a class of convex fuzzy mappings based on the concept of ordering defined by goetschel and voxman fuzzy sets and systems 18 1996 31 in this paper we present the concepts of epigraph and convexity of the fuzzy mappings and describe characteristics of the convex fuzzy mappings and quasi convex fuzzy mappings then we discuss the properties of convex fuzzy optimizations

preinvexity and φ 1 convexity of fuzzy mappings through a linear ordering the preinvexity prequasiinvexity 1 convexity and 1 quasiconvexity of fuzzy mappings are defined based on a linear ordering on the set of fuzzy numbers characterizations for these fuzzy mappings are obtained the local global minimum properties of real valued preinvex functions and 1 convex functions are extended to preinvex fuzzy mappings and 1 convex fuzzy mappings respectively it is also proved that every strict local minimizer of a prequasiinvex fuzzy mapping is a strict global minimizer and that every strict local minimizer of a 1 quasiconvex fuzzy mapping is a strict global minimizer

on integration of fuzzy mappings abstract in this paper we introduce the concept of an integral of a fuzzy mapping this generalises the integral introduced by matloka 10 and also extends riemann stieltjes integral over a closed interval to fuzzy mappings

thermomechanical homogenization analysis of axisymmetric inelastic solids at finite strains based on an incremental minimization principle the paper presents an attempt to extend homogenization analysis to axisymmetric solids under thermo mechanical loading in axisymmetric solids under axisymmetric thermomechanical loading and or torsion on both scales macro and micro the displacement and rotation response is by definition independent of the cylindrical angle co ordinate in homogenization analysis the deformation of the micro structure is driven by the deformation gradient f of the macro structure and enhanced by a micro scale fluctuation field such that x f x and in consequence f f f what is new on the micro scale the fact of independence of the cylindrical angle co ordinate imposes the homogeneous or taylor assumption on the fluctuation field of the r epresentative v olume e lement in the radial direction whereas the other two fluctuation fields the torsional angle and the axial displacement are not affected the thermomechanical problem on the macroscale is solved via a split approach an isentropic mechanical phase an isogeometrical thermal phase and in case of inelasticity an update phase of the internal micro variables the homogenization of inelastic solid materials at finite strains is based on an incremental minimization principle recently introduced by miehe et al j mech phys solids 2002 50 2123 2167 two finite element examples demonstrate the viability of the proposed approach copyright 2006 john wiley sons ltd

finite element analysis of a class of problems in finite elastoplasticity based on the thermodynamical theory of materials of type n abstract this paper describes a finite element approximation scheme for computing finite elastoplastic deformations of a hypothetical class of materials referred to as materials of type n kinematical descriptions of motion and strain rate and rotationally invariant stress rates measures appropriate for finite elastoplastic deformations are derived and the thermodynamical theory for materials of type n is briefly reviewed finite element approximations of the governing nonlinear equations are derived along with numerical algorithms for solving the associated discrete nonlinear system these new methods are applied to some representative plane strain problem including problems with unloading and reloading and the problem of upsetting of a partially constrained rectangular billet

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

existence of solutions to fuzzy differential equations with generalized hukuhara derivative via contractive like mapping principles abstract we study the existence and uniqueness of solution for fuzzy initial value problems in the setting of a generalized hukuhara derivative and by using some recent results of fixed point of weakly contractive mappings on partially ordered sets

finite deformation plasticity based on the additive split of the rate of deformation and hyperelasticity abstract finite deformation plasticity formulation based on additive split of rate of deformation and hyperelasticity is presented this approach is valid for finite elastic and plastic strains while rendering the choice and numerical integration of objective stress rates superfluous as the results are automatically objective for small elastic strains our method reduces to the classical hypoelastic corotational formulation provided that the dienes objective stress rate is employed while in the absence of inelastic deformation it coincides with the hyper elastic formulation the validity of the model has been examined on four test problems and the numerical results were found to be in good agreement with either the exact solution or experimental data

invex and generalized convex fuzzy mappings in this paper we introduce the concepts of pseudo convexity invexity and pseudo invexity for fuzzy mappings of one variable based on the notion of differentiability proposed by goetschel and voxman 4 and investigate the relationship between convex fuzzy mappings preinvex fuzzy mappings and these classes of fuzzy mappings we shall prove that pseudo convex fuzzy mappings and invex fuzzy mappings are pseudo invex and that a differentiable convex resp preinvex fuzzy mapping is pseudo convex resp invex in addition sufficient optimality conditions are obtained for pseudo convex invex and pseudo invex fuzzy mappings

