satisficing solutions and duality in interval and fuzzy linear programming in this paper we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions the necessary tools for dealing with such problems in this way we show that the class of crisp classical lp problems can be embedded into the class of flp ones moreover for flp problems we define the concept of duality and prove the weak and strong duality theorems further we define a class of interval linear programming problems as a special subclass of flp problems and apply the previous results to this special case

comparison of dynamic response of structures with uncertain but bounded parameters using non probabilistic interval analysis method and probabilistic approach abstract now by combining the finite element analysis and interval mathematics a new non probabilistic set theoretical models that is interval analysis method is being developed in scientific and engineering communities to predict the variability or uncertainty resulting from the unavoidable scatter in structural parameters and the external excitations as an alternative to the classical probabilistic approaches interval analysis methods of uncertainty were developed for modeling uncertain parameters of structures in which bounds on the magnitude of uncertain parameters are only required not necessarily knowing the probabilistic distribution densities instead of conventional optimization studies where the minimum possible response is sought here an uncertainty modeling is developed as an anti optimization problem of finding the least favorable response and the most favorable response under the constraints within the set theoretical description in this study we presented the non probabilistic interval analysis method for the dynamical response of structures with uncertain but bounded parameters under the condition of the interval vector which contains the uncertain but bounded parameters determined from probabilistic statistical information or stochastic sample test through comparing between the non probabilistic interval analysis method and the probabilistic approach in the mathematical proof and the numerical examples we can see that the region of the dynamical response of structures with uncertain but bounded parameters obtained by the interval analysis method contains that produced by the probabilistic approach in other words the width of the maximum or upper and minimum or lower bounds on the dynamical responses yielded by the probabilistic approach is tighter than those produced by the interval analysis method this kind of results is coincident with the meaning of the probabilistic theory and interval mathematics

robust fuzzy structural safety assessment using mathematical programming approach this paper presents a robust safety assessment for engineering structures involving fuzzy uncertainties uncertain applied loads and yielding capacities of structural elements are modelled as fuzzy variables with associated membership functions representing possibility distributions a new computation orientated methodology namely the level collapse assessment level ca approach is developed to provide structural safety profile by constructing membership function of the structural collapse load limit accommodating fuzzy uncertainties the proposed method firstly utilizes the level strategy to transform the fuzzy limit analysis into a series of interval limit analyses by implementing the concept of robust and optimistic optimizations a mathematical programming mp scheme is proposed to explicitly capture the upper and lower bounds of the collapse load limit at each sublevel subsequently the membership function of the collapse load limit is established by using the upper and lower bounds obtained from the series of sublevel calculations the proposed level mathematical programming scheme preserves the quality of sharpness of the bounds of collapse load limit at each sublevel which consequently provides a rigorous evaluation on the fuzzy profile of the safety of engineering structures against structural collapse numbers of numerical examples motivated by real world engineering applications have been investigated to illustrate the accuracy efficiency and applicability of the proposed method

finite element analysis with uncertain probabilities this paper presents a general method for the finite element analysis of linear mechanical systems by taking into account probability density functions whose parameters are affected by fuzziness within this framework the standard perturbation based stochastic finite element method is relaxed in order to incorporate uncertain probabilities in static dynamic and modal analyses general formulae are provided for assessing the fuzzy structural reliability and several typologies of optimization problems reliability based design robust design robust reliability based design are formalized in doing this the credibility theory is extensively used to extract qualified crisp data from the available set of fuzzy results so that standard optimizers can be adopted to solve the most important design problems it is shown that the proposed methodology is a general and versatile tool for finite element analyses because it is able to consider both probabilistic and non probabilistic sources of uncertainties such as randomness vagueness ambiguity and imprecision

random homogenization analysis in linear elasticity based on analytical bounds and estimates in this work random homogenization analysis of heterogeneous materials is addressed in the context of elasticity where the randomness and correlation of components properties are fully considered and random effective properties together with their correlation for the two phase heterogeneous material are then sought based on the analytical results of homogenization in linear elasticity when the randomness of bulk and shear moduli the volume fraction of each constituent material and correlation among random variables are considered simultaneously formulas of random mean values and mean square deviations of analytical bounds and estimates are derived from random factor method results from the random factor method and the monte carlo method are compared with each other through numerical examples and impacts of randomness and correlation of random variables on the random homogenization results are inspected by two methods moreover the correlation coefficients of random effective properties are obtained by the monte carlo method the random factor method is found to deliver rapid results with comparable accuracy to the monte carlo approach

