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Modified Bases of PHT-Splines

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Abstract

Recently, it was found that during the process of certain refinement of hierarchical T-meshes, some basis functions of PHT-splines decay severely, which is not expected in solving numerical PDEs and in least square data fitting since the matrices assembled by these basis functions are likely to be ill-conditioned. In this paper, we present a method to modify the basis functions of PHT-splines in the case that the supports of the original truncated basis functions are rectangular domains to overcome the decay problem. The modified basis functions preserve the same nice properties of the original PHT-spline basis functions such as partition of unity, local support, linear independency. Numerical examples show that the modified basis functions can greatly decrease the condition numbers of the stiffness matrices assembled in solving Poisson’s equation with Dirichlet boundary conditions.

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References

  1. Sederberg, T.W., Cardon, D.L., Finnigan, G.T., North, N.S., Zheng, J., Lyche, T.: T-spline simplification and local refinement. ACM Trans. Graph. 23, 276–283 (2004)

    Article  Google Scholar 

  2. Dörfel, M., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local h-refinement with T-splines. Comput. Methods Appl. Mech. Eng. 199, 264–275 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bazilevs, Y., Calo, V.M., Cottrell, J.A., Evans, J.A., Hughes, T.J.R., Lipton, S., Scott, M.A., Sederberg, T.W.: Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Eng. 199, 229–263 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buffa, A., Cho, D., Sangalli, G.: Linear independence of the T-spline blending functions associated with some particular T-meshes. Comput. Methods Appl. Mech. Eng. 199, 1437–1445 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Scott, M.A., Li, X., Sederberg, T.W., Hughes, T.J.R.: Local refinement of analysis-suitable T-splines. Comput. Methods Appl. Mech. Eng. 213–216, 206–222 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Li, X., Scott, M.A.: Analysis-suitable T-splines: characterization, refineability, and approximation. Math. Models Methods Appl. Sci. 24, 1141–1164 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang, J.J., Li, X.: On the linear independence and partition of unity of arbitrary degree analysis-suitable T-splines. Commun. Math. Stat. 3, 353–364 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Forsey, D., Bartels, R.: Hierarchical B-spline refinement. Comput. Graph. 22, 205–212 (1988)

    Article  Google Scholar 

  9. Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29, 485–498 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Des. 30, 331–356 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Johannessen, K.A., Remonato, F., Kvamsdal, T.: On the similarities and differences between classical hierarchical, truncated hierarchical and LR B-splines. Comput. Methods Appl. Mech. Eng. 291, 64–101 (2015)

    Article  MathSciNet  Google Scholar 

  12. Deng, J.S., Chen, F.L., Feng, Y.Y.: Dimensions of spline spaces over T-meshes. J. Comput. Appl. Math. 194, 267–283 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deng, J.S., Chen, F.L., Li, X., Hu, C.Q., Tong, W.H., Yang, Z.W., Feng, Y.Y.: Polynomial splines over hierarchical T-meshes. Graph. Models 70, 76–86 (2008)

    Article  Google Scholar 

  14. Li, X., Deng, J.S., Chen, F.L.: Surface modeling with polynomial splines over hierarchical T-meshes. Vis. Comput. 23, 1027–1033 (2007)

    Article  Google Scholar 

  15. Tian, L., Chen, F.L., Du, Q.: Adaptive finite element methods for elliptic equations over hierarchical T-meshes. J. Comput. Appl. Math. 236, 878–891 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nguyen-Thanh, N., Nguyen-Xuan, H., Bordas, S.P.A., Rabczuk, T.: Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids. Comput. Methods Appl. Mech. Eng. 200, 1892–1908 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nguyen-Thanh, N., Kiendl, J., Nguyen-Xuan, H., Wüchner, R., Bletzinger, K.U., Bazilevs, Y., Rabczuk, T.: Rotation free isogeometric thin shell analysis using PHT-splines. Comput. Methods Appl. Mech. Eng. 200, 3410–3424 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, P., Xu, J.L., Deng, J.S., Chen, F.L.: Adaptive isogeometric analysis using rational PHT-splines. Comput. Aided Des. 43, 1438–1448 (2011)

    Article  Google Scholar 

  19. Deng, F., Zeng, C., Deng, J.S.: Boundary-mapping parametrization in isogeometric analysis. Commun. Math. Stat. 4, 203–216 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kang, H.M., Xu, J.L., Chen, F.L., Deng, J.S.: A new basis for PHT-splines. Graph. Models 82, 149–159 (2015)

    Article  Google Scholar 

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Acknowledgements

We thank the anonymous reviewers for their valuable remarks for improvements. The research is supported by the National Natural Science Foundation of China (Nos. 11571338, 11626253) and Postdoctoral Science Foundation of China (2015M571931).

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  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Yuanpeng Zhu & Falai Chen

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  1. Yuanpeng Zhu
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  2. Falai Chen
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Correspondence to Falai Chen.

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Zhu, Y., Chen, F. Modified Bases of PHT-Splines. Commun. Math. Stat. 5, 381–397 (2017). https://doi.org/10.1007/s40304-017-0116-7

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  • DOI: https://doi.org/10.1007/s40304-017-0116-7

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