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Research Papers: Design Automation

An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses

[+] Author and Article Information
Po Ting Lin

Intelligent Unmanned System R&D Group,
Department of Mechanical Engineering,
Chung Yuan Christian University,
Chungli, Taoyuan 32023, Taiwan
e-mail: potinglin@cycu.edu.tw

Shu-Ping Lin

Intelligent Unmanned System R&D Group,
Department of Mechanical Engineering,
Chung Yuan Christian University,
Chungli, Taoyuan 32023, Taiwan

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received August 20, 2015; final manuscript received May 1, 2016; published online May 18, 2016. Assoc. Editor: Nam H. Kim.

J. Mech. Des 138(7), 071403 (May 18, 2016) (14 pages) Paper No: MD-15-1587; doi: 10.1115/1.4033548 History: Received August 20, 2015; Revised May 01, 2016

Reliability-based design optimization (RBDO) algorithms have been developed to solve design optimization problems with existence of uncertainties. Traditionally, the original random design space is transformed to the standard normal design space, where the reliability index can be measured in a standardized unit. In the standard normal design space, the modified reliability index approach (MRIA) measured the minimum distance from the design point to the failure region to represent the reliability index; on the other hand, the performance measure approach (PMA) performed inverse reliability analysis to evaluate the target function performance in a distance of reliability index away from the design point. MRIA was able to provide stable and accurate reliability analysis while PMA showed greater efficiency and was widely used in various engineering applications. However, the existing methods cannot properly perform reliability analysis in the standard normal design space if the transformation to the standard normal space does not exist or is difficult to determine. To this end, a new algorithm, ensemble of Gaussian reliability analyses (EoGRA), was developed to estimate the failure probability using Gaussian-based kernel density estimation (KDE) in the original design space. The probabilistic constraints were formulated based on each kernel reliability analysis for the optimization processes. This paper proposed an efficient way to estimate the constraint gradient and linearly approximate the probabilistic constraints with fewer function evaluations (FEs). Some numerical examples with various random distributions are studied to investigate the numerical performances of the proposed method. The results showed that EoGRA is capable of finding correct solutions in some problems that cannot be solved by traditional methods. Furthermore, experiments of image processing with arbitrarily distributed photo pixels are performed. The lighting of image pixels is maximized subject to the acceptable limit. Our implementation showed that the accuracy of the estimation of normal distribution is poor while the proposed method is capable of finding the optimal solution with acceptable accuracy.

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Figures

Grahic Jump Location
Fig. 1

Determination of MPFP of each kernel analysis in the original design space

Grahic Jump Location
Fig. 2

Estimation of MPFP of each kernel reliability analysis in the original design space

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Fig. 3

Generated random distributions: (a) a heart-shaped distribution, (b) a like-shaped distribution, (c) a star-shaped distribution, and (d) a corona-shaped distribution. (Small dots represent the sampling points; the circle at the center represents the mean value of randomly distributed points; and dashed ellipses represent the ranges of 1σ, 2σ, and 3σ if treating the random points as normal distributions).

Grahic Jump Location
Fig. 4

Iteration path of solving example 2 with corona-shaped distribution and Pf=30% using EoGRA with N=50,000

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Fig. 5

Iteration path of solving example 3 with like-shaped distribution and Pf=30% using EoGRA with N=50,000

Grahic Jump Location
Fig. 6

Using EoGRA with N=50,000 to solve example 1 with Pf=1% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star-shaped, and (f) corona-shaped distributions

Grahic Jump Location
Fig. 7

Using EoGRA with N=50,000 to solve example 1 with Pf=30% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star-shaped, and (f) corona-shaped distributions

Grahic Jump Location
Fig. 8

Using EoGRA with N=50,000 to solve example 2 with Pf=1% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star-shaped, and (f) corona-shaped distributions

Grahic Jump Location
Fig. 9

Using EoGRA with N=50,000 to solve example 2 with Pf=30% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star-shaped, and (f) corona-shaped distributions

Grahic Jump Location
Fig. 12

Results of photo enhancements: (a) original photo, (b) pixel distribution in CIELAB, (c) enhanced photo by treating image pixels as normally distributed (Pnorm  = 0.2811%), and (d) enhanced photo by EoGRA (PEoGRA  = 4.9485%)

Grahic Jump Location
Fig. 11

Using EoGRA with N=50,000 to solve example 3 with Pf=30% and (a) normal, (b) uniform, (c) heart-shaped, (d) like-shaped, (e) star-shaped, and (f) corona-shaped distributions

Grahic Jump Location
Fig. 10

Using EoGRA with N=50,000 to solve example 3 with Pf=1% and (a) normal and (b) uniform distributions

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