Computational implementation of stress integration in FE analysis of elasto-plastic large deformation problems

In this paper, we compare different numerical implementation algorithms for the rate type constitutive equation and present an integration scheme based on the physical meaning of the stress. Numerical implementation of various schemes is investigated in conjunction with the return mapping algorithm and the conditions to maintain plastic consistency. Jaumann and Truesdell rates are taken as the objective stress rates in the constitutive equation. An alternative numerical treatment for rate of deformation tensor D_ij is presented and is shown to maintain incremental objectivity. Numerical examples included a single element under rigid body rotation, a necking bifurcation of a bar in tension and a punch indentation process. It is shown that the use of Truesdell stress rate with specific numerical integration procedure gives more accurate results than other procedures presented.     

Geometrical super-elements for elasto-plastic shells with large deformation

This paper presents the application of the so-called Geometrical Elements Method (Lukasiewicz and Szyszkowski, 1974; Pogorelov, 1967) to the solution of elasto-plastic problems of shells. The approach is based on the observation that, during large deformations, the shell structure deforms in a nearly isometrical manner. Therefore, its deformed shape can be determined and analysed making use of the Gauss theorem according to which the Gaussian curvature of the isometrically deformed surface remains unchanged. The shell structure is subdivided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. The equilibrium of the whole structure is defined by the stationary value of the Hamiltonian function which requires the calculation of the strain energy in the elements. This can easily be obtained if we recognize that the isometrically deformed elements contain only bending energy. Using the method described, we are able to significantly the number of unknown values defining the shape of the deformed structure. The problem is reduced to the numerical evaluation of the minimum of a function of many variables. The elasto-plastic state of stress of the plastic material in the structure canbe determined by using the deformation theory of plasticity or the theory of plastic flow. Also, the strains and stresses in the plastic regions are the only functions of the assumed displacements field. The corresponding energy of the plastic deformation can easily be evaluated and added to the minimized functionals. For example, the elasto-plastic behaviour of a spherical shell under a concentrated load is studied. The solution obtained defines the large deformation behaviour and the motion of the plastic zones on the surface of the shell.     

Coupled problems of thermoviscoelasticity

The author derives a heat flux equation with allowance for the dissipation of internal forces on the basis of the fundamental equations of continuum mechanics and the thermodynamics of irreversible processes.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, No. 3, pp. 415–421, May–June, 1969.     

On the h-adaptive coupling of FE and BE for viscoplastic and elasto-plastic interface problems

This paper presents a posteriori error estimates for the symmetric finite element and boundary element coupling for a nonlinear interface problem: A bounded body with a viscoplastic or plastic material behaviour is surrounded by an elastic body. The nonlinearity is treated by the finite element method while large parts of the linear elastic body are approximated using the boundary element method. Based on the a posteriori error estimates we derive an algorithm for the adaptive mesh refinement of the boundary elements and the finite elements. Its implementation is documented and numerical examples are included.     

Scalarization of the fuzzy optimization problems

Scalarization of the fuzzy optimization problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Two solution concepts are proposed by considering two convex cones. The set of all fuzzy numbers can be embedded into a normed space. This motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to solve the fuzzy optimization problems. By applying scalarization to the optimization problem with fuzzy coefficients, we obtain its corresponding scalar optimization problem. Finally, we show that the optimal solution of its corresponding scalar optimization problem is the optimal solution of the original fuzzy optimization problem.     

Balanced optimization problems

Suppose we are given a finite set E, a family F of ‘feasible’ subsets of E and a real weight c(e) associated with every e?E. We consider the problem of finding S?F for which max {c(e)?c(e′): e, e′ ?S} is minimized. In other words, the differenc value between the largest and smallest value used should be as small as possible. We show that if we can efficiently answer the feasibility question then we can efficiently solve the optimization problem. We specialize these results to assignment problems and thereby obtain on O(n⁴) algorithm for ‘balanced’ assignment problems.     

Mixed equilibrium problems and optimization problems

In this paper, we introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings. Furthermore, we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions. We also discuss some special cases. Results obtained in this paper improve the previously known results in this area.     

On the reciprocal vector optimization problems

A necessary and sufficient condition is established for an optimal solution of a primal vector optimization problem to be an optimal solution of its reciprocal. Such a condition is developed and analyzed in the Pareto case, the strong case, and the lexicographic case. We detail these results for ordinary (i.e., scalar) optimization problems.     

Scalarizing vector optimization problems

A scalarization of vector optimization problems is proposed, where optimality is defined through convex cones. By varying the parameters of the scalar problem, it is possible to find all vector optima from the scalar ones. Moreover, it is shown that, under mild assumptions, the dependence is differentiable for smooth objective maps defined over reflexive Banach spaces. A sufficiency condition of optimality for a general mathematical programming problem is also given in the Appendix.     

