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Research Papers

Probabilistic Sensitivity Analysis With Respect to Bounds of Truncated Distributions

[+] Author and Article Information
H. Millwater

Department of Mechanical Engineering,  University of Texas at San Antonio, San Antonio, TX 78246harry.millwater@utsa.edu

Y. Feng

Department of Mechanical Engineering,  University of Texas at San Antonio, San Antonio, TX 78246yusheng.feng@utsa.edu

J. Mech. Des 133(6), 061001 (Jun 15, 2011) (10 pages) doi:10.1115/1.4003819 History: Received February 19, 2010; Revised January 03, 2011; Published June 15, 2011; Online June 15, 2011

Bounds on variables are often implemented as a part of a quality control program to ensure a sufficient pedigree of a product component, and these bounds may significantly affect the product’s design through constraints such as cost, manufacturability, and reliability. Thus, it is useful to determine the sensitivity of the product reliability to the imposed bounds. In this work, a method to compute the partial derivatives of the probability-of-failure and the response moments, such as mean and the standard deviation, with respect to the bounds of truncated distributions are derived for rectangular truncation. The sensitivities with respect to the bounds are computed using a supplemental “flux” integral that can be combined with the probability-of-failure or response moment information. The formulation is exact in the sense that the accuracy depends only upon the numerical algorithms employed. The flux integral is formulated as a special case of the probability integral for which the sensitivities are being computed. As a result, the methodology can be implemented with any probabilistic method, such as sampling, first order reliability method, conditional expectation, etc. Moreover, the maximum and minimum values of the sensitivities can be obtained without any additional computational cost. The methodology is quite general and can be applied to both component and system reliability. Several numerical examples are presented to demonstrate the advantages of the proposed method. In comparison, the examples using Monte Carlo sampling demonstrated that the flux-based methodology achieved the same accuracy as a standard finite difference approach using approximately 4 orders of magnitude fewer samples. This is largely due to the fact that this method does not rely upon subtraction of two near-equal numbers.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Description of velocity and unit normal along bounds

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Figure 2

Flux of the JPDF in the failure region over the surface S

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Figure 3

Two dimensional problems with the same flux values with respect to lower and upper bounds

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Figure 4

Joint PDF over failure domain (direct view of the face of S=1)

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Figure 5

Joint PDF over failure domain (direct view of the face of S=0)

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Figure 6

(a) 95% confidence limits (100 trials) for ∂Pf/∂a as a function of the number of samples (dashed - finite difference; solid – flux based). (b) 95% confidence limits (100 trials) for ∂Pf/∂b as a function of the number of samples (dashed - finite difference; solid – flux based).

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Figure 7

(a) Plot of JPDF times Z (direct view of the face S=1). (b) Plot of JPDF times Z (direct view of the face S=0).

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Figure 8

(a) Plot of JPDF times Z2 (direct view of the face S=0). (b) Plot of JPDF times Z2 (direct view of the face S=1).

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Figure 9

JPDF of series system

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Figure 10

(a) Flux along bound S=0 for series system. (b) Flux along bound S=1 for series system.

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Figure 11

JPDF of parallel system

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Figure 12

Flux along bound a of JPDF along bound a for parallel system

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