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


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.


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 



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 ( 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”.


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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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