过渡面生成算法及曲面光顺技术研究
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摘要
过渡曲面生成是计算机辅助几何设计和计算机图形学的一项重要课题,主要研究在计算机图形系统的环境下对曲面的表示、设计、显示和分析。随着CAD/CAM技术的迅猛发展,曲面造型技术在产品设计中的应用日益广泛,与此同时过渡曲面的构造理论与方法是当前的研究热点之一。
曲面光顺也是CAGD中的重要课题,在航空、宇航、汽车和船舶等设计制造业中有着重要的应用,因此受到普遍重视。但由于光顺处理的复杂性,直到现在,此问题还没有得到彻底的解决,对它的研究仍在进行中。
本文首先对典型过渡曲面生成算法的原理、特点、应用范围、优缺点,如滚球法、偏微分方程法、能量法、基于脊线、裁剪线的过渡等进行了研究和分析,探讨了过渡面设计中存在的问题。重点研究了基于偏微分方程法构造过渡曲面,从传统静态PDE和动态PDE两个方面来系统研究了过渡面的构造。利用Pseudo-Navier法实现了构造C~1连续过渡面。
针对传统静态偏微分方程构造过渡面存在的不足,提出基于动态偏微分方程构造C~1连续的过渡面,并用迭代有限差分法求解PDE的数值解,讨论了形状控制因子、密度、阻尼系数等物理参数的变化对曲面形状的影响,使过渡面构造具有更高的灵活性,大大提高了工业几何设计的交互性。
此外,本文总结介绍了曲面光顺的原理,曲面光顺的检查准则,以及实现曲面光顺的几种比较常用的方法,如小波分析法、能量法。比较了各种方法的特点、优缺点及适用场合。将非均匀B样条小波分解用于曲面光顺,根据整体光顺度量确定NURBS曲面整体光顺时的光顺方向。根据曲面在节点对处的光顺性选择曲面上的坏点,通过局部小波分解实现曲面的局部光顺。该方法有效可行,能较好的实现对曲面的光顺处理。
Blending surfaces generation is an extremely important part in Computer Aided Geometric Design and Computer Graphics.It's primary research objects are surfaces expression,design,display and analyze under Computer Graphics system.With the rapid development of CAD/CAM,technology of surface modeling has been used in more and more products design.At the same time,the theory and methods of blending surfaces generation is one of attractive fields.
Surface fairing is also one of important topics in CAGD research.Because of its significant application in CAD/CAM domains such as aviation,space navigation,autos and shipping,this issue become a research focus.The research on surface fairing has been continued to the present because of the complexity.
In this paper,firstly,we research blending surfaces generation methods,analyzed and compared all these methods on the aspects of their theories,characteristics,scope,advantages and disadvantages.The research focus on generating blending surfaces base on partial differential equations.From traditional static PDE-based and dynamic PDE-based,analyzed their theory and methods.We generated blending surface with C'continuity using Pseudo-Navier method.
To solve the shortage that traditional static PDE method has,proposes dynamic PDE to generate blending surfaces.It uses iterative finite difference algorithm to solve dynamic PDE, and investigates how different user handles affect dynamic deformations of surfaces,such as parametric variables,density and damping coefficient on.The modeling samples illustrate that the new dynamic PDE method brings more and better flexibility.It is very valuable for aided geometric design.
Finally,we research the theory and rule of surface fairing and several classical faring methods,moreover analyze their advantage and disadvantage.In this paper,non-uniform B-spline multi-resolution wavelet decomposition was applied to the fairing of NURBS surface. The global faring direction was determined by global faring measure of two parametrical directions,and then the global fairing of surface was realized though global decomposition of surface.This algorithm is effective,and it will be one of research focuses in the future.
