Additive manufacturing involves creating three-dimensional objects by depositing materials layer-by-layer. The freeform nature of the method permits the production of components with complex geometry. Deposition processes provide one more capability, which is the addition of multiple materials in a discrete manner to create “heterogeneous” objects with locally controlled composition. The result is direct digital manufacturing (DDM) by which dissimilar materials are added voxel-by-voxel (a voxel is volumetric pixel) following a predetermined tool-path. A typical example is functionally-graded material such as a gear with a tough core and a wear resistant surface. The inherent complexity of DDM processes is such that process modeling based on direct physics-based theory is difficult, especially due to a lack of temperature-dependent thermo-physical properties and particularly when dealing with melt-deposition processes. To overcome this difficulty the inverse problem approach is adopted to develop thermal models for multi-material, direct digital melt-deposition. This approach is based on the construction of a numerical-algorithmic framework for modeling anisotropic diffusivity such as that which would occur during energy deposition within a heterogeneous work-piece. This framework consists of path-weighted integral formulations of heat diffusion according to spatial variations in material composition and requires consideration of parameter sensitivity issues.