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

complementarity framework for non linear dynamic analysis of skeletal structures with softening plastic hinges in this paper we describe an algorithm for the incremental state update of elasto plastic systems with softening the algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem in developing the algorithm we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance and hence the ability to compute numerical solutions to practical size problems for example the notion of a tangent stiffness matrix arises numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm the numerical examples also raise some interesting questions about multiplicity of the solutions copyright 2010 john wiley sons ltd

spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

linearly constrained global optimization a general solution algorithm with applications this paper presents an efficient enumerative approach to solve general linearly constrained optimization problems this class of optimization problems includes fractional nonlinear network models quadratic convex and non convex programs the unified approach is accomplished by converting the constrained optimization problem to an unconstrained optimization problem through a parametric representation of its feasible region the proposed solution algorithm consists of three phases in phase 1 it finds all interior critical points in phase 2 the parametric representation of the feasible region is constructed to identify any critical points on the edges and faces of the feasible region this is done by a modified version of an algorithm for finding the v representation of the polyhedron then in phase 3 the global optimal value of the objective function is found by evaluating the objective function at the critical points as well as at the vertices for an illustration of the algorithm and a comparison with the existing methods small size numerical examples are presented

numerical modelling of longitudinal vibrations of a sucker rod string a new technique for analyzing the dynamic behavior of a sucker rod string used in the oil well industry is presented the main difficulty in the numerical calculation of the examined structure is a multivalued velocity force relation determined by coulomb s friction and by loads generated during operation of pump valves both the monotonic and nonmonotonic velocity force relations are considered a quasi variational inequality formulation of the problem is proposed the solution of the inequality amounts to finding the minimum of a convex nonsmooth functional at each time step by means of the newmark difference time scheme successive iterations and finite element discretization the problem of functional minimization is reduced to construction of a sequence of smooth nonlinear programming problems by introducing the auxiliary variables and applying the augmented lagrangian method the proposed approach is used to study the longitudinal vibrations of sucker rod strings under near real conditions in such systems the most commonly occurring vibration modes are the stick slip vibrations and the vibrations with natural force excited twice a cycle the nonmonotonic character of the friction law leads to intensification of these vibrations in the case of nonmonotonic friction law the stick slip vibrations can occur even under the action of constant external forces

damage integration in the strain space the focus is on isotropic elastodamaging softening materials where the damage parameter is expressed as a function of the total strain by integrating the mechanical work in the strain space along a stepwise holonomic loading history an incremental strain energy is obtained a coaxiality condition for the incremental strain energy to be potential is identified and its implications on the associativity of the damage evolution are discussed under some hypotheses the increment of the mechanical work is shown to be minimum along strain radial paths these results are used to construct a multifield variational framework supporting finite element nonlocal formulations

a vanishing viscosity approach to quasistatic evolution in plasticity with softening we deal with quasistatic evolution problems in plasticity with softening in the framework of small strain associative elastoplasticity the presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate independent problems to the case of a nonconvex energy functional we argue that in this case the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view thus we analyse a different selection criterion for the solutions of the quasistatic evolution problem based on a viscous approximation this leads to a generalized formulation in terms of young measures developed in the first part of the paper in the second part we apply our approach to some concrete examples

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

a simplex algorithm for piecewise linear programming i derivation and proof the simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise linear objective subject to linear constraints this three part paper develops and analyzes a general computationally practical simplex algorithm for piecewiselinear programming

a basis deficiency allowing variation of the simplex method for linear programming h one of the most important and fundamental concepts in the simplex methodology basis is restricted to being a square matrix of the order exactly equal to the number of rows of the coefficient matrix such inflexibility might have been the source of too many zero steps taken by the simplex method in solving real world linear programming problems which are usually highly degen erate to dodge this difficulty we first generalize the basis to allow the deficient case characterized as one that has columns fewer than rows of the coefficient matrix variations of the primal and dual simplex procedures are then made and used to form a two phase method based on such a basis the number of whose columns varies dynamically in the solution process generally speaking the more degenerate a problem to be handled is the fewer columns the basis will have so thii renders the possibility of efficiently solving highly degenerate problems we also provide a valuable insight into the distinctive and favorable behavior of the proposed method by reporting our computational experiments 1998 elsevier science ltd all rights reserved

multilevel formulations in the limit analysis and design of structures with bilateral contact constraints in this paper we study the rich class of formulations that arise in the limit analysis and design of elastic plastic structures in the presence of contact constraints it is well known that in the absence of contacts both the limit analysis and limit design problems can be written as linear programs however when contact constraints are present the structure effectively exhibits both softening and stiffening behaviour under monotonically increasing loading the resulting limit analysis and limit design problems are non convex and are difficult to solve due to the presence of complementary type of equality constraints we show that by using a mixed form of the minimum principle we can restate the limit analysis and limit design problems as two and three level formulations respectively further under a strong assumption on the problem and solution data we can take advantage of the underlying convexity to reduce both these multilevel formulations to equivalent linear programs while it may not be possible to always verify this assumption in practice we show that a two step iterative procedure is effective in reaching a solution to the limit design problem

