a sequential quadratically constrained quadratic programming method with an augmented lagrangian line search function based on an augmented lagrangian line search function a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems compared to quadratic programming solved in the traditional sqp methods a convex quadratically constrained quadratic programming is solved here to obtain a search direction and the maratos effect does not occur without any other corrections the active set strategy used in this subproblem can avoid recalculating the unnecessary gradients and approximate hessian matrices of the constraints under certain assumptions the proposed method is proved to be globally superlinearly and quadratically convergent as an extension general problems with inequality and equality constraints as well as nonmonotone line search are also considered

sequential systems of linear equations algorithm for nonlinear optimization problems general constrained problems in ref j comput math 20 3 2002 301 a new superlinearly convergent algorithm of sequential systems of linear equations for nonlinear optimization problems with inequality constraints was proposed since the new algorithm only needs to solve four systems of linear equations having a same coefficient matrix per iteration the computation amount of the algorithm is much less than that of the existing sequential quadratic programming algorithms per iteration under some mild assumptions the new algorithm is globally convergent and its rate of convergence is one step superlinearly in this paper it is shown that the new algorithm also can be used to deal with nonlinear optimization problems having nonlinearly equality and inequality constraints by solving an auxiliary problem some numerical results are reported

an a posteriori error estimator for the p and hp versions of the finite element method an a posteriori error estimator is proposed in this paper for the p and hp versions of the finite element method in two dimensional linear elastostatic problems the local error estimator consists in an enhancement of an error indicator proposed by bertoti and szabo int j numer meth engng 1998 42 561 587 which is based on the minimum complementary energy principle in order to obtain the local error estimate this error indicator is corrected by a factor which depends only on the polynomial degree of the element the proposed error estimator shows a good effectivity index in meshes with uniform and non uniform polynomial distributions especially when the global error is estimated furthermore the local error estimator is reliable enough to guide p and hp adaptive refinement strategies

a nonmonotone trust region algorithm with nonmonotone penalty parameters for constrained optimization in this paper we present a nonmonotone trust region algorithm with nonmonotone penalty parameters for the solution of optimization problems with nonlinear equality constraints and bound constraints the proposed algorithm combines an sqp approach with a trust region strategy to globalize the process each step is obtained through the computation of a normal step to reduce infeasibility and a tangential step to decrease some merit function the algorithm makes use of an augmented lagrangian function as merit function and allows the value of this merit function and the penalty parameter involved in it to decrease non monotonically the global convergence theory for the proposed algorithm is developed without regularity assumption and shows that any limit point of the sequence generated by the algorithm is a stationary point while at least one limit point under the suitable assumptions is a substationary point and a stationary point if it is feasible some preliminary numerical experiments are also reported

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

an rqp algorithm using a differentiable exact penalty function for inequality constrained problems in this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function the algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an armijo type line search procedure that avoids the need to evaluate second order derivatives of the problem functions we prove that the algorithm possesses global and superlinear convergence properties numerical results are reported

an sqp method for general nonlinear programs using only equality constrained subproblems in this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints compared with an older version p spellucci han s method without solving qp in a auslender w oettli j stoer eds optimization and optimal control lecture notes in control and information sciences vol 30 springer berlin 1981 pp 123 141 it is much simpler to implement and allows any kind of changes of the working set in every step our method relies on a strong regularity condition as far as it is applicable the new approach is superior to conventional sqp methods as demonstrated by extensive numcrical tests 1998 the mathematical programming society inc published by elsevier science b v

error estimation and adaptive meshing in strongly nonlinear dynamic problems we present work aimed at developing a general framework for mesh adaption in strongly nonlinear possibly dynamic problems we begin by showing that the solutions of the incremental boundary value problem for a wide class of materials including nonlinear elastic materials compressible newtonian fluids and viscoplastic solids obey a minimum principle provided that the constitutive updates are formulated appropriately this minimum principle can be taken as a basis for asymptotic error estimation in particular we chose to monitor the error of a lower order projection of the finite element solution the optimal mesh size distribution then follows from a posteriori error indicators which are purely local i e can be computed element by element we demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application