unified lagrangian formulation for solid and fluid mechanics and fsi problems abstract we present a lagrangian monolithic strategy for solving fluid structure interaction fsi problems the formulation is called unified because fluids and solids are solved using the same solution scheme and unknown variables the method is based on a mixed velocity pressure formulation each time step increment is solved via an iterative partitioned two step procedure the particle finite element method pfem is used for solving the fluid parts of the domain while for the solid ones the finite element method fem is employed both velocity and pressure fields are interpolated using linear shape functions for quasi incompressible materials the solution scheme is stabilized via the finite calculus fic method the stabilized elements for quasi incompressible hypoelastic solids and newtonian fluids are called vps s element and vps f element respectively other two non stabilized elements are derived for hypoelastic solids one is based on a velocity formulation v element and the other on a mixed velocity pressure scheme vp element the algorithms for coupling the solid elements with the vps f fluid element are explained in detail the unified formulation is validated by solving benchmark fsi problems and by comparing the numerical solution to the ones published in the literature

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

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

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

a note on generalized convexity for fuzzy mappings through a linear ordering in de campos ibanez and gonzalez munoz fuzzy sets syst 29 145 154 1989 6 goestschel and voxman fuzzy sets syst 18 31 43 1986 7 the authors considered a linear ordering on the space of fuzzy intervals for each fuzzy mapping fuzzy interval valued mapping f based on the aforementioned linear ordering they introduced a real valued function t f on the domain of the fuzzy mapping f recently chalco cano et al fuzzy sets syst 231 70 83 2013 4 have studied the relationship between the generalized hukuhara differentiability of a fuzzy mapping f g differentiability for short and the differentiability of t f and some properties of local global minima this paper studies such properties for fuzzy mappings using new concepts which generalize the existing ones

consistent derivation of the constitutive algorithm for plane stress isotropic plasticity part i theoretical formulation abstract a derivation of the projected algorithm for general isotropic three invariant plasticity models under plane stress conditions is presented it is obtained by consistently specializing the 3d formulation to the 2d subspace defined by the plane stress condition closed form intrinsic algorithm linearization and a novel expression of the consistent tangent tensor are provided these are also shown to directly emanate from the analogous quantities pertaining to the fully 3d case a detailed discussion of the proposed implementation along with a representative set of numerical examples is provided in the second part of this paper valoroso n rosati l 2008 consistent derivation of the constitutive algorithm for plane stress isotropic plasticity part ii computational issues international journal of solids and structures 46 92 124

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

an implicit integration algorithm with a projection method for viscoplastic constitutive equations robinson s viscoplastic model a representative of the so called overstress models is integrated by use of the generalized midpoint rule the solution of the non linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with newton s method consistent tangent moduli are calculated and the quadratic convergence of the global newton equilibrium iteration is shown the time increment size is controlled by the convergence behaviour of the equilibrium iteration and the accuracy of the numerical integration various numerical examples are considered to demonstrate the efficiency of the methods

on the equivalence between the multiplicative hyper elasto plasticity and the additive hypo elasto plasticity based on the modified kinetic logarithmic stress rate abstract in two theorems presented herein we prove and subsequently demonstrate in several numerical examples involving homogeneous deformation that for isotropic materials hyper elasto plasticity models based on the multiplicative decomposition of the deformation gradient coincide with an additive hypo elasto plasticity model see section 3 2 that employs the spin tensor based on the modified kinetic logarithmic rate in the absence of strain induced anisotropy characterized by kinematic hardening herein this objective stress rate coincides with the kinetic logarithmic rate recently developed by jiao and fish 2017 we also show that other well known additive decomposition models such as those based on the jaumann and logarithmic rates may considerably deviate from the multiplicative model

on the newton method for solving fuzzy optimization problems in this article we consider optimization problems where the objectives are fuzzy functions fuzzy valued functions for this class of fuzzy optimization problems we discuss the newton method to find a non dominated solution for this purpose we use the generalized hukuhara differentiability notion which is the most general concept of existing differentiability for fuzzy functions this work improves and corrects the newton method previously proposed in the literature

a hybrid mixed model for non linear shell analysis and its applications to large rotation problems adopting an updated lagrange approach the general framework for the fully non linear analysis of curved shells is developed using a simple quadrilateral c0 model hmsh5 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified hellinger reissner type with independent assumptions for displacement and strain fields emphasis is placed on devising effective solution procedures to deal with large rotations in space finite stretches and generalized rate type material models in particular a geometrically exact scheme for configuration update is developed by making use of the so called exponential mapping algorithm and the resulting element was shown to exhibit a quadratic rate of asymptotic convergence in solving practical shell problems with newton raphson type iterative schemes for the purpose of updating the spatial stress field of the element an objective generalized midpoint integration rule is utilized which relies crucially on the concept of polar decomposition for the deformation gradient and is in keeping with the underlying mixed method finally the effectiveness and practical usefulness of the hmsh5 element are demonstrated through a number of test cases involving beams plates and shells undergoing very large displacements and rotations

velocity based formulations for standard and quasi incompressible hypoelastic plastic solids this is the accepted version of the following article franci a onate e and carbonell j m 2016 velocity based formulations for standard and quasi incompressible hypoelastic plastic solids int j numer meth engng 107 970 990 doi 10 1002 nme 5205 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 5205 abstract

