基于边界网格模型的T样条实体重建
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摘要
为解决零亏格边界网格模型的T样条实体重建问题,提出一种基于八叉树细分和渐进迭代最小二乘拟合算法的T样条实体构建算法.首先给出一种基于体-面-边-点4层几何拓扑的T样条实体数据结构和节点矢量计算算法;接着对边界网格进行参数化,在单位参数立方体和网格模型之间建立参数映射关系,并且采用MVC方法保证参数化结果的单射无自交性;最后实现T样条实体的渐进迭代最小二乘拟合.对sphere模型, head模型和bunny模型进行测试,实现了基于边界网格模型的T样条实体重建,提高了T样条实体构建的效率,并且使得该算法在处理大规模数据时更具优势
A T-spline solid reconstruction algorithm based on octree subdivision and least square progressive iterative approximation(LSPIA) is proposed to solve the problem of reconstruction of T-spline solid from genus-zero boundary mesh. Firstly, this paper presents a cube-face-edge-vertex based four-layer geometry topology T-spline solid data structure and its knot vector calculation algorithm. Then, the parameterization of boundary mesh is realized to build the parametric mapping between the unit cube and the boundary mesh. The MVC parameterization method is also implemented to guarantee the injective mapping and no self-intersection property of the parameterization results. Later, the least square progressive iterative approximation of T-spline solid is implemented. The algorithm presented in this paper has been tested in the case study, which realizes the reconstruction of T-spline solid from boundary mesh. The proposed algorithm improves the efficiency of the reconstruction of T-spline solid and has the advantage in handling large amount of data.
A T-spline solid reconstruction algorithm based on octree subdivision and least square progressive iterative approximation(LSPIA) is proposed to solve the problem of reconstruction of T-spline solid from genus-zero boundary mesh. Firstly, this paper presents a cube-face-edge-vertex based four-layer geometry topology T-spline solid data structure and its knot vector calculation algorithm. Then, the parameterization of boundary mesh is realized to build the parametric mapping between the unit cube and the boundary mesh. The MVC parameterization method is also implemented to guarantee the injective mapping and no self-intersection property of the parameterization results. Later, the least square progressive iterative approximation of T-spline solid is implemented. The algorithm presented in this paper has been tested in the case study, which realizes the reconstruction of T-spline solid from boundary mesh. The proposed algorithm improves the efficiency of the reconstruction of T-spline solid and has the advantage in handling large amount of data.
引文
[1]Cottrell J A,Hughes T J R,Reali A.Studies of refinement and continuity in isogeometric structural analysis[J].Computer Methods in Applied Mechanics and Engineering,2007,196(41-44):4160-4183
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[9]Buffa A,Rivas J,Sangalli G.Some estimates for h-p-k-refinement in isogeometric analysis[J].Numerische Mathematik,2011,118(2):271-305
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[12]Escobar J M,Cascón J M,Rodríguez E,et al.A new approach to solid modeling with trivariate T-splines based on mesh optimization[J].Computer Methods in Applied Mechanics and Engineering,2011,200(45/46):3210-3222
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[14]Floater M S.Parametrization and smooth approximation of surface triangulations[J].Computer Aided Geometric Design,1997,14(3):231-250
[15]Liu L,Zhang Y J,Liu Y.Feature-preserving T-mesh construction using skeleton-based polycubes[J].Computer-Aided Design,2015,58:162-172
[16]Liu L,Zhang Y,Hughes T J R,et al.Volumetric T-spline construction using Boolean operations[J].Engineering with Computers,2014,30(4):425-439
[17]Deng J,Chen F,Feng Y.Dimensions of spline spaces over T-meshes[J].Journal of Computational and Applied Mathematics,2006,194(2):267-283
