Semi-standard Local Refinement Algorithm of Uniform T-grid Splines Surface
-
摘要: 为解决T样条曲面造型中控制点插入的半规范问题,提出了一种基于均匀T网格样条曲面的半规范局部细分算法。首先根据T样条曲面定义,将B样条曲面转换为均匀T网格样条曲面;然后在任意单一网格上进行控制点插入,或将网格进行嵌套细分。为满足曲面局部特征细分要求,建立超定方程组,用神经网络求解额外系数,添加于特定控制点上,以保证T样条曲面的半规范性;最后,通过额外系数计算新的混合函数,并构造最终曲面。并通过实例证明该算法的可行性,与现有的T样条的相比,曲面在达到相同精度时,该算法减少了多余控制点。Abstract: In order to solve the semi-standard problem of control point insertion in the T-splines surface modeling, a semi-standard local refinement algorithm based on the uniform t-grid splines surface is proposed. Firstly, according to the definition of T-splines surface, B-splines surface is transformed into uniform T-grids splines surface, then control points are inserted on any single grids, and the grids can also be nested and subdivided. In order to meet the refinement requirements of local surface features, the additional coefficients need to be added to some control points to ensure the semi-standard of T-splines surface. The least square method is used to solve the extra coefficients by establishing the overdetermined equation, and the blending function of the initial uniform T-grid is transformed into the blending function of the final T-grids by the transformation matrix. The algorithm is proved to be feasible through examples. Comparing with the local refinement of T-splines surface, when the curved surface reaches the same precision, this method reduces redundant control points.
-
Key words:
- T-spline /
- semi-standard /
- B-spline /
- blending function
-
表 1 n=1时额外系数值
1 2 ${ { {\rm{15} } } }/{ { {\rm{16} } } }$ ${{{\rm{15}}}}/{{{\rm{16}}}}$ ${{{\rm{15}}}}/{{{\rm{16}}}}$ ${{{\rm{15}}}}/{{{\rm{16}}}}$ 表 2 n=2时额外系数值
1 2 3 4 5 ${{{\rm{15}}}}/{{{\rm{16}}}}$ ${{{\rm{63}}}}/{{{\rm{64}}}}$ ${{{\rm{127}}}}/{{{\rm{128}}}}$ ${{{\rm{63}}}}/{{{\rm{64}}}}$ ${{{\rm{15}}}}/{{{\rm{16}}}}$ ${{{\rm{63}}}}/{{{\rm{64}}}}$ ${{{\rm{255}}}}/{{{\rm{256}}}}$ ${{{\rm{511}}}}/{{{\rm{512}}}}$ ${{{\rm{255}}}}/{{{\rm{256}}}}$ ${{{\rm{63}}}}/{{{\rm{64}}}}$ ${{{\rm{127}}}}/{{{\rm{128}}}}$ ${{{\rm{511}}}}/{{{\rm{512}}}}$ ${ { {\rm{1\;023} } } }/{ { {\rm{1\;024} } } }$ ${{{\rm{511}}}}/{{{\rm{512}}}}$ ${{{\rm{127}}}}/{{{\rm{128}}}}$ ${{{\rm{63}}}}/{{{\rm{64}}}}$ ${{{\rm{255}}}}/{{{\rm{256}}}}$ ${{{\rm{511}}}}/{{{\rm{512}}}}$ ${{{\rm{255}}}}/{{{\rm{256}}}}$ ${{{\rm{63}}}}/{{{\rm{64}}}}$ ${{{\rm{15}}}}/{{{\rm{16}}}}$ ${{{\rm{63}}}}/{{{\rm{64}}}}$ ${{{\rm{127}}}}/{{{\rm{128}}}}$ ${{{\rm{63}}}}/{{{\rm{64}}}}$ ${{{\rm{15}}}}/{{{\rm{16}}}}$ 表 3 n=3时额外系数值
1 2 3 4 5 6 7 8 9 ${{{\rm{527}}}}/{{{\rm{576}}}}$ ${{{\rm{1\;487}}}}/{{{\rm{1\;536}}}}$ ${{{\rm{1\;003}}}}/{{{\rm{1\;024}}}}$ ${{{\rm{505}}}}/{{{\rm{512}}}}$ ${{{\rm{3\;037}}}}/{{{\rm{3\;072}}}}$ ${{{\rm{505}}}}/{{{\rm{512}}}}$ ${{{\rm{1\;003}}}}/{{{\rm{1\;024}}}}$ ${{{\rm{1\;487}}}}/{{{\rm{1\;536}}}}$ ${{{\rm{527}}}}/{{{\rm{576}}}}$ ${{{\rm{1\;487}}}}/{{{\rm{1\;536}}}}$ ${{{\rm{4\;047}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;129}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;157}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;129}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;047}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{1\;487}}}}/{{{\rm{1\;536}}}}$ ${{{\rm{1\;003}}}}/{{{\rm{1\;024}}}}$ ${{{\rm{8\;129}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;303}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{8\;165}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;339}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{8\;165}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;303}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{1\;003}}}}/{{{\rm{1\;024}}}}$ ${{{\rm{505}}}}/{{{\rm{512}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;065}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;087}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;177}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;087}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;157}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{505}}}}/{{{\rm{512}}}}$ ${{{\rm{3\;037}}}}/{{{\rm{3\;072}}}}$ ${{{\rm{8\;157}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;339}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{8\;177}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;359}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{8\;177}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;339}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{8\;157}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{3\;037}}}}/{{{\rm{3\;072}}}}$ ${{{\rm{505}}}}/{{{\rm{512}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;165}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;087}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;177}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;087}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;165}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{505}}}}/{{{\rm{512}}}}$ ${{{\rm{1\;003}}}}/{{{\rm{1\;024}}}}$ ${{{\rm{8\;129}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;303}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{8\;165}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;339}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{8\;165}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{16\;303}}}}/{{{\rm{16\;384}}}}$ ${{{\rm{8\;129}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{1\;003}}}}/{{{\rm{1\;024}}}}$ ${{{\rm{1\;487}}}}/{{{\rm{1\;536}}}}$ ${{{\rm{4\;047}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;129}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;157}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;075}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{8\;129}}}}/{{{\rm{8\;192}}}}$ ${{{\rm{4\;047}}}}/{{{\rm{4\;096}}}}$ ${{{\rm{1\;487}}}}/{{{\rm{1\;536}}}}$ ${{{\rm{527}}}}/{{{\rm{576}}}}$ ${{{\rm{1\;487}}}}/{{{\rm{1\;536}}}}$ ${{{\rm{1\;003}}}}/{{{\rm{1\;024}}}}$ ${{{\rm{505}}}}/{{{\rm{512}}}}$ ${{{\rm{3\;037}}}}/{{{\rm{3\;072}}}}$ ${{{\rm{505}}}}/{{{\rm{512}}}}$ ${{{\rm{1\;003}}}}/{{{\rm{1\;024}}}}$ ${{{\rm{1\;487}}}}/{{{\rm{1\;536}}}}$ ${{{\rm{527}}}}/{{{\rm{576}}}}$ -
[1] SEDERBERG T W, ZHENG J M, BAKENOV A, et al. T-splines and T-NURCCs[J]. ACM Transactions on Graphics, 2003, 22(3): 477-484 doi: 10.1145/882262.882295 [2] SEDERBERG T W, CARDON D L, FINNIGAN G T, et al. T-spline simplification and local refinement[J]. ACM Transactions on Graphics, 2004, 23(3): 276-283 doi: 10.1145/1015706.1015715 [3] GUO M Y, ZHAO G, WANG W, et al. T-splines for isogeometric analysis of two-dimensional nonlinear problems[J]. Computer Modeling in Engineering & Sciences, 2020, 123(2): 