interval process model and non random vibration analysis abstract this paper develops an interval process model for time varying or dynamic uncertainty analysis when information of the uncertain parameter is inadequate by using the interval process model to describe a time varying uncertain parameter only its upper and lower bounds are required at each time point rather than its precise probability distribution which is quite different from the traditional stochastic process model a correlation function is defined for quantification of correlation between the uncertain but bounded variables at different times and a matrix decomposition based method is presented to transform the original dependent interval process into an independent one for convenience of subsequent uncertainty analysis more importantly based on the interval process model a non random vibration analysis method is proposed for response computation of structures subjected to time varying uncertain external excitations or loads the structural dynamic responses thus can be derived in the form of upper and lower bounds providing an important guidance for practical safety analysis and reliability design of structures finally two numerical examples and one engineering application are investigated to demonstrate the feasibility of the interval process model and corresponding non random vibration analysis method

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

maximum loads of imperfect systems corresponding to stable bifurcation abstract a simple and computationally inexpensive approach is presented for obtaining the maximum load factor of an elastic structure considering reduction of load carrying capacity due to inevitable initial imperfections the structure has a stable bifurcation point if no initial imperfection exists an antioptimization problem is formulated for minimizing the maximum loads reduced by the most sensitive imperfection within the convex bounds on the imperfections of nodal locations and nodal loads the maximum loads may be defined by bifurcation points or deformation constraints a problem of simultaneous analysis and design with energy method is formulated to avoid laborious nonlinear path following analysis the stable bifurcation point is located by minimizing the load factor under constraint on the lowest eigenvalue of the stability matrix it is shown in the examples that a minor imperfection that is usually dismissed is very important in evaluating the maximum load of a flexible structure

correlation analysis of non probabilistic convex model and corresponding structural reliability technique compared with the probability approach the non probabilistic convex model only requires a small amount of samples to obtain the variation bounds of the imprecise parameters and whereby makes the reliability analysis very convenient and economical in this paper we attempt to propose and create a correlation analysis technique mathematically for the non probabilistic convex model and based on it develop an effective method to construct the multidimensional ellipsoids on the uncertainty a marginal convex model is defined to describe the variation range of each uncertain parameter and a covariance is defined to represent the correlation degree of two uncertain parameters for a multidimensional problem the covariance matrix and correlation matrix can be created through all marginal convex models and covariances based on which the required ellipsoid on the uncertainty can be conveniently achieved by combining the correlation analysis technique and the reliability index approach a non probabilistic reliability analysis method is also developed for uncertain structures six numerical examples are presented to demonstrate the effectiveness of the present method

interval dynamic response analysis of vehicle bridge interaction system with uncertainty abstract interval dynamic response analysis of vehicle bridge interaction system with uncertain parameters is studied in this paper the bridge s and vehicle s parameters are considered as interval variables a half car model is used to represent a moving vehicle and the bridge is assumed as a simply supported euler bernoulli beam by introducing the interval algorithm into the dynamic analysis of vehicle bridge interaction system the expressions of the midpoint value interval width lower and upper bounds of interval bridge dynamic response are developed the lower and upper bounds are respectively the minimum and maximum values of bridge dynamic response which can also be determined by optimization approaches a heuristic optimization method a new particle swarm optimization method with low discrepancy sequence initialized particles and high order nonlinear time varying inertia weight and constant acceleration coefficients lhnpso is then developed to find the bounds of bridge displacement examples are used to illustrate the effectiveness of the presented methods and monte carlo simulations are implemented to validate the computational results the effects of individual system parameters on bridge response are also investigated

fuzzy stochastic fuzzy time series and its models in this paper as an extension of the concept of time series we will present the definition and models of fuzzy stochastic fuzzy time series fsfts both of whose values and probabilities with which the fsfts assumes its values are fuzzy sets and which may not be modeled properly by the concept of time series to investigate fsfts the definition of fuzzy valued probability distributions is considered and discussed when the fsfts is time invariant several preliminary conclusions are derived

robust optimization in discrete design space for constrained problems robust design in discrete design space is defined as a discrete design that is insensitive to external uncertainties or variations the application of robust discrete design is not prevalent yet due to high computational cost a relatively simple method is proposed to select discrete and robust optimum at first the discrete design is achieved as the postprocess of conventional optimization an orthogonal array is established around a conventional optimum and the characteristic functions are evaluated the characteristic function is defined by considering the robustness of the objective and constraints the parameter design of the taguchi method is introduced to obtain the robust solution in discrete space the present method has insensitive performance to variations of the design variables within the selected discrete values enhancing the feasibility of constraints to enhance feasibility ranking the estimators of the characteristic function is developed several structural problems are solved to show the usefulness of the present method

hybrid uncertain analysis for structural acoustic problem with random and interval parameters abstract in this paper hybrid perturbation monte carlo method hpmcm and hybrid perturbation vertex method hpvm are developed for the hybrid uncertain structural acoustic problem with random and interval variables the random variables are used to treat the uncertain parameters with sufficient information whereas the interval variables are used to treat the uncertain parameters with limited information in hpmcm the intervals of expectation and variance of the frequency response vector are calculated by a combination of random interval matrix perturbation method random interval moment method and monte carlo method in hpvm a monotonicity analysis is conducted for the frequency response vector based on the random interval matrix perturbation method and then the intervals of expectation and variance of the frequency response vector are calculated by employing the random interval moment method and the vertex method numerical results on a shell structural acoustic problem verify the effectiveness and high efficiency of hpvm compared with hpmcm

