semi definite programming for topology optimization of trusses under multiple eigenvalue constraints abstract topology optimization problem of trusses for specified eigenvalue of vibration is formulated as semi definite programming sdp and an algorithm is presented based on the semi definite programming algorithm sdpa which utilizes extensively the sparseness of the matrices since the sensitivity coefficients of the eigenvalues with respect to the design variables are not needed the sdpa is especially useful for the case where the optimal design has multiple fundamental eigenvalues global and local modes are defined and a procedure is presented for generating optimal topology from the practical point of view it is shown in the examples that sdpa has advantage over existing methods in view of computational efficiency and accuracy of the solutions and an optimal topology with five fold fundamental eigenvalue is found without any difficulty

second order linear differential equations with piecewise constant arguments subject to nonlocal boundary conditions abstract we analyze the existence of solution for the class of second order linear differential equations with functional dependence x t ax t bx t cx t dx t t t j 0 t where a b c d r t 0 and is a piecewise continuous function subject to a certain type of nonlocal mixed boundary conditions which include the value of the derivative of x at the initial point x 0

the generalized eigenvalue problem for tridiagonal symmetric interval matrices an algorithm to compute the eigenintervals of the generalized eigenvalue problem for tridiagonal symmetric matrices a b with interval entries is developed the algorithm is based on the sturm sequences and requires twice as much computational effort as for point matrices the algorithm is applied to the problem of computing the natural frequencies of a vibrating beam where the manufacturing tolerance is taken into consideration

dynamic response analysis of stochastic truss structures under non stationary random excitation using the random factor method abstract the dynamic characteristics and responses of stochastic truss structures under non stationary random excitation are investigated using a new method called the random factor method rfm using the rfm the structural physical parameters and geometry can be considered as random variables based on the assumption that the randomness on each element is the same for all elements the structural stiffness and mass matrices can then respectively be described by the product of two parts corresponding to the random factor and the deterministic matrix from the expressions of structural random response in the frequency domain computational expressions for the mean value standard deviation and variation coefficient of the mean square value of the non stationary random displacement and stress response are developed by means of the random variable s functional moment method and the algebra synthesis method the influences of the randomness of the structural parameters on the dynamic characteristics structural displacement and stress responses are demonstrated using a truss structure

larger lower bounds on the obdd complexity of integer multiplication integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams obdds are one of the most common dynamic data structures for boolean functions recently the question whether the obdd complexity of the most significant bit of integer multiplication is exponential has been answered affirmatively in this paper a larger general lower bound is presented using a simpler proof furthermore we prove a larger lower bound for the variable order assumed to be one of the best ones for the most significant bit moreover the best known lower bound on the obdd complexity for the so called graph of integer multiplication is improved

on optimal solution of interval linear equations for interval linear algebraic systems bf a x bf b we consider the problem of component wise evaluation of the set sigma exists exists bf a bf b a 1 b mid a in bf a b in bf b formed by all solutions of ax b when a and b vary independently in bf a and bf b respectively an iterative pss algorithm is introduced that computes optimal exact componentwise estimates of sigma exists exists and its convergence is proved under fairly general conditions on the interval system we introduce the concept of a sequentially guaranteeing algorithm as a reasonable compromise between the requirements for the interval result to be guaranteed and to be obtained in a practically acceptable time

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

green s function for second order periodic boundary value problems with piecewise constant arguments we obtain under suitable conditions the green s function to express the unique solution for a second order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function this expression is given in terms of the solutions for certain associated problems the sign of the solution is determined taking into account the sign of that green s function

interval static displacement analysis for structures with interval parameters in this paper a new method to solve the uncertain static displacement problem of structures with interval parameters is presented it is difficult to obtain all possible solutions with sharp bounds even if an optimum scheme is adopted when there are many uncertain parameters with the interval mathematics the interval finite element equation is developed based on the perturbation and the interval extension the upper and lower bounds of the static displacements are obtained in which the sharp bounds are guaranteed by the interval calculation operators two numerical examples a box cantilever beam and an automobile frame are given to illustrate the validity of the present method copyright 2001 john wiley sons ltd