an algorithm for shakedown analysis with nonlinear yield functions abstract this paper is devoted to propose an algorithm to solve the discrete form of the shakedown analysis problem with a nonlinear yield function firstly variational formulations for shakedown analysis of structures under variable loads are considered secondly the corresponding discrete forms are briefly recalled then the algorithm is derived by combining a newton formula based on the discrete equality conditions with a return mapping procedure focusing plastic admissibility the proposed numerical procedure can be applied to finite element models with large number of degrees of freedom because the algorithm is specially designed to take advantage of the structure of such models the numerical procedure is applied to a square plate with a circular hole under variable traction in two directions and the obtained approximations are compared with available results finally a rectangular plate with two small holes is considered in both plane strain and plane stress conditions all application use the mises condition without linearization

on wolfe s method for resolving degeneracy in linearly constrained optimization wolfe j soc indust appl math 11 1963 pp 205 211 describes a novel and very useful method for resolving degeneracy in the simplex method for linear programming lp the simplicity and reliability of the method makes it an excellent choice in this author s opinion the method is described here in the context of an active set method asm format for lp the method solves recursively generated subproblems that are smaller than the original which contributes to efficiency data structures are described that enable the recursive aspect of the method to be implemented very easily with minimal storage overhead the method is extended to solve a general form of linearly constrained optimization problem that includes quadratic programming qp and allows simple bounds on the variables and both equation and inequality general linear constraints issues of round off error are discussed and heuristics are described that have proved to be very reliable in practice the method is further extended to qp pr

computation of collapse states with von mises type yield condition a new computational approach to the problem of limit analysis with quadratic yield condition is developed and tested the problem is solved with the exact convex yield condition and the general case of unbounded yield set corresponding to unrestricted hydrostatic pressure is treated the solution method simultaneously approximates the collapse fields for stresses and flow the approach consists of two separate steps first the continuous problem is discretized by the finite element method into a finite dimensional optimization problem which turns out to be of a very specific form then this problem is solved by a suitable optimization method this way it is not necessary to be an expert in optimization in order to solve collapse problems and we can take advantage of the rapid development in optimization methods for large sparse problems without changing the discretization program the efficiency and accuracy of the method for large problems is demonstrated by solving a classical problem in the plane strain model approximately 90000 finite element nodes with 3 stress components and 2 velocity components at each node in two space dimensions this may be overkill but it shows that we are able to solve problems in three space dimensions

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 general non linear optimization algorithm for lower bound limit analysis the non linear programming problem associated with the discrete lower bound limit analysis problem is treated by means of an algorithm where the need to linearize the yield criteria is avoided the algorithm is an interior point method and is completely general in the sense that no particular finite element discretization or yield criterion is required as with interior point methods for linear programming the number of iterations is affected only little by the problem size some practical implementation issues are discussed with reference to the special structure of the common lower bound load optimization problem and finally the efficiency and accuracy of the method is demonstrated by means of examples of plate and slab structures obeying different non linear yield criteria copyright 2002 john wiley sons ltd

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

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

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

a projected newton algorithm for the dual convex program of elastoplasticity summary in this paper we study a dual optimization problem that arises when taking a mathematical programming approach to incremental state update in nonlinear problems the mathematical programming approach stems from energy based descriptions of constitutive models we describe a projected newton algorithm to solve the dual optimization problem this algorithm requires solution of an unconstrained optimization problem at the integration point level rather than a constrained one as carried out by classical return mapping schemes especially implementation of multi surface plasticity models is no more involved than that of single surface models we explore characteristics and performance of the projected newton algorithm through numerical examples insights gained from such a further exploration of mathematical programming algorithms are likely to aid in development of successive convex programming approaches to geometric nonlinear and other non convex problems such as non associated flow models copyright 2014 john wiley sons ltd

on the spectral radius of a 0 1 matrix related to merten s function abstract define n n matrices dn dij and cn cij by dij 1 if i j 0 otherwise and cn 0 1 1 1 t 1 0 0 0 let an dn cn we use the directed graph of an in to obtain the characteristic polynomial of an then we show that all but log2n 1 of the eigenvalues of an are equal to 1 and that an is asymptotically equal to n as n

on the mathematical modeling for elastoplastic contact problem and its solution by quadratic programming in this paper a variational inequality model is created for the elastoplastic contact problem in which the constraint of the constitutional relation for elastoplastic material and the contact condition are relaxed totally it gives an effective and strict mathematical modeling for the problem the quadratic programming is used here for the numerical solution this algorithm has great advantages of convergence and computational effectiveness over the conventional methods avoiding the tedious procedure of iterations