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

linearized stability in periodic functional differential equations with state dependent delays in this paper we study stability of periodic solutions of a class of nonlinear functional differential equations fdes with state dependent delays using the method of linearization we show that a periodic solution of the nonlinear fde is exponentially stable if the zero solution of an associated linear periodic linear homogeneous fde is exponentially stable

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

a new feasible descent algorithm combining sqp with generalized projection for optimization problems without strict complementarity in this paper optimization problems with nonlinear inequality constraints are discussed by combining the sequential quadratic programming sqp with an new generalized projection technique a new feasible descent algorithm for solving the problems is presented at each iteration of the new algorithm a convex quadratic program qp is solved and a master direction is obtained and an improved feasible descent direction is yielded by updating the master direction with an explicit formula and in order to avoid the maratos effect a height order correction direction is computed by another explicit formula of the master direction and the improved direction both this two correction formulas contain a new generalized projection technique under weaker conditions without the strict complementarity the new algorithm is proved to possess global convergence and superlinear convergence furthermore the quadratic convergence rate of the algorithm is obtained when the twice derivatives of the objective function and constrained functions are adopted

partitioned quasi newton methods for nonlinear equality constrained optimization we examine the nonlinear equality constrained problem nep in a projected or partitioned form to reveal the basic structure of the hessian matrix the concepts of quasi newton methods are then introduced by considering unconstrained optimization problems drawing on this background we present quasi newton methods for nep and explore properties of the approximate hessian matrix which insure good performance of sequential quadratic programming sqp type algorithms positivity quasi newton relation symmetry and the least change property current quasi newton methods and the strategies they use to provide these properties are then examined although it is particularly desirable to maintain positivity of the null space projection of the hessian matrix we show that existing methods fail to do this in a simple and reliable manner consequently our goal in this thesis is to find new updates which incorporate these desirable properties of the hessian matrix using a variational approach we derive several new quasi newton updates for nep these updates are derived using the partitioned forms of the hessian matrix allowing us to incorporate the desired properties for each partition two of the update formulas the lower half update lhu and the partitioned bfgs update pbu are shown to be particularly successful the lhu and pbu formulas are of the least change type and satisfy a standard quasi newton equation retain symmetry of the full hessian and keep the null space projection of the hessian positive definite in special cases these new updates simplify to projected forms of the broyden powell symmetric broyden and broyden fletcher goldfarb shanno formulas these new updates are particularly attractive since they require only one gradient evaluation per iteration they may be used at nearly every iteration even when we are far from the solution without requiring special line search conditions or step size restrictions next we discuss implementation of the lhu and pbu formulas in particular we examine the cubic equation which must be solved at every iteration its accurate solution and the selection of the cubic root we prove r superlinear convergence and present extensive computational results including comparisons with existing quasi newton methods for the nonlinear equality problem

the kidney model as an inverse problem the mammalian kidney is modeled by a multipoint boundary value problem for a system of nonlinear ordinary differential equations a corresponding inverse problem is presented which allows the rigorous judgement of the potential of the given modeling technique for its numerical solution a discretization is proposed which is tailor made for kidney models it leads to a nonlinear programming problem with nonlinear equality and inequality constraints the suggested methods are applied to current research problems in renal physiology

a recursive quadratic programming algorithm that uses differentiable exact penalty functions in this paper a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied the line search functions used are approximations to fletcher s differentiable exact penalty function global convergence and local superlinear convergence results are proved and some numerical results are given

a robust sequential quadratic programming method the sequential quadratic programming method developed by wilson han and powell may fail if the quadratic programming subproblems become infeasible or if the associated sequence of search directions is unbounded this paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems furthermore questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed

an asymptotically exact pointwise a posteriori error estimator for linear elliptic one dimensional problems with possible singularities in the solution abstract when the finite element solution of a variational problem possesses certain superconvergence properties it is possible very inexpensively to obtain a correction term providing an additional order of approximation of the solution the correction can be used for error estimation locally or globally in whatever norm is preferred or if no error estimation is wanted it can be used for postprocessing of the solution to improve the quality in this paper such a correction term is found and its convergence properties proved in the special case of one dimensional linear elliptic problems special attention is given to the effects of the appearance of singularities and zeros of derivatives in the exact solution

fully reliable error control for first order evolutionary problems abstract this work is focused on the application of functional type a posteriori error estimates and corresponding indicators to a class of time dependent problems we consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results this paper examines two different methods of solution approximation for evolutionary models i e a time marching technique and a space time approach the first part of the study presents an algorithm for global minimisation of the majorant on each of discretisation time cylinders time slabs the effectiveness of this approach to error estimation is confirmed by extensive numerical tests in the second part of the publication the application of functional error estimates is discussed with respect to a space time approach it is followed by a set of extensive numerical tests that demonstrates the efficiency of proposed error control method the numerical results obtained in this paper rely on the implementation carried out using open source software which allows formulating the problem in a weak setting to work in a variational framework appears to be very natural for functional error estimates due to their derivation method the search for the optimal parameters for the majorant is done by a global functional minimisation which to the authors knowledge is the first work using this technique in an evolutionary framework

a sequential quadratic programming method for potentially infeasible mathematical programs abstract the paper describes a modification to the wilson han powell sequential quadratic programming technique that is applicable to potentially infeasible nonlinear programs the method is designed to locate stationary points of a nonlinear program in a generalized sense definition 2 8 these stationary points include the familiar kuhn tucker and fritz john points but also include what we call external and strong external stationary points the external and strong external stationary points are not feasible thus the problem may be infeasible and yet have stationary points loosely speaking an external stationary point is one that satisfies a certain first order necessary optimality condition for being as close to feasibility as possible the proposed technique is similar to those described by j v burke and s p han a modified sequential quadratic programming method manuscript and m sahba j optim theory appl 52 1987 291 309 however it is shown to possess substantially enhanced stability properties the global convergence properties of the method are described along with some continuity results for the quadratic programming subproblems

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

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

element by element a posteriori error estimation and improvement of stress solutions for two dimensional elastic problems a quantitative method of a posteriori error estimation based on finite element solutions is developed the proposed method can estimate the error of solutions obtained by the finite element method for two dimensional elastic problems with 4 node elements since the error is estimated element by element the proposed method does not require a large memory or long computing time not only rectangular elements but also arbitrarily shaped 4 node elements can be used in this method for estimating the error in the finite element computation with high accuracy the finite element solutions are improved by adding the estimated errors onto the original solutions the proposed method can be utilized for any type of linear problem if an isoparametric finite element method is used

sequentially decomposed programming model based decomposition is a powerful tool for breaking design problems into smaller subproblems establishing hierarchical structure and analyzing the interrelations in engineering design problems however the theoretical foundation for solving decomposed nonlinear optimization problems requires further work we show that theformulation of the coordination problem is critical in quickly identifying thecorrectactiveconstraints and that solving subproblems independently may hinder the local convergence of algorithms tailored to hierarchical coordination yet hierarchical decomposition algorithms can have excellent global convergence properties and can be expected to exhibit superior improvement in the rst few iterations when compared to the undecomposed case basedontheseinsights agenericsequentially decomposedprogramming sdp algorithmisoutlined sdphastwo phases far from the solution rst phase decomposition is used close to the solution second phase subproblems are not solved separately the generic sdp is applied to sequential quadratic programming sqp to de ne an sdp sqp implementation a global convergence proof and a simple example are given

