Design optimization considering the presence of uncertainties in parameters poses an extremely challenging problem. The source of difficulties comes with reliability-based formulations, where addressing the probabilistic problem exhausts the large computing efforts for failure estimations of the structure violating limit-state functions (LSFs). This paper proposes a novel decoupled approach for effectively solving reliability-based design optimization (RBDO) problems, namely an invertible cross-entropy (iCE) method advantageously combined with a Gaussian process regression (GPR) model, termed as iCE-GPR. The GPR model is applied to approximate the spectrum of LSFs under random parameters. Furthermore, to enhance the accurate prediction of the system failure probability, an active learning process is applied to systematically refine the GPR model by adding new learning points in the region with the largest uncertainty and high-reliability sensitivity through the maximization of an expected feasibility function (EFF). Based on the updated GPR model, the failure probability is estimated by a cost-effective cross-entropy (CE) method without any calls to the actual performance function. To perform the decoupling optimization process with the reliability analysis, the novel iCE, based on the CE method, is developed to update the most probable point (MPP) assigned for the next deterministic optimization process in determining the new optimal design. The method iteratively performs the deterministic optimization process based on the MPP underpinning LSFs sequentially updated by the active learning process. The proposed iCE-GPR method fast-converges the optimal design and significantly alleviates computational burdens associated with reliability analyses. The proposed method is also applied to solve a reliability-based topology optimization (RBTO) problem. Four numerical examples for both the RBDO and RBTO problems are provided to illustrate efficiency and robustness of the proposed iCE-GPR method.

中文翻译：

---

一种新颖的解耦方法，将可逆交叉熵方法与高斯过程建模相结合，用于基于可靠性的设计和拓扑优化

考虑到参数不确定性的存在，设计优化提出了一个极具挑战性的问题。困难的根源在于基于可靠性的公式，其中解决概率问题耗尽了对违反极限状态函数 (LSF) 的结构的故障估计的大量计算工作。本文提出了一种有效解决基于可靠性的设计优化（RBDO）问题的新型解耦方法，即可逆交叉熵（iCE）方法，与高斯过程回归（GPR）模型有利地结合，称为iCE-GPR。 GPR 模型用于近似随机参数下 LSF 的频谱。此外，为了增强对系统故障概率的准确预测，采用主动学习过程，通过最大化预期可行性函数，在不确定性最大和高可靠性灵敏度的区域添加新的学习点，系统地细化探地雷达模型（电子前线）。基于更新的探地雷达模型，通过具有成本效益的交叉熵（CE）方法来估计故障概率，而无需调用实际性能函数。为了通过可靠性分析执行解耦优化过程，开发了基于CE方法的新型iCE，以更新为下一个确定性优化过程分配的最可能点（MPP），以确定新的最优设计。该方法基于主动学习过程顺序更新的 MPP 支撑 LSF 迭代执行确定性优化过程。所提出的 iCE-GPR 方法可以快速收敛最优设计，并显着减轻与可靠性分析相关的计算负担。所提出的方法还应用于解决基于可靠性的拓扑优化（RBTO）问题。提供了 RBDO 和 RBTO 问题的四个数值示例，以说明所提出的 iCE-GPR 方法的效率和鲁棒性。