dynamic response analysis of structure under time variant interval process model abstract due to the aggressiveness of the environmental factor the variation of the dynamic load the degeneration of the material property and the wear of the machine surface parameters related with the structure are distinctly time variant typical model for time variant uncertainties is the random process model which is constructed on the basis of a large number of samples in this work we propose a time variant interval process model which can be effectively used to deal with time variant uncertainties with limit information and then two methods are presented for the dynamic response analysis of the structure under the time variant interval process model the first one is the direct monte carlo method dmcm whose computational burden is relative high the second one is the monte carlo method based on the chebyshev polynomial expansion mcm cpe whose computational efficiency is high in mcm cpe the dynamic response of the structure is approximated by the chebyshev polynomials which can be efficiently calculated and then the variational range of the dynamic response is estimated according to the samples yielded by the monte carlo method to solve the dependency phenomenon of the interval operation the affine arithmetic is integrated into the chebyshev polynomial expansion the computational effectiveness and efficiency of mcm cpe is verified by two numerical examples including a spring mass damper system and a shell structure

vertex method for computing functions of fuzzy variables abstract this paper is to introduce a new approach the vertex method for computing functions of fuzzy variables the method is based on the cut concept and interval analysis the vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi occurrence of variables in the functional expression by conventional interval analysis methods the algorithm is very easy to implement and can be applied to many practical problems some examples are given to illustrate the applications

an interdependency index for the outputs of uncertain systems the study of mechanical systems with uncertain parameters is gaining increasing interest in the field of system analysis to provide an expedient model for the prediction of the system behavior making use of the transformation method the uncertain parameters of the system are modeled by fuzzy numbers in contrast to random numbers used in stochastic approaches as a result of this analysis a quantification of the overall uncertainty of the system outputs including a worst case scenario is obtained the inputs of the resulting fuzzy valued model are a priori uncorrelated but after the uncertainties are propagated through the model interdependency or interaction between the outputs may arise if such interdependency is neglected a misinterpretation of the results may occur for example in the case of applying uncertainty analysis in the early design phase of a product to determine the relevant design parameter space the interdependency between the design variables may reduce significantly the available part of the design space this paper proposes a measure of interdependency between the uncertain system outputs the interdependency index can be derived by a postprocessing of the data gained by the analysis with the transformation method such information can be obtained by a negligible amount of extra computation time

fuzzy finite element method for frequency response function analysis of uncertain structures a concept is presented for incorporating fuzzy uncertainties in dynamic e nite element analyses of uncertain structures the objective is twofold the e rst goal is to clarify and extend the classical fuzzy e nite element ffe method as it was introduced for static analyses the shortcomings of the classical approach are described and an extension to a generalized approach is proposed this generalized approach is proven to be a more realistic and therefore more reliable concept for taking uncertainty into account the second goal is to illustrate the applicability of the method for dynamic analyses the classical and the generalized approach are compared using an eigenvalue analysis of a simple numerical example the ffe method is also applied to the calculation of the total envelope frequency response function frf using the modal superposition principle this method requires safe approximations of the individual mode envelope frequency response functions for this purpose a number of safe approximate optimization strategies are introduced the numerical example shows that useful results are obtained using this ffe approach for frf calculations

unified finite element method for engineering systems with hybrid uncertainties the finite element method fem is established as one of the most popular and powerful tools in analysis and design of engineering systems however the method can only be used at an advanced or detail design stage when a designer has developed sufficient confidence in the design scheme in the early or conceptual preliminary design phase the method can not be used directly due to the existence of uncertainties of various types in this work a procedure is proposed for the fem to handle engineering systems in the presence of hybrid uncertainties that are characterized by randomness or stochastic uncertainty and fuzziness or design imprecision the proposed scheme utilizes the techniques of stochastic and fuzzy fems in a unified manner to investigate the effects of both input random and fuzzy uncertainties on the output system response interpretations of the generated solution space and guidelines in selection of an appropriate solution according to specific design requirements are discussed two numerical examples are presented to illustrate the computational aspects of the proposed scheme the current procedure is expected to enhance the understanding of designers regarding the effects of uncertainties in an evolutionary design process especially in the conceptual or preliminary design phase

nonlocal boundary value problems for second order functional differential equations abstract we study the existence of a solution to a nonlocal boundary value problem for a class of second order functional differential equations with piecewise constant arguments the equation studied includes terms depending on the derivative as well as delay arguments in the derivative

