×

Shape optimization by free-form deformation: existence results and numerical solution for Stokes flows. (English) Zbl 1303.49017

Summary: Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from the literature.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49J20 Existence theories for optimal control problems involving partial differential equations
65K10 Numerical optimization and variational techniques
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D55 Flow control and optimization for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allaire, G.: Conception Optimale de Structures. Springer, Berlin (2007) · Zbl 1132.49033
[2] Amoiralis, E.I., Nikolos, I.K.: Freeform deformation versus B-spline representation in inverse airfoil design. J. Comput. Inf. Sci. Eng. 8(2), 1-13 (2008)
[3] Andreoli, M., Janka, A., Désidéri, J.A.: Free-form-deformation parametrization for multilevel 3D shape optimization in aerodynamics. Technical Report 5019, INRIA Sophia Antipolis (2003)
[4] Bello, J.A., Fernández-Cara, E., Lemoine, J., Simon, J.: The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow. SIAM J. Control Optim. 35, 626-640 (1997) · Zbl 0873.76019
[5] Bertsekas, D.P.: On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans. Autom. Control 21(2), 174-184 (1976) · Zbl 0326.49025
[6] Bourot, J.-M.: On the numerical computation of the optimum profile in Stokes flow. J. Fluid Mech. 65(3), 513-515 (1974) · Zbl 0284.76025
[7] Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence. Numer. Math. 107(1), 39-77 (2007) · Zbl 1117.76032
[8] Campolongo, F., Cariboni, J., Saltelli, A.: An effective screening design for sensitivity analysis of large models. Environ. Model. Softw. 22(10), 1509-1518 (2007)
[9] Céa, J.: Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. Math. Model. Num. Anal. 20(3), 371-402 (1986) · Zbl 0604.49003
[10] Farin, G.: Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. Morgan Kaufmann, Los Altos (2001) · Zbl 0702.68004
[11] Gain, J., Bechmann, D.: A survey of spatial deformation from a user-centered perspective. ACM Trans. Graph. 27(4), 107:1-107:21 (2008)
[12] Gain, J.E., Dodgson, N.A.: Preventing self-intersection under free-form deformation. IEEE Trans. Vis. Comput. Graph. 7(4), 289-298 (2001)
[13] Gunzburger, M.D.: Perspectives in Flow Control and Optimization. SIAM, Philadelphia (2003) · Zbl 1088.93001
[14] Gunzburger, M.D., Hou, L., Svobodny, T.P.: Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30, 167 (1992) · Zbl 0756.49004
[15] Gunzburger, M.D., Kim, H., Manservisi, S.: On a shape control problem for the stationary Navier-Stokes equations. ESAIM Math. Model. Numer. Anal. 34(6), 1233-1258 (2000) · Zbl 0981.76027
[16] Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization: Theory, Approximation, and Computation. SIAM, Philadelphia (2003) · Zbl 1020.74001
[17] Henrot, A., Pierre, M.: Variation et Optimisation de Formes: Une Analyse Géométrique. Springer, Berlin (2005) · Zbl 1098.49001
[18] Henrot, A., Privat, Y.: What is the optimal shape of a pipe? Arch. Ration. Mech. Anal. 196(1), 281-302 (2010) · Zbl 1304.76022
[19] Jameson, A.: Aerodynamic design via control theory. J. Sci. Comput. 3(3), 233-260 (1988) · Zbl 0676.76055
[20] Jameson, A.: Optimum aerodynamic design using CFD and control theory. In: Proceedings of the 12th AIAA Computational Fluid Dynamics Conference, pp. 926-949. AIAA Paper 95-1729 (1995)
[21] Lamousin, H.J., Waggenspack, W.N.: NURBS-based free-form deformations. IEEE Comput. Graph. Appl. 14(6), 59-65 (1994)
[22] Lassila, T., Rozza, G.: Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng. 199(23-24), 1583-1592 (2010) · Zbl 1231.76245
[23] Lehnhäuser, T., Schäfer, M.: A numerical approach for shape optimization of fluid flow domains. Comput. Methods Appl. Mech. Eng. 194, 5221-5241 (2005) · Zbl 1092.76024
[24] Lombardi, M., Parolini, N., Quarteroni, A., Rozza, G.: Numerical simulation of sailing boats: dynamics, FSI, and shape optimization. In: Buttazzo, G., Frediani, A. (eds.) Variational Analysis and Aerospace Engineering: Mathematical Challenges for Aerospace Design. Contributions from a Workshop Held at the School of Mathematics in Erice, Italy, volume 66 of Springer Optimization and Its Applications (2012) · Zbl 0604.49003
[25] Manzoni, A.: Reduced models for optimal control, shape optimization and inverse problems in haemodynamics. Ph.D. Thesis, N. 5402, École Polytechnique Fédérale de Lausanne, 2012.
[26] Manzoni, A., Quarteroni, A., Rozza, G.: Shape optimization of cardiovascular geometries by reduced basis methods and free-form deformation techniques. Int. J. Numer. Methods Fluids 70(5), 646-670 (2012) · Zbl 1412.76031
[27] Mohammadi, B., Pironneau, O.: Optimal shape design for fluids. Annu. Rev. Fluids Mech. 36, 255-279 (2004) · Zbl 1076.76020
[28] Mohammadi, B., Pironneau, O.: Applied shape optimization for fluids. Numerical Mathematics and Scientific Computation. Oxford Univ. Press, New York (2010)
[29] Morris, M.D.: Factorial sampling plans for preliminary computational experiments. Technometrics 33(2), 161-174 (1991)
[30] Murat, F., Simon, J.: Sur le contrôle par un domaine géométrique. Internal Report No. 76 015, Laboratoire d’Analyse Numérique de l’Université Paris 6, (1976) · Zbl 0326.49025
[31] Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999) · Zbl 0930.65067
[32] Ogawa, Y., Kawahara, M.: Shape optimization of body located in incompressible viscous flow based on optimal control theory. Int. J. Comput. Fluid Dyn. 17, 243-251 (2003) · Zbl 1043.76504
[33] Pironneau, O.: On optimum profiles in Stokes flow. J. Fluid Mech. 59(1), 117-128 (1973) · Zbl 0274.76022
[34] Pironneau, O.: Optimal Shape Design for Elliptic Systems, Springer Series in Computational Physics. Springer, New York (1984) · Zbl 0496.93029
[35] Richardson, S.: Optimum profiles in two-dimensional Stokes flow. Proc. Math. Phys. Sci. 450(1940), 603-622 (1995) · Zbl 0924.76024
[36] Samareh, J.A.: Aerodynamic shape optimization based on free-form deformation. Proc. 10th AIAA/ISSMO Multidiscip. Anal. Optim. Conf. 6, 3672-3683 (2004)
[37] Sarakinos, S.S., Amoiralis, E., Nikolos, I.K.: Exploring freeform deformation capabilities in aerodynamic shape parameterization. Proc. Int. Conf. Comput. Tool 1, 535-538 (2005)
[38] Sederberg, T.W., Parry, S.R.: Free-form deformation of solid geometric models. Comput. Graph. 20(4), 151-160 (1986)
[39] Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, New York (1992) · Zbl 0761.73003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.