基于有限元法的曲面变形技术研究
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摘要
飞机、汽车等复杂产品的数字化设计是一个反复编辑、不断修改的过程。因此，灵活精确、自然高效的曲面变形技术一直是该领域的研究热点。本文基于有限元法，针对不同的曲面模型和变形需求，对曲面变形技术进行了研究，主要研究内容和成果如下：
借鉴骨架驱动生物运动的机制，提出了一种骨架-刚架模型驱动网格曲面变形算法。算法采用光线求交法求取由直线段组成的骨架，通过将直线段定义为梁单元创建骨架-刚架模型，建立网格顶点与骨架-刚架模型的映射，基于有限元法分别建立载荷作用下和几何约束下的骨架-刚架模型变形控制方程，求解该方程获得骨架-刚架模型的变形，进而带动附着在骨架上的网格曲面发生变形；通过梁单元形函数插值创建旋转场进一步修正计算结果。该算法适用于具有明显骨架语义的网格变形设计，与已有的物理变形技术相比，无需将变形过程划分多步，提高了使用方便性。
为实现由网格曲面描述的产品表面的变形设计，提出了一种网格-刚架模型驱动网格曲面保特征变形算法。算法通过对初始网格进行一定程度的简化得到相应的基网格，将基网格顶点定义为网格-刚架模型节点，确立初始网格顶点与网格-刚架模型之间的映射关系，基于有限元法分别建立载荷作用下和几何约束下的网格-刚架模型的变形控制方程，求解该方程获得网格-刚架模型的变形，根据映射关系计算初始网格的变形。在变形计算中，通过增大网格曲面特征区域所关联单元体材料的弹性模量，达到整体变形中局部形状特征的保持。该算法兼有载荷作用下的自由变形和满足几何约束的精确设计功能，同时，采用基网格创建网格-刚架模型，保证了曲面变形计算的高效性。
以有限元六面体单元变形理论为基础，提出了一种六面体栅格模型驱动网格曲面变形算法。该算法采用栅格法剖分网格曲面的包围盒创建六面体栅格模型，建立网格顶点与六面体栅格模型之间的映射关系，基于有限元法建立载荷作用下六面体栅格模型的变形控制方程，进而采用“变形累加法”计算六面体栅格模型的变形，根据映射关系计算网格曲面变形。该算法适用于载荷作用下的网格曲面保特征自由变形设计。
针对由多张B-Spline曲面拼接构成的几何变形设计问题，提出了刚架模型驱动多张B-Spline曲面变形算法。通过合并曲面片的控制顶点建立刚架模型，将曲面片之间的光滑拼接条件抽象为刚架模型节点间的位置约束，结合点约束、法矢约束共同组成约束条件，基于有限元法建立刚架模型的变形控制方程，求解方程获得刚架模型的变形，进而计算曲面变形。变形后的多张B样条曲面不仅满足给定的几何约束，而且保持变形前的光滑拼接状态。
It is well known that shape modeling of any complicated objects is a time-consuming process withrepeated modification and editing. Therefore, flexible and natural deformation technology has beenwidely concerned. Based on FEM, research on deformation technique of surface is investigated in thispaper. The main research contents and achievements are as follows:
In order to solve the deformation of skeleton semantic mesh, a deformation algorithm driven byskeleton-rigid frame model is proposed. The skeleton of the mesh extracted with light intersectionmethod is composed of straight line segments. The skeleton-rigid frame model is constructed bydefining each line segment as a beam element. The mapping between mesh vertex and skeleton-rigidframe model is established. By finite element method, the control equations of deformation under loadand geometric constraint are constructed. The deformation of skeleton-rigid frame model is computedby solving the equations. The mesh surface attached to the skeleton is deformed with skeleton. Arotation field is created by shape function of beam element to revise the deformation result. Comparedwith the existing physically based deformation technology, this algorithm is without sub-stepcalculation and improves the convenient usability.
For the deformation of the shape of product surface, a mesh deformation algorithm driven bymesh-rigid frame model is proposed. The original mesh is simplified to a coarse base mesh at a certainextent, and the mapping relationship between original mesh and base mesh is established. Themesh-rigid frame model is constructed by defining each vertex of base mesh as a node. By finiteelement method, the control equations of deformation under load and geometric constraint areconstructed. The deformation of skeleton-rigid frame model is computed by solving the equations.According to the mapping relationship, the deformation result of initial mesh is calculated out. Theelastic modulus of elements corresponding to features of mesh is increased to preserve form feature indeformation process. The algorithm is not only suitable for freedom design under load, but alsosuitable for fine design under geometry constraint. Using mesh simplification method to create basemesh, the algorithm can improve the calculation efficiency.
With the computational theory of hexahedral element FEM, the method of hexahedral meshgeneration is carried out. A mesh deformation algorithm driven by hexahedral grid model is putforward. The hexahedral grid model of deformable model is constructed by subdividing the boundingbox of surface model. The mapping between mesh vertex and hexahedral grid model is established.Through imposing external loads on the node of hexahedral element, the deformation of hexahedral grid model is computed by deformation acculation method. The mesh deforms with hexahedral gridmodel simultaneously. This algorithm is able to realize the freedom design under load.
The appearance of complex products is described by many free-form surfaces. For the shapemodification of surface connected by multi B-spline surfaces, a deformation algorithm based on thedeformation principle of rigid frame under force is presented. A uniform rigid frame model is createdthrough merger of control mesh topology among the surfaces. Smooth connection relationship amongdifferent surfaces and geometric constraint (point constraint, point and normal constraint) areextracted as constraint equations. Shape deformation of multi-surfaces under geometric constraints isachieved by finite element method. The deformed surfaces can not only subject to specified geometricconstraints, but also keep their smooth connectivity relationship.