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

generalized convexity in fuzzy vector optimization through a linear ordering in this article we study efficiency and weakly efficiency in fuzzy vector optimization after formulating the problem we introduce two new concepts of generalized convexity for fuzzy vector mappings based on the generalized hukuhara differentiability pseudoinvexity i and pseudoinvexity ii we prove that pseudoinvexity is the necessary and sufficient condition for a stationary point to be a solution of a fuzzy vector optimization problem we give conditions to insure that a fuzzy vector mapping is invex and pseudoinvex i and ii moreover we present some examples to illustrate the results lastly we use these results to study the class of problems which have uncertainty and inaccuracies in the objective function coefficients of mathematical programming models

convex fuzzy mapping with differentiability and its application in fuzzy optimization abstract the concepts of differentiability convexity generalized convexity and minimization of a fuzzy mapping are known in the literature the purpose of this present paper is to extend and generalize these concepts to fuzzy mappings of several variables using buckley feuring approach for fuzzy differentiation and derive karush kuhn tucker condition for the constrained fuzzy minimization problem

nonconvex fuzzy mappings and the fuzzy pre variational inequality in this paper we introduce the concepts of fuzzy pseudoconvex fuzzy invex fuzzy pseudoinvex and fuzzy preinvex mapping from r n to the set of fuzzy numbers and put forward the concepts of fuzzy extended directional differential subgradient and subdifferential some properties are obtained for fuzzy pseudoconvex fuzzy invex fuzzy pseudoinvex and fuzzy preinvex mapping relations are discussed between the fuzzy pre variational inequality and the fuzzy vector optimization problem

on convex and concave fuzzy mappings abstract in this paper we introduce the concepts of convex subsets of f 0 the set of all fuzzy numbers and monotonic fuzzy mappings on r2 and show that a monotonic fuzzy mapping f on a convex set c r1 or r2 is quasi convex and quasi concave we establish two characterizations for convex fuzzy mappings and two characterizations for concave fuzzy mappings we also prove the least upper bound property

a hyperelastic based large strain elasto plastic constitutive formulation with combined isotropic kinematic hardening using the logarithmic stress and strain measures summary this paper addresses the formulation of a set of constitutive equations for finite deformation metal plasticity the combined isotropic kinematic hardening model of the infinitesimal theory of plasticity is extended to the large strain range on the basis of three main assumptions i the formulation is hyperelastic based ii the stress strain law preserves the elastic constants of the infinitesimal theory but is written in terms of the hencky strain tensor and its elastic work conjugate stress tensor and iii the multiplicative decomposition of the deformation gradient is adopted since no stress rates are present the formulation is of course numerically objective in the time integration it is shown that the model gives adequate physical behaviour and comparison is made with an equivalent constitutive model based on the additive decomposition of the strain tensor

some computational aspects of large deformation rate dependent plasticity problems abstract the governing equations for a finite element formulation of boundary value problems for large deformation metal forming processes are derived using a principle of virtual work formulated in a lagrangian reference system the updated lagrangian method is used to simplify the equations the resulting nonlinear equilibrium equations are solved using newton s method a constitutive model for large deformation rate dependent plasticity is developed the model incorporates a one parameter implicit integration operator for stability and accuracy the stressrate strain rate relation is written in terms of the jaumann rate of stress numerous example problems are solved to demonstrate the effectiveness of the numerical algorithms

duality theory in fuzzy mathematical programming problems with fuzzy coefficients in this paper the notions of subgradient subdifferential and differential with respect to convex fuzzy mappings are investigated which provides the basis for the fuzzy extremum problem theory we consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and or sufficient optimality conditions furthermore the concept of saddle points and minimax theorems under fuzzy environment is discussed the results obtained are used to formulate the lagrangian dual of fuzzy programming under certain fuzzy convexity assumptions kkt conditions for fuzzy programming are derived and the perturbed convex fuzzy programming is considered finally these results are applied to fuzzy linear programming and fuzzy quadratic programming

