stochastic approximations of perturbed fredholm volterra integro differential equation arising in mathematical neurosciences this paper extends the results of synaptically generated wave propagation through a network of connected excitatory neurons to a continuous model defined by a fredholm volterra integro differential equation fvide which includes memory effects of the past in the propagation stochastic approximation and numerical simulations are discussed

unified response probability distribution analysis of two hybrid uncertain acoustic fields abstract for the response analysis of uncertain acoustic fields with limited information two hybrid uncertain models have been developed one is the hybrid probability and interval model in which the probability and interval variables exist simultaneously the other one is the hybrid interval probability model in which some distribution parameters of probability variables are expressed as interval variables for the unified response probability distribution analysis of acoustic fields under two hybrid uncertain models an inverse mapping hybrid perturbation method imhpm is proposed in imhpm the responses of two hybrid uncertain acoustic fields are converted to invertible functions of probability variables based on the taylor series expansion according to the inverse mapping relationships between responses and probability variables the formal expressions of the response probability distributions are yielded on the basis of the change of variable technique the variational ranges of the response probability distributions are estimated by the interval perturbation technique the effectiveness and efficiency of the proposed method for the unified response probability distribution analysis of two hybrid uncertain acoustic fields are investigated by a numerical example

combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters abstract this paper is devoted to a fundamental problem of accounting for parameter uncertainties in random vibrations of structures in contrast to the overwhelming majority of random vibration studies where perfect knowledge is assumed for the parameters of the excitation this crucial conjecture is dispensed with the probabilistic characteristics of the excitation are assumed to be given as depending on some parameters which are not known in advance we postulate that some imprecise knowledge is available namely that these parameters belong to a bounded convex set in the case where this convex set is represented by an ellipsoid closed form solutions are given for the upper and lower bounds of the mean square displacement of the structures for the first time in the literature the system uncertainty in the random vibrations is dealt with as an anti optimization problem of finding the least favorable values of the mean square response the approach developed here opens a new avenue for tackling parameter uncertainty which is often encountered in various branches of engineering

fuzzy interval finite element statistical energy analysis for mid frequency analysis of built up systems with mixed fuzzy and interval parameters abstract this paper introduces mixed fuzzy and interval parametric uncertainties into the fe components of the hybrid finite element statistical energy analysis fe sea model for mid frequency analysis of built up systems thus an uncertain ensemble combining non parametric with mixed fuzzy and interval parametric uncertainties comes into being a fuzzy interval finite element statistical energy analysis fife sea framework is proposed to obtain the uncertain responses of built up systems which are described as intervals with fuzzy bounds termed as fuzzy bounded intervals fbis in this paper based on the level cut technique a first order fuzzy interval perturbation fe sea ffipfe sea and a second order fuzzy interval perturbation fe sea method sfipfe sea are developed to handle the mixed parametric uncertainties efficiently ffipfe sea approximates the response functions by the first order taylor series while sfipfe sea improves the accuracy by considering the second order items of taylor series in which all the mixed second order items are neglected to further improve the accuracy a chebyshev fuzzy interval method cfim is proposed in which the chebyshev polynomials is used to approximate the response functions the fbis are eventually reconstructed by assembling the extrema solutions at all cut levels numerical results on two built up systems verify the effectiveness of the proposed methods

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

an improved perturbation method for stochastic finite element model updating in this paper an improved perturbation method is developed for the statistical identification of structural parameters by using the measured modal parameters with randomness on the basis of the first order perturbation method and sensitivity based finite element fe model updating two recursive systems of equations are derived for estimating the first two moments of random structural parameters from the statistics of the measured modal parameters regularization technique is introduced to alleviate the ill conditioning in solving the equations the numerical studies of stochastic fe model updating of a truss bridge are presented to verify the improved perturbation method under three different types of uncertainties namely natural randomness measurement noise and the combination of the two the results obtained using the perturbation method are in good agreement with although less accurate than those obtained using the monte carlo simulation mcs method it is also revealed that neglecting the correlation of the measured modal parameters may result in an unreliable estimation of the covariance matrix of updating parameters the statistically updated fe model enables structural design and analysis damage detection condition assessment and evaluation in the framework of probability and statistics copyright 2007 john wiley sons ltd

galois closure and lagrangian varieties abstract let x be a complex projective variety and consider the morphism k k h 0 x x 1 h 0 x x k we use galois closures of finite rational maps to introduce a new method for producing varieties such that k has non trivial kernel we then apply our result to the two dimensional case and we construct a new family of surfaces which are lagrangian in their albanese variety moreover we analyze these surfaces computing their chern invariants and proving that they are not fibred over curves of genus g 2 the topological index of these surfaces is negative and this provides a counterexample to a conjecture on lagrangian surfaces formulated in barja et al 2007 3