probabilistic safety analysis of structures under hybrid uncertainty the probabilistic and the possibilistic methods of safety evaluation of structure under uncertain parameters have been developed independently when the structural system is defined with some of the input parameters as possibilistic and others are sufficient enough to model as probabilistic available literatures normally start with either probabilistic or possibilistic description of all the variables this may pose restriction on necessary flexibility to the designer at early stage of modelling of the structural system the primary objective of the present work is to critically examine various emerging methods of transformation of the possibilistic variables to equivalent probabilistic variables so that probabilistic safety evaluation approach becomes compatible with the nature and quality of the input data relying on the fundamental concept of equivalent transformations i e the entropy based transformation and the scaling of fuzzy membership function the reliability analysis is proposed in the framework of second moment format in doing so the bounds on the reliability indices based on the evidence theory are also obtained encompassing the first order reliability analysis for consistent comparison among alternative transformations finally the reliability computation under hybrid uncertainty is elucidated numerically with examples for comparative study on the suitability of the transformation alternatives

fuzzy based robust structural optimization robust optimization is conventionally defined as the collection of the possible problem solutions that can ensure acceptable performances and sufficient immunity against the effects of uncertain parameter variability methods proposed until now use a probabilistic way to model uncertainty and to quantify the final sensitivity in this work a fuzzy uncertainty modellization is adopted for structural engineering in particular to define solution performance scattering the fuzzy entropy is used as a global measure of variable dispersion the final formulation of the problem deals with two antithetical objective functions the fuzzy expected value of structural performance and its fuzzy entropy this fuzzy based approach in robust design is able to give a set of pareto optimal solutions in terms of structural efficiency and sensitivities regarding uncertainty and represents a suitable tool in supporting the decision maker finally different applications have been developed to demonstrate the applicability of the proposed method

a random fuzzy analysis of existing structures abstract two approaches are proposed to estimate the reliability of existing structures by considering both the randomness in some of the design parameters and the fuzzy imprecision in some other parameters representing the in place condition of the aged structures in the first approach the fuzzy imprecision is transformed into random uncertainty using the entropy concept and the reliability of existing structures is estimated using well established theories of probability in the second approach a hybrid approach in the random fuzzy domain is used to evaluate the reliability using an level concept the multiple fuzzy variables case is also considered both discrete and continuous fuzzy variables are considered the results obtained from the proposed approaches are compared with other techniques available in the literature whenever possible both methods are applied to civil structural engineering problems in this paper the results obtained are very encouraging and demonstrate the applicability and robustness of the algorithms

a class of linear differential dynamical systems with fuzzy initial condition this paper investigates linear first order fuzzy differential dynamical systems where the initial condition is described by a fuzzy number we use a complex number representation of the a level sets of the fuzzy system and prove theorems that provide the solutions under such representation which is applicable to practical computations and also has some implications for theory then the paper shows some properties of the two dimensional dynamical systems and their phase portraits are described by means of examples there may be a significant difference between the solutions according to whether the matrix is nonnegative or not finally the paper points out future research on the fuzzy dynamical systems

fuzzy behavior of mechanical systems with uncertain boundary conditions the problem of structures sensitivities to uncertain boundary conditions is presented in this paper and the effect of uncertain prescribed displacements on the static response of structures is discussed the imprecise or uncertain prescribed displacements are treated as fuzzy quantities with known membership functions for numerical computations a fuzzy quantity can be approximated by sets of closed intervals with respect to specified cuts this transforms the problem into a set of interval equations at each level a numerical method is developed to solve the system of fuzzy equations simple stress analysis is also used to establish and illustrate the method a numerical example is presented to demonstrate the capability of the method

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

spectral stochastic isogeometric analysis of free vibration abstract a novel spectral stochastic isogeometric analysis ssiga is proposed for the free vibration analysis of engineering structures involving uncertainties the proposed ssiga framework treats the stochastic free vibration problem as a stochastic generalized eigenvalue problem the stochastic young s modulus and material density of the structure are modelled as random fields with gaussian and non gaussian distributions the basis functions the non uniform rational b spline nurbs and t spline within computer aided design cad system are adopted within the ssiga which can eliminate geometric errors between design model and uncertainty analysis model the arbitrary polynomial chaos apc expansion is implemented to investigate the stochastic responses i e eigenvalues and eigenvectors of the structure a galerkin based method is freshly proposed to solve the stochastic generalized eigenvalue problems the statistical moments probability density function pdf and cumulative distribution function cdf of the eigenvalues can be effectively obtained two numerical examples with irregular geometries are investigated to illustrate the applicability accuracy and efficiency of the proposed ssiga for free vibration analysis of engineering structures

a fuzzy finite element procedure for the calculation of uncertain frequency response functions of damped structures part 2 numerical case studies this work introduces a numerical algorithm to calculate frequency response functions of damped finite element models with fuzzy uncertain parameters part 1 of this paper focusses on the numerical procedure for the solution of the underlying interval finite element problem based on the undamped procedure and the principle of rayleigh damping part 2 of this paper illustrates the applicability of the methodology through four case studies the concepts of the interval and the fuzzy finite element frequency response function analysis are illustrated for different types of uncertainties the obtained results are compared with the results of monte carlo simulations