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

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

green s function for second order differential equations with piecewise constant arguments abstract in this paper the second order nonautonomous differential equations with piecewise constant arguments x t t t x t t t t x t c t x t d t x t t t 0 t x 0 x t x 0 x t is considered we obtain the green s function to express the unique solution for the system

constructing membership functions using statistical data abstract determination of the membership functions is vital in practical applications of the fuzzy set theory this paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function pdf in the universe of discourse the method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf it is show that for any pdf the method is capable of generating membership functions in accordance with the possibility probability consistency principle membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented

confidence structural robust optimization by non linear semidefinite programming based single level formulation structural robust optimization problems are often solved via the so called bi level approach this solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non smooth nature of the bi level formulation another problem associated with the traditional bi level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically in the present paper confidence single level non linear semidefinite programming nlsdp formulations for structural robust optimization problems under stiffness uncertainties are proposed this is achieved by using some tools such as s procedure and quadratic embedding for convex analysis the resulted nlsdp problems are solved using the modified augmented lagrange multiplier method which has sound mathematical properties numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively copyright 2010 john wiley sons ltd

fuzzy number theory to obtain conservative results with respect to probability fuzzy number and possibility theories are used for problems where uncertainties in the definition of input data do not allow for a treatment by means of probabilistic methods starting from a scarce uncertain body of information fuzzy numbers are used to define possibility distributions as well as upper and lower bounds for a wide class of probability distributions compatible with available data it is investigated if relations between possibility distributions and probability measures are preserved also when the fuzzy number represents an output variable computed making use of extended fuzzy operations general real one to one and binary operations are considered asymptotic expressions for small large fractiles for the membership function of the fuzzy number and for cdfs given by probability theory are obtained it is shown that fuzzy number theory gives conservative bounds with respect to probability for characteristic values corresponding to prescribed occurrence expectations these results are of special interest for computational applications in fact it is easier to define fuzzy variables than random variables when no or few statistical data are available as in the case of structural design stages moreover extended fuzzy operations are much simpler than analogous operations required in the framework of probability especially when several variables are involved

semidefinite programming for uncertain linear equations in static analysis of structures abstract this paper presents a method for computing a minimal bound that contains the solution set to the uncertain linear equations particularly we aim at finding a bounding ellipsoid for static response of structures where both external forces and elastic moduli of members are assumed to be imprecisely known and bounded by using the s lemma we formulate a semidefinite programming sdp problem which provides an outer approximation of the minimal bounding ellipsoid bounding ellipsoids are computed for nodal displacements of uncertain braced frames as the solutions of the presented sdp problems by using the primal dual interior point method

robustness analysis of trusses with separable load and structural uncertainties this paper discusses evaluation techniques of the robustness function of trusses which is regarded as one of measures of structural robustness under the uncertainties of member stiffnesses and external forces by using quadratic embedding of the uncertainty and the s procedure we formulate a quasiconvex optimization problem which provides lower bounds of the robustness functions a bisection method is proposed where we solve a finite number of semidefinite programming problems in order to obtain a global optimal solution to the proposed quasiconvex optimization problem the lower bounds of the robustness functions are computed for various trusses under several uncertainty circumstances

ellipsoidal bounds for uncertain linear equations and dynamical systems abstract in this paper we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations the proposed technique is based on the combination of a quadratic embedding of the uncertainty and the s procedure this formulation leads to convex optimization problems that can be essentially solved in o n3 n being the size of unknown vector by means of suitable interior point barrier methods as well as to closed form results in some particular cases we further show that the uncertain linear equations paradigm can be directly applied to various state bounding problems for dynamical systems subject to set valued noise and model uncertainty

