多面体融合与多管道融合研究及应用
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摘要
在计算机辅助几何设计（CAGD）领域，“过渡曲面”是指光滑连接两个或多个曲面的中间曲面，被连接的曲面称为“基曲面”。生成过渡曲面的操作则称为对基曲面进行“融合”（Blending）。过渡曲面因而也称为融合曲面。多面体融合和多管道融合一直是曲面造型中研究热点之一，其工程应用非常广泛，如：机械零件上的凸尖点和边容易磨损、碰出毛刺、刺伤人，凹尖点和边难以加工，且产生应力集中，所以必须倒圆角。对于模具制造的零件，也有利于它与模具型腔/型芯分离；在构造内燃机进气或排气歧管时，过渡曲面能有效地减小管内气体流通阻力，提高进排气效率，从而改善内燃机性能；在船体外形设计中，船壳和船首之间也需要过渡曲面来减小水流阻力，提高航行速度；在飞机的机身和机翼拼接时（业内称“翼身整流”），也需要过渡曲面来降低飞行时气流阻力、减少涡流，同时使连接处压力分布均匀，减小振动和噪声；此外，根据CT或MRI成像精确重建人体三维血管网络或器官（这是医生诊断病情、明确手术方案的依据）以及为修补骨损伤用生物材料制作支架，也需要过渡曲面进行造型。因此，融合曲面在航空航天、舰船、机械乃至现代建筑业的几何形状设计中重要性不言而喻。生物医学工程更是融合曲面崭露头角的广阔天地。尽管当前三维几何造型的软件不少，它们也能生成某些过渡曲面。但现有方法还存在缺陷、不足与局限性，主要有以下四个方面：（1）融合多面体顶点的参数方法会出现扭矢兼容性问题，导致几何连续性仅为近似、构造过程复杂、高阶几何连续性难以实现；（2）融合多管道（准线为自由曲线）的隐式方法难以得到明确的隐式代数曲面方程，几何连续性近似，管道半径不可变；（3）姿态任意的多个自由截面管道还不能融合；（4）考虑使用功能及工艺对过渡曲面的外形进行优化设计，还刚刚起步。以上均为学科前沿的难题，具有较大的现实意义与理论价值。本文迎难而上，立足于对相关基础理论的深入钻研与创新，提出了诸多关于多面体融合和多管道融合的崭新方法，进而为优化其工作性能而设计了内燃机某单通排气管道和多通进气歧管中过渡曲面的外形，还提出了一种给NURBS过渡曲面自动添加拔模斜度的新视角与新算法。全文共分十章。主要进展如下：
鉴于当前绝大多数用参数曲面融合多面体的方法只能达到G1连续性（切平面连续），提出了顶点优先法。即首先以三边Bézier曲面片，张量积Bézier曲面或m边S-曲面片为工具，巧妙地通过合理放置它们的控制点到多面体的顶点及周围相关边和面上，分别对3边，4边及m边的顶点进行高阶融合。接着再利用（本文改进的）Hartmann融合函数法高阶融合多面体的边。所用的融合函数是本文提出的含多个设计参数的多项式，使得（顶点过渡曲面为三边Bézier曲面片或张量积Bézier曲面时）边过渡曲面可表示为张量积Bézier形式。融合过程中保留了足够的自由度，可用于调节顶点过渡曲面的丰满度及边过渡曲面的宽度。此外，还推导了m边S-曲面片的高阶跨界导矢。（第３章3.2节，即论文［1］，请查本学位论文最后一页）
定义了多面体上的边串。它是由多面体上多条边经多个顶点连接在一起形成的几何结构。进而将顶点优先法拓展到边串法。边串法将顶点优先法中多次顶点和边的融合转化为一次边串融合，将融合曲面从多片简化为一片，大大提高了效率。此外，还推导了m边S-曲面片可以融合所有m边边串结构的充要条件，并展示了4、5、6和7边边串的各种具体结构。（第３章3.3节［9］）
鉴于当前用多个四边参数曲面拼接成含缩进的多面体顶点过渡曲面的方法过程复杂，并且出现扭矢兼容性问题，提出用一张深度3边数m的有理S-曲面片直截了当地解决了这问题。整个构造过程被简化为一些用于确定S-曲面片控制点的线性运算（如求重心坐标）和集合运算（如求集合M）。与现有使用非有理S-曲面片的方法相比，不仅多了缩进，而且边过渡曲面可用有理张量积Bézier形式，同时使用S-曲面片的深度从5降到3。此外，推导了有理S-曲面片的一阶跨界导矢，并给出了它和平面保持G1连续性的几何约束条件。（第4章［2］）
鉴于当前融合多个法向环面的封包法未能真正解决基管道的隐式化，提出了辅助球法。封包法是Hartmann把隐函数样条法用于多个法向环面的一大贡献。隐函数样条法要求被融合的基曲面必须是隐式表示，但基管道的法向环面是参数式表示的。为此，他将基管道的准线作数值隐式化来满足这要求，所以只适用于半径不变的管道，而且只能得到近似的数值解，不能给出过渡曲面隐式代数方程。与基管道的连续性也是近似的。本文在基管道的端部，加一个或两个与之相切的辅助球，替代基管道作为隐函数样条的基曲面来构造封包曲面。因为球是可以隐式表示的，所以问题迎刃而解。这样，计算效率和精度都大大提高。另外，还提供了足够的设计参数来交互式地调节过渡曲面的形状，避免曲面造型时可能出现的爆裂、鼓包、颈缩和突出等奇异现象。如果过渡曲面形状不满足工艺和美观要求，可设定一些基准点，通过遗传算法最小化过渡曲面和基准点之间代数距离之和，来优化曲面形状。（第5章5.2节［3］）
鉴于辅助球法仍有二点不足：生成的过渡曲面只具有G1连续性，以及当给定法向环面的半径函数在接口处一阶导数不为零时要求其准线在接口处曲率为零，提出了添加Gn（n阶的几何连续性）封包法。即先对法向环面进行有理参数化，然后基于隐式代数曲面和有理参数曲面间的参数连续性条件，借助Hermite插值法直接给法向环面添加Gn封包。为了避免构造法向环面封包时出现多分支和爆裂，同时按基准点及其法向量来优化确定设计参数。此外，还为空间任意姿态的圆柱和圆锥构造了具有高阶几何连续性的封包，并将添加Gn封包法推广至多个法向截面为椭圆的管道。（第5章5.3节［5］）
鉴于当前方法只能融合多个平行自由截面管道或构造圆截面二分叉，提出依次使用三个积木块（侧曲面片、缩进曲面片和填洞曲面片）来融合任意姿态的多个自由截面管道。其中，侧曲面片由本文修改过的Hartmann融合函数法生成，用于连接相邻两管道；缩进曲面片由Coons曲面生成，用于修整所填洞的外形；填洞曲面片由Gordon-Coons曲面片生成，用于完成过渡曲面构造。与连接三个圆截面管道的传统二分叉构造方法相比，新算法可以光滑地连接三个自由截面的管道；与目前只能连接多个平行管道的方法相比，各种拓扑结构都可融合。要害在于构造缩进曲面片和填洞曲面片时的边界兼容性和扭矢兼容性，本文都予以很好解决。（第6章［6］）
