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

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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.

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Notes

  1. Nevertheless, the FFD technique presents some drawbacks; for instance, it is not possible to change the topology of the control points lattice, FFD it is not interpolatory, and the choice of enabled displacements is application dependent (as it happens with other common parameterizations).

  2. The set \({\mathcal {O}}\) can be endowed with the convergence in the sense of the characteristic functions or in the sense of Hausdorff—see e.g. [17] for definitions.

  3. A notion of derivative of a cost functional with respect to the domain can be introduced for the shape deformation with perturbation of identity maps. In particular, recall [17, 34, 39] that the Eulerian derivative of \(J: {\mathcal {O}}_{\mathcal {T}}(\varOmega ) \rightarrow \mathbb {R}\), in \(\varOmega \) and direction \(\varvec{\theta }\) is defined as

    $$\begin{aligned} dJ(\varOmega ; \varvec{\theta }) = \lim _{t \rightarrow 0} \frac{1}{t}[{J(({{\varvec{I}}} + t \varvec{\theta })(\varOmega ))-J(\varOmega )}]. \end{aligned}$$

  4. A large absolute value for the mean \(E_i\) implies that \(\mu _i\) has an important overall effect on the output, whereas a high standard deviation \(S_i\) indicates that the effect of \(\mu _i\) is not constant, which may be implied by a parameter interacting with other parameters.

  5. Thanks to the formulation (7) and the definition of Bernstein polynomials, it is easy to check that the boundary of \(D\) can be deformed only by control points belonging to \(\partial \widehat{D}\).

  6. Most of the CPU time is spent in the backtracking procedure to seek a step-length \(\alpha ^{(n)}\) fulfilling Armijo’s rule (33), which, once a near-optimal shape is found at the first iteration, is verified only for very small \(\alpha ^{(n)}\). This explains the large number of evaluations of \(j(\tilde{\varvec{\mu }})\).

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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”.

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Authors and Affiliations

  1. MOX-Modellistica e Calcolo Scientifico, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 , Milano, Italy

    Francesco Ballarin

  2. SISSA Mathlab, Scuola Internazionale Superiore di Studi Avanzati, via Bonomea 265, 34136 , Trieste, Italy

    Andrea Manzoni & Gianluigi Rozza

  3. Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 , Milano, Italy

    Sandro Salsa

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  1. Francesco Ballarin
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  2. Andrea Manzoni
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Correspondence to Andrea Manzoni.

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Ballarin, F., Manzoni, A., Rozza, G. et al. Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows. J Sci Comput 60, 537–563 (2014). https://doi.org/10.1007/s10915-013-9807-8

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