parity graph driven read once branching programs and an exponential lower bound for integer multiplication branching programs are a well established computation model for boolean functions especially read once branching programs have been studied intensively exponential lower bounds for read once branching programs are known for a long time on the other hand the problem of proving superpolynomial lower bounds for parity read once branching programs is still open in this paper restricted parity read once branching programs are considered and an exponential lower bound on the size of the so called well structured parity graph driven read once branching programs for integer multiplication is proven this is the first strongly exponential lower bound on the size of a parity nonoblivious read once branching program model for an explicitly defined boolean function in addition more insight into the structure of integer multiplication is yielded

risk averse structural topology optimization under random fields using stochastic expansion methods abstract this work proposes a level set based approach for solving risk averse structural topology optimization problems considering random field loading and material uncertainty the use of random fields increases the dimensionality of the stochastic domain which poses several computational challenges related to the minimization of the excess probability as a measure of risk awareness this problem is addressed both from the theoretical and numerical viewpoints first an existence result under a typical geometrical constraint on the set of admissible shapes is proved second a level set continuous approach to find the numerical solution of the problem is proposed since the considered cost functional has a discontinuous integrand the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic polynomial chaos pc approach with a monte carlo mc sampling method for uncertainty propagation furthermore to address the increment of dimensionality induced by the random field an anisotropic sparse grid stochastic collocation method is used for the efficient computation of the pc coefficients a key point is that the non intrusive nature of such an approach facilitates the use of high performance computing hpc to alleviate the computational burden of the problem several numerical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal

fuzzy boundary element method for the analysis of imprecisely defined systems many engineering systems contain uncertainties that cannot be described by either deterministic or probabilistic approaches the uncertainties present may be associated with parameters that are vague imprecise or linguistic for example the geometry material properties external actions loads and boundary conditions may be imprecise in a practical system a fuzzy boundary element method for the analysis of imprecisely defined systems is developed starting from the basic concepts of fuzzy sets and fuzzy arithmetic the various steps of the boundary element method are redefined using fuzzy concepts the resulting fuzzy equations are solved using a fuzzified version of gaussian elimination procedure coupled with truncation the truncation method limits the growth of interval ranges of response parameters to obtain realistic and accurate solutions the procedure is illustrated by considering the analysis of a potential flow problem the methodology is applicable to uncertain systems that are described in linguistic terms as well as those that are described by incomplete information the approach represents a unique methodology that enables the analysis and design of many engineering systems more realistically

robust shape optimization of continuous structures via the level set method abstract this work proposes a stochastic shape optimization method for continuous structures using the level set method such a method aims to minimize the expected compliance and its variance as measures of the structural robustness the behavior of continuous structures is modeled by linear elasticity equations with uncertain loading and material this uncertainty can be modeled using random variables with different probability distributions as well as random fields the proper problem formulation is ensured by the proof of the existence colorrev of solution under certain geometrical constraints on the set of admissible shapes the proposed method addresses the stochastic linear elasticity problem in its weak form obtaining the explicit expressions for the continuous shape derivatives some numerical examples are presented to show the effectiveness of the proposed approach

a novel interval finite element method based on the improved interval analysis abstract static analysis of linear elastic structures with uncertain parameters subjected to deterministic loads is addressed the uncertain structural properties are modeled as interval variables with assigned lower bound and upper bound a novel interval finite element method is formulated in the framework of the improved interval analysis via extra unitary interval recently proposed to limit the conservatism affecting the classical interval analysis the key idea of the novel method is to associate an extra unitary interval to each uncertain parameter in order to keep physical properties linked to the finite elements in both the assembly and solution phases this allows one to reduce overestimation and perform standard assembly of the interval element matrices the lower bound and upper bound of interval displacements and stresses are evaluated by applying two different strategies both based on the so called interval rational series expansion for deriving the approximate explicit inverse of the interval global stiffness matrix numerical examples concerning 2d and 3d structures with uncertain young s modulus are presented to demonstrate the accuracy and efficiency of the proposed procedure