曲面光顺也是CAGD中的重要课题,在航空、宇航、汽车和船舶等设计制造业中有着重要的应用,因此受到普遍重视。但由于光顺处理的复杂性,直到现在,此问题还没有得到彻底的解决,对它的研究仍在进行中。
本文首先对典型过渡曲面生成算法的原理、特点、应用范围、优缺点,如滚球法、偏微分方程法、能量法、基于脊线、裁剪线的过渡等进行了研究和分析,探讨了过渡面设计中存在的问题。重点研究了基于偏微分方程法构造过渡曲面,从传统静态PDE和动态PDE两个方面来系统研究了过渡面的构造。利用Pseudo-Navier法实现了构造C~1连续过渡面。
针对传统静态偏微分方程构造过渡面存在的不足,提出基于动态偏微分方程构造C~1连续的过渡面,并用迭代有限差分法求解PDE的数值解,讨论了形状控制因子、密度、阻尼系数等物理参数的变化对曲面形状的影响,使过渡面构造具有更高的灵活性,大大提高了工业几何设计的交互性。
此外,本文总结介绍了曲面光顺的原理,曲面光顺的检查准则,以及实现曲面光顺的几种比较常用的方法,如小波分析法、能量法。比较了各种方法的特点、优缺点及适用场合。将非均匀B样条小波分解用于曲面光顺,根据整体光顺度量确定NURBS曲面整体光顺时的光顺方向。根据曲面在节点对处的光顺性选择曲面上的坏点,通过局部小波分解实现曲面的局部光顺。该方法有效可行,能较好的实现对曲面的光顺处理。
Blending surfaces generation is an extremely important part in Computer Aided Geometric Design and Computer Graphics.It's primary research objects are surfaces expression,design,display and analyze under Computer Graphics system.With the rapid development of CAD/CAM,technology of surface modeling has been used in more and more products design.At the same time,the theory and methods of blending surfaces generation is one of attractive fields.
Surface fairing is also one of important topics in CAGD research.Because of its significant application in CAD/CAM domains such as aviation,space navigation,autos and shipping,this issue become a research focus.The research on surface fairing has been continued to the present because of the complexity.
In this paper,firstly,we research blending surfaces generation methods,analyzed and compared all these methods on the aspects of their theories,characteristics,scope,advantages and disadvantages.The research focus on generating blending surfaces base on partial differential equations.From traditional static PDE-based and dynamic PDE-based,analyzed their theory and methods.We generated blending surface with C'continuity using Pseudo-Navier method.
To solve the shortage that traditional static PDE method has,proposes dynamic PDE to generate blending surfaces.It uses iterative finite difference algorithm to solve dynamic PDE, and investigates how different user handles affect dynamic deformations of surfaces,such as parametric variables,density and damping coefficient on.The modeling samples illustrate that the new dynamic PDE method brings more and better flexibility.It is very valuable for aided geometric design.
Finally,we research the theory and rule of surface fairing and several classical faring methods,moreover analyze their advantage and disadvantage.In this paper,non-uniform B-spline multi-resolution wavelet decomposition was applied to the fairing of NURBS surface. The global faring direction was determined by global faring measure of two parametrical directions,and then the global fairing of surface was realized though global decomposition of surface.This algorithm is effective,and it will be one of research focuses in the future.
引文
[1]周儒荣,张丽艳,苏旭,周来水.海量数据点的曲面重建算法研究[J].软件学报,2001,41(2):249-252.
[2]Joseph Pegna,Franz-Erich Wolter.Geometricla criteria to guarantee curvature continuity of blend surfaces[J].Journal of mechanical design.2002,114(4):201-210.
[3]Janos Vida,Ralph R Martin and Tamas Varady.A survey of blending methods that use parametric surfaces[J].Computer-aided Design,1994,26(5):341-365.
[4]Erich Hartmann.Parametric G~n blending of curves and surfaces[J].The Visual Computer,2001,17:1-13.
[5]刘金义,候宝明.STL 格式实体的快速拓扑重建[J].工程图学学报,2003,24(4):34-39.
[6]Hosaka.M,Theory of Curve and Surface Synthesis and Their Smooth Fitting[J].Information Processing in Japan,1969,9,60-68.