fuzzy multiobjective optimization of mechanical structures this paper proposes a method that takes imperfections in multiobjective optimization problems into account the main objective is to determine a feasible design space for fuzzy design variables that respects fuzzy restriction rules and optimizes the defuzzified objective functions the proposed method uses the fuzzy finite element method ffem to calculate fuzzy solutions namely displacements and frequencies and genetic algorithms for exploring the design space of fuzzy design variables these elements are explained in detail and then applied to design a drop tower impactor sled defined with imperfections

time variant random interval natural frequency analysis of structures abstract this paper presents a new robust method namely unified interval chebyshev based random perturbation method to tackle hybrid random interval structural natural frequency problem in the proposed approach random perturbation method is implemented to furnish the statistical features i e mean and standard deviation and chebyshev surrogate model strategy is incorporated to formulate the statistical information of natural frequency with regards to the interval inputs the comprehensive analysis framework combines the superiority of both methods in a way that computational cost is dramatically reduced this presented method is thus capable of investigating the day to day based time variant natural frequency of structures accurately and efficiently under concrete intrinsic creep effect with probabilistic and interval uncertain variables the extreme bounds of the mean and standard deviation of natural frequency are captured through the embedded optimization strategy within the analysis procedure three particularly motivated numerical examples with progressive relationship in perspective of both structure type and uncertainty variables are demonstrated to justify the computational applicability accuracy and efficiency of the proposed method

bector chandra type duality in fuzzy linear programming with exponential membership functions in this paper we study a pair of fuzzy primal dual linear programming problems and calculate duality results using an aspiration level approach we use an exponential membership function which is in contrast to the earlier works that relied on a linear membership function as the fuzzy environment causes a duality gap we investigate how choosing the exponential membership function impacts this gap this issue is particularly important for fuzzy linear programming where in general the primal and dual objective values may not be bounded

a complete method for efficient fuzzy modal analysis the objective of this paper is to determine both the fuzzy eigenvalues and eigenvectors of a finite element model defined with fuzzy parameters the proposed method introduces the concepts of mode shape pairing and the functional dependence of eigensolutions with respect to design parameters high order approximations are then introduced to limit the computational cost associated with variability management numerical test cases are used to highlight the abilities of this method to predict behaviour modifications due to variations in the physical parameters

closed characteristics on compact convex hypersurfaces in r 2n for any given compact c 2 hypersurface in r 2n bounding a strictly convex set with nonempty interior in this paper an invariant n is defined and satisfies n n 2 1 where a denotes the greatest integer which is not greater than a r the following results are proved in this paper there always exist at least n geometrically distinct closed characteristics on if all the geometrically distinct closed characteristics on are nondegenerate then n n if the total number of geometrically distinct closed characteristics on is finite there exists at least an elliptic one among them and there exist at least n 1 of them possessing irrational mean indices if this total number is at most 2 n 2 there exist at least two elliptic ones among them

a new approach to fuzzy finite element analysis in this paper a new finite element analysis method for fuzzy structure is proposed by using the information entropy concept according to this method the fuzzy variables can be transferred into random variables so that the fuzzy structure can be treated as random structure when the random variables transferred from fuzzy structure are small the perturbation finite element method can be used to obtain the mean values and the variances of the structural responses

index theory nontrivial solutions and asymptotically linear second order hamiltonian systems abstract in this paper we consider the existence and multiplicity of solutions of second order hamiltonian systems we propose a generalized asymptotically linear condition on the gradient of hamiltonian function classify the linear hamiltonian systems prove the monotonicity of the index function and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear hamiltonian systems by global analysis methods such as the leray schauder degree theory the morse theory the ljusternik schnirelman theory etc

periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems clarke has shown that the problem of findingt periodic solutions for a convex hamiltonian system is equivalent to the problem of finding critical points to a certain functional dual to the classical action functional in this paper we relate the morse index of the critical point to the minimal period of the correspondingt periodic solution in particular we show that if the critical point is obtained by the ambrosetti rabinowitz mountain pass theorem the corresponding solution has minimal periodt that is it cannot bet k periodic withk integer k 2 as a consequence we prove that if the hamiltonian is flat near an equilibrium and superquadratic near infinity then for anyt 0 the corresponding hamiltonian system has a periodic solution with minimal periodt

multiplicity of closed characteristics on symmetric convex hypersurfaces in mathbb r 2n let sigma be a compact c 2 hypersurface in mathbb r 2n bounding a convex set with non empty interior in this paper it is proved that there always exist at least n geometrically distinct closed characteristics on sigma if sigma is symmetric with respect to the origin

