Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 685826, 13 pages

http://dx.doi.org/10.1155/2015/685826

## Parametric Sensitivity Analysis for Importance Measure on Failure Probability and Its Efficient Kriging Solution

Institute of Aircraft Reliability Engineering, Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710129, China

Received 27 May 2014; Revised 23 September 2014; Accepted 23 September 2014

Academic Editor: Shaomin Wu

Copyright © 2015 Yishang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The moment-independent importance measure (IM) on the failure probability is important in system reliability engineering, and it is always influenced by the distribution parameters of inputs. For the purpose of identifying the influential distribution parameters, the parametric sensitivity of IM on the failure probability based on local and global sensitivity analysis technology is proposed. Then the definitions of the parametric sensitivities of IM on the failure probability are given, and their computational formulae are derived. The parametric sensitivity finds out how the IM can be changed by varying the distribution parameters, which provides an important reference to improve or modify the reliability properties. When the sensitivity indicator is larger, the basic distribution parameter becomes more important to the IM. Meanwhile, for the issue that the computational effort of the IM and its parametric sensitivity is usually too expensive, an active learning Kriging (ALK) solution is established in this study. Two numerical examples and two engineering examples are examined to demonstrate the significance of the proposed parametric sensitivity index, as well as the efficiency and precision of the calculation method.

#### 1. Introduction

Uncertainties existing in engineering analysis and design are inherently unavoidable in nature associated with the manufacturing error, material property, loads, and so forth. Fortunately, reliability analysis and sensitivity analysis are now available to deal with the uncertainty existing in design variables to improve the performance of a mechanical or structural system [1–3]. Reliability analysis aims at predicting the failure probability (or reliability) of the structure under the effects of random uncertainties. On the other hand, sensitivity analysis focuses on the contribution of each uncertainty or distribution parameters of the input variables [4–7]. It is reasonable and practicable to obtain reliability sensitivity analysis for quantifying and ranking the effects of random uncertainties on the failure probability. In this paper, the reliability sensitivity analysis is concerned.

Sensitivity analysis (SA) is the study of how the output response of a model (numerical or otherwise) is affected by the input uncertainty, which can be classified into two groups: local SA and global SA [8]. The local SA investigates how small variation of the distribution parameters near a reference point changes the output value. The classical local SA is defined as the partial derivative of the output with respect to the distribution parameters of inputs [9]. The global SA, also named as importance measure (IM), gives consideration to measure the effect of the output uncertainty on the uncertainty of the input parameters, covering their variation range space as opposed to local SA using partial derivatives.

Saltelli and Marivoet [10] and Helton and Davis [11] proposed the nonparametric techniques (input-output correlation), but this method lacks model independence. With the advantage of “global, quantitative and model free,” the variance-based importance measures are gaining the increasing attention of practitioners and have been used extensively for quantitative analysis [12–16]. However, Borgonovo addressed the following fact: “the premise of variance-based GSA technique that the variance is sufficient to identify the variability of model output is not always true” [17]. The “moment-independent” importance measures have been presented [6, 16–19]. They are also global, quantitative, model-free, and additionally moment-independent, thus attracting more and more attention of practitioners recently. Generally speaking, in reliability analysis, researchers often pay the most attention to the failure probability. With this respect, Cui et al. [6] introduced a moment-independent importance measure of the basic variable on the failure probability which was further developed by Li et al. [20].

This moment-independent importance measure on failure probability is applied to quantify the average effect of the basic variables on the reliability of the model and obtain the importance ranking. The IM on the failure probability can be used in the priordesign stage for variables screening when a reliability design solution is yet identified and the postdesign stage for uncertainty reduction after an optimal design has been determined. Uncertain inputs inherent in most engineering problems are assumed as random variables obeying probabilistic distributions. Obviously, system reliability and reliability IM on failure probability are decided by distribution parameters. One can directly change the input’s IMs by controlling or modifying some input’s distribution parameters; namely, changing the input’s distribution parameters can also influence the failure probability, which would facilitate its use under various scenarios of design under uncertainty, for instance, in reliability-based design. It is necessary to further recognize effects of the distribution parameters within system reliability on the importance ranking. At present, Cui et al. [7] defined the parametric sensitivities to illustrate the influences of the distribution parameters on the importance measures.

Combined with the local SA technique of input parameters, the effects of the distribution parameters on the IM on failure probability can be introduced, by which IMs of the inputs can be controlled or modified by changing the distribution parameters. This can provide important guidance for robust design, reliability-based design, and reliability-based optimization in engineering. However, its solution still relies on the corresponding method for failure probability and the computation of the derivative operation on failure probability existing in the parametric sensitivity of IM.

The Monte Carlo simulation (MCS) procedure is easy to implement and is available for computing the parametric sensitivity of IM based purely on model evaluation [15, 21], but it has to face the problem of “curse of computational cost” for the problem with small failure probability (10^{−3}–10^{−4} or smaller). Thus, to deal with this problem, the Kriging approach is widely used for deterministic optimization problems [22] and reliability analysis [23] has been intensively investigated. Furthermore, an advanced Kriging method, named active learning Kriging (ALK), has been proved to be highly efficient in reliability analysis problems [24–26]. This work would employ the ALK method to compute the parametric sensitivity of IM on failure probability. In the ALK method, the Kriging model is updated by adding new training points to the design of experiment (DOE) in iterations by active learning until the Kriging model satisfies necessary accuracy. The computational efficiency of the Kriging method can be validated by several numerical and engineering examples.

The remainder of this work is organized as follows. Section 2 reviews the definition of the moment-independent importance measure of the basic variable on the failure probability. And the parametric sensitivity of IM on failure probability is firstly presented. In Section 3, the established ALK solution can effectively solve the problem that the computational cost of the parametric sensitivity of IM relies on small failure probability. The effectiveness of the proposed parametric sensitivity of IMs and efficiency of the ALK method are demonstrated by several examples in Section 4. The discussions and conclusions are given at the end of this paper.

#### 2. Definition of the Parametric Sensitivity of IM on Failure Probability

##### 2.1. Review of the Importance Measure on the Failure Probability

Consider a probabilistic reliability model , where is the performance function, is the model output, and is the -dimensional vector of random input variables with joint probability density function (PDF) . Denote by the unconditional failure probability; that is, . When the th input variable is fixed at one given value, the conditional failure probability can be obtained.

Based on the idea of the moment-independent importance analysis, the importance measure of basic variable on the failure probability is defined by Cui et al. [6] aswhere represents a random basic variable or a set of random basic variables , where . is the operator of expectation.

As the absolute value in (1) is difficult to compute, it is transformed into square operation by Li et al. [20]. The modified version of importance measures on the failure probability can be expressed as follows:

In the reliability analysis, the failure domain of this structure system is defined as

Suppose the indicator function of this failure domain is given as ; that is,

Then, the unconditional failure probability and conditional failure probability on can be expressed asHere, is the conditional joint PDF on . According to the probability theory, is defined aswhere is the PDF of .

##### 2.2. The Parametric SA of IM on Failure Probability

For the influential distribution parameter, it is significant to identify how it influences IM on failure probability. We suppose that each input only depends on one distribution parameter in order to simplify the notation in the following.

As stated above, is the th distribution parameter of input which influences the unconditional failure probability , but not the conditional failure probability . It is also noticed that is not the distribution parameter of input , but it still influences and . is a vector containing all distribution parameters of the input variables but . The contributions of distribution parameter on and can be shown in Figure 1.