Dimensions of Tri-Quadratic

Abstract

A 3D T-mesh is generally a cuboid grid which allows hanging vertices. Here, a hanging vertex is an interior vertex, but it is not a corner point of eight cells. Spline function spaces with high order smoothness over 3D T-meshes have great application prospect due to their local refinement and relatively low degrees of freedom, for example, 3D isogeometric analysis and implicit representation of surfaces. However, there are still no available dimension formulae of those kinds of spline spaces for application. In this paper, we explore the dimensions of trivariate quadratic spline spaces with

C^1

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continuity over hierarchical 3D T-meshes. By using space embedding method, the problem is converted into a system of linear constraints, and then a lower bound on the dimension of the spline space over a hierarchical 3D T-mesh is provided. For a special type of hierarchical 3D T-meshes, the explicit dimension formula is obtained. In addition, a topological explanation of the dimension is given, which presents a way to construct basis functions.

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Funding

National Natural Science Foundation of China(12171453)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature