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Research Papers: Design Automation

Reliability Analysis Using Second-Order Saddlepoint Approximation and Mixture Distributions

[+] Author and Article Information
Dimitrios I. Papadimitriou

Mechanical Engineering Department,
Oakland University,
Rochester, MI 48309

Zissimos P. Mourelatos

Mechanical Engineering Department,
Oakland University,
Rochester, MI 48309
e-mail: mourelat@oakland.edu

Zhen Hu

Department of Industrial and Manufacturing
Systems Engineering,
University of Michigan-Dearborn,
Dearborn, MI 48128

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 9, 2018; final manuscript received June 18, 2018; published online December 20, 2018. Assoc. Editor: Xiaoping Du.

J. Mech. Des 141(2), 021401 (Dec 20, 2018) (10 pages) Paper No: MD-18-1204; doi: 10.1115/1.4041370 History: Received March 09, 2018; Revised June 18, 2018

This paper proposes a new second-order saddlepoint approximation (SOSA) method for reliability analysis of nonlinear systems with correlated non-Gaussian and multimodal random variables. The proposed method overcomes the limitation of current available SOSA methods, which are applicable to problems with only Gaussian random variables, by employing a Gaussian mixture model (GMM). The latter is first constructed using the expectation maximization (EM) method to approximate the joint probability density function (PDF) of the input variables. Expressions of the statistical moments of the response variables are then derived using a second-order Taylor expansion of the limit-state function and the GMM. The standard SOSA method is finally integrated with the GMM to effectively analyze the reliability of systems with correlated non-Gaussian random variables. The accuracy of the proposed method is compared with existing methods including a SOSA based on Nataf transformation. Numerical examples demonstrate the effectiveness of the proposed approach.

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Figures

Grahic Jump Location
Fig. 1

(a) Example 1: sample points in the X1-X2 space, (b) example 1: sample points in the X1-X3 space, (c) example 1: sample points in the X1-X4 space, (d) example 1: sample points in the X2-X3 space, (e) example 1: sample points in the X2-X4 space, and (f) example 1: sample points in the X3-X4 space

Grahic Jump Location
Fig. 2

Example 1: CDF of Y from different methods, (b) example 1: CDF of Y from different methods. Close-up view at left tail, and (c) example 1: CDF of Y from different methods. Close-up view at right tail.

Grahic Jump Location
Fig. 3

Logarithm of error of CDF of Y for different methods for example 1

Grahic Jump Location
Fig. 4

(a) Example 2: Sample points in the X1-X2 space, (b) example 2: sample points in the X1-X3 space, (c) example 2: sample points in the X1-X4 space, (d) example 2: sample points in the X2-X3 space, (e) example 2: sample points in the X3-X4 space, and (f) example 2: sample points in the X4-X5 space

Grahic Jump Location
Fig. 5

CDF of Y from different methods for example 2

Grahic Jump Location
Fig. 6

Logarithm of error of CDF from different methods for example 2

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