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

interval k l expansion of interval process model for dynamic uncertainty analysis abstract the interval process model describes a time variant or dynamic uncertain parameter by the upper and lower bounds rather than the precise probability distribution at each time point providing an effective structural dynamic uncertainty quantification model with insufficient sample information by reference to the karhunen loeve k l expansion for stochastic process and random field models a novel expansion method for the interval process model namely the interval k l expansion is proposed in this work the interval k l expansion describes the continuous uncertainty of the interval process in time domain by superposition of infinite deterministic time related functions with uncorrelated interval coefficients which makes the interval process model much more convenient to use by introducing a splicing technique the interval k l expansion method for multiple correlated interval processes is also presented based on the interval k l expansion the vibration of structures subject to interval process excitations is analyzed where the analytic formulation of dynamic response bounds for linear elastic systems and continuum structures are derived finally several numerical examples are investigated to demonstrate the effectiveness of the proposed method

nonprobabilistic convex theoretic modeling of scatter in material properties nonprobabilistic convex modeling of uncertain material properties for viscoelastic structures is developed in this paper in particular the problem of forced vibrations of viscoelastic beams is studied first the analytic solution by inman is generalized for a deterministic set of variables describing material properties next these variables are treated as varying in a solid ball in the four dimensional space thus modeling the scatter in material properties the least favorable response needed for the design of the structure is determined

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

stability of elastic bars on uncertain foundations using a convex model a method is presented for evaluating the buckling load of weightless prismatic rigid bars resting on uncertain foundations which are modeled using linear extensional elastic springs the uncertainty in the spring elements is expressed in terms of a nonprobabilistic convex model an ellipsoidal bound is used which defines the uncertainty of the foundation in terms of a size parameter and the deviations of the elastic spring constants from their nominal values the size parameter represents the size of the ellipsoid and is analogous to the standard deviation magnitude in probabilistic analyses the semiaxes of the ellipsoid are the deviations of the foundation spring constants from their nominal values and they determine the shape of the ellipsoid a first order analysis shows that the reduction in the buckling load when uncertainty in the foundation s spring stiffnesses is present is a linear function of the size parameter and a nonlinear function of the semiaxes of the uncertainty ellipsoid for the same uncertainty in the spring elements different reductions in the buckling load result for beams with multimode buckling

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

transient waves in composite laminated plates with uncertain load and material property a method is suggested to deal with the wave propagation problems in composite laminated plates subjected to uncertainty in load and material property based on the interval analysis method and the hybrid numerical method hnm the uncertain parameters are treated as intervals in which only their bounds of the uncertainty are needed using the first order taylor expansion the transient responses can be approximated as a linear function of the uncertain parameters in this function the transient responses at the midpoints of the uncertain parameters can be obtained directly through the hnm a sensitivity analysis technique is suggested to calculate the first derivative of the transient responses with respect to each uncertain parameter based on two cases that the parameter exists in load or material property applying the interval extension in interval mathematics the lower and upper bounds of the transient responses caused by the uncertainty can be finally obtained the present method is applied to a numerical example in which the uncertainty of the load elastic constants of the layer material and ply orientations are all investigated and the results demonstrate the effectiveness of the present method

interval uncertain method for multibody mechanical systems using chebyshev inclusion functions this study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on chebyshev inclusion functions the interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain but bounded parameters which only requires lower and upper bounds of uncertain parameters without having to know probability distributions a chebyshev inclusion function based on the truncated chebyshev series rather than the taylor inclusion function is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions to effectively handle the overestimation caused by the wrapping effect intrinsic to interval computations the mehler integral is used to evaluate the coefficients of chebyshev polynomials in the numerical implementation the multibody dynamics of mechanical systems are governed by index 3 differential algebraic equations daes including a combination of differential equations and algebraic equations responsible for the dynamics of the system subject to certain constraints the proposed interval method with chebyshev inclusion functions is applied to solve the daes in association with appropriate numerical solvers this study employs hht i3 as the numerical solver to transform the daes into a series of nonlinear algebraic equations at each integration time step which are solved further by using the newton raphson iterative method at the current time step two typical multibody dynamic systems with interval parameters the slider crank and double pendulum mechanisms are employed to demonstrate the effectiveness of the proposed methodology the results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect as cosine functions are incorporated to sharpen the range of non monotonic interval functions copyright 2013 john wiley sons ltd

hybrid uncertain natural frequency analysis for structures with random and interval fields abstract this paper presents a robust non deterministic free vibration analysis for engineering structures involving hybrid yet spatially dependent uncertain system parameters distinguished from the conventional hybrid uncertain eigenvalue problem the concept of interval field is enclosed with random field model such that both the stochastic and non stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non deterministic free vibration analysis in order to determine the probabilistic characteristics i e means and standard deviations of the extremities of structural natural frequencies an extended unified interval stochastic sampling x uiss method is implemented for the purpose of effective hybrid uncertain free vibration analysis by meticulously blending sharpness promised interval eigenvalue analysis with stochastic sampling techniques the stochastic profiles i e probability density functions pdfs and the cumulative distribution functions cdfs of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods the applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval an optimization method for uncertain structures is suggested based on convex model and a satisfaction degree of interval in the investigated problem the uncertainty only exists in constraints convex model is used to describe the uncertainty in which the intervals of the uncertain parameters are only needed not necessarily to know the precise probability distributions a satisfaction degree of interval which represents the possibility that one interval is smaller than another is employed to deal with the uncertain constraints based on a predetermined satisfaction degree level the uncertain constraints are transformed to deterministic ones and the transformed optimization problem can be solved by traditional optimization methods for complex structural problems that the optimization model cannot be expressed in an explicit form the interval analysis method is adopted to calculate the intervals of the constraints efficiently and whereby eliminate the optimization nesting two numerical examples have been presented to demonstrate the efficiency of the suggested method

