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Volume 40, Issue 2
On Distributed $H^1$ Shape Gradient Flows in Optimal Shape Design of Stokes Flows: Convergence Analysis and Numerical Applications

Jiajie Li & Shengfeng Zhu

DOI: 10.4208/jcm.2009-m2020-0020

J. Comp. Math., 40 (2022), pp. 231-257.

Published online: 2022-01

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  • Abstract

We consider optimal shape design in Stokes flow using $H^1$ shape gradient flows based on the distributed Eulerian derivatives. MINI element is used for discretizations of Stokes equation and Galerkin finite element is used for discretizations of distributed and boundary $H^1$ shape gradient flows. Convergence analysis with a priori error estimates is provided under general and different regularity assumptions. We investigate the performances of shape gradient descent algorithms for energy dissipation minimization and obstacle flow. Numerical comparisons in 2D and 3D show that the distributed $H^1$ shape gradient flow is more accurate than the popular boundary type. The corresponding distributed shape gradient algorithm is more effective.

  • Keywords

Shape optimization, Stokes equation, Distributed shape gradient, Finite element, MINI element, Eulerian derivative.

  • AMS Subject Headings

49Q10, 49Q12, 76D55, 76M10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

lijiajie20120233@163.com (Jiajie Li)

sfzhu@math.ecnu.edu.cn (Shengfeng Zhu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-231, author = {Li , Jiajie and Zhu , Shengfeng}, title = {On Distributed $H^1$ Shape Gradient Flows in Optimal Shape Design of Stokes Flows: Convergence Analysis and Numerical Applications}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {2}, pages = {231--257}, abstract = {

We consider optimal shape design in Stokes flow using $H^1$ shape gradient flows based on the distributed Eulerian derivatives. MINI element is used for discretizations of Stokes equation and Galerkin finite element is used for discretizations of distributed and boundary $H^1$ shape gradient flows. Convergence analysis with a priori error estimates is provided under general and different regularity assumptions. We investigate the performances of shape gradient descent algorithms for energy dissipation minimization and obstacle flow. Numerical comparisons in 2D and 3D show that the distributed $H^1$ shape gradient flow is more accurate than the popular boundary type. The corresponding distributed shape gradient algorithm is more effective.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009-m2020-0020}, url = {http://global-sci.org/intro/article_detail/jcm/20185.html} }
TY - JOUR T1 - On Distributed $H^1$ Shape Gradient Flows in Optimal Shape Design of Stokes Flows: Convergence Analysis and Numerical Applications AU - Li , Jiajie AU - Zhu , Shengfeng JO - Journal of Computational Mathematics VL - 2 SP - 231 EP - 257 PY - 2022 DA - 2022/01 SN - 40 DO - http://doi.org/10.4208/jcm.2009-m2020-0020 UR - https://global-sci.org/intro/article_detail/jcm/20185.html KW - Shape optimization, Stokes equation, Distributed shape gradient, Finite element, MINI element, Eulerian derivative. AB -

We consider optimal shape design in Stokes flow using $H^1$ shape gradient flows based on the distributed Eulerian derivatives. MINI element is used for discretizations of Stokes equation and Galerkin finite element is used for discretizations of distributed and boundary $H^1$ shape gradient flows. Convergence analysis with a priori error estimates is provided under general and different regularity assumptions. We investigate the performances of shape gradient descent algorithms for energy dissipation minimization and obstacle flow. Numerical comparisons in 2D and 3D show that the distributed $H^1$ shape gradient flow is more accurate than the popular boundary type. The corresponding distributed shape gradient algorithm is more effective.

Li , Jiajie and Zhu , Shengfeng. (2022). On Distributed $H^1$ Shape Gradient Flows in Optimal Shape Design of Stokes Flows: Convergence Analysis and Numerical Applications. Journal of Computational Mathematics. 40 (2). 231-257. doi:10.4208/jcm.2009-m2020-0020
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