基于网格数据的CAD到CAE无损映射
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摘要
基于特征的实体建模技术的CAD系统主要进行产品的结构设计,CAE系统则用来进行产品的结构力学性能分析,CAD与CAE系统的高度集成能够更高效地完成产品的设计过程。现有的CAD、CAE系统往往采用不同造型引擎和系统精度,因此不存在既能用于设计又能用于分析的统一模型。以几何模型为桥梁实现不同系统之间模型转换的方法往往出现数据的丢失与冗余。
鉴于几何模型实现模型转换的种种弊端,本文采用网格模型代替几何模型实现从CAD到CAE系统的无损数据映射。针对CAD系统生成的网格质量不高的问题,本文提出了一种新的四面体网格优化方法。在网格优化过程中需要保留边界网格包含的特征信息,因此研究网格模型特征信息的提取显得非常必要。本文的研究内容主要体现为:
1.回顾了CAD/CAE集成的国内外研究现状,分析了各种方法的优点与不足,提出了本文的研究内容和结构安排。
2.深入分析了采用几何模型实现模型转换的失败原因,介绍了基于网格模型的CAD到CAE的无损映射的基本原理、基本算法;以CAD平台UG NX5为例介绍了其单元和节点信息的提取方法;分析了常用CAE系统的网格信息的基本格式,并成功实现从UG到ANSYS、Abaqus等大型CAE系统的数据映射。
3.使用几何单元转换原理对四面体网格进行优化,介绍了四面体单元质量的度量准则,详细描述了单元转换原理的并行算法和顺序算法,并用单位球模型、胯关节模型进行验证。对同一模型使用不同的四面体网格优化算法进行优化,通过结果对比分析各种方法的优缺点。
4.介绍了微分几何的基本理论,通过分析常用方法不足,提出脊线提取的改进算法,包括脐区检测、脐点检测、脊点检测以及脊类型确定的基本算法。最后通过实例来验证本方法。
Feature-based solid modeling techniques CAD system has been used for product structure design, while CAE system has been used for analysis of product mechanical properties. Obviously, the high integration of CAD and CAE will efficiently accomplish the product design process. Unfortunately, the existing CAD and CAE system inherently using different geometry engine and modeling accuracy exists no generic, unified model which could be used both for design and for analysis. Model transformation based on geometric model among different system usually lead to data loss and data redundancy.
Aiming to the drawbacks of geometry model-based transformation, this paper adopt mesh model instead of geometry model to realise lossless data mapping from CAD to CAE system. Meanwhile it proposes a mesh optimization method toward meshes generated from CAD system, and extracts boundary feature information from the mesh. The research is primarily reflected in:
1. Giving a review of research status about CAD/CAE integration at home and abroad, the merits and faults are analyed, then puting forward the research content and structure arrangement in this paper.
2. Reseaching failure cause of the geometry model-based transformation in depth, introducing basic principle of lossless data mapping for mesh model-based transformation form CAD to CAE system. It also presents how to extract node and element information from CAD platform-UG NX5. It analyzes file format of mesh data in exiting CAE system and successfully realize data transformation from UG to ANSYS、Abaqus and so on.
3. Using element transformation theory smooths tetrahedron mesh, introducing tetrahedron element quality measure rule, describing the simultaneous smoothing algorithm sequential smoothing algorithm, and verifying this method using unit sphere and modular hip endoprosthesis model. According to result compared with traditonal method shows the superiority of this method.
4. Introducing basic theories of differential geometry, proposing the corrective method about ridge extracting including detection region of umbilicus, umbilicus detection, ridge detection and classification of ridge. Finally, using sevesal examples verify this method.