on the spectral radius of 0 1 matrices we determine the maximum spectral radius for 0 1 matrices with k2 andk2 1 1 s respectively and for symmetric 0 1 matrices with zero trace and e k21 s graphs with e edges in all cases equality is characterized

an enumerative method for the solution of linear complementarity problems abstract in this paper an enumerative method for the solution of the linear complementarity problem lcp is presented this algorithm either finds all the solutions of any general lcp that is no assumption is made concerning the class of the matrix or it establishes that no such solution exists the method is extended to the second linear complementarity problem slcp a new problem which has been introduced for the solution of a general quadratic program

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

three dimensional quasi static frictional contact by using second order cone linear complementarity problem a new formulation is presented for the three dimensional incremental quasi static problems with unilateral frictional contact under the assumptions of small rotations and small strains a second order cone linear complementarity problem is formulated which consists of complementarity conditions defined by bilinear functions and second order cone constraints the equilibrium configurations are obtained by using a combined smoothing and regularization method for the second order cone complementarity problem copyright 2005 john wiley sons ltd

an fe mpec approach for limit load evaluation in the presence of contact and displacement constraints abstract this paper describes a mathematical programing based approach for the direct limit load evaluation of a structural system under simultaneous contact and limited displacement conditions the contact model we adopt can simulate either a classical unilateral nonassociative coulomb friction situation or a cohesive fracture idealization at the potential discontinuity interface between contacting bodies the discrete fe model is constructed using locking free mixed finite elements the main feature and novelty of our proposed approach is to compute in a single step the maximum load capacity of the structure such that both the imposed displacement limitations and the contact conditions are satisfied in essence the formulation is a nontrivial extension of classical limit analysis the analysis is cast in its most natural form namely in mixed static kinematic variables and leads to what is known in the mathematical programing literature as a mathematical program with equilibrium constraints or mpec a nonlinear programing nlp based algorithm is proposed to solve the mpec two examples are provided to illustrate application of the proposed scheme and some comments regarding the various advantages of the adopted mathematical programing framework are made

a displacement like finite element model for j2 elastoplasticity variational formulation and finite step solution abstract the displacement like finite element formulation for finite step j2 elastoplasticity is revisited in this paper the classical computational strategy according to which plastic loading is tested at the gauss points of each element and an independent return mapping algorithm is performed for given incremental displacements is consistently derived from a suitably discretized version of a min max variational principle the sequence of solution phases to be performed within each load step adopting a full newton s method is illustrated in detail and the importance of a correct update of the plastic strains is emphasized it is further shown that in order to increase the rate of convergence and the stability properties of the newton s method the consistent elastoplastic tangent operator must be exploited even at the first iteration of each load step subsequent to the first yielding of the structural model this is in contrast with the traditional implementation according to which the elastic operator is used at the first iteration of each load step the effectiveness of the present approach is shown by a set of numerical examples referred to plane strain problems

newton methods for large scale linear inequality constrained minimization newton methods of the linesearch type for large scale minimization subject to linear inequality constraints are discussed the purpose of the paper is twofold i to give an active set type method with the ability to delete multiple constraints simultaneously and ii to give a relatively short general convergence proof for such a method it is also discussed how multiple constraints can be added simultaneously the approach is an extension of a previous work by the same authors for equality constrained problems it is shown how the search directions can be computed without the need to compute the reduced hessian of the objective function the convergence analysis states that every limit point of a sequence of iterates satisfies the second order necessary optimality conditions

a primal deficient basis simplex algorithm for linear programming abstract the standard basis which plays a fundamental role in simplex methodology was recently extended to include a deficient case with fewer columns than rows by taking advantage of primal degeneracy computational results have been favorable with dense implementations in this paper we propose a primal simplex algorithm using sparse lu factors of deficient bases amenable to real world linear programming problems which are often degenerate or even highly degenerate the algorithm would solve them with potentially improved stability compared to the simplex algorithm the proposed algorithm has been implemented and tested on a set of 50 netlib test problems as well as a set of 15 much larger real world problems including 8 kennington and 5 bpmpd problems it significantly outperformed minos 5 3 in terms of both iteration counts and run time in particular these results reveal that there is no inevitable correlation between an algorithm s inefficiency and degeneracy contradicting common belief