借鉴骨架驱动生物运动的机制，提出了一种骨架-刚架模型驱动网格曲面变形算法。算法采用光线求交法求取由直线段组成的骨架，通过将直线段定义为梁单元创建骨架-刚架模型，建立网格顶点与骨架-刚架模型的映射，基于有限元法分别建立载荷作用下和几何约束下的骨架-刚架模型变形控制方程，求解该方程获得骨架-刚架模型的变形，进而带动附着在骨架上的网格曲面发生变形；通过梁单元形函数插值创建旋转场进一步修正计算结果。该算法适用于具有明显骨架语义的网格变形设计，与已有的物理变形技术相比，无需将变形过程划分多步，提高了使用方便性。
为实现由网格曲面描述的产品表面的变形设计，提出了一种网格-刚架模型驱动网格曲面保特征变形算法。算法通过对初始网格进行一定程度的简化得到相应的基网格，将基网格顶点定义为网格-刚架模型节点，确立初始网格顶点与网格-刚架模型之间的映射关系，基于有限元法分别建立载荷作用下和几何约束下的网格-刚架模型的变形控制方程，求解该方程获得网格-刚架模型的变形，根据映射关系计算初始网格的变形。在变形计算中，通过增大网格曲面特征区域所关联单元体材料的弹性模量，达到整体变形中局部形状特征的保持。该算法兼有载荷作用下的自由变形和满足几何约束的精确设计功能，同时，采用基网格创建网格-刚架模型，保证了曲面变形计算的高效性。
以有限元六面体单元变形理论为基础，提出了一种六面体栅格模型驱动网格曲面变形算法。该算法采用栅格法剖分网格曲面的包围盒创建六面体栅格模型，建立网格顶点与六面体栅格模型之间的映射关系，基于有限元法建立载荷作用下六面体栅格模型的变形控制方程，进而采用“变形累加法”计算六面体栅格模型的变形，根据映射关系计算网格曲面变形。该算法适用于载荷作用下的网格曲面保特征自由变形设计。
针对由多张B-Spline曲面拼接构成的几何变形设计问题，提出了刚架模型驱动多张B-Spline曲面变形算法。通过合并曲面片的控制顶点建立刚架模型，将曲面片之间的光滑拼接条件抽象为刚架模型节点间的位置约束，结合点约束、法矢约束共同组成约束条件，基于有限元法建立刚架模型的变形控制方程，求解方程获得刚架模型的变形，进而计算曲面变形。变形后的多张B样条曲面不仅满足给定的几何约束，而且保持变形前的光滑拼接状态。
It is well known that shape modeling of any complicated objects is a time-consuming process withrepeated modification and editing. Therefore, flexible and natural deformation technology has beenwidely concerned. Based on FEM, research on deformation technique of surface is investigated in thispaper. The main research contents and achievements are as follows:
In order to solve the deformation of skeleton semantic mesh, a deformation algorithm driven byskeleton-rigid frame model is proposed. The skeleton of the mesh extracted with light intersectionmethod is composed of straight line segments. The skeleton-rigid frame model is constructed bydefining each line segment as a beam element. The mapping between mesh vertex and skeleton-rigidframe model is established. By finite element method, the control equations of deformation under loadand geometric constraint are constructed. The deformation of skeleton-rigid frame model is computedby solving the equations. The mesh surface attached to the skeleton is deformed with skeleton. Arotation field is created by shape function of beam element to revise the deformation result. Comparedwith the existing physically based deformation technology, this algorithm is without sub-stepcalculation and improves the convenient usability.
For the deformation of the shape of product surface, a mesh deformation algorithm driven bymesh-rigid frame model is proposed. The original mesh is simplified to a coarse base mesh at a certainextent, and the mapping relationship between original mesh and base mesh is established. Themesh-rigid frame model is constructed by defining each vertex of base mesh as a node. By finiteelement method, the control equations of deformation under load and geometric constraint areconstructed. The deformation of skeleton-rigid frame model is computed by solving the equations.According to the mapping relationship, the deformation result of initial mesh is calculated out. Theelastic modulus of elements corresponding to features of mesh is increased to preserve form feature indeformation process. The algorithm is not only suitable for freedom design under load, but alsosuitable for fine design under geometry constraint. Using mesh simplification method to create basemesh, the algorithm can improve the calculation efficiency.
With the computational theory of hexahedral element FEM, the method of hexahedral meshgeneration is carried out. A mesh deformation algorithm driven by hexahedral grid model is putforward. The hexahedral grid model of deformable model is constructed by subdividing the boundingbox of surface model. The mapping between mesh vertex and hexahedral grid model is established.Through imposing external loads on the node of hexahedral element, the deformation of hexahedral grid model is computed by deformation acculation method. The mesh deforms with hexahedral gridmodel simultaneously. This algorithm is able to realize the freedom design under load.
The appearance of complex products is described by many free-form surfaces. For the shapemodification of surface connected by multi B-spline surfaces, a deformation algorithm based on thedeformation principle of rigid frame under force is presented. A uniform rigid frame model is createdthrough merger of control mesh topology among the surfaces. Smooth connection relationship amongdifferent surfaces and geometric constraint (point constraint, point and normal constraint) areextracted as constraint equations. Shape deformation of multi-surfaces under geometric constraints isachieved by finite element method. The deformed surfaces can not only subject to specified geometricconstraints, but also keep their smooth connectivity relationship.
引文
[1] Dietz R, Hoschek J, Juttler B. An algebraic approach to curves and surfaces on the sphere and onother quadrics. Computer Aided Geometric Design,1993,10(3-4):211-229.
[2] Sederberg T W. Techniques for cubic algebraic surfaces. IEEE Computer Graphics&Application,1990,10(5):12-21.
[3] Sederberg T W, Chen F L. Implicitization using moving curves and surfaces. Proceedings of the22nd Annual Conference on Computer Graphics and Interactive Techniques, New York: ACMPress,1995:301-308.
[4] Akkouche S, Galin E. Implicit surface reconstruction from contours. The Visual Computer,2004,20(6):392-401.
[5]李道伦,卢德唐,孔祥言,等.径向基函数网络的隐式曲面方法.计算机辅助设计与图形学学报,2006,18(8):1142-1148.
[6]莫堃.基于隐式函数的曲面重构方法及其应用,[博士学位论文].武汉：华中科技大学,2010.
