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arXiv · 2608.29545

Combinatorial maps for hierarchical splines

Caleb B. Goates·Kendrick M. Shepherd·Derek C. Thomas
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Abstract

Hierarchical splines are an important part of multiscale and adaptive isogeometric analysis formulations. The B\'ezier meshes of these splines are an essential part of their definition and of several important hierarchical spline algorithms, such as adaptive refinement and B\'ezier extraction. Topological data associated with the B\'ezier 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 B\'ezier meshes of hierarchical splines over cubical cell complexes where the refinement levels have conforming B\'ezier 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.

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Caleb B. Goates, Kendrick M. Shepherd, Derek C. Thomas. 2026-08-30. Combinatorial maps for hierarchical splines. https://arxiv.org/abs/2608.29545

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