limit analysis of plates using the efg method and second order cone programming the meshless element free galerkin efg method is extended to allow computation of the limit load of plates a kinematic formulation that involves approximating the displacement field using the moving least squares technique is developed only one displacement variable is required for each efg node ensuring that the total number of variables in the resulting optimization problem is kept to a minimum with far fewer variables being required compared with finite element formulations using compatible elements a stabilized conforming nodal integration scheme is extended to plastic plate bending problems the evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly the formulation can be expressed as the problem of minimizing a sum of euclidean norms subject to a set of equality constraints this non smooth minimization problem can be transformed into a form suitable for solution using second order cone programming the procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper bound solutions for benchmark problems

effects of softening in elastic plastic structural dynamics softening understood as unstable behaviour of material or structural components is considered herein for its possible consequences on the overall behaviour of discrete dynamic models of elastic plastic beam structures in the absence of geometric effects it is shown that multiplicity of incremental solutions response bifurcations and manifestations of overall instability may occur the bifurcated responses may exhibit different scenarios for the same excitation e g shakedown or local damage up to failure an insight into and criteria for such occurrences are achieved by formulating the rate and the finite increment problem as linear complementarity problems period doubling and deterministic chaos under harmonic excitation are observed when reversible holonomic softening behaviour is assumed the strong sensitivity of the phenomena investigated with respect to the choice of the structural model is pointed out the findings are elucidated by illustrative examples

pivot versus interior point methods pros and cons linear optimization lo is the fundamental problem of mathematical optimization it admits an enormous number of applications in economics engineering science and many other fields the three most significant classes of algorithms for solving lo problems are pivot ellipsoid and interior point methods because ellipsoid methods are not efficient in practice we will concentrate on the computationally successful simplex and primal dual interior point methods only and summarize the pros and cons of these algorithm classes

implicit integration of plasticity models for granular materials a stress integration algorithm for granular materials based on fully implicit integration with explicit updating is presented in the implicit method the solution makes use of the gradient to the potential surface at the final stress state which is unknown the final stress and hardening parameters are determined solving the non linear equations iteratively so that the stress increment fulfills the consistency condition the integration algorithm is applicable for models depending on all the three stress invariants and it is applied to a characteristic state model for granular material since tensile stresses are not supported the functions and their derivatives are not representative outside the compressive octant of the principal stress space the elastic predictor is therefore preconditioned in order to ensure that the first predictor is within the valid region capability and robustness of the integration algorithm are illustrated by simulating both drained and undrained triaxial tests on sand the algorithm is developed in a standard format which can be implemented in several general purpose finite element codes it has been implemented as an abaqus subroutine and a traditional geotechnical problem of a flexible strip footing resting on a surface of sand is investigated in order to demonstrate the global accuracy and stability of the numerical solution

on the solution of a minimum weight elastoplastic problem involving displacement and complementarity constraints this paper deals with a special class of minimum weight design problem involving discretized structures holonomic reversible plasticity and constraints on displacements a key feature of the optimization problem is the presence of complementarity conditions involving the orthogonality of two sign constrained vectors this problem falls within the important class of so called mathematical programs with equilibrium constraints mpecs two simple algorithms are proposed to solve our synthesis problem and application is illustrated by some examples concerning truss like structures

two aggregate function based algorithms for analysis of 3d frictional contact by linear complementarity problem formulation three dimensional frictional contact is formulated as linear complementarity problem lcp by using the parametric variational principle and quadratic programming method two aggregate function based algorithms called respectively as self adjusting interior point algorithm and aggregate function smoothing algorithm are proposed for the solution of the lcp derived from the contact problems a nonlinear finite element code is developed for numerical analysis of 3d multi body contact problems four numerical examples are computed to demonstrate the applicability and computational efficiency of the methods proposed

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

uniqueness loss of ellipticity and localization for the time discretized rate dependent boundary value problem with softening the discretization in time of the initial boundary value problem for rate dependent elastic viscoplastic solid materials in presence of softening is investigated in this paper the emphasis is put on uniqueness loss of ellipticity and localization it is found that the time discretized problem resembles the incremental problem for rate independent materials and softening may lead to ill posedness loss of ellipticity if the time step is greater than a critical value it is well established that the implication of loss of ellipticity for the numerical simulations after spatial discretization is the pathological mesh dependency of the computed results we furnish a method to compute the critical time step and demonstrate its use for a simple example problem copyright 2010 john wiley sons ltd

efficient computation of multiple solutions in quasibrittle fracture analysis we propose a novel and efficient computational scheme for capturing the multiplicity of solutions that may exist at any loading step of a quasibrittle fracture analysis formulated as a linear complementarity problem lcp the algorithm proposed is based on successively augmenting the current lcp by complementarity constraints that effectively remove previously found solutions from the solution set we focus on the single step or so called holonomic analysis for mode i behavior the performance of the proposed approach is illustrated by means of physically meaningful benchmark problems

on the minimum spectral radius of matrices of zeros and ones abstract we consider the minimum spectral radius for an n n matrix of 0 s and 1 s having a specified number of 0 s we determine this minimum spectral radius when n 2 n 2 and bound it between two consecutive integers for all other values of

