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 - <p style="text-align: justify;">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.</p>