استفاده از روش کریگینگ در تجزیه و تحلیل حساسیت جهانی با عدم قطعیت پارامتر

For structural systems with both epistemic and aleatory uncertainties, the effect of epistemic uncertainty on failure probability is measured by the variance based sensitivity analysis, which generally needs a “triple-loop” crude sampling procedure to solve and is time consuming. Thus, the Kriging method is employed to avoid the complex sampling procedure and improve the computational efficiency. By utilizing the Kriging predictor model, the conditional expectation of failure probability on the given epistemic uncertainty can be calculated efficiently. Compared with the Sobol’s method, the proposed one can ensure reasonable accuracy of results but with lower computational cost. Three examples are employed to demonstrate the reasonability and efficiency of the proposed method.

Sensitivity analysis (SA) is widely used in engineering design, which can be classified into two groups: local SA and global SA [1]. Local SA techniques are usually investigated how small variations of parameters around a nominal point change the value of the output. The main disadvantages of them are that they depend on the choice of the nominal point in the parameters space. Global SA takes into account all the variation range of the parameters, and apportions the output uncertainty to the uncertainty of the input parameters, covering their entire range space [2]. At present, a number of measures have been suggested, such as, Helton and Saltelli [3] and [4] proposed the nonparametric techniques (input–output correlation), Sobol, Iman and Saltelli [4], [5] and [6] proposed a series of variance based importance measures, Chun, Liu and Borgonovo [7] and [8] proposed moment independent sensitivity indicators. But those indicators are all proposed for structural system with epistemic input uncertainty. Hofer and Krzykaca-Hansmann [9] and [10] investigated another situation that the input uncertainty of a model is only aleatory uncertainty described by the probability distribution and the distributional parameters of inputs are not known precisely which are subject to epistemic uncertainty. In their works, they proposed the variance-based sensitivity measures in the presence of epistemic and aleatory uncertainties which can be used to identify the most influential distribution parameters. Based on this idea, we proposed the variance-based sensitivity measures of failure probability in the presence of epistemic and aleatory uncertainties, which can be used to identify the most influential distribution parameters on the safety of a system. The variance based measures generally require a large number of function evaluations to achieve reasonable convergence and can become impractical for most engineering problems. Recently, several works estimate the variance based sensitivity measure with a group of given samples [11] and [12], but they still face the problem of “curse of dimensionality”. Thus this paper employs the Kriging method to overcome this problem, which has been widely used for deterministic optimization problems [13] and reliability analysis [14]. The Kriging method can be represented as an improved linear regression technique [15]. It consists of a parametric linear regression model and a nonparametric stochastic process, which can approximate the failure probability of a test point by the weighted average of the failure probability of training points surrounding the test point. The mapping relationship of epistemic parameters and failure probability can be obtained directly by the Kriging method, and then the conditional expectation of failure probability can be calculated conveniently which avoids the complex sampling procedure. The computational efficiency of the Kriging method can be validated by several numerical and engineering examples. This paper is organized as follows: Section 2 analyzes the propagation of the uncertainty of the structural system and distinguishes the epistemic and aleatory uncertainties. Then the “black box” model of epistemic parameters and failure probability is given. Section 3 first employs the standard Sobol’s method to solve the variance based sensitivity measure, and then a novel method based on the Kriging model is proposed. Three examples, including a numerical example, a roof truss structure and an automobile front axle structure, are employed to validate the reasonability and efficiency of the proposed method in Section 4. Finally, conclusions are drawn in Section 5.

This paper investigates the use of the Kriging method for global sensitivity analysis with both epistemic and aleatory uncertainties. At first, the propagation of uncertainty in the structural system is sealed in a “black box” model which can be represented as the mapping relationship of epistemic parameters and failure probability. Then a variance based sensitivity measure of failure probability is investigated to illustrate the effect of epistemic uncertainty on the failure probability. The Kriging method is used to solve this measure which generally requires large computational cost. From the above examples, the following points can be deduced to indicate some advantages of the variance based sensitivity measure and the Kriging method: 1. The variance based sensitivity measure of failure probability can be used to identify which epistemic parameters are much more influential on the failure probability than others. By accumulating the data and improving the understanding of those parameters, the safety of structural system can be enhanced in a maximum extent. 2. For decreasing the computational cost, the Kriging method is employed to calculate the variance based measure. It can be seen in the examples that the Kriging method is more efficient than Sobol’s method, let alone the crude sampling method (Due to the large computational cost, the results are not listed in this paper for comparison). 3. Since the training points used in the Kriging method is essential to the computational accuracy, the low discrepancy sampling method is employed to obtain those points. It is notice that only a small quantity of points used to construct the Kriging predictor model can ensure the computational accuracy and the additional computational cost is acceptable. It is significant to investigate the effect of epistemic uncertainty on the failure probability of failure probability. Compared with the Sobol’s method, the proposed method is validated to be rationality and efficiency.