Abstract
In this paper, we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph (CVR graph). We propose a novel structure, by which we offer an effective and easy operative method for constructing the basis functions of the biquadratic spline space. The mapping we construct is an isomorphism. The basis functions of the biquadratic spline space hold the properties such as linear independence, completeness and the property of partition of unity, which are the same as the properties for the basis functions of piecewise constant space over the CVR graph. To demonstrate that the new basis functions are efficient, we apply the basis functions to fit some open surfaces.
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Berdinsky, D., Oh, M., Kim, T., Mourrain, B.: On the problem of instability in the dimension of a spline space over a T-mesh. Comput. Graph. 36(5), 507–513 (2012)
Bernard, M.: On the dimension of spline spaces on planar T-meshes. Math. Comput. 83(286), 847–871 (2014)
Bressan, A.: Some properties of LR-splines. Comput. Aided Geometr. Design 30, 778–794 (2013)
Buffa, A., Cho, D., Sangalli, G.: Linear independence of the T-spline blending functions associated with some particular T-meshes. Comput. Methods Appl. Mech. Engi. 199(23–24), 1437–1445 (2010)
Deng, J., Chen, F., Feng, Y.: Dimensions of spline spaces over T-meshes. J. Comput. Appl. Math. 194(2), 267–283 (2006)
Deng, J., Chen, F., Li, X., Hu, C., Tong, W., Yang, Z., Feng, Y.: Polynomial splines over hierarchical T-meshes. Graph. Models 70(4), 76–86 (2008)
Deng, F., Zeng, C., Meng, W., Deng, J.: Bases of biquadratic polynomial spline spaces over hierarchical T-meshes. J. Comput. Math. 35(1), 91–120 (2017)
Deng, J., Chen, F., Jin, L.: Dimensions of biquadratic spline spaces over T-meshes. J. Comput. Appl. Math. 238, 68–94 (2013)
Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geometr. Design 30(3), 331–356 (2013)
Feng, Y., Zeng, F., Deng, J.: Spline Functions and Approximaition Theory. USTC Press, London (2013)
Floater, M.S.: Parameterization and smooth approximation of surface triangulations. Comput. Aided Geometr. Design 14(3), 231–250 (1997)
Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geometr. Design 29(7), 485–498 (2012)
Giannelli, C., Jüttler, B.: Bases and dimensions of bivariate hierarchical tensor-product splines. J. Comput. Appl. Math. 239, 162–178 (2013)
Li, X., Zheng, J., Sederberg, T.W., Hughes, T.J.R., Scott, M.A.: On linear independence of T-spline blending functions. Comput. Aided Geometr. Design 29(1), 63–76 (2012)
Li, X.: On the instability in the dimension of spline space over particular T-meshes. Comput. Aided Geometr. Design 28(7), 420–426 (2011)
Li, X., Deng, J.: On the dimension of splines spaces over T-meshes with smoothing cofactor-conformality method. Comput. Aided Geometr. Design 41, 76–86 (2016)
Mokriš, D., Jüttler, B.: TDHB-splines: the truncated decoupled basis of hierarchical tensor-product splines. Comput. Aided Geometr. Design 31(7–8), 531–544 (2014)
Mokriš, D., Jüttler, B., Giannelli, C.: On the completeness of hierarchical tensor-product B-splines. J. Comput. Appl. Math. 271, 53–70 (2014)
Meng, W., Deng, J., Chen, F.: Dimension of spline spaces with highest order smoothness over hierarchical T-meshes. Comput. Aided Geometr. Design 30(1), 20–34 (2013)
Patrizi, F., Dokken, T.: Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes. Comput. Aided Geometr. Design 2020, 77 (2020)
Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCs. ACM Trans. Graph. 22(3), 477–484 (2003)
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(3), 276–283 (2004)
Schumaker, L.L., Wang, L.: Approximation power of polynomial splines on T-meshes. Comput. Aided Geometr. Design 29(8), 599–612 (2012)
Thanh, N.N., Li, W.: Static and free-vibration analyses of cracks in thin-shell structures based on an isogeometric-meshfree coupling approach. Comput. Mech. 62(6), 1287–1309 (2018)
Thanh, N.N., Li, W., Huang, J., Zhou, K.: An adaptive isogeometric analysis meshfree collocation method for elasticity and frictional contact problems. Int. J. Numer. Methods Eng. 120(2), 209–230 (2019)
Thanh, N.N., Zhou, K.: Extended isogeometric analysis based on PHT-splines for crack propagation near inclusions. Int. J. Numer. Methods Eng. 112(12), 1777–1800 (2017)
Toshniwala, D., Mourrain, B., Hughes, Thomas, J.R.: Polynomial spline spaces of non-uniform bi-degree on T-meshes: combinatorial bounds on the dimension. arXiv preprint arXiv:1903.05949 (2019)
Toshniwala, D., Villamizar, N.: Dimension of polynomial splines of mixed smoothness on T-meshes. Comput. Aided Geometr. Design 80, 101880 (2020)
Vuong, A.-V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200(49–52), 3554–3567 (2011)
Zeng, C., Deng, F., Li, X., Deng, J.: Dimensions of biquadratic and bicubic spline spaces over hierarchical T-meshes. J. Comput. Appl. Math. 287, 162–178 (2015)
Zeng, C., Meng, W., Deng, F., Deng, J.: Dimensions of spline spaces over non-rectangular T-meshes. Adv. Comput. Math. 42(6), 1259–1286 (2016)
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The authors are supported by the NSF of China (No. 11771420, No. 12001197).
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Liu, J., Deng, F. & Deng, J. Space Mapping of Spline Spaces over Hierarchical T-meshes. Commun. Math. Stat. 11, 403–438 (2023). https://doi.org/10.1007/s40304-021-00258-3
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DOI: https://doi.org/10.1007/s40304-021-00258-3