[7] Xu J B, Song H C, Wu L D, et al. A seamless representation of multi-scale clouds. Proceedings ofthe2009Fifth International Joint Conference on INC, IMS and IDC, Piscataway: IEEE,2009:1255-1260.
[8] Chang J W, Lei S I E, Chang C F, et al. Real-time rendering of splashing stream water.Proceedings of3rd International Conference on Intelligent Information Hiding and MultimediaSignal Processing, IIHMSP2007, Piscataway: IEEE,2007:337~340.
[9] Ferguson J. Multivariable curve interpolation. Journal of the ACM,1964,11(2):221-228.
[10] Coons S A. Surface for computer-aided design of space figures. Technical Report,MIT,1964:Project MAC-TR-255.
[11] Forrest A R. Interactive interpolation and approximation by Bézier polynomial. Journal ofComputer,1972,15(1):71-79.
[12] Gordon W J, Riesenfeld R F. Bernstein-Bézier methods for the computer aided design offree-form curves and surfaces. Journal of the ACM,1974,21(2):293-310.
[13]苏步青,刘鼎元.论Bézier曲线的仿射不变量.计算数学,1980,2:289-298.
[14]刘鼎元.有理Bézier曲线.应用数学学报,1985,8(1):70-83.
[15]汪国昭,沈金福.有理Bézier曲线的离散和几何性质.浙江大学学报,1985,19(3):123-130.
[16]施法中,吴骏恒. Bézier作图定理与三次Bézier曲线的几何性质.航空学报,1982,1(3):97-104.
[17] de Boor C W. On calculation with B-spline. Journal of Approximation Theory,1972,(6):50-62.
[18] Cox M G. The numerical evaluation of B-splines. Journal of Applied Mathematics,1971,10(2):134-149.
[19] Boehm W. Inserting new knots into B-spline curves. Computer Aided Design,1980,12(4):199-201.
[20] Prautzsch H. Degree elevation of B-spline curves. Computer Aided Geometric Design,1984,1(2):193-198.
[21] Vergeest J S M. CAD surface data exchange using STEP. Computer Aided Design,1991,23(4):269~281.
[22] Huang J, Chen L, Liu X G, et al. Efficient mesh deformation using tetrahedron control mesh.Proceedings of the2008ACM symposium on Solid and physical modeling, New York: ACMPress,2008:241-247.
[23] Yoshizawa S, Belyaev A, Seidel H P. Skeleton-based variational mesh deformations. ComputerGraphics Forum,2007,26(3):255-264
[24] Kho Y, Garland M. Sketching mesh deformations. Proceedings of the2005Symposium onInteractive3D Graphics and Games, New York: ACM Press,2005:147-154.
[25]董洪伟.网格变形综述.中国图象图形学报,2011,16(12):2095-2014.
[26]许秋儿.网格变形技术研究,[博士学位论文].杭州:浙江大学,2009.
[27] Barr Alan H. Global and local deformations of solid primitives. Computer Graphics,1984,18(3):21-30.
[28] Sederberg T W, Parry S R. Free-form deformations of solid geometric models. Proceedings ofComputer Graphics, Annual Conference Series, ACM SIGGRAPH, New York: ACM Press,1986:151-160.
[29] Griessmair J, Purgathofer W. Deformation of solids with trivariate B-Splines. Proeeedings ofEurographics, North-Holland: Elsevier Science,1989:137-148.
[30] Kalra P, Mangili A, Thalmann N M, et al. Simulation of facial muscle actors based on rationalfree-form deformation. Computer Graphics Forum,1992,11(3):59-69.
[31] Lamousin H J, Waggenspack W N. NURBS-based free-form deformation. IEEE ComputerGraphics and Applications,1994,14(6):59-65.
[32] Coquillart S. Extended free-form deformation: A sculpting tool for3D geometric modeling.Computer Graphics,1990,24(4):187~196.
[33] MacCracken R, Joy K I. Free-form deformations with lattices of arbitrary topology.Proceedingsof the23rdannual conference on Computer graphics and interactive techniques, NewYork: ACM SIGGRAPH,1996:181~188.
[34] Ju T, Schaefer S, Warren J. Mean value coordinates for closed triangular meshes. ACMTransactions on Graphics,2005,24(3):561-566.
[35]张湘玉,廖文和,刘浩.基于细分曲面的泊松网格编辑.计算机辅助设计与图形学学报,2010,22(1):76-84.
[36]王仁芳,许秋儿,汪沁,等.基于最小二乘网格的模型变形算法.计算机辅助设计与图形学学报,2010,22(5):777-783.
[37] Feng J Q, Peng Q S, Ma L Z. A new free-form deformation through the control of parametricsurfaces. Computers&Graphics,1996,20(4):531-539.
[38] Chen B Y, Ono B, Johan H, et al.3D model deformation along a parametric surface. Proceedingsof International Conference on Visualization, Imaging and Image Processing, Berlin: Springer,2002:282-287.
[39] Wang X P, Ye Z L, Meng Y Q, et al. Space deformation of parametric surface based on extensionfunction. International Journal of CAD/CAM,2004,17(4):251-259.
[40] Wang X P, Zhou R R, Ye Z L, et al. Shape modification and deformation of parametric surface.Computer Aided Drafting, Design and Manufacture,2002,12(1):1-7.
[41]陆友太,周来水,王志国,等.基于乘幂伸缩因子的参数曲线自由变形.南京航空航天大学学报,2011,43(6):793-798.
[42]陆友太,周来水,王志国,等.基于多项式伸缩因子的参数曲面自由变形.中国机械工程,2012,23(5):609-613.
[43] Lazarus F, Coquillart S, Jancene P. Axial deformation: an intuitive deformationtechnique.Computer Aided Design,1994,26(8):607~613.
[44] Chang Y K, Rockwood A P. A generalized de casteljua approach to3D free-form deformation.Proceedings of21st International SIGGRAPH Conference on Computer Graphics, New York:ACM SIGGRAPH,1994:257~260.
[45] Singh K, Fiume E. Wires: A geometric deformation technique. Computer Graphics,1998,32(3):405-414.
[46]王隽,张宏鑫,许栋,等.勾画式泊松网格编辑.计算机辅助设计与图形学学报,2006,18(11):1723-1729.
