Previous Articles     Next Articles

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.

CLC Number: 

[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.
[1] Baiying Dong, Xiufeng Feng, Zhilin Li. AN L SECOND ORDER CARTESIAN METHOD FOR 3D ANISOTROPIC INTERFACE PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(6): 882-912.
[2] Bei Zhang, Jikun Zhao, Minghao Li, Hongru Chen. STABILIZED NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY ON RECTANGULAR OR CUBIC MESHES [J]. Journal of Computational Mathematics, 2022, 40(6): 865-881.
[3] Yanping Chen, Qiling Gu, Qingfeng Li, Yunqing Huang. A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(6): 936-954.
[4] Tianliang Hou, Chunmei Liu, Chunlei Dai, Luoping Chen, Yin Yang. TWO-GRID ALGORITHM OF H1-GALERKIN MIXED FINITE ELEMENT METHODS FOR SEMILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(5): 667-685.
[5] Kai Wang, Na Wang. ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA [J]. Journal of Computational Mathematics, 2022, 40(5): 777-793.
[6] Linshuang He, Minfu Feng, Qiang Ma. PENALTY-FACTOR-FREE STABILIZED NONCONFORMING FINITE ELEMENTS FOR SOLVING STATIONARY NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(5): 728-755.
[7] Huan Liu, Xiangcheng Zheng, Hongfei Fu. ANALYSIS OF A MULTI-TERM VARIABLE-ORDER TIME-FRACTIONAL DIFFUSION EQUATION AND ITS GALERKIN FINITE ELEMENT APPROXIMATION [J]. Journal of Computational Mathematics, 2022, 40(5): 814-834.
[8] Hanzhang Hu, Yanping Chen. A CHARACTERISTIC MIXED FINITE ELEMENT TWO-GRID METHOD FOR COMPRESSIBLE MISCIBLE DISPLACEMENT PROBLEM [J]. Journal of Computational Mathematics, 2022, 40(5): 794-813.
[9] Abdelhamid Zaghdani, Sayed Sayari, Miled EL Hajji. A NEW HYBRIDIZED MIXED WEAK GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC PROBLEMS* [J]. Journal of Computational Mathematics, 2022, 40(4): 499-516.
[10] Xinjiang Chen, Yanqiu Wang. A CONFORMING QUADRATIC POLYGONAL ELEMENT AND ITS APPLICATION TO STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(4): 624-648.
[11] Xiaonian Long, Qianqian Ding. A SECOND ORDER UNCONDITIONALLY CONVERGENT FINITE ELEMENT METHOD FOR THE THERMAL EQUATION WITH JOULE HEATING PROBLEM [J]. Journal of Computational Mathematics, 2022, 40(3): 354-372.
[12] Noelia Bazarra, José R. Fernández, MariCarme Leseduarte, Antonio Magaña, Ramón Quintanilla. NUMERICAL ANALYSIS OF A PROBLEM INVOLVING A VISCOELASTIC BODY WITH DOUBLE POROSITY [J]. Journal of Computational Mathematics, 2022, 40(3): 415-436.
[13] Ram Manohar, Rajen Kumar Sinha. ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR FULLY DISCRETE SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(2): 147-176.
[14] Weijie Huang, Wei Jiang, Yan Wang. A θ-L APPROACH FOR SOLVING SOLID-STATE DEWETTING PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(2): 275-293.
[15] Pengzhan Huang, Yinnian He, Ting Li. A FINITE ELEMENT ALGORITHM FOR NEMATIC LIQUID CRYSTAL FLOW BASED ON THE GAUGE-UZAWA METHOD [J]. Journal of Computational Mathematics, 2022, 40(1): 26-43.
Viewed
Full text


Abstract