鉴于几何模型实现模型转换的种种弊端,本文采用网格模型代替几何模型实现从CAD到CAE系统的无损数据映射。针对CAD系统生成的网格质量不高的问题,本文提出了一种新的四面体网格优化方法。在网格优化过程中需要保留边界网格包含的特征信息,因此研究网格模型特征信息的提取显得非常必要。本文的研究内容主要体现为:
1.回顾了CAD/CAE集成的国内外研究现状,分析了各种方法的优点与不足,提出了本文的研究内容和结构安排。
2.深入分析了采用几何模型实现模型转换的失败原因,介绍了基于网格模型的CAD到CAE的无损映射的基本原理、基本算法;以CAD平台UG NX5为例介绍了其单元和节点信息的提取方法;分析了常用CAE系统的网格信息的基本格式,并成功实现从UG到ANSYS、Abaqus等大型CAE系统的数据映射。
3.使用几何单元转换原理对四面体网格进行优化,介绍了四面体单元质量的度量准则,详细描述了单元转换原理的并行算法和顺序算法,并用单位球模型、胯关节模型进行验证。对同一模型使用不同的四面体网格优化算法进行优化,通过结果对比分析各种方法的优缺点。
4.介绍了微分几何的基本理论,通过分析常用方法不足,提出脊线提取的改进算法,包括脐区检测、脐点检测、脊点检测以及脊类型确定的基本算法。最后通过实例来验证本方法。
Feature-based solid modeling techniques CAD system has been used for product structure design, while CAE system has been used for analysis of product mechanical properties. Obviously, the high integration of CAD and CAE will efficiently accomplish the product design process. Unfortunately, the existing CAD and CAE system inherently using different geometry engine and modeling accuracy exists no generic, unified model which could be used both for design and for analysis. Model transformation based on geometric model among different system usually lead to data loss and data redundancy.
Aiming to the drawbacks of geometry model-based transformation, this paper adopt mesh model instead of geometry model to realise lossless data mapping from CAD to CAE system. Meanwhile it proposes a mesh optimization method toward meshes generated from CAD system, and extracts boundary feature information from the mesh. The research is primarily reflected in:
1. Giving a review of research status about CAD/CAE integration at home and abroad, the merits and faults are analyed, then puting forward the research content and structure arrangement in this paper.
2. Reseaching failure cause of the geometry model-based transformation in depth, introducing basic principle of lossless data mapping for mesh model-based transformation form CAD to CAE system. It also presents how to extract node and element information from CAD platform-UG NX5. It analyzes file format of mesh data in exiting CAE system and successfully realize data transformation from UG to ANSYS、Abaqus and so on.
3. Using element transformation theory smooths tetrahedron mesh, introducing tetrahedron element quality measure rule, describing the simultaneous smoothing algorithm sequential smoothing algorithm, and verifying this method using unit sphere and modular hip endoprosthesis model. According to result compared with traditonal method shows the superiority of this method.
4. Introducing basic theories of differential geometry, proposing the corrective method about ridge extracting including detection region of umbilicus, umbilicus detection, ridge detection and classification of ridge. Finally, using sevesal examples verify this method.
引文
[1] Armstrong, C. G. Modelling requirements for finite-element analysis. CAD Computer Aided Design, 1994, 26 (7): 573-578.
[2] De Martino, Teresa. Design and engineering process integration through a multiple view intermediate modeller in a distributed object-oriented system environment. CAD Computer Aided Design, 1998, 30 (6): 437-452.
[3] Gordon S. An analyst's view: STEP-enabled CAD-CAE integration. Presentation materials of NASA's STEP for aerospace workshop, Pasadena, 2001.
[4] R. Bidarra, W.F. Bronsvoort. Semantic feature modelling. Computer-Aided Design, 2000, 32 (1000): 201-225.
[5] R. Bidarra, K.J. de Kraker, W.F. Bronsvoort. Representation and management of feature information in a cellular model. Computer-Aided Design, 1998, 30 (4): 301-313.
[6] J. S, C. TC. Graph-based heuristics for recognition of machined features from a 3D solid model. Computer-Aided Design, 1988, 20 (2): 58-66.
[7] K. YS. Recognition of form features using convex decomposition. Computer Aided Design, 1992, 24 (9): 461-476.
[8] T. SJ, J. GEM. Recognising symmetry in solid models. Computer Aided Design, 2003, 35 (7): 673-692.
[9] V. S, S. M, K. V. A graph-based framework for feature recognition. Proceedings of sixth ACM symposium on solid modeling and applications, 2001.
[10] S.H. Lee. A CAD–CAE integration approach using feature-based multi-resolution and multi-abstraction modelling techniques. Computer-Aided Design, 2005, 37 (9): 941-955.
[11] Y.M. Deng, G.A. Britton, Y.C. Lam, et al. A Feature-Based CAD-CAE Integration Model for Injection Molded Product Design. International Journal of Production Research, Vol. 40, 2002, pp. 3737-3750.
[12] S. Zeng, R.S. Peak. ZAP: a knowledge-based FEA modeling method for highly coupled variable topology multi-body problems. Engineering with Computers, 2008, 24 (4): 359-381.
