Topology optimization of heat transfer and fluid flow – O-TO-TT-FU

The current project wish to develop new mathematical and numerical methods to solve topology optimization problems for heat transfers in fluid flow. Such kind of problems are mathematically equivalent to optimization problems whose constraints are given by a set of Partial-Differential-Equations and aim at finding the location and the physical characteristics of a given material in order to enhance a given physical phenomenon. This topic has thus several applications in engineering and the applied science. An important example, thanks to the geographical location of La Réunion, is the design of a permanent air intake maximizing air flow in some specific location of a building allowing to cool a room without using air-conditioner hence saving energy. Having such applications in mind, this project is also going to be concerned with topology optimization problems for heat transfer in buoyancy-driven flows like those arising in natural and/or forced convection problems.

Classically, the location of the solid inside the fluid is done by adding a penalization term in the Navier-Stokes system. The latter ideally taking only two values, 0 indicating presence of fluid and infinity indicating presence of solid, is then interpolated by introducing a new continuous variable. Doing this allows to recover the existence of an optimal solution to the topology optimization problem and also to favor the use of gradient-based algorithms to solve the latter. The so-called adjoint method is usually used to compute the gradient of the considered cost function since it only needs to solve the "direct" problem (given by the constraints) and an "adjoint" problem which has the same number of unknowns as the direct problem.

We identified in the state of the art two important barriers to be lifted. First, the adjoint method applied to unsteady flow yield memory storage problems and/or requires prohibitive computation time. Time-parallel algorithms can still be used but their convergence has not been mathematically analysed yet. Secondly, the most part of works dealing with topology optimization in fluid mechanics only treat the case where the optimal solid is characterized by a constant thermal conductivity. The few contributions aiming at finding optimal solid with anisotropic and/or spatially-varying conductivity can actually only get piecewise-constant conductivity and have to introduce a new optimization variable for each constant. These techniques can thus not be used for conductivity that massively depend on the spatial variable.

This project is thus going to be structured around two tasks that have been identified as potential progress with respect to the state of the art.
The first task will design and study the convergence of parallel-in-time algorithms for topology optimization problem for unsteady flow modeled by the Navier-Stokes system under the Boussinesq assumption coupled to an energy equation. The second task will study topology optimization problems that wish to find optimal materials having anisotropic and/or spatially-dependent thermal conductivity. To reach this goal, we will used fluid/solid interpolation techniques that does not suffer from the limitations of the current methods used.