Path-Weighted Diffusivity Functions for Parameterization of Heat Deposition Processes

Abstract

General parameterizations are constructed for spatial modulation of heat diffusion patterns according to energy deposition characteristics occurring within a volume of material where there exist inhomogeneous or anisotropic thermal diffusivity. These parameterizations are formulated in terms of path-weighted diffusivity functions. The construction of a general parameterization of energy deposition processes where spatially dependent thermal diffusivity exists is necessary for their inverse analysis. The structure of such a parameterization follows from the concepts of model and data spaces that imply the existence of an optimal parametric representation for a given class of inverse problems. Accordingly, the optimal parametric representation is determined by the characteristics of the available data, which, in principle, can contain both experimental measurements and numerical simulation data. Parameterizations for spatial modulation of heat diffusion follow from the observation that many different types of energy deposition processes can be represented by weighted sums of basis functions whose general forms are that of spatially modulated diffusion. The parameterizations presented are constructed according to a specific definition of the inverse heat deposition problem that provides a rigorous foundation for a highly flexible and general parameterization of energy deposition processes, which is essential for their inverse analysis.