localization analysis via a geometrical method a geometrical technique is proposed in order to solve explicitly the critical conditions at localization for a quite general constitutive behaviour with isotropic elastic properties these critical conditions are shown to be closely related to the spectral properties eigenvalues and eigenvectors of the sum and difference of two tensors describing the inelastic effects when these two tensors are coaxial it is shown that the normal to a potential localization plane always lies in one of their principal planes it is also demonstrated that depending on the constitutive behaviour and the loading conditions several expressions for the critical hardening modulus at localization are available and their respective domain of validity well defined the roles and interactions of both deviatoric and hydrostatic non associativities in the critical conditions of localization are emphasized

a bound on the spectral radius of matrices of zeros and ones abstract we consider the minimum spectral radius for an n n matrix of 0 s and 1 s having a specified number of 0 s we determine this minimum spectral radius when n 2 4

the maximum determinant of an n n lower hessenberg 0 1 matrix abstract let a aij be an n n 0 1 matrix which is lower hessenberg i e aij 0 for j i 1 there are 2n 1 possibly nonzero terms in the determinant of an n n lower hessenberg 0 1 matrix so this is a triavial upper bound for the determinant we define an n n 0 1 matrix dn and show that this upper bound is det d n 1 5 1 5 2 n 1 5 2 n here det dn is the nth filbonacci number i e det d n det d n 1 det d n 2 and det d 1 det d 2 1 one has det dn as n this answers positively a question due to w w barrett

feasible descent algorithms for mixed complementarity problems in this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems the main features of this framework are a it is well defined for an arbitrary mixed complementarity problem b it generates only feasible iterates c it has a strong global convergence theory and d it is locally fast convergent under standard regularity assumptions this framework is applied to the path solver in order to show viability of the approach numerical results for an appropriate modification of the path solver indicate that this framework leads to substantial computational improvements

an efficient linear model and optimisation algorithm for multi site combined heat and power production abstract combined heat and power chp production is an important energy production technology which can help to improve the efficiency of energy production and to reduce the emission of co2 cost efficient operation of a chp system can be planned using an optimisation model based on hourly load forecasts a long term planning model decomposes into hourly models which can be formulated as linear programming lp problems such problems can be solved efficiently using the specialized power simplex algorithm however power simplex can only manage one heat and one power balance since heat cannot be transported over long distances power simplex applies only for local chp planning in this paper we formulate the hourly multi site chp planning problem with multiple heat balances as an lp model with a special structure we then develop the extended power simplex eps algorithm for solving such models efficiently even though the problem can be quite large as the number of demand sites increases eps demonstrates very good efficiency in test runs with realistic models eps is from 29 to 85 times faster than an efficient sparse simplex code using the product form of inverse pfi furthermore the relative efficiency of eps improves as the problem size grows

elasto plasticity revisited numerical analysis via reproducing kernel particle method and parametric quadratic programming aiming to simplify the solution process of elasto plastic problems this paper proposes a reproducing kernel particle algorithm based on principles of parametric quadratic programming for elasto plasticity the parametric quadratic programming theory is useful and effective for the assessment of certain features of structural elasto plastic behaviour and can also be exploited for numerical iteration examples are presented to illustrate the essential aspects of the behaviour of the model proposed and the flexibility of the coupled parametric quadratic programming formulations with the reproducing kernel particle method copyright 2002 john wiley sons ltd

on the solution of highly degenerate linear programs the solution of scheduling problems often gives rise to highly degenerate linear programmes which cause significant computational difficulties for the revised simplex method wolfe s highly effective ad hoc method for overcoming the cycling or stalling problems associated with degeneracy is described here it is given a geometric interpretation in terms of finding a direction of recession for a reduced problem which is fundamental to a full understanding of the procedure an example of an aircrew scheduling problem is used to illustrate the effectiveness of the method

uniqueness and localization i associative and non associative elastoplasticity abstract localization of deformation into a planar band in the incremental response of elasto plastic material is studied in the case of small strains and rotations the critical hardening modulus for localization is given in an explicit form uncoupled from the band normal for an arbitrary rate independent non associative plasticity loss of uniqueness of the response is investigated in terms of positiveness of the second order work density criteria for loss of second order work positiveness and localization are compared for plane stress and plane strain in these cases for the associative flow rule the threshold for the second order work positiveness coincides with the threshold for shear band formation this coincidence may not however occur if localization into splitting mode is attained

the simplest examples where the simplex method cycles and conditions where expand fails to prevent cycling this paper introduces a class of linear programming examples that cause the simplex method to cycle and that are the simplest possible examples showing this behaviour the structure of examples from this class repeats after two iterations cycling is shown to occur for both the most negative reduced cost and steepest edge column selection criteria in addition it is shown that the expand anti cycling procedure of gill et al is not guaranteed to prevent cycling

a practical anti cycling procedure for linearly constrained optimization a procedure is described for preventing cycling in active set methods for linearly constrained optimization including the simplex method the key ideas are a limited acceptance of infeasibilities in all variables and maintenance of a working feasibility tolerance that increases over a long sequence of iterations the additional work per iteration is nominal and stalling cannot occur with exact arithmetic the method appears to be reliable based on computational results for the first 53 linear programming problems in thenetlib set

