Research Papers

Intricate Interrelation Between Robustness and Probability in the Context of Structural Optimization

[+] Author and Article Information
Oded Amir

Faculty of Civil and Environmental Engineering, Technion—Israel Institute of Technology, Haifa 3200003, Israel

Isaac Elishakoff

Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431

Manuscript received September 23, 2014; final manuscript received January 21, 2015; published online July 1, 2015. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 1(3), 031003 (Jul 01, 2015) (7 pages) Paper No: RISK-14-1059; doi: 10.1115/1.4030456 History: Received September 23, 2014; Accepted April 27, 2015; Online July 01, 2015

In this study, we deal with the problem of structural optimization under uncertainty. In previous studies, either of three philosophies were adopted: (a) probabilistic methodology, (b) fuzzy-sets-based design, or (c) nonprobabilistic approach in the form of given bounds of variation of uncertain quantities. In these works, authors are postulating knowledge of either involved probability densities, membership functions, or bounds in the form of boxes or ellipsoids, where the uncertainty is assumed to vary. Here, we consider the problem in its apparently pristine setting, when the initial raw data are available and the uncertainty model in the form of bounds must be constructed. We treat the often-encountered case when scarce data are available and the unknown-but-bounded uncertainty is dealt with. We show that the probability concepts ought to be invoked for predicting the worst- and best-possible designs. The Chebyshev inequality, applied to the raw data, is superimposed with the study of the robustness of the associated deterministic optimal design. We demonstrate that there is an intricate relationship between robustness and probability.

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Fig. 1

Problem of a two-segment cantilever, following Christensen and Klarbring [14]. Design variables are the side lengths y1 and y2, and the uncertain parameter is L2.

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Fig. 2

Minimum weight of a two-segment cantilever based on limited data and Chebyshev’s inequality

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Fig. 3

Hypothesized performance function f(α)=1α+0.01+110.01−α; “robust” performance is obtained for the raw data points (diamonds) but not for the bounds corresponding to the inclusion of 90% of the data according to Chebyshev’s inequality (circles)

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Fig. 4

Results of robust topology optimization of a compliant mechanism; a worst-case scenario was considered for 0.3≤η≤0.7. Robust performance is achieved for the specified bounded range, but it is lost once the parameter η exceeds the upper bound. The numerical results are based on Amir et al. [19].

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Fig. 5

von Mises truss problem; the design variable is the (uniform) cross-sectional area A and the uncertain parameter is the height a

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Fig. 6

Force–displacement responses of a von Mises truss for several deviations from the mean value of the height a

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Fig. 7

Minimum weight of a von Mises truss based on limited data and Chebyshev’s inequality

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Fig. 8

Three-bar truss problem; the design variables are the cross-sectional areas x1 and x2, and the uncertain parameter is the imperfection factor α

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Fig. 9

Graphical solution of the three-bar truss under stress constraints only. At optimum, the active constraint is related to the tensile stress in bar #3

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Fig. 10

Graphical solution of the three-bar truss under buckling constraints. At optimum, the active constraints are related to the buckling in bar #1 and to the tensile stress in bar #3. It can be seen that the value of the objective function (linear contours) does not rise sharply when considering high immunity against uncertainty in the form of 2Q and 3.5Q designs.



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