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Topology Optimization for Steady-State Anisothermal Flow Targeting Solids with Piecewise Constant Thermal Diffusivity

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Abstract

Several engineering problems result in a PDE-constrained optimization problem that aims at finding the shape of a solid inside a fluid which minimizes a given cost function. These problems are categorized as Topology Optimization (TO) problems. In order to tackle these problems, the solid may be located with a penalization term added in the constraints equations that vanishes in fluid regions and becomes large in solid regions. This paper addresses a TO problem for anisothermal flows modelled by the steady-state incompressible Navier–Stokes system coupled to an energy equation, with mixed boundary conditions, under the Boussinesq approximation. We first prove the existence and uniqueness of a solution to these equations as well as the convergence of its finite element discretization. Next, we show that our TO problem has at least one optimal solution for cost functions that satisfy general assumptions. The convergence of discrete optimum toward the continuous one is then proved as well as necessary first order optimality conditions. Eventually, all these results let us design a numerical algorithm to solve a TO problem approximating solids with piecewise constant thermal diffusivities also refered as multi-materials. A physical problem solved numerically for varying parameters concludes this paper.

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  1. https://osur-devspot.univ-reunion.fr/avieira/tossaf_pctd

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Acknowledgements

All the authors are supported by the “Agence Nationale de la Recherche” (ANR), Project O-TO-TT-FU number ANR-19-CE40-0011. The used code for this article is available at https://osur-devspot.univ-reunion.fr/avieira/tossaf_pctd.

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  1. Physique et Ingénierie Mathématique pour l’Énergie et l’Environnement (PIMENT), Université de la Réunion, 2 rue Joseph Wetzell, 97490, Sainte-Clotilde, France

    Alexandre Vieira & Alain Bastide

  2. Laboratoire des Sciences de l’Ingénieur Appliquées à la Mécanique et au Génie Electrique (SIAME), E2S-UPPA, Université de Pau et des Pays de l’Adour, 64000, Pau, France

    Pierre-Henri Cocquet

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  1. Alexandre Vieira
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Vieira, A., Bastide, A. & Cocquet, PH. Topology Optimization for Steady-State Anisothermal Flow Targeting Solids with Piecewise Constant Thermal Diffusivity. Appl Math Optim 85, 41 (2022). https://doi.org/10.1007/s00245-022-09828-5

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