Study On Polyhedron Blending And Multi-pipe Blending With Applications

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 maincontributions 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])...