Shape and Topology Optimization in Stokes Flow with a Phase Field Approach

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• 作者：Harald Garcke ; Claudia Hecht
• 关键词：Shape and topology optimization ; Phase field method ; Diffuse interfaces ; Stokes flow ; Fictitious domain ; 35R35 ; 35Q35 ; 49Q10 ; 49Q20 ; 76D07
• 刊名：Applied Mathematics and Optimization
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：73
• 期：1
• 页码：23-70
• 全文大小：803 KB
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• 作者单位：Harald Garcke (1)
  Claudia Hecht (1)
  
  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Systems Theory and Control
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  
• 出版者：Springer New York
• ISSN：1432-0606

文摘

In this paper we introduce a new formulation for shape optimization problems in fluids in a diffuse interface setting that can in particular handle topological changes. By adding the Ginzburg–Landau energy as a regularization to the objective functional and relaxing the non-permeability outside the fluid region by introducing a porous medium approach we hence obtain a phase field problem where the existence of a minimizer can be guaranteed. This problem is additionally related to a sharp interface problem, where the permeability of the non-fluid region is zero. In both the sharp and the diffuse interface setting we can derive necessary optimality conditions using only the natural regularity of the minimizers. We also pass to the limit in the first order conditions. Keywords Shape and topology optimization Phase field method Diffuse interfaces Stokes flow Fictitious domain