Reliability-based design optimization under dependent random variables by a generalized polynomial chaos expansion

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.

Appendices

Appendix 1: Statistical first-moment

Consider the mth-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

$$\begin{aligned} \mathbb{E}_{\mathbf {g}}[h_m(\mathbf {Z};\mathbf {r})] =C_1(\mathbf {r})=\mathbb{E}_{\mathbf {g}}[h(\mathbf {Z};\mathbf {r})] \end{aligned}$$

(28)

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 kth 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

$$\begin{aligned} s_{k}(\mathbf {Z};\mathbf {g})=\displaystyle \sum _{i=2}^{\infty }D_{k,i}(\mathbf {g})\varPsi _i(\mathbf {Z};\mathbf {g}), \end{aligned}$$

with its GPCE coefficients

$$\begin{aligned} D_{k,i}(\mathbf {g}) = \int _{\mathbb{\bar{A}}^N}s_{k}(\mathbf {z};\mathbf {g})\varPsi _i(\mathbf {z};\mathbf {g}) f_{\mathbf {Z}}(\mathbf {z};\mathbf {g}) d\mathbf {z},~i=2,3,\ldots ,\infty . \end{aligned}$$

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 mth-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:

$$\begin{aligned} \displaystyle \frac{\partial \mathbb{E}_{\mathbf {g}}[h_m(\mathbf {Z};\mathbf {r})]}{\partial d_k}= \displaystyle \frac{\partial g_k}{\partial d_k} \displaystyle \sum _{i=2}^{L_{\min }} C_i(\mathbf {r}) D_{k,i}(\mathbf {g}), \end{aligned}$$

(29)

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 $m' \rightarrow \infty$.