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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 [1] G. Allaire, F. Jouve, A. Toader, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194(2004), 363-393.[2] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.[3] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson, Shape optimization using the cut finite element method, Comput. Methods Appl. Mech. Engrg., 328(2018), 242-261.[4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.[5] F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J. Control Optim., 45(2006), 343-367.[6] M.C. Delfour and J.P. Zolésio, Shapes and Geometries:Metrics, Analysis, Differential Calculus, and Optimization, 2nd ed., SIAM, Philadelphia, 2011.[7] X. Duan, Y. Ma, and R. Zhang, Shape-topology optimization for NavierCStokes problem using variational level set method, J. Comput. Appl. Math., 222(2008), 487-499.[8] A. Ern and J.L. Guermond, Theory and Practice of Finite Elements, Springer, Berlin, 2004.[9] P. Gangl, U. Langer, A. Laurain, H. Meftahi, and K. Sturm, Shape optimization of an electric motor subject to nonlinear magnetostatics, SIAM J. Sci. Comput., 37(2015), B1002-B1025.[10] V. Girault, R.H. Nochetto, and L.R. Scott, Maximum-norm stability of the finite element Stokes projection, J. Math. Pures Appl., 84(2005), 279-330.[11] V. Girault, R.H. Nochetto, L.R. Scott, Max-norm estimates for Stokes and NavierCStokes approximations in convex polyhedra, Numer. Math., 131(2015), 771-822.[12] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms, Springer-Verlag, Berlin, 1986.[13] F. Hecht, New development in FreeFem++, J. Numer. Math., 20(2012), 251-265.[14] A. Henrot and Y. Privat, What is the optimal shape of a pipe, Arch. Rational Mech. Anal., 196(2010), 281-302.[15] M. Hintermller, A. Laurain, and I. Yousept, Shape sensitivities for an inverse problem in magnetic induction tomography based on the eddy current model, Inverse Problems, 31(2015) 065006, 25.[16] R. Hiptmair, A. Paganini, and S. Sargheini, Comparison of approximate shape gradients, BIT Numer. Math., 55(2015), 459-485.[17] A. Laurain and K. Sturm, Distributed shape derivative via averaged adjoint method and applications, ESAIM Math. Model. Numer. Anal., 50(2016), 1241-1267.[18] J. Li and S. Zhu, Shape identification in Stokes flow with distributed shape gradients, Appl. Math. Lett., 95(2019), 165-171.[19] B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, 2nd ed., Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2010.[20] O. Pironneau, On optimum profiles in Stokes flow, J. Fluid Mech., 59(1973), 117-128.[21] A. Quarteroni, A. Manzoni, and C. Vergara, The cardiovascular system:mathematical modelling, numerical algorithms and clinical applications, Acta Numer., 26(2017), 365-590.[22] S. Schmidt, M. Schtte, and A. Walther, Efficient numerical solution of geometric inverse problems involving Maxwell's equations using shape derivatives and automatic code generation, SIAM J. Sci. Comput., 40(2018), B405-B428.[23] V. Schulz, M. Siebenborn, and K. Welker, Structured inverse modeling in parabolic diffusion problems, SIAM J. Control Optim., 53(2015), 3319-3338.[24] J. Soko lowski and J.P. Zolésio, Introduction to Shape Optimization:Shape Sensitivity Analysis, Springer, Heidelberg, 1992.[25] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Springer, 2007.[26] W. Yan and Y. Ma, Shape reconstruction of an inverse Stokes problem, J. Comput. Appl. Math., 216(2008), 554-562.[27] S. Zhu, Effective shape optimization of Laplace eigenvalue problems using domain expressions of Eulerian derivatives, J. Optim. Theory Appl., 176(2018), 17-34.[28] S. Zhu and Z. Gao, Convergence analysis of mixed finite element approximations to shape gradients in the Stokes equation, Comput. Methods Appl. Mech. Engrg., 343(2019), 127-150.[29] S. Zhu, X. Hu, and Q. Wu, On accuracy of approximate boundary and distributed H1 shape gradient flows for eigenvalue optimization, J. Comput. Appl. Math., 365(2020), 112374[30] S. Zhu, X. Hu, and Q. Liao, Convergence analysis of Galerkin finite element approximations to shape gradients in eigenvalue optimization, BIT. Numer. Math., 60(2020), 853-878.
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