Zhenzhou Lu

Structural Reliability Analysis Using Combined Space Partition Technique and Unscented Transformation

AbstractIn this paper, a new approach for estimating the probability of failure is proposed. The method works by dividing the input space into subspaces with the Voronoi diagram and applying the unscented transformation in each subspace. The probability of failure is defined as the sum of the products of the expectation of the failure domain indicator function and the corresponding weight in subspaces. A small number of samples is needed for this method to get a failure probability estimation with high accuracy. The efficiency and robustness of the proposed method are investigated by solving several examples. The results are compared with results of other reliability methods, and they demonstrate the efficiency and robustness of the proposed method.

Multivariate sensitivity analysis based on the direction of eigen space through principal component analysis

In this paper, a new kind of sensitivity indices based on the principal component analysis (PCA) is proposed to measure the effects of input variables on multivariate outputs. Through PCA, the outputs are projected onto a new coordinate system (eigen space), which is constructed by the eigenvector (principal components). The existent sensitivity indices based on PCA focus on the variance of principal components, which can be considered as a magnitude of the uncertainty in the corresponding coordinate axes. In addition, the direction of the coordinate axes in the eigen space also contains another part of uncertainty of outputs (the direction of the uncertainty). The new sensitivity indices measure the effect of input variables on the direction of the coordinates axes through the angles between the unconditional and conditional eigenvectors. Thus, the new sensitivity indices can reflect different effect of input variables on the output compared to the existent sensitivity indices. The results of three numerical examples and an environmental model show the difference between the new sensitivity indices and the existent sensitivity indices. Since the new sensitivity indices measure the effects of input variables on the multivariate outputs from a different perspective compared to the existent sensitivity indices, they should be mutually complementary to each other.

A new effective screening design for structural sensitivity analysis of failure probability with the epistemic uncertainty

Abstract In this paper, two new sampling strategies are proposed to estimate the Morris’ screening sensitivity measure and its improved version. The two new sampling strategies, which employ random sampling and quasi-random sampling respectively, compute the elementary effects of each factor at the same initial point and with a same step size in the input space. The new quasi-random sampling strategy performs better than the radial based sampling strategy and the new random sampling strategy performs almost the same with the radial based sampling strategy. Then, the improved version of the Morris’ screening sensitivity measure is applied to estimate the effects of the epistemic uncertainty of random variables’ distribution parameters on the failure probability using the new quasi-random sampling strategy. During this process, the principle of maximum entropy, fractional moments and dimension reduction method are used to estimate the failure probability with a good accuracy and a low computational demand. Several examples are employed to demonstrate the reasonability and the efficiency of the proposed strategy.

Copula-based decomposition approach for the derivative-based sensitivity of variance contributions with dependent variables

Abstract Variance-based sensitivity analysis with dependent variables represents how the uncertainties and dependence of variables influence the output uncertainty. Since the distribution parameters of variables are difficult to be given precisely, this work defines the derivative-based sensitivity of variance contribution with respect to the distribution parameters, which reflects how small variation of distribution parameters influences the variance contributions. By introducing the copula functions to describe the dependence of variables, the derivative of variance contributions can be transformed into those of marginal PDF and copula function, which can be defined by kernel function and copula kernel function. Then the derivative-based sensitivity of variance contributions can be decomposed into the independent part and dependent part. Since the derivatives of marginal PDF and copula function can be given analytically, the proposed derivative-based sensitivity can be computed with no additional computational cost, which is seen as the ‘by-product’ of variance-based sensitivity analysis. To calculate the proposed sensitivity, two computational methods, numerical method and SDP (state dependent parameter) method are presented for comparison. Several examples are used to demonstrate the reasonability of the proposed sensitivity and the accuracy of the applied method.

Reliability Analysis by Combining Higher-Order Unscented Transformation and Fourth-Moment Method

AbstractIn this work, the fourth-moment method and the higher-order unscented transformation is combined to calculate the failure probability. The higher-order unscented transformation is used to e...

Bayesian Inference and Prediction Analysis of the Power Law Process Based on a Gamma Prior Distribution

Under a Gamma prior distribution, the importance sampling (IS) technique is applied to the Bayesian analysis of the Power Law Process (PLP). Samples of important parameters in the PLP are obtained from IS. Based on these samples, not only the posterior analyses of parameters and some functions of the parameter in the PLP can be performed conveniently, but also single-sample and two-sample predictions are constructed easily by the transformation formula of double integral. The sensitivity of the posterior mean of the parameter functions in the PLP is studied with respect to the prior moments in the Gamma prior distribution, and it can guide the selections of the prior moments. After some numerical experiments illustrate the rationality and feasibility of the proposed methods, an engineering example demonstrates its application.

Variance-based sensitivity analysis with the uncertainties of the input variables and their distribution parameters

ABSTRACT For the structural systems with both the uncertainties of input variables and their distribution parameters, three sensitivity indices are proposed to measure the influence of input variables, distribution parameters and their interactive effects. With those sensitivity indices, analysts can make a decision that whether it is worth to accumulate data of one distribution parameter to reduce its uncertainty. Due to the large computational cost, the analytical solutions are derived for quadratic polynomial output responses. Whereas for the complex models, state dependent parameter (SDP) method is utilized to solve the proposed sensitivity indices efficiently.

Estimation of the Generalized Sobol’s Sensitivity Index for Multivariate Output Model Using Unscented Transformation

AbstractGlobal sensitivity analysis is frequently applied to models with multivariate output. The generalized Sobol’s sensitivity index has been defined to evaluate the importance of input variable...