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

construction and application of an ellipsoidal convex model using a semi definite programming formulation from measured data abstract as a set theory based convex model the ellipsoidal model provides an attractive framework for treating uncertain but bounded variations in the structural reliability analysis and design optimization however improper modeling of the uncertainties may give rise to misleading non probabilistic reliability analysis thus result in either unsafe or over conservative designs this paper presents a systematic study on the mathematical formulation for constructing the minimum volume ellipsoidal convex model using a given set of sample data and shows its application in existing methods of non probabilistic reliability analysis and design optimization of structures with bounded uncertainties in this method the uncertain parameters are first divided into groups according to their sources for each individual group of uncertainties the minimum volume ellipsoid problem is reformulated into a semi definite programming sdp problem and thus can be efficiently solved to its global optimum further a linear transformation based on the eigenvalue analysis is employed to map the ellipsoidal model into a standard uncertainty space this uncertainty modeling technique enables a compact and differentiable bound description of the parameter variations moreover it has another useful property the affine invariance which is shown to be necessary for meaningful definition of a non probabilistic reliability index the effectiveness and efficiency of the present techniques for convex model construction and the corresponding reliability analysis are demonstrated with numerical examples of structural topology optimization problems with bounded variations arising from different sources

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

interval and subinterval homogenization based method for determining the effective elastic properties of periodic microstructure with interval parameters abstract the main issue of this paper is to present an interval homogenization based method ihm for the estimation of the effective elastic tensor for microscopic material properties with interval uncertainty based on the finite element method fem for homogenization analysis the interval effective elastic tensor is expanded by the first or second order taylor series for obtaining its upper and lower bounds in order to guarantee the accuracy of ihm the subinterval technique is introduced into the ihm for estimating the effective elastic tensor numerical examples of a unit rectangular voids cell and a unit particle filled polymers cell are implemented through comparing the numerical results with the results obtained by the monte carlo simulation the effectiveness and the efficiency of the ihm for homogenization analysis with interval uncertainty are verified the influence of the microscopic interval uncertainty in material properties on a homogenized macroscopic elastic property of an inhomogeneous material has also been investigated

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

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

comparison of static response of structures using convex models and interval analysis method in this paper by combining the finite element analysis and non probabilistic convex models we present the numerical algorithm of non probabilistic convex models and interval analysis method for the static displacement of structures with uncertain but bounded parameters under the condition of the box or interval vector determined from the ellipsoid of the uncertain but bounded structural parameter vector by comparing the numerical algorithm of non probabilistic convex models and the interval analysis method in the mathematical proof and the numerical example we can see that the width of the maximum or upper and minimum or lower bounds on the static displacement yielded by the numerical algorithm of non probabilistic convex models is tighter than those produced by the interval analysis method copyright 2003 john wiley sons ltd

confidence ellipsoids for static response of trusses with load and structural uncertainties abstract this paper presents a semidefinite relaxation technique for computing a minimal bounding ellipsoid that contains the set of static responses of an uncertain truss we assume that the parameters both of member stiffnesses and external forces are uncertain but bounded by using a combination of the quadratic embedding technique of the uncertainty and the s procedure we formulate a semidefinite programming sdp problem which provides an outer approximation of the minimal bounding ellipsoid our approach has polynomial computational complexity of the problem size if the sdp problem presented is solved by using the primal dual interior point method the minimum bounding ellipsoids are computed for various trusses under several uncertain circumstances

finite element analysis of flexible multibody systems with fuzzy parameters abstract a computational procedure is presented for predicting the dynamic response and evaluating the sensitivity coefficients of flexible multibody systems whose characteristics include fuzzy parameters time histories of the possibility distributions of the response and the sensitivity coefficients are generated these coefficients measure the sensitivity of the dynamic response to variations in the material geometric and external force parameters of the system the five key components of the procedure are a a corotational frame approach used in conjunction with a total lagrangian formulation b beam and shell elements with the cartesian coordinates of the nodes selected as degrees of freedom and with continuous inter element slopes c use of an approximate method of extension based on the cut representation called the vertex method for generating the possibility distributions of the desired response quantities and their sensitivity coefficients d semi explicit temporal integration technique for generating the dynamic response and e direct differentiation approach for evaluating the sensitivity coefficients the effectiveness of the procedure and the usefulness of the fuzzy output are demonstrated through numerical examples including an articulated space structure consisting of beams shells and revolute joints