鉴于当前连接两圆柱/圆锥的方法所生成的G2连续过渡曲面不能用NURBS曲面表示，所以很多现成的CAD/CAE/CAM软件用不上，本文提出用一辅助法向环面连接两管道，再构造蒙皮NURBS曲面插值该法向环面上多个截面圆，并和两基管道有G2连续性。先用应变能最小来优化法向环面的准线，最后借助CFD（计算流体力学）软件优化其传输功能。以某摩托车排气管道为案例，计算结果表明，经准线应变能优化得到的过渡曲面，其质流率（单位：千克/秒，越大流通性越好）比传统方法高26.46%,而经CFD优化得到的过渡曲面，其质流率又进一步提高18.58%。（第7章［7］）
从CFD优化某柴油机进气歧管的形状，以四个支管质流率之和（越大流通性越好）、支管质流率间最大差值（越小进气均匀性越好）和支管质流率中最小值与内腔表面积之比（越大材料利用率越高）为目标，基于多项式响应面法，分别构造联系三个指标和设计参数的代理模型。进一步利用统一目标法，建立了多目标优化函数，从而完成了进气歧管中过渡曲面外形的多目标优化设计。经优化后，三个指标相对原有设计均有不同程度提高。（第8章［8］）
鉴于当前用NURBS曲面进行过渡曲面的模具CAD设计时缺乏有效工具来添加拔模斜度以满足模具生产工艺要求，提出一个自动给NURBS曲面添加任意大小拔模斜度的算法。此外，还利用凸组合的特性，改正了现有的双线性插值曲面和B样条曲面上处处具有足够拔模斜度的充分条件。并将NURBS曲面上拔模斜度的分布用“地形”和“水平面”来可视化。比较它们之间高度，就凸现拔模斜度够不够和差多少。再借助等高线算法使曲面斜度可视化，以帮助设计者识别曲面上的陡面。（第9章［4］）
In Computer Aided Geometric Design (CAGD), a transitional surface is the surfacesmoothly connecting two or more surfaces, which are called base surfaces. Such anoperation is called blending the base surfaces, hence the transitional surface is also knownas the blending surface, or simply the blend. In surface modeling, various approaches togenerating blending surfaces for polyhedra and multiple pipes have been ardentlyinvestigated due to their extensive applications. For example, convex sharp vertices andedges in mechanical parts cause wear, burr, or harm to human, while concave ones causestechnological difficulties and stress concentration, hence rounding them are necessary. Asfor the molded parts, the blending surfaces facilitate the work of pulling them out of themould core and cavity. In constructing intake or exhaust manifolds of internal combustionengines, blending surfaces can reduce the flow resistance in the manifolds effectively andenhance the intake/exhaust efficiency so as to improve the performance. In ship building,the blending surfaces between the bow and the hull reduce the sailing resistance. Inairplanes, the blending surfaces between the fuselage and airfoil (called wing-to-bodyfairing in engineering), may decrease the resistance of airflow and vortex, make thepressure distribution uniformly, and reduce the vibration and noise. Besides, thereconstruction of3D human vascular networks or organs for diagnosis and planning theoperation based on CI or MRI images, as well as making the scaffold for repairing thebone injury using biomaterial also needs blending surfaces to model them.. Therefore, theimportance of blending surfaces goes without saying in shape design of aeronautic,astronautic, marine, mechanical and modern architecture engineering. They also play anincreasingly important role in biomedical engineering. Although there are a number ofcommercially available software packages for3D