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

a new interval pattern analysis method of array antennas based on taylor expansion in this communication a new taylor expansion based interval pattern analysis method is developed to predict the impact of the small excitation amplitude uncertainty on the radiated array pattern the dependence problem of the classical interval arithmetic which may lead to overestimation of performance intervals is addressed in this communication the proposed method approximates by approximating the antenna power values to the first order with respect to the uncertain amplitude the new method is simpler and can produce more accurate bounds of the antenna pattern and the pattern features than the existing techniques several representative numerical examples are investigated to demonstrate the validity and efficiency of the proposed method

topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system abstract this paper proposes a topology optimization for a linear elasticity design problem subjected to an uncertain load the design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors the idea of aggregation used in the field of control is introduced to assess the value of the robust compliance the aggregation solution technique provides the direct relationship between the uncertain input load and output displacement as a small linear system composed of these vectors and the reduced size of a symmetric matrix in the context of a discretized linear elasticity problem using the finite element method introducing the constraint that the euclidean norm of the uncertain load set is fixed the robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix according to the rayleigh ritz theorem for symmetric matrices moreover the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix the proposed structural optimization method is implemented using topology optimization and the method of moving asymptotes mma the numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures

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

development of a simple and efficient method for robust optimization robust optimization problems are newly formulated and an efficient computational scheme is proposed both design variables and design parameters are considered as random variables about their nominal values to ensure the robustness of objective performance we introduce a new performance index bounding the performance together with a constraint limiting the performance variation the constraint variations are regulated by considering the probability of feasibility each probability constraint is transformed into a sub optimization problem by the advanced first order second moment afosm method for computational efficiency the proposed robust optimization method has the advantages that the mean value and the variation of the performance function are controlled simultaneously and rationally and the second order sensitivity information is not required even in case of gradient based optimization process the suggested method is examined by solving three examples and the results are compared with those for the deterministic case and those available in the literature copyright 2002 john wiley sons ltd

reliability analysis of structures with interval uncertainties under stationary stochastic excitations reliability analysis of linear discretized structural systems with interval uncertainties subjected to stationary gaussian random excitation is addressed under the assumption of independent up crossings of a specified threshold an efficient procedure for the evaluation of the bounds of the interval reliability function of the generic response process is presented the first step of the proposed approach is to derive approximate expressions of the interval mean value and spectral moments of the response along with the associated bounds to this aim the improved interval analysis via extra unitaryinterval is applied in conjunction with a novel series expansion of the inverse of an interval matrix with modifications called interval rational series expansion irse then the lower bound and upper bounds of the interval reliability function are readily evaluated by properly combining the bounds of the interval mean value and spectral moments of the response two numerical examples are provided to demonstrate the accuracy of the proposed procedure and its usefulness in view of decision making in engineering practice

interval analysis method for damage identification of structures description was adopted for the measured natural frequencies in this paper via the first order taylor series expansion the interval bounds of the elemental stiffness parameters of both undamaged and damaged structures were derived by using the model updating based on the initial analytical finite element model the damage can be identified by comparing the differences between the two models where the quantitative measure of the possibility of damage existence in the elements is introduced a larger value of possibility of damage existence implied a higher possibility of damage occurrence the damage identifications for a steel cantilever beam and a steel cantilever plate were performed by the presented method which is validated by monte carlo simulation moreover the case of the multidamage identification the number of the used natural frequencies and the effects of damage level and uncertainty level on the damage detection were studied as well the numerical results proved the validity and applicability of the presented interval analysis method

minimal complexity of semi bounded components in bifurcation theory abstract in this paper we use the topological degree to estimate the minimal number of solutions of the slices of the semi bounded and bounded components of a general class of fixed point equations depending on a real parameter by means of the signature of the component the signature is defined as the set of bifurcation intervals from a given state together with all changes of the local index of the state calculating the minimal number of solutions of the component slices enables us to ascertain its minimal complexity in terms of graph theory at least for a generic family of non degenerate situations where the general theory of this paper provides the minimal canonical structure of the component

first passage of uncertain single degree of freedom nonlinear oscillators the first passage of uncertain single degree of freedom nonlinear oscillators subjected to random excitation is examined statistical fourth moment method is developed to determine first four moments of system response and state function distribution function of the system state function is approximately determined by the standard normal distribution functions using edgeworth series technique and the reliability is obtained

extreme structural response analysis of truss structures under material uncertainty via linear mixed 0 1 programming this paper concerns the extreme structural response analysis of truss structures taking the possible uncertainties of young s moduli of the bars into consideration the uncertainty of young s modulus is described by a non probability but bounded interval model with the use of this uncertain model it is shown that the structural responses such as nodal displacement and bar stress will attain their extreme values when young s moduli of the bars take either their upper or lower bound value based on this conclusion and simultaneous analysis and design formulation a linear mixed 0 1 programming problem can be constructed to find the extreme values of the structural response by the well established branch and bound approach with global optimality the proposed approach is applied to several test problems for the demonstration of its validity

