probability distributions of natural frequencies of uncertain dynamic systems of random eigenvalues commonly encountered in dynamic systems the method involves a hierarchical decomposition of a multivariate function in terms of variables with increasing dimensions a broad range of orthonormal polynomial bases consistent with the probability measure for a fourier polynomial expansion of component functions and an innovative dimension reduction integration for calculating the expansion coefficients the new decomposition does not require sample points yet it generates a convergent sequence of lower variate estimates of theprobabilitydistributionsofeigensolutions numericalresults includingfrequencydistributionsofapiezoelectric transducer indicate that the decomposition method developed provides accurate convergent and computationally efficient estimates of the tail probabilistic characteristics of eigenvalues

coupling finite element and reliability analysis through proper generalized decomposition model reduction summary the fem is the main tool used for structural analysis when the design of the mechanical system involves uncertain parameters a coupling of the fem with reliability analysis algorithms allows to compute the failure probability of the system however this coupling leads to successive finite element analysis of parametric models involving high computational effort over the past years model reduction techniques have been developed in order to reduce the computational requirements in the numerical simulation of complex models the objective of this work is to propose an efficient methodology to compute the failure probability for a multi material elastic structure where the young moduli are considered as uncertain variables a proper generalized decomposition algorithm is developed to compute the solution of parametric multi material model this parametrized solution is used in conjunction with a first order reliability method to compute the failure probability of the structure applications to multilayered structures in two dimensional plane elasticity are presented copyright 2013 john wiley sons ltd

novel computational methods for high dimensional stochastic sensitivity analysis summary this paper presents three new computational methods for calculating design sensitivities of statistical moments and reliability of high dimensional complex systems subject to random input the first method represents a novel integration of the polynomial dimensional decomposition pdd of a multivariate stochastic response function and score functions applied to the statistical moments the method provides mean square convergent analytical expressions of design sensitivities of the first two moments of a stochastic response the second and third methods relevant to probability distribution or reliability analysis exploit two distinct combinations built on pdd the pdd saddlepoint approximation spa or pdd spa method entailing spa and score functions and the pdd monte carlo simulation mcs or pdd mcs method utilizing the embedded mcs of the pdd approximation and score functions for all three methods developed the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation numerical examples including a 100 dimensional mathematical problem indicate that the new methods developed provide not only theoretically convergent or accurate design sensitivities but also computationally efficient solutions a practical example involving robust design optimization of a three hole bracket illustrates the usefulness of the proposed methods copyright 2014 john wiley sons ltd

reliability based optimization of stochastic systems using line search this contribution presents an approach for solving reliability based optimization problems involving structural systems under stochastic loading the associated reliability problems to be solved during the optimization process are high dimensional 1000 or more random variables a standard gradient based algorithm with line search is used in this work subset simulation is adopted for the purpose of estimating the corresponding failure probabilities the gradients of the failure probability functions are estimated by an approach based on the local behavior of the performance functions that define the failure domains numerical results show that only a moderate number of reliability estimates has to be performed during the entire design process two numerical examples showing the effectiveness of the approach reported herein are presented

unified uncertainty analysis under probabilistic evidence fuzzy and interval uncertainties abstract the uncertainty analysis of structures generally involves uncertain parameters of different types in order to derive predictions regarding uncertain structural responses it is crucial to represent the uncertainty appropriately according to the underlying information available this paper presents a unified framework for uncertainty analysis under probabilistic evidence fuzzy and interval uncertainties by which the quantities with sufficient data sparse data and subjective information can be simultaneously considered a taylor expansion based unified uncertainty analysis t uua method is first proposed for small uncertainty problems by temporarily neglecting the evidence fuzzy and interval variables the probability evidence interval fuzzy model is degraded into a random problem in which the expectations and variances of responses can be obtained as functions in terms of evidence interval and fuzzy uncertainties then through dealing with the evidence variables the previous expectations and variances are further expressed as a summation of functions in terms of fuzzy and interval variables with basic probability assignments bpas the fuzziness is then discretized by using cut technique and thus the expectations and variances are further expressed as functions of only intervals afterwards by reconsidering the interval uncertainties the bounds of the expectations and variances are computed via combining taylor expansion with interval arithmetic in addition a dimensional reduction dr efficient global optimization ego based unified uncertainty analysis dr ego uua method is also presented to solve the large uncertainty problems the framework of dr ego uua is similar as t uua however in dr ego uua the second moments of responses are computed by dr integrations and their upper and lower bounds are calculated by the ego finally three numerical examples are investigated to demonstrate the effectiveness of the proposed methods

a simple method to compute economic order quantities abstract in this note a simple method by using the arithmetic geometric mean inequality theorem is proposed to computer the global minimum economic order quantities without taking complex differential calculus or using tedious algebraic manipulations in contrast to minner s 2007 a note on how to compute economic order quantity without derivatives by cost comparisons international journal of production economics 105 293 296 wee h m wang w t chung c j 2009 a modified method to computer economic order quantities without derivatives by cost difference comparisons european journal of operational research based on a local cost minimum initially to derive the solution and then proven it s the global minimum the proposed method yields the global minimum cost immediately and explicitly without using the cost comparisons and letting the time horizon to infinity