modeling, which can generate certainblending surfaces nowadays, deficiencies, inadequacies and limitations in the existingmethods have been found in four aspects:(1) the twist compatibility problem occurs in theparametric method for polyhedral vertex blending, resulting in the inaccuracy in thegeometric continuity, complication in the construction process and difficulty in realizinghigher-order geometric continuity;(2) the implicit method for blending multiple pipes with freeform directrices can not get exact analytical implicit equations, which leads to theinaccuracy in the geometric continuity and restricts the pipe radii to be constant;(3)multiple pipes with freeform cross sections and arbitrary poses can not be blended;(4)optimizing the blend design from viewpoint of working performance and technology juststarted. All the above are hard problems on the discipline front, having greater practicalmeaning and theoretical significance.Despite the difficulty, based on assiduous study of the relevant basic theory andinnovation, this thesis presents a number of novel methods for polyhedron blending andmulti-pipe blending. Furthermore, the shape of an exhaust pipe and an intake manifold forinternal combustion engines are designed aiming to optimize their performance. Inaddition, a new algorithm and a new visualization method for adding draft angle to theNURBS surfaces are proposed. The thesis is composed of ten chapters. The main
contributions are as follows:Propose the vertex-first algorithm to blend polyhedron with higher order geometriccontinuity, viewing that the majority of existing parametric methods have only G1continuity (tangent plane continuity). It first makes use of triangular Bézier patches,tensor product Bézier surfaces or m-sided S-patches to blend3-edge,4-edge andm-edge polyhedral vertices respectively with higher-order continuity by a clever trick—properly placing the control points on the vertex, surrounding edges and faces.Then, the Hartmann blending function method (improved herein) is adopted toconstruct the edge blending surface with higher order geometric continuity, in whicha polynomial blending function with enough design parameters is proposed, so thatthe edge blending surfaces can be represented in tensor product Bézier form when thevertex blending surfaces are triangular Bézier or tensor product Bézier surfaces.During the blending process, enough freedoms are left to adjust the fullness of vertexblending surfaces and width of edge blending surfaces. By the way, the higher-ordercross boundary derivatives of m-sided S-patches are derived.(Chapter3, Section3.2,namely paper [1]. Please refer to the last page of this thesis.)