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

a consistent dxdr method for elastic plastic problems this paper proposed a new efficient method the consistent dxdr method for analysing general elastic plastic problems two important factors in computational plasticity the convergence and stability were addressed the method was constructed through a natural combination of two vital components the dxdr algorithm which had been proved to be a powerful equation solver developed by the authors and a stable consistent algorithm for the integration of the constitutive equations of plasticity numerical examples demonstrated in detail the efficiency accuracy and stability attainable in solving various engineering problems

a new fictitious time for the dynamic relaxation dxdr method this paper addresses the development of the dxdr method by introducing a modified fictitious time mft increment the mft is determined by minimizing the residual force after each iteration the rank of the convergence rate shows the advantage of the new method the results obtained from plate and truss analyses demonstrate the potential of the new method it is shown that compared with a unit fictitious time the mft is more efficient especially during the initial iterations moreover mft does not impose any additional constraints on the dxdr method

directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming abstract in this paper we put forward the concepts of directional derivative differential and subdifferential of fuzzy mappings from r n into e 1 and discuss the characterizations of directional derivative and differential by respectively using the directional derivative and differential of two crisp functions that are determined by the fuzzy mapping and we also consider the problem of existence of directional derivative for convex fuzzy mappings and discuss the relations among directional derivative differential and subdifferential of fuzzy mappings at last we give two results of application in convex fuzzy programming

diffusions on symmetric spaces of type a iii and random matrix theories for rectangular matrices isotropic diffusion processes on cosets and and their zero curvature limit are studied from a unified viewpoint as indicated in our previous works the projection of the fokker planck equation onto the maximal commutative subgroup of these cosets can be described by using the radial part of the laplace beltrami operator by taking the zero curvature limit an integral which is a natural extension of the itzykson zuber integral to the rectangular matrices is explicitly evaluated the probability density function obtained from the diffusion on is studied in detail which can be applied to the quantum transport problem the explicit expressions for the probability density function in the metallic and insulating regimes are obtained for the metallic regime the integral representation for the hypergeometric function is used and the results are exact furthermore by using the orthogonal polynomial method n point correlation functions are obtained exactly for arbitrary n

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

operator split methods for the numerical solution of the elastoplastic dynamic problem the elastoplastic dynamic problem is first formulated in a form that facilitates the application of product formula techniques the additive decomposition of the dynamic equations into elastic and plastic parts is taken as a basis for the definition of product algorithms that exploit such decomposition in the context of a finite element discretization these product algorithms entail for every time step the solution of an elastic problem followed by the application of plastic algorithms that operate on the stresses and internal variables at the integration points and bring in the plastic constitutive relations suitable plastic algorithms are discussed for the cases of perfect and hardening plasticity and viscoplasticity the proposed formalism does not depend on any notion of smoothness of the yield surface and is applicable to arbitrary convex elastic regions with or without corners the stability properties of the product algorithm are identical to those of the elastic algorithm used whereas the computational expense is practically equal to that of an elastic problem

random matrices beyond the cartan classification it is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of lie groups or else with tangent spaces of the same this results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables in this paper we show that a large number of non hermitean random matrix ensembles defined by physically motivated symmetries chiral symmetry time reversal invariance space rotation invariance particle hole symmetry or different reality conditions can likewise be identified with symmetric spaces we give explicit representations of the random matrix ensembles identified with lateral algebra subspaces and of the corresponding symmetric subalgebras spanning the group of invariance among the ensembles listed we identify as special cases all the hermitean ensembles identified with the cartan classes of symmetric spaces and the three ginibre ensembles with complex eigenvalues

numerical analysis of large deformation processes at elevated temperatures abstract the authors describe a theoretical model and kinematic approach suitable for the numerical analysis of problems at large deformation finite strains and rotations the presented formulations are implemented incrementally with a non linear and rate dependent constitutive model adequate to the simulation of high temperature processes the large deformation kinematic model is based on the additive decomposition of the transformation gradient and the strain rate tensor in thermoelastic and viscoplastic parts stress strain relations leading to the constitutive equations governing finite deformations are formulated on an unrotated frame of reference these relations are obtained either by polar decomposition of the transformation gradient or by the rotation increment method both approaches and algorithms are thoroughly compared and discussed the flow rule is a function of the equivalent stress and the deviatoric stress tensor of the temperature field and of a set of internal state variables the increments of the state variables are calculated with a forward gradient time integration procedure based on a numerical estimation of the integral of the strain rate tensor the model was implemented in a three dimensional finite element code and tested with a set of tensile and shear tests on aluminium alloy specimens in order to evaluate the performance of the numerical algorithms with elevated temperature processes tests were done at temperatures ranging from room temperature to 580 k numerical simulation results are compared with experimental results and shown to be in good agreement real material behaviour is fairly well predicted and matches the experimental results obtained with different loading conditions at different temperatures

