T. Borrvall and J. Petersson [Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77--107] developed the first model for topology optimization of fluids in Stokes flow. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this work, we prove novel regularity results and extend their numerical analysis. In particular, given an isolated local minimizer to the infinite-dimensional problem, we show that there exists a sequence of finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to it. We also provide the first numerical investigation into convergence rates.
翻译:T. Borrvall和J. Petersson[斯托克斯流流流液体的理学优化,《国际流体数字方法杂志》41(1)(2003年)77-107] 开发了斯托克斯流流液体表面学优化的第一个模型,这些模型证明在无限维度环境中存在最小化器,并表明一个适当选择的有限元素方法在弱(-*)意义上会汇合到一个不确定的解决方案。在这项工作中,我们证明了新颖的规律性结果,并扩展了它们的数字分析。特别是,鉴于一个孤立的局部最小化器,我们发现存在一系列有限的元素解决方案,满足了必要的一阶最佳性条件,这些条件与它非常接近。我们还对趋同率进行了首次数字调查。