[7]苏步青 刘鼎元 计算几何[M]上海科技出版社,1981
[8]Kjellander J A P.Smoothing of cubic Parametric splines[J].CAD,1983,15(3):175-179.
[9]G.Farin,G.Rein,N.Sapidis,A.J.Worsey Faring cubic B-spline curves[J].CAGD 1987,4:91-103.
[10]Sapidis N,Farin G Automatic fairing algorithm for B-spline curves[J].CAD,1989,22(2):121-129
[11]Hans Hagen,Variational design of smoooth rational Bezier curves[J].CAGD.1991,8:393-399
[12]Poliakoff JF.An improved algorithm for automatic fairing of non-uniform parametric cubic splines[J].CAD 1996,28:59-66.
[13]Nowaki.H and Resse.D Design and fairing of ship surface,in:Barnihill and Boehmends[C],Surface in CAGD,North-Holland,Amsterdem,1983:121-134.
[14]Kjellander J A P.Smoothing of cubic Parametric splines[J].CAD,1983,15(5):288-293.
[15]Lott.NJ and Fullon.DJ.Method for fairing B-spline surface[J].CAD,1988,Vol.20 597-604
[16]Rando.T and Roulier.J.A.Designing faired parametric surface[J].CAD,1991,VOL.23
[17]S.A.Coons.Surfaces for Computer-Aided Design of Space Form[C].technical report MAC-TR-41,Massachusetts of Technology,Cambridge,Mass,1967.
[18]R.E.Bamhill,G.Birkhoff,and W.J.Gordon.Smooth Interpolation in Triangles[J].J.Approximation Theory,1973,8:114-128.
[19]J.A.Gregory.N-Sided Surface Pathches[C].The Mathematics of Surfaces,J.A.Gregory,ed.,Clarendon Press,Oxford,England,1986:217-232
[20]Varady,T,Rockwood,A.Geometric construction for setback vertex blending[J].Computer-Aided Design.1993,21:871-878.
[21]Farouki.RAMT,Sverrission.R.Approximation of rolling-ball blends for free-form parametric surfaces[J].Computer-Aided Design,1993,21:871-878.
[22]Gabor Lukacs.Differential geometry of G~1 variable radius rolling ball blends surfaces[J].Computer Aided Geometric Design,1998,15(6):585-613.
[23]Braid.IC.Non-local blending of boundary models[J].Computer-Aided Design,1997,29(2):89-100
[24]Chuang.JH.Hwang,WC.Variable-radius blending by constrained spine generation[J].Visual Computer,1997,13(7):316-329.
[25]Wallner,J,pottmann,H.Rational blending surfaces between quadrics[J].Computer Aided Geometric Design,1997,14:407-419.
[26]Hoffmann,C,Hopcroft,J..Automatic surface generation in computer aided design[J].The visual Computer,1985,1:92-100.
[27]Hoffmann,C,Hopcroft,J..Quadric blending surfaces[J],Computer-Aided Design,1986,18:301-306.
[28]Rockwood,A.P.Owen,J.C.Blending surfaces in solid modeling in Farin,G.(ED.)[J],Geometric Modelin g,SIAM,Philadelphia,1987:367-383.
[29]Warren,J..Blending algebraic surfaces[J].ACM Trans.On Graphics,1989,8,263-278.
[30]Erich Hartmann.Implicit G~n-blending of vertices[J].Computer Aided Geometric Design,2001,18:267-285
[31]马岭,张鲜,朱心雄.用偏微分方程构造过渡曲面[J].工程图学学报,1995,NO.1:1-8.
[32]周来水,张乐年.NURBS 曲面的过渡曲面生成[J].工程图学学报,1995,NO.2:52-57.
[33]朱汉东,金东光.过渡曲面的母线构造[J].工程图学学报,1998,NO.3:45-48.
[34]张三元,梁友栋.G~1管状曲面的整体造型方法[J].计算机辅助设计与图形学学报,1999,11(1):4-7.
[35]陈发来,陈长松,邓建松.用分片代数曲面构造管道曲面的过渡曲面[J].计算机学报,2000,23(9):911-916.