a survey of non probabilistic uncertainty treatment in finite element analysis the objective of this paper is to critically review the emerging non probabilistic approaches for uncertainty treatment in finite element analysis the paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis first the applicability of the non probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint the necessary conditions for a useful application of the non probabilistic concepts are determined and are proven to be complementary rather than competitive to the classical probabilistic approach the second part of the paper focuses on numerical aspects of the interval finite element method it describes two principal strategies for the implementation i e the anti optimisation and the interval arithmetic approach and gives a state of the art of the interval finite element algorithms available from literature it is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output and the main sources of this conservatism are identified a numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built up benchmark structure

multiplicity results for periodic solutions to delay differential equations via critical point theory abstract by the critical point theory we study the existence and multiplicity of periodic solutions to the following system of delay differential equations x t f x t r where x r n f c r n r n and r 0 is a given constant a sufficient condition on the existence and multiplicity of periodic solutions for eq is obtained to the authors knowledge this is the first time the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

the uncertainty importance measures of the structural system in view of mixed uncertain variables abstract the uncertainty importance measure i e global sensitivity analysis of the basic variable is used for investigating the influence of model input uncertainty on model output uncertainty there are two kinds of uncertainty importance measures of the structural reliability and response with respect to both random and fuzzy valued input variables are investigated for problems with aleatory uncertainty and epistemic uncertainty first the structural reliability and probability distribution of the response at each membership level are analyzed then the differences between the unconditional and conditional membership functions mfs of reliability and the unconditional and conditional probability density functions pdfs of response at all membership levels are measured for the presented uncertainty importance measures the mathematical properties of the presented importance measures are discussed and proven in this study the defined uncertainty importance measures are easy to apprehend and are not restricted to the distribution form of random variables or to the membership function of fuzzy valued variables all evaluations are based on the pdf of random variables and the mf of fuzzy valued variables and thus the established importance measures are global sensitivity indicators that consider the influence of random and fuzzy valued input uncertainty on the structural reliability and response the results of the examples show that the proposed sensitivity indicators of uncertainty importance measure can intuitively describe the effects of random and fuzzy valued input variables on the reliability and the probability distribution of response for single and multiple failure modes

risk assessment system of natural hazards a new approach based on fuzzy probability this paper presents a system for assessing the risk of natural disasters particularly under highly uncertain conditions i e where neither the statistical data nor the physical knowledge required for a purely probabilistic risk analysis are sufficient the theoretical foundation of this study is based on employing fuzzy set theory to complement the probability theory with an additional dimension of uncertainty this would allow for expressing the likelihood of natural hazards by fuzzy probability the fuzzy probability is characterized in terms of possibility probability distributions for which a new approach has been developed the new approach has been compared with an alternative approach moreover uncertainties about the correlation of the parameters of hazard intensity damage and loss i e vulnerability relations have been considered by means of fuzzy relations the composition of fuzzy probability of hazard and fuzzy vulnerability relation yields the fuzzy probability of damage or loss the system has been applied for assessing the earthquake risk in istanbul metropolitan area

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

probabilistic interval reliability of structural systems the probabilistic reliability approach is the most widely used method for reliability analysis the recent research shows that the reliabilities of structural systems strongly depend on the parameters of the probability model it is possible that the little error in the estimation of the parameters may lead to the remarkable error of the resulting probability in this study we introduce the interval approach into the conventional reliability theory we present a novel approach which allows us to obtain the system failure probability interval from the statistical parameter intervals of the basic variables this approach is a combination of the two techniques namely the classical reliability theory and the interval analysis in the end of this paper we show the feasibility of the proposed approach through two examples of the truss systems

the bird s eye view on finite element method for structures with large stochastic variations abstract extensive motivation to do an additional work in the finite element method in stochastic problems femsp is discussed the qualitative comparison of femsp with the state of the art of the deterministic fem is given these observations and thoughts are then realized in several manners we first present the exact inverse femsp which however cannot serve as a general tool for stochastic analysis of complex structures the new variational principles for stochastic beams for the mean response function as well as response s auto correlation function are formulated and the fem based on the variational principles is presented finally a general non perturbative fem for stochastic problems with large variations is developed based on the element level flexibility it is concluded that much work needs to be done in order femsp to be at the level compared to that of the deterministic fem the advantage of the main methods presented here over the conventional ones lies in their non perturbational nature numerical examples are presented

non probabilistic reliability based topology optimization of geometrically nonlinear structures using convex models this paper describes a non probabilistic reliability based topology optimization method for the design of continuum structures undergoing large deformations the variation of the structural system is treated with the multi ellipsoid convex model which is a realistic description of the parameters being inherently uncertain but bounded or lacking sufficient probabilistic data the formulation of the optimal design is established as a volume minimization problem with non probabilistic reliability constraints on the geometrically nonlinear structural behaviour in order to circumvent numerical difficulties in solving the nested double loop optimization problem a performance measure based approach is employed to transform the constraint on the reliability index into one on the concerned performance in conjunction with an efficient adjoint variable scheme for the sensitivity analysis of reliability constraints the optimization problem is solved by gradient based mathematical programming methods three numerical examples for the optimization design of planar structures are presented to illustrate the validity and applicability of the proposed method the obtained optimal solutions show the importance of incorporating various uncertainties in the design problem moreover it is also revealed that the geometrical nonlinearity needs to be accounted for to ensure satisfaction of the reliability constraints in the optimal design of structures with large deformation

the effect of material and thickness variability on the buckling load of shells with random initial imperfections abstract the effect of material and thickness imperfections on the buckling load of isotropic shells is investigated in this paper for this purpose the concept of an initial imperfect structure is introduced involving not only geometric deviations of the shell structure from its perfect geometry but also a spatial variability of the modulus of elasticity as well as the thickness of the shell the initial geometric imperfections are described as a two dimensional uni variate 2d 1v stochastic field with statistical properties that are either based on an available data bank of measured initial imperfections or assumed in cases where no experimental data is available in order to describe the non homogeneous characteristics of the initial imperfections the spectral representation method is used in conjunction with an autoregressive moving average model with evolutionary power spectra based on a statistical analysis of the experimentally measured imperfections in cases where no experimental results is available the initial imperfections are assumed to be homogeneous and their impact on the buckling load is investigated on the basis of worst case scenarios with respect to the correlation length parameters of the stochastic fields the elastic modulus and the shell thickness are described as 2d 1v non correlated homogeneous stochastic fields while the stochastic stiffness matrix of the shell elements is formulated using the local average method the monte carlo simulation method is used to calculate the variability of the buckling load while for the determination of the limit load of the shell a stochastic formulation of the elastoplastic and geometrically non linear tric facet triangular shell element is implemented

coupled fuzzy interval model and method for structural response analysis with non probabilistic hybrid uncertainties abstract in many engineering practices with uncertainty the non probabilistic methods play an increasing important role to overcome the limitation of traditional single uncertainty modeling methods in handling coupled uncertain problems this paper develops a more general hybrid uncertainty analysis framework the non probabilistic hybrid uncertainties are expressed as coupled fuzzy interval variables where the bounds of interval are interpreted as fuzzy sets instead of deterministic values by means of the cut set strategy and decomposition theorem in fuzzy set theory the hybrid uncertain problem is transformed into a series of dual interval problems the conservative and radical extreme value predictions in different variable subspaces are adopted to characterize the coupled uncertainty in output response to further improve the computational efficiency of extreme value prediction an adaptive response surface model using radial basis function is proposed where the potential global optimum points are introduced as the new sample points in the sequential sampling process finally the effectiveness of the proposed method on dealing with non probabilistic hybrid uncertainties is validated by two examples

analysis of mean and mean square response of general linear stochastic finite element systems a general finite element based formulation is presented for the analysis of the mean and mean square response of stochastic structural systems whose material properties are described by random fields a flexibility based formulation is followed that does not involve any approximations closed formed integral expressions for the mean and mean square value of the response displacement of statically indeterminate stochastic structures are introduced in this paper these integral expressions involve the spectral density function of a stochastic field modeling the uncertain material properties and two new quantities named the mean response function mrf and the variability response function for the mean square response vrf1 the mrf and the vrf1 have many similarities with the classical variability response function vrf used to estimate the variance of the response displacement the existence of the mrf and the vrf1 depends on a conjecture that is validated numerically using a brute force monte carlo simulation approach a so called finite element method based fast monte carlo simulation procedure fem fmcs is introduced for the accurate and efficient numerical evaluation of the mrf and the vrf1 this methodology is established for the analysis of stochastic beam frame structures as well as for more general stochastic finite element systems i e plane stress and shell problems numerical examples are provided including a statically indeterminate beam a plane stress problem and a shell structure the mrf and vrf1 are used to perform sensitivity parametric analyses with respect to various probabilistic characteristics involved in the problem i e correlation distance standard deviation and to establish realizable upper bounds on the mean value of the response displacement throughout this paper the vrf is computed in parallel to the mrf and the vrf1 it should be noted that although the concept of the vrf has been introduced in an earlier work this is the first time that a flexibility based approach has been used to estimate the vrf for two dimensional problems

