Some Researches On Spline Method Over Hierarchical Meshes

In the fields of numerical approximation,geometric modeling and engineering calcula*tion etc.,spline is a generally used method.The researches in these fields raise new questions for the theory of multivariate spline method.For example,local modification algorithm is introduced to the standard NURBS method to remove the limitation of rectangular grid and improve the new basic theory of spline spaces over T-meshes.In addition,it needs to extend the method of local refined splines to irregular meshes.Combined with the method of mult.ivariate spline,we find tha.t the adaptive refined spline method based on hierarchical mesh has good applicability and can obtain satisfactory surface fitting results.The main problem and difficulty of multivariate spline theory is the singularity of spline space di-mension and the construction of basis function with local support.This thesis discuss the problems of the dimension of spline space,especially the singularity of dimension,explicit dimension formula,construction of basis functions,spline interpolation etc..We focus on solving the problem of fitting scattered data points by adaptively local refined spline surface over rectangular and quadrilateral meshes.The main work includes the following several aspects:1.The dimension of spline space is a basic and difficult problem,we study the insta-bility in the dimensions of spline spaces over T-meshes with nested T-cycles.The modified dimension formulas of spline spaces over T-meshes with N-nested T-cycles are also presented.Moreover,a possible degeneration for a case of parallel T-cycles is illustrated.2.We present a new surface reconstruction algorithm of polynomial spline surface of S(3,3,1,1,T)over arbitrary hierarchical T-mesh.This surface algorithm is piecewise constructed by interpolation of the 16 parameters of four vertices on each rectangular cell of hierarchical T-meshes.Moreover,we give an adaptively refined surface algorith-m for fitting scattered data points based on piecewise Coons surface.The experimental results show that the proposed adaptive algorithm is efficient in fitting scattered data points.3.We present a new surface reconstruction algorithm of cubic spline surface over local refined hierarchical quadrilateral mesh.This surface algorithm is piecewise construct-ed by interpolation of the 12 parameters of four vertices on each quadrilateral cell of hierarchical quadrilateral mesh.Moreover,we give an adaptively refined surface algo-rithm for fitting scattered data points based on cubic spline with 12 parameters.The experimental results show that the proposed adaptive algorithm is efficient in fitting scattered data points on irregular domain.