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Some Studies About Splines With High Order Smoothness On T-meshes

Posted on:2017-04-19Degree:DoctorType:Dissertation
Country:ChinaCandidate:C CengFull Text:PDF
GTID:1220330485451569Subject:Computational Mathematics
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Because NURBS cannot be refined adaptively, promoted by the development of isogeometric analysis and other questions, the splines that can be locally refined have drawn many researcher’s attention in the past ten years. Many outstanding results have appeared. Based on this background, we discuss several questions on splines defined on T-meshes.In chapter 1, we review the history of the development of spline, and list some results about splines defined on T-meshes. Some basic definitions and results, which will be used in the following chapters, are introduced in chapter 2.In geometry modelling, splines with high order smoothness are used widely. There-fore, computing the dimensions of splines with high order smoothness is very important. However, in’1-2’, some examples have been given to show that the dimensions of splines with high order smoothness may be unstable. Therefore, it is impossible to give a for-mula for the dimensions of splines defined on general T-meshes. We should give some restrictions on T-meshes, i.e., we consider splines defined on some special T-meshes. In a more practical way, hierarchical T-meshes are a type of very important and useful T-meshes. Therefore, giving the dimensions of spline spaces defined on hierarchical T-meshes is the first thing we should solve. In chapter 3, we first present a general method for computing the dimensions of spline spaces with the highest order smoothness. This method is useful for splines defined on T-meshes with hierarchial structures. Then, we list the formulae of dim S2(J) and dim S3(J). As an application of this method, we give the formula of dim S3(J) when J is a 3 × 3 hierarchal T-mesh.There are so many discussions about spline functions defined on hierarchial T-meshes. One essential question is the completeness of the basis functions. That is, can all the basis functions span the spline space defined on the T-mesh? In chapter 4, we discuss the completeness of bicubic hierarchial B-splines and construct a basis. Some properties of the basis are discussed and some applications are listed.Spline spaces over rectangular T-meshes have been discussed in many papers. In chapter 5, we consider spline spaces over non-rectangular T-meshes. The dimension formulae of spline spaces over special simply connected T-meshes have been obtained. For T-meshes with holes, we discover a new type of dimension instability. We construct a relationship between the dimension of the spline space over a T-mesh J with holes and the dimension of the spline space over a simply connected T-mesh associated with J.In chapter 6, we discuss the dimensions of trivairate spline spaces on 3D T-meshes. The smoothing cofactor method for trivairate splines is explored. We obtain the confor-mality conditions. For the 3D T-meshes with some restrictions, a dimension formula is obtained.At last, we conclude all the context, and list several considerable questions.
Keywords/Search Tags:T-mesh, 3D T-mesh, non-rectangular T-mesh, dimension, basis
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