[47] Hsu W M, Hughes J F, Kaufman H. Direct manipulation of free-form deformations. ComputerGraphics,1992,26(2):177-184.
[48] Hu S M, Zhang H, Tai C L, et al. Direct manipulation of FFD: Efficient explicit solutions anddecomposable multiple point constraints. Visual Computer,2001,17(6):370-379.
[49] McDonnell K T, Qin H. PB-FFD: A point-based technique for free-form deformation. Journal ofGraphics Tools,2007,12(3):25-41.
[50] Xu D, Chen W, Zhang H X, et al. Multi-level differential surface representation based on localtransformations. The Visual Computer,2006,22(7):493-505.
[51]张湘玉.基于细分的曲线曲面变形技术研究,[博士学位论文].南京:南京航空航天大学,2010.
[52]李登高.基于小波的多分辨率数字几何造型技术的研究,[博士学位论文].北京:清华大学,2006.
[53]张亚萍,熊华,姜晓红,等.大型网格模型简化和多分辨率技术综述.计算机辅助设计与图形学学报,2010,22(4):559-568.
[54] Zorin D, Schroder P, Sweldens W. Interactive multiresolution mesh editing. Proeeedings of the24th Annual Conference on Computer Grapphics and Interaetive Techniques, NewYork: ACMPress,1998:259-268.
[55] Lee A, Moreton H, Hoppe H. Displaced subdivision surfaces. Proceedings of the ACMSIGGRAPH Conference on Computer Graphics, New York: ACM Press,2000:85-94.
[56] Feng J Q, Shao J, Jin X G, et al. Multiresolution free-form deformation with subdivision surfaceof arbitrary topology. The Visual Computer,2005,22(1):28-42.
[57] Finkelstein A, Salesin D H. Multiresolution curves. Proceedings of the21stAnnual Conferenceon Computer Graphics and Interactive Techniques, New York: ACM Press,1994:261-268.
[58] Lounsbery M, DeRose T D, Warren J. Multiresolution analysis for surfaces of arbitrarytopological type. ACM Transactions on Graphics1997,16(1):34-73.
[59] Hoppe H. Progressive meshes. Proceedings of the23rd Annual Conference on ComputerGraphics and Interactive Techniques New York: ACM Press,1996:99-108.
[60] Botsch M, Kobbelt L. Multiresolution surface representation based on displacement volumes.Comupter Graphics Forum,2003,22(3):483-491.
[61] Blum H. Biological shape and visual science(part I). Journal of Theoretical Biology,1973,38(2):205-287.
[62] Capell S, Green S, Curless B, et al. Interactive skeleton-driven dynamic deformation. ACMTransactions on Graphics,2002,21(3):586-593.
[63] Yoshizawa S, Belyaev A G, Seidel H P. Free-form skeleton driven mesh deformations.Proceedings of the8th ACM Symposium Solid Modeling and Application, New York: ACMPress,2003:247-253.
[64] Magnenat N, Laperriere R. Joint-dependent local deformations for hand animation and objectgrasping. Proceedings on Graphics interface '88, Edmonton: Canadian Information ProcessingSociety Toronto,1989:26-33.
[65] Lewis J P, Cordner M. Pose space deformation: A unified approach to shape interpolation andskeleton-drvien deformation. Proceedings of the27th Annual Conference on Computer Graphicsand Interactive Techniques, New York: ACM Press,2000:165-172.
[66] Allen B, Curless B, Popovic Z. Articulated body deformation from range scan data. ACMTransactions on Graphics,2002,21(3):612-619.
[67]许秋儿,谭光华,张三元,等.保持几何特征的均值骨架子空间网格变形.计算机辅助设计与图形学学报,2009,21(3):289-294.
[68] Wang X C, Phillips C. Multi-weight enveloping: least-squares approximation techniques for skinanimation. Proceedings of the2002ACM SIGGRAPH/Eurographics Symposium on ComputerAnimation, NewYork: ACM Press,2002:129-138.
[69] Kavan L, Zara J. Spherical blend skinning: a real-time deformation of articulated models.Proceedings of the2005symposium on Interactive3D graphics and games, New York: ACMPress,2005:9-16.
[70] Yan H B, Hu S M, Martin R R. Shape deformation using a skeleton to drive simplextransformations. IEEE Transactions on Visualization and Computer Graphics,2008,14(3):693-706.
[71] Terzopoulos D, Platt J, Barr A. Elastically deformable models. Computer Graphics,1987,21(4):205-214.
[72] Celniker G, Welch W. Linear constraints for deformable non-uniform B-spline surfaces.Proceedings of the Symposium on Interactive3D Graphics, New York: ACM Press,1992:61-68.
[73] Wang Z G, Zhou L S, Wang X P. Direct Manipulation of B-spline surfaces. Chinese Journal ofMechanical Engineering,2005,18(1):103-108.
[74] Terzopoulos D, Qin H. Dynamic NURBS with geometric constraints for interactive sculpting.ACM Translations on Graphics,1994,13(2):103-136.
[75] Qin H, Terzopoulos D. Dynamic NURBS swung surfaces for physics-based shape design.Computer Aided Design, l995,27(2):111-127.
[76] Qin H, Terzopoulos D. Triangular NURBS and their dynamic generalizations. Compter AidedGeometric Design,1997,14(4):325-347.
[77] Qin H, Mandal C, Vemuri B C. Dynamic Catmull-Clark subdivision surface. Visualization andComputer Graphics,1998,4(3):215-229.
[78]李永林,王启付,钟毅芳,等.基于物理的准D-NURBS造型方法.计算机仿真,1998,15(2):46-50.
[79] Nealen A, Müller M, Keiser R, et al. Physically based deformable models in computer graphics.Computer Graphics Forum,2006,25(4):809-836.
[80] Waters K. A muscle model for animation three-dimensional facial expression. ACM SIGGRAPHComputer Graphics,1987,21(4):17-24.
[81] Miller G S P. The motion dynamics of snakes and worms. ACM SIGGRAPH Computer Graphics,1988,22(4):169-173.
