A functional optimization formulation and implementation of an inverse natural convection problem

A formulation and a FEM implementation for the solution of an ill-posed inverse/design natural convection problem is proposed. In particular, we consider an incompressible viscous liquid material occupying a given domain Omega where convection is driven by buoyancy. Thermal boundary conditions are only prescribed in the part (Gamma - Gamma(h0)) of the boundary Gamma. In addition, the temperature distribution is also prescribed in the part Gamma(I) of the boundary Gamma(h1) where the heat flux is known, i.e. Gamma(I) is a boundary with overspecified thermal boundary conditions. The inverse convection problem is posed as an optimization problem in L-2(Gamma(I) x [0, t(max)]) for the calculation of the boundary heat flux q(o)(x, t), with (x, t) is an element of (Gamma(h0) x [0,t(max)]) The optimization scheme minimizes the discrepancy parallel to theta(m)(x,t) - theta(x,t;q(o)) parallel to L-2(Gamma(I) x[0,t(max)]) between the temperature theta (x, t; q(o)) calculated from the solution of a direct problem for each flux q(o) and the measured or desired temperature theta(m)(x, t) at the boundary Gamma(I). The gradient of the cost functional is obtained from the adjoint fields. The adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem. The conjugate gradient method is employed to perform the optimization process. The FEM is used for the calculation of the direct, adjoint and sensitivity thermal and fluid flow fields. Finally, the method is demonstrated with the solution of some two-dimensional inverse problems with known solutions. The numerical results are discussed and future work with potential applications is addressed.