Much work has been done in topology optimization of multiscale structures for maximum stiffness or minimum compliance design. Such approaches date back to the original homogenization-based work by Bends{\o}e and Kikuchi from 1988, which lately has been revived due to advances in manufacturing methods like additive manufacturing. Orthotropic microstructures locally oriented in principal stress directions provide for highly efficient stiffness optimal designs, whereas for the pure stiffness objective, porous isotropic microstructures are sub-optimal and hence not useful. It has, however, been postulated and exemplified that isotropic microstructures (infill) may enhance structural buckling stability but this has yet to be directly proven and optimized. In this work, we optimize buckling stability of multiscale structures with isotropic porous infill. To do this, we establish local density dependent Willam-Warnke yield surfaces based on local buckling estimates from Bloch-Floquet-based cell analysis to predict local instability of the homogenized materials. These local buckling-based stress constraints are combined with a global buckling criterion to obtain topology optimized designs that take both local and global buckling stability into account. De-homogenized structures with small and large cell sizes confirm validity of the approach and demonstrate huge structural gains as well as time savings compared to standard singlescale approaches.

翻译：关于多尺度结构拓扑优化以实现最大刚度或最小柔顺性设计的研究已有大量工作。此类方法可追溯到1988年Bendsøe与Kikuchi基于均匀化的开创性研究，近年来因增材制造等制造技术的进步而复兴。沿主应力方向局部定向的正交各向异性微结构能够实现高效的刚度最优设计，而对于纯刚度目标，多孔各向同性微结构并非最优，因此不具实用性。然而，已有假设和实例表明，各向同性微结构（填充物）可能增强结构屈曲稳定性，但这尚未得到直接证明和优化。本文针对含各向同性多孔填充物的多尺度结构，优化其屈曲稳定性。为此，我们基于Bloch-Floquet单元分析得到的局部屈曲估计，建立与局部密度相关的Willam-Warnke屈服面，以预测均匀化材料的局部失稳。将这些基于局部屈曲的应力约束与全局屈曲准则相结合，得到同时考虑局部与全局屈曲稳定性的拓扑优化设计。具有小尺寸与大尺寸单元的退均匀化结构验证了该方法的有效性，并展示了相较于标准单尺度方法的巨大结构性能提升及时间节省。