fuzzy sets based interpretation of the safety factor abstract safety factor is a universally utilized concept in several branches of engineering on one hand most engineers as it were neglect uncertainty but on the other hand the allowable stress level was introduced long time ago as a ratio of the yield stress to the so called safety factor to provide the region for the safe utilization of the structure thus the uncertainty is introduced into practice by the back door this observation led to a considerable literature dedicated to the probabilistic interpretation of the safety factor the present paper deals with the novel aspect of elucidation of the concept of safety factor through the theory of fuzzy sets apparently for the first time in the literature the aim of the paper is to present the safety factor that is uniformly employed by engineers but in the new light the safety factor in the fuzzy setting is introduced the ideas are illustrated on two strength of material problems simple examples are chosen so as to allow for clearer illustration of ideas

robust duality for generalized convex programming problems under data uncertainty abstract in this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization we establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain lagrangian dual by showing strong duality between the deterministic counterparts robust counterpart of the primal model and the optimistic counterpart of its dual problem a robust strong duality theorem is given whenever the lagrangian function is convex we provide classes of uncertain non convex programming problems for which robust strong duality holds under a constraint qualification in particular we show that robust strong duality is guaranteed for non convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized slater condition numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems we further show that robust duality continues to hold under a weakened convexity condition

systems of fuzzy equations in structural mechanics systems of linear and nonlinear equations with fuzzy parameters are relevant to many practical problems arising in structure mechanics electrical engineering finance economics and physics in this paper three methods for solving such equations are discussed method for outer interval solution of systems of linear equations depending linearly on interval parameters fuzzy finite element method proposed by rama rao and sensitivity analysis method the performance and advantages of presented methods are described with illustrative examples extended version of the present paper can be downloaded from the web page of the utep i skalna m v rama rao a pownuk systems of fuzzy equations in structural mechanics the university of texas at el paso department of mathematical sciences research reports series texas research report no 2007 01 2007

an interval finite element approach for the calculation of envelope frequency response functions this paper focusses on the application of the interval finite element method in dynamic analyses it describes a methodology for calculating frequency response function envelopes from a finite element model containing imprecise parameters defined as interval uncertainties the resulting envelope functions give a conservative approximation of the possible range of the frequency response function taking into account that the uncertain parameters in the model can adopt any value in their presumed uncertainty intervals the methodology is based on the modal superposition principle it consists of an interval aritmethic algorithm which processes the results of a preliminary global optimization performed on the modal parameters the algorithm is constructed such that it optimally combines the advantages of both the anti optimization and the interval arithmetic strategy for general numerical interval calculations in the first stage of the development the modal parameter ranges of each individual mode are independently combined in the modal response contributions this yields the modal rectangle mr method in order to remedy the high conservatism inherent to the mr method the exact eigenfrequency ranges are added to the analysis this results in the modal rectangle method with eigenfrequency interval correction mre a second improvement consists of adding extra delimiters to the mre modal parameter range approximation this is achieved by performing an extra optimization on the modal response contributions at discrete frequencies the method is referred to as the locally optimized modal rectangle method with eigenfrequency interval correction omre finally a numerical example illustrates the different algorithms copyright 2004 john wiley sons ltd

exact bounds for the static response set of structures with uncertain but bounded parameters abstract the numerical estimation of the static displacement bounds of structures with uncertain but bounded parameters is considered in this paper by representing each uncertain but bounded parameter as an interval number or vector a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations in which the coefficient matrix and the right hand side term are respectively the interval matrix and the interval vector in this study we present two new simple mathematical proofs of the vertex solution theorem using cramer s rule for solving linear interval equations different from the other proof methods to find the upper and lower bounds on the set of solutions the first takes advantage of optimization theory while the second is based on interval extension by means of a typical example considered first by hansen it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the oettli prager criterion examples of a three stepped beam and a 10 bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method

limit load and shakedown analysis of plastic structures under stochastic uncertainty abstract problems from plastic analysis are based on the convex linear or linearised yield strength condition and the linear equilibrium equation for the stress state vector in practice one has to take into account stochastic variations of several model parameters hence in order to get robust maximum load factors the structural analysis problem with random parameters must be replaced by an appropriate deterministic substitute problem a direct approach is proposed based on the primary costs for missing carrying capacity and the recourse costs e g costs for repair compensation for weakness within the structure damage failure etc based on the mechanical survival conditions of plasticity theory a quadratic error loss criterion is developed the minimum recourse costs can be determined then by solving an optimisation problem having a quadratic objective function and linear constraints for each vector a of model parameters and each design vector x one obtains then an explicit representation of the best internal load distribution f moreover also the expected recourse costs can be determined explicitly consequently an explicit stochastic nonlinear program results for finding a robust maximal load factor the analytical properties and possible solution procedures are discussed

