### ON DISTRIBUTED H1 SHAPE GRADIENT FLOWS IN OPTIMAL SHAPE DESIGN OF STOKES FLOWS: CONVERGENCE ANALYSIS AND NUMERICAL APPLICATIONS

Jiajie Li, Shengfeng Zhu

1. School of Mathematical Sciences, East China Normal University, Shanghai 200241, China
• Received:2020-01-19 Revised:2020-06-12 Online:2022-03-15 Published:2022-03-29
• Contact: Shengfeng Zhu,Email:sfzhu@math.ecnu.edu.cn
• Supported by:
This work was supported in part by the National Natural Science Foundation of China under grants (No.11571115 and No.12071149),Natural Science Foundation of Shanghai (No.19ZR1414100),and Science and Technology Commission of Shanghai Municipality (No.18dz2271000).

Jiajie Li, Shengfeng Zhu. ON DISTRIBUTED H1 SHAPE GRADIENT FLOWS IN OPTIMAL SHAPE DESIGN OF STOKES FLOWS: CONVERGENCE ANALYSIS AND NUMERICAL APPLICATIONS[J]. Journal of Computational Mathematics, 2022, 40(2): 231-257.

We consider optimal shape design in Stokes flow using H1 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 H1 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 H1 shape gradient flow is more accurate than the popular boundary type. The corresponding distributed shape gradient algorithm is more effective.

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