Define the edge cluster on the polyhedron that means a series of polyhedral edgesconnected together by polyhedral vertices. Further, the vertex-first algorithm isextended to the edge-cluster algorithm, so that repeated vertex and edge blendingoperations for an m-edge cluster can be replaced by one edge cluster blending, andmultiple blending patches are replaced by a single patch. Thus, the blendingefficiency is greatly raised. Moreover, the necessary and sufficient condition for anm-sided S-patch to blend all the m-edge clusters is deduced, and the concreteconfigurations of4,5,6, and7-edge clusters are displayed.(Chapter3, Section3.3, [9])
Propose a whole rational S-patch of depth3to blend polyhedral vertex with setbacks,in view of the complication of the current method, which achieves setback vertexblending by joining several quadrilateral parametric surfaces with the emerging twistcompatibility problem. Thus, without twist incompatibility, the overall blendingconstruction is simplified to determining the relevant control points of a regularrational S-patch by simple linear computation and some set operations. Comparedwith the existing method using non-rational S-patches for blending polyhedral vertex,the new method not only produces setback, but also reduces the depth of the resultingregular S-patch from5to3. Moreover, the available form of edge blending surfacesis extended to rational tensor product Bézier surfaces. In addition, the first-ordercross boundary derivative of rational S-patches is deduced and the geometricconstraint condition of G1continuity between rational S-patches and planes is given.(Chapter4,[2])
Propose the auxiliary sphere based algorithm to blend multiple normal ringedsurfaces with definite implicit equations, because the existing closing basedalgorithm can not make the base pipe surfaces really implicit. Closings were animportant contribution of Hartmann using the functional spline method to blendmultiple normal ringed surfaces. This method requires the base surfaces to be implicit,but the base pipes are expressed parametrically. To meet this requirement, heimplicitized the directrices of the pipes numerically. Consequently he can blend onlythe pipes with constant radii, while the result is numerical and the geometriccontinuity is only approximate. No implicit algebraic equation of the blendingsurface is yielded. This thesis adds one or two auxiliary spheres tangent to the pipeend, employed as the base surface(s) instead of the base pipe for functional splines toconstruct closings. Since the spheres can be implicitly defined, everything is OK. Thecomputational efficiency and accuracy are both enhanced significantly. Moreover,enough design parameters are provided to adjust the blending shape interactively, inorder that different unwanted configurations, such as burst, bulge, necking andprotrusion, are avoided in surface modeling. If the generated shape does not meet thetechnical and aesthetic requirements, we can optimize it by assigning certain fiducialpoints first and then minimizing the sum of algebraic distances between the blendingsurface and fiducial points using genetic algorithm.(Chapter5, Section5.2,[3])
Aiming to remedy the two weaknesses of the above auxiliary sphere based algorithm,namely that the blending surface has only G1continuity and the directrices of initialpipes need zero curvature at their ends when the first-order derivatives of radius functions are nonzero, the algorithm of adding Gnclosing is put forward. It firstfinishes the rational parametrization of normal ringed surfaces, then adds closingsGn-continuous with the base pipes directly by Hermite interpolation method based onthe parametric continuity conditions between implicit algebraic surfaces and rationalparametric surfaces. To avoid burst and unwanted branches in the construction ofclosings, not only fiducial points but also their associated normal vectors are chosenin optimization to determine the design parameters. Moreover, closingsGn-continuous with cylinders and cones in arbitrary poses are constructed.Furthermore, this method can blend several pipe surfaces with elliptic sections aswell.(Chapter5, Section5.3,[5])
