Abstract In this paper, a new computational framework based on the topology derivative concept is presented for evaluating stochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with high dimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculate topology sensitivities of the first three stochastic moments which are often required in robust topology optimization (RTO). On another hand, it offers embedded Monte Carlo Simulation (MCS) and finite difference formulations to estimate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases, the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochastic analysis. Moreover, an original example of two random variables is developed for the first time to obtain analytical solutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructed for analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity of failure probabilities in order to verify the accuracy and efficiency of the proposed method for high-dimensional scenarios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivities of existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achieves better accuracies for stochastic topology sensitivities than for the stochastic quantities themselves.

中文翻译：

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一种基于拓扑导数的多项式维数分解框架，用于高维复杂系统的随机拓扑敏感性分析和一类基准问题

摘要 本文提出了一种基于拓扑导数概念的新计算框架，用于评估复杂系统的随机拓扑敏感性。所提出的框架旨在处理高维随机输入，与多元随机响应函数和确定性拓扑导数的多项式维数分解 (PDD) 相吻合。一方面，它提供了分析表达式来计算前三个随机矩的拓扑敏感性，这在稳健拓扑优化 (RTO) 中通常是必需的。另一方面，它提供嵌入式蒙特卡罗模拟 (MCS) 和有限差分公式，以估计基于可靠性的拓扑优化 (RBTO) 的故障概率的拓扑敏感性。对于这两种情况，不确定性的量化及其拓扑敏感性是通过单个随机分析同时确定的。此外，首次开发了两个随机变量的原始示例，以获得矩的拓扑敏感性和失效概率的解析解。为了验证所提方法在高维场景下的准确性和有效性，构建了另一个53维实例，分别构建了矩拓扑敏感性解析解和失效概率拓扑敏感性半解析解。这些例子是新的，使研究人员可以对现有算法或新算法的随机拓扑敏感性进行基准测试。此外，