[36]黄正东,王启付,周济,余俊.曲面G~1拼接中的变分法[J].计算数学,1996,18(4):367-376.
[37]罗宏志,李志刚,肖景容.用有理 Bezier 曲面生成过渡曲面[J].华中科技大学学报,23(9):911-916.
[38]石雄辉.高阶几何连续过渡曲面的造型研究与应用[J].西北工业大学硕士论文,2003
[39]HaradaT,Variable-radius blending by using Gregorypat chesin geometric modeling[J].Proc.Eurographics'91,North-Holland,2001:507-518
[40]Celniker.G.and Gossard,D,Dcformable Curvcand Surface finite-Elements for Free Forem Shape Design[J].ACM.Computer Graphics,Vol.25.No.4.1997,257-266
[41]彭芳瑜,周云飞,周济.基于广义能量法的曲面光顺[J].华中科技大学学报(自然科学版),2002,30(2):5-8.
[42]赵作智,林亨,时晓明.采用能量方程优化零件数据模型[J].机械设计与制造,2000,(2):24-26.
[43]MIG Bloorand M.J.Wilson.Generating blend usingpartial differential equations[J].Computer-Aided design,1999,21(4):165-171.
[44]MIG Bloorand M.J.Wilson.Using partial differential equations generating free form surfaces [J].Computer-Aided design,2000,22(4):221-234.
[45]Rossignac J.R,and Requicha A.A.Constant-Radius blending in solid modeling[J].Comput Mech.Eng,1984,(3):65-73
[46]Tiller W.Rational B-spline for curves and surface Representation[C].IEEE CG and A,1983,3(4):61-69.
[47]Woodward C.D.Cross-Scctional design of B-spline surface[J].Computer Graphics,1987,11(2):193-201.
[48]Sanglikar N.A,Koparkar P,Joshi V.N,Modeling Rolling Ball blends for computer aided geometric design[J].Computer aided geometric design,1990,7(7):399-414.
[49]Alfeld,P.Derivative generation from multivariate scattered data by functional minimization[J],CAGD2,1985:281-296
[50]Williams,C.J.K.Use of structural analogy in generation of smooth surfaces for engineering purposes[J],Computer-Aided Design 19,1987:310-322.
[51]Celniker,G and gossard,D.Energy-based bodels for free-form surfaces shape design[C].In:ASME Design Automation Conference,ASME,New York,1988:107-112.
[52]L.H.You,Jian J.Zhang,P.Comninos,Generating Blending Surfaces with a Pseudo-Le'vy Series Solution to Fourth Order Partial Differential Equations[J],Computing 71,353-373(2003)
[53]L.H.You,P.Comninos,Jian J.Zhang,PDE blending surfaces with C~2 continuity[J],Computers &Graphics 28(2004)895-906.
[54]Du,H.,Qin,H.,Dynamic PDE-based surface design using geometric and physical constraints[J],Graphical Models,67(2005)43-71.
[55]马岭.偏微分方程曲面造型方法及其应用研究[D]北京航空航天大学博士论文,1997.
[56]熊一奇等,关于网格光顺问题[J],武汉大学学报,1987,3,9-22.
[57]刘鼎元,苏文荣,船体线型的三向光顺法和艏艉部光顺法[C],沪东造船厂技术资料,1979.
[58]沂云龙,双三次样条函数及其在曲面光顺中的应用[J],复旦学报(自然科学版),1977,63-68.
[59]满家巨 胡事民 雍俊海 孙家广 B 样条曲线的节点去除与光顺[J]软件学报2001Vol.12.No.1:143-147.
[60]王国瑾 王振武 寿华好 B 样条曲面在严格约束条件下的光顺拟和[J]软件学报1998Vol.9,No.9:696-698.
[2]Joseph Pegna,Franz-Erich Wolter.Geometricla criteria to guarantee curvature continuity of blend surfaces[J].Journal of mechanical design.2002,114(4):201-210.
[3]Janos Vida,Ralph R Martin and Tamas Varady.A survey of blending methods that use parametric surfaces[J].Computer-aided Design,1994,26(5):341-365.
[4]Erich Hartmann.Parametric G~n blending of curves and surfaces[J].The Visual Computer,2001,17:1-13.
[5]刘金义,候宝明.STL 格式实体的快速拓扑重建[J].工程图学学报,2003,24(4):34-39.
[6]Hosaka.M,Theory of Curve and Surface Synthesis and Their Smooth Fitting[J].Information Processing in Japan,1969,9,60-68.
[7]苏步青 刘鼎元 计算几何[M]上海科技出版社,1981
[8]Kjellander J A P.Smoothing of cubic Parametric splines[J].CAD,1983,15(3):175-179.
[9]G.Farin,G.Rein,N.Sapidis,A.J.Worsey Faring cubic B-spline curves[J].CAGD 1987,4:91-103.
[10]Sapidis N,Farin G Automatic fairing algorithm for B-spline curves[J].CAD,1989,22(2):121-129
[11]Hans Hagen,Variational design of smoooth rational Bezier curves[J].CAGD.1991,8:393-399
[12]Poliakoff JF.An improved algorithm for automatic fairing of non-uniform parametric cubic splines[J].CAD 1996,28:59-66.
[13]Nowaki.H and Resse.D Design and fairing of ship surface,in:Barnihill and Boehmends[C],Surface in CAGD,North-Holland,Amsterdem,1983:121-134.
[14]Kjellander J A P.Smoothing of cubic Parametric splines[J].CAD,1983,15(5):288-293.
[15]Lott.NJ and Fullon.DJ.Method for fairing B-spline surface[J].CAD,1988,Vol.20 597-604
[16]Rando.T and Roulier.J.A.Designing faired parametric surface[J].CAD,1991,VOL.23
[17]S.A.Coons.Surfaces for Computer-Aided Design of Space Form[C].technical report MAC-TR-41,Massachusetts of Technology,Cambridge,Mass,1967.
[18]R.E.Bamhill,G.Birkhoff,and W.J.Gordon.Smooth Interpolation in Triangles[J].J.Approximation Theory,1973,8:114-128.
[19]J.A.Gregory.N-Sided Surface Pathches[C].The Mathematics of Surfaces,J.A.Gregory,ed.,Clarendon Press,Oxford,England,1986:217-232
[20]Varady,T,Rockwood,A.Geometric construction for setback vertex blending[J].Computer-Aided Design.1993,21:871-878.
[21]Farouki.RAMT,Sverrission.R.Approximation of rolling-ball blends for free-form parametric surfaces[J].Computer-Aided Design,1993,21:871-878.
[22]Gabor Lukacs.Differential geometry of G~1 variable radius rolling ball blends surfaces[J].Computer Aided Geometric Design,1998,15(6):585-613.
[23]Braid.IC.Non-local blending of boundary models[J].Computer-Aided Design,1997,29(2):89-100
[24]Chuang.JH.Hwang,WC.Variable-radius blending by constrained spine generation[J].Visual Computer,1997,13(7):316-329.
[25]Wallner,J,pottmann,H.Rational blending surfaces between quadrics[J].Computer Aided Geometric Design,1997,14:407-419.
[26]Hoffmann,C,Hopcroft,J..Automatic surface generation in computer aided design[J].The visual Computer,1985,1:92-100.
[27]Hoffmann,C,Hopcroft,J..Quadric blending surfaces[J],Computer-Aided Design,1986,18:301-306.
[28]Rockwood,A.P.Owen,J.C.Blending surfaces in solid modeling in Farin,G.(ED.)[J],Geometric Modelin g,SIAM,Philadelphia,1987:367-383.
[29]Warren,J..Blending algebraic surfaces[J].ACM Trans.On Graphics,1989,8,263-278.
