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Conservative reliability-based design optimization method with insufficient input data

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Abstract

Reliability analysis and reliability-based design optimization (RBDO) require an exact input probabilistic model to obtain accurate probability of failure (PoF) and RBDO optimum design. However, often only limited input data is available to generate the input probabilistic model in practical engineering problems. The insufficient input data induces uncertainty in the input probabilistic model, and this uncertainty forces the PoF to be uncertain. Therefore, it is necessary to consider the PoF to follow a probability distribution. In this paper, the probability of the PoF is obtained with consecutive conditional probabilities of input distribution types and parameters using the Bayesian approach. The approximate conditional probabilities are obtained under reasonable assumptions, and Monte Carlo simulation is applied to calculate the probability of the PoF. The probability of the PoF at a user-specified target PoF is defined as the conservativeness level of the PoF. The conservativeness level, in addition to the target PoF, will be used as a probabilistic constraint in an RBDO process to obtain a conservative optimum design, for limited input data. Thus, the design sensitivity of the conservativeness level is derived to support an efficient optimization process. Using numerical examples, it is demonstrated that the conservativeness level should be involved in RBDO when input data is limited. The accuracy and efficiency of the proposed design sensitivity method is verified. Finally, conservative RBDO optimum designs are obtained using the developed methods for limited input data problems.

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Abbreviations

CDF:

Cumulative distribution function

PDF:

Probability density function

PoF, p F :

Probability of failure

G(x):

Performance measure

Ω F :

Failure domain of a performance measure

X, x :

Input random variable vector and its realization

N :

Number (dimension) of input random variables

X i :

i-th input random variable

f X (x; ζ, ψ):

Joint PDF of X

\( {f}_{X_i}\left({x}_i;{\zeta}_i,{\mu}_i,{\sigma}_i^2\right) \) :

Marginal PDF of X i

Z, ζ :

Input distribution types and their realizations

ζ i :

Marginal distribution type of X i

Ψ, ψ :

Input distribution parameters and their realizations

M i , μ i :

Input mean of \( {X}_i \) and its realization

Σ 2 i , σ 2 i :

Input variance of \( {X}_i \) and its realization

*x :

Input data set, *x = {*x 1, …, *x N }

*x (j) i , *x i :

Input data and data set for X i

ND :

Number of input data

\( {}^{*}{\overline{x}}_i{,}^{*}{\overline{\mathbf{x}}}_i \) :

Mean of input data and its vector form

\( {}^{*}{\tilde{x}}_i{,}^{*}{\tilde{\mathbf{x}}}_i \) :

Dispersion of input data and its vector form

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Acknowledgments

Research was supported by the Automotive Research Center (ARC) in accordance with Cooperative Agreement W56HZV-04-2-0001 U.S. Army Tank Automotive Research, Development and Engineering Center (TARDEC). This research was partially supported by high-performance computer time and resources from the DOD High Performance Computing Modernization Program, and the Technology Innovation Program (10048305, Launching Plug-in Digital Analysis Framework for Modular System Design) funded by the Ministry of Trade, Industry & Energy (MI, Korea). These supports are greatly appreciated.

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA, 52242, USA

    Hyunkyoo Cho & K. K. Choi

  2. RAMDO Solutions, LLC, Iowa City, IA, 52240, USA

    Nicholas J. Gaul

  3. Mechanical Engineering Department, Korea Institute of Science and Technology (KAIST), Daejeon, Republic of Korea

    Ikjin Lee

  4. US Army RDECOM/TARDEC, Warren, MI, 48397-5000, USA

    David Lamb & David Gorsich

Authors
  1. Hyunkyoo Cho
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  2. K. K. Choi
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  3. Nicholas J. Gaul
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  4. Ikjin Lee
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  5. David Lamb
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  6. David Gorsich
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Corresponding author

Correspondence to K. K. Choi.

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Cho, H., Choi, K.K., Gaul, N.J. et al. Conservative reliability-based design optimization method with insufficient input data. Struct Multidisc Optim 54, 1609–1630 (2016). https://doi.org/10.1007/s00158-016-1492-4

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