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

a twist condition and multiple solutions of unbounded self adjoint operator equation with symmetries abstract by using the index theory for unbounded self adjoint operator equations and the symmetric mountain pass theorem we investigate the existence of multiple solutions for nonlinear operator equations with twist conditions we prove an abstract theorem and give some applications to first order hamiltonian systems with sturm liouville boundary conditions and delay differential equations

analysis of structure in fuzzy linear programs abstract fuzzy linear programming problems are analyzed within the fuzzy set context to uncover redundancies infeasibilities variables whose values are fixed and implied bounds on rows and columns applications include pre analysis of perturbed constraint matrices and post optimization analysis fuzzy linear programming problems

a twist condition and periodic solutions of hamiltonian systems abstract in this paper we investigate existence of nontrivial periodic solutions to the hamiltonian system hs j z h t z z r 2 n under a general twist condition for the hamiltonian function in terms of the difference of the conley zehnder index at the origin and at infinity we establish existence of nontrivial periodic solutions compared with the existing work in the literature our results do not require the hamiltonian function to have linearization at infinity our results allow interactions at infinity between the hamiltonian and the linear spectra the general twist condition raised here seems to resemble more the spirit of poincare s last geometric theorem

hybrid uncertain static analysis with random and interval fields abstract uncertain static analysis of an engineering structure with diverse type of non deterministic system parameter is investigated in this study unlike the traditional hybrid uncertain static analysis involving random and interval variables the concept of random and interval fields has been implemented to model the spatially dependent uncertainties associated with the system inputs a novel computational approach namely the extended unified interval stochastic sampling x uiss method is proposed to calculate the statistical characteristics i e mean and standard deviation of the extreme bounds i e lower and upper bounds of the concerned responses e g displacement and stress of engineering structure involving hybrid spatially dependent uncertainties subsequently by utilizing either parametric or nonparametric statistical analysis the probability density functions pdfs as well as the cumulative distribution functions cdfs of the extreme bounds of the concerned structural responses can be effectively established consequently the upper and lower bounds of either the concerned responses of the engineering structure at any particular percentile of probability or the structural reliability against any specified capacities can be effectively secured the applicability and effectiveness of the proposed computational analysis framework are illustrated through the numerical investigations on various examples

robust assessment of collapse resistance of structures under uncertain loads based on info gap model abstract the paper proposes a pair of novel mathematical programming based approaches which directly determine the worst collapse load limit in one case and the best limit in the other case of rigid perfectly plastic structures subjected to uncertain but bounded applied loads using an info gap model the methods take advantage of the important properties in which the worst collapse load limit defined for an uncertain static formulation is equivalent to the most favourable solution of the uncertain kinematic limit analysis problem and vice versa for the best collapse load limit the formulation for capturing the worst collapse load limit robust worst case solution takes the form as a mathematical program with equilibrium constraints that is processed using a penalty algorithm whilst that for the best collapse load limit is a standard linear programming problem the efficiency and robustness of the proposed schemes are evidenced from a number of successfully solved examples

convex hamiltonian energy surfaces and their periodic trajectories in this paper we introduce symplectic invariants for convex hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations in particular we show that they can be used to prove multiplicity results for the number of periodic trajectories

fuzzy modal finite element analysis of structures with imprecise material properties this paper extends the fuzzy set theory to a dynamic finite element analysis of engineering systems which have uncertainties in material properties a general algorithm which resolves the uncertain eigenvalue problem by using a reanalysis approach is considered this algorithm is applied to the study of the modal behaviour of structures presenting uncertain material properties some indexes which determine the more sensitive eigenvalue to several uncertainty sources are also put forward finally a plate structure as numerical path test is analysed the results of such a calculation determine the sensitivity of the modal behaviour to multiple simultaneous material parameters

interval analysis of structures with uncertain but bounded axial stiffness abstract a method to evaluate the static response of structures with interval axial stiffnesses is proposed the procedure overcomes the drawbacks arising in traditional interval analysis due to dependency phenomenon the inversion of the interval stiffness matrix is pursued by means of a novel approximate relationship based on an interval valued sherman morrison woodbury formula numerical results on truss and frame structures evidence the accuracy and the effectiveness of the proposed approach to evaluate bounds of displacements and internal forces

multiple solutions of generalized asymptotical linear hamiltonian systems satisfying sturm liouville boundary conditions abstract this paper consider the multiple solutions for even hamiltonian systems satisfying sturm liouville boundary conditions the gradient of hamiltonian function is generalized asymptotically linear the solutions obtained are shown to coincide with the critical points of a dual functional thanks to the index theory for linear hamiltonian systems by dong 2010 1 we find critical points of this dual functional by verifying the assumptions of a lemma about multiple critical points given by chang 1993 2

