Shape gradient methods for shape optimization of an unsteady multiscale fluid-structure interaction model. (English) Zbl 1542.65121
Summary: We consider numerical shape optimization of a fluid-structure interaction model. The constrained system involves multiscale coupling of a two-dimensional unsteady Navier-Stokes equation and a one-dimensional ordinary differential equation for fluid flows and structure, respectively. We derive shape gradients for both objective functionals of least-squares type and energy dissipation. The state and adjoint state equations are numerically solved on the time-dependent domains using the Arbitrary-Lagrangian-Eulerian method. Numerical results are presented to illustrate effectiveness of algorithms.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65K10 | Numerical optimization and variational techniques |
| 49M41 | PDE constrained optimization (numerical aspects) |
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
Software:
FreeFem++References:
| [1] | Brügger, R.; Harbrecht, H.; Tausch, J., On the numerical solution of a time-dependent shape optimization problem for the heat equation, SIAM J. Control Optim., 59, 931-953, 2021 · Zbl 1461.49051 · doi:10.1137/19M1268628 |
| [2] | Correa, R.; Seeger, A., Directional derivative of a min-max function, Nonlinear Anal., 9, 13-22, 1985 · Zbl 0556.49007 · doi:10.1016/0362-546X(85)90049-5 |
| [3] | Delfour, MC; Zolesio, JP, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2011, Philadelphia: SIAM, Philadelphia · Zbl 1251.49001 · doi:10.1137/1.9780898719826 |
| [4] | Haslinger, J.; Mäkinen, R., Introduction to Shape Optimization: Theory, Approximation, and Computation, 2003, Philadelphia: SIAM, Philadelphia · Zbl 1020.74001 · doi:10.1137/1.9780898718690 |
| [5] | Dziri, R.; Zolesio, JP, Drag reduction for non-cylindrical Navier-Stokes flows, Optim. Methods. Softw., 26, 575-600, 2011 · Zbl 1230.49034 · doi:10.1080/10556788.2010.516434 |
| [6] | Feppon, F.: Shape and topology optimization of multiphysics systems, PhD thesis, Universite Paris Saclay, Paris (2019) |
| [7] | Feppon, F.; Allaire, G.; Dapogny, C.; Jolivet, P., Topology optimization of thermal fluid-structure systems using body-fitted meshes and parallel computing, J. Comput. Phys., 417, 2020 · Zbl 1437.74021 · doi:10.1016/j.jcp.2020.109574 |
| [8] | Forti, D.; Dede, L., Semi-implicit BDF time discretization of the Navier-Stokes equations with VMS and LES modeling in a high performance computing framework, Comput. Fluids, 117, 168-182, 2015 · Zbl 1390.76149 · doi:10.1016/j.compfluid.2015.05.011 |
| [9] | Girault, V.; Raviart, PA, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, 1986, Berlin: Springer-Verlag, Berlin · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5 |
| [10] | Haslinger, J.; Makinen, RAE, Introduction to Shape Optimization, 2003, Philadelphia: SIAM, Philadelphia · Zbl 1020.74001 · doi:10.1137/1.9780898718690 |
| [11] | Iglesias, JA; Sturm, K.; Wechsung, F., Two-dimensional shape optimization with nearly conformal transformations, SIAM J. Sci. Comput., 40, 3807-3830, 2018 · Zbl 1403.49043 · doi:10.1137/17M1152711 |
| [12] | Haubner, J.; Ulbrich, M.; Ulbrich, S., Analysis of shape optimization problems for unsteady fluid-structure interaction, Inverse Probl., 36, 2020 · Zbl 1440.49044 · doi:10.1088/1361-6420/ab5a11 |
| [13] | Heners, JP; Radtke, L.; Hinze, M.; Düster, A., Adjoint shape optimization for fluid-structure interaction of ducted flows, Comput. Mech., 61, 259-276, 2018 · Zbl 1461.76152 · doi:10.1007/s00466-017-1465-5 |
| [14] | Hecht, F., New developments in freefem++, J. Numer. Math., 20, 251-265, 2012 · Zbl 1266.68090 · doi:10.1515/jnum-2012-0013 |
