| In this thesis, the concept of modeling shapes has been extended to model continuously varying material properties, where material properties may be modeled by prescribing their values and rates at a collection of material features, along with additional global constants. The central notion of the representation scheme is a parametrization of space by distance from material features---either exactly or approximately. Distance fields are employed widely in material modeling to construct models, to represent models, or to perform both. The proposed distance field based representation subsumes almost all existing representation schemes for continuously varying material properties and generalizes the concept of feature to any shapes and dimensions, in addition to ensuring smoothness on constructed material functions.;The two key components of the representation scheme which depend on normalized distance fields with guaranteed differential properties are: generalized Taylor series with distance fields, for specifying material properties at an individual feature; and inverse distance interpolation to combine specified material functions at multiple features. In this thesis, it will be demonstrated that normalized distance fields of any shape can be constructed from their polygonal approximations. Viability of the representation scheme has been demonstrated for bone remodeling problems, where significant variation in feature shapes and material properties can be observed.;In addition, the representation scheme is theoretically complete in the sense that it allows representation of all material property functions. The numerous advantages of this representation scheme include: precise and intuitive control using explicit, analytic, differential, and integral constraints specified on the native geometry; guaranteed smoothness and analytic properties without meshing; and applicability to material features of arbitrary dimension, shape, and topology. |