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

the maximal eigenvalue of 0 1 matrices with prescribed number of ones abstract we determine the maximum spectral radius for 0 1 matrices with m 2 l ones for l 2 m 2 m 3 for all m and for a fixed l and m m l similar results are obtained for symmetric 0 1 matrix with zero diagonal in all cases equality is characterized

quadratic programming method in numerical simulation of metal forming process abstract in this paper a quadratic programming qp model based on a parametric variational principle is proposed for elastic plastic ep finite element analysis of metal forming processes the contact problem with friction between blank and tools is treated in the same way as in plastic analysis the penalty factors which are normally introduced into the algorithm for contact analysis have a direct influence on accuracy of solution there is no available rule for choosing a reasonable value of these factors for simulation of metal forming and they are therefore cancelled through a special technique so that the numerical results can be of high accuracy the algorithms for contact analysis and plastic analysis are established in one frame and consistent with each other compared with the conventional ep fem the newly developed method requires no tedious iterative procedures and has no convergence problems to apply this method easily to simulation of metal forming detailed forms of some key matrices or vectors for 2d fem and 3d fem are presented and a parametric loading algorithm for the qp model is developed which is suitable for qp problem with free variables and can decrease memory cost by avoiding the introduction of additional slack variables and improve the solution efficiency to some extent finally the proposed qp model is validated by two examples analysis of v notched tension test and analysis of the drawing of a square box one of the benchmarks proposed at numisheet93 it can be seen that the accuracy of solution of the new ep fem based on qp is better than that of the conventional ep fem based on iteration to make the new ep fem more applicable to metal forming industries it is necessary to develop a more efficient qp algorithm that is suitable for large scale problems

mixed variational principles for dynamic response of thermoelastic and poroelastic continua abstract the primary objective of the present work is to make further connections between variational methods on the one hand and reversible and irreversible thermodynamics on the other this begins with the development of a new stationary principle involving mixed field variables for continuum problems in infinitesimal dynamic thermoelasticity by defining lagrangian and dissipation functions in terms of physically relevant contributions and invoking the rayleigh formalism for damped systems we are able to recover the governing equations of thermoelasticity as the euler lagrange equations this includes the balance laws of linear momentum and entropy energy the constitutive models for elastic response and heat conduction and the natural boundary conditions by including energy contributions associated with second sound phenomena one eliminates the paradox of infinite thermal propagation speeds and the resulting set of governing equations has an elegant symmetry which is most easily seen in the fourier wave number domain a related formulation for dynamic poroelasticity yields two new stationary mixed variational principles depending upon the selection of primary field variables these governing equations can also exhibit an elegant structure which can deepen our understanding of the underlying phenomena and the thermoelastic poroelastic analogy in addition to the theoretical significance the variational formulations developed here can provide the basis for a class of optimization based methods for computational mechanics

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

a multiparametric quadratic programming algorithm with polyhedral computations based on nonnegative least squares model predictive control mpc is one of the most successful techniques adopted in industry to control multivariable systems under constraints on input and output variables to circumvent the main drawback of mpc i e the need to solve a quadratic program qp on line to compute the control action explicit mpc was proposed in the past to precompute the control law off line using multiparametric qp mpqp the resulting form of the mpc law is piecewise affine which is extremely easy to code can be computed online by simple arithmetic operations and requires a maximum number of iterations that can be exactly determined a priori on the other hand the offline computations to solve the mpqp problem require detecting emptiness full dimensionality and minimal hyperplane representations of polyhedra and other computational geometric operations while most of the existing methods solve such operations via linear programming the approach proposed in this paper relies on a nonnegative least squares nnls solver that is very simple to code and fast to execute in addition the new approach exploits qp duality to identify and construct critical regions and to handle degeneracy issues

the interior point revolution in optimization history recent developments and lasting consequences interior methods are a pervasive feature of the optimization landscape today but it was not always so although interior point techniques primarily in the form of barrier methods were widely used during the 1960s for problems with nonlinear constraints their use for the fundamental problem of linear programming was unthinkable because of the total dominance of the simplex method during the 1970s barrier methods were superseded nearly to the point of oblivion by newly emerging and seemingly more efficient alternatives such as augmented lagrangian and sequential quadratic programming methods by the early 1980s barrier methods were almost universally regarded as a closed chapter in the history of optimization this picture changed dramatically in 1984 when narendra karmarkar announced a fast polynomial time interior method for linear programming in 1985 a formal connection was established between his method and classical barrier methods since then interior methods have continued to transform both the theory and practice of constrained optimization we present a condensed unavoidably incomplete look at classical material and recent research about interior methods