second order reliability method based inverse reliability analysis using hessian update for accurate and efficient reliability based design optimization summary first order reliability method form has been mostly utilized for solving reliability based design optimization rbdo problems efficiently however second order reliability method sorm is required in order to estimate a probability of failure accurately in highly nonlinear performance functions despite accuracy of sorm its application to rbdo is quite challenging due to unaffordable numerical burden incurred by a hessian calculation for reducing the numerical efforts a quasi newton approach to approximate the hessian is introduced in this study instead of calculating the true hessian the proposed sorm with the approximated hessian requires computations only used in form leading to very efficient and accurate reliability analysis the proposed sorm also utilizes a generalized chi squared distribution in order to achieve better accuracy furthermore sorm based inverse reliability method is proposed in this study an accurate reliability index corresponding to a target probability of failure is updated using the proposed sorm two approaches in terms of finding an accurate most probable point using the updated reliability index are proposed the proposed sorm based inverse analysis is then extended to rbdo in order to obtain a reliability based optimum design satisfying probabilistic constraints more accurately even for a highly nonlinear system the numerical study results show that the proposed reliability analysis and rbdo achieve efficiency of form and accuracy of sorm at the same time copyright 2014 john wiley sons ltd

uncertainty quantification of high dimensional complex systems by multiplicative polynomial dimensional decompositions summary the central theme of this paper is multiplicative polynomial dimensional decomposition pdd methods for solving high dimensional stochastic problems when a stochastic response is dominantly of multiplicative nature the standard pdd approximation predicated on additive function decomposition may not provide sufficiently accurate probabilistic solutions of a complex system to circumvent this problem two multiplicative versions of pdd referred to as factorized pdd and logarithmic pdd were developed both versions involve a hierarchical multiplicative decomposition of a multivariate function a broad range of orthonormal polynomial bases for fourier polynomial expansions of component functions and a dimension reduction or sampling technique for estimating the expansion coefficients three numerical problems involving mathematical functions or uncertain dynamic systems were solved to corroborate how and when a multiplicative pdd is more efficient or accurate than the additive pdd the results show that indeed both the factorized and logarithmic pdd approximations can effectively exploit the hidden multiplicative structure of a stochastic response when it exists since a multiplicative pdd recycles the same component functions of the additive pdd no additional cost is incurred finally the random eigensolutions of a sport utility vehicle comprising 40 random variables were evaluated demonstrating the ability of the new methods to solve industrial scale problems copyright 2013 john wiley sons ltd received 31 january 2012 revised 6 september 2012 accepted 20 september 2012

higher order failure probability calculation using nonlinear approximations use of approximations for calculating the structural failure probability under uncertainties is a necessary aspect but efficient use of suitable approximations at different stages of reliability prediction and design makes this valuable tool practical for many large scale structures the objective of this paper is to address the modern approximation concepts and their utility in structural reliability estimation emphasis is placed on building the higher order approximations using intervening variables to closely represent the nonlinear limit state functions the most probable failure point mpp search can be carried either in x space or u space for a wide variety of random variable distributions several test problems with highly nonlinear performance functions are used to demonstrate the accuracy of the proposed method

an imbedded family of fully symmetric numerical integration rules we consider an imbedded family of m 1 generator 2m 1 degree fully symmetric rules for numerical integration over an n dimensional hypercube the rules are shown to exist in theory for arbitrary degree in practice the rules are most economical and useful for 3 leqq m leqq 6 3 leqq n leqq 3m

stochastic isogeometric analysis in linear elasticity abstract a new stochastic method integrating spline dimensional decomposition sdd and isogeometric analysis iga is proposed for solving stochastic boundary value problems from linear elasticity the method referred to as sdd iga involves galerkin isogeometric analysis as a deterministic solver for governing partial differential equations and a novel fourier like orthogonal spline expansion generated from the analysis of variance decomposition of a high dimensional function for the stochastic part of the sdd iga method an innovative dimension reduction integration technique is presented for efficiently calculating the expansion coefficients analytical formulae have been derived to calculate the second moment properties of an sdd iga approximation for a general output random variable of interest numerical examples demonstrate the capability of a low order sdd iga in efficiently delivering probabilistic solutions with an approximation quality as good as if not better than that obtained from a high order polynomial dimensional decomposition the proposed sdd iga method is most suitable in the presence of locally nonlinear or nonsmooth behavior commonly found in applications

an adaptive high dimensional stochastic model representation technique for the solution of stochastic partial differential equations a computational methodology is developed to address the solution of high dimensional stochastic problems it utilizes high dimensional model representation hdmr technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower order to higher order component functions hdmr is efficient at capturing the high dimensional input output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower order terms an adaptive version of hdmr is also developed to automatically detect the important dimensions and construct higher order terms using only the important dimensions the newly developed adaptive sparse grid collocation asgc method is incorporated into hdmr to solve the resulting sub problems by integrating hdmr and asgc it is computationally possible to construct a low dimensional stochastic reduced order model of the high dimensional stochastic problem and easily perform various statistic analysis on the output several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method the cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large input variability the efficiency of the proposed method is examined by comparing with monte carlo mc simulation

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

sequential subspace reliability method with univariate revolving integration a new computational framework called the sequential subspace reliability method ssrm is presented this method decomposes the multidimensional random space into multiple two dimensional subspaces in this manner ssrm is able to approximate bivariate interaction effects when the reliability estimate contribution is calculated subspace by subspace the final assessment is updated in a progressive manner the iterative history of sequential reliability assessment can be used to understand the complexity and convergence behavior of the limit state function of interest in a decision making situation the flexibility of the proposed ssrm to provide iterative updates on reliability estimation becomes especially valuable in dealing with large scale and complex problems under the constraints of limited time and resources to calculate the individual subspace contributions a novel univariate revolving integration uri method is proposed the uri method takes advantage of the axisymmetric nature of a joint pr

enriched performance measure approach for reliability based design optimization an enriched performance measure approach is presented for reliability based design optimization to substantially improve computational efficiency when applied to large scale applications in the enriched performance measure approach four improvements are made over the original performance measure approach as a way to launch reliability based design optimization at a deterministic optimum design as a new enhanced hybrid mean value method as an efficient probabilistic feasibility check and as a fast reliability analysis under the condition of design closeness it is found that deterministic design optimization helps improve numerical efficiency by reducing some reliability based design optimization iterations in reliability based design optimization a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first order method and by carrying out the refined reliability analysis using the enhanced hybrid mean value method for e active and violated constraints the enhanced hybrid mean value method is developed to handle nonlinear and or nonmonotonic constraints in reliability analysis the fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness moreover two numerical examples are provided to compare the enriched performance measure approach to existing reliability based design optimization methods from a numerical efficiency and stability point of view