[18]Zhao Xiangjun.Research on mesh surface modeling[D].Hangzhou:Zhejiang University,2006(in Chinese)(赵向军.网格曲面造型技术研究[D].杭州:浙江大学,2006)
[19]Tong Weihua,Feng Yuyu,Chen Falai.A surface reconstruction algorithm based on implicit T-spline surface[J].Journal of Computer-Aided Design&Computer Graphics,2006,18(3):358-365(in Chinese)(童伟华,冯玉瑜,陈发来.基于隐式T样条的曲面重构算法[J].计算机辅助设计与图形学学报,2006,18(3):358-365)
[20]Tang Yuehong,Li Xiujuan,Cheng Zeming,et al.Closed surface reconstruction based on implicit T-splines[J].Journal of Computer-Aided Design&Computer Graphics,2011,23(2):270-275(in Chinese)(唐月红,李秀娟,程泽铭,等.隐式T样条实现封闭曲面重建[J].计算机辅助设计与图形学学报,2011,23(2):270-275)
[21]Xu Gang,Li Xin,Huang Zhangjin,et al.Geometric computing for isogeometric analysis[J].Journal of Computer Aided Design&Computer Graphics,2015,27(4):570-581(in Chinese)(徐岗,李新,黄章进,等.面向等几何分析的几何计算[J].计算机辅助设计与图形学学报,2015,27(4):570-581)
[22]Xu G,Mourrain B,Duvigneau R,et al.Parameterization of computational domain in isogeometric analysis:methods and comparison[J].Computer Methods in Applied Mechanics and Engineering,2010,200(23):2021-2031
[23]Xu G,Mourrain B,Duvigneau R,et al.Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method[J].Journal of Computational Physics,2013,252(11):275-289
[24]Xu G,Li M,Mourrain B,et al.Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization[J].Computer Methods in Applied Mechanics and Engineering,2018,328:175-200
[25]Xu G,Mourrain B,Duvigneau R,et al.Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications[J].Computer-Aided Design,2013,45(2):395-404
[26]Dijkstra E W.A note on two problems in connexion with graphs[J].Numerische Mathematik,1959,1(1):269-271
[27]Guo Fenghua,Zhang Caiming,Jiao Wenjiang.Research progress on mesh parameterization[J].Journal of Software,2016,27(1):112-135(in Chinese)(郭凤华,张彩明,焦文江.网格参数化研究进展[J].软件学报,2016,27(1):112-135)
[28]Zhou Kun.Digital geometry processing:theory and applications[D].Hangzhou:Zhejiang University,2002(in Chinese)(周昆.数字几何处理:理论与应用[D].杭州:浙江大学,2002)
[29]Hormann K,Polthier K,Sheffer A.Mesh parameterization:theory and practice[C]//Proceedings of ACM SIGGRAPH Asia2008 Courses.New York:ACM Press,2008:Article No.12
[30]Floater M S.Mean value coordinates[J].Computer Aided Geometric Design,2003,20(1):19-27
[31]Tang Rongxi,Wang Jiaye,Peng Qunsheng,et al.Computer graphics tutorial[M].Beijing:Science Press,1990(in Chinese)(唐荣锡,汪嘉业,彭群生,等.计算机图形学教程[M].北京:科学出版社,1990)
[32]Deng C Y,Lin H W.Progressive and iterative approximation for least squares B-spline curve and surface fitting[J].Computer-Aided Design,2014,47:32-44
[2]Shi Fazhong.CAGD&NURBS[M].Beijing:Higher Education Press,2001(in Chinese)(施法中.计算机辅助几何设计与非均匀有理B样条[M].北京:高等教育出版社,2001)
[3]Piegl L,Tiller W.The NURBS book[M].Beijing:Tsinghua University Press,2010(in Chinese)(Piegl L,Tiller W.非均匀有理B样条[M].赵罡,等译.北京:清华大学出版社,2010)
[4]Sederberg T W,Zheng J,Bakenov A,et al.T-splines and T-NURCCs[J].ACM Transactions on Graphics,2003,22(3):477-484
[5]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
[6]Sederberg T W,Finnigan G T,Li X,et al.Watertight trimmed NURBS[J].ACM Transactions on Graphics,2008,27(3):Article No.79
[7]D?rfel M R,Jüttler B,Simeon B.Adaptive isogeometric analysis by local h-refinement with T-splines[J].Computer Methods in Applied Mechanics and Engineering,2010,199(s5-8):264-275
[8]Scott M A,Li X,Sederberg T W,et al.Local refinement of analysis-suitable T-splines[J].Computer Methods in Applied Mechanics and Engineering,2012,213-216(1):206-222
[9]Buffa A,Rivas J,Sangalli G.Some estimates for h-p-k-refinement in isogeometric analysis[J].Numerische Mathematik,2011,118(2):271-305
[10]Hughes T J R,Cottrell J A,Bazilevs Y.Isogeometric analysis:CAD,finite elements,NURBS,exact geometry and mesh refinement[J].Computer Methods in Applied Mechanics and Engineering,2005,194(39-41):4135-4195