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

shakedown analysis of structures made of materials with temperature dependent yield stress in this work the shakedown of structures made of materials with temperature dependent yield stress is considered under some restrictions on the thermal loading condition the yield stress is linearized and shakedown theorems are established based on these linearized shakedown theorems the shakedown limit is formulated as a problem of convex optimization an algorithm is built to compute shakedown limits numerical tests show good agreement with analytic solutions and experimental data

numerical aspects in developing lp softwares lpako and lpabo we have developed two public domain linear programming programs for several years lpako and lpabo which can solve large scale sparse lp problems stably and fast in this paper several important numerical aspects which were considered in developing lpako and lpabo are presented common issues are scaling tolerances and presolving for the lpako lu factorization and pivoting rule are important aspects in case of lpabo cholesky factorization ordering and dense column handling are important in the end of this paper several issues to be considered in the future development are proposed

making progress during a stall in the simplex algorithm abstract all the standard methods for avoiding cycling in the simplex algorithm use row selection rules for resolving degeneracy the method proposed in this paper does not instead the gass saaty parametric method applied to the objective cost form is used to choose the incoming column the pivot row choice among blocking rows i e those tied for pivot is arbitrary a simple anticycling device is used which avoids dual degeneracy of the parametrized objective with probability one tests were run on nine highly degenerate practical test problems ranging in size from small to large using minos software the standard simplex method required 31 195 iterations of which 20 504 or 66 were blocked pivots using the parametric scheme required 13 812 iterations of which 67 were blocked the reduction in the number of iterations for this set of highly degenerate test problems was 56 the cpu time reduction was 48

on the numerical integration of three invariant elastoplastic constitutive models abstract we investigate the performance of a numerical algorithm for the integration of isotropically hardening three invariant elastoplastic constitutive models with convex yield surfaces the algorithm is based on a spectral representation of stresses and strains for infinitesimal and finite deformation plasticity and a return mapping in principal stress directions smooth three invariant representations of the mohr coulomb model such as the lade duncan and matsuoka nakai models are implemented within the framework of the proposed algorithm among the specific features incorporated into the formulation are the hardening softening responses and the tapering of the yield surfaces toward the hydrostatic axis with increasing confining pressure isoerror maps are generated to study the local accuracy of the numerical integration algorithm finally a boundary value problem involving loading of a strip foundation on a soil is analyzed with and without finite deformation effects to investigate the performance of the integration algorithm in a full scale non linear finite element simulation

numerical analysis of augmented lagrangian algorithms in complementary elastoplasticity the main subject of the paper is the investigation of augmented lagrangian algorithms and update formulas in the solution of elastoplastic problems a stress rate formulation for elastoplastic models with internal variables and its finite increment form is employed to state the mechanical problem in this formulation the augmented lagrangian is used to enforce the constraint of plastic admissibility directly on the stresses and thermodynamic forces this is not a limitation of the augmented lagrangian approach and the same framework can be built on more classical displacement formulations as well the meaning and the derivation of various first and second order lagrangian multipliers update formulas and iterative schemes is shown a new diagonal iteration algorithm and the introduction of a scale factor for the augmented lagrangian term are proposed numerical examples compare the efficiency of several forms of augmented lagrangian algorithms and illustrate the influence of the scale factor and of the penalty parameter copyright 2004 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

a linear programming decomposition focusing on the span of the nondegenerate columns abstract the improved primal simplex ips was recently developed by elhalaloui et al to take advantage of degeneracy when solving linear programs with the primal simplex it implements a dynamic constraint reduction based on the compatible columns i e those that belong to the span of a given subset of basic columns including the nondegenerate ones the identification of the compatible variables may however be computationally costly and a large number of linear programs are solved to enlarge the subset of basic variables in this article we first show how the positive edge criterion of raymond et al can be included in ips for a fast identification of the compatible variables our algorithm then proceeds through a series of reduction and augmentation phases until optimality is reached in a reduction phase we identify compatible variables and focus on them to make quick progress toward optimality during an augmentation phase we compute one greatest normalized improving direction and select a subset of variables that should be considered in the reduced problem compared with ips the linear program that is solved to find this direction involves the data of the original constraint matrix this new algorithm is tested over mittelmann s benchmark for linear programming and on instances arising from industrial applications the results show that the new algorithm outperforms the primal simplex of cplex on most highly degenerate instances in which a sufficient number of nonbasic variables are compatible in contrast ips has difficulties on the eleven largest mittelmann instances

