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

Alexandre Vieira (1) , Alain Bastide (1) , Pierre-Henri Cocquet (1)

1 PIMENT - Physique et Ingénierie Mathématique pour l'Énergie, l'environnemeNt et le bâtimenT

A Topology Optimization (TO) problem is a PDE-constrained optimization problem that aims at finding the shape of a solid inside a fluid which minimizes a given cost function. The solid is 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 assumption. We first prove the existence and uniqueness of solution to this problem as well as the convergence of 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 provided as well as necessary first order optimality conditions. Eventually, all these results let us design a numerical algorithm to solve a TO problem that looks for solid with piecewise constant thermal diffusivities. A physical problem solved numerically with this method concludes this paper. 1. Introduction. Finding the shape of a solid located inside a fluid that either minimizes or maximizes a given physical effect has several applications in engineering and the applied science (see e.g. [39, 40, 43, 44] for several different applications). There exists various mathematical methods to deal with such problems that fall into the class of PDE-constrained optimization. The topological asymptotic expansion [4, 16, 41] consider the solid as a hole or an inhomogeneity with characteristic size ε. The so-called topological gradient is then defined as the first order term in the asymptotic expansion of the cost function as ε → 0. The shape optimization method [25, 39, 40] computes the gradient of the cost function with respect to perturbation of the boundary of the solid also referred as shape derivative. We emphasize that the two aforementioned methods exactly locate the solid. Nevertheless, once the gradient of the cost function is computed, the geometry of the computational domain changes and thus these two methods usually need some specific techniques to follow the evolution of the mesh during the numerical solving. In this paper, we choose to locate the solid thanks to a penalization term added in the Navier-Stokes equation. The latter vanishes in the fluid zone and goes to infinity in a solid region of the computational domain (see [5] for the mathematical justification of this method). Using such model as constraint in the shape optimization problem is referred as a topology optimization (TO) problem. It is worth noting that the major drawback of this approach is that the solid is only located when the velocity of the fluid is smaller that a given tolerance. Nevertheless,the numerical solving of a TO problem do not need specific remeshing techniques. We refer to the review papers [2, 20] for many references that deals with numerical res

Mots clés: 49Q10; 93C20; 65K99; Topology optimization; Navier-Stokes equations; anisothermal flows; Multi-materials AMS subject classifications 76D55

Domaines: Optimisation et contrôle [math.OC]; Dynamique des Fluides [physics.flu-dyn]

Alexandre Vieira, Alain Bastide, Pierre-Henri Cocquet. Topology Optimization for Steady-state anisothermal flow targeting solid with piecewise constant thermal diffusivity. 2020. ⟨hal-02569142v1⟩