robust topology optimization minimization of expected and variance of compliance robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies this paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude using exact formulations and analytically derived sensitivities it shows that only a few additional load cases are required which scales in polynomial time with the number of uncertain load cases the approach is implemented using the level set topology optimization method shape sensitivities are derived using the adjoint method several examples are used to investigate the effect of including variance in robust compliance optimization

random sets and fuzzy interval analysis abstract a general expression of function with random set valued arguments is stated which encompasses zadeh s extension principle as well as functions of random variables and interval analysis a monotonicity property is derived for algebraic operations performed between random set valued variables this property extends the monotonicity of functions with set valued arguments in the sense of inclusion the application of this result to the approximate calculation of functions of random variables is suggested and illustrated by an example

on the second order differential equation with piecewise constant argument and almost periodic coefficients in this paper we study the existence of almost and quasi periodic solutions to two classes of second order differential equations as a corollary it is shown that periodic and unbounded solutions can coexist for the equation x t 2x t bx t f t which is different from the case b 0 this phenomena is due to the piecewise constant argument and illustrates a crucial difference between ordinary differential equations and differential equations with piecewise constant argument the results are extended to nonlinear equations

a comparison of some methods for solving linear interval equations certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined we extend a proposed technique of hansen and rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix bf a underline a overline a is an h matrix when bf a is centered about a diagonal matrix these bounds are the smallest possible i e the bounds are then the solution hull hansen s scheme also computes the solution hull when the linear interval system bf a bf x bf b underline b overline b is such that bf a is inverse positive and underline b overline b neq 0 earlier results of others also imply that when bf a is an m matrix and bf b geq 0 bf b leq 0 or 0 in bf b interval gaussian elimination iga computes the hull we also give a method of computing the solution hull inexpensively in many instances when bf a is inverse positive given an outer approximation such as that obtained from iga examples are used to compare these schemes under various conditions

overestimations in bounding solutions of perturbed linear equations abstract it is proved that some classical bounds on solutions of perturbed systems of linear equations may yield arbitrarily large overestimations for arbitrarily narrow perturbations the proofs are constructive

on quasi periodic solutions of differential equations with piecewise constant argument we study the existence of quasi periodic solutions to differential equations with piecewise constant argument epca for short it is shown that epca with periodic perturbations possess a quasi periodic solution and no periodic solution the appearance of quasi periodic rather than periodic solutions is due to the piecewise constant argument this new phenomenon illustrates a crucial difference between ode and epca the results are extended to nonlinear equations

approximating a cuts with the vertex method if f rn r is continuous and monotonic in eachvariable and if i is a fuzzy number on the ith coordinate then the membership on r induced by and by the membership onfrn given by x min 1 x1 n xn can be evaluated by determining the membership at the endpoints of the level cuts of each i here more general conditions are given for both the function and the manner in which the fuzzy numbers i are combined so that this simple method for computing induced membership may be used in particular a geometric condition is given so that the cuts computed when the fuzzy numbers are combined using min is an upper bound for the actual induced membership

dynamics of multiple cracked prismatic beams with uncertain but bounded depths under deterministic and stochastic loads abstract dynamic response of cracked beams changes with respect to undamaged ones due to the change in mechanical properties moreover usually the crack characteristics crack size crack position applied stress etc are assumed to be known however in practice they possess considerable scatter of uncertainty the aim of this paper is to evaluate the time varying upper and lower bounds of the response of damaged beams with multiple cracks subjected to deterministic or stochastic excitations assuming uncertain crack depths as interval variables in this study the analysis is performed adopting a finite element model for the damaged beam extended to the case where multiple non propagating cracks are presented in order to provide the bounds of the response the authors propose an approach developed in the time domain starting from the derivation of the bounds of the interval eigenvalues evaluated as solution of two appropriate deterministic eigenvalue problems to assess the efficiency and the accuracy of the presented method a damaged cantilever prismatic aluminum beam with multiple uncertain but bounded cracks is examined the bounds of the free end displacement and variance are calculated and compared with the exact solution via combinatorial procedure the proposed results show a high level of accuracy for both increasing amplitude value of interval deviation and increasing number of uncertain parameters

