用相场方法计算Navier-Stokes流中表面泛函的形状optimization
Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach
摘要：
我们考虑流体的形状和拓扑优化由Navier-斯托克斯方程。利用相场方法对固体进行了建模，并将固体松弛为多孔介质。相场法使用Ginzburg-Landau泛函来近似周长惩罚。我们专注于表面泛函，并仔细介绍了一个新的建模变量，显示存在的极小，并推导出一阶必要条件。这些条件与经典的形状衍生物通过识别尖锐的界面极限的帮助下，正式匹配的渐近展开。最后，我们提出了数值计算的基础上的Cahn-Hilliard型梯度下降，这表明该方法可以用来解决形状最优化问题的流体与新的方法的帮助。
英文摘要：
We consider shape and topology optimization for fluids which are governed by the Navier--Stokes equations. Shapes are modelled with the help of a phase field approach and the solid body is relaxed to be a porous medium. The phase field method uses a Ginzburg--Landau functional in order to approximate a perimeter penalization. We focus on surface functionals and carefully introduce a new modelling variant, show existence of minimizers and derive first order necessary conditions. These conditions are related to classical shape derivatives by identifying the sharp interface limit with the help of formally matched asymptotic expansions. Finally, we present numerical computations based on a Cahn--Hilliard type gradient descent which demonstrate that the method can be used to solve shape optimization problems for fluids with the help of the new approach.
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