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Stability Analysis and Convergence Control of Iterative Algorithms for Reliability Analysis and Design Optimization

[+] Author and Article Information
Dixiong Yang

Associate Professor
e-mail: yangdx@dlut.edu.cn

Hui Xiao

e-mail: vihuixiao@163.com
Department of Engineering Mechanics,
Dalian University of Technology,
State Key Laboratory of Structural Analysis for Industrial Equipment,
Dalian 116023, China

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received January 14, 2011; final manuscript received December 12, 2012; published online January 17, 2013. Assoc. Editor: Wei Chen.

J. Mech. Des 135(3), 034501 (Jan 17, 2013) (8 pages) Paper No: MD-11-1037; doi: 10.1115/1.4023327 History: Received January 14, 2011; Revised December 12, 2012

Iterative algorithms are widely applied in reliability analysis and design optimization. Nevertheless, phenomena of failed convergence, such as periodic oscillation, bifurcation, and chaos, are oftentimes observed in iterative procedures of solving some nonlinear problems. In the present paper, the essential causes of numerical instabilities including periodic oscillation and chaos of iterative solutions are revealed by the eigenvalue-based stability analysis of iterative schemes. To understand and control these instabilities, the stability transformation method (STM), which is capable of tackling numerical instabilities of iterative algorithms in reliability analysis and design optimization, is proposed. Finally, several benchmark examples of convergence control of PMA (performance measure approach) for probabilistic analysis and the SORA (sequential optimization and reliability assessment) for reliability-based design optimization (RBDO) are presented. The observations from the benchmark examples indicate that the STM is a promising approach to achieve convergence control for iterative algorithms in reliability analysis and design optimization.

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Figures

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Fig. 1

Eigenvalues of the Henon map at larger fixed point

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Fig. 2

Bifurcation plot of the Henon map

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Fig. 3

Eigenvalues of STM scheme of the Henon map

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Fig. 4

STM chaos control of the Henon map

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Fig. 5

Iterative history of the Henon map

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Fig. 6

Iterative history of STM scheme of the Henon map

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Fig. 7

(a) Bifurcation plot of the AMV formula and (b) Local zooming view of bifurcation plot of the AMV formula

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Fig. 8

Eigenvalues of the AMV scheme at the fixed point

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Fig. 9

STM control of the AMV formula

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Fig. 10

Eigenvalues of STM scheme of the AMV formula

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Fig. 11

Iterative history of the AMV formula

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Fig. 12

Iterative history of STM control for the AMV formula

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Fig. 13

CDF of performance function

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Fig. 14

PDF of performance function

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