a globally convergent stabilized sqp method sequential quadratic programming sqp methods are a popular class of methods for nonlinearly constrained optimization they are particularly effective for solving a sequence of related problems such as those arising in mixed integer nonlinear programming and the optimization of functions subject to differential equation constraints recently there has been considerable interest in the formulation of stabilized sqp methods which are specifically designed to handle degenerate optimization problems existing stabilized sqp methods are essentially local in the sense that both the formulation and analysis focus on the properties of the methods in a neighborhood of a solution a new sqp method is proposed that has favorable global convergence properties yet under suitable assumptions is equivalent to a variant of the conventional stabilized sqp method in the neighborhood of a solution the method combines a primal dual generalized augmented lagrangian function with a flexible line search to obtain a seque

error estimation and adaptivity in elastodynamics abstract the paper discusses adaptive procedures for elastodynamic problems using semi discrete finite elements the adaptivity in space may be done by mesh refinement h version by increase of order of the approximation polynomials p version or a combination of these hp version in time the integration is either made by mode superposition of characteristic functions or by direct integration the adaptation in time may either be made by change of the global time step h version or increase of the order of approximation polynomials p version crucial parts of the analysis are the error estimator and the error indicator which are based either on interpolation theory or on total energy obtained from postprocessed stresses for the mesh generation and regeneration an isoline technique is utilized by which the mesh can be smoothed by interpolation theory the isolines are created for some characteristic scalar function for time discretization a new simple a posteriori local error estimator for the newmark scheme is described it is derived by a post processing technique without solving new equations so the additional computational cost is small and the implementation is convenient the paper also describes a sequence of nested time integration methods for dynamic problems formulated as two first order equations in time these nested integration methods in time based on a hierarchical formulation in time are a stable schemes of order 2k and l stable schemes of order 2k 1 a considerable improvement in terms of accuracy as well as effectiveness is obtained compared to currently available methods

convergence properties of algorithms for nonlinear optimization this paper reviews some of the most successful methods for unconstrained constrained and nondifferentiable optimization calculations particular attention is given to the contribution that theoretical analysis has made to the development of algorithms it seems that practical considerations provide the main new ideas and that subsequent theoretical studies give improvements to algorithms coherence to the subject and better understanding

on the techniques for constructing admissible stress fields in model verification performances on engineering examples robust global goal oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation in the context of computational mechanics the construction of admissible stress fields ie stress tensors which verify the equilibrium equations is required to set up strict and guaranteed error bounds using residual based error estimators and plays an important role in the quality of the error estimates this work focuses on the different procedures used in the calculation of admissible stress fields which is a crucial and technically complicated point the three main techniques that currently exist called the element equilibration technique eet the star patch equilibration technique spet and the element equilibration star patch technique eespt are investigated and compared with respect to three different criteria namely the quality of associated error estimators computational cost and easiness of practical implementation into commercial finite element codes the numerical results which are presented focus on industrial problems they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators which have the same convergence rate as the exact global error is consistent two and three dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches the analysis of the results reveals that the spet is more accurate than eet and eespt methods but the corresponding computational cost is higher overall the numerical tests prove the interest of the hybrid method eespt and show that it is a correct compromise between quality of the error estimate practical implementation and computational cost furthermore the influence of the cost function involved in the eet and the eespt is studied in order to optimize the estimators

a recovery type error estimator for the extended finite element method based on singular smooth stress field splitting a new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method xfem is presented the procedure is an adaptation of the superconvergent patch recovery spr technique for the xfem framework it is based on three fundamental aspects a the use of a singular smooth stress field decomposition technique involving the use of different recovery methods for each field standard spr for the smooth field and reconstruction of the recovered singular field using the stress intensity factor k for the singular field b direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement and c assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack the method was validated by testing it on problems with an exact solution in mode i mode ii and mixed mode and on a problem without analytical solution the results obtained showed the accuracy of the proposed error estimator copyright 2008 john wiley sons ltd

a penalty free type nonmonotone trust region method for nonlinear constrained optimization in this paper we propose a penalty free type nonmonotone trust region method for solving general nonlinear programming problem the algorithmic framework yields global convergence without using any penalty function we analyze the global convergence of the main algorithm for the degenerate problems and also give the globally convergent results under the linear independence constraint qualification the preliminary numerical tests are reported

