Discrete and Continuous Adjoint Approaches to Estimate Boundary Heat Fluxes in Falling Films

A wavy falling film simulation is considered in which a liquid travels along one side of a thin metal foil that is heated electrically from the opposite side. The direct problem consists of a three-dimensional heat conduction equation on a cuboid domain representing the foil with suitable initial and boundary conditions. The inverse problem consists of determining the heat flux on the film side of the foil from a given distribution of the temperature on the heating side. Two different adjoint approaches for the solution of this inverse problem are compared. In the continuous adjoint approach, the adjoint problem is analytically derived from the direct problem and then discretized. In the discrete adjoint approach, the direct problem is discretized from which an adjoint code is generated by means of the reverse mode of automatic differentiation. Numerical experiments are reported demonstrating the advantages and disadvantages of the two approaches