This paper presents a consistent topology optimization formulation for mass minimization with local stress constraints by means of the augmented Lagrangian method. To solve problems with a large number of constraints in an effective way, we modify both the penalty and objective function terms of the augmented Lagrangian function. The modification of the penalty term leads to consistent solutions under mesh refinement and that of the objective function term drives the mass minimization towards black and white solutions. In addition, we introduce a piecewise vanishing constraint, which leads to results that outperform those obtained using relaxed stress constraints. Although maintaining the local nature of stress requires a large number of stress constraints, the formulation presented here requires only one adjoint vector, which results in an efficient sensitivity evaluation. Several 2D and 3D topology optimization problems, each with a large number of local stress constraints, are provided.

本文提出了一种一致的拓扑优化公式，用于通过增强拉格朗日方法对具有局部应力约束的质量进行最小化。为了有效地解决具有大量约束的问题，我们修改了增强拉格朗日函数的惩罚和目标函数项。惩罚项的修改导致网格细化下的一致解，而目标函数项的修改则将质量最小化推向黑白解决方案。此外，我们引入了分段消失约束，其结果要优于使用松弛应力约束获得的结果。尽管维持应力的局部性质需要大量的应力约束，但此处介绍的公式仅需要一个伴随向量，这样可以进行有效的灵敏度评估。提供了几个2D和3D拓扑优化问题，每个问题都有大量的局部应力约束。