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On the dimension of trivariate spline spaces with the highest order smoothness on 3D T-meshes

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Abstract

T-meshes are a type of rectangular partitions of planar domains which allow hanging vertices. Because of the special structure of T-meshes, adaptive local refinement is possible for splines defined on this type of meshes, which provides a solution for the defect of NURBS. In this paper, we generalize the definitions to the three-dimensional (3D) case and discuss a fundamental problem – the dimension of trivariate spline spaces on 3D T-meshes. We focus on a special case where splines are C d−1 continuous for degree d. The smoothing cofactor method for trivariate splines is explored for this situation. We obtain a general dimension formula and present lower and upper bounds for the dimension. At last, we introduce a type of 3D T-meshes, where we can give an explicit dimension formula.

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Acknowledgements

The authors would like to thank Dr. Chunlin Wu and the anonymous referees for the careful reading of the manuscript and providing valuable suggestions that helped improve this paper. This work is supported by Postdoctoral Science Foundation of China (2016M601248) and the NSF of China (11371341).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of NanKai, Tianjin, China

    Chao Zeng

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, China

    Jiansong Deng

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  1. Chao Zeng
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  2. Jiansong Deng
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Correspondence to Chao Zeng.

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Communicated by: Larry L. Schumaker

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Zeng, C., Deng, J. On the dimension of trivariate spline spaces with the highest order smoothness on 3D T-meshes. Adv Comput Math 44, 423–451 (2018). https://doi.org/10.1007/s10444-017-9551-y

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  • DOI: https://doi.org/10.1007/s10444-017-9551-y

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