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

statistical volume element method for predicting microstructure constitutive property relations abstract there is an inevitable need to establish multilength scale statistical microstructure constitutive property relations in materials design in this paper we have developed a statistical volume element method to analyze quantify and calibrate such microstructure constitutive property relations by statistical means statistical volume element simulations are adopted to predict material constitutive properties corresponding to various realizations of random microstructure configurations a computing framework that links random configuration generators and finite element analysis has been developed a statistical cause effect analysis approach is proposed to study the influence of random material microstructure on material constitutive properties within the proposed approach statistically significant microstructure parameters are first identified based on their linear impacts on material constitutive properties global sensitivity analysis is then employed to provide a more comprehensive importance ranking of these critical microstructure parameters considering both main and interaction effects the uncertainties in material constitutive properties due to random microstructure configurations are quantified in terms of distributions statistical moments and correlations the obtained probabilistic constitutive relations are used to calibrate the model parameters in a constitutive relation model following a statistical calibration process the proposed approach is applied to examine a porous steel alloy material for demonstrative purposes

random field finite elements the probabilistic finite element method pfem is formulated for linear and non linear continua with inhomogeneous random fields analogous to the discretization of the displacement field in finite element methods the random field is also discretized the formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization furthermore it is shown that a reduced set of the uncorrelated variables is sufficient for the second moment analysis based on the linear formulation of the pfem the method is then extended to transient analysis in non linear continua the accuracy and efficiency of the method is demonstrated by application to a one dimensional elastic plastic wave propagation problem and a two dimensional plane stress beam bending problem the moments calculated compare favourably with those obtained by monte carlo simulation also the procedure is amenable to implementation in deterministic fem based computer programs

a generalized dimension reduction method for multidimensional integration in stochastic mechanics summary a new generalized multivariate dimension reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads material properties and geometry the method involves an additive decomposition of an n dimensional response function into at most s dimensional functions where s n an approximation of response moments by moments of input random variables and a moment based quadrature rule for numerical integration a new theorem is presented which provides a convenient means to represent the taylor series up to a specific dimension without involving any partial derivatives a complete proof of the theorem is given using two lemmas also proved in this paper the proposed method requires neither the calculation of partial derivatives of response as in commonly used taylor expansion perturbation methods nor the inversion of random matrices as in the neumann expansion method eight numerical examples involving elementary mathematical functions and solid mechanics problems illustrate the proposed method results indicate that the multivariate dimension reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods such as first and second order taylor expansion methods statistically equivalent solutions quasi monte carlo simulation and the fully symmetric interpolatory rule while the accuracy of the dimension reduction method is comparable to that of the fourth order neumann expansion method a comparison of cpu time suggests that the former is computationally far more efficient than the latter copyright 2004 john wiley sons ltd

a perturbation method for stochastic meshless analysis in elastostatics a stochastic meshless method is presented for solving boundary value problems in linear elasticity that involves random material properties the material property was modelled as a homogeneous random field a meshless formulation was developed to predict stochastic structural response unlike the finite element method the meshless method requires no structured mesh since only a scattered set of nodal points is required in the domain of interest there is no need for fixed connectivities between nodes in conjunction with the meshless equations classical perturbation expansions were derived to predict second moment characteristics of response numerical examples based on one and two dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method a good agreement is obtained between the results of the proposed method and monte carlo simulation since mesh generation of complex structures can be a far more time consuming and costly effort than the solution of a discrete set of equations the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems copyright 2001 john wiley sons ltd

accuracy improvement of the most probable point based dimension reduction method using the hessian matrix summary this paper proposes a most probable point mpp based dimension reduction method drm using the hessian matrix called hedrm to improve accuracy of reliability analysis in existing mpp based drm methods conventional mpp based drms contain two types of errors 1 error due to eliminating cross terms of a performance function by using the univariate drm 2 error because of dependency of an axis direction after a rotational transformation the proposed method minimizes the aforementioned errors by utilizing the hessian matrix of a performance function by performing an orthogonal transformation using the eigenvectors of the hessian matrix the cross term effect of the performance function is minimized and the axis direction that results in the most accurate calculation is obtained because the gaussian quadrature points for numerical integration are arranged along the eigenvector directions in this way the error incurred by exiting mpp based drms can be reduced that leads to more accurate probability of failure estimation in addition this paper proposes to allocate the gaussian quadrature points using the magnitude of the eigenvalues of the hessian matrix this allocation makes it possible to predetermine the number of function evaluations required to estimate the probability of failure accurately and efficiently copyright 2016 john wiley sons ltd

a stochastic algorithm for high dimensional integrals over unbounded regions with gaussian weight abstract details are given for a fortran implementation of an algorithm that uses stochastic spherical radial rules for the numerical computation of multiple integrals over unbounded regions with gaussian weight the implemented rules are suitable for high dimensional problems a high dimensional example from a computational finance application is used to illustrate the use of the rules

first order saddlepoint approximation for reliability analysis in the approximation methods of reliability analysis nonnormal random variables are transformed into equivalent standard normal random variables this transformation tends to increase the nonlinearity of a limit state function and hence results in less accurate reliability approximation the first order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis by the approximation of a limit state function at the most likelihood point in the original random space and employment of the accurate saddlepoint approximation the proposed method reduces the chance of an increase in the nonlinearity of the limit state function this approach generates more accurate reliability approximation than the first order reliability method without an increase in the computational effort the effectiveness of the proposed method is demonstrated with two examples and is compared with the first and second order reliability methods

a univariate approximation at most probable point for higher order reliability analysis this paper presents a new univariate method employing the most probable point as the reference point for predicting failure probability of structural and mechanical systems subject to random loads material properties and geometry the method involves novel decomposition at the most probable point that facilitates a univariate approximation of a general multivariate function response surface generation of the univariate function and monte carlo simulation in addition to the effort of identifying the most probable point the method requires a small number of exact or numerical evaluations of the performance function at selected input results of four numerical examples involving elementary mathematical functions and structural solid mechanics problems indicate that the proposed method provides accurate and computationally efficient estimates of probability of failure finally the fatigue failure of lever arm in a wheel loader has been evaluated demonstrating the ability of the new method in solving industrial scale fatigue reliability problems

fully symmetric interpolatory rules for multiple integrals over infinite regions with gaussian weight fully symmetric interpolatory integration rules are constructed for multidimensional integrals over infinite integration regions with a gaussian weight function the points for these rules are determined by successive extensions of the one dimensional three point gauss hermite rule the new rules are shown to be efficient and only moderately unstable

an efficient dimension adaptive uncertainty propagation approach abstract in recent years there has been a growing interest in uncertainty propagation and a wide variety of uncertainty propagation methods exist in literature in this paper an uncertainty propagation approach is developed by using high dimensional model representation hdmr and dimension reduction dr method technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower order to higher order component functions to save the computational cost a dimension adaptive version of the additive decomposition is proposed to detect the important component functions to reduce the terms the proposed method requires neither the calculation of partial derivatives of response as in commonly used taylor expansion perturbation methods nor the inversion of random matrices as in the neumann expansion method two numerical examples show the efficiency and accuracy of the proposed method

a generalized l1 greedy algorithm for image reconstruction in ct the sparse vector solutions for an underdetermined system of linear equations ax b have many applications in signal recovery and image reconstruction in tomography under certain conditions the sparsest solution can be found by solving a constrained l 1 minimization problem min x 1 subject to ax b recently the reweighted l 1 minimization and l 1 greedy algorithm have been introduced to improve the convergence of the l 1 minimization problem as an extension a generalized l 1 greedy algorithm for computerized tomography ct is proposed in this paper it is implemented as a generalized total variation minimization for images with sparse gradients in ct numerical experiments are also given to illustrate the advantage of the new algorithm

sensitivity analyses of form based and drm based performance measure approach pma for reliability based design optimization rbdo in a gradient based design optimization it is necessary to know sensitivities of the constraint with respect to the design variables in a reliability based design optimization rbdo the constraint is evaluated at the most probable point mpp and called the probabilistic constraint thus it requires the sensitivities of the probabilistic constraints at mpp this paper presents the rigorous analytic derivation of the sensitivities of the probabilistic constraint at mpp for both first order reliability method form based performance measure approach pma and dimension reduction method drm based pma numerical examples are used to demonstrate that the analytic sensitivities agree very well with the sensitivities obtained from the finite difference method fdm however since the sensitivity calculation at the true drm based mpp requires the second order derivatives and additional mpp search the sensitivity derivation at the approximated drmbased mpp which does not require the second order derivatives and additional mpp search to find the drm based mpp is proposed in this paper a convergence study illustrates that the sensitivity at the approximated drm based mpp converges to the sensitivity at the true drm based mpp as the design approaches the optimum design hence the sensitivity at the approximated drm based mpp is proposed to be used for the drm based rbdo to enhance the efficiency of the optimization

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

inverse analysis method using mpp based dimension reduction for reliability based design optimization of nonlinear and multi dimensional systems abstract there are two commonly used analytical reliability analysis methods linear approximation first order reliability method form and quadratic approximation second order reliability method sorm of the performance function the reliability analysis using form could be acceptable in accuracy for mildly nonlinear performance functions whereas the reliability analysis using sorm may be necessary for accuracy of nonlinear and multi dimensional performance functions even though the reliability analysis using sorm may be accurate it is not as much used for probability of failure calculation since sorm requires the second order sensitivities moreover the sorm based inverse reliability analysis is rather difficult to develop this paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second order sensitivities for reliability based design optimization rbdo of nonlinear and multi dimensional systems for the inverse reliability analysis the most probable point mpp based dimension reduction method drm is developed since the form based reliability index is inaccurate for the mpp search of the nonlinear performance function a three step computational procedure is proposed to improve accuracy of the inverse reliability analysis probability of failure calculation using constraint shift reliability index update and mpp update using the three steps a new drm based mpp is obtained which estimates the probability of failure of the performance function more accurately than form and more efficiently than sorm the drm based mpp is then used for the next design iteration of rbdo to obtain an accurate optimum design even for nonlinear and or multi dimensional system since the drm based rbdo requires more function evaluations the enriched performance measure approach pma with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations

uncertainty quantification under dependent random variables by a generalized polynomial dimensional decomposition abstract this paper is concerned with uncertainty quantification analysis of complex systems subject to dependent input random variables the analysis focuses on a new generalized version of polynomial dimensional decomposition pdd referred to as gpdd entailing hierarchically ordered measure consistent multivariate orthogonal polynomials in dependent variables under a few prescribed assumptions gpdd exists for any square integrable output random variable and converges in mean square to the correct limit new analytical formulae are proposed to calculate the mean and variance of a gpdd approximation of a general output variable in terms of the expansion coefficients and second moment properties of multivariate orthogonal polynomials however unlike in pdd calculating the coefficients of gpdd requires solving a coupled system of linear equations besides the variance formula of gpdd contains extra terms due to statistical dependence among input variables the extra terms disappear when the input variables are statistically independent reverting gpdd to pdd two numerical examples the one derived from a stochastic boundary value problem and the other entailing a random eigenvalue problem illustrate second moment error analysis and estimation of the probabilistic characteristics of eigensolutions

