Geodesic
An adjoint-based method for the numerical approximation of shape optimization problems in presence of fluid-structure interaction
Manzoni, Andrea1   ; Ponti, Luca2
1 CMCS-MATHICSE-SB, Ecole Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland
2 Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1501-1532

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In this work, we propose both a theoretical framework and a numerical method to tackle shape optimization problems related with fluid dynamics applications in presence of fluid-structure interactions. We present a general framework relying on the solution to a suitable adjoint problem and the characterization of the shape gradient of the cost functional to be minimized. We show how to derive a system of (first-order) optimality conditions combining several tools from shape analysis and how to exploit them in order to set a numerical iterative procedure to approximate the optimal solution. We also show how to deal efficiently with shape deformations (resulting from both the fluid-structure interaction and the optimization process). As benchmark case, we consider an unsteady Stokes flow in an elastic channel with compliant walls, whose motion under the effect of the flow is described through a linear Koiter shell model. Potential applications are related e.g. to design of cardiovascular prostheses in physiological flows or design of components in aerodynamics.

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DOI : 10.1051/m2an/2017006
Classification : 49Q10, 49J20, 65K10, 65N30, 74F10, 76D55
Keywords: PDE-constrained optimization, shape optimization, fluid-structure interaction, adjoint problem

Manzoni, Andrea 1 ; Ponti, Luca 2

1 CMCS-MATHICSE-SB, Ecole Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland
2 Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy 
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     title = {An adjoint-based method for the numerical approximation of shape optimization problems in presence of fluid-structure interaction},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1501--1532},
     publisher = {EDP-Sciences},
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     year = {2018},
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Manzoni, Andrea; Ponti, Luca. An adjoint-based method for the numerical approximation of shape optimization problems in presence of fluid-structure interaction. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1501-1532. doi: 10.1051/m2an/2017006

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