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

a new penalty free type algorithm based on trust region techniques abstract in this paper we propose a new penalty free type method for nonlinear equality constrained problems the new algorithm uses trust region framework and feasibility safeguarding technique moreover it has no choice of penalty parameter and penalty function as a merit function and it does not use the filter technique to avoid the penalty function either we analyze the global convergence of the main algorithm under the standard assumptions the preliminary numerical tests are reported

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

mesh adaptations for linear 2d finite element discretizations in structural mechanics especially in thin shell analysis abstract in this paper we describe an automatic refinement of finite element meshes in structural analysis including shell problems at first the adaptation to curved boundaries and surfaces is discussed central point is a topological description which is independent of the discretization the second part delas with local refinements adapted to the solution by error estimation three a posteriori indicators are compared by examples

time and space adaptive methods applied to localization phenomena in empty and saturated micropolar and standard porous materials localization phenomena as e g shear bands occur as a result of local concentrations of plastic strains in small bands of finite width the reason for this behaviour lies in the basic properties of elasto plastic frictional materials where variations of the lode angle together with the non associated plastic dilatation result in local softening effects proceeding from non viscous empty porous materials it is well known that the computation of shear bands reveals an ill posed problem to overcome this behaviour two different regularization strategies are considered which are 1 the inclusion of independent rotational degrees of freedom micropolar formulation and 2 the inclusion of viscosity effects standard formulation by taking into consideration viscoplastic properties of the solid matrix or a viscous pore fluid respectively the numerical treatment furthermore proceeds from time and space adaptive strategies to refine and to coarsen both the time step and the mesh size considering quasi static problems the space discretization yields a dae system of index 1 in the time domain which can be successfully treated by sdirk methods thus allowing for an efficient estimation of the time error and as a consequence for an adaptive time step control in the space domain the present investigation proceeds from an error indicator of zienkiewicz zhu type where smoothened values of the l2 norms of characteristic quantities are compared to the respective discrete values the efficiency of this procedure is demonstrated by the computation of the biaxial experiment on an empty and a liquid saturated porous soil material and of the slope failure problem of a fluid saturated soil copyright 2001 john wiley sons ltd

an adaptive algorithm for the crank nicolson scheme applied to a time dependent convection diffusion problem an a posteriori upper bound is derived for the nonstationary convection diffusion problem using the crank nicolson scheme and continuous piecewise linear stabilized finite elements with large aspect ratio following lozinski et al 2009 13 a quadratic time reconstruction is used a space and time adaptive algorithm is developed to ensure the control of the relative error in the l 2 h 1 norm numerical experiments illustrating the efficiency of this approach are reported it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step even in the convection dominated regime and in the presence of boundary layers

mid node admissible spaces for quadratic triangular 2d finite elements with one edge curved a robust element distortion metric based on the new concept of mid node admissible spaces for two dimensional quadratic triangular finite elements is developed the metric is based on the jacobian determinant over the entire element without requiring that it actually be computed everywhere on the element the metric is relatively inexpensive to compute especially for mildly distoted elements the metric is able to detect elementsof poor quality that other distortion metrics fall to detect it also has the ability to approve elements of good quality regardless of the extent to which they may appear geometrically distorted copyright 2001 john wiley sons ltd

sharp constants in the sobolev embedding theorem and a derivation of the brezis gallouet interpolation inequality sharp estimates are obtained for the constants appearing in the sobolev embedding theorem on the two dimensional torus furthermore a version of the brezis gallouet interpolation inequality is obtained with an explicit but not necessarily optimal constant in the leading term namely the logarithmic term the constants are expressed in terms of the riemann zeta function and the dirichlet beta series