wiener hermite polynomial expansion for multivariate gaussian probability measures abstract this paper introduces a new transformation free generalized polynomial chaos expansion pce comprising multivariate hermite orthogonal polynomials in dependent gaussian random variables the second moment properties of hermite polynomials reveal a weakly orthogonal system when obtained for a general gaussian probability measure still the exponential integrability of norm allows the hermite polynomials to constitute a complete set and hence a basis in a hilbert space the completeness is vitally important for the convergence of the generalized pce to the correct limit the optimality of the generalized pce and the approximation quality due to truncation are discussed new analytical formulae are proposed to calculate the mean and variance of a generalized pce approximation of a general output variable in terms of the expansion coefficients and statistical properties of hermite polynomials however unlike in the classical pce calculating the coefficients of the generalized pce requires solving a coupled system of linear equations besides the variance formula of the generalized pce contains additional terms due to statistical dependence among gaussian variables the additional terms vanish when the gaussian variables are statistically independent reverting the generalized pce to the classical pce numerical examples illustrate the generalized pce approximation in estimating the statistical properties of various output variables

a convex hull approach for the reliability based design optimization of nonlinear transient dynamic problems nonlinear problems such as transient dynamic problems exhibit structural responses that can be discontinuous due to numerous bifurcations this hinders gradient based or response surface based optimization this paper proposes a novel approach to split the design space into regions where the response is continuous this makes traditional optimization viable a convex hull approach is adopted to isolate the points corresponding to unwanted bifurcations in the design space the proposed approach is applied to a tube impacting a rigid wall representing a transient dynamic problem since nonlinear behavior is highly sensitive to small variations in design reliability based design optimization is performed the proposed method provides the designer an optimal design with a prescribed dynamic behavior

asymptotic distribution method for structural reliability analysis in high dimensions in the reliability analysis of safety critical complex engineering structures a very large number of the system parameters can be considered as random variables the difficulty in computing the failure probability using the classical first and second order reliability methods form and sorm increases rapidly with the number of variables or dimension there are mainly two reasons behind this the first is the increase in computational time with the increase in the number of random variables in principle this problem can be handled with superior computational tools the second reason which is perhaps more fundamental is that there are some conceptual difficulties typically associated with high dimensions this means that even when one manages to carry out the necessary computations the application of existing form and sorm may still lead to incorrect results in high dimensions this paper is aimed at addressing this issue based on the asymptotic distribution of quadratic form in gaussian random variables two formulations for the case when the number of random variables n n is provided the first is called strict asymptotic formulation and the second is called weak asymptotic formulation both approximations result in simple closed form expressions for the probability of failure of an engineering structure the proposed asymptotic approximations are compared with existing approximations and monte carlo simulations using numerical examples

optimization based on reliability and confidence interval design for the structural acoustic system with interval probabilistic variables abstract in practice the uncertain parameters whose unique distributions cannot be identified on the basis of limited data existed in the structural acoustic system extensively the uncertain parameters with limited data can be modeled as interval probabilistic variables whose distribution parameters are expressed as interval variables instead of unique values for the optimization of the structural acoustic system with interval probabilistic variables a new optimization technique named as the optimization based on reliability and confidence interval design o rcid is developed o rcid is a nested loop optimization whose computational cost during the evaluation of the confidence interval of sound pressure response and the system reliability is considerable to improve the computational efficiency of o rcid a change of variable interval probabilistic perturbation method cv ippm which is based on the perturbation approach and change of variable technique is proposed to calculate the confidence interval of sound pressure response and the system reliability by introducing cv ippm the nested loop o rcid is converted into an approximate single loop one which can be efficiently solved numerical examples demonstrate the effectiveness and efficiency of o rcid based on cv ippm for the optimization of the structural acoustic system with interval probabilistic variables

adaptive sparse polynomial dimensional decomposition methods for high dimensional stochastic computing this article presents two novel adaptive sparse polynomial dimensional decomposition pdd methods for solving high dimensional uncertainty quantification problems in computational science and engineering the methods entail global sensitivity analysis for retaining important pdd component functions and a full or sparse grid dimension reduction integration or quasi monte carlo simulation for estimating the pdd expansion coefficients a unified algorithm endowed with two distinct ranking schemes for grading component functions was created for their numerical implementation the fully adaptive sparse pdd method is comprehensive and rigorous leading to the second moment statistics of a stochastic response that converges to the exact solution when the tolerances vanish a partially adaptive sparse pdd method obtained through regulated adaptivity and sparsity is economical and is therefore expected to solve practical problems with numerous variables compared with past developments the adaptive sparse pdd methods do not require their truncation parameter s to be assigned a priori or arbitrarily the numerical results reveal that an adaptive sparse pdd method achieves a desired level of accuracy with considerably fewer coefficients compared with existing pdd approximations for a required accuracy in calculating the probabilistic response characteristics the new bivariate adaptive sparse pdd method is more efficient than the existing bivariately truncated pdd method by almost an order of magnitude finally stochastic dynamic analysis of a disk brake system was performed demonstrating the ability of the new methods to tackle practical engineering problems