Provide the constructive approach using three building blocks (side patch, setbackpatch and hole-filling patch) sequentially to blend multiple freeform pipes in almostarbitrary poses with G1continuity, considering that the existing methods can onlyblend multiple parallel freeform pipes or construct bifurcations with circular sections.Side patches are constructed by (revised herein) Hartmann blending method forconnecting arbitrary two adjacent pipes. Setback patches are generated by Coonssurfaces for shaping the holes to be filled. Hole-filling patches are generated byGordon-Coons surfaces for accomplishing the construction of blending surface.Compared with the traditional bifurcation algorithm, the new one can connect threefreeform pipes. Compared with the current method for connecting multiple parallelpipes, miscellaneous topological configurations are blended. In constructing thesetback patches and hole-filling patches, the cruxes of this algorithm—boundarycompatibility and twist compatibility are well coped with.(Chapter6,[6])
Propose a method for blending two cylinders/cones using NURBS surfaces with G2continuity and optimized shape, in contrast to the existing method by which thenormal ringed surfaces can not be represented in NURBS form, so that manyready-to-use CAD/CAE/CAM software packages can not apply. The new methodfirst connects two base pipes using a reference normal ringed surface whose directrixstrain energy is minimized, and then a skinning NURBS surface with G2continuity isconstructed to interpolate the cross section circles of the reference normal ringedsurface. Finally the conveying capability of the NURBS surfaces is maximized withCFD (Computational Fluid Dynamics) software The computation results of a casestudy—a motorcycle exhaust pipe show that the strain energy optimization canraise the mass flow rate (in kg/sec, the larger the better)26.46%greater than theconventional method, and the CFD optimization further increases it by18.58%.(Chapter7,[7])
Optimize the blending surface shape in a diesel’s intake manifold from CFDviewpoint. After computing the mass flow rates of the four branching pipes of theintake manifold based on CFD simulation respectively, the sum of them (itsmaximum implying the minimum flow resistance), the maximum difference betweenthem (its minimum implying the best uniformity) and the ratio of the minimum massflow rate to the internal surface area (its maximum implying the highest materialutilization) are taken as the objectives. The polynomial response surfacemethodology is utilized to build three surrogate models that associate the above threeobjectives with the design parameters respectively. Then, multi-objectiveoptimization function is constructed by unification-object method. Compared withthe original design, the performances of three objectives are improved in differentextent after optimization.(Chapter8,[8])
Propose an automatic algorithm for adding arbitrary draft angle to NURBS surfacesin mould manufacturing, as there is lacking such an effective tool in mould CAD.Moreover, the sufficient conditions for bilinear interpolation and B-spline surfaces tohave draft angle everywhere are corrected based on the convex combination. Thedraft angle distribution all over a NURBS surface is visualized by a terrain and alevel (a horizontal plane). By comparing their height, the draft eligibility of thesurface becomes obvious qualitatively (yes or no) and quantitatively (how much).Additionally, the isocline curves can be found by computing the terrain-levelintersection points numerically using a contouring algorithm, so that the steep surfacecan be viewed by designers easily.(Chapter9,[4])
鉴于当前绝大多数用参数曲面融合多面体的方法只能达到G1连续性（切平面连续），提出了顶点优先法。即首先以三边Bézier曲面片，张量积Bézier曲面或m边S-曲面片为工具，巧妙地通过合理放置它们的控制点到多面体的顶点及周围相关边和面上，分别对3边，4边及m边的顶点进行高阶融合。接着再利用（本文改进的）Hartmann融合函数法高阶融合多面体的边。所用的融合函数是本文提出的含多个设计参数的多项式，使得（顶点过渡曲面为三边Bézier曲面片或张量积Bézier曲面时）边过渡曲面可表示为张量积Bézier形式。融合过程中保留了足够的自由度，可用于调节顶点过渡曲面的丰满度及边过渡曲面的宽度。此外，还推导了m边S-曲面片的高阶跨界导矢。（第３章3.2节，即论文［1］，请查本学位论文最后一页）
定义了多面体上的边串。它是由多面体上多条边经多个顶点连接在一起形成的几何结构。进而将顶点优先法拓展到边串法。边串法将顶点优先法中多次顶点和边的融合转化为一次边串融合，将融合曲面从多片简化为一片，大大提高了效率。此外，还推导了m边S-曲面片可以融合所有m边边串结构的充要条件，并展示了4、5、6和7边边串的各种具体结构。（第３章3.3节［9］）
鉴于当前用多个四边参数曲面拼接成含缩进的多面体顶点过渡曲面的方法过程复杂，并且出现扭矢兼容性问题，提出用一张深度3边数m的有理S-曲面片直截了当地解决了这问题。整个构造过程被简化为一些用于确定S-曲面片控制点的线性运算（如求重心坐标）和集合运算（如求集合M）。与现有使用非有理S-曲面片的方法相比，不仅多了缩进，而且边过渡曲面可用有理张量积Bézier形式，同时使用S-曲面片的深度从5降到3。此外，推导了有理S-曲面片的一阶跨界导矢，并给出了它和平面保持G1连续性的几何约束条件。（第4章［2］）
鉴于当前融合多个法向环面的封包法未能真正解决基管道的隐式化，提出了辅助球法。封包法是Hartmann把隐函数样条法用于多个法向环面的一大贡献。隐函数样条法要求被融合的基曲面必须是隐式表示，但基管道的法向环面是参数式表示的。为此，他将基管道的准线作数值隐式化来满足这要求，所以只适用于半径不变的管道，而且只能得到近似的数值解，不能给出过渡曲面隐式代数方程。与基管道的连续性也是近似的。本文在基管道的端部，加一个或两个与之相切的辅助球，替代基管道作为隐函数样条的基曲面来构造封包曲面。因为球是可以隐式表示的，所以问题迎刃而解。这样，计算效率和精度都大大提高。另外，还提供了足够的设计参数来交互式地调节过渡曲面的形状，避免曲面造型时可能出现的爆裂、鼓包、颈缩和突出等奇异现象。如果过渡曲面形状不满足工艺和美观要求，可设定一些基准点，通过遗传算法最小化过渡曲面和基准点之间代数距离之和，来优化曲面形状。（第5章5.2节［3］）