a sequential nonlinear interval number programming method for uncertain structures a sequential nonlinear interval number programming sninp method is suggested to deal with the uncertain optimization problems a general uncertain optimization model is investigated in which the objective function and constraints are both nonlinear and uncertain a nonlinear interval number programming ninp method is employed to transform the uncertain optimization problem into a deterministic two objective optimization problem then using the linear combination method and the constraint penalty function method a deterministic single objective and non constraint optimization problem is formulated in terms of a penalty function combining this ninp method with a modified approximation management framework amf an efficient sninp method is then constructed at each iterative step an approximation optimization problem is created based on the latin hypercube design lhd and the quadratic polynomial response surface approximation rsa and it can then be solved by the ninp method efficiently the trust region method is used to manage the sequence of the approximation optimization problems based on a reliability index an efficient method is suggested to calculate the actual penalty function and whereby the reliability index and based on it the current design space can be updated two numerical examples are presented to demonstrate the effectiveness of the present method

the time response of structures with bounded parameters and interval initial conditions uncertainty plays an important role in the performance of structures in this paper we focus on the dynamic response of structures with bounded parameters and interval initial conditions and present a new method to determine the supremum and infimum of the time response the method is based on the vertex solution theorem for the first order deviation of the dynamic response from its central value and avoids interval extension problems present in current methods where the length of the interval increases significantly due to the intermediate calculations the method is more accurate than existing perturbation methods and provides tighter bounds on the response the approach neglects the second order terms in the equation of motion and care should be exercised when the parameter variations are large the other advantage of this method is its ability to solve problems with uncertainties in the initial conditions

interval field model and interval finite element analysis uncertain parameters with inherent spatial variability are commonly encountered in engineering modeling of this kind of spatial uncertainty plays a fundamental role in structural uncertainty analysis which provides a necessary basis for subsequent uncertainty propagation through the system an interval field model for quantification of spatial uncertain parameters is proposed by which only the upper and lower bounds of the spatial uncertain parameters rather than their precise probability distributions are required the dependency can be fully considered by the proposed interval field model with the information of dependency an interval k l expansion is presented as a combination of deterministic functions with uncorrelated standard interval variables through which the continuous spatial interval field with dependency can be expressed only by very limited intervals necessary mathematical illustrations are provided for the proposed interval field model and the interval k l expansion the sampling method for the interval field model is given providing a robust numerical analysis basis for subsequent structural uncertainty analysis when the interval field model is applied in finite element analysis of structures with spatial uncertain parameters the non deterministic equilibrium equations with interval factors is then formulated the mcs method and the perturbation method are developed for solution of the derived interval equilibrium equations the feasibility and validity of the proposed interval field model and corresponding interval finite element methods are verified by numerical examples where the upper and lower bounds of the responses such as the displacements and the stresses of structures with spatial uncertain parameters are computed and compared

hybrid finite element statistical energy method for mid frequency analysis of structure acoustic systems with interval parameters abstract the hybrid finite element statistical energy analysis fe sea model for mid frequency analysis consists of two parts the fe components and the sea components which are called as the master system and the subsystem respectively the hybrid finite element statistical energy analysis approach is established on the assumption that the fe components have fully deterministic properties while the sea subsystems have a high degree of randomness this method has been recently extended to allow the properties of the fe components of built up structures to be uncertain like interval or probabilistic and the hybrid model with parametric and non parametric uncertainties is obtained by dealing with the non parametric uncertainty analytically and the parametric uncertainty with monte carlo simulations the distribution of the responses of the hybrid model can be obtained in this paper the interval parametric uncertainty is introduced into the hybrid fe sea framework for structure acoustic systems and the interval dynamic equilibrium equations are established thus a hybrid model with non parametric and interval parametric uncertainties for structure acoustic systems is obtained to improve the computational efficiency the second order interval perturbation finite element method is introduced into the hybrid fe sea framework and the second order interval perturbation finite element statistical energy analysis sipfem sea method is proposed for the structure acoustic system with interval parameters modeled by fe sea the parameters of fe components are considered as interval parameters instead of deterministic ones thus the expectations of the second order quantities such as the vibrational energy and the response cross spectrum in the fe sea framework become interval variables too in the sipfem sea the expectations of the second order response quantities of a structure acoustic system are expanded with the second order taylor series at the nominal values of interval parameters and for the sake of simplicity and efficiency the non diagonal elements of the hessian matrices are neglected then by searching the target positions of interval parameters that maximize or minimize the objective functions the bounds of the vibrational energy and the response cross spectrum can be obtained because of the neglect of the higher order terms of taylor series sipfem sea is limited to the interval analysis with narrow parameter intervals for larger parameter intervals the sub interval perturbation method based on the sipfem sea is introduced the proposed methods are illustrated by applications to two example structure acoustic systems in which the acoustic cavities are modeled by using the interval fe method and the structures are modeled by using the sea reference comparisons are made with the monte carlo simulations of the hybrid fe sea model the numerical results verify the accuracy and efficiency of the proposed methods

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

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 topology optimization under multiple independent unknown but bounded loads abstract the optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable or even lead to catastrophic failures a two level optimization formulation is often used in the robust topology optimization rto under uncertain loads various approaches have been reported to identify the critical loads associated with the worst structure responses because convex model approaches apply convex approximations to the original non convex model at the lower level the optimal designs obtained by these methods are greatly dependent on the quality of the approximation in this paper a new formulation based on the wolfe duality for the rto problems with multiple independent unknown but bounded loads is proposed following the two level formulation the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the wolfe duality both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non convexity associated with the lower level formulation numerical examples are also presented to demonstrate the proposed approach

a mixed integer programming for robust truss topology optimization with stress constraints this paper presents a mixed integer programming mip formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load a design dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization for a truss with the discrete member cross sectional areas it is shown that the robust topology optimization problem can be reduced to an mip problem which is solved globally numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty

monogamy equality in 2 2 d quantum systems there is an interesting property about multipartite entanglement called the monogamy of entanglement the property can be shown by the monogamy inequality called the coffman kundu wootters inequality phys rev a 61 052306 2000 coffman kundu woottersphys rev lett 96 220503 2006 and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance ca bc 2 cab2 caca 2 in the three qubit system in this paper we consider the monogamy equality in 2 2 d quantum systems we show that ca bc cab if and only if caca 0 and also show that if ca bc caca then cab 0 while there exists a state in a 2 2 d system such that cab 0 but ca bc caca

confidence structural robust design and optimization under stiffness and load uncertainties abstract most of the structural behavior constraints involved in structural robust design and optimization are non convex in nature therefore if local optimality criteria based optimization algorithms are employed to find the worst case structural responses that are used for examining the feasibility of a given design it is highly possible that the optimization process will get stuck in a local optimum if this is the case the reliability of a robust design cannot be guaranteed at least theoretically the aim of the present paper is to develop some new formulations and the corresponding numerical algorithms to find the confidence optimal design under non probabilistic stiffness and load uncertainties to this end two bi level program formulations for confidence robust design are proposed in order to ensure the strict feasibility of the optimal solution in the lower level of program the constraints are imposed on the confidence upper bounds of the structural responses which can be obtained efficiently by solving some convex linear semi definite programs lsdps based on the sensitivity analysis of the lower level lsdp problem the upper level programs are then solved by employing the classical gradient based nonlinear optimization algorithms furthermore for the case of stiffness uncertainty a single level nonlinear semi definite programming nsdp formulation is also proposed and its mathematical properties are analyzed numerical examples show that confidence robust optimal design can be obtained via the proposed approaches effectively without resorting to too many computational efforts

convex models for optimizing diagnosis of uncertain slender obstacles on surfaces abstract blockage of a flowing compressible fluid has in the past been assessed by measuring shifts in acoustic eigenfrequencies this method has proven successful in many situations but measurements in complex geometries can present ambiguities partial blockage of the flow over a surface can be assessed by measuring pressure oscillations detection of small obstacles requires careful measurement design in this paper a technique for establishing the optimal deployment of pressure sensors is presented the primary uncertainty with respect to which the design is optimized is the spatial uncertainty in the shape and location of the blockage simulations are presented for subconic flow

convex models and interval analysis method to predict the effect of uncertain but bounded parameters on the buckling of composite structures a new non probabilistic set theoretical models that is interval analysis method is developed in this study to predict the variability or uncertainty in buckling loads of composite structures resulting from the unavoidable scatter in structural parameters by mathematically analyzing and proving the width of the upper and lower bounds on the critical buckling loads calculated by the interval analysis method is sharper than those that are obtained by convex models method moreover the interval analysis method has less computational cost than convex models method comparison between convex models and interval analysis method is performanced through numerical examples for the specific cases considered the results of numerical examples indicate that uncertainties in structural parameters have significant effect on the critical buckling loads of composite structures

the static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem abstract in practical engineering on account of physical imperfections model inaccuracies and system complexities almost all structures have physical and geometrical uncertainties in some degrees in this paper we focus on the static response problem and the stress distribution of structures with bounded uncertainties in terms of the vertex notation in interval analysis a new mathematical proof of the vertex solution theorem which is used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties is given then we extend this theorem for calculating the interval stress and the deformation distributions which was only used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties the basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations the resulting deterministic linear equations are then solved by using the familiar techniques like the gauss elimination method or the gauss seidel iteration method another advantage of this method is that the increase in number of the uncertain parameters in a structure element will not result in the increase in number of the uncertain elements in the stiffness matrix for practical engineering problem with bounded uncertainties the chief thing is to avoid the disadvantages of large calculation and much runtime of the vertex solution in this paper we present the parallel arithmetic of the vertex solution theorem which can be used in large scale computations in practical engineering to avoid much runtime on the other hand we can use the symmetrical characteristic of the stiffness matrix and the substructure method to reduce the computational effort in practical engineering the numerical results of the typical example presented by deif calculated by the proposed vertex solution theorem show that the width between the supremum and the infimum on the static response yielded by the vertex solution theorem is tighter than those produced by the kcn s method presented by koyluoglu cakmak and nielson the numerical examples of the two dimensional shell with an edge fixed and the two dimensional shell with surrounding fixed are used to illustrate the computational feasibility of the vertex solution theorem