821-843 [4] LI X, SCOTT M A. Analysis-suitable T-splines: characterization, refineability, and approximation[J]. Mathematical Models and Methods in Applied Sciences, 2014, 24(6): 1141-1164 doi: 10.1142/S0218202513500796 [5] ZHANG J J, LI X. On the linear independence and partition of unity of arbitrary degree analysis-suitable T-splines[J]. Communications in Mathematics and Statistics, 2015, 3(3): 353-364 doi: 10.1007/s40304-015-0064-z [6] CHEN L, DE BORST R. Locally refined T-splines[J]. International Journal for Numerical Methods in Engineering, 2018, 114(6): 637-659 doi: 10.1002/nme.5759 [7] DENG J S, CHEN F L, LI X, et al. Polynomial splines over hierarchical T-meshes[J]. Graphical Models, 2008, 70(4): 76-86 doi: 10.1016/j.gmod.2008.03.001 [8] LI X, DENG J S, CHEN F L. Polynomial splines over general T-meshes[J]. The Visual Computer, 2010, 26(4): 277-286 doi: 10.1007/s00371-009-0410-9 [9] ATRI H R, SHOJAEE S. Meshfree truncated hierarchical refinement for isogeometric analysis[J]. Computational Mechanics, 2018, 62(6): 1583-1597 doi: 10.1007/s00466-018-1580-y [10] WEI X D, ZHANG Y J, LIU L, et al. Truncated T-splines: fundamentals and methods[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 349-372 doi: 10.1016/j.cma.2016.07.020 [11] GIANNELLI C, JÜTTLER B, SPELEERS H. THB-splines: the truncated basis for hierarchical splines[J]. Computer Aided Geometric Design, 2012, 29(7): 485-498 doi: 10.1016/j.cagd.2012.03.025 [12] SCOTT M A, LI X, SEDERBERG T W, et al. Local refinement of analysis-suitable T-splines[J]. Methods in Applied Mechanics and Engineering, 2012, 213-216: 206-222 doi: 10.1016/j.cma.2011.11.022 [13] LI X, ZHANG J J. AS++ T-splines: Linear independence and approximation[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 333: 462-474 doi: 10.1016/j.cma.2018.01.041 [14] DOKKEN T, LYCHE T, PETTERSEN K F. Polynomial splines over locally refined box-partitions[J]. Computer Aided Geometric Design, 2013, 30(3): 331-356 doi: 10.1016/j.cagd.2012.12.005 [15] BRESSAN A. Some properties of LR-splines[J]. Computer Aided Geometric Design, 2013, 30(8): 778-794 doi: 10.1016/j.cagd.2013.06.004 [16] JOHANNESSEN K A, KVAMSDAL T, DOKKEN T. Isogeometric analysis using LR B-splines[J]. Computer Methods in Applied Mechanics and Engineering, 2014, 269: 471-514 doi: 10.1016/j.cma.2013.09.014 [17] EVANS E J, SCOTT M A, LI X, et al. Hierarchical T-splines: analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 1-20 doi: 10.1016/j.cma.2014.05.019 [18] BRACCO C, GIANNELLI C, VÁZQUEZ R. Refinement algorithms for adaptive isogeometric methods with hierarchical splines[J]. Axioms, 2018, 7(3): 43 doi: 10.3390/axioms7030043 [19] VUONG A V, GIANNELLI C, JÜTTLER B, et al. A hierarchical approach to adaptive local refinement in isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(49-52): 3554-3567 doi: 10.1016/j.cma.2011.09.004 [20] NGUYEN-THANH N, NGUYEN-XUAN H, BORDAS S P A, et al. Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(21-22): 1892-1908 doi: 10.1016/j.cma.2011.01.018 [21] 胡海涛, 纪小刚, 张溪溪, 等. 均匀T网格样条曲面的局部细分算法[J]. 机械科学与技术, 2020, 39(5): 743-750HU H T, JI X G, ZHANG X X, et al. local subdivision algorithm for uniform T-grid spline surface[J]. Mechanical Science and Technology for Aerospace Engineering, 2020, 39(5): 743-750 (in Chinese)