[82] Li L, Volkov V. Cloth animation with adaptively refined meshes. Proceedings of the28thAustralasian conference on Computer Science, Darlinghurst: Australian Computer Society,2005:107-113.
[83] Vassilev T I. Comparison of parallel algorithms for modelling mass-springs systems with severalAPIs on modern GPU. Proceedings of the12th International Conference on Computer Systemsand Technologies, New York: ACM Press,2011:204-209.
[84] Thierry S E B. A particle-spring approach to geometric constraints solving. Proceedings of the2011ACM Symposium on Applied Computing, New York: ACM Press,2011:1100-1105.
[85]王峰军,邓克文,魏发远.基于物理的变形曲线曲面造型技术综述.计算机仿真,2006,23(4):41-45.
[86]梁醒培,王辉.应用有限元分析.北京:清华大学出版社,2010.
[87] Clough R W. The Finite Element Method in Plane Stress Analysis. The2nd ASME Conferenceon Electronic Computation, New York: ACM Press,1960:1114-1131.
[88] Turner M J, Clougb R W, Martin H C, et al. Stiffness and deflection analyses of complexstructures. J. Aero. Sci,1956,23(7):805-823.
[89] Courant R. Variational method for solutions of problems of equilibriumand vibrations. Bull. Am.Math. SoC.,1943,49(3):1~23.
[90]朱菊芬,汪海,徐胜利.非线性有限元及其在飞机结构设计中的应用.上海:上海交通大学出版社,2012.
[91] Celniker G, Gossard D. Deformable curve and surface finite elments for free-form shape design.Computer Graphics,1991,25(4):257-266.
[92] Collier J R, Collier B J, O'Toole G, et al. Drape prediction by means of finite-element analysis.Journal o f the Textile Insitute,1991,82(1):96-107.
[93] Chen D, Zeltzer D. Pump it up: computer animation of a biomechanically based model of muscleusing the finite element method. ACM SIGGRAPH Computer Graphics,1992,26(2):89-98.
[94] Guan Z D, Jing L, Ning T, et al. Study and application of physics-based deformable curves andsurfaces. Computers&Graphics,1997,21(3):305-313.
[95]经玲,席平,唐荣锡.有限元方法在变形曲线曲面造型中的应用.计算机学报,1998,21(3):245-251.
[96]于超升.基于刚架模型的复杂多曲面光滑协调变形算法研究,[硕士学位论文].南京：南京航空航天大学,2012.
[97]成思源,张湘伟.可变形B样条曲线曲面模型及其有限元求解.重庆大学学报,2003,26(2):75-77.
[98]程仙国,刘伟军,卞宏友.基于外载荷的B样条曲面变形.机械工程学报,2011,47(09):146-150.
[99]程仙国,刘伟军.外载荷的B样条曲线变形.中国图像图形学报,2011,16(5):898-902.
[100] Yu C S, Zhou L S, Wang Z G, et al. Shape modification of B-spline surfaces based on rigidframe via geometric constraint optimization. Proceedings of the conference on intelligentHuman-machine Systems and Cybernetices, Hangzhou：IEEE,2011:120-124.
[101] O'Brien J F, Hodgins J K. Graphical modeling and animation of brittle fracture. Proceedings ofSIGGRAPH1999, NewYork: ACM Press,1999:137-146.
[102] O'Brien J F, Hodgins J K, Bargteil A M. Graphical modeling and animation of ductile fracture.ACM Transactions on Graphics,2002,21(3):291-294.
[103] Capell S, Green S, Curless B, et al. A multi-resolution framework for dynamic deformations.Proceedings of the2002ACM SIGGRAPH Symposium on Computer Animation, NewYork:ACM Press,2002:41-47.
[104] Grinspun E, Krysl P, Schroder P. CHARMS: a simple framework for adaptive simulation. ACMTransactions on Graphics,2002,21(3):281-290.
[105] Müller M, Dorsey J, McMillan L, et al. Stable real-time deformations. Proceedings of the2002ACM SIGGRAPH/eurographics symposium on computer animation, New York: ACM Press,2002:49-54.
[106] Müller M, Gross M. Interactive virtual materials. Proceedings of Graphics Interface '04,Waterloo: Canadian Human-Computer Communications Society School of Computer Science,2004:239-246.
[107] Huang J, Liu X G, Bao H J, et al. An efficient large deformation method using domaindecomposition. Computers&Graphics,2006,30(6):927-935.
[108] Terzopoulos D, Witkin A. Physically based models with rigid and deformable components.IEEE Computer and Graphic Application,1988,8(6):41-51.
[109]王志国,周来水,王小平.基于材料力学模型的变形造型新方法.机械工程学报,2005,41(9):155-160.
[110] Wang Z G, Wang X P, Bao Y D, et al. Shape modification by beam model in FEM. ChineseJournal of Aeronautics,2010,23(2):246-251.
[111] Dick C, Georgii J, Westermann R. A hexahedral multigrid approach for simulating cuts indeformable objects. IEEE Transactions on visualization and computer graphics,2011,17(11):1663-1675.
[112] Song C, Zhang H X, et al Wu Y. Cutting and fracturing models without remeshing. Proceedingsof the5th international conference on Advances in geometric modeling and processing, Berlin:Springer-Verlag,2008,
[113]李静,陈健云,周晶.改进的SSOR-PCG迭代法在接触问题研究中应用.大连理工大学,2006,46(4):533-537.
[114] Jin X G, Li Y F, Peng Q S. General constrained deformation based on generalized metaballs.Computers&Graphics,2000,24(2):219-231.
[115]施法中.计算机辅助几何设计与非均匀有理B样条.北京:高等教育出版社,2001.
[2] Sederberg T W. Techniques for cubic algebraic surfaces. IEEE Computer Graphics&Application,1990,10(5):12-21.
[3] Sederberg T W, Chen F L. Implicitization using moving curves and surfaces. Proceedings of the22nd Annual Conference on Computer Graphics and Interactive Techniques, New York: ACMPress,1995:301-308.
[4] Akkouche S, Galin E. Implicit surface reconstruction from contours. The Visual Computer,2004,20(6):392-401.
