Research On Some Properties And Methods In Spline Approximation

Spline approximation is an indispensable technique in fields such as CAGD(Computer-Aided Geometric Design)and reverse engineering.With the rapid development of geometric modeling technology and isogeometric analysis,NURBS(Non-Uniform Rational B-splines)can no longer meet the current needs,which is the the only standard mathematical expression in industrial products.Therefore,locally refined splines came into being.Locally refined splines have flexible modeling capabilities and analysis-suitable properties,which are widely used in geometric modeling and engineering fields.At present,the research on global and local approximation methods in B-splines has been relatively mature,but the research on the local approximation method of locally refined splines is not perfect.How to design an efficient and robust approximation method for locally refined splines is very important for its application in geometric modeling and isogeometric analysis.This thesis studies some properties and methods in the approximation of B-splines,AST-splines(Analysis-Suitable T-splines),and PHT-splines(Polynomial splines over Hierarchical T-meshes),which are divided into the following three parts:In the first part,this thesis presents a proof of unimodality in the initial B-splines approximation,which provides a theoretical guarantee for the practical application of unimodality.B-splines are not only the most commonly used curve/surface representation in CAGD but also one of the underlying core representations of most current CAD(Computer-Aided Design)systems.The knot calculation problem is still a challenging problem in B-splines approximation.Suitable knots can not only significantly improve the fitting quality but also reduce the generation of redundant knots.In recent years,scholars have found that the unimodality of initial cubic B-splines approximation of discrete data points plays a key role in knot calculation problem,especially the determination of the knot number.Here,the unimodality of initial cubic B-splines approximations refers to the existence of local maximum values in the jumps,which are defined as the absolute difference of the left and right third derivatives of the B-splines at initial knots.This thesis proves that if the data points are sampled from cubic B-splines.and the initial knots are uniformly and densely chosen.then there exists local maximum for the jumps of the initial B-splines approximations at the two endpoints of the initial knot interval which contains the real knot of the sampling cubic B-splines in Chapter 3.In the second part,this thesis presents the AST-splines local approximation method,and proposes an arbitrary degree AST-splines quasi-interpolation method for the first time,and proves AST-splines quasi-interpolation optimal approximation properties.AST-splines have flexible modeling capabilities and their basis functions have the properties of linear independence,partition of unity and other analysis-suitable properties,so AST-splines are widely used in geometric modeling and isogeometric analysis.At present,most of the AST-splines approximation methods are global,such as least squares and spline interpolation.Global approximation is not only time-consuming to solve,but also has low numerical stability,especially for large-scale data approximation.Quasi-interpolation is simple and efficient local approximation method.Quasiinterpolation has developed into a very important approximation method in the spline field.This thesis proposes a simple AST-spline quasi-interpolation scheme and proves the optimal approximation property of AST-splines quasi-interpolation in Chapter 4.This quasi-interpolation scheme is a local tensor product generalization of the B-spline quasi-interpolation scheme without additional computation.This local approximation operator will make AST-splines more attractive in geometric modeling and isogeometric analysis.In the third part,this thesis presents two weight calculation methods for the basis functions defined by support mesh in PHT-splines approximation,such that weighted PHT-splines satisfy the partition of unity.This work further strengthens the PHT-splines approximation properties and geometric modeling capabilities which defined by support mesh.PHT-splines are another type of locally refined splines that differ from T-splines.Their efficient subdivision algorithm and inherent good properties make them also widely used in geometric modeling and isogeometric analysis.However,the original PHT-splines suffer from attenuation phenomenon.Therefore,scholars have proposed a PHT-spline basis function construction method based on support mesh,which effectively avoided the attenuation phenomenon and improved the performance in practical applications.However,the support mesh at the basis vertices of the PHT-splines may overlap,this would lose the partition of unity,which is very important in design and analysis.Therefore,based on the definition of the support mesh,this thesis proposes two weight calculation methods for the purpose of the PHT-splines basis functions satisfies the partition of unity in Chapter 5.On the one hand,with the help of the PHT-splines geometric information operator,this thesis transforms the basis function weight calculation task of the PHT-spline satisfying partition of unity into the solution of the classical Hermite interpolation problem,and obtains the PHTsplines basis function weight calculation formula.On the other hand,according to the B-spline knot insertion algorithm and the subdivision method,this thesis deduces the subdivision template that the PHT-splines basis function satisfies the partition of unity.