Numerical methods for the solution of a three-dimensional anisotropic inverse heat conduction problem

The research topic of this thesis is closely related to subjects studied in the collaborative research center (SFB) 540 at the RWTH Aachen University. One goal in SFB 540 is the systematic modelling of selected (kinetic) phenomena, as e.g. the heat and mass transfer in wavy falling films, leading to an improved physical understanding of the related transport mechanisms. The research in this thesis is motivated by the question how the heat transfer in falling films is influenced by the wave characteristics. The approach used for answering this question is based on high resolution temperature measurements provided by the Chair of Heat and Mass Transfer (RWTH Aachen University). These measurements were performed in a falling film experiment. In the experiment a laminar wavy film travels along a thin foil that is heated electrically from the back side. We estimate the space- and time-dependent heat flux on the inaccessible film side of the foil using infrared temperature measurements on the foil back side, which are influenced by the transport phenomena in the falling film and the wavy film surface. For the solution of this inverse heat conduction problem (IHCP) efficient and robust numerical methods are needed. The three-dimensional IHCP in falling films is formulated as an optimal control problem. The optimization is performed in cooperation with the Chair of Process Systems Engineering (RWTH Aachen University). The idea in this approach is to consider the unknown heat flux on the estimation boundary as a decision variable to minimize a specified defect functional on the measurement boundary of the heat conductor. In the defect functional the computed temperature for an approximation of the searched for boundary heat flux is needed, which is obtained from the solution of a direct heat conduction problem. For the solution of the optimal control problem we apply the conjugate gradient type method CGNE. Since the optimal control problem based on realistic measurement data is ill-posed, we use iterative regularization. For the choice of the regularization parameter we use the heuristic L-curve method which turned out to be satisfactory for our problem class. Simulation studies show that a significant reduction of the computational time can be achieved, if the optimization is performed on a hierarchy of nested space grids. In the CGNE algorithm well-posed direct heat conduction problems have to be solved. However, the very different length scales of the heat conductor, i.e. the thin heating foil, result in an anisotropy effect. For the numerical treatment we use an implicit time integration method and a special space discretization based on (linear) anisotropic tetrahedral finite elements. In order to obtain a bound for the discretization error we use the Cea-lemma and error bounds for standard nodal Lagrangian interpolation. Different approaches for the derivation of these error bounds are considered. We give a uniform presentation of results for anisotropic (triangular and) tetrahedral finite elements that can be found at different places in the literature. From these results we can conclude that anisotropic tetrahedral elements which satisfy a maximum angle condition are suitable for the space discretization of the direct heat conduction problems. Important phenomena concerning anisotropic finite elements are illustrated by means of systematic numerical experiments. In each step of the implicit time integration method linear anisotropic reaction-diffusion problems have to be solved, which contain two important parameters: there is a parameter which describes the anisotropy of the domain and one that measures the size of the reaction term relative to that of the diffusion term. After discretization we have a third parameter, namely the mesh width. For the efficient solution of the corresponding discrete problems the convergence of a multigrid method with a special (symmetric line Gauss-Seidel) smoother is analyzed. In the analysis there is a need for other (better) interpolation operators than the standard Lagrangian interpolation operator. Thus, a modified Scott-Zhang interpolation operator is studied. Both, for the W- and the V-cycle method we derive contraction number bounds smaller than one uniform with respect to all three parameters. For the multigrid W-cycle the convergence is analyzed in the framework of the approximation and smoothing property