Lexicographic balanced optimization problems

We introduce the lexicographic balanced optimization problem (LBaOP) and show that it can be solved efficiently if an associated lexicographic bottleneck problem can be solved efficiently. For special cases of cuts in a graph and base system of a matroid, improved algorithms are proposed. A generalization of LBaOP is also discussed.     

On minimax optimization problems

We give a short proof that in a convex minimax optimization problem ink dimensions there exist a subset ofk + 1 functions such that a solution to the minimax problem with thosek + 1 functions is a solution to the minimax problem with all functions. We show that convexity is necessary, and prove a similar theorem for stationary points when the functions are not necessarily convex but the gradient exists for each function.     

Equilibrium constrained optimization problems

We consider equilibrium constrained optimization problems, which have a general formulation that encompasses well-known models such as mathematical programs with equilibrium constraints, bilevel programs, and generalized semi-infinite programming problems. Based on the celebrated KKM lemma, we prove the existence of feasible points for the equilibrium constraints. Moreover, we analyze the topological and analytical structure of the feasible set. Alternative formulations of an equilibrium constrained optimization problem (ECOP) that are suitable for numerical purposes are also given. As an important first step for developing efficient algorithms, we provide a genericity analysis for the feasible set of a particular ECOP, for which all the functions are assumed to be linear.     

Infinitely constrained optimization problems

A generalized cutting-plane algorithm designed to solve problems of the form min{f(x) :x X andg(x,y) 0 for ally Y} is described. Convergence is established in the general case (f,g continuous,X andY compact). Constraint dropping is allowed in a special case [f,g(·,y) convex functions,X a convex set]. Applications are made to a variety of max-min problems. Computational considerations are discussed.Dr. Falk's research was supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under AFOSR Contract No. 73–2504.     

Implicitly defined optimization problems

This paper considers the solution of the problem: inff[y, x(y)] s.t.y [y, x(y)] E ^k , wherex(y) solves: minF(x, y) s.t.x R(x, y) E ⁿ . In order to obtain local solutions, a first-order algorithm, which uses {dx(y)/dy} for solving a special case of the implicitly definedy-problem, is given. The derivative is obtained from {dx(y, r)/dy}, wherer is a penalty function parameter and {x(y, r)} are approximations to the solution of thex-problem given by a sequential minimization algorithm. Conditions are stated under whichx(y, r) and {dx(y, r)/dy} exist. The computation of {dx(y, r)/dy} requires the availability of _y F(x, y) and the partial derivatives of the other functions defining the setR(x, y) with respect to the parametersy.Research sponsored by National Science Foundation Grant ECS-8709795 and Office of Naval Research Contract N00014-89-J-1537. We thank the referees for constructive comments on an earlier version of this paper.     

Extended well-posedness of optimization problems

The well-posedness concept introduced in Ref. 1 for global optimization problems with a unique solution is generalized here to problems with many minimizers, under the name of extended well-posedness. It is shown that this new property can be characterized by metric criteria, which parallel to some extent those known about generalized Tikhonov well-posedness.This work was partially supported by MURST, Fondi 40%, Rome, Italy.     

Scalarization of vector optimization problems

In this paper, we investigate the scalar representation of vector optimization problems in close connection with monotonic functions. We show that it is possible to construct linear, convex, and quasiconvex representations for linear, convex, and quasiconvex vector problems, respectively. Moreover, for finding all the optimal solutions of a vector problem, it suffices to solve certain scalar representations only. The question of the continuous dependence of the solution set upon the initial vector problems and monotonic functions is also discussed.The author is grateful to the two referees for many valuable comments and suggestions which led to major imporvements of the paper.     

An exact finite deformation elasto-plastic solution for the outside-in free eversion problem of a tube of elastic linear-hardening material

An exact solution is obtained in this paper for the elasto-plasticoutside-in free eversion problem of a tube of elastic linear-hardeningmaterial using a tensorial formulation. The solution is basedon a finite-strain version of Hencky's deformation theory, thevon Mises yield criterion, and the assumptions of volume incompressibilityand axial length constancy. All expressions for the stress,strain distributions and the eversion load are derived in anexplicit form. In addition, with both the linear-elastic andstrain-hardening-plastic responses of the material being includedand with the thickness effect of the tube being incorporated,this solution provides a rigorous and complete theoretical analysisof the elasto-plastic eversion problem, unlike existing solutions.Two specific solutions are also presented as limiting casesof the solution. Also provided are some numerical results andthe related observations to show quantitatively applicationsof the solution.     

Reducibility of bilevel programming problems to vector optimization problems

Reductions are studied of the bilevel programming problems to vector (multicriteria) optimization problems. A general framework is proposed for constructing these reductions. Some particular cases of bilevel problems are considered.