reliability based design optimization with dependent interval variables summary reliability based design optimization rbdo ensures design requirements are satisfied at required reliability levels in the presence of uncertainty the recent rbdo methodologies can accommodate two types of uncertainty aleatory type represented by random variables and epistemic type represented by interval variables however only independent interval variables can be handled the objective of this research is to introduce dependent interval variables into rbdo the dependency of interval variables comes from equality and inequality constraints with the intervals the reliability is also an interval and its lower bound is considered to preserve conservativeness an efficient optimization method is developed to solve rbdo problems the method is an extension of the existing framework of sequential optimization and reliability analysis with the inclusion of inequality constraints of interval variables the new method is efficient because it decouples the optimization process from the process of reliability and interval analyses the new method is applied to the rbdo of a slider mechanism with joint clearances which are known as dependent interval variables copyright 2012 john wiley sons ltd

stochastic dynamic systems with complex valued eigensolutions a dimensional decomposition method is presented for calculating the probabilistic characteristics of complex valued eigenvalues and eigenvectors of linear stochastic dynamic systems the method involves a function decomposition allowing lower dimensional approximations of eigensolutions lagrange interpolation of lower dimensional component functions and monte carlo simulation compared with the commonly used perturbation method neither the assumption of small input variability nor the calculation of the derivatives of eigensolutions is required by the method developed results of numerical examples from linear stochastic dynamics indicate that the decomposition method provides excellent estimates of the moments and or probability densities of eigenvalues and eigenvectors for various cases including large statistical variations of input copyright 2007 john wiley sons ltd

anchored anova petrov galerkin projection schemes for parabolic stochastic partial differential equations abstract we present a numerical scheme based on the combination of hoeffding s functional analysis of variance anova decomposition with stochastic galerkin projection for solving a class of high dimensional parabolic stochastic partial differential equations spdes the central idea underpinning the proposed approach is to approximate the spde solution using a hoeffding anova decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure we show that when a dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately the original stochastic weak formulation can be decoupled into low dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes as a result the proposed approach scales very well to spdes with large number of random variables we theoretically analyze the proposed formulation and provide a priori error estimates as a function of the spatial stochastic and temporal discretization parameters and the anova expansion order that hold under appropriate stochastic regularity assumptions numerical studies are presented for a set of time dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic galerkin projection scheme gpc and the generalized spectral decomposition scheme these studies show that the proposed approach provides accuracy comparable to classical gpc galerkin projection schemes while incurring significantly lower computational cost

revisiting lot sizing for an inventory system with product recovery this note investigates the study by teunter 2004 1 on lot sizing for inventory systems with product recovery where lot sizing formulae for two recovery policies 1 r and p 1 are derived instead of applying the classical optimization technique we develop an integrated solution procedure for each of the two policies using algebraic approaches numerical analysis show that our examples result in a lower total cost for both policies

advanced probabilistic structural analysis method for implicit performance functions in probabilistic structural analysis the performance or response functions usually are implicitly defined and must be solved by numerical analysis methods such as finite elemen t methods in such cases the commonly used probabilistic analysis tool is the mean based second moment method which provides only the first two statistical moments this paper presents an advanced mean based method which is capable of establishing the full probability distributions to provide additional information for reliability design the method requires slightly more computations than the mean based second moment method but is highly efficient relative to the other alternative methods several examples are presented to demonstrate the method in particular the examples show that the new mean based method can be used to solve problems involving nonmonotonic functions that result in truncated distributions

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

an improved response function based stochastic meshless method for problems in elasto statics abstract the current study proposes an improved response function irf based stochastic element free galerkin method sefgm for the analysis of problems in elasto statics wherein young s modulus is modelled as a homogeneous random field with symmetric distribution characteristics the proposed sefgm approximates displacement as the sum of a deterministic part and a stochastic part the stochastic part is modelled with the help of an irf which is a function of discretized set of random variables moving least square shape functions are employed to discretize the random field utilizing taylor series expansions of stiffness matrix and force vector and irf approximation of displacement explicit expressions for system responses in terms of random variables are derived stochastic informations of system responses are evaluated by employing monte carlo simulation mcs on the response function which eliminates the need of construction and simulation of system matrices at each set of sample generation 1d and 2d numerical examples in elasto statics are solved using proposed method results are validated with those obtained by direct simulation of system of equations using mcs and also compared with other methods like second order perturbation and ad hoc response function based sefgm normalized computational times required for all the methods are also compared it is found that the proposed method is computationally efficient and can produce accurate results even for higher coefficient of variation of input random fields

a polynomial dimensional decomposition for stochastic computing this article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences the method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions a broad range of orthonormal polynomial bases consistent with the probability measure for fourier polynomial expansion of component functions and an innovative dimension reduction integration for calculating the coefficients of the expansion the new decomposition method does not require sample points as in the previous version yet it generates a convergent sequence of lower variate estimates of the probabilistic characteristics of a generic stochastic response the results of five numerical examples indicate that the proposed decomposition method provides accurate convergent and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems

reduction of ordering effect in reliability based design optimization using dimension reduction method in reliability based design optimization problems with correlated input variables a joint cumulative distribution function needs to be used to transform the correlated input variables into independent standard gaussian variables for the inverse reliability analysis to obtain a true joint cumulative distribution function a very large number of data if not infinite needs to be used which is impractical in industry applications in this paper a copula is proposed to model the joint cumulative distribution function using marginal cumulative distribution functions and correlation parameters obtained from samples using the joint cumulative distribution function modeled by the copula the transformation and the first order reliability method can be carried out however the first order reliability method may yield different reliability analysis results for different transformation ordering of input variables thus the most probable point based dimension reduction method which is more accurate than the first order reliability method and more efficient than the second order reliability method is proposed for the inverse reliability analysis to reduce the effect of transformation ordering

an analytic method for sensitivity analysis of complex systems abstract sensitivity analysis is concerned with understanding how the model output depends on uncertainties variances in inputs and identifying which inputs are important in contributing to the prediction imprecision uncertainty determination in output is the most crucial step in sensitivity analysis in the present paper an analytic expression which can exactly evaluate the uncertainty in output as a function of the output s derivatives and inputs central moments is firstly deduced for general multivariate models with given relationship between output and inputs in terms of taylor series expansion a order relative uncertainty for output denoted by r v is introduced to quantify the contributions of input uncertainty of different orders on this basis it is shown that the widely used approximation considering the first order contribution from the variance of input variable can satisfactorily express the output uncertainty only when the input variance is very small or the input output function is almost linear two applications of the analytic formula are performed to the power grid and economic systems where the sensitivities of both actual power output and economic order quantity models are analyzed the importance of each input variable in response to the model output is quantified by the analytic formula

