This work presents an extension of the highly efficient de-homogenization method for obtaining high-resolution, near-optimal 2D topologies optimized for minimum compliance subjected to multiple load cases. We perform a homogenization-based topology optimization based on stiffness optimal Rank-$N$ microstructure parameterizations to obtain stiffness optimal multi-scale designs on relatively coarse grids. To avoid relatively thin microstructure features, we regularize the design by introducing a material indicator field which results in well-defined widths and structural boundaries. In order to avoid singularities from the multiple load case problem, the orientations of the microstructures are further regularized. Subsequently, we derive a single-scale interpretation of stiffness optimal multi-scale designs on a fine grid using de-homogenization. The single-scale interpretation can be derived without costly postprocessing analysis on the fine grid, as an implicit boundary formulation is used.

The effect of starting guesses is discussed, as they are non-trivial for Rank-$N$ microstructures. Different numerical examples verify the performance of the inexpensive high-resolution solutions, both in comparison to the Rank-$N$ based homogenization solutions, to equivalent density-based large-scale solutions, as well as to strict isotropic microstructure solutions. Depending on starting guesses, the approach consistently delivers structural performance values within a few percent of density-based large-scale solutions with a CPU time reduction factor of more than 300. Finally, we confirm that isotropic as well as orthogonal Rank-2 microstructure models are inferior to stiffness optimal anisotropic microstructure models for minimum compliance problems subjected to multiple load cases.

这项工作提出了高效去均质化方法的扩展，用于获得高分辨率、近乎最优的 2D 拓扑，该拓扑针对多个负载情况下的最小顺应性进行了优化。我们基于刚度优化Rank-执行基于同质化的拓扑优化$ñ$微观结构参数化以获得刚度最佳的多尺度设计相对粗糙的网格。为了避免相对较薄的微结构特征，我们通过引入材料指示字段来规范设计，从而产生明确定义的宽度和结构边界。为了避免多工况问题的奇异性，进一步规范了微观结构的方向。随后，我们使用去均匀化在精细网格上推导出刚度最优多尺度设计的单尺度解释。由于使用了隐式边界公式，因此无需对精细网格进行昂贵的后处理分析即可得出单尺度解释。

讨论了开始猜测的影响，因为它们对于 Rank-$ñ$微结构。不同的数值示例验证了廉价高分辨率解决方案的性能，两者都与 Rank-$ñ$基于均匀化的解决方案，基于等效密度的大规模解决方案，以及严格的各向同性微观结构解决方案。根据最初的猜测，该方法始终如一地在基于密度的大规模解决方案的百分之几内提供结构性能值，CPU 时间减少因子超过 300。最后，我们确认各向同性以及正交 Rank-2 微观结构模型对于在多个载荷情况下的最小柔顺性问题，刚度最佳各向异性微观结构模型是次要的。