[30]Erich Hartmann.Implicit G~n-blending of vertices[J].Computer Aided Geometric Design,2001,18:267-285
[31]马岭,张鲜,朱心雄.用偏微分方程构造过渡曲面[J].工程图学学报,1995,NO.1:1-8.
[32]周来水,张乐年.NURBS 曲面的过渡曲面生成[J].工程图学学报,1995,NO.2:52-57.
[33]朱汉东,金东光.过渡曲面的母线构造[J].工程图学学报,1998,NO.3:45-48.
[34]张三元,梁友栋.G~1管状曲面的整体造型方法[J].计算机辅助设计与图形学学报,1999,11(1):4-7.
[35]陈发来,陈长松,邓建松.用分片代数曲面构造管道曲面的过渡曲面[J].计算机学报,2000,23(9):911-916.
[36]黄正东,王启付,周济,余俊.曲面G~1拼接中的变分法[J].计算数学,1996,18(4):367-376.
[37]罗宏志,李志刚,肖景容.用有理 Bezier 曲面生成过渡曲面[J].华中科技大学学报,23(9):911-916.
[38]石雄辉.高阶几何连续过渡曲面的造型研究与应用[J].西北工业大学硕士论文,2003
[39]HaradaT,Variable-radius blending by using Gregorypat chesin geometric modeling[J].Proc.Eurographics'91,North-Holland,2001:507-518
[40]Celniker.G.and Gossard,D,Dcformable Curvcand Surface finite-Elements for Free Forem Shape Design[J].ACM.Computer Graphics,Vol.25.No.4.1997,257-266
[41]彭芳瑜,周云飞,周济.基于广义能量法的曲面光顺[J].华中科技大学学报(自然科学版),2002,30(2):5-8.
[42]赵作智,林亨,时晓明.采用能量方程优化零件数据模型[J].机械设计与制造,2000,(2):24-26.
[43]MIG Bloorand M.J.Wilson.Generating blend usingpartial differential equations[J].Computer-Aided design,1999,21(4):165-171.
[44]MIG Bloorand M.J.Wilson.Using partial differential equations generating free form surfaces [J].Computer-Aided design,2000,22(4):221-234.
[45]Rossignac J.R,and Requicha A.A.Constant-Radius blending in solid modeling[J].Comput Mech.Eng,1984,(3):65-73
[46]Tiller W.Rational B-spline for curves and surface Representation[C].IEEE CG and A,1983,3(4):61-69.
[47]Woodward C.D.Cross-Scctional design of B-spline surface[J].Computer Graphics,1987,11(2):193-201.
[48]Sanglikar N.A,Koparkar P,Joshi V.N,Modeling Rolling Ball blends for computer aided geometric design[J].Computer aided geometric design,1990,7(7):399-414.
[49]Alfeld,P.Derivative generation from multivariate scattered data by functional minimization[J],CAGD2,1985:281-296
[50]Williams,C.J.K.Use of structural analogy in generation of smooth surfaces for engineering purposes[J],Computer-Aided Design 19,1987:310-322.
[51]Celniker,G and gossard,D.Energy-based bodels for free-form surfaces shape design[C].In:ASME Design Automation Conference,ASME,New York,1988:107-112.
[52]L.H.You,Jian J.Zhang,P.Comninos,Generating Blending Surfaces with a Pseudo-Le'vy Series Solution to Fourth Order Partial Differential Equations[J],Computing 71,353-373(2003)
[53]L.H.You,P.Comninos,Jian J.Zhang,PDE blending surfaces with C~2 continuity[J],Computers &Graphics 28(2004)895-906.
[54]Du,H.,Qin,H.,Dynamic PDE-based surface design using geometric and physical constraints[J],Graphical Models,67(2005)43-71.
[55]马岭.偏微分方程曲面造型方法及其应用研究[D]北京航空航天大学博士论文,1997.
[56]熊一奇等,关于网格光顺问题[J],武汉大学学报,1987,3,9-22.
[57]刘鼎元,苏文荣,船体线型的三向光顺法和艏艉部光顺法[C],沪东造船厂技术资料,1979.
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