finite elastoplastic deformations of space curved membranes abstract under restriction of an isotropic elastic response of deformed lattice a covariant theory of finite elastoplasticity is developed in principal axes of a pair of deformation tensors in material description the tensor pair consists of the plastic deformation tensor and the total deformation cauchy green tensor the proposed theory is applied to elastoplastic membranes whose reference and current configurations can be arbitrary space curved surfaces pressure insensitive von mises yield criterion and isotropic hardening are considered as a model problem with a particular form of the strain energy function given in terms of elastic principal stretches through an explicit enforcement of the plane stress condition we arrive at a reduced two dimensional problem representation which is set in the membrane tangent plane numerical implementation details are given to show an important role of the operator split methodology in simplifying the state update computation and the computation of the consistent tangent modulus a set of numerical examples illustrates the performance of the presented theory and indicates some of the possible areas of application

evaluate fuzzy optimization problems based on biobjective programming problems abstract the embedding theorem shows that the set of fuzzy numbers can be embedded into a banach space inspired by this embedding theorem we propose a solution concept of the fuzzy optimization problem by solving a biobjective optimization problem which is transformed from the original fuzzy optimization problem using the embedding theorem we then also consider the fuzzy optimization problem with fuzzy coefficients i e the coefficients are assumed to be fuzzy numbers under a setting of core values of fuzzy numbers we show that the optimal solution of its corresponding crisp optimization problem the usual optimization problem is also an optimal solution of the original fuzzy optimization problem

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

convexity and local lipschitz continuity of fuzzy valued mappings abstract two concepts of a convexity and a local lipschitz continuity are introduced into the class of fuzzy valued mappings the convexity is defined through the fuzzy max order on the class of fuzzy numbers the local lipschitz continuity of the fuzzy valued mapping cannot be defined in the same manner as in the case of the real valued function since the space of values of the fuzzy valued mapping is not a linear space owing to this reason the local lipschitz continuity is defined in a new formula through the fuzzy max order a fundamental theorem concerning the local lipschitz continuity of the fuzzy valued convex mapping is given

finite time lyapunov exponents for products of random transformations i show how continuous products of random transformations constrained by a generic group structure can be studied by using iwasawa s decomposition into angular diagonal and shear degrees of freedom in the case of a gaussian process a set of variables adapted to the iwasawa decomposition and still having a gaussian distribution is introduced and used to compute the statistics of the finite time lyapunov spectrum of the process the variables also allow to show the exponential freezing of the shear degrees of freedom which contain information about the lyapunov eigenvectors

generalized convex fuzzy mappings and fuzzy variational like inequality in this paper we introduce the concepts of some generalized convex fuzzy mappings from r n to the set of fuzzy numbers under the condition of upper or lower semicontinuity a new criterion be obtained for the existence of a fuzzy preinvex mapping a fuzzy preinvex mapping can be achieved when the invex subset of r n satisfies certain conditions the relations between generalized convex fuzzy mappings are discussed and some properties are obtained that relate the concepts of fuzzy pseudoinvex and fuzzy prequasiinvex mapping fuzzy invex and fuzzy preinvex mapping fuzzy preinvex and fuzzy prequasiinvex mapping we also discuss the relationship between the fuzzy variational like inequality the fuzzy weak variational like inequality and fuzzy optimization problems and an application example of a fuzzy variational like inequality the fuzzy variational like inequality representation of a fuzzy transportation equilibrium problem

formulation of implicit finite element methods for multiplicative finite deformation plasticity some constitutive and computational aspects of finite deformation plasticity are discussed attention is restricted to multiplicative theories of plasticity in which the deformation gradients are assumed to be decomposable into elastic and plastic terms it is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non symmetric even for associated plastic flow both explicit i e no equilibrium iteration and implicit finite element formulations are considered an assumed strain formulation is used to accommodate the near incompressibility associated with fully developed isochoric plastic flow as an example of explicit integration the rate tangent modulus method is reviewed in some detail an implicit scheme is derived for which the consistent tangents resulting in quadratic convergence of the equilibrium iterations can be written out in closed form for arbitrary material models all the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method

fuzzy weirstrass theorem and convex fuzzy mappings the convexity and continuity of fuzzy mappings are defined through a linear ordering and a metric on the set of fuzzy numbers the local global minimum property of real valued convex functions is extended to convex fuzzy mappings it is proved that a strict local minimizer of a quasiconvex fuzzy mapping is also a strict global minimizer characterizations for convex fuzzy mappings and quasiconvex fuzzy mappings are given in addition the weirstrass theorem is extended from real valued functions to fuzzy mappings

