The Isomorphic Mapping On Spline Spaces Over Hierarchical T-meshes And Its Application

In the applications of geometric modeling and isogeometric analysis,the tensor product meshes generated by NURBS method can not be refined locally,to avoid the superfluous control points generated by global refinement,many splines that can be locally refined have been extensively studied.By discussing splines with local subdivision capabilities from the perspective of spline spaces,we can obtain the basis functions that hold many good properties.The dimension formulas are the fundamental work of the study of spline spaces.As the local refinement of the hierarchical T-meshes is very natural and practical,splines defined over hierarchical T-meshes are widely studied and applied in practice.In order to give a concise topological explanation of the dimension formulas and make the dimension formulas well used in the construction of the spline basis functions,in this thesis,we use the CVR graph(crossing-vertex-relationship graph)to propose a method,which is called the isomorphic mapping method,for studying dimensions of the spline spaces.By the isomorphic mapping method and spline basis functions generated by the isomorphic mapping method,we mainly do the following four works.For univariate spline spaces,we give the bijective mapping between the spline space of degree m that defined over the knot sequence T and the spline space of degree m-2 that defined over the knot sequence G(m=2,3),where G is consisted of the inner knots of T.By the basis functions of the spline space of degree m-2 that defined over G,we obtain the corresponding basis functions of the spline space of degree m that defined over T.By the bijective mapping,we obtain the conclusion that the spline space of degree m that defined over T is isomorphic to the spline space of degree m-2 that defined over G.By the isomorphic conclusion,we obtain that the basis functions of the spline space that defined over T hold the same properties with the basis functions of the spline space that defined over G,such as linear independence,completeness and partition of unity.The idea of using low-order spline space to study high-order spline space is realized.For the biquadratic spline space over the hierarchical T-meshes,we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph(CVR graph).We give the topological explanation of the dimension:the dimension is the number of the cells in the CVR graph.We offer an effective method for constructing the basis functions of the biquadratic spline space by the piecewise constant space.We obtain the conclusion that the two spaces are isomorphic.By the isomorphic mapping method,the constructed basis functions of the biquadratic spline space hold the properties such as linear independence,completeness and the property of partition of unity,which are the same as the properties for the basis functions of piecewise constant space over the CVR graph.For the bicubic spline space over the hierarchical T-meshes,we also construct a bijective mapping between a bicubic spline space over the hierarchical T-mesh and the bilinear space over the corresponding CVR graph.We give the topological explanation of the dimension:the dimension is the number of the base mesh in the CVR graph.We also construct the basis functions of the bicubic spline space by the basis function of the bilinear space over the corresponding CVR graph.We obtain the conclusion that the two spaces are isomorphic.By the isomorphic mapping method,the constructed basis functions of the bicubic spline space hold the properties such as linear independence and completeness,which are the same as the properties for the basis functions of bilinear space over the CVR graph.By the isomorphic mapping method,we have overcome the limitations of the level difference when constructing the basis functions over the hierarchical T-meshes.Without the limitations for level difference,the cells in the parameter space of the biquadratic polynomial splines can be subdivided adapt to the given tolerance freely.Instead of subdividing additional cells to ensure the level difference,the refinement for each level hierarchical T-mesh is more local.Thus,both the degree of freedom and the amount of calculation can be reduced.Based on the results of the above research,this thesis provides a new method for the theoretical research of the spline space over the hierarchical T-mesh by the CVR graph.Under the guidance of this isomorphic mapping method,we can get a concise topological explanation of the dimensionality formula of the spline space on the hierarchical T-mesh.Our research provides a new impetus for the study of multivariate splines.