Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization. The goal of our two-part work is to extend lattice optimization to stress minimization problems two-dimensionally. The present first part is devoted to the choice of a parametrized periodicity cell that will be used for structural optimization in the second part of our work. In order to avoid stress concentration, we propose a square cell microstructure with a super-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimization. This type of cell is parametrized two-dimensionally by one orientation angle, two semi-axis and a corner smoothing parameter. We first analyse their influence on the stress amplification factor by performing some numerical experiments. Second, we compute the optimal corner smoothing parameter for each possible microstructure and macroscopic stress. Then, we average (with specific weights) the optimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz microstructure which is known to be optimal for stress minimization in some special cases.

This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

晶格结构是周期性的多孔体，由于它们是刚性和重量之间的良好折衷，并且可以通过增材制造技术建造，因此逐渐变得流行。基于均质化方法，它们的优化最近引起了一些关注，主要用于最小化合规性。我们两部分工作的目标是将晶格优化扩展到二维应力最小化问题。当前的第一部分致力于选择参数化的周期性单元，该单元将在我们工作的第二部分中用于结构优化。为了避免应力集中，我们提出了具有超椭圆形孔的方形单元微观结构，而不是经常用于最小化柔量的标准矩形孔。通过一个定向角，两个半轴和一个角平滑参数对二维类型的单元进行二维参数化。我们首先通过执行一些数值实验来分析它们对应力放大因子的影响。其次，我们为每种可能的微观结构和宏观应力计算最佳的角平滑参数。然后，针对宏观应力，我们平均（使用特定权重）最佳平滑指数。最后，为了验证结果，我们将最佳的超椭圆形孔与Vigdergauz微结构进行了比较，该结构在某些特殊情况下对于最小化应力是最佳的。我们为每种可能的微观结构和宏观应力计算最佳的角平滑参数。然后，针对宏观应力，我们平均（使用特定权重）最佳平滑指数。最后，为了验证结果，我们将最佳的超椭圆形孔与Vigdergauz微结构进行了比较，该结构在某些特殊情况下对于最小化应力是最佳的。我们为每种可能的微观结构和宏观应力计算最佳的角平滑参数。然后，针对宏观应力，我们平均（使用特定权重）最佳平滑指数。最后，为了验证结果，我们将最佳的超椭圆形孔与Vigdergauz微结构进行了比较，该结构在某些特殊情况下对于最小化应力是最佳的。

本文是主题“复杂材料的数学设计主题”的一部分。