Polynomial Splines Over Hierachical Meshes And Applications

Free-form curve and surface modeling is one of the core parts of computer aided geometric design,and NURBS geometric modeling method is an important means and method of free-form curve and surface modeling.Because NURBS curves and surfaces do not support meshes with T-vertex,a large number of redundant control points will inevitably be generated in complex geometric mod-eling,which will cause problems in geometric modeling design.At the same time,NURBS geometric modeling method does not have the local refinement proper-ties,which also limits its further development in the field of geometric analysis.In this way,new technologies and methods are called for in the field of design and analysis.T-splines,Hierarchical B-splines,PHT-splines and other design and analysis techniques have emerged as the times require.Among these techniques and methods,the hierarchical spline method is one of the important means of de-sign and analysis.In this dissertation,the theory and application of polynomial splines on hierarchical meshes are studied.The main contents of this paper are as follows seven chapters.:In the first chapter,the background and research status of Computer Aided Geometric Design are introduced.In the second chapter,it introduces some important concepts and theories involved in the follow up work of this paper.It mainly includes the concepts of hierarchical T-mesh,dimension formula,PHT-spline and its construction strat-egy.In chapter 3,this dissertation discusses the evaluation of PHT-spline sur-faces.PHT-spline surfaces are polynomial surfaces defined on hierarchical T-meshes.Because of its perfect local refinement property,it has a wide range of applications in geometric processing and analysis.However,PHT-spline basis functions are defined in Bezier ordinate form.According to the specific hierarchi-cal T-mesh,we need to compute the Bezier ordinate of each basis function on its supported cell chamber and save it.As the number of hierarchy increases,a large amount of storage space is needed.In this dissertation,we propose a de Boor like algorithm to evaluate PHT-splines provided that only the information about the control coefficients and the hierarchical mesh structure is given.The basic idea is to represent a PHT-spline locally in a tensor product B-spline,and then apply the de-Boor algorithm to evaluate the PHT-spline at a certain parameter pair.We perform analysis about computational complexity and memory costs.The results show that our algorithm takes about the same order of computational costs while requires much less amount of memory compared with the Bezier representations.In chapter 4,polynomial splines with control network on T-grid are given.Because each basis vertex on the hierarchical T-mesh corresponds to four basis functions for PHT-spline surface,this results in the fact that the basis verices on the T-mesh do not correspond to the control points corresponding to the basis functions of the PHT-spline surface one by one,which makes it difficult to use PHT splines to model and edit the model.At the same time,PHT-spline surfaces do not define basis functions for T-vertices,which weakens the editing ability of PHT spline surfaces at T-junctions.In this dissertation,polynomial splines with control nets on T-meshes are proposed.The basic idea is to extend T-vertices such that those T-vertices are interior cross vertices or boundary vertices.For each T-vertex and basis vertex on the original T-mesh,the corresponding basis functions are defined on the extended T-mesh.Then the index net of extended grid is introduced to make the basis functions and vertices correspond one to one.In Chapter 5,this dissertation discusses the conversion between the PHT-spline surfaces and the hierarchical B-spline surfaces.This mutual conversion is essentially the representation of the same spline surface under different basis functions.The key strategy of the conversion is to compute the Hermite geometry informations of the spline surfaces at each basis vertex on the hierarchical T-mesh(?).Then the PHT-spline surfaces or the hierarchical B-spline surfaces require interpolation of Hermite geometric informations at each basis vertex and calculation of four control coefficients corresponding to each basis vertex.Thus we obtain the conversion between them.In Chapter 6,we first discuss the solution of the PDE problems with non-homogeneous boundary on the B-splines basis functions space based on type Ⅱtriangulation.In paper[56],Kang etc.discuss the PDE problem with homoge-neous boundaries on the B-splines basis functions space S21,0(?).However,when the PDE problem with inhomogeneous boundaries constraints is solved directly on the spline spaces S21,0(?),the numerical solutions may not be convergent.In this paper,by means of S21,0(?)and S21(?)based on type Ⅱ triangulation,a blending B-spline space is obtained by putting the basis functions in S21,0(?)together with the basis functions in S21(?)whose support centers are not in Ω.In this blending B-spline space over type Ⅱ triangulation,the PDE problem with inhomogeneous boundary is solved and the solution is convergent.Experiments show that under these blending B-spline basis functions over type II triangulation,the PDE problem with inhomogeneous boundary is solved,and the numerical so-lution converges to the true solution rapidly.This dissertation also discusses the hierarchical B-splines basis functions construction on regular type II triangula-tion.The basis functions of this hierarchical B-spline spaces possess many good properties,such as non-negative,local support,polynomial completeness,linear independence,nestalitity and unit partition property.In the last chapter,the work of this dissertation is summarized,and the follow-up work is prospected.