high dimensional model representation method for fuzzy structural dynamics abstract uncertainty propagation in multi parameter complex structures possess significant computational challenges this paper investigates the possibility of using the high dimensional model representation hdmr approach when uncertain system parameters are modeled using fuzzy variables in particular the application of hdmr is proposed for fuzzy finite element analysis of linear dynamical systems the hdmr expansion is an efficient formulation for high dimensional mapping in complex systems if the higher order variable correlations are weak thereby permitting the input output relationship behavior to be captured by the terms of low order the computational effort to determine the expansion functions using the cut method scales polynomically with the number of variables rather than exponentially this logic is based on the fundamental assumption underlying the hdmr representation that only low order correlations among the input variables are likely to have significant impacts upon the outputs for most high dimensional complex systems the proposed method is first illustrated for multi parameter nonlinear mathematical test functions with fuzzy variables the method is then integrated with a commercial finite element software adina modal analysis of a simplified aircraft wing with fuzzy parameters has been used to illustrate the generality of the proposed approach in the numerical examples triangular membership functions have been used and the results have been validated against direct monte carlo simulations it is shown that using the proposed hdmr approach the number of finite element function calls can be reduced without significantly compromising the accuracy

on the complexity of vlsi implementations and graph representations of boolean functions with application to integer multiplication lower bound results on boolean function complexity under two different models are discussed the first is an abstraction of tradeoffs between chip area and speed in very large scale integrated vlsi circuits the second is the ordered binary decision diagram obdd representation used as a data structure for symbolically representing and manipulating boolean functions the lower bounds demonstrate the fundamental limitations of vlsi as an implementation medium and that of the obdd as a data structure it is shown that the same technique used to prove that any vlsi implementation of a single output boolean function has area time complexity at sup 2 omega n sup 2 also proves that any obdd representation of the function has omega c sup n vertices for some c 1 but that the converse is not true an integer multiplier for word size n with outputs numbered 0 least significant through 2n 1 most significant is described for the boolean function representing either output i 1 or output 2n i 1 where 1

antioptimization of structures with large uncertain but non random parameters via interval analysis non probabilistic interval modeling of uncertain but non random parameters for structures is developed in this paper for antioptimization analysis consisting in determining the least favorable responses the uncertain but non random parameter is considered to be a deterministic variable belonging to a set modeled as an interval the least favorable static displacement bound estimation for structures with uncertain but non random parameters is transformed into solving interval linear equations for small interval parameters the width of interval being small the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters by means of the interval matrix central notation and the natural interval extension interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by qiu et al 1 for large interval parameters a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study the numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method

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

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

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

semidefinite programming for dynamic steady state analysis of structures under uncertain harmonic loads this paper presents semidefinite programming approaches for computing confidential bounds for the dynamic steady state responses of a damped structure subjected to uncertain driving loads we assume that amplitudes of harmonic driving loads obey a non probabilistic uncertainty model the semidefinite programming problems are formulated for finding confidential bounds for various characteristic amounts of dynamic steady state response including the modulus and phase angle of the complex amplitude of the displacement and stress numerical examples demonstrate that sufficiently tight bounds can be obtained by solving the presented semidefinite programming problems

elasto plastic finite element analysis with fuzzy parameters in this paper an elastoplastic analysis based on fuzzy mathematics and using finite element method will be described the drucker prager yield criterion and the elasto plastic matrix are fuzzified for the non linear analyses which adopt the initial stress method for the solution a numerical example is given to illustrate the possible variations in the displacements and the extent of plastic zones at the discrete membership functions a reliability index of a membership grade based on the concept of non probabilistic entropy is also proposed

toward verifying nonlinear integer arithmetic we eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using cdcl sat solvers by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits such short proofs were conjectured not to exist more precisely we give no 1 size regular resolution proofs for arbitrary degree 2 identities on array diagonal and booth multipliers and no log n size proofs for these identities on wallace tree multipliers