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

a derivative based sensitivity measure of failure probability in the presence of epistemic and aleatory uncertainties in order to analyze the effect of the epistemic uncertainty of random variables distribution parameters on the safety of the structure system a novel sensitivity measure of the failure probability is constructed by integrating the derivative of failure probability in the parameter space compared with the variance based sensitivity index the new derivative based sensitivity measure can be evaluated with less computational cost it is noticed that the ranking of the new derivative based sensitivity measure is the same as that of the variance based one for the problem of the variance based sensitivity index with large computational cost the standard sobol s method is employed and the quasi monte carlo method and double loop point estimate method are then utilized to compute the derivative based sensitivity measure for comparison four examples are employed to demonstrate the reasonability of the proposed sensitivity measure and the efficiency of the proposed method

uncertainty quantification by alternative decompositions of multivariate functions this article advocates factorized and hybrid dimensional decompositions fdd hdd as alternatives to analysis of variance dimensional decomposition add for second moment statistical analysis of

new decoupled framework for reliability based design optimization traditionally reliability based design optimization rbdo has been formulated as a nested optimization problem the inner loop generally involves the solution to optimization problems for computing the probabilities of failure of the critical failure modes and the outer loop performs optimization by varying the decision variables such formulations are by nature computationally intensive requiring numerous function and constraint evaluations to alleviate this problem researchers have developed iterative decoupled rbdo approaches these methods perform deterministic optimization and reliability assessment in a sequential manner until a consistent reliability based design is obtained the sequential methods are attractive because a consistent reliable design can be obtained at considerably lower computational cost however the designs obtained by using these decoupled approaches cannot guarantee production of the true solution a new decoupled method for rbdo is developed in this investigation postoptimal sensitivities of the most probable point mpp of failure with respect to the decision variables are introduced to update the mpps during the deterministic optimization phase of the proposed approach a damped broyden fletcher goldfarb shanno method is used to significantly reduce the cost of obtaining these sensitivities it is the use of postoptimal sensitivities that differentiates this new decoupled rbdo approach from previous efforts

a polynomial chaos expansion in dependent random variables abstract this paper introduces a new generalized polynomial chaos expansion pce comprising measure consistent multivariate orthonormal polynomials in dependent random variables unlike existing pces whether classical or generalized no tensor product structure is assumed or required important mathematical properties of the generalized pce are studied by constructing orthogonal decomposition of polynomial spaces explaining completeness of orthogonal polynomials for prescribed assumptions exploiting whitening transformation for generating orthonormal polynomial bases and demonstrating mean square convergence to the correct limit analytical formulae are proposed to calculate the mean and variance of a truncated generalized pce for a general output variable in terms of the expansion coefficients an example derived from a stochastic boundary value problem illustrates the generalized pce approximation in estimating the statistical properties of an output variable for 12 distinct non product type probability measures of input variables

a full row rank system matrix generated along two directions in discrete tomography abstract a full row rank system matrix generated by the strip based projection model along one scanning direction was studied recently in 9 in this paper we generalize the result to multiple directions let c u h be a reduced binary linear system generated along two distinct scanning directions by the strip based projection model in discrete tomography where c is row rank deficient we identify all the linearly dependent rows explicitly through a partition of the rows of c into minimal linearly dependent sets the removal of these linearly dependent rows results in a full rank matrix consequently the computational cost for image reconstruction is reduced

a modified method to compute economic order quantities without derivatives by cost difference comparisons this note presents a modified method to compute economic order quantities without derivatives by cost difference comparisons extensions to allow backorders are done for the eoq epq models in contrast to previous literatures limiting values on a finite planning horizon are used rather than algebraic manipulations for the cost function comparisons

polynomial chaos expansion with latin hypercube sampling for estimating response variability a computationally efficient procedure for quantifying uncertainty and finding significant parameters of uncertainty models is presented to deal with the random nature of input parameters of structural models several efficient probabilistic methods are investigated specifically the polynomial chaos expansion with latin hypercube sampling is used to represent the response of an uncertain system latin hypercube sampling is employed for evaluating the generalized fourier coefficients of the polynomial chaos expansion because the key challenge in uncertainty analysis is to find the most significant components that drive response variability analysis of variance is employed to find the significant parameters of the approximation model several analytical examples and a large finite element model of a joined wing are used to verify the effectiveness of this procedure

error analysis of reweighted l 1 greedy algorithm for noisy reconstruction sparse solutions for an underdetermined system of linear equations x u can be found more accurately by l 1 minimization type algorithms such as the reweighted l 1 minimization and l 1 greedy algorithms than with analytical methods in particular in the presence of noisy data recently a generalized l 1 greedy algorithm was introduced and applied to signal and image recovery numerical experiments have demonstrated the convergence of the new algorithm and the superiority of the algorithm over the reweighted l 1 minimization and l 1 greedy algorithms although the convergence has not yet been proven theoretically in this paper we provide an error bound for the reweighted l 1 greedy algorithm a type of the generalized l 1 greedy algorithm in the noisy case and show its improvement over the reweighted l 1 minimization