鉴于辅助球法仍有二点不足：生成的过渡曲面只具有G1连续性，以及当给定法向环面的半径函数在接口处一阶导数不为零时要求其准线在接口处曲率为零，提出了添加Gn（n阶的几何连续性）封包法。即先对法向环面进行有理参数化，然后基于隐式代数曲面和有理参数曲面间的参数连续性条件，借助Hermite插值法直接给法向环面添加Gn封包。为了避免构造法向环面封包时出现多分支和爆裂，同时按基准点及其法向量来优化确定设计参数。此外，还为空间任意姿态的圆柱和圆锥构造了具有高阶几何连续性的封包，并将添加Gn封包法推广至多个法向截面为椭圆的管道。（第5章5.3节［5］）
鉴于当前方法只能融合多个平行自由截面管道或构造圆截面二分叉，提出依次使用三个积木块（侧曲面片、缩进曲面片和填洞曲面片）来融合任意姿态的多个自由截面管道。其中，侧曲面片由本文修改过的Hartmann融合函数法生成，用于连接相邻两管道；缩进曲面片由Coons曲面生成，用于修整所填洞的外形；填洞曲面片由Gordon-Coons曲面片生成，用于完成过渡曲面构造。与连接三个圆截面管道的传统二分叉构造方法相比，新算法可以光滑地连接三个自由截面的管道；与目前只能连接多个平行管道的方法相比，各种拓扑结构都可融合。要害在于构造缩进曲面片和填洞曲面片时的边界兼容性和扭矢兼容性，本文都予以很好解决。（第6章［6］）
鉴于当前连接两圆柱/圆锥的方法所生成的G2连续过渡曲面不能用NURBS曲面表示，所以很多现成的CAD/CAE/CAM软件用不上，本文提出用一辅助法向环面连接两管道，再构造蒙皮NURBS曲面插值该法向环面上多个截面圆，并和两基管道有G2连续性。先用应变能最小来优化法向环面的准线，最后借助CFD（计算流体力学）软件优化其传输功能。以某摩托车排气管道为案例，计算结果表明，经准线应变能优化得到的过渡曲面，其质流率（单位：千克/秒，越大流通性越好）比传统方法高26.46%,而经CFD优化得到的过渡曲面，其质流率又进一步提高18.58%。（第7章［7］）
从CFD优化某柴油机进气歧管的形状，以四个支管质流率之和（越大流通性越好）、支管质流率间最大差值（越小进气均匀性越好）和支管质流率中最小值与内腔表面积之比（越大材料利用率越高）为目标，基于多项式响应面法，分别构造联系三个指标和设计参数的代理模型。进一步利用统一目标法，建立了多目标优化函数，从而完成了进气歧管中过渡曲面外形的多目标优化设计。经优化后，三个指标相对原有设计均有不同程度提高。（第8章［8］）
鉴于当前用NURBS曲面进行过渡曲面的模具CAD设计时缺乏有效工具来添加拔模斜度以满足模具生产工艺要求，提出一个自动给NURBS曲面添加任意大小拔模斜度的算法。此外，还利用凸组合的特性，改正了现有的双线性插值曲面和B样条曲面上处处具有足够拔模斜度的充分条件。并将NURBS曲面上拔模斜度的分布用“地形”和“水平面”来可视化。比较它们之间高度，就凸现拔模斜度够不够和差多少。再借助等高线算法使曲面斜度可视化，以帮助设计者识别曲面上的陡面。（第9章［4］）
In Computer Aided Geometric Design (CAGD), a transitional surface is the surfacesmoothly connecting two or more surfaces, which are called base surfaces. Such anoperation is called blending the base surfaces, hence the transitional surface is also knownas the blending surface, or simply the blend. In surface modeling, various approaches togenerating blending surfaces for polyhedra and multiple pipes have been ardentlyinvestigated due to their extensive applications. For example, convex sharp vertices andedges in mechanical parts cause wear, burr, or harm to human, while concave ones causestechnological difficulties and stress concentration, hence rounding them are necessary. Asfor the molded parts, the blending surfaces facilitate the work of pulling them out of themould core and cavity. In constructing intake or exhaust manifolds of internal combustionengines, blending surfaces can reduce the flow resistance in the manifolds effectively andenhance the intake/exhaust efficiency so as to improve the performance. In ship building,the blending surfaces between the bow and the hull reduce the sailing resistance. Inairplanes, the blending surfaces between the fuselage and airfoil (called wing-to-bodyfairing in engineering), may decrease the resistance of airflow and vortex, make thepressure distribution uniformly, and reduce the vibration and noise. Besides, thereconstruction of3D human vascular networks or organs for diagnosis and planning theoperation based on CI or MRI images, as well as making the scaffold for repairing thebone injury using biomaterial also needs blending surfaces to model them.. Therefore, theimportance of blending surfaces goes without saying in shape design of aeronautic,astronautic, marine, mechanical and modern architecture engineering. They also play anincreasingly important role in biomedical engineering. Although there are a number ofcommercially available software packages for3D modeling, which can generate certainblending surfaces nowadays, deficiencies, inadequacies and limitations in the existingmethods have been found in four aspects:(1) the twist compatibility problem occurs in theparametric method for polyhedral vertex blending, resulting in the inaccuracy in thegeometric continuity, complication in the construction process and difficulty in realizinghigher-order geometric continuity;(2) the implicit method for blending multiple pipes with freeform directrices can not get exact analytical implicit equations, which leads to theinaccuracy in the geometric continuity and restricts the pipe radii to be constant;(3)multiple pipes with freeform cross sections and arbitrary poses can not be blended;(4)optimizing the blend design from viewpoint of working performance and technology juststarted. All the above are hard problems on the discipline front, having greater practicalmeaning and theoretical significance.Despite the difficulty, based on assiduous study of the relevant basic theory andinnovation, this thesis presents a number of novel methods for polyhedron blending andmulti-pipe blending. Furthermore, the shape of an exhaust pipe and an intake manifold forinternal combustion engines are designed aiming to optimize their performance. Inaddition, a new algorithm and a new visualization method for adding draft angle to theNURBS surfaces are proposed. The thesis is composed of ten chapters. The main