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

optimality conditions for fuzzy constrained programming problems abstract this paper solves optimization problems where both the objective and constraints are given by fuzzy functions in order to get it we first prove that these problems are equivalent to optimization problems where the constraints functions are non fuzzy functions and we introduce a new and wider stationary point concept that generalizes all existing concepts so far this new stationary point concept is based on the gh differentiability and has many computational advantages that we describe it is well known that obtain a useful differentiability notion for fuzzy functions is a difficult task without linearity and we are in that case due to the fact that the fuzzy numbers intervals space is a nonlinear one in this direction the gh derivative for fuzzy functions is a concept that is more general than hukuhara and level wise derivatives that are usually used in fuzzy optimization so far in the sense that they can be applied to a wider number of fuzzy function classes than above concepts with this new differentiability concept we prove a necessary optimality condition for fuzzy optimization problems that is more operational and less restrictive that the few ones we can find in the literature so far moreover due to the fact that we do not have a linear space for fuzzy numbers the convex concepts and generalized convex fuzzy function notion are very restrictive also this implies that the sufficiency optimality conditions for fuzzy problems published so far are not useful

a subjective approach for ranking fuzzy numbers abstract starting from a subjective assignation of weights related to the relative importance of the different level sets we define a new method of comparison of fuzzy numbers this procedure is an extension of some well known indices adamo tsumura et al yager some properties of our index are studied in this paper as well as its behaviour on several particular cases

design sensitivity analysis of elastoplastic structures an iterative and incremental algorithm is presented for the design sensitivity analysis dsa of elastoplastic structures geometrical non linearity is included in the formulation and the structure may be subjected to a cyclic loading the key concept is discussed using an elastoplastic truss it is shown that the design sensitivity coefficients dscs are path dependent and discontinuous at the time at which yielding takes place in a member yield time in the proposed algorithm the yield time is considered to be a function of the design variables in this way the discontinuity in the dscs at the material transition points can be overcome in a simple and routine way incremental response analysis and dsa are carried out simultaneously using the newton raphson type iterative procedure since the proposed algorithm is completely consistent with the analysis procedure it can be implemented into an existing code for structural analysis application to distributed parameter structures with kinematic and or isotropic hardening is discussed using the von mises yield condition and the elastic predictor radial return method dsa is carried out for a ten bar truss with a piecewise linear constitutive relation it is shown that the dscs found using the proposed method agree quite well with those calculated by the central difference method finally an alternate method without any iteration for dsa is proposed and the results from the two methods are compared

an eulerian projection method for quasi static elastoplasticity a well established numerical approach to solve the navier stokes equations for incompressible fluids is chorin s projection method 1 whereby the fluid velocity is explicitly updated and then an elliptic problem for the pressure is solved which is used to orthogonally project the velocity field to maintain the incompressibility constraint in this paper we develop a mathematical correspondence between newtonian fluids in the incompressible limit and hypo elastoplastic solids in the slow quasi static limit using this correspondence we formulate a new fixed grid eulerian numerical method for simulating quasi static hypo elastoplastic solids whereby the stress is explicitly updated and then an elliptic problem for the velocity is solved which is used to orthogonally project the stress to maintain the quasi staticity constraint we develop a finite difference implementation of the method and apply it to an elasto viscoplastic model of a bulk metallic glass based on the shear transformation zone theory we show that in a two dimensional plane strain simple shear simulation the method is in quantitative agreement with an explicit method like the fluid projection method it is efficient and numerically robust making it practical for a wide variety of applications we also demonstrate that the method can be extended to simulate objects with evolving boundaries we highlight a number of correspondences between incompressible fluid mechanics and quasi static elastoplasticity creating possibilities for translating other numerical methods between the two classes of physical problems

convergence analysis of solution sets for fuzzy optimization problems abstract in this paper we first consider fuzzy optimization problem in short fop and sequences of fuzzy optimization problems in short fop n then we establish the concept of d convergence of a sequence of fuzzy mappings and its characterizations finally the painleve kuratowski lower convergence and the painleve kuratowski convergence of approximate solution sets for these problems are proposed using d convergence of a sequence of fuzzy mappings some examples are provided to illustrate the essentialness of the imposed assumptions

convex fuzzy mappings abstract in this note the concept of convexity and logarithmic convexity for fuzzy mappings are introduced and some applications to optimization are discussed

disordered wires from a geometric viewpoint the mello pereyra kumar theory is extended to arbitrary spin classification yields three universality classes the average over disorder is related to geometrical concepts the solution of the resulting differential equation is discussed localization lengths and universal conductance fluctuations are computed

duality theorems in fuzzy mathematical programming problems based on the concept of necessity the fuzzy valued lagrangian function for the fuzzy mathematical programming problem via the concept of the fuzzy scalar inner product is proposed a solution concept of fuzzy optimization problems which is essentially similar to the notion of pareto solution in multiobjective programming problems is also introduced by ranking the fuzzy numbers using the necessity indices under these settings the weak and strong duality theorems can be elicited we show that the primal and dual fuzzy mathematical programming problems have no duality gap under suitable convexity assumptions for fuzzy valued functions

