algorithm based on the classical primal dual interior point method is developed several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem the resulting method is compared to state of the art elastoplastic procedures for which both similarities and differences are found finally a number of examples are solved demonstrating the capabilities of the algorithm when applied to standard perfect plasticity hardening multisurface plasticity and problems involving softening copyright 2006 john wiley sons ltd

on the formulation of closest point projection algorithms in elastoplasticity part i the variational structure we present in this paper the characterization of the variational structure behind the discrete equations defining the closest point projection approximation in elastoplasticity rate independent and viscoplastic formulations are considered in the infinitesimal and the finite deformation range the later in the context of isotropic finite strain multiplicative plasticity primal variational principles in terms of the stresses and stress like hardening variables are presented first followed by the formulation of dual principles incorporating explicitly the plastic multiplier augmented lagrangian extensions are also presented allowing a complete regularization of the problem in the constrained rate independent limit the variational structure identified in this paper leads to the proper framework for the development of new improved numerical algorithms for the integration of the local constitutive equations of plasticity as it is undertaken in part ii of this work

comparison of different implementation algorithms for hiss constitutive models in fem relative accuracy stability and efficiency of four popular algorithms to implement elasto plastic constitutive models in non linear finite element procedures have been compared elastic predictor plastic corrector ep pc method plastic predictor plastic corrector pp pc method implicit integration implicit method and modified euler me method wereused to implement hierarchical single surface hiss 0 model into the general purpose finite element program abaqus first these algorithms were used outside the finite element program to simulate various triaxial stress paths after that they were used in abaqus to simulate two strip footings a displacement controlled rigid footing and b load controlled flexible footing the effect of the sub strain increment size on the accuracy stability and efficiency of each algorithm during these simulations have been compared implicit method and me performed well for some problems but performed poorly on others ep pc and pp pc methods performed equally well for all the problems this also showed the importance of testing algorithms under various stress paths and boundary value problems to assess their relative performance

formulation of an implicit algorithm for finite deformation viscoplasticity abstract within the framework of additive plasticity an objective stress update algorithm has been proposed the procedure is implemented in such way that the extension from a standard small strain fe code to the finite strain range is straightforward and objectivity can be retained for any choice of the intermediate configuration the additional computational cost only includes some geometrical manipulations for the newton raphson iteration method a closed form solution of the consistent tangent is derived by direct linearization of the stress update algorithm numerical examples show a quadratic rate of convergence with the proposed viscoplastic model the analysis of a tensile test first shows a shear band with a finite thickness independent of the finite element size at large deformation the shear band pattern transforms into a necking failure mode as a second example a thin sheet tensile test is analyzed a necking failure mode leads to global softening even though locally the material is still hardening although viscoplastic regularization is used the results still show a mesh dependence for deformations over 40

a robust and efficient substepping scheme for the explicit numerical integration of a rate dependent crystal plasticity model the hood doors and other panels of a car can be made by aluminium sheets through complex forming operations design of forming processes requires trustable models and simulations forming properties are different in different directions there are two principle ways to describe this plastic anisotropy of metals the first one is the continuum theories for materials the flow properties in various directions are described by the yield function it is a parametric function calibrated by experimental tests unfortunately only a few mechanical tests can be directly used the second approach takes into account that metals consist of many small crystals the properties of each crystal are determined by crystal planes but crystals are rotated in many directions the crystal models either resolve all the details of thousands of crystals in a small volume element requiring huge computer capacity or simplified statistical models can be applied in both cases the models are much more demanding than continuum models in practice these models can be used to perform virtual experiments for calibration of continuum models in his phd work kai zhang kai zhang ntnu no implemented and tested both type of models more efficient numerical algorithms for computer simulations were developed and programmed for the detailed crystal plasticity approach three different cases were investigated experimentally in one of the cases it was found that even for the detailed computer implementation the predictions could not replace experiments for design purposes a combination of real and virtual experiments was recommended for calibration for the other two cases the virtual experiments could be trusted it was found that the best statistical models that are more than thousand times faster can replace detailed simulations of the crystal structure with high accuracy

finite element implementation of non linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control implicit stress integration algorithms have been demonstrated to provide a robust formulation for finite element analyses in computational mechanics but are difficult and impractical to apply to increasingly complex non linear constitutive laws this paper discusses the performance of fully explicit local and global algorithms with automatic error control used to integrate general non linear constitutive laws into a non linear finite element computer code the local explicit stress integration procedure falls under the category of return mapping algorithm with standard operator split and does not require the determination of initial yield or the use of any form of stress adjustment to prevent drift from the yield surface the global equations are solved using an explicit load stepping with automatic error control algorithm in which the convergence criterion is used to compute automatically the coarse load increment size the proposed numerical procedure is illustrated here through the implementation of a set of elastoplastic constitutive relations including isotropic and kinematic hardening as well as small strain hysteretic non linearity a series of numerical simulations confirm the robustness accuracy and efficiency of the algorithms at the local and global level published in 2001 by john wiley sons ltd

an adaptive approach to limit analysis abstract the main objective of this paper is to propose an adaptive mesh refinement procedure for finite element models in limit analysis we use an a posteriori indicator based on the local directional interpolation error and a recovering scheme to compute second derivatives of the finite element solution the proposed mesh adaptation process gives improved results in localizing regions of rapid or abrupt variations of the variables whose location is not known a priori limit analysis of bodies in plane strain and plane stress is considered in the applications

adaptive return mapping algorithms for j2 elasto viscoplastic flow in this paper a new class of integration algorithms for elasto viscoplastic constitutive equations is proposed it is based on the generalized trapezoidal rule which is a weighted combination of the start and end of the increment but instead of taking constant weights the weights are a function of the plastic multiplier in this way the magnitude of the plastic strain increment determines the way the integration is performed the stress update and consistent tangent are derived for the case of j2 flow several candidates within the class of adaptive return mapping algorithms are investigated it is shown numerically that some of the proposed algorithms are more accurate than commonly used algorithms such as mean normal and radial return

on the application of multi step integration methods to infinitesimal elastoplasticity this article proposes an implicit algorithm for time integration of the differential algebraic equations of small deformation elastoplasticity in strain space the algorithm employs a two step backwards differentiation formula and a standard operator split procedure second order accuracy and non linear stability are established under appropriate constitutive restrictions the algorithm is implemented for the model of j2 plasticity and numerical simulations are conducted that illustrate its applicability and accuracy

cam clay plasticity part v a mathematical framework for three phase deformation and strain localization analyses of partially saturated porous media we present a mathematical framework for deformation and strain localization analyses of partially saturated granular media using three phase continuum mixture theory first we develop conservation laws governing a three phase mixture to identify energy conjugate expressions for constitutive modeling energy conjugate expressions identified relate a certain measure of effective stress to the deformation of the solid matrix the degree of saturation to the matrix suction the pressure in each phase to the corresponding intrinsic volume change of this phase and the seepage forces to the corresponding pressure gradients from the second of law of thermodynamics we obtain the dissipation inequality from the principle of maximum plastic dissipation we derive a condition for the convexity of the yield function then we formulate expressions describing conditions for the onset of tabular deformation bands under locally drained and locally undrained conditions finally we cast a specific constitutive model for partially saturated soils within the proposed mathematical framework and implement it in the context of return mapping algorithm of computational plasticity the proposed constitutive model degenerates to the classical modified cam clay model of soil mechanics in the limit of full saturation numerical examples are presented to demonstrate the performance of the return mapping algorithm as well as illustrate the localization properties of the model as functions of imposed deformation and matrix suction histories

accurate stress updating algorithm based on constant strain rate assumption abstract the calculation of the stresses for path dependent materials possesses a delicate problem here the constitutive equations are reformulated using the assumption of constant strain rate within an increment this assumption allows the constitutive laws to be rewritten as ordinary differential equations and thus a large set of tools is available to solve the constitutive equations within this concept the possibility to form an algorithmic stiffness tensor is investigated and it turns out that the true tensor is expensive to calculate but it is of interest that an inexpensive well behaving approximative algorithmic stiffness tensor can be formulated to investigate the properties of the solution method both with respect to stress calculation and convergence properties in a newton raphson method an elasto plastic damage model based on von mises plasticity is considered an interesting observation is that for the usual closest point projection method the damage evolution is very sensitive to the size of load step

