A response surface approach for structural reliability analysis using evidence theory
Introduction
Uncertainties related to the material property, bounding condition, load, etc. widely exist in practical engineering problems. With intensive requirements of high product quality and reliability, understanding, identifying, controlling and managing various uncertainties have become imperative. Uncertainty refers to the difference between the present state of knowledge and the complete knowledge. Based on this view, uncertainty can be described as two distinct types – aleatory (random) and epistemic (subjective) uncertainty [1]. Aleatory uncertainty is irreducible and describes the inherent variability of a physical system, which can be modeled as random variables or processes using probability theory. Many probability-based reliability analysis techniques have been well established and successfully applied to varieties of industrial fields [2], [3], [4], [5]. However, when data are scarce, the probability theory becomes not so useful because the key probability distributions cannot be obtained. In this case, the epistemic uncertainty will be involved. Epistemic uncertainty is defined as the lack of knowledge or information in some phases or activities of the modeling process. Therefore, it can be reduced with the collection of more information or an increase of knowledge. Some representative theories, including convex models [6], [7], [8], [9], [10], [11], possibility theory [12], [13], [14], fuzzy sets [15] and evidence theory [16], [17], [18], [19], [20], [21], can be used to deal with the epistemic uncertainty.
Among the above theories for epistemic uncertainty, evidence theory employs a much more flexible framework with respect to the body of evidence and its measures [22]. Under some special situations, it can provide equivalent descriptions to the probability theory, convex models, possibility theory and fuzzy sets, respectively. Hence, in recent years evidence theory has been introduced to conduct reliability analysis and design for engineering structures and mechanical systems. Oberkampf and Helton [22] compared the similarities and differences between evidence theory and probability theory in reliability analysis through a simple algebraic function. Helton et al. [23] explored several approaches (probability model, evidence theory, possibility theory and interval analysis) in the representation of the uncertainty in model prediction and thereby gave a unified framework. Soundappan et al. [24] compared evidence theory with Bayesian theory in aspects of uncertainty modeling and decision making under epistemic uncertainty. Du [25] formulated a new reliability analysis model to handle the epistemic and aleatory mixed uncertainty. Tonon et al. [26] employed evidence theory to quantify the parameter uncertainty in rock engineering and whereby carried out a reliability-based design of tunnels. Through creating a multi-point approximation at a certain point on the limit-state surface, Bae et al. [27], [28] proposed an efficient reliability analysis method for structures with epistemic uncertainty. Jiang et al. [29] proposed a structural reliability method using evidence theory by introducing a non-probabilistic reliability index approach. Agarwal et al. [30] proposed an evidence-theory-based multidisciplinary design optimization (EBDO) algorithm through a sequential approximate strategy. Alyanak et al. [31] adopted a gradient projection technique to conduct a reliability-based design optimization (RBDO) for structures with epistemic uncertainty. Helton et al. [32] developed a sampling-based approach for sensitivity analysis of the uncertainty propagation problems using evidence theory. Mourelatos and Zhou [33] proposed a RBDO method based on evidence theory. Guo et al. [34] developed a RBDO method by combining evidence theory and interval analysis. Bai et al. [35] compared three metamodeling techniques for evidence-theory-based reliability analysis through six numerical examples.
Despite the above achievements, presently evidence theory has been barely applied to conduct reliability analysis for complex engineering problems. One main reason is the high computational cost caused by the discontinuous nature of uncertainty quantification for the evidence variable [26]. Unlike the probability density function (PDF) in probability model, the uncertainty modeled by evidence theory is propagated through a discrete basic probability assignment (BPA), which cannot be expressed by any explicit function but generally described by a series of discontinuous subsets. This will in general lead to a combination explosion difficulty for a multidimensional problem. By using the response surface of the actual limit-state function, the high computational cost of evidence-theory-based reliability analysis can be significantly reduced. Some numerical methods [27], [28] have been developed to reduce the computational cost by introducing the response surface technique, however, it seems not always an easy job to construct a sufficiently accurate response surface for a practical engineering problem using the existing methods. Therefore, to improve the applicability of evidence theory in practical applications, it seems necessary to develop some more robust and efficient reliability analysis methods.
In this paper, a new response surface method is proposed to significantly improve the computational efficiency of evidence-theory-based reliability analysis, in which the analysis precision can be well guaranteed through a design of experiments technique. The remainder of this paper is organized as follows. The conventional reliability analysis using evidence theory is introduced in Section 2. An efficient algorithm is formulated to assess the reliability in Section 3. Four numerical examples are investigated in Section 4. Finally some conclusions are summarized in Section 5.
Section snippets
Conventional reliability analysis using evidence theory
In this section, a simple problem is used to show the conventional reliability analysis using evidence theory, in which some fundamentals of evidence theory will also be introduced.
Consider the following two-dimensional limit-state function:where X = (X1, X2) is the vector of two independent uncertain input parameters; g0 denotes an allowable value of the structural responses. For this problem, the safety region G is defined as:
In this paper, the uncertain parameters X will be
A response surface method
As introduced above, the high computational cost significantly restricted the application of evidence theory in engineering problems. To solve this problem, the response surface (RS) technique can be adopted to approximate the black-box limit-state function. As indicated in [38], [39], [40], design of experiments (DOE) plays a significant role in improving the accuracy of the established RS. Therefore, in this paper, a new DOE technique is developed to construct a sufficiently accurate RS for
A mathematical problem
The following limit-state function is considered:
The FDs of X1, X2 and X3 are all [2, 4]. Two different cases that the BPA structure contains 4 and 8 subintervals are investigated, as shown in Table 1, Table 2.
Firstly, we make the allowable value α vary and a series of different limit-state functions are generated. For each limit-state function, the Newton’s method is employed to search the control point on each edge, according to which the
Conclusions
In this paper, a response surface method is developed for the evidence-theory-based reliability analysis to resolve its low efficiency problem and hence expands its engineering application range. In the proposed method, a new DOE technique is developed, which includes the search of the important control points and the deployment of expansion sample points. Based on them, a high precise RBF RS is established. Thus, the present method has a fine accuracy. Also, the obtained RBF RS is used to
Acknowledgements
This work is supported by the National Outstanding Youth Science Foundation of China (51222502), the National Science Foundation of China (11172096), the Fok Ying-Tong Education Foundation, China (131005) and the program for Century Excellent Talents in University (NCET-11-0124).
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