contributions are as follows:Propose the vertex-first algorithm to blend polyhedron with higher order geometriccontinuity, viewing that the majority of existing parametric methods have only G1continuity (tangent plane continuity). It first makes use of triangular Bézier patches,tensor product Bézier surfaces or m-sided S-patches to blend3-edge,4-edge andm-edge polyhedral vertices respectively with higher-order continuity by a clever trick—properly placing the control points on the vertex, surrounding edges and faces.Then, the Hartmann blending function method (improved herein) is adopted toconstruct the edge blending surface with higher order geometric continuity, in whicha polynomial blending function with enough design parameters is proposed, so thatthe edge blending surfaces can be represented in tensor product Bézier form when thevertex blending surfaces are triangular Bézier or tensor product Bézier surfaces.During the blending process, enough freedoms are left to adjust the fullness of vertexblending surfaces and width of edge blending surfaces. By the way, the higher-ordercross boundary derivatives of m-sided S-patches are derived.(Chapter3, Section3.2,namely paper [1]. Please refer to the last page of this thesis.)
Define the edge cluster on the polyhedron that means a series of polyhedral edgesconnected together by polyhedral vertices. Further, the vertex-first algorithm isextended to the edge-cluster algorithm, so that repeated vertex and edge blendingoperations for an m-edge cluster can be replaced by one edge cluster blending, andmultiple blending patches are replaced by a single patch. Thus, the blendingefficiency is greatly raised. Moreover, the necessary and sufficient condition for anm-sided S-patch to blend all the m-edge clusters is deduced, and the concreteconfigurations of4,5,6, and7-edge clusters are displayed.(Chapter3, Section3.3, [9])
Propose a whole rational S-patch of depth3to blend polyhedral vertex with setbacks,in view of the complication of the current method, which achieves setback vertexblending by joining several quadrilateral parametric surfaces with the emerging twistcompatibility problem. Thus, without twist incompatibility, the overall blendingconstruction is simplified to determining the relevant control points of a regularrational S-patch by simple linear computation and some set operations. Comparedwith the existing method using non-rational S-patches for blending polyhedral vertex,the new method not only produces setback, but also reduces the depth of the resultingregular S-patch from5to3. Moreover, the available form of edge blending surfacesis extended to rational tensor product Bézier surfaces. In addition, the first-ordercross boundary derivative of rational S-patches is deduced and the geometricconstraint condition of G1continuity between rational S-patches and planes is given.(Chapter4,[2])
Propose the auxiliary sphere based algorithm to blend multiple normal ringedsurfaces with definite implicit equations, because the existing closing basedalgorithm can not make the base pipe surfaces really implicit. Closings were animportant contribution of Hartmann using the functional spline method to blendmultiple normal ringed surfaces. This method requires the base surfaces to be implicit,but the base pipes are expressed parametrically. To meet this requirement, heimplicitized the directrices of the pipes numerically. Consequently he can blend onlythe pipes with constant radii, while the result is numerical and the geometriccontinuity is only approximate. No implicit algebraic equation of the blendingsurface is yielded. This thesis adds one or two auxiliary spheres tangent to the pipeend, employed as the base surface(s) instead of the base pipe for functional splines toconstruct closings. Since the spheres can be implicitly defined, everything is OK. Thecomputational efficiency and accuracy are both enhanced significantly. Moreover,enough design parameters are provided to adjust the blending shape interactively, inorder that different unwanted configurations, such as burst, bulge, necking andprotrusion, are avoided in surface modeling. If the generated shape does not meet thetechnical and aesthetic requirements, we can optimize it by assigning certain fiducialpoints first and then minimizing the sum of algebraic distances between the blendingsurface and fiducial points using genetic algorithm.(Chapter5, Section5.2,[3])