on the implicit integration of rate dependent inelastic constitutive models although there are many different algorithms for the integration of inelastic constitutive models the fully implicit backward euler method has become the most popular one in this study further investigations on the accuracy of the backward euler method have been carried out also the performance of the discontinuous galerkin family and some implicit runge kutta time integrators is evaluated by using a simple scalar model problem accuracy of some integrators is studied when a single finite time step is applied conclusions drawn from this scalar model problem has been verified to apply also to a full six dimensional strain space formulation by numerical means special emphasis is placed on rate dependent inelastic creep models copyright 2005 john wiley sons ltd

an iterative algorithm for limit analysis with nonlinear yield functions abstract a mathematical programming algorithm is proposed for the general limit analysis problem plastic behavior is described by a set of linear or nonlinear yield functions abstract formulations of limit analysis are first considered and a quasi newton strategy for solving the optimality conditions is then sketched the structure of the problem arising from a finite element discretization is taken into account in order that the algorithm should be able to solve large scale problems

general solution procedures in elasto viscoplasticity constitutive relations and numerical integration algorithms for elasto viscoplastic problems are investigated a fully implicit integration scheme is adopted and the relevant expression of the consistent tangent operator for yield criteria and flow functions of arbitrary type is derived by suitably generalizing the approach exploited for rate independent plasticity this is achieved by replacing the consistency condition of plasticity with a relation between the viscoplastic consistency parameter and the flow function of the constitutive model in use the methodology adopted simplifies alternative procedures for the evaluation of consistent tangent operators which necessitate the inversion of a viscoplastic compliance operator the general expression of the consistent tangent operator is then specialized to the von mises yield criterion endowed with commonly adopted viscoplastic constitutive models numerical examples are finally presented

consistent integration of the constitutive equations of gurson materials under adiabatic conditions in this paper a consistent integration procedure for the thermoviscoplastic version of the complete gurson model is proposed with adiabatic conditions considered and with the use of the backward euler integration scheme a numerical algorithm implicit in all variables as well as the corresponding algorithmic operator have been developed the proposed algorithm was implemented in a finite element code and its performance is demonstrated with the numerical simulation of different examples

a coupled fem dem approach for hierarchical multiscale modelling of granular media summary a hierarchical multiscale framework is proposed to model the mechanical behaviour of granular media the framework employs a rigorous hierarchical coupling between the fem and the discrete element method dem to solve a bvp the fem is used to discretise the macroscopic geometric domain into an fem mesh a dem assembly with memory of its loading history is embedded at each gauss integration point of the mesh to serve as the representative volume element rve the dem assembly receives the global deformation at its gauss point from the fem as input boundary conditions and is solved to derive the required constitutive relation at the specific material point to advance the fem computation the dem computation employs simple physically based contact laws in conjunction with coulomb s friction for interparticle contacts to capture the loading history dependence and highly nonlinear dissipative response of a granular material the hierarchical scheme helps to avoid the phenomenological assumptions on constitutive relation in conventional continuum modelling and retains the computational efficiency of fem in solving large scale bvps the hierarchical structure also makes it ideal for distributed parallel computing to fully unleash its predictive power importantly the framework offers rich information on the particle level with direct link to the macroscopic material response which helps to shed lights on cross scale understanding of granular media the developed framework is first benchmarked by a simulation of single element drained test and is then applied to the predictions of strain localisation for sand subject to monotonic biaxial compression as well as the liquefaction and cyclic mobility of sand in cyclic simple shear tests it is demonstrated that the proposed method may reproduce interesting experimental observations that are otherwise difficult to be captured by conventional fem or pure dem simulations such as the inception of shear band under smooth symmetric boundary conditions non coaxial granular response large dilation and rotation at the edges of shear band and critical state reached within the shear band copyright 2014 john wiley sons ltd

assessments of methods for locating the point of initial yield numerical methods for locating the intersection point of the loading path with the temperature sensitive yield surface are discussed the newton method is found to be suitable for this purpose it is concluded that the iterates should begin at the plastic state rather than at the stress point estimated by a linear interpolation the influence of temperature changes on the iterative process is analyzed

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

locating and computing arbitrarily distributed zeros the problem of locating and computing with certainty all the simple roots of a twice continuously differentiable function f colon a b subset r to r is studied when some additional information on the distribution of the roots in the interval is available the framework is the one proposed by siam j sci comput 17 1996 pp 1232 1248 where only the uniform case was examined this paper settles some of the problems posed there and generalizes some of its results by considering an arbitrary distribution of the roots in a b the theoretical results are accompanied by simulations in a number of problems of various size