primal and dual active set methods for convex quadratic programming computational methods are proposed for solving a convex quadratic program qp active set methods are defined for a particular primal and dual formulation of a qp with general equality constraints and simple lower bounds on the variables in the first part of the paper two methods are proposed one primal and one dual these methods generate a sequence of iterates that are feasible with respect to the equality constraints associated with the optimality conditions of the primal dual form the primal method maintains feasibility of the primal inequalities while driving the infeasibilities of the dual inequalities to zero the dual method maintains feasibility of the dual inequalities while moving to satisfy the primal inequalities in each of these methods the search directions satisfy a kkt system of equations formed from hessian and constraint components associated with an appropriate column basis the composition of the basis is specified by an active set strategy that guarantees the nonsingularity of each set of kkt equations each of the proposed methods is a conventional active set method in the sense that an initial primal or dual feasible point is required in the second part of the paper it is shown how the quadratic program may be solved as a coupled pair of primal and dual quadratic programs created from the original by simultaneously shifting the simple bound constraints and adding a penalty term to the objective function any conventional column basis may be made optimal for such a primal dual pair of shifted penalized problems the shifts are then updated using the solution of either the primal or the dual shifted problem an obvious application of this approach is to solve a shifted dual qp to define an initial feasible point for the primal or vice versa the computational performance of each of the proposed methods is evaluated on a set of convex problems from the cutest test collection

a direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures this paper presents a direct complementarity approach for carrying out the elastoplastic analysis of plane stress and plane strain structures founded on a traditional finite step formulation our approach however avoids the typically cumbersome implementation of iterative predictorcorrector procedures associated with the ubiquitous return mapping algorithm instead at each predefined step the governing formulationcast in its most natural mathematical programming format known as a mixed complementarity problemis directly solved by using a complementarity solver run from within a mathematical modeling system we have chosen the industry standard general algebraic modeling system path mixed complementarity problem solver that is called from within the general algebraic modeling system environment we consider both von mises and tresca materials with perfect or hardening kinematic and isotropic behaviors our numerical tests five benchmark examples of which are presented in this paper have been carried out using models constructed from the mixed finite element of capsoni and corradi comput methods appl mech eng 1997 141 6793 which beneficially offers a locking free behavior and coarse mesh accuracy the results indicate in addition to an isochoric locking free behavior good accuracy and the ability to circumvent the difficult singularity problem associated with the corners of tresca yield surfaces

nonholonomic elastoplastic analysis involving unilateral frictionless contact as a mixed complementarity problem this paper deals with the elastoplastic analysis of suitably discretized elastoplastic structures under frictionless unilateral contact conditions whilst the mathematical programming formulation we adopt can cater equally for hardening as well as softening path dependent nonholonomic plasticity the state problem involving softening is by far the more difficult to solve in view of the inherent nonconvex nature of the underlying mathematical program we propose a novel description of a fairly general class of piecewise linear softening laws and investigate use of an approximate stepwise holonomic path independent approach for capturing the evolution of the structure with softening and or hardening constitutive laws this finite step problem is formulated in a specific way to take advantage of the use of a robust and industry standard complementarity solver named path from within the gams modeling environment we illustrate its application on truss like structures

a phase 1 approach for the generalized simplex algorithm abstract a new simplex variant allowing basis deficiency has recently been proposed to attack the degeneracy 1 as a generalization of the simplex algorithm it uses a phase 1 procedure solving an auxiliary problem with piecewise linear sums of infeasibilities as its objective in this paper we develop another phase 1 approach that only introduces a single artificial variable unlike the former which needs a crash procedure to supply an initial basis the proposed phase 1 is able to get itself started from scratch with an artificial basis having a single column computational results with a set of standard test problems from netlib are also reported

a finite element procedure for rigorous numerical enclosures on the limit load in the analysis of multibody structures a rigorous finite element numerical procedure is proposed for the computation of guaranteed lower and upper bounds for the limit load of failure in a system of linear elastic blocks in mutual non penetrative contact with given friction first the static and kinematic principles are formulated as continuous optimization problems and existence of a solution to the corresponding limit load problems at the infinite dimensional level is established two numerical approaches are devised one for each limit load problem to obtain actual numerical bounds on the unique critical load the first approach uses the static limit load problem involving stresses in conjunction with a non standard conforming finite element method to obtain a linear program from which one can derive a lower safe bound for the limit load and an expression for the corresponding stress field the second approach uses the kinematic limit load problem to obtain a linear optimization problem from which one can determine an upper unsafe bound for the limit load and an expression for the failure mode together these procedures give rise to rigorous numerical enclosures on the limit load