Multivariate Splines On Special Triangulations And Their Applications

Multivariate splines are applied widely in approximation theory, computer aided geometric design and finite element method. In this thesis, we study some important spline spaces defined on special triangulations. We focus on uniform type-2 triangulation, triangulated quadrangulation and some special tetrahedral elements in 3D. Moreover, we consider the application of those splines in CAGD and FEM. The thesis is organized as follows:In chapter 2, we discuss some spline spaces on uniform type-2 triangulation. It is a special crosscut partition, or a four-directional mesh, which is used widely because of its simple construction and good symmetry. We consider different smoothness on different grid segments. By using the smoothing cofactor-conformality method, we obtain spline bases of cubic and quartic spline spaces, respectively. Furthermore, The corresponding quasi-interpolation operators with high approximation power are constructed, and their approximation properties are discussed.In chapter 3, we construct a kind of bivariate quadratic, cubic and quartic NURBS surfaces by using the bivariate B-spline bases systematically. As we know, the Non-Uniform Rational B-Splines (NURBS) have become the standard for the representation, design, and data exchange of geometric information processed by computers. Usually, the NURBS surfaces are obtained by using the tensor product B-splines. For example, a p × q B-spline surface is of degree p or q in the u or v direction. However, it should be a surface of degree p + q in other ways. As a result, the curves on the surface are of high degree, and there may be some inflection points on the surface. Besides, A serious weakness with NURBS models is that NURBS control points must lie topologically in a rectangular grid. This means that typically, a large number of NURBS control points serve no purpose other than to satisfy topological constraints. They carry no significant geometric information. The NURBS control points are, in this sense, superfluous.In order to resolve these problems, we construct a kind of non-tensor product NURBS surfaces by using the bivariate B-spline bases. The new surfaces are of lower degree than the degree of the usual NURBS surfaces. Meanwhile, the former retains many good properties and consistency with the latter. Since the B-spline bases satisfy the partition of unity and have high approximation power, these surfaces have convex hull property and transformation invariance, and also have good approximation properties. Especially, each bivariate B-spline base is a piecewise polynomial and has its single support based on the type-2 trian-gulation. So, the surfaces can be constructed on irregular parametric domains without trimming. Furthermore, the boundaries (including diagonal grid segments) of the bivariate NURBS surfaces are univariate NURBS or Bezier curves with corresponding control points. So that the surfaces have similar properties to those of usual NURBS surfaces, and the surfaces can be controlled conveniently as well. Moreover, the control points can be locally refined by using the decomposition of B-spline bases.In chapter 4, we discuss the spline finite element method. Up to now, the univariate B-splines and tensor product B-splines are considered in FEM. However, as for multivariate spline, there are only a few papers which studied the simple application of the bivariate quadratic B-splines defined on type-2 triangulation. Besides, for pyramid element in 3D case, element shape function in terms of polynomial, which satisfies both compatibility and non-singularity conditions, has not been constructed.By using the bivariate quadratic splines on the triangulated quadrangula-tion, we construct a new 8-node quadrilateral element, which reproduces polynomials of degree 2. The computation is simplified greatly by using their Bezier coefficients on each triangle cell. Some appropriate examples are employed to evaluate the performance of the proposed element, and the numerical results are satisfied. Furthermore, we construct 13-node pyramid element an...