[11]Austin Cottrell J,Hughes T J R,Bazilevs Y.Isogeometric analysis:toward integration of CAD and FEA[M].New York:John Wiley&Sons,2009
[12]Escobar J M,Cascón J M,Rodríguez E,et al.A new approach to solid modeling with trivariate T-splines based on mesh optimization[J].Computer Methods in Applied Mechanics and Engineering,2011,200(45/46):3210-3222
[13]Zhang Y,Wang W,Hughes T J R.Solid T-spline construction from boundary representations for genus-zero geometry[J].Computer Methods in Applied Mechanics and Engineering,2012,249-252:185-197
[14]Floater M S.Parametrization and smooth approximation of surface triangulations[J].Computer Aided Geometric Design,1997,14(3):231-250
[15]Liu L,Zhang Y J,Liu Y.Feature-preserving T-mesh construction using skeleton-based polycubes[J].Computer-Aided Design,2015,58:162-172
[16]Liu L,Zhang Y,Hughes T J R,et al.Volumetric T-spline construction using Boolean operations[J].Engineering with Computers,2014,30(4):425-439
[17]Deng J,Chen F,Feng Y.Dimensions of spline spaces over T-meshes[J].Journal of Computational and Applied Mathematics,2006,194(2):267-283
[18]Zhao Xiangjun.Research on mesh surface modeling[D].Hangzhou:Zhejiang University,2006(in Chinese)(赵向军.网格曲面造型技术研究[D].杭州:浙江大学,2006)
[19]Tong Weihua,Feng Yuyu,Chen Falai.A surface reconstruction algorithm based on implicit T-spline surface[J].Journal of Computer-Aided Design&Computer Graphics,2006,18(3):358-365(in Chinese)(童伟华,冯玉瑜,陈发来.基于隐式T样条的曲面重构算法[J].计算机辅助设计与图形学学报,2006,18(3):358-365)
[20]Tang Yuehong,Li Xiujuan,Cheng Zeming,et al.Closed surface reconstruction based on implicit T-splines[J].Journal of Computer-Aided Design&Computer Graphics,2011,23(2):270-275(in Chinese)(唐月红,李秀娟,程泽铭,等.隐式T样条实现封闭曲面重建[J].计算机辅助设计与图形学学报,2011,23(2):270-275)
[21]Xu Gang,Li Xin,Huang Zhangjin,et al.Geometric computing for isogeometric analysis[J].Journal of Computer Aided Design&Computer Graphics,2015,27(4):570-581(in Chinese)(徐岗,李新,黄章进,等.面向等几何分析的几何计算[J].计算机辅助设计与图形学学报,2015,27(4):570-581)
[22]Xu G,Mourrain B,Duvigneau R,et al.Parameterization of computational domain in isogeometric analysis:methods and comparison[J].Computer Methods in Applied Mechanics and Engineering,2010,200(23):2021-2031
[23]Xu G,Mourrain B,Duvigneau R,et al.Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method[J].Journal of Computational Physics,2013,252(11):275-289
[24]Xu G,Li M,Mourrain B,et al.Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization[J].Computer Methods in Applied Mechanics and Engineering,2018,328:175-200
[25]Xu G,Mourrain B,Duvigneau R,et al.Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications[J].Computer-Aided Design,2013,45(2):395-404
[26]Dijkstra E W.A note on two problems in connexion with graphs[J].Numerische Mathematik,1959,1(1):269-271
[27]Guo Fenghua,Zhang Caiming,Jiao Wenjiang.Research progress on mesh parameterization[J].Journal of Software,2016,27(1):112-135(in Chinese)(郭凤华,张彩明,焦文江.网格参数化研究进展[J].软件学报,2016,27(1):112-135)
[28]Zhou Kun.Digital geometry processing:theory and applications[D].Hangzhou:Zhejiang University,2002(in Chinese)(周昆.数字几何处理:理论与应用[D].杭州:浙江大学,2002)
[29]Hormann K,Polthier K,Sheffer A.Mesh parameterization:theory and practice[C]//Proceedings of ACM SIGGRAPH Asia2008 Courses.New York:ACM Press,2008:Article No.12
[30]Floater M S.Mean value coordinates[J].Computer Aided Geometric Design,2003,20(1):19-27
[31]Tang Rongxi,Wang Jiaye,Peng Qunsheng,et al.Computer graphics tutorial[M].Beijing:Science Press,1990(in Chinese)(唐荣锡,汪嘉业,彭群生,等.计算机图形学教程[M].北京:科学出版社,1990)
[32]Deng C Y,Lin H W.Progressive and iterative approximation for least squares B-spline curve and surface fitting[J].Computer-Aided Design,2014,47:32-44