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Sampling-based weighted reliability-based design optimization

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Abstract

Reliability-based design optimization (RBDO) derives an optimum design that satisfies target reliability and minimizes an objective function by introducing probabilistic constraints that take into account failures caused by uncertainties in random inputs. However, existing RBDO studies treat all failures equally and derive the optimum design without considering the magnitude of failure exceeding the limit state. Since the damage caused by failure varies according to the magnitude of failure, a probabilistic framework that considers the magnitude of failure differently is necessary. Therefore, this study proposes a weighted RBDO (WRBDO) framework that assigns a different weight to each failure according to the magnitude of failure and derives an optimum design that quantitatively reflects the magnitude of failures. In the WRBDO framework, the weight function is modeled based on warranty cost or damage cost according to the magnitude of failure, and the weighted failure is determined by assigning different weights according to the magnitude of failure through the weight function. Then, weighted probabilistic constraints reflecting the weighted failure are evaluated. Sampling-based reliability analysis using the direct Monte Carlo simulation (MCS) is performed to evaluate the weighted probabilistic constraints. Stochastic sensitivity analysis that calculates the sensitivities of the weighted probabilistic constraints is derived, and it is verified through numerical examples that the stochastic sensitivity analysis is more accurate and efficient than the sensitivity analysis using the finite difference method (FDM). To enable the practical application of WRBDO, AK-MCS for WRBDO in which the Kriging model is updated to identify both the limit state and the magnitude of failures in the failure region is proposed. The results of various WRBDO problems show that the WRBDO yields conservative designs than a conventional RBDO, and more conservative designs are derived as the slope of weight functions and the nonlinearity of constraint functions increase. The optimum results of a 6D arm model show that the cost increases by 3.39% and the number of failure samples decreases by 88.48% in WRBDO and the weighted failures of WRBDO are averagely 9.1 times larger than those of RBDO. The results of applying AK-MCS for WRBDO to the 6D arm model verify that the AK-MCS for WRBDO enables practical application of WRBDO with a small number of function evaluations.

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Abbreviations

d :

Design variable vector

X :

Random variable vector

\(P[ \cdot ]\) :

Probability measure

g(X):

Constraint function

\(P_{F}^{{{\text{Target}}}}\) :

Target probability of failure

d L :

Lower design bound

d U :

Upper design bound

P F :

Probability of failure

\({{\varvec{\upmu}}}\) :

Vector of the mean of the random input

\(f_{{\mathbf{X}}} ({\mathbf{x}};\,{{\varvec{\upmu}}})\) :

Joint probability density function of X

\(E[ \cdot ]\) :

Expectation operator

\(\Omega_{F}\) :

Failure set

\(I_{{\Omega_{F} }} ({\mathbf{x}})\) :

Indicator function for the failure set

\(f_{W}\) :

Weight function

\(P_{F,W}\) :

Weighted probability of failure

\(\Omega_{S}\) :

Safe set

\(I_{{\Omega_{S} }} ({\mathbf{x}})\) :

Indicator function for the safe set

\(P_{F,W}^{{{\text{Target}}}}\) :

Target-weighted probability of failure

S :

Monte Carlo population

S F :

Monte Carlo population in the failure region

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Acknowledgements

This research was supported by the KAIST-funded Global Singularity Research Program for 2021 (N11210060).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 34141, South Korea

    Ungki Lee & Ikjin Lee

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  1. Ungki Lee
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  2. Ikjin Lee
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Correspondence to Ikjin Lee.

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Lee, U., Lee, I. Sampling-based weighted reliability-based design optimization. Struct Multidisc Optim 65, 20 (2022). https://doi.org/10.1007/s00158-021-03133-5

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