| [15] | Katamine, E.; Kawai, R.; Takahashi, M., Shape optimization for stiffness maximization of geometrically nonlinear structure by considering fluid-structure-interaction, Mech. Eng. Lett., 7, 1-8, 2021 · doi:10.1299/mel.21-00048 |
| [16] | Lasiecka, I.; Szulc, K.; Zochowski, A., Reducing drag of the obstacle in the channel by boundary control: theory and numerics, IFAC-PapersOnline, 52, 2, 168-173, 2019 · doi:10.1016/j.ifacol.2019.08.030 |
| [17] | Lasiecka, I.; Szulc, K.; Zochowski, A., Boundary control of small solutions to fluid-structure interactions arsing in coupling of elasticity with Navier-Stokes equation under mixed boundary conditions, Nonlinear Anal. Real World Appl., 44, 54-85, 2018 · Zbl 1406.35229 · doi:10.1016/j.nonrwa.2018.04.004 |
| [18] | Laurain, A.; Walker, SW, Optimal control of volume-preserving mean curvature flow, J. Comput. Phys., 438, 2021 · Zbl 07505967 · doi:10.1016/j.jcp.2021.110373 |
| [19] | Leugering, G.; Novotny, AA; Menzala, GP; Sokołłowski, J., On shape optimization for an evolution coupled system, Appl. Math. Optim., 64, 441-466, 2011 · Zbl 1242.49089 · doi:10.1007/s00245-011-9148-7 |
| [20] | Li, J.; Zhu, S., Shape optimization of Navier-Stokes flows by a two-grid method, Comput. Methods Appl. Mech. Eng., 400, 2022 · Zbl 1507.49038 · doi:10.1016/j.cma.2022.115531 |
| [21] | Li, J.; Zhu, S.; Shen, X., On mixed finite element approximations of shape gradients in shape optimization with the Navier-Stokes equation, Numer. Methods Partial Differ. Equ., 39, 1604-1634, 2023 · Zbl 1537.65169 · doi:10.1002/num.22947 |
| [22] | Li, J.; Zhu, S., Shape optimization of the Stokes eigenvalue problem, SIAM J. Sci. Comput., 45, A798-A828, 2023 · Zbl 1512.65254 · doi:10.1137/21M1451543 |
| [23] | Mohammadi, B.; Pironneau, O., Applied Shape Optimization for Fluids, 2010, Oxford: Oxford University Press, Oxford |
| [24] | Moubachir, M.; Zolesio, JP, Moving Shape Analysis and Control Applications to Fluid Structure Interactions. Pure and Application in Mathematics, 2006, Boca Raton, FL: Chapman and Hall/CRC, Boca Raton, FL · Zbl 1117.49003 |
| [25] | Plotnikov, P.; Sokołłiowski, J., Compressible Navier-Stokes Equations Theory and Shape Optimization, 2012, Basel: Birkähser/Springer, Basel · Zbl 1260.35002 · doi:10.1007/978-3-0348-0367-0 |
| [26] | Raja, D., Moubachir, M., Zolesio, J. P.: Navier-Stokes dynamical shape control: from state derivative to min-max principle. INRIA., N 4610 1-57 (2002) |
| [27] | Scheid, JF; Sokolowski, J., Shape optimization for a fluid elasticity system, Pure. Appl. Funct. Anal., 3, 1, 193-217, 2018 · Zbl 1474.35540 |
| [28] | Sigmund, O., Design of multiphysics actuators using topology optimization—part II: two-material structures, Comput. Methods Appl. Mech. Eng., 190, 6605-6627, 2001 · Zbl 1116.74407 · doi:10.1016/S0045-7825(01)00252-3 |
| [29] | Sokolowski, J.; Zolesio, JP, Introduction to Shape Optimization: Shape Sensitivity Analysis, 1992, Heidelberg: Springer, Heidelberg · Zbl 0761.73003 · doi:10.1007/978-3-642-58106-9 |
| [30] | Yan, W.; Li, Y.; Hou, J., Shape optimization for an obstacle located in incompressible Boussinesq flow, Comput. Fluids, 240, 2022 · Zbl 1521.76126 · doi:10.1016/j.compfluid.2022.105431 |
| [31] | Yagi, H.; Kawahara, M., Optimal shape determination of a body located in incompressible viscous fluid flow, Comput. Methods Appl. Mech. Eng., 196, 5084-5091, 2007 · Zbl 1173.76317 · doi:10.1016/j.cma.2007.07.008 |
| [32] | Zhu, S.; Gao, Z., Convergence analysis of mixed finite element approximations to shape gradients in the Stokes equation, Comput. Methods Appl. Mech. Eng., 343, 127-150, 2019 · Zbl 1440.76093 · doi:10.1016/j.cma.2018.08.024 |
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