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Journal of Scientific Computing

, Volume 60, Issue 3, pp 537–563 | Cite as

Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows

  • Francesco Ballarin
  • Andrea Manzoni
  • Gianluigi Rozza
  • Sandro Salsa
Article

Abstract

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 literature.

Keywords

Shape optimization Computational fluid dynamics Free-form deformations Perturbation of identity Finite elements method Stokes equations 

Mathematics Subject Classification

49Q10 49J20 65K10 65N30 76D55 78M34 

Notes

Acknowledgments

The authors gratefully acknowledge the collaboration with Prof. Alfio Quarteroni (CMCS, EPFL and MOX, Politecnico di Milano) and Dr. Toni Lassila (CMCS, EPFL) for their insights, useful discussions and support. We acknowledge the use of the finite element library LifeV (www.lifev.org) as a basis for the numerical simulations presented in this paper. Computational support from Consorzio Interuniversitario Lombardo per l’Elaborazione Automatica (CILEA) computing facilities under the LISA initiative is also acknowledged. This work has been partially funded by the Swiss National Science Foundation (Projects 122136 and 135444) and by the SHARM 2012–2014 SISSA post-doctoral research grant on the Project “Reduced Basis Methods for shape optimization in computational fluid dynamics”.

References

  1. 1.
    Allaire, G.: Conception Optimale de Structures. Springer, Berlin (2007)zbMATHGoogle Scholar
  2. 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)CrossRefGoogle Scholar
  3. 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)Google Scholar
  4. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bertsekas, D.P.: On the Goldstein–Levitin–Polyak gradient projection method. IEEE Trans. Autom. Control 21(2), 174–184 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bourot, J.-M.: On the numerical computation of the optimum profile in Stokes flow. J. Fluid Mech. 65(3), 513–515 (1974)CrossRefzbMATHGoogle Scholar
  7. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 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)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Farin, G.: Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. Morgan Kaufmann, Los Altos (2001)Google Scholar
  11. 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)CrossRefGoogle Scholar
  12. 12.
    Gain, J.E., Dodgson, N.A.: Preventing self-intersection under free-form deformation. IEEE Trans. Vis. Comput. Graph. 7(4), 289–298 (2001)CrossRefGoogle Scholar
  13. 13.
    Gunzburger, M.D.: Perspectives in Flow Control and Optimization. SIAM, Philadelphia (2003)zbMATHGoogle Scholar
  14. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization: Theory, Approximation, and Computation. SIAM, Philadelphia (2003)CrossRefGoogle Scholar
  17. 17.
    Henrot, A., Pierre, M.: Variation et Optimisation de Formes: Une Analyse Géométrique. Springer, Berlin (2005)CrossRefGoogle Scholar
  18. 18.
    Henrot, A., Privat, Y.: What is the optimal shape of a pipe? Arch. Ration. Mech. Anal. 196(1), 281–302 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Jameson, A.: Aerodynamic design via control theory. J. Sci. Comput. 3(3), 233–260 (1988)CrossRefzbMATHGoogle Scholar
  20. 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)Google Scholar
  21. 21.
    Lamousin, H.J., Waggenspack, W.N.: NURBS-based free-form deformations. IEEE Comput. Graph. Appl. 14(6), 59–65 (1994)CrossRefGoogle Scholar
  22. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lehnhäuser, T., Schäfer, M.: A numerical approach for shape optimization of fluid flow domains. Comput. Methods Appl. Mech. Eng. 194, 5221–5241 (2005)CrossRefzbMATHGoogle Scholar
  24. 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)Google Scholar
  25. 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.Google Scholar
  26. 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)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Mohammadi, B., Pironneau, O.: Optimal shape design for fluids. Annu. Rev. Fluids Mech. 36, 255–279 (2004)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Mohammadi, B., Pironneau, O.: Applied shape optimization for fluids. Numerical Mathematics and Scientific Computation. Oxford Univ. Press, New York (2010)Google Scholar
  29. 29.
    Morris, M.D.: Factorial sampling plans for preliminary computational experiments. Technometrics 33(2), 161–174 (1991)CrossRefGoogle Scholar
  30. 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)Google Scholar
  31. 31.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  32. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Pironneau, O.: On optimum profiles in Stokes flow. J. Fluid Mech. 59(1), 117–128 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Pironneau, O.: Optimal Shape Design for Elliptic Systems, Springer Series in Computational Physics. Springer, New York (1984)CrossRefGoogle Scholar
  35. 35.
    Richardson, S.: Optimum profiles in two-dimensional Stokes flow. Proc. Math. Phys. Sci. 450(1940), 603–622 (1995)CrossRefzbMATHGoogle Scholar
  36. 36.
    Samareh, J.A.: Aerodynamic shape optimization based on free-form deformation. Proc. 10th AIAA/ISSMO Multidiscip. Anal. Optim. Conf. 6, 3672–3683 (2004)Google Scholar
  37. 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)Google Scholar
  38. 38.
    Sederberg, T.W., Parry, S.R.: Free-form deformation of solid geometric models. Comput. Graph. 20(4), 151–160 (1986)CrossRefGoogle Scholar
  39. 39.
    Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, New York (1992)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Francesco Ballarin
    • 1
  • Andrea Manzoni
    • 2
  • Gianluigi Rozza
    • 2
  • Sandro Salsa
    • 3
  1. 1.MOX-Modellistica e Calcolo Scientifico, Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanoItaly
  2. 2.SISSA MathlabScuola Internazionale Superiore di Studi AvanzatiTriesteItaly
  3. 3.Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoMilanoItaly

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