[5]李道伦,卢德唐,孔祥言,等.径向基函数网络的隐式曲面方法.计算机辅助设计与图形学学报,2006,18(8):1142-1148.
[6]莫堃.基于隐式函数的曲面重构方法及其应用,[博士学位论文].武汉：华中科技大学,2010.
[7] Xu J B, Song H C, Wu L D, et al. A seamless representation of multi-scale clouds. Proceedings ofthe2009Fifth International Joint Conference on INC, IMS and IDC, Piscataway: IEEE,2009:1255-1260.
[8] Chang J W, Lei S I E, Chang C F, et al. Real-time rendering of splashing stream water.Proceedings of3rd International Conference on Intelligent Information Hiding and MultimediaSignal Processing, IIHMSP2007, Piscataway: IEEE,2007:337~340.
[9] Ferguson J. Multivariable curve interpolation. Journal of the ACM,1964,11(2):221-228.
[10] Coons S A. Surface for computer-aided design of space figures. Technical Report,MIT,1964:Project MAC-TR-255.
[11] Forrest A R. Interactive interpolation and approximation by Bézier polynomial. Journal ofComputer,1972,15(1):71-79.
[12] Gordon W J, Riesenfeld R F. Bernstein-Bézier methods for the computer aided design offree-form curves and surfaces. Journal of the ACM,1974,21(2):293-310.
[13]苏步青,刘鼎元.论Bézier曲线的仿射不变量.计算数学,1980,2:289-298.
[14]刘鼎元.有理Bézier曲线.应用数学学报,1985,8(1):70-83.
[15]汪国昭,沈金福.有理Bézier曲线的离散和几何性质.浙江大学学报,1985,19(3):123-130.
[16]施法中,吴骏恒. Bézier作图定理与三次Bézier曲线的几何性质.航空学报,1982,1(3):97-104.
[17] de Boor C W. On calculation with B-spline. Journal of Approximation Theory,1972,(6):50-62.
[18] Cox M G. The numerical evaluation of B-splines. Journal of Applied Mathematics,1971,10(2):134-149.
[19] Boehm W. Inserting new knots into B-spline curves. Computer Aided Design,1980,12(4):199-201.
[20] Prautzsch H. Degree elevation of B-spline curves. Computer Aided Geometric Design,1984,1(2):193-198.
[21] Vergeest J S M. CAD surface data exchange using STEP. Computer Aided Design,1991,23(4):269~281.
[22] Huang J, Chen L, Liu X G, et al. Efficient mesh deformation using tetrahedron control mesh.Proceedings of the2008ACM symposium on Solid and physical modeling, New York: ACMPress,2008:241-247.
[23] Yoshizawa S, Belyaev A, Seidel H P. Skeleton-based variational mesh deformations. ComputerGraphics Forum,2007,26(3):255-264
[24] Kho Y, Garland M. Sketching mesh deformations. Proceedings of the2005Symposium onInteractive3D Graphics and Games, New York: ACM Press,2005:147-154.
[25]董洪伟.网格变形综述.中国图象图形学报,2011,16(12):2095-2014.
[26]许秋儿.网格变形技术研究,[博士学位论文].杭州:浙江大学,2009.
[27] Barr Alan H. Global and local deformations of solid primitives. Computer Graphics,1984,18(3):21-30.
[28] Sederberg T W, Parry S R. Free-form deformations of solid geometric models. Proceedings ofComputer Graphics, Annual Conference Series, ACM SIGGRAPH, New York: ACM Press,1986:151-160.
[29] Griessmair J, Purgathofer W. Deformation of solids with trivariate B-Splines. Proeeedings ofEurographics, North-Holland: Elsevier Science,1989:137-148.
[30] Kalra P, Mangili A, Thalmann N M, et al. Simulation of facial muscle actors based on rationalfree-form deformation. Computer Graphics Forum,1992,11(3):59-69.
[31] Lamousin H J, Waggenspack W N. NURBS-based free-form deformation. IEEE ComputerGraphics and Applications,1994,14(6):59-65.
[32] Coquillart S. Extended free-form deformation: A sculpting tool for3D geometric modeling.Computer Graphics,1990,24(4):187~196.
[33] MacCracken R, Joy K I. Free-form deformations with lattices of arbitrary topology.Proceedingsof the23rdannual conference on Computer graphics and interactive techniques, NewYork: ACM SIGGRAPH,1996:181~188.
[34] Ju T, Schaefer S, Warren J. Mean value coordinates for closed triangular meshes. ACMTransactions on Graphics,2005,24(3):561-566.
[35]张湘玉,廖文和,刘浩.基于细分曲面的泊松网格编辑.计算机辅助设计与图形学学报,2010,22(1):76-84.
[36]王仁芳,许秋儿,汪沁,等.基于最小二乘网格的模型变形算法.计算机辅助设计与图形学学报,2010,22(5):777-783.
[37] Feng J Q, Peng Q S, Ma L Z. A new free-form deformation through the control of parametricsurfaces. Computers&Graphics,1996,20(4):531-539.
[38] Chen B Y, Ono B, Johan H, et al.3D model deformation along a parametric surface. Proceedingsof International Conference on Visualization, Imaging and Image Processing, Berlin: Springer,2002:282-287.
[39] Wang X P, Ye Z L, Meng Y Q, et al. Space deformation of parametric surface based on extensionfunction. International Journal of CAD/CAM,2004,17(4):251-259.
[40] Wang X P, Zhou R R, Ye Z L, et al. Shape modification and deformation of parametric surface.Computer Aided Drafting, Design and Manufacture,2002,12(1):1-7.
[41]陆友太,周来水,王志国,等.基于乘幂伸缩因子的参数曲线自由变形.南京航空航天大学学报,2011,43(6):793-798.
[42]陆友太,周来水,王志国,等.基于多项式伸缩因子的参数曲面自由变形.中国机械工程,2012,23(5):609-613.
[43] Lazarus F, Coquillart S, Jancene P. Axial deformation: an intuitive deformationtechnique.Computer Aided Design,1994,26(8):607~613.
