Combinatorial maps for hierarchical splines
Computational Geometry
Summary
The authors focus on hierarchical splines, which are mathematical tools used in detailed computer modeling. They note that existing methods store only simple lists of small parts, missing helpful information about how these parts connect. To fix this, they developed a new way to organize these connections using a structure called a combinatorial map. Their approach speeds up tasks that rely on these splines by making it easier to understand their layout. They also tested their method to show it can make computations much faster.
Authors
Caleb B. Goates, Kendrick M. Shepherd, Derek C. Thomas
Abstract
Hierarchical splines are an important part of multiscale and adaptive isogeometric analysis formulations. The Bézier meshes of these splines are an essential part of their definition and of several important hierarchical spline algorithms, such as adaptive refinement and Bézier extraction. Topological data associated with the Bézier mesh-such as adjacency information-can be used to improve the performance of many of these algorithms as well as downstream applications of the splines, but typical hierarchical spline formulations do not compute the topological data, storing instead just a list of elements. In this work we present algorithms to build a performant topological data structure, namely the combinatorial map, to represent Bézier meshes of hierarchical splines over cubical cell complexes where the refinement levels have conforming Bézier meshes. This includes hierarchical and truncated hierarchical B-splines, as well as subsets of other hierarchical spline formulations. We show the performance characteristics of the construction algorithms of these hierarchical combinatorial maps, as well as an example use case, showing that the topological information can provide up to an order of magnitude reduction in computation time in downstream applications of the splines.