Threshold shift method for reliability-based design optimization

Abstract

We present a novel approach, referred to as the “threshold shift method” (TSM), for reliability-based design optimization (RBDO). The proposed approach is similar in spirit with the sequential optimization and reliability analysis (SORA) method where the RBDO problem is decoupled into an optimization and a reliability analysis problem. However, unlike SORA that utilizes shift vector to shift the design variables within a constraint (independently), in TSM, we propose to shift the threshold of the constraints. We argue that modifying a constraint, either by shifting the design variables (SORA) or by shifting the threshold of the constraints (TSM), influences the other constraints of the system. Therefore, we propose to determine the thresholds for all the constraints by solving a single optimization problem. Additionally, the proposed TSM is equipped with an active-constraint determination scheme. To make the method scalable, a practical algorithm for TSM that utilizes two surrogate models is proposed. Unlike the conventional RBDO methods, the proposed approach has the ability to handle highly non-linear probabilistic constraints. The performance of the proposed approach is examined on six benchmark problems selected from the literature. The proposed approach yields excellent results outperforming other popular methods in literature. As for the computational efficiency, the proposed approach is found to be highly efficient, indicating it’s future application to other real–life problems.

Appendix: PC-Kriging

PC-Kriging is novel surrogate model that combines polynomial chaos expansion (PCE) (Pascual and Adhikari 2012; Xiu and Karniadakis 2002; Sudret 2008; Hu and Youn 2010) with Kriging (Bilionis et al. 2013; Biswas et al. 2018; Biswas et al. 2017; Mukhopadhyay et al. 2016). To be specific, PCE is used to model the mean function within the Kriging surrogate. This allows one to surrogate a computational model more accurately than PCE or Kriging taken separately.

In PC-Kriging, the trend function, for the P th order (P > 0) PCE of the function, is mathematically presented as:

$$ \mathcal{M}(\boldsymbol{x}) \approx \mathcal{M}_{P}^{(PCK)}(\boldsymbol{x}) = \underbrace {\sum\limits_{\left| \mathbf i = 0 \right|}^{P} \gamma_{\mathbf i} \psi_{\mathbf i}}_{PCE} + \underbrace{{\sigma^{2}}Z(\boldsymbol{x},\omega )}_{Kriging}, $$

(24)

where σ² and Z(x,ω) are the Gaussian process variance and the zero-mean, unit variance stationary Gaussian process, respectively, and ω ∈Ω denotes an elementary event in the probability space $(\Omega ,\mathcal {F},\mathbb {P})$. In (24), γ_i represent the unknown coefficients and ψ_i represents the orthogonal polynomial of degree i. Under limiting conditions, the expression of PC-Kriging in (24) converges to either PCE or Kriging.

Despite the superiority of PC-Kriging over PCE and Kriging, it suffers from the curse of dimensionality. As the number of input variables N increases, the number of of unknown coefficients to be estimated grows factorially. This limitation stems from the PCE part of PC-Kriging. To address this issue, an optimal PCE-Kriging (OPC-Kriging) was proposed, which utilizes the least angle regression–based model selection technique to retain only the important PCE components in the PC-Kriging model. To be specific, the LAR algorithm results in a list of ranked polynomials which are chosen depending on their correlation to the current residual at each iteration in decreasing order. OPC-Kriging consists of an iterative algorithm where each polynomial is added one-by-one to the trend part. A flowchart depicting the steps involved in OPC-Kriging is shown in Fig. 8. For further details on OPC-Kriging, interested readers may refer (Schobi et al. 2015) and (Kersaudy et al. 2015).

Fig. 8

Flowchart depicting the steps involved in OPC-Kriging