structural reliability analysis using non probabilistic convex model abstract by using the convex model approach the bounds of the uncertain parameters are only required rather than their precise probability distributions based on which it can be made possible to conduct the reliability analysis for many complex engineering problems with limited information in this paper a non probabilistic reliability model is given for structures with convex model uncertainty which is defined as a ratio of the multidimensional volume falling into the reliability domain to the one of the whole convex model a monte carlo simulation is then formulated for the reliability model to obtain a reference solution a first order approximation method foam is proposed to solve the reliability model based on a linear approximation of the failure surface a second order approximation method soam is further formulated to improve the reliability analysis precision for limit state functions with relatively strong nonlinearity by introducing the suggested reliability model a convex model based system reliability method is also formulated eight numerical examples are provided to demonstrate the effectiveness of the present methods

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

a spectral approach for fuzzy uncertainty propagation in finite element analysis abstract uncertainty propagation in complex engineering systems with fuzzy variables constitutes a significant challenge this paper proposes a polynomial chaos type spectral approach based on orthogonal function expansion a fuzzy variable is represented as a set of interval variables via the membership function the interval variables are further transformed into the standard interval 1 1 smooth nonlinear functions of standard interval variables are projected in the basis of legendre polynomials by exploiting its orthogonal properties over the interval 1 1 the coefficients associated with the basis functions are obtained by a galerkin type of error minimisation the method is first illustrated using scalar functions of multiple fuzzy variables later the method is proposed for elliptic type finite element problems where the technique is extended to vector valued functions with multiple fuzzy variables the response of such systems can be expressed in the complete basis of multivariate legendre polynomials the coefficients obtained by galerkin type of error minimisation can be calculated from the solution of an extended set of linear algebraic equations an eigenfunction based model reduction technique is proposed to obtain the coefficient vectors in an efficient way a numerical example of axial deformation of a rod with fuzzy axial stiffness is considered to illustrate the proposed methods linear and nonlinear membership functions are used and the results are compared with direct numerical simulation results

fuzzy linear programming using a penalty method in this paper we begin with a standard form of the linear programming problem we replace each constant in the problem with a fuzzy number we then reformat the objective and constraints into an unconstrained fuzzy function by penalizing the objective for possible constraint violations the range of this fuzzy function lies in the space of fuzzy numbers the objective is then redefined as optimizing the expected midpoint of the image of this fuzzy function we show that this objective defines a concave function which therefore can be maximized globally we present an algorithm for finding the optimum

solving processes for a system of first order fuzzy differential equations abstract in this third paper of a series of reports on fuzzy differential equations we consider the system of first order inhomogeneous fuzzy differential equations d d t x t a t x t f t 1 with x 0 x 0 where d x t t is an n dimensional vector of first same order or reverse order derived functions of an n dimensional vector x t x 1 t x n t t of unknown fuzzy set valued functions that is d dt x t d dt x 1 t d dt x n t t f t is an n dimensional vector f 1 t f n t t of known fuzzy set valued functions a t is an n n matrix of known real functions we introduce the time domain and frequency domain methods for the solutions of the system of first order fuzzy differential equations 1 the solving processes of time domain and frequency domain for the system of first order fuzzy differential equations with constant coefficients and variable coefficients are put forward one example is considered in order to demonstrate the rationality and validity of the methods the work provides an indispensable mathematical tool for setting up the theories of fuzzy stochastic differential equations 7 fuzzy dynamical systems 3 fuzzy random vibration 8 fuzzy stochastic dynamical systems 10 11 14 16 and fuzzy stochastic systems 17 19

middle convolution of fuchsian systems and the construction of rigid differential systems abstract in m dettweiler s reiter an algorithm of katz and its application to the inverse galois problem j symbolic comput 30 2000 761 798 a purely algebraic analogon of katz middle convolution functor see n m katz rigid local systems ann of math stud vol 139 princeton university press 1997 is given in this paper we find an explicit riemann hilbert correspondence for this functor this leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the riemann hilbert correspondence

fuzzy finite element approach for the analysis of imprecisely defined systems many engineering systems are too complex to be defined in precise mathematical terms they often contain information and features that are vague imprecise qualitative linguistic or incomplete the traditional deterministic and probabilistic techniques are not adequate to analyze such systems this paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems the development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation manipulation and solution of the finite element equations simple stress analysis problems involving vaguely defined geometry material properties external loads and boundary conditions are solved to establish and to illustrate the new procedure the approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information if complete data are known the method handles the information similar to that of a probabilistic approach the present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

a differential equation with monodromy group 2 j2 abstract we construct a sixth order differential equation having the central extension of c 2 by the hall janko group j 2 as monodromy group moreover it arises from an iterated application of tensor products and convolution operations from a first order differential equation