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

reliability stochastic optimization for a series system with interval component reliability via genetic algorithm this paper deals with chance constraints based reliability stochastic optimization problem in the series system this problem can be formulated as a nonlinear integer programming problem of maximizing the overall system reliability under chance constraints due to resources the assumption of traditional reliability optimization problem is that the reliability of a component is known as a fixed quantity which lies in the open interval 0 1 however in real life situations the reliability of an individual component may vary due to some realistic factors and it is sensible to treat this as a positive imprecise number and this imprecise number is represented by an interval valued number in this work we have formulated the reliability optimization problem as a chance constraints based reliability stochastic optimization problem with interval valued reliabilities of components then the chance constraints of the problem are converted into the equivalent deterministic form the transformed problem has been formulated as an unconstrained integer programming problem with interval coefficients by big m penalty technique then to solve this problem we have developed a real coded genetic algorithm ga for integer variables with tournament selection uniform crossover and one neighborhood mutation to illustrate the model two numerical examples have been solved by our developed ga finally to study the stability of our developed ga with respect to the different ga parameters sensitivity analyses have been done graphically

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

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

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

unified polynomial expansion for interval and random response analysis of uncertain structure acoustic system with arbitrary probability distribution abstract for structure acousticsystem with uncertainties the interval model the random model and the hybrid uncertain model have been introduced in the interval model and the random model the uncertain parameters are described as either the random variable with well defined probability density function pdf or the interval variable without any probability information whereas in the hybrid uncertain model both interval variable and random variable exist simultaneously for response analysis of these three uncertain models of structure acoustic problem involving arbitrary pdfs a unified polynomial expansion method named as the interval and random arbitrary polynomial chaos method irapcm is proposed in irapcm the response of the structure acoustic system is approximated by apc expansion in a unified form particularly only the weight function of polynomial basis is required to be changed to construct the apc expansion for the response of different uncertain models through the unified apc expansion the uncertain properties of the response of three uncertain models can be efficiently obtained as the apc expansion can provide a free choice of the polynomial basis the optimal polynomial basis for the random variable with arbitrary pdfs can be obtained by using the proposed irapcm the irapcm has been employed to solve a mathematical problem and a structure acoustic problem and the effectiveness of the unified irapcm for response analysis of three uncertain models is demonstrated by fully comparing it with the hybrid first order perturbation method and several existing polynomial chaos methods

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

stochastic second order perturbation approach to the stress based finite element method abstract the paper is devoted to the application of the second order perturbation second probabilistic moment method to the stress based finite element method fem the approach is introduced for the linear elastic heterogeneous medium up to the second order variational equations of the complementary energy principle are presented together with an additional stochastic finite element discretization based on airy and prandtl stress functions the numerical examples shown in this paper illustrate the probabilistic stress and strain tensors in the cantilever beam under shear loading and torsioned square beam with randomly defined material and geometrical parameters the results obtained in the tests can be applied in probabilistic analyses of the boundary value problems having any closed form mathematical solutions as well as in the stress based stochastic fem analysis of solids and structures

perturbation based stochastic fe analysis and robust design of inelastic deformation processes the study addresses the perturbation based stochastic finite element analysis and the robust design optimization of deformation processes of inelastic solids the perturbation equations for the stochastic moment analysis of both steady state and non stationary processes are presented an iteration scheme based on the secant system operator is given for the solution of the perturbation equations in case that a tangent matrix is not available the robust design of deformation processes is stated as a two criteria optimal design problem that attempts best mean properties of the outcome at minimum variability the task is solved using optimization techniques based on sequential quadratic programming in conjunction with the stochastic finite element analysis the proposed method is applied to the design of an extrusion die for robustness with respect to friction variability and to a workpiece preform design problem the numerical results show the potential of the method for applications regarding the design of robust deformation processes