reliability sensitivity analysis of stochastic finite element models abstract this contribution presents a scheme for integrating a model reduction technique into a simulation based method for reliability sensitivity analysis of a class of medium large nonlinear finite element models under stochastic excitation the solution of this type of problems requires a large number of finite element model re analyses to be performed over the space of system parameters a component mode synthesis technique is implemented to carry out the sensitivity analysis in a reduced space of generalized coordinates the reliability sensitivity analysis is performed by an approach based on a simple post processing of an advanced sampling based reliability analysis the feasibility and effectiveness of the proposed scheme is demonstrated on a bridge finite element model under stochastic ground excitation

reliability analysis of stiffened aircraft panels using adjusting mean value method with regard to real world engineering structures an adjusting mean value called the self adjusting mean value smv is proposed for improving the efficiency and robustness of a maximum performance

hybrid dimension reduction method for robust design optimization robust design optimization has gained increasing concern in the engineering design process because it can provide an economical design that is insensitive to variations in the input variables without eliminating their causes robustness assessment which estimates the mean values and standard deviations of the objective function and constraint functions is an important and inevitable component in robust design optimization a hybrid dimension reduction method is proposed for efficient and accurate robustness assessment in this paper it is a combination of the univariate dimension reduction method and the bivariate dimension reduction method for the balance of efficiency and accuracy the significant random variables are identified by their variation contributions to the output variation of the objective function and constraints using the univariate dimension reduction method the approximate part of the significant random variables in the approximate function is then extended to use the bivariate dimens

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

an easy method to derive eoq and epq inventory models with backorders recently a cost minimization method to determine the lot size for the eoq epq models with backorders was published this method is based on the well known arithmetic geometric mean inequality although the cost minimization method is correct and interesting it does not focus on deriving the backorders level this paper proposes another simple approach the proposed method finds both the lot size and the backorders level

sensitivity analysis using anchored anova expansion and high order moments computation an anchored analysis of variance anova method is proposed in this paper to decompose the statistical moments compared to the standard anova with mutually orthogonal component functions the anchored anova with an arbitrary choice of the anchor point loses the orthogonality if employing the same measure however an advantage of the anchored anova consists in the considerably reduced number of deterministic solver s computations which renders the uncertainty quantification of real engineering problems much easier different from existing methods the covariance decomposition of the output variance is used in this work to take account of the interactions between non orthogonal components yielding an exact variance expansion and thus with a suitable numerical integration method provides a strategy that converges this convergence is verified by studying academic tests in particular the sensitivity problem of existing methods to the choice of anchor point is analyzed via the ishigami case and we point out that covariance decomposition survives from this issue also with a truncated anchored anova expansion numerical results prove that the proposed approach is less sensitive to the anchor point the covariance based sensitivity indices si are also used compared to the variance based si furthermore we emphasize that the covariance decomposition can be generalized in a straightforward way to decompose higher order moments for academic problems results show the method converges to exact solution regarding both the skewness and kurtosis finally the proposed method is applied on a realistic case that is estimating the chemical reactions uncertainties in a hypersonic flow around a space vehicle during an atmospheric reentry

a solution of the random eigenvalue problem by a dimensional decomposition method summary this paper presents a dimensional decomposition method for obtaining probabilistic descriptors of realvalued eigenvalues of positive semi definite random matrices the method involves a novel function decomposition allowing lower variate approximations of eigenvalues lower dimensional numerical integration for statistical moments and lagrange interpolation facilitating efficient monte carlo simulation for probability density functions compared with commonly used perturbation and recently developed asymptotic methods no derivatives of eigenvalues are required by the new method developed results of numerical examples from structural dynamics indicate that the decomposition method provides excellent estimates of moments and probability densities of eigenvalues for various cases including closely spaced modes and large statistical variations of input copyright 2006 john wiley sons ltd

the economic lot size of the integrated vendor buyer inventory system derived without derivatives a simple derivation abstract in this paper we first complement an inappropriate mathematical error on the total cost in two previous published research papers in inventory field then an arithmetic geometric inequality method is proposed not only to simplify the algebraic method of completing prefect squares but also to complement their shortcomings in addition we also provide a closed form optimal solution to the integrated vendor buyer inventory system without using complex derivatives finally the proposed method seems to be easy to understand and simple to apply by individuals who are not familiar with differential calculus

some comments on a simple method to compute economic order quantities abstract in this note we emphasize that the arithmetic geometric mean inequality approach proposed by teng teng j t 2008 a simple method to compute economic order quantities european journal of operational research doi 10 1016 j ejor 2008 05 019 is not a general solution method teng s approach happens to work and give the correct results when the two terms in an objective function are any functions such that their product is a constant the classical eoq model works fine since the product of the two terms is indeed a constant when the product is not a constant teng s approach is of little use this is exemplified in comment 1 via solving the eoq model with complete backorders where the model is regarded as having two decision variables comment 2 is generally valid for an algebraic method when it is used to solve an objective function with two decision variables

uncertainty based structural optimization using universal grey number theory several types of uncertainties are encountered in the design of machines and structures the geometric parameters or dimensions are specified in terms of nominal values and tolerances the experime

reliability sensitivity estimation of nonlinear structural systems under stochastic excitation a simulation based approach abstract this work explores the feasibility of using advanced simulation based methods for reliability sensitivity analysis of nonlinear structural systems subject to stochastic excitation reliability is defined in terms of a first excursion probability which measures the chances that uncertain structural responses exceed in magnitude prescribed threshold levels within a specified time interval the proposed reliability sensitivity analysis consists of applying a simple post processing step associated with an advanced sampling based reliability analysis it does not require any additional system response evaluations in addition the method provides information about the whole trend of the reliability sensitivity estimates in terms of the threshold levels the approach is validated on two nonlinear structural models under stochastic ground excitation