We study the finite element approximation of the solid isotropic material with penalization (SIMP) model for the topology optimization of the compliance of a linearly elastic structure. To ensure the existence of a minimizer to the infinite-dimensional problem, we consider two popular restriction methods: $W^{1,p}$-type regularization and density filtering. Previous results prove weak(-*) convergence in the solution space of the material distribution to an unspecified minimizer of the infinite-dimensional problem. In this work, we show that, for every isolated local or global minimizer, there exists a sequence of finite element minimizers that strongly converges to the minimizer in the solution space. As a by-product, this ensures that there exists a sequence of unfiltered discretized material distributions that does not exhibit checkerboarding.
翻译:我们研究了固态异地材料的有限元素近似值,以惩罚性模型优化线性弹性结构的合规性。为了确保存在一个最小化器,我们考虑两种流行的限制方法:$W+1,p}美元类型规范化和密度过滤。以往的结果证明,材料分配的解决方案空间(-*)向未具体说明的最小度问题最小化器的融合程度不强。在这项工作中,我们显示,对于每一个孤立的局部或全球最小化器来说,都有一系列有限元素最小化器,它们与溶液空间的最小化器高度趋同。作为副产品,这确保了存在一系列非过滤离散材料分布的序列,而没有显示检查盘。