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Abstract

Based on Grimm and Hughes (1995) we introduce and study a mathematical framework for analysis-suitable unstructured B-spline spaces. In this setting the parameter domain has a manifold structure which allows for the definition of function spaces such as, for instance, B-splines over multi-patch domains with extraordinary points or analysis-suitable unstructured T-splines. Within this framework, we generalize the concept of dual-compatible B-splines (developed for structured T-splines in Beirão da Veiga et al. (2013)). This allows us to prove the key properties that are needed for isogeometric analysis, such as linear independence and optimal approximation properties for h-refined meshes.

6. Conclusion and possible extensions

We have introduced a general mathematical framework, based on manifolds, for the definition and the analysis of unstructured spline spaces. As it is done in Grimm and Hughes (1995) the main idea is to decompose the domain into charts, which are meshed with quadrilaterals or hexahedra. Then spline basis functions and dual functionals can be defined locally on each chart. Unstructured charts are necessary to cover extraordinary vertices and edges of the domain. We have used this framework to generalize the dual-compatibility condition of Beirão da Veiga et al. (2012a, 2013) to unstructured spline spaces, and in particular to analyze the approximation properties of splines with high continuity everywhere except in the vicinity of extraordinary vertices and edges, where the continuity is only C0. Although the analysis was restricted to the low continuity case, the framework allows for the definition of spline functions with higher smoothness, and their analysis will be the aim of future work. In our definitions the physical domain is necessarily a manifold. However, since we are defining the charts in the parametric domain, and not in the physical domain, it is possible to extend our framework to non-manifold domains using special bifurcation charts (such as T-shaped or X-shaped charts for curves, etc.) This could be of interest for certain beam or shell formulations, or for the proper representation of the medial axis or medial surface of an object, for instance. Finally, for the sake of simplicity we have restricted ourselves to B-splines and T-splines on quadrilateral/hexahedral meshes. The framework can be generalized to other spline spaces, such as NURBS or trigonometric splines.