[13] O. Hamri, L.C. Lèon. Interoperability between CAD and CAE models for cooperative design. Methods and Tools for Co-operative and Integrated Design, 2004 451-462. 67
[14] D. Su, M. Wakelam. Intelligent hybrid system for integration in design and manufacture. Journal of Materials Processing Technology, 1998, 76 (1-3): 23-28.
[15] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics And Engineering, 2005, 194 (39-41): 4135-4195.
[16] B.W. Cao, J.J. Chen, Z.G. Huang, et al. CAD/CAE integration framework with layered software architecture. Proceedings - 2009 11th IEEE International Conference on Computer-Aided Design and Computer Graphics, 2009, pp. 410-415.
[17] M.S. Shephard, M.W. Beall, R.M. O'Bara, et al. Toward simulation-based design. Finite Elements in Analysis And Design, 2003, 40 (12): 1575-1598.
[18] G. Foucault, J. Cuillière, V.F. Ois, et al. Adaptation of CAD model topology for finite element analysis. Computer-Aided Design, 2008, 40 (2): 176-196.
[19] M. Novak, B. Dol?ak. Intelligent FEA-based design improvement. Engineering Applications Of Artificial Intelligence, 2008, 21 (8): 1239-1254.
[20] G.P. Gujarathi, Y.S. Ma. Generative CAD and CAE Integration Using Common Data Model. 6th annual IEEE Conference on Automation Science and Engineering, 2010.
[21] http://www.spatial.com/cn.
[22] http://www.transcendata.com/products/cadfix.
[23]常智,勇当松,常智勇. CAD/CAE集成中的几何模型转换方法研究.现代制造工程, 2007 (12): 47-50.
[24]李晓宁,吴爱萍. CAD/CAE集成中几何模型的自动修复问题的研究.计算机辅助工程, 2003, 9 (3): 1-5.
[25]王威信,吴延江,张凤军.以STL为接口的CAD/CAE集成应用.计算机辅助设计与图形学学报, 2005, 17 (8): 1878-1882.
[26]段丽,郑建靖,孙力胜, et al.一类基于ACIS的CAD/CAE数据转换接口实现.计算机工程与应用, 2010, 46 (25).
[27]韩玲莉,黄俊.基于特征建模技术的CAD/CAE集成方法研究.机械设计与研究, 2006, 22 (6): 81-83.
[28]谢世坤,黄菊花,杨国泰. CAD/CAE集成中的有限元模型转换之研究.中国机械工程, 2005, 16 (5): 428-431.
[29]戴磊,基于CAD/CAE集成技术的开放式参数化结构形式优化设计平台,博士论文,大连理工大学, 2008.
[30]李佩林,基于STEP的CAD/CAE系统集成技术的研究,硕士论文,哈尔滨工程大学, 2009.
[31]王爱华, CAD/CAE集成中多分辨率建模技术的研究,硕士论文,哈尔滨理工大学, 2009.
[32] R. Lohner, P. P, Uhh. Generation of three-dimensional unstructured grids by the advancing-front method. International Journal For Numerical Methods In Fluids, 1988, 8 (10): 1135-1149.
[33] I. Y, M.S. A, K.S. B. Reliable Isotropic Tetrahedral Mesh Generation Based on an Advancing Front Method. Proceedings of the 13th International Meshing Roundtable, 2004, pp. 95-106.
[34] M.S. Shephard, M.K. Georges. Automatic three-dimensional mesh generation by the finite octree technique. International Journal for Numerical Methods in Engineering, 1991, 32 (4): 709-749.
[35] Shewchuk, J. R. Tetrahedral Mesh Generation by Delaunay Refinement. Proceedings of the 14th Annual Symposium on Computational Geometry, 1998, pp. 86-95.
[36] M.S. Smit, W.F. Bronsvoort. Efficient Tetrahedral Remeshing of Feature Models for Finite Element Analysis. Engineering with Computers, 2009, 25 (4): 327-344.
[37] L.A. Freitag, C. Ollivier-Gooch. Tetrahedral Mesh Improvement Using Swapping and Smoothing. International Journal For Numerical Methods In Engineering, 1997, 40 (21): 3979-4002.
[38] J. Escobar, R. Montenegro, E. Rodriguez. Smoothing and Local Refinement Techniques for Improving Tetrahedral Mesh Quality. Computer & Structures, 2005, 83 (28-30): 2430-2432.