Aiming to remedy the two weaknesses of the above auxiliary sphere based algorithm,namely that the blending surface has only G1continuity and the directrices of initialpipes need zero curvature at their ends when the first-order derivatives of radius functions are nonzero, the algorithm of adding Gnclosing is put forward. It firstfinishes the rational parametrization of normal ringed surfaces, then adds closingsGn-continuous with the base pipes directly by Hermite interpolation method based onthe parametric continuity conditions between implicit algebraic surfaces and rationalparametric surfaces. To avoid burst and unwanted branches in the construction ofclosings, not only fiducial points but also their associated normal vectors are chosenin optimization to determine the design parameters. Moreover, closingsGn-continuous with cylinders and cones in arbitrary poses are constructed.Furthermore, this method can blend several pipe surfaces with elliptic sections aswell.(Chapter5, Section5.3,[5])
Provide the constructive approach using three building blocks (side patch, setbackpatch and hole-filling patch) sequentially to blend multiple freeform pipes in almostarbitrary poses with G1continuity, considering that the existing methods can onlyblend multiple parallel freeform pipes or construct bifurcations with circular sections.Side patches are constructed by (revised herein) Hartmann blending method forconnecting arbitrary two adjacent pipes. Setback patches are generated by Coonssurfaces for shaping the holes to be filled. Hole-filling patches are generated byGordon-Coons surfaces for accomplishing the construction of blending surface.Compared with the traditional bifurcation algorithm, the new one can connect threefreeform pipes. Compared with the current method for connecting multiple parallelpipes, miscellaneous topological configurations are blended. In constructing thesetback patches and hole-filling patches, the cruxes of this algorithm—boundarycompatibility and twist compatibility are well coped with.(Chapter6,[6])
Propose a method for blending two cylinders/cones using NURBS surfaces with G2continuity and optimized shape, in contrast to the existing method by which thenormal ringed surfaces can not be represented in NURBS form, so that manyready-to-use CAD/CAE/CAM software packages can not apply. The new methodfirst connects two base pipes using a reference normal ringed surface whose directrixstrain energy is minimized, and then a skinning NURBS surface with G2continuity isconstructed to interpolate the cross section circles of the reference normal ringedsurface. Finally the conveying capability of the NURBS surfaces is maximized withCFD (Computational Fluid Dynamics) software The computation results of a casestudy—a motorcycle exhaust pipe show that the strain energy optimization canraise the mass flow rate (in kg/sec, the larger the better)26.46%greater than theconventional method, and the CFD optimization further increases it by18.58%.(Chapter7,[7])
Optimize the blending surface shape in a diesel’s intake manifold from CFDviewpoint. After computing the mass flow rates of the four branching pipes of theintake manifold based on CFD simulation respectively, the sum of them (itsmaximum implying the minimum flow resistance), the maximum difference betweenthem (its minimum implying the best uniformity) and the ratio of the minimum massflow rate to the internal surface area (its maximum implying the highest materialutilization) are taken as the objectives. The polynomial response surfacemethodology is utilized to build three surrogate models that associate the above threeobjectives with the design parameters respectively. Then, multi-objectiveoptimization function is constructed by unification-object method. Compared withthe original design, the performances of three objectives are improved in differentextent after optimization.(Chapter8,[8])
Propose an automatic algorithm for adding arbitrary draft angle to NURBS surfacesin mould manufacturing, as there is lacking such an effective tool in mould CAD.Moreover, the sufficient conditions for bilinear interpolation and B-spline surfaces tohave draft angle everywhere are corrected based on the convex combination. Thedraft angle distribution all over a NURBS surface is visualized by a terrain and alevel (a horizontal plane). By comparing their height, the draft eligibility of thesurface becomes obvious qualitatively (yes or no) and quantitatively (how much).Additionally, the isocline curves can be found by computing the terrain-levelintersection points numerically using a contouring algorithm, so that the steep surfacecan be viewed by designers easily.(Chapter9,[4])
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