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 引用本文: 王选, 时元昆, 杨博, 程长征, 龙凯. 基于响应面方法的破损−安全结构可靠性拓扑优化. 力学学报, 2023, 55(5): 1206-1216.
Wang Xuan, Shi Yuankun, Yang Bo, Cheng Changzheng, Long Kai. Reliability-based topology optimization of fail-safe structures using response surface method. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(5): 1206-1216.
 Citation: Wang Xuan, Shi Yuankun, Yang Bo, Cheng Changzheng, Long Kai. Reliability-based topology optimization of fail-safe structures using response surface method. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(5): 1206-1216.

## RELIABILITY-BASED TOPOLOGY OPTIMIZATION OF FAIL-SAFE STRUCTURES USING RESPONSE SURFACE METHOD

• 摘要: 传统结构由于缺少冗余, 忽略了不确定性因素的影响, 更容易受到局部刚度损失的影响, 文章针对载荷不确定性下破损-安全结构的设计问题提出了一种有效的基于响应面的可靠性拓扑优化方法, 以提高结构的安全性, 确保结构在发生局部破损时仍能满足服役性能及可靠性要求. 为此, 建立了柔度概率约束下的结构体积比最小化的双循环可靠性拓扑优化模型, 其中内层循环实施可靠性分析, 外层循环实施拓扑优化. 为了有效处理可靠性分析中响应函数关于随机变量的导数计算高成本问题, 基于响应面方法建立了响应函数关于随机变量的显式表达式. 详细推导了响应函数关于设计变量和随机变量的解析灵敏度列式, 并采用移动渐近线方法(method of moving asymptotes, MMA)对优化问题进行求解. 将基于响应面的可靠性拓扑优化方法与基于解析导数的方法作对比, 并实施蒙特卡洛仿真验证了所提方法的有效性和优越性, 讨论了随机载荷标准差对优化结果的影响. 结果表明, 本文方法可以有效设计满足指定可靠性水平的破损-安全结构, 优化后结构可靠性指标的相对误差不超过1.3%, 另外基于响应面的可靠性设计方法相对于基于解析导数的可靠性设计方法可节省约74%的可靠性分析时间.

Abstract: Traditional structures are more susceptible to local stiffness loss due to the lack of redundancy and problem of ignoring the influence of uncertain factors. This paper proposes an effective reliability-based topology optimization methodology for design problem of fail-safe structures under load uncertainty basing on response surface method, to improve the structural safety and ensure that the structure can still meet the service performance and reliability requirements even when local damage occurs. To this end, a double-loop reliability-based topology optimization model of minimizing the volume fraction while satisfying the probabilistic compliance constraint is established, in which the inner loop is used for reliability analysis and the outer loop is used for topology optimization. To solve the problem of high calculation cost of the derivative of response function with respect to random variables in reliability analysis, an explicit expression of response function with respect to random variables was established based on response surface method. The analytic sensitivity formulations of the response function with respect to design variables and random variables are deduced in detail, and the method of moving asymptotes (MMA) is used to solve the optimization problem. The response surface-based reliability design method is compared with the method based on analytic derivative, and Monte Carlo simulation is also carried out to verify the effectiveness and superiority of the proposed method, discussing the influence of standard deviation of random load on the optimization results. The optimization results show that the proposed method can effectively design the fail-safe structures that meets the specified reliability level, and the relative error of the reliability index of the optimized structures does not exceed 1.3%. In addition, the response surface-based reliability design method can save about 74% of the reliability analysis time compared with the reliability design method based on analytical derivative.

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