[44] Chang Y K, Rockwood A P. A generalized de casteljua approach to3D free-form deformation.Proceedings of21st International SIGGRAPH Conference on Computer Graphics, New York:ACM SIGGRAPH,1994:257~260.
[45] Singh K, Fiume E. Wires: A geometric deformation technique. Computer Graphics,1998,32(3):405-414.
[46]王隽,张宏鑫,许栋,等.勾画式泊松网格编辑.计算机辅助设计与图形学学报,2006,18(11):1723-1729.
[47] Hsu W M, Hughes J F, Kaufman H. Direct manipulation of free-form deformations. ComputerGraphics,1992,26(2):177-184.
[48] Hu S M, Zhang H, Tai C L, et al. Direct manipulation of FFD: Efficient explicit solutions anddecomposable multiple point constraints. Visual Computer,2001,17(6):370-379.
[49] McDonnell K T, Qin H. PB-FFD: A point-based technique for free-form deformation. Journal ofGraphics Tools,2007,12(3):25-41.
[50] Xu D, Chen W, Zhang H X, et al. Multi-level differential surface representation based on localtransformations. The Visual Computer,2006,22(7):493-505.
[51]张湘玉.基于细分的曲线曲面变形技术研究,[博士学位论文].南京:南京航空航天大学,2010.
[52]李登高.基于小波的多分辨率数字几何造型技术的研究,[博士学位论文].北京:清华大学,2006.
[53]张亚萍,熊华,姜晓红,等.大型网格模型简化和多分辨率技术综述.计算机辅助设计与图形学学报,2010,22(4):559-568.
[54] Zorin D, Schroder P, Sweldens W. Interactive multiresolution mesh editing. Proeeedings of the24th Annual Conference on Computer Grapphics and Interaetive Techniques, NewYork: ACMPress,1998:259-268.
[55] Lee A, Moreton H, Hoppe H. Displaced subdivision surfaces. Proceedings of the ACMSIGGRAPH Conference on Computer Graphics, New York: ACM Press,2000:85-94.
[56] Feng J Q, Shao J, Jin X G, et al. Multiresolution free-form deformation with subdivision surfaceof arbitrary topology. The Visual Computer,2005,22(1):28-42.
[57] Finkelstein A, Salesin D H. Multiresolution curves. Proceedings of the21stAnnual Conferenceon Computer Graphics and Interactive Techniques, New York: ACM Press,1994:261-268.
[58] Lounsbery M, DeRose T D, Warren J. Multiresolution analysis for surfaces of arbitrarytopological type. ACM Transactions on Graphics1997,16(1):34-73.
[59] Hoppe H. Progressive meshes. Proceedings of the23rd Annual Conference on ComputerGraphics and Interactive Techniques New York: ACM Press,1996:99-108.
[60] Botsch M, Kobbelt L. Multiresolution surface representation based on displacement volumes.Comupter Graphics Forum,2003,22(3):483-491.
[61] Blum H. Biological shape and visual science(part I). Journal of Theoretical Biology,1973,38(2):205-287.
[62] Capell S, Green S, Curless B, et al. Interactive skeleton-driven dynamic deformation. ACMTransactions on Graphics,2002,21(3):586-593.
[63] Yoshizawa S, Belyaev A G, Seidel H P. Free-form skeleton driven mesh deformations.Proceedings of the8th ACM Symposium Solid Modeling and Application, New York: ACMPress,2003:247-253.
[64] Magnenat N, Laperriere R. Joint-dependent local deformations for hand animation and objectgrasping. Proceedings on Graphics interface '88, Edmonton: Canadian Information ProcessingSociety Toronto,1989:26-33.
[65] Lewis J P, Cordner M. Pose space deformation: A unified approach to shape interpolation andskeleton-drvien deformation. Proceedings of the27th Annual Conference on Computer Graphicsand Interactive Techniques, New York: ACM Press,2000:165-172.
[66] Allen B, Curless B, Popovic Z. Articulated body deformation from range scan data. ACMTransactions on Graphics,2002,21(3):612-619.
[67]许秋儿,谭光华,张三元,等.保持几何特征的均值骨架子空间网格变形.计算机辅助设计与图形学学报,2009,21(3):289-294.
[68] Wang X C, Phillips C. Multi-weight enveloping: least-squares approximation techniques for skinanimation. Proceedings of the2002ACM SIGGRAPH/Eurographics Symposium on ComputerAnimation, NewYork: ACM Press,2002:129-138.
[69] Kavan L, Zara J. Spherical blend skinning: a real-time deformation of articulated models.Proceedings of the2005symposium on Interactive3D graphics and games, New York: ACMPress,2005:9-16.
[70] Yan H B, Hu S M, Martin R R. Shape deformation using a skeleton to drive simplextransformations. IEEE Transactions on Visualization and Computer Graphics,2008,14(3):693-706.
[71] Terzopoulos D, Platt J, Barr A. Elastically deformable models. Computer Graphics,1987,21(4):205-214.
[72] Celniker G, Welch W. Linear constraints for deformable non-uniform B-spline surfaces.Proceedings of the Symposium on Interactive3D Graphics, New York: ACM Press,1992:61-68.
[73] Wang Z G, Zhou L S, Wang X P. Direct Manipulation of B-spline surfaces. Chinese Journal ofMechanical Engineering,2005,18(1):103-108.
[74] Terzopoulos D, Qin H. Dynamic NURBS with geometric constraints for interactive sculpting.ACM Translations on Graphics,1994,13(2):103-136.
[75] Qin H, Terzopoulos D. Dynamic NURBS swung surfaces for physics-based shape design.Computer Aided Design, l995,27(2):111-127.
[76] Qin H, Terzopoulos D. Triangular NURBS and their dynamic generalizations. Compter AidedGeometric Design,1997,14(4):325-347.
[77] Qin H, Mandal C, Vemuri B C. Dynamic Catmull-Clark subdivision surface. Visualization andComputer Graphics,1998,4(3):215-229.
[78]李永林,王启付,钟毅芳,等.基于物理的准D-NURBS造型方法.计算机仿真,1998,15(2):46-50.