is an additive decomposition of a rate of deformation and objective stress rates passe abstract the manuscript critically examines a half a century old idea of decomposing the rate of deformation into elastic and plastic parts we show that even the most recent additive variant based on the logarithmic rate is inconsistent with the notion of elasticity in so called unloading stress ratchetting obstacle test while the earlier corotational variants based on the jaumann and green naghdi rates are well known not to be integrable we then propose a stress dependent so called kinetic logarithmic stress rate that is both integrable and passes the unloading stress ratchetting obstacle test we also demonstrate that the additive hypo elasto plasticity model based on the proposed kinetic logarithmic rate is weakly invariant under isochoric reference change for simple materials in the sense of noll

a study of the ranking function approach through mean values abstract the average value was introduced to help in the ordering of fuzzy numbers and it was defined by means of an integrating process of a parametric function representing the position of every cut in the real line by considering the relation between a particular case of the average value and the notion of expectation for fuzzy numbers we interpret the general ranking function approach as a ranking process using a mean value and an interval relation some well known indices are included in this schema using the general average value a more general mean value for a fuzzy number is defined finally some properties of the general mean value are studied as well as its behaviour on several particular cases

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

diffusion in a one dimensional random medium and hyperbolic brownian motion classical diffusion in a random medium involves an exponential functional of brownian motion this functional also appears in the study of brownian diffusion on a riemann surface of constant negative curvature we analyse in detail this relationship and study various distributions using stochastic calculus and functional integration

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

a family of integration algorithms for constitutive equations in finite deformation elasto viscoplasticity a two parameter family of incrementally objective integration schemes is proposed for the analysis of a broad range of unified rate dependent viscoplastic constitutive models in large deformation problems a similar scheme can be applied to rate independent solids as well these algorithms are a generalization of the mid point integration rule full linearization of the principle of virtual work is performed in an updated lagrangian framework together with a calculation of the consistent linearized moduli some details of the finite element implementation are given for plane strain and axisymmetric problems the method is compared with other objective integration schemes and is tested with several examples where large strains and rotations occur

an analysis of metal forming processes using large deformation elastic plastic formulations a concise survey of the literature related to the large deformation elasto plasticity problems including unilateral contact and friction is presented together with an extension of the friction law for large deformation analysis starting from the principle of virtual work the so called total lagrangian and updated lagrangian formulations are derived based on some fundamental assumptions in linearizing the nonlinear equations by introducing the zaremba jaumann co rotational increment to the cauchy stress tensor the classical prandtl reuss equations are generalized for describing the elastic plastic material behavior to allow a proper consideration of the contact conditions in the incremental analysis a general friction law with an associated isotropic coulomb sliding rule is obtained by the similarity between dry friction and plasticity finite element discretizations and approximations are applied to the resulting formulation of the updated lagrangian approach four example problems are solved to test the formulations developed in this paper the emphasis is made toward the numerical accuracy of the finite element solutions

modelling and numerical analysis of heat treatments on aluminium parts a theoretical model suitable for the modelling and analysis of heat treatment problems is described the presented formulations are implemented incrementally with a non linear rate dependent constitutive model adequate to the simulation of a wide range temperature processes the large deformation model is based on the additive decomposition of the strain rate tensor in elastic thermal and viscoplastic parts the flow rule is a function of the equivalent stress and the deviatoric stress tensor of the temperature field and of a set of internal state variables the thermal problem is solved by a prediction correction algorithm the prediction which uses a semi implicit integration provides a first estimation of the temperature field the correction stage performed with a newton raphson implicit scheme improves the solution until a satisfactory result is found the thermomechanical coupled problem is solved with a staggered approach the finite element formulations for coupled analysis of deformation and heat transfer given in this article are validated and applied to the process of quenching of aluminium parts into boiling water copyright 2006 john wiley sons ltd

on the accuracies of numerical integration algorithms for gurson based pressure dependent elastoplastic constitutive models abstract a class of generalized mid point algorithms for pressure dependent elastoplastic models is formulated in the paper the accuracies of the formulated generalized mid point algorithms including the euler backward algorithm and the one step euler forward algorithm are systematically analyzed against the exact solution for gurson based pressure dependent elastoplastic model the accuracies of the algorithms are assessed by means of iso error maps results show that the formulated generalized mid point algorithms are reasonably accurate in both small and large increment steps it is found that in all the cases considered the maximum errors in the presence of volumetric strain increments are less than those without volumetric strain increments when the deviatoric strain increments are given in the radial direction the true mid point algorithm is the most accurate one furthermore the optimal value of the generalized mid point algorithms in terms of maximum errors is observed between 0 5 and 1 for both small and large increment steps the one step euler forward algorithm gives the worst accuracy