[39] D.A. Field. Laplacian Smoothing and Delaunay Triangulations. Communications in Applied Numerical Methods, 1988, 4 (6): 709-712.
[40] N. Amenta, M. Bern, D. Eppstein. Optimal Point Placement for Mesh Smoothing. Journal Of Algorithms, 30 (1999): 302-322.
[41] Freitag, A. Lori. On Combining Laplacian and Optimization-Based Mesh Smoothing Techniques. Proceedings of the 1997 Joint ASME/ASCE/SES Summer Meeting, 1997, pp. 37-43.
[42] T. Xia, E. Shaffer. Streaming Tetrahedral Mesh Optimization. Proceedings of the 2008 ACM symposium on Solid and physical modeling, Vol. SPM'08, 2008, pp. 281-286.
[43] D. Vartziotis, T. Athanasiadis, I. Goudas. Mesh Smoothing Using the Geometric Element Transformation Method. Computer Methods In Applied Mechanics And Engineering, 2008, 197 (45-48): 3760-3767.
[44] D. Vartziotis, J. Wipper. The Geometric Element Transformation Method for Mixed Mesh Smoothing. Engineering With Computers, 2009, 25 (3): 287-301.
[45] L.A.F. Diachin, P.M. Knupp, T. Munson, et al. A Comparison of Two Optimization Methods for Mesh Quality Improvement. Engineering With Computers, 2006, 22 (2): 61-74.
[46] P.M. Knupp. Algebraic Mesh Quality Metrics. SIAM Journal on Scientific, 2001, 23 (1): 193-218.
[47] L.A. Freitag. On Combining Laplacian and Optimization-Based Mesh Smoothing Techniques. Trends in Unstructured Mesh Generation, 1997 37-43.
[48] http://www.cs.sandia.gov/optimization/knupp/Mesquite.html.
[49] http://tetgen.berlios.de.
[50] D. Vartziotis, A. Poulis, C. Vartziotis, et al. Integrated Digital Engineering Methodology for Virtual Orthopedics Surgery Planning. Proceedings of the International Special Topic Conference on Information Technology in Biomedicine, 2006.
[51] D. Vartziotis. Modular Endoprosthesis for Total Hip Arthroplasty. 2006.
[52]朱心雄.自由曲线曲面造型技术.北京:科学出版社, 2000.
[53] P.W. Hallinan, G. Gordon, A.L. Yuille, et al. Two-and Three-Dimensional Patterns of the Face, 1999.
[54] F. Cazals, M. Pouget. Smooth surfaces,umbilics,lines of curvatures,foliations,ridges and the medial axis: a concise overview. Computation Geometry and Applications, 2005.
[55] F. Cazals, M. Pouget. Estimating Differential Quantities Using Polynomial Fitting of Osculating Jets. Computer Aided Geometric Design, 2005, 22 (2).
[56] J.P. Thirion. The extremal mesh and the understanding of 3d surfaces. International Journal Of Computer Vision, 1996, 19 (2): 115-128.
[57] K.H. Ko, T. Maekawa, N.M. Patrikalakis, et al. Shape intrinsic fingerprints for free-form object matching. SM '03 Proceedings of the eighth ACM symposium on Solid modeling and applications, 2003.
[58] K.H. Ko, T. Maekawa, N.M. Patrikalakis. Algorithms for optimal partial matching of free-form objects with scaling effects. Graphical Models, 2005, 67 (2): 120-148.
[59] M. Berger, B. Gostiaux. Differential geometry: curves - surfaces - manifolds, 1988.
[60] Y. Ohtake, A. Belyaev, H. Seidel. Ridge-valley lines on meshes via implicit surface fitting. ACM Transactions on Graphics (TOG) - Proceedings of ACM SIGGRAPH 2004, Vol. 23, 2004.
[61] F. Cazals, J.C. Faugeres, M. Pouget, et al. Certified detection of umbilics, parabolic curves and ridges on polynomial parametric surfaces. 2005.
[2] De Martino, Teresa. Design and engineering process integration through a multiple view intermediate modeller in a distributed object-oriented system environment. CAD Computer Aided Design, 1998, 30 (6): 437-452.
[3] Gordon S. An analyst's view: STEP-enabled CAD-CAE integration. Presentation materials of NASA's STEP for aerospace workshop, Pasadena, 2001.
[4] R. Bidarra, W.F. Bronsvoort. Semantic feature modelling. Computer-Aided Design, 2000, 32 (1000): 201-225.
