Abstract It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: (1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; (2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; (3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Results are computed for two reference problems: similarities and differences between optimized topologies obtained with the three formulations are exploited and discussed.

中文翻译：

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不确定载荷和应力约束下稳健、基于可靠性和非概率拓扑优化的比较

摘要 现在人们普遍认为，优化结构设计应该在载荷和材料参数的不确定性方面具有鲁棒性。然而，在结构优化问题中考虑这种不确定性有多种选择。本文综合比较了三种不同方法在不确定载荷下拓扑优化的结果，考虑了应力约束：(1) 稳健公式，它只需要每个单元应力的平均值和标准偏差；(2) 基于可靠性的公式，它对计算应力施加可靠性约束；(3) 非概率公式，它考虑了由不确定载荷引起的应力的最坏情况。每种方法所需的关于不确定载荷的信息，并且在每种情况下使用的不确定性传播方法完全不同。稳健公式只需要不确定载荷的均值和标准差；应力是通过一阶微扰方法计算的。基于可靠性的公式需要随机载荷的全概率分布，可靠性约束是通过一阶性能测量方法计算的。非概率公式适用于有界不确定载荷；只使用下限和上限，最坏情况的应力是通过带有反优化的嵌套优化计算的。这三种方法在处理不确定性方面有很大不同；然而，基本的拓扑优化框架是相同的：采用传统的密度方法进行材料参数化，同时采用增广拉格朗日方法来解决由此产生的问题，以处理大量的应力约束。计算了两个参考问题的结果：利用和讨论了使用三种公式获得的优化拓扑之间的异同。