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

non probabilistic stability reliability measure for active vibration control system with interval parameters abstract a systematic non probabilistic reliability analysis procedure for structural vibration active control system with unknown but bounded parameters is proposed the state space representation of active vibration control system with uncertain parameters is presented compared with the robust control theory which is always over conservative the reliability based analysis method is more suitable to deal with uncertain problem stability is the core of the closed loop feedback control system design so stability criterion is adopted to act as the limited state function for reliability analysis the uncertain parameters without enough samples are modeled as interval variables interval perturbation method is employed to estimate the interval bounds of eigenvalues which can be used to characterize the stability of the closed loop active control system formulation of defining the reliability of active control system based on stability is discussed a novel non probabilistic reliability measurement index is discussed and used to determine the probability of the stability based on the area ratio the feasibility and efficiency of the proposed method are demonstrated by two numerical examples

comparison of statistical and interval analysis for the energy flow uncertainties in structural vibrating systems in this paper the energy flow confidence between two structural multimodal systems coupled by a joint with uncertain parameters is computed by two different methods the first one assumes that the joint parameters perturbed randomly the statistical moments of the energy flow are calculated by an analytical procedure the second one uses interval analysis the joint parameters are considered interval variables and the interval of the energy flow is determined the properties of the statistical and interval solution are investigated and compared

the encounter of interval and probabilistic approaches to structural reliability at the design point abstract the consideration of uncertainties in structural analysis has traditionally been made in the field of structural reliability within the framework of classical probability theory in many cases however there is no sufficient probabilistic information to substantiate such an approach in recent years several non probabilistic methods have been proposed as alternatives one of them is the interval analysis oriented to estimate the range of variation of output structural variables given the ranges of the input ones the method however requires the use of special arithmetics which renders its application unpractical for structural analysis based on finite elements or other numerical techniques in this paper a monte carlo method that can be applied to probabilistic as well as to interval approaches to reliability analysis is proposed it is based on the optimal properties of the classical design point of probabilistic structural reliability in particular it is shown that the order statistics of the output variable is concealed in a plot defined by the design point vector this facilitates the selection of the relevant samples for either interval or probability based analysis the examples show that the desired interval of the response or the failure probability can be accurately estimated on such a basis the paper also discusses the use of monte carlo methods for both reliability and interval analysis from the point of view of the proposed representation

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

modal analysis of structures with uncertain but bounded parameters via interval analysis in this paper the modal interval analysis method to estimate modal parameters frequency response function frf and mode shapes of structures with uncertain but bounded is presented although the system parameters or properties are uncertain in many engineering problems but their probable range of values i e upper and lower bounds can be provided from practical experience and engineering knowledge moreover to avoid the resonance of a structure and to consider the dynamic response of an uncertain one and also for reliability and stability analysis we often need the bounds of the ranges of structural characteristic parameters such as natural frequency and normal mode by using modal analysis and interval calculus we investigate the method of computing upper and lower bounds of paramenters such as natural frequencies modal shapes and frfs theoretically it is possible to analyze the uncertain but bounded of a structure by using modal analysis and interval calculus they can be estimated by modal interval analysis on the basis of the estimated intervals the engineering structure parameters can be applied into engineering design a numerical example is presented for a tower structure and the results illustrate that the proposed method is effective a comparison of the modal interval method with the results of monte carlo simulation serves to validate the solutions and to identify the bounded ranges of parameters

fuzzy time series and its models based upon the works by zadeh 1973 1975 in this paper a special dynamic process with linguistic values as its observations is defined and studied this dynamic process is called fuzzy time series to describe fuzzy time series fuzzy relational equations are employed as the models there are two classes of fuzzy time series time variant and time invariant the definition of fuzzy time series is given some properties of fuzzy time series are explored and procedures to develop fuzzy time series models are discussed finally a procedure for using fuzzy time series for foracasting is suggested

forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions abstract in this study a general frame of the forward and inverse structural uncertainty propagations ups based on the dimension reduction dr method and the derivative lambda probability density function pdf is proposed to handle arbitrary probability distribution for the forward up a dr method is applied to decompose a multivariable system into multiple univariate subsystems and a derivative pdf is adopted to transform the arbitrary probability distribution of each subsystem then the statistical moments and a fitting region are mathematically derived to analyze the fitting ability of the derivative pdf according to whether the kurtosis skewness point lies in or out the fitting region two different strategies combining the gauss gegenbauer quadrature are proposed to implement the forward propagation compared with the conventional methods the proposed method has advantages in higher accuracy stability and efficiency for the inverse propagation because the unknown variable may be arbitrary distribution the general frame based on the derivative pdf and the gauss gegenbauer quadrature are utilized to convert the uncertainty propagation into multiple deterministic calculations based on this optimization method is adopted to accurately obtain the statistical moments and pdfs of the unknown stochastic variables five examples are provided to verify the accuracy and efficiency of the proposed general frame for the forward and inverse ups

parameter perturbation method for dynamic responses of structures with uncertain but bounded parameters based on interval analysis the study was intended to evaluate the range of dynamic responses of structures with uncertain but bounded parameters by using the parameter perturbation method the uncertain parameters were modeled as an interval vector the first order perturbation quantities of responses of the perturbed system were obtained through the parameter perturbation method and then taking advantage of interval mathematics a new algorithm to estimate the response interval was presented comparisons between the parameter perturbation method and the probabilistic approach from mathematical proofs and numerical simulations were performed the numerical results are in agreement with the mathematical proofs the response range given by the parameter perturbation method encloses that obtained by the probabilistic approach the results also show good robustness of the proposed method

non probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty effects of uncertainties on the dynamic response of the nonlinear vibration systems with general form are investigated based on interval mathematics modeling the uncertain parameters as interval numbers a non probabilistic interval analysis method which estimates the range of the nonlinear dynamic response with the help of taylor series expansion is presented where the partial derivatives of the dynamic response with respect to uncertain parameters are considered to be interval numbers the sensitivity matrices of dynamic response with the uncertain parameters are derived for the presented method only the bounds on uncertain parameters are needed instead of probabilistic density distribution or statistical quantities numerical examples are used to illustrate the validity and feasibility of the presented method

nonprobabilistic reliable lqr design method for active vibration control of structures with uncertainties the linear quadratic regulator lqr which allows a natural consideration of the constraints on state variables and control effort is widely used in modern control theory for the active control o

dual monogamy inequality for entanglement we establish duality for monogamy of entanglement whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system as quantified by linear entropy we prove that the same quantity namely linear entropy provides a lower bound for distribution of bipartite entanglement in a multipartite system our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest

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

non probabilistic convex model process a new method of time variant uncertainty analysis and its application to structural dynamic reliability problems in this paper we propose a method for time variant uncertainty analysis namely the non probabilistic convex model process which provides an effective mathematical tool for the analysis of structural dynamic uncertainty when lacking relevant information in the convex model process we express the variables at any time with intervals and establish the corresponding auto covariance function and correlation coefficient function to depict the correlation between variables at different times we also define several important characteristic parameters for the uni and bi dimensional convex model processes including the mid value function variance function auto covariance function and cross covariance function we provide the definition for the stationary convex model process and its ergodicity then by combining the convex model process with the first passage failure mechanism we propose a non probabilistic analysis model of structural dynamic reliability and formulate the solving algorithm based on monte carlo simulation finally through the analysis of numerical examples we verify the effectiveness of the convex model process and the model of dynamic reliability analysis proposed in this paper

maximal stiffness design of two material structures by topology optimization with nonprobabilistic reliability basedon the multi ellipsoid convex modelandthe quantified measure of the nonprobabilistic reliability topology optimization of two material structures in the presence of parameter uncertainties is investigated the task of the optimal design problem is to distribute a given amount of two candidate materials into the design domain for acquiring the maximal stiffness while satisfying the reliability requirement the extended power law interpolation scheme for material properties is employed for relaxing the two material topological design problem into a continuous valued optimization problem in addition through transforming the minimax type optimization problemintoaseriesofdeterministiconesbyusingasequentialapproximateprogrammingstrategy thispaperaims tomaketheoptimizationdesignnumericallytractable theresultingmathematicalprogrammingproblemsarethen efficientlysolvedbytheassociationofthemethodofmovingasymptoteswithaheuristiciterativemanner numerical investigations reveal that system uncertainties may have considerable effects on the optimal material layout of twomaterial structures the proposed topology optimization methodology could yield a more reasonable two material structuraldesignthan theconventional deterministic counterpart whenthe samereliabilityrequirements need tobe achieved

uncertain buckling its past present and future an authoritative review on stochastic buckling of structures was written by amazigo some quarter century ago the present review summarizes some of the developments which took place since then a brief overview of the effect of uncertainty in the initial geometric imperfections elastic moduli applied forces and thickness variation is given for the benefit of the thoughtful reader the review is of critical nature

a general framework for robust topology optimization under load uncertainty including stress constraints we present a worst case approach for topology optimization under load uncertainty based on a general problem formulation involving maxima of quadratic functions of an uncertain load vector as both

multidimensional parallelepiped model a new type of non probabilistic convex model for structural uncertainty analysis summary non probabilistic convex models need to be provided only the changing boundary of parameters rather than their exact probability distributions thus such models can be applied to uncertainty analysis of complex structures when experimental information is lacking the interval and the ellipsoidal models are the two most commonly used modeling methods in the field of non probabilistic convex modeling however the former can only deal with independent variables while the latter can only deal with dependent variables this paper presents a more general non probabilistic convex model the multidimensional parallelepiped model this model can include the independent and dependent uncertain variables in a unified framework and can effectively deal with complex multi source uncertainty problems in which dependent variables and independent variables coexist for any two parameters the concepts of the correlation angle and the correlation coefficient are defined through the marginal intervals of all the parameters and also their correlation coefficients a multidimensional parallelepiped can easily be built as the uncertainty domain for parameters through the introduction of affine coordinates the parallelepiped model in the original parameter space is converted to an interval model in the affine space thus greatly facilitating subsequent structural uncertainty analysis the parallelepiped model is applied to structural uncertainty propagation analysis and the response interval of the structure is obtained in the case of uncertain initial parameters finally the method described in this paper was applied to several numerical examples copyright 2015 john wiley sons ltd

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

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

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

structure of solutions set of nonlinear eigenvalue problems abstract in this paper by using the global bifurcation theory we obtain some results for structure of the solution set of some nonlinear equations with parameters we show a result concerning the existence of a connected component which either has a loop or is unbounded both from left and right hand side especially in this paper we also give some sufficient conditions for a bounded connected component of solution set being a loop

