Multiscale shape-material modeling by composition

We propose a formal framework for modeling multiscale material structures by recursive composition of two-scale material structures. The framework comprises three components: (1) single scale shape-material models, supported by single scale queries, to represent the geometry and spatial distribution of material property on each coarse and fine scales, (2) mechanisms to link the scales by establishing an explicit relationship between shape-material properties at fine scale and material properties at the coarse scale, and (3) multiscale queries abstracting fundamental multiscale operations by recursive composition. While the first component is consistent with classical solid heterogeneous material modeling, the second component manifests itself as a pair of conceptually new upscaling and downscaling functions. We show that classical solid modeling queries, exemplified by point membership testing, distance computation, and material evaluation, generalize to the corresponding multiscale queries that support implicit representations of multiscale structures as a composition of distinct single scale solid material models. The concept of neighborhood is indispensable in all three components. The framework provides a formal and consistent extension of solid modeling framework that underlies most commercial systems in use today, encompasses the variety of different approaches to multiscale modeling, identifies open issues and research problems with existing two-scale modeling methods, and provides foundations for next-generation systems by identifying key objects, classes, representation schemes, and API queries.