on the adaptive finite element method of steady state rolling contact for hyperelasticity in finite deformations in this paper we present an adaptive finite element method for steady state rolling contact in finite deformations along with a residual based a posteriori error estimator for rolling contact problem with coulomb friction a general formulation of rolling contact geometry is derived from the point of view of differential geometry solvability conditions for the rolling contact problems are discussed we use newton s method to solve variational equations derived from a penalty regularization of the finite element approximation of the rolling contact problem we provide a numerical example to illustrate the method

an sqp feasible descent algorithm for nonlinear inequality constrained optimization without strict complementarity in this paper a kind of nonlinear optimization problems with nonlinear inequality constraints are discussed and a new sqp feasible descent algorithm for solving the problems is presented at each iteration of the new algorithm a convex quadratic program qp which always has feasible solution is solved and a master direction is obtained then an improved feasible descent direction is yielded by updating the master direction with an explicit formula and in order to avoid the maratos effect a height order correction direction is computed by another explicit formula of the master direction and the improved direction the new algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity furthermore the quadratic convergence rate of the algorithm is obtained when the twice derivatives of the objective function and constrained functions are adopted finally some numerical tests are reported

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

adaptive variational multiscale methods for incompressible flow based on two local gauss integrations we consider variational multiscale vms methods with h adaptive technique for the stationary incompressible navier stokes equations the natural combination of vms with adaptive strategy retains the best features of both methods and overcomes many of their deficits a reliable a posteriori projection error estimator is derived which can be computed by two local gauss integrations at the element level finally some numerical tests are presented to illustrate the method s efficiency

a stabilization procedure by line search computation for first order approximation strategies in structural optimization the paper is concerned with first order approximation methods to solve non linear optimization problems as they arise in structural optimization a modification of the original method of moving asymptotes mma is presented to prevent oscillation or divergence by incorporating a line search into the global iteration algorithm in this line search a merit function of augmented lagrangian type is decreased by using a backtracking strategy a proof is presented to show the descent property of the proposed merit function for mma numerical examples demonstrate how the new technique stabilizes the iteration process

on error estimates and adaptivity in elastoplastic solids applications to the numerical simulation of strain localization in classical and cosserat continua the a posteriori error estimates based on the post processing approach are introduced for elastoplastic solids the standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy this is known to represent a natural metric for the class of elastoplastic problems of evolution in addition the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates a posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem as a model problem the constitutive equations describing a generalization of standard j2 elastoplasticity within the cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post instability region the proposed a posteriori error estimates are appropriately modified to account for the cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems superior behaviour of the cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material resulting in convergent solutions with an h refinement and almost uniform error distribution in all considered error norms

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

a subject specific finite element musculoskeletal framework for mechanics analysis of a total knee replacement abstract concurrent use of finite element fe and musculoskeletal ms modeling techniques is capable of considering the interactions between prosthetic mechanics and subject dynamics after a total knee replacement tkr surgery is performed however it still has not been performed in terms of favorable prediction accuracy and systematic experimental validation in this study we presented a methodology to develop a subject specific fe ms model of a human right lower extremity including the interactions among the subject specific ms model the knee joint model with ligament bundles and the deformable fe prosthesis model in order to evaluate its accuracy the fe ms model was compared with a traditional hinge constraint ms model and experimentally verified over a gait cycle both models achieved good temporal agreement between the predicted muscle force and the electromyography results though the magnitude on models is different a higher predicted accuracy quantified by the root mean square error rmse and the squared pearson correlation coefficient r2 was found in the fe ms model rmse 177 2 n r2 0 90 when compared with the ms model rmse 224 1 n r2 0 81 on the total tibiofemoral contact force the contact mechanics including the contact area pressure and stress were synchronously simulated and the maximum contact pressure 22 06 mpa occurred on the medial side of the tibial insert without exceeding the yield strength of the ultra high molecular weight polyethylene 24 79 mpa the approach outlines an accurate knee joint biomechanics analysis and provides an effective method of applying individualized prosthesis design and verification in tkr

a posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem the reciprocal theorem of betti and rayleigh or in other notions duality arguments are used to derive error estimators for the finite element approximation of various quantities including local variables like single displacements and stresses the proposed error estimator is evaluated solving the set of equations for an additional right hand side simply applying the energy norm error estimators two times furthermore a general h adaptive algorithm is introduced which allows us to optimize meshes with respect to different user specified variables the efficiency of the current approach is demonstrated for plate and shell examples

automated adaptive two dimensional system for the hp version of the finite element method this paper presents an automated adaptive finite element modelling system based on the hp version of the finite element method error estimation and correction indication procedures with an automatic mesh generator and a p version analysis code are combined to provide a computationally efficient modelling environment error prediction techniques are employed to find the best combination of finite element discretization h and overall polynomial order p needed to achieve the prescribed level of accuracy the automated adaptive modelling results demonstrate the capability of the modelling system of predicting discretizations to perform a p version finite element analysis to achieve prescribed levels of accuracy in the most efficient manner

minimal periods of semilinear evolution equations with lipschitz nonlinearity it is known that any periodic orbit of a lipschitz ordinary differential equation must have period at least 2 l where l is the lipschitz constant of f in this paper we prove a similar result for the semilinear evolution equation du dt au f u for each with 0 1 2 there exists a constant k such that if l is the lipschitz constant of f as a map from d a into h then any periodic orbit has period at least k l 1 1 as a concrete application we recover a result of kukavica giving a lower bound on the period for the 2d navier stokes equations with periodic boundary conditions

adaptive topology optimization of shell structures to solve generalized shape optimization problems material topology optimization is applied to determine the basic layout i e to insert and eliminate holes in the design space in addition the inner and outer contours are optimized in detail by shape optimization to improve the quality of the results and to increase the numerical efe ciency the design model is adapted during the optimization process two aspects of design adaptivity are considered on the one hand the parametrization of the design space is varied during the material topology optimization process and on the other hand the kind of design model is adapted during the overall optimization process switching between material topology and shape optimization the adaptive techniques are discussed and adaptive design modeling is verie ed for topology optimization problems of shell structures

recent developments in constrained optimization constrained optimization problems occur in many applications of science engineering social science and medicine and hence it is useful for practitioners in these areas to be informed about numerical methods for solving such problems however improvements in optimization typically do not spread quickly to the non specialist literature this paper contains a brief summary of two recent developments in the solution of nonlinearly constrained problems direct use of the lagrangian function and linearization of nonlinear constraints in order to highlight the new approaches we also discuss techniques used in older methods to illustrate the capabilities of modern optimization methods a brief description is given of the development of methods to solve large nonlinearly constrained problems that arise in the electrical power industry the general nonlinear programming problem may be stated in the following form

iterative method for constrained and mixed approximation an inexpensive improvement of f e m performance a general iterative procedure for solving mixed lagrangian finite element equations is introduced the method allows a dramatic improvement in accuracy of simple irreducible formulations to be achieved in a few resolutions applications to incompressible flow stress analysis and metal forming problems are given the procedures can be easily implemented into standard programs

a variant of sqp method for inequality constrained optimization and its global convergence in this paper a variant of sqp method for solving inequality constrained optimization is presented this method uses a modified qp subproblem to generate a descent direction as each iteration and can overcome the possible difficulties that the qp subproblem of the standard sqp method is inconsistency furthermore the method can start with an infeasible initial point under mild conditions we prove that the algorithm either terminates as kkt point within finite steps or generates an infinite sequence whose accumulation point is a kkt point or satisfies certain first order necessary condition finally preliminary numerical results are reported

a trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

a robust algorithm for optimization with general equality and inequality constraints an algorithm for general nonlinearly constrained optimization is presented which solves an unconstrained piecewise quadratic subproblem and a quadratic programming subproblem at each iterate the algorithm is robust since it can circumvent the difficulties associated with the possible inconsistency of qp subproblem of the original sqp method moreover the algorithm can converge to a point which satisfies a certain first order necessary optimality condition even when the original problem is itself infeasible which is a feature of burke and han s methods math programming 43 1989 pp 277 303 unlike burke and han s methods our algorithm does not introduce additional bound constraints the algorithm solves the same subproblems as the han powell sqp algorithm at feasible points of the original problem under certain assumptions it is shown that the algorithm coincides with the han powell method when the iterates are sufficiently close to the solution some global convergence results are proved and locally superlinear convergence results are also obtained preliminary numerical results are reported