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

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

bounds analysis of a beam based on the convex model of uncertain foundation a methodology of bounds analysis based on the convex set of uncertainty is presented an infinitely long beam on an uncertain foundation is subjected to a moving force with constant speed the steady state vibration is analysed in the context of the finite element method uncertainty variables representing the uncertain fluctuation of foundation stiffness is confined within a convex hull given as a hyperellipse the fluctuation of dynamic responses of the beam that is the vertical deflection and the bending moment are approximated to a first order with respect to the uncertainty variables the upper and lower bounds of the response fluctuations are identified on the boundary of the convex model by means of the lagrange multiplier method numerical analysis demonstrates the validity of the proposed formulation for various constant speeds of the moving force the analysis is summarized in terms of the worst response which is the maximum absolute value of the response within the identified bounds

a general frame for uncertainty propagation under multimodally distributed random variables abstract uncertainty propagation under multimodally distributed random variables called multimodal distribution propagation for short is a challenging problem due to the complicate probability density function of the random variables in this paper a general frame based on a new finite mixture model mm constructed by derivative lambda probability density function and polynomial chaos expansion method is put forward to efficiently solve the multimodal distribution propagation problem first mm with high accuracy and extensive applicability is proposed to represent the arbitrary unimodal distributions and multimodal distributions second a pseudo em method proved strictly is proposed to estimate the parameters of this mm third the statistical moments of the response by polynomial chaos expansion method is derived mathematically since the new mixture model can be decomposed into several unimodal probability distributions the multimodal distribution propagation can be successfully converted into several unimodal distribution propagations which further can be easily solved by polynomial chaos expansion method finally the maximum entropy principle is adopted to evaluate the probability density function of the result of every unimodal distribution propagation which is then superimposed into the final probability density function of the system response the proposed frame only requires the first four statistical moments to evaluate the probability density function and avoid calculating the higher statistical moments especially for the situation that the probability distribution of the system response is multimodal four examples are presented to verify the high accuracy and efficiency of the proposed general frame for uncertainty propagation

safety margin analysis of buckling for structures with unknown but bounded uncertainties abstract a safety margin of buckling for structures with unknown but bounded uncertainties is proposed in this paper the perturbation matrixes consisting of the elastic stiffness matrix and the geometric stiffness matrix are introduced to derive the concept of safety margin the concept of safety margin of buckling is first proposed to quantify the influence of uncertain parameters on structural buckling based on the spectrum norm and the singular value analysis the mathematical basis of the safety margin is expounded furthermore to obtain a more accurate estimation of safety margin the correction factor for the safety margin is introduced through the interval extension of interval analysis two engineering examples compared with traditional interval analysis method are utilized to demonstrate the validity of the proposed safety margin and numerical results indicate that the safety margin can provide a relatively accurate index to indicate the allowable perturbation range of the perturbation matrix of a structure under given loads

a novel method of non probabilistic reliability based topology optimization corresponding to continuum structures with unknown but bounded uncertainties abstract a new non probabilistic reliability based topology optimization nrbto method for detailed design of continuum structures is proposed in this paper in which the unknown but bounded ubb uncertainties existing in material and external loads are considered simultaneously based on the interval set theoretic model traditional displacement constraint is transformed into the non probabilistic reliability format in which a novel measuring index namely the optimization feature distance is defined by a combination of the area ratio ideas thus the optimal layout issue is converted by the lightweight design problem with displacement reliability constraints moreover the interval parametric vertex approach and the adjoint vector method are also employed to obtain the sensitivity information between the reliability measurement and design variables by utilizing the method of moving asymptotes mma the optimization problem that is investigated can be solved iteratively two engineering examples are eventually presented to demonstrate the validity and effectiveness of the methodology that has been developed and numerical results show that considering the ubb uncertainty effects during the whole procedure of topology optimization may have a significant influence on the final structural configurations

a stronger entanglement monogamy inequality in a 2 2 3 system in this paper we prove a stronger entanglement monogamy inequality in a 2 2 3 system s pure state abc specifically we show that the linear entropy of a which is the entanglement between a and bc is always larger than the sum of the square of concurrence between a and b and the square of concurrence of assistance between a and c our proof is based on direct generalizations of the qubit system s results our inequality is stronger than the known monogamy inequality of concurrence and shows that the entanglement of assistance always comes from the existing entanglement however our inequality also shows that unlike the three qubit case in higher dimensional systems the entanglement between a and bc cannot be completely transformed into bipartite entanglement with assistance through our proof we also give some cases when the inequality reduces to an equality

a deterministic method to predict the effect of unknown but bounded elastic moduli on the buckling of composite structures abstract a new non probabilistic method is developed in this study to predict the variability in buckling loads of composite plates and shells stemming from the unavoidable scatter in elastic moduli the available measurements of elastic moduli are fitted by the four dimensional uncertainty box and appropriate ellipsoid for the specific cases considered the maximum buckling load of the composite plate exceeds by 11 the value of the buckling load based on the calculation using the nominal values of elastic moduli in the case of the cylindrical shell an appropriate exceedance constitutes 9

the vertex solution theorem and its coupled framework for static analysis of structures with interval parameters summary this work gives new statement of the vertex solution theorem for exact bounds of the solution to linear interval equations and its novel proof by virtue of the convex set theory the core idea of the theorem is to transform linear interval equations into a series of equivalent deterministic linear equations then the important theorem is extended to find the upper and lower bounds of static displacements of structures with interval parameters following discussions about the computational efforts a coupled framework based on vertex method vm is established which allows us to solve many large scale engineering problems with uncertainties using deterministic finite element software compared with the previous works the contribution of this work is not only to obtain the exact bounds of static displacements but also lay the foundation for development of an easy to use interval finite element software numerical examples demonstrate the good accuracy of vm meanwhile the implementation of vm and availability of the coupled framework are demonstrated by engineering example copyright 2017 john wiley sons ltd