## Abstract

This article brings forward a new computational method for reliability-based design optimization (RBDO) of complex mechanical systems subject to input random variables following arbitrary, dependent probability distributions. It involves a generalized polynomial chaos expansion (GPCE) for reliability analysis subject to dependent input random variables, a novel fusion of the GPCE approximation and score functions for estimating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point single-step design process. The method, designated as the multi-point single-step GPCE method or simply the MPSS-GPCE method, yields analytical formulae for computing the failure probability and its design sensitivities concurrently from a single stochastic simulation or analysis. For this reason, the MPSS-GPCE method affords the ability to solve industrial-scale problems with large design spaces. Numerical results stemming from mathematical functions or elementary engineering problems indicate that the new method provides more accurate or computationally efficient design solutions than existing methods or reference solutions. Furthermore, the shape design optimization of a jet engine compressor blade root was successfully conducted, demonstrating the power of the new method in confronting practical RBDO problems.

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## Notes

Here, the symbol \(\sim\) represents equality in a weaker sense, such as equality in mean square, but not necessarily pointwise, nor almost everywhere.

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## Acknowledgements

The authors acknowledge the financial support from the US National Science Foundation under Grant No. CMMI-1933114.

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### Replication of results

The results for Examples 1â€“4 provided in this paper were generated by MATLAB codes developed by the authors. In Example 4, the CAD model of the blade/disk assembly was created by CREO parametric. Then, the FE model was built and solved by ABAQUS/CAE. Finally, the fatigue crack initiation life was analyzed using an in-house MATLAB code. The CAD and FEA processes were fully integrated with the proposed RBDO algorithm for design automation by using MACRO functions and batch files in the MATLAB platform. The interested reader is encouraged to contact the corresponding author for further implementation details by e-mail.

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## Appendices

### Appendix 1: Statistical first-moment

Consider the *m*th-order GPCE approximation \(h_m(\mathbf {Z};\mathbf {r})\) of \(h(\mathbf {Z};\mathbf {r})\), presented in (12). Applying the expectation operator on \(h_m(\mathbf {Z};\mathbf {r})\) and recognizing (7), its mean

coincides with the exact mean of \(h(\mathbf {Z};\mathbf {r})\) for any \(m\in \mathbb{N}_0\).

### Appendix 2: Sensitivities of the first-moment

For \(k=1,\ldots ,M\), consider the *k*th first-order score function \(s_k(\mathbf {Z};\mathbf {g})\) in (17). Then, applying the full GPCE to \(s_k(\mathbf {Z};\mathbf {g})\) leads to

with its GPCE coefficients

Similarly, as shown in (16), by interchanging differential and integral operators and by replacing \(h(\mathbf {Z};\mathbf {r})\) and \(s_k(\mathbf {Z};\mathbf {g})\) with their *m*th-order and \(m'\)th-order GPCE approximations, respectively, for \(m,m'\in \mathbb{N}_0\), the sensitivity of the first-moment with respect to \(d_k\) is formulated as follows:

where \(L_{\min }:=\min (L_{N,m},L_{N,m'})\). The approximate sensitivity in (29) converges to \({\partial \mathbb{E}_{\mathbf {g}}[h(\mathbf {Z};\mathbf {r})]}\mathbin {/}{\partial d_k}\) when \(m \rightarrow \infty\) and \(m' \rightarrow \infty\).

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Lee, D., Rahman, S. Reliability-based design optimization under dependent random variables by a generalized polynomial chaos expansion.
*Struct Multidisc Optim* **65**, 21 (2022). https://doi.org/10.1007/s00158-021-03123-7

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DOI: https://doi.org/10.1007/s00158-021-03123-7