[5] R. Bidarra, K.J. de Kraker, W.F. Bronsvoort. Representation and management of feature information in a cellular model. Computer-Aided Design, 1998, 30 (4): 301-313.
[6] J. S, C. TC. Graph-based heuristics for recognition of machined features from a 3D solid model. Computer-Aided Design, 1988, 20 (2): 58-66.
[7] K. YS. Recognition of form features using convex decomposition. Computer Aided Design, 1992, 24 (9): 461-476.
[8] T. SJ, J. GEM. Recognising symmetry in solid models. Computer Aided Design, 2003, 35 (7): 673-692.
[9] V. S, S. M, K. V. A graph-based framework for feature recognition. Proceedings of sixth ACM symposium on solid modeling and applications, 2001.
[10] S.H. Lee. A CAD–CAE integration approach using feature-based multi-resolution and multi-abstraction modelling techniques. Computer-Aided Design, 2005, 37 (9): 941-955.
[11] Y.M. Deng, G.A. Britton, Y.C. Lam, et al. A Feature-Based CAD-CAE Integration Model for Injection Molded Product Design. International Journal of Production Research, Vol. 40, 2002, pp. 3737-3750.
[12] S. Zeng, R.S. Peak. ZAP: a knowledge-based FEA modeling method for highly coupled variable topology multi-body problems. Engineering with Computers, 2008, 24 (4): 359-381.
[13] O. Hamri, L.C. Lèon. Interoperability between CAD and CAE models for cooperative design. Methods and Tools for Co-operative and Integrated Design, 2004 451-462. 67
[14] D. Su, M. Wakelam. Intelligent hybrid system for integration in design and manufacture. Journal of Materials Processing Technology, 1998, 76 (1-3): 23-28.
[15] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics And Engineering, 2005, 194 (39-41): 4135-4195.
[16] B.W. Cao, J.J. Chen, Z.G. Huang, et al. CAD/CAE integration framework with layered software architecture. Proceedings - 2009 11th IEEE International Conference on Computer-Aided Design and Computer Graphics, 2009, pp. 410-415.
[17] M.S. Shephard, M.W. Beall, R.M. O'Bara, et al. Toward simulation-based design. Finite Elements in Analysis And Design, 2003, 40 (12): 1575-1598.
[18] G. Foucault, J. Cuillière, V.F. Ois, et al. Adaptation of CAD model topology for finite element analysis. Computer-Aided Design, 2008, 40 (2): 176-196.
[19] M. Novak, B. Dol?ak. Intelligent FEA-based design improvement. Engineering Applications Of Artificial Intelligence, 2008, 21 (8): 1239-1254.
[20] G.P. Gujarathi, Y.S. Ma. Generative CAD and CAE Integration Using Common Data Model. 6th annual IEEE Conference on Automation Science and Engineering, 2010.
[21] http://www.spatial.com/cn.
[22] http://www.transcendata.com/products/cadfix.
[23]常智,勇当松,常智勇. CAD/CAE集成中的几何模型转换方法研究.现代制造工程, 2007 (12): 47-50.
[24]李晓宁,吴爱萍. CAD/CAE集成中几何模型的自动修复问题的研究.计算机辅助工程, 2003, 9 (3): 1-5.
[25]王威信,吴延江,张凤军.以STL为接口的CAD/CAE集成应用.计算机辅助设计与图形学学报, 2005, 17 (8): 1878-1882.
[26]段丽,郑建靖,孙力胜, et al.一类基于ACIS的CAD/CAE数据转换接口实现.计算机工程与应用, 2010, 46 (25).
[27]韩玲莉,黄俊.基于特征建模技术的CAD/CAE集成方法研究.机械设计与研究, 2006, 22 (6): 81-83.
[28]谢世坤,黄菊花,杨国泰. CAD/CAE集成中的有限元模型转换之研究.中国机械工程, 2005, 16 (5): 428-431.
[29]戴磊,基于CAD/CAE集成技术的开放式参数化结构形式优化设计平台,博士论文,大连理工大学, 2008.
[30]李佩林,基于STEP的CAD/CAE系统集成技术的研究,硕士论文,哈尔滨工程大学, 2009.
[31]王爱华, CAD/CAE集成中多分辨率建模技术的研究,硕士论文,哈尔滨理工大学, 2009.
