Temperature Distribution Reconstruction by Eigenfunction Interpolation of Boundary Measurement Data

This paper deals with the inverse problem of evaluating the temperature distribution over time in a 3-D composite solid material having an arbitrary geometry. This approach is capable of evaluating the temperature distribution within the domain of the nonhomogeneous object under observation at each time instance. In this paper, we propose to use the eigenfunctions of the heat equation model, representing the heat problem under observation, as a basis for reconstructing the 3-D temperature distribution. This choice of basis functions has the advantage of incorporating the physics of the problem, making the temperature reconstruction inverse problem more robust. Because of the geometry complexity, the eigenfunctions have been computed numerically using a finite-element method. In principle, the method uses temperature measurements in just a few points of the object domain. To consider the practical aspect, here we focus our attention on a noninvasive approach taking the observation points only on the available boundary surfaces. The proper weighting of the eigenfunction basis used as temperature interpolators is achieved inverting the collected measured data. The two critical problems of selecting the best subset of eigenfunctions from the set of infinitely many available ones and the optimization of numbering and positioning the boundary measurement spots are studied as well.