We study the finite element approximation of the solid isotropic material with penalization (SIMP) model for the topology optimization problem of minimizing the compliance of a linearly elastic structure. To ensure the existence of a local minimizer to the infinite-dimensional problem, we consider two popular regularization methods: $W^{1,p}$-type penalty methods and density filtering. Previous results prove weak(-*) convergence in the space of the material distribution to a local minimizer of the infinite-dimensional problem. Notably, convergence was not guaranteed to \emph{all} the isolated local minimizers. In this work, we show that, for every isolated local or global minimizer, there exists a sequence of finite element local minimizers that strongly converges to the minimizer in the appropriate space. As a by-product, this ensures that there exists a sequence of unfiltered discretized material distributions that does not exhibit checkerboarding.

翻译：我们研究了固体各向同性材料惩罚（SIMP）模型在线性弹性结构柔顺度最小化拓扑优化问题中的有限元逼近。为确保无穷维问题局部极小值点的存在性，我们考虑了两种常用正则化方法：$W^{1,p}$型惩罚方法和密度滤波。先前的结果证明了材料分布在空间中的弱(-*)收敛于无穷维问题的局部极小值点。值得注意的是，收敛性并不能保证对\emph{所有}孤立局部极小值点成立。在本文中，我们证明：对每一个孤立局部或全局极小值点，存在一列有限元局部极小值点在相应空间中强收敛于该极小值点。作为推论，这保证了存在一列无棋盘格现象的未滤波离散材料分布。