[32] R. Lohner, P. P, Uhh. Generation of three-dimensional unstructured grids by the advancing-front method. International Journal For Numerical Methods In Fluids, 1988, 8 (10): 1135-1149.
[33] I. Y, M.S. A, K.S. B. Reliable Isotropic Tetrahedral Mesh Generation Based on an Advancing Front Method. Proceedings of the 13th International Meshing Roundtable, 2004, pp. 95-106.
[34] M.S. Shephard, M.K. Georges. Automatic three-dimensional mesh generation by the finite octree technique. International Journal for Numerical Methods in Engineering, 1991, 32 (4): 709-749.
[35] Shewchuk, J. R. Tetrahedral Mesh Generation by Delaunay Refinement. Proceedings of the 14th Annual Symposium on Computational Geometry, 1998, pp. 86-95.
[36] M.S. Smit, W.F. Bronsvoort. Efficient Tetrahedral Remeshing of Feature Models for Finite Element Analysis. Engineering with Computers, 2009, 25 (4): 327-344.
[37] L.A. Freitag, C. Ollivier-Gooch. Tetrahedral Mesh Improvement Using Swapping and Smoothing. International Journal For Numerical Methods In Engineering, 1997, 40 (21): 3979-4002.
[38] J. Escobar, R. Montenegro, E. Rodriguez. Smoothing and Local Refinement Techniques for Improving Tetrahedral Mesh Quality. Computer & Structures, 2005, 83 (28-30): 2430-2432.
[39] D.A. Field. Laplacian Smoothing and Delaunay Triangulations. Communications in Applied Numerical Methods, 1988, 4 (6): 709-712.
[40] N. Amenta, M. Bern, D. Eppstein. Optimal Point Placement for Mesh Smoothing. Journal Of Algorithms, 30 (1999): 302-322.
[41] Freitag, A. Lori. On Combining Laplacian and Optimization-Based Mesh Smoothing Techniques. Proceedings of the 1997 Joint ASME/ASCE/SES Summer Meeting, 1997, pp. 37-43.
[42] T. Xia, E. Shaffer. Streaming Tetrahedral Mesh Optimization. Proceedings of the 2008 ACM symposium on Solid and physical modeling, Vol. SPM'08, 2008, pp. 281-286.
[43] D. Vartziotis, T. Athanasiadis, I. Goudas. Mesh Smoothing Using the Geometric Element Transformation Method. Computer Methods In Applied Mechanics And Engineering, 2008, 197 (45-48): 3760-3767.
[44] D. Vartziotis, J. Wipper. The Geometric Element Transformation Method for Mixed Mesh Smoothing. Engineering With Computers, 2009, 25 (3): 287-301.
[45] L.A.F. Diachin, P.M. Knupp, T. Munson, et al. A Comparison of Two Optimization Methods for Mesh Quality Improvement. Engineering With Computers, 2006, 22 (2): 61-74.
[46] P.M. Knupp. Algebraic Mesh Quality Metrics. SIAM Journal on Scientific, 2001, 23 (1): 193-218.
[47] L.A. Freitag. On Combining Laplacian and Optimization-Based Mesh Smoothing Techniques. Trends in Unstructured Mesh Generation, 1997 37-43.
[48] http://www.cs.sandia.gov/optimization/knupp/Mesquite.html.
[49] http://tetgen.berlios.de.
[50] D. Vartziotis, A. Poulis, C. Vartziotis, et al. Integrated Digital Engineering Methodology for Virtual Orthopedics Surgery Planning. Proceedings of the International Special Topic Conference on Information Technology in Biomedicine, 2006.
[51] D. Vartziotis. Modular Endoprosthesis for Total Hip Arthroplasty. 2006.
[52]朱心雄.自由曲线曲面造型技术.北京:科学出版社, 2000.
[53] P.W. Hallinan, G. Gordon, A.L. Yuille, et al. Two-and Three-Dimensional Patterns of the Face, 1999.
[54] F. Cazals, M. Pouget. Smooth surfaces,umbilics,lines of curvatures,foliations,ridges and the medial axis: a concise overview. Computation Geometry and Applications, 2005.
[55] F. Cazals, M. Pouget. Estimating Differential Quantities Using Polynomial Fitting of Osculating Jets. Computer Aided Geometric Design, 2005, 22 (2).
[56] J.P. Thirion. The extremal mesh and the understanding of 3d surfaces. International Journal Of Computer Vision, 1996, 19 (2): 115-128.
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