A functional optimization approach to an inverse magneto-convection problem
A formulation and an FEM implementation for the solution of an ill-posed inverse/design magneto-convection problem is proposed. In particular, an incompressible, viscous, electrically conducting liquid material occupying a given domain ? is considered. Convection is driven by buoyancy effects as well as a Lorentz force generated due to the applied magnetic field. Thermal boundary conditions are only prescribed in the part (???h0) of the boundary ?. In addition, the temperature distribution is also prescribed in the part ?I of the boundary ?h1, where the heat flux is known, i.e. ?I is a boundary with overspecified thermal boundary conditions. The inverse magneto-convection problem is posed as an optimization problem in L2(?I[0,tmax]) for the calculation of the boundary heat flux qo(x,t), with (x,t)?(?h0[0,tmax]). The optimization scheme minimizes the discrepancy ??m(x,t)??(x,t;qo)?L2(?I[0,tmax]) between the temperature ?(x,t;qo) calculated from the solution of a direct problem for each flux qo and the desired (or measured) temperature ?m(x,t) on the boundary ?I. The exact L2 gradient of the cost functional is obtained from appropriately defined adjoint fields. The adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem. The standard SUPG/PSPG stabilized FEM formulation for incompressible fluid flow simulation is here extended to thermo-magnetically driven flows. The proposed FEM formulation is used for the calculation of the direct, adjoint and sensitivity thermal, fluid flow and electric potential fields. The entire optimization algorithm is solved using the conjugate gradient method. Finally, the method is demonstrated through the solution of a few inverse problems with known results. The need for regularization is identified in one of the example problems with uniformly distributed random errors in the temperature data ?m(x,t) and a H1 regularized formulation is introduced to obtain stable solutions. Finally, the numerical results are elucidated and potential applications are addressed.
functional optimization inverse problems regularization stabilized finite element methods mhd adjoint techniques