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

response variance prediction for uncertain vibro acoustic systems using a hybrid deterministic statistical method imperfections during the manufacturing process can cause significant variations in the noise and vibration levels exhibited by nominally identical structures any response calculations employed during the design process should ideally take account of these uncertainties and predict the expected range in performance recently a hybrid method has been developed to predict the ensemble average response of a built up system by combining a deterministic model of parts of the system with a statistical model of other components shorter p j and langley r s 2005 j sound vib 288 669 700 in this paper the method is extended to predict the ensemble variance of the response expressions are derived for the variance of the vibrational energies in the statistical components and for the variance of the cross spectrum of the response of the deterministic components which augment the mean values of these quantities predicted by the original theory the method employs a nonparametric model of uncertainty in the sense that the statistical components are taken to carry diffuse wave fields and this obviates the requirement for a detailed description of the system uncertainties the method is validated by application to a range of coupled plate structures and good agreement with detailed monte carlo simulations is found

samba sparse approximation of moment based arbitrary polynomial chaos a new arbitrary polynomial chaos apc method is presented for moderately high dimensional problems characterised by limited input data availability the proposed methodology improves the algorithm of apc and extends the method that was previously only introduced as tensor product expansion to moderately high dimensional stochastic problems the fundamental idea of apc is to use the statistical moments of the input random variables to develop the polynomial chaos expansion this approach provides the possibility to propagate continuous or discrete probability density functions and also histograms data sets as long as their moments exist are finite and the determinant of the moment matrix is strictly positive for cases with limited data availability this approach avoids bias and fitting errors caused by wrong assumptions in this work an alternative way to calculate the apc is suggested which provides the optimal polynomials gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the hankel matrix of moments it can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions the extension to more input variables suggested in this work is an anisotropic and adaptive version of smolyak s algorithm that is solely based on the moments of the input probability distributions it is referred to as samba pc which is short for sparse approximation of moment based arbitrary polynomial chaos it is illustrated that for moderately high dimensional problems up to 20 different input variables or histograms samba can significantly simplify the calculation of sparse gaussian quadrature rules samba s efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2 5 and 10 different input distributions or histograms

stochastic wave finite element for random periodic media through first order perturbation abstract wave propagation features in random guided elastic media are considered in this paper a formulation named stochastic wave finite element swfe is offered this formulation allows wave characteristics definition by means of a stochastic finite element model the case of spatially homogeneous random properties is dealt with through a parametric probabilistic technique numerical experiments show the effectiveness of the proposed approach to predict the statistics of guided wave propagation characteristics means and standard deviations when compared to monte carlo calculations

modified sub interval perturbation finite element method for 2d acoustic field prediction with large uncertain but bounded parameters abstract based on the sub interval perturbation analysis a modified sub interval perturbation finite element method is proposed to determine the bounds of sound pressure in the 2d acoustic field with large uncertain but bounded parameters in the proposed method the inversion of the invertible sub interval dynamic stiffness matrix is approximated by a modified approximate interval value sherman morrison woodbury formula to overcome the drawbacks arising from the dependency phenomenon of parameters and the unpredictable effect of neglecting the higher order terms in neumann series the modified sub interval perturbation procedures are implemented in a numerical finite element framework numerical results on a 2d acoustic tube and a 2d acoustic cavity of a car with large uncertain but bounded parameters evidence the remarkable accuracy and effectiveness of the proposed method

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

interval and random analysis for structure acoustic systems with large uncertain but bounded parameters abstract for the response analysis of the structure acoustic system with uncertain but bounded parameters three bounded uncertain models are introduced one is the bounded random model in which all of the uncertain but bounded parameters are described as bounded random variables with well defined probability distribution the second one is the interval model in which all of the uncertain but bounded parameters are described as interval variables due to the limited information the third one is the bounded hybrid uncertain model in which the interval variables and the bounded random variables exist simultaneously based on the parametric gegenbauer polynomial which is formulated for bounded random model recently the gegenbauer series expansion method gsem is developed for the response prediction of the structure acoustic system under these three bounded uncertain models within gsem the response of these three bounded uncertain models of the structure acoustic system can be approximated by the unified gegenbauer series with different values of polynomial parameter then the interval and random analysis for these three bounded uncertain models of the structure acoustic system are conducted on the basis of gegenbauer series owing to the orthogonal property of gegenbauer polynomial the analytical solution of the expectation and variance of gegenbauer series with respect to the bounded random variables can be readily obtained the bounds of gegenbauer series with respect to the interval variables are determined by the monte carlo simulation the gsem is applied to solve a shell structure acoustic system under these three bounded uncertain models inspired by the convergence behavior of gsem the relative improvement criterion is established to estimate the required retained order of gegenbauer series for large uncertain problems the results on numerical examples show that gsem with the estimated retained order can achieve a prescribed accuracy and good efficiency for structure acoustic systems with large uncertain but bounded parameters