on the spectrum of almost periodic solution of second order differential equations with piecewise constant argument abstract we study the spectrum containment of almost periodic solution to second order differential equations with piecewise constant argument some known periodic quasi periodic results would be expanded as a corollary it is shown that such equations with periodic perturbations possess a quasi periodic solution and no periodic solution this phenomena is due to the piecewise constant argument the results are extended to nonlinear equations

homogenization based interval analysis for structural acoustic problem involving periodical composites and multi scale uncertain but bounded parameters this paper presents a homogenization based interval analysis method for the prediction of coupled structural acoustic systems involving periodical composites and multi scale uncertain but bounded parameters in the structural acoustic system the macro plate structure is assumed to be composed of a periodically uniform microstructure the equivalent macro material properties of the microstructure are computed using the homogenization method by integrating the first order taylor expansion interval analysis method with the homogenization based finite element method a homogenization based interval finite element method hifem is developed to solve a periodical composite structural acoustic system with multi scale uncertain but bounded parameters the corresponding formulations of the hifem are deduced a subinterval technique is also introduced into the hifem for higher accuracy numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented method for a periodical composite structural acoustic system with multi scale uncertain but bounded parameters

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

a second order delay differential equation with multiple periodic solutions abstract the equation x t 2 x t bx t 1 where designates the greatest integer function can be described in brief by two amazing properties first for certain values of the coefficients some or all of its solutions are monotone although the corresponding homogeneous equation is clearly oscillatory second for a specific relation between and b there exist periodic solutions with different periods

exact statistics of systems with uncertainties an analytical theory of rank one stochastic dynamic systems this paper concerns the response of uncertain vibro acoustic and structural dynamic systems here exact expressions are presented for the statistics of systems with a random rank one component the expressions are derived using the sherman morrison update formula that gives the exact expression of the disturbed response for any magnitude of the disturbance it is shown that the probability density function pdf of any transfer function is a simple function of the pdf of the disturbance magnitude of the random component the expressions for the mean variance and covariance of any transfer function and at any frequency of a random system necessitate non trivial integrals exact including closed form expressions of these integrals are derived in the particular cases of a real or complex normal disturbance magnitude and qualitative differences between these two cases are highlighted the theoretical and practical advantages of the theory are discussed and applied to a model of a bladed disk subjected to random damage the comparison with monte carlo simulations demonstrates that the statistics can be evaluated efficiently and precisely the theory derived formally in the context of discretised systems is directly applicable to continuous systems

analysis of uncertain structural systems using interval analysis the imprecision or uncertainty present in many engineering analysis design problems can be modeled using probabilistic fuzzy or interval methods this work considers the modeling of uncertain structural systems using interval analysis by representing each uncertain input parameter as an interval number a static structural analysis problem can be expressed in the form of a system of linear interval equations in addition to the direct and gaussian elimination based solution approaches a combinatorial approach based on an exhaustive combination of the extreme values of the interval numbers and an inequality hased method are presented for finding the solution of interval equations the range or interval of the solution vector response parameters is found to increase with increasing size of the problem in all of the methods an interval truncation approach is proposed to limit the growth of intervals of response parameters so that realistic and accurate solutions can be obtained in the presence of large amounts of uncertainty numerical examples are presented to illustrate the computational aspects of the methods and also to indicate the importance of the truncation approach in practical problems the utility of interval methods in predicting the extreme values of the response parameters of structures is discussed

analysis of mechanical systems using interval computations applied to finite element methods this paper addresses the problem of mechanical systems in which parameters are uncertain and bounded interval calculation is used to find an envelope of transfer functions for mechanical systems modelled with finite elements within this context a new formulation has been developed for finite element problems involving bounded parameters to avoid the problems of overestimation an iterative algorithm is introduced which leads to a conservative solution for linear mechanical problems a method to ensure the convergence of this algorithm is also proposed this new algorithm has been tested on simple mechanical systems and leads to a conservative envelope of the transfer functions