Topology Optimization for Steady-state anisothermal flow targeting solid with piecewise constant thermal diffusivity

• Published in: Applied mathematics & optimization 2022, Vol.85 (3)
• Main Authors: Vieira, Alexandre, Bastide, Alain, Cocquet, Pierre-Henri
• Format: Article
• Language: English
• Subjects: Fluid Dynamics; Mathematics; Optimization and Control; Physics
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-022-09828-5

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-Stokessystem 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 numericallyfor varying parameters concludes this paper.