A new uncertainty propagation method for problems with parameterized probability-boxes

https://doi.org/10.1016/j.ress.2017.12.004Get rights and content

Highlights

  • A new uncertainty propagation method for problems with parameterized probability-boxes is proposed.

  • The proposed method can significantly improve the computational efficiency of calculating the probability bounds of output response.

  • The proposed method can provide sufficient accuracy for reliability analysis or risk assessment of a structure or system.

Abstract

This paper proposes a new uncertainty propagation method for problems with parameterized probability-boxes (p-boxes), which could efficiently compute the probability bounds of system response function. In practical engineering, these probability bounds are often very important for reliability analysis or risk assessment of a structure or system. First, based on the univariate dimension reduction method (UDRM), an optimized UDRM (OUDRM) is presented to calculate the bounds on statistical moments of response function. Then, utilizing the bounds on moments, a family of Johnson distributions fitting to the distribution function of response can be acquired using the moment matching method. Finally, by using an optimization approach based on percentiles, the probability bounds of the response function can be successfully obtained. Four numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Introduction

Uncertainties existing in practical engineering problems are often associated with the material property, load, manufacturing error, etc. The probabilistic model has been widely used to deal with the uncertainty existing in engineering problems. Uncertainty propagation analysis aims to calculate the probability distributions of output responses by propagating the input uncertain parameters, which is important for structural reliability analysis and safety design. Many uncertainty propagation approaches have been developed with probability model, including the Monte Carlo simulation method (MCS) [1], the stochastic finite element method [2], the polynomial chaos method [3], [4], [5], the most probable point-based method [6], the Taylor expansion method [7], the dimension reduction method [8], [9], [10], [11], [12], [13], etc. The probability model has currently become a principal means for uncertainty propagation analysis, which has been applied in a wide variety of industrial fields. The main disadvantage of probability model is that the precise probability distributions of the uncertain parameters should be known beforehand. However, constructing a precise probability distribution for a practical parameter generally requires numerous experimental samples, which unfortunately is infeasible in many engineering problems due to the limitations from testing conditions or cost.

In recent years, a kind of probability and interval hybrid uncertainty model known as probability-boxes (p-boxes) has been developed for uncertainty modeling [14], [15], [16], [17], in which the uncertainty of a parameters is quantified through two bounds of the cumulative distribution function (CDF). P-boxes combine probability and interval models and can simultaneously consider the inherent randomness and imprecision of an uncertain parameter [18], and hence could be used to effectively deal with the problems lacking sufficient information on the uncertainty. The p-boxes can be classified into parameterized and non-parameterized representations [19], [20], [21]. For a parameterized p-box, the distribution type of a random variable can be determined, while some of its distribution parameters could only be given intervals due to limited information. Recently, various methods have been developed to conduct uncertainty propagation analysis for parameterized p-boxes. Through the combination of the outer sampling of interval distribution parameters and the inner MCS for random variables, a double-loop sampling (DLS) method was proposed to compute the probability bounds of response function [21], [22]. The optimized parameter sampling method (OPS) was then developed to improve DLS by replacing the outer loop with an optimization approach [19], [20]. The optimization-based interval estimation and stochastic expansion method were used to improve the sampling of the outer and inner loops of DLS [23]. To improve the performance of the optimized parameter sampling, the sparse grid numerical integration was used to replace its inner-loop MCS [24]. By integrating the interval sampling technique with interval finite element method (FEM), the interval Monte Carlo method (IMC) was proposed for uncertain structural analysis [25], [26]. Based on the monotonicity and vertex analysis, a vertex-based DLS (V-DLS) method was developed to improve the performance of DLS [27]. On the other hand, a non-parameterized p-box is only known as the upper and lower probability bounds of a random variable, which is more general than a parameterized p-box. Theoretically, a parameterized p-box can be transformed into a non-parameterized p-box by some approximation treatments, such as the discrete approximation method [28] and the iterative rescaling method [29]. The uncertainty propagation methods for non-parameterized p-boxes mainly include the dependency bound convolution (DBC) [21], [22], [30], [31], the distribution envelope determination (Denv) [32], the Dempster–Shafer belief function (DSBF) [33], etc.

Although several uncertainty propagation methods have been successfully developed for p-boxes, the low efficiency problem still needs to be focused and it actually has been a main difficulty in this area [34]. Generally, a two-layer nesting loop is involved in the propagation analysis of p-boxes, such as in the DLS method and OPS method [20–22], leading to a large amount of repetitive computations of response function. For the single-layer sampling methods, such as the IMC, a large number of interval operations are involved, which also causes a heavy computational burden especially for complex engineering problems. Therefore, to utilize p-boxes in practical applications, the computational efficiency of the existing uncertainty propagation methods should be significantly improved.

This paper aims to develop a new uncertainty propagation method for parameterized p-boxes, which provides an effective tool for probability bounds computation of uncertain response function with limited information. This paper is further divided into the following sections. Section 2 presents an introduction of the parameterized p-boxes and several existing methods for its uncertainty propagation analysis. Section 3 gives the formulation of the proposed method. Section 4 provides several numerical examples and Section 5 provides concluding remarks.

Section snippets

Fundamentals of parameterized p-box model

For many practical engineering problems, sufficient information cannot be obtained to construct precise probability distributions for some random variables. In this case, the p-box model can be used to quantify these parameters. The p-box of an uncertain quantity X defined by two probability bounds can be expressed as F̲X(x)FX(x)F¯X(x) or [F̲X(x),F¯X(x)], where F̲X(x) and F¯X(x) are the lower and upper cumulative distribution bounds, respectively. The p-box can be further classified into

The proposed uncertainty propagation method

In this paper, an efficient uncertainty propagation method is proposed for parameterized p-boxes. Firstly, an optimized univariate dimension reduction method (OUDRM) is provided to calculate the bounds on the first four moments of a response function. Then, based on the bounds on moments, a family of Johnson distributions fitted to the response distribution function is obtained using the moment matching method. Finally, the probability bounds of the response function can be conveniently

Analytical function problems

Two analytical function problems with p-box variables are considered, as shown in Table 1. Firstly, OURDM is compared with OPS to calculate the bounds on the first four moments in terms of computational cost. Here, the number of function calls is used to quantify the computational cost. For both OURDM and OPS, since there are 8 times outer-loop optimizations in computing the bounds on the first four moments, the number of function calls is obtained by multiplying the total iterations required

Conclusion

This paper proposes a new uncertainty propagation method to calculate the probability bounds of a response function with input parameterized p-boxes, which provides an efficient method for reliability analysis or risk assessment of structures or systems. An optimized univariate dimension reduction method is presented to calculate the bounds on the first four moments. Using the bounds on moments, we fit a family of Johnson distributions to the response distribution functions. Utilizing an

Acknowledgments

This work is supported by the National Science Fund for Distinguished Young Scholars (51725502), Major Projects of the National Natural Science Foundation of China (51490662) and the National Key Research and Development Plan (2016YFD0701105).

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