a dual active set algorithm for positive semi definite quadratic programming because of the many important applications of quadratic programming fast and efficient methods for solving quadratic programming problems are valued goldfarb and idnani 1983 describe one such method well known to be efficient and numerically stable the goldfarb and idnani method suffers only from the restriction that in its original form it cannot be applied to problems which are positive semi definite rather than positive definite in this paper we present a generalization of the goldfarb and idnani method to the positive semi definite case and prove finite termination of the generalized algorithm in our generalization we preserve the spirit of the goldfarb and idnani method and extend their numerically stable implementation in a natural way

minimal periods of semilinear evolution equations with lipschitz nonlinearity revisited abstract we show that when a is a self adjoint sectorial operator on a hilbert space for 0 1 there exists a constant k depending only on such that if f d a x satisfies f u f v x l a u v x then any periodic orbit of the equation u a u f u has period at least k l 1 1 this generalises our previous result j c robinson a vidal lopez minimal periods of semilinear evolution equations with lipschitz nonlinearity j differential equations 220 2006 396 406 which was restricted to 0 1 2 and a 1 compact

finite element mesh control by integer programming a novel technique is presented for the control of mapped quadrilateral and hexahedral meshes generated inside a collection of connected subregions target division numbers on the subregion edges are given as input by allowing these numbers to vary meshes can be found which satisfy both the subregion constraints imposed by the mesh patterns being employed and the compatibility constraints on edges shared by two or more adjacent subregions among all such feasible solutions the optimal one is obtained by minimizing the changes to the division numbers using an integer programming technique

application of sequential quadratic programming software program to an actual problem we produced a nonlinear optimization software program which is based on a sequential quadratic programming sqp method schittkowski 1981a our program has several original characteristics 1 automatic choice between two qp solvers the goldfarb idnani gi method goldfarb and idnani 1983 and the least squares ls method schittkowski 1981b 2 an augmented lagrange function schittkowski 1981a as the objective function for line search 3 adaptive armijo method for line search 4 direct definition of upper and lower bounds for variables and constraint functions and 5 accurate numerical differentials these characteristics ensure the reliability of our program for solving standard problems in this paper point 3 is described in detail then the program is applied to an actual problem the optimal placement of blocks a model for this problem has been suggested by sha and dutton 1984 but it was unsuited to treatment by the sqp method thus we modify it to ensure program applicability

a successive quadratic programming method that uses new corrections for search directions abstract in this paper a successive quadratic programming method for solving general nonlinear programming problems is proposed and studied in order to avoid the maratos effect we present new corrections for the search directions global convergence and local superlinear convergence results are proved and some numerical results are reported

on the augmented lagrangian it is well known that for a nonlinear program any karush kuhn tucker point that satisfies the second order sufficient condition is also a strict local minimum of the augmented lagrangian with an underlying penalty parameter sufficiently large a main theme of the paper is to strengthen the theorem and bring to light an interesting connection among general critical points of a nonlinear program and its augmented lagrangian when a penalty parameter is sufficiently large it will be shown that a karush kuhn tucker point of a nonlinear program is also a critical point of its augmented lagrangian with the same index and the same nullity the aforementioned theorem is just the special case when the index and the nullity are zeros this result is established through an extension of a lemma due to finsler and debreu which is interesting in its own right consequently under a constraint qualification any karush kuhn tucker point can also be shown to satisfy a second order necessary condition for a local maximum of the generalized wolfe dual induced from the augmented lagrangian interestingly this relation holds without requiring the karush kuhn tucker point itself to satisfy any second order optimality condition

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner

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