on the computation of design derivatives for huber mises plasticity with non linear hardening this paper concerns design sensitivity analysis dsa for an elasto plastic material with material parameters depending on or serving as design variables the considered constitutive model is huber mises deviatoric plasticity with non linear isotropic kinematic hardening one which is applicable to metals the standard radial return algorithm for linear hardening is generalized to account for non linear hardening functions two generalizations are presented in both the non linearity is treated iteratively but the iteration loop contains either a scalar equation or a group of tensorial equations it is proven that the second formulation which is the one used in some parallel codes can be equivalently brought to a scalar form more suitable for design differentiation the design derivatives of both the algorithms are given explicitly enabling thus calculation of the explicit design derivative of stresses entering the global sensitivity equation the paper addresses several issues related to the implementation and testing of the dsa module among them the concept of verification tests both outside and inside a fe code as well as the data handling implied by the algorithm the numerical tests which are used for verification of the dsa module are described they shed light on a the accuracy of the design derivatives by comparison with finite difference computations and b the effect of the finite element formulation on the design derivatives for an isochoric plastic flow copyright 2003 john wiley sons ltd

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

embedding of level continuous fuzzy sets on banach spaces abstract in this paper we present an extension of the minkowski embedding theorem showing the existence of an isometric embedding between the class f c x of compact convex and level continuous fuzzy sets on a real separable banach space x and c 0 1 b x the banach space of real continuous functions defined on the cartesian product between 0 1 and the unit ball b x in the dual space x also by using this embedding we give some applications to the characterization of relatively compact subsets of f c x in particular an ascoli arzela type theorem is proved and applied to solving the cauchy problem x t f t x t x t0 x0 on f c x

numerical integration of rate constitutive equations in finite deformation analysis abstract in analysis of finite deformation problems the use of constitutive equations in rate form is often required in a spatial setting these equations may express a relationship between some objective rate of spatial stress tensor and the rate of deformation constitutive equations of this type characterize a variety of material models including hyperelasticity hypoelasticity and elastoplasticity employing geometrical concepts a family of unconditionally stable and incrementally objective algorithms is proposed for the integration of such rate constitutive equations these algorithms which are appropriate for finite deformation analysis are applicable to any choice of stress rate and in most cases employ quantities that arise naturally in the context of finite element analysis examples illustrate the objectivity and accuracy of the algorithms

on the numerical integration of a class of pressure dependent plasticity models an unconditionally stable algorithm for the numerical integration of elastoplastic pressure dependent constitutive relations is analysed in detail in this paper the application of the method to plane stress problems in which the out of plane strain component is not defined kinematically is discussed the tangent moduli resulting from this integration algorithm are obtained by consistent linearization of the elastoplastic constitutive equations the algorithm is applied to gurson s constitutive model some one dimensional problems are solved and comparisons with exact solutions are made the paper closes with a numerical study of the necking of an axi symmetric specimen using gurson s plasticity model to describe the constitutive behaviour of the material

a framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition part ii computational aspects computational aspects of the formulation of elastoplasticity proposed in part i of this work employing a multiplicative decomposition and based on the principle of maximum plastic dissipation are examined in detail in particular the implementation of the proposed hyperelastic j2 flow theory is shown to reduce to a trivial modification of the classical radial return algorithm which is amenable to exact linearization this results in a closed form expression for the consistent elastoplastic tangent moduli the algorithmic treatment of the general case is considered subsequently it is emphasized that objective rates and incrementally objective algorithms play no role in the present approach a number of numerical experiments are presented that illustrate the excellent performance of the proposed formulation

generalized derivative and π derivative for set valued functions in this paper we study the generalized derivative and the p derivative for interval valued functions we show the connections between these derivatives some illustrative examples and applications to interval differential equations and fuzzy functions are presented

karush kuhn tucker conditions for interval and fuzzy optimization in several variables under total and directional generalized differentiability abstract we extend the interval and fuzzy gh differentiability to consider interval and fuzzy valued functions of several variables and to include directional gh differentiability the proposed setting is more general than the existing definitions in the literature and allows a unified view of total and direction gh differentiability and for the computation of partial gh derivatives directional gh derivative and level wise gh differentiability in the fuzzy valued case a concept of gh differential is then deduced and its values are used to define an abstract local tangency property for a gh differentiable function similar to the well known tangency between a differentiable function and its tangent plane the proposed new setting allows an analysis of conditions for local optimality dominance with respect to interval and level wise partial orders well known in the literature in terms of directional gh derivatives including concepts of local convexity and to formulate kkt like conditions for non dominated solutions in constrained optimization problems