[79] Nealen A, Müller M, Keiser R, et al. Physically based deformable models in computer graphics.Computer Graphics Forum,2006,25(4):809-836.
[80] Waters K. A muscle model for animation three-dimensional facial expression. ACM SIGGRAPHComputer Graphics,1987,21(4):17-24.
[81] Miller G S P. The motion dynamics of snakes and worms. ACM SIGGRAPH Computer Graphics,1988,22(4):169-173.
[82] Li L, Volkov V. Cloth animation with adaptively refined meshes. Proceedings of the28thAustralasian conference on Computer Science, Darlinghurst: Australian Computer Society,2005:107-113.
[83] Vassilev T I. Comparison of parallel algorithms for modelling mass-springs systems with severalAPIs on modern GPU. Proceedings of the12th International Conference on Computer Systemsand Technologies, New York: ACM Press,2011:204-209.
[84] Thierry S E B. A particle-spring approach to geometric constraints solving. Proceedings of the2011ACM Symposium on Applied Computing, New York: ACM Press,2011:1100-1105.
[85]王峰军,邓克文,魏发远.基于物理的变形曲线曲面造型技术综述.计算机仿真,2006,23(4):41-45.
[86]梁醒培,王辉.应用有限元分析.北京:清华大学出版社,2010.
[87] Clough R W. The Finite Element Method in Plane Stress Analysis. The2nd ASME Conferenceon Electronic Computation, New York: ACM Press,1960:1114-1131.
[88] Turner M J, Clougb R W, Martin H C, et al. Stiffness and deflection analyses of complexstructures. J. Aero. Sci,1956,23(7):805-823.
[89] Courant R. Variational method for solutions of problems of equilibriumand vibrations. Bull. Am.Math. SoC.,1943,49(3):1~23.
[90]朱菊芬,汪海,徐胜利.非线性有限元及其在飞机结构设计中的应用.上海:上海交通大学出版社,2012.
[91] Celniker G, Gossard D. Deformable curve and surface finite elments for free-form shape design.Computer Graphics,1991,25(4):257-266.
[92] Collier J R, Collier B J, O'Toole G, et al. Drape prediction by means of finite-element analysis.Journal o f the Textile Insitute,1991,82(1):96-107.
[93] Chen D, Zeltzer D. Pump it up: computer animation of a biomechanically based model of muscleusing the finite element method. ACM SIGGRAPH Computer Graphics,1992,26(2):89-98.
[94] Guan Z D, Jing L, Ning T, et al. Study and application of physics-based deformable curves andsurfaces. Computers&Graphics,1997,21(3):305-313.
[95]经玲,席平,唐荣锡.有限元方法在变形曲线曲面造型中的应用.计算机学报,1998,21(3):245-251.
[96]于超升.基于刚架模型的复杂多曲面光滑协调变形算法研究,[硕士学位论文].南京：南京航空航天大学,2012.
[97]成思源,张湘伟.可变形B样条曲线曲面模型及其有限元求解.重庆大学学报,2003,26(2):75-77.
[98]程仙国,刘伟军,卞宏友.基于外载荷的B样条曲面变形.机械工程学报,2011,47(09):146-150.
[99]程仙国,刘伟军.外载荷的B样条曲线变形.中国图像图形学报,2011,16(5):898-902.
[100] Yu C S, Zhou L S, Wang Z G, et al. Shape modification of B-spline surfaces based on rigidframe via geometric constraint optimization. Proceedings of the conference on intelligentHuman-machine Systems and Cybernetices, Hangzhou：IEEE,2011:120-124.
[101] O'Brien J F, Hodgins J K. Graphical modeling and animation of brittle fracture. Proceedings ofSIGGRAPH1999, NewYork: ACM Press,1999:137-146.
[102] O'Brien J F, Hodgins J K, Bargteil A M. Graphical modeling and animation of ductile fracture.ACM Transactions on Graphics,2002,21(3):291-294.
[103] Capell S, Green S, Curless B, et al. A multi-resolution framework for dynamic deformations.Proceedings of the2002ACM SIGGRAPH Symposium on Computer Animation, NewYork:ACM Press,2002:41-47.
[104] Grinspun E, Krysl P, Schroder P. CHARMS: a simple framework for adaptive simulation. ACMTransactions on Graphics,2002,21(3):281-290.
[105] Müller M, Dorsey J, McMillan L, et al. Stable real-time deformations. Proceedings of the2002ACM SIGGRAPH/eurographics symposium on computer animation, New York: ACM Press,2002:49-54.
[106] Müller M, Gross M. Interactive virtual materials. Proceedings of Graphics Interface '04,Waterloo: Canadian Human-Computer Communications Society School of Computer Science,2004:239-246.
[107] Huang J, Liu X G, Bao H J, et al. An efficient large deformation method using domaindecomposition. Computers&Graphics,2006,30(6):927-935.
[108] Terzopoulos D, Witkin A. Physically based models with rigid and deformable components.IEEE Computer and Graphic Application,1988,8(6):41-51.
[109]王志国,周来水,王小平.基于材料力学模型的变形造型新方法.机械工程学报,2005,41(9):155-160.
[110] Wang Z G, Wang X P, Bao Y D, et al. Shape modification by beam model in FEM. ChineseJournal of Aeronautics,2010,23(2):246-251.
[111] Dick C, Georgii J, Westermann R. A hexahedral multigrid approach for simulating cuts indeformable objects. IEEE Transactions on visualization and computer graphics,2011,17(11):1663-1675.
[112] Song C, Zhang H X, et al Wu Y. Cutting and fracturing models without remeshing. Proceedingsof the5th international conference on Advances in geometric modeling and processing, Berlin:Springer-Verlag,2008,
[113]李静,陈健云,周晶.改进的SSOR-PCG迭代法在接触问题研究中应用.大连理工大学,2006,46(4):533-537.
[114] Jin X G, Li Y F, Peng Q S. General constrained deformation based on generalized metaballs.Computers&Graphics,2000,24(2):219-231.
[115]施法中.计算机辅助几何设计与非均匀有理B样条.北京:高等教育出版社,2001.