In traditional topology optimization, the computing time required to iteratively update the material distribution within a design domain strongly depends on the complexity or size of the problem, limiting its application in real engineering contexts. This work proposes a multi-stage machine learning strategy that aims to predict an optimal topology and the related stress fields of interest, either in 2D or 3D, without resorting to any iterative analysis and design process. The overall topology optimization is treated as regression task in a low-dimensional latent space, that encodes the variability of the target designs. First, a fully-connected model is employed to surrogate the functional link between the parametric input space characterizing the design problem and the latent space representation of the corresponding optimal topology. The decoder branch of an autoencoder is then exploited to reconstruct the desired optimal topology from its latent representation. The deep learning models are trained on a dataset generated through a standard method of topology optimization implementing the solid isotropic material with penalization, for varying boundary and loading conditions. The underlying hypothesis behind the proposed strategy is that optimal topologies share enough common patterns to be compressed into small latent space representations without significant information loss. Results relevant to a 2D Messerschmitt-B\"olkow-Blohm beam and a 3D bridge case demonstrate the capabilities of the proposed framework to provide accurate optimal topology predictions in a fraction of a second.

翻译：在传统拓扑优化中，迭代更新设计域内材料分布所需的计算时间高度依赖于问题的复杂度或规模，这限制了其在实际工程中的应用。本文提出了一种多阶段机器学习策略，旨在无需任何迭代分析与设计过程的情况下，预测二维或三维空间中的最优拓扑及其相关应力场。整个拓扑优化被视作低维潜在空间中的回归任务，该潜在空间编码了目标设计的可变性。首先，采用全连接模型来替代表征设计问题的参数输入空间与相应最优拓扑的潜在空间表示之间的函数映射关系。随后，利用自编码器的解码器分支从潜在表示中重构期望的最优拓扑。深度学习模型基于通过固体各向同性材料惩罚（SIMP）标准拓扑优化方法生成的数据集进行训练，该数据集包含不同的边界条件和载荷条件。该策略的基本假设是：最优拓扑共享足够多的共同模式，从而可被压缩为小维度潜在空间表示，且无明显信息损失。针对二维梅塞施米特-伯尔科-布洛姆（MBB）梁和三维桥梁算例的数值结果表明，所提框架能够在亚秒级时间内提供精确的最优拓扑预测。