Some Researches On Multivariate Splines And Computation Of Piecewise Algebraic Varieties

In this thesis, we mainly study some applications of multivariate splines and computation of piecewise algebraic varieties. On one hand, we study the applications of multivariate splines, primarily including finite element method, singular integral and approximate implicitization with bivariate splines; On the other hand, we study the computation of piecewise algebraic varieties, mainly including the interval iterative algorithm for computing the zero-dimensional piecewise algebraic variety and the real root isolation of zero-dimensional piecewise algebraic variety. Our primary work is organized as follows:Firstly, we discuss the bivariate B-spline finite element methods. In finite element methods, multivariate splines are mainly used to construct all kinds of model functions. A kind of quadratic B-spline bases which interpolate the boundary function on uniform (non-uniform) type-2 triangulations is constructed. Therefore, the Poisson's equation on any rectangular or parallelogram region is solved with combination of spline space S₂^1,0(â–³_mn⁽²⁾) satisfying homogeneous boundary condition and the constructed B-spline bases by using bivariate B-spline finite element methods. Moreover, we discuss the applications of bivariate splines on two-dimensional singular integral. In recent years, many researchers study the numerical integration by using multivariate splines. In particular, bivariate spline quasi-interpolation operators are used to solve all types of singular integral, mainly including Cauchy singular integral and singular integral defined in the Hadamard finite part sense etc. It has been widely used in solving singular integral equations. Since the bivariate B-splines on type-2 triangulations have the advantages of simple configuration, good symmetry and quasi-interpolation operators, it has wide use in practical fields. Using two kinds of quasi-interpolation operators possessing good approximation behavior on spline space S₄^2,3(â–³_mn⁽²⁾) , we construct the integration formulas and their applications on the evaluation of 2-D singular integral defined in the Hadamard finite part sense. Secondly, we discuss the approximate implicitization of parametric curves by using cubic algebraic splines. For a general parametric curve/surface, we usually cannot compute its exact implicit form. Even if the exact form can be computed, it is not necessary to do in many cases. This is partly due to the fact that the exact implicitization always involves relatively complicated computation and the resulted implicit form might have large number of coefficients. Another difficulty is that implicit curves/surfaces may have unwanted components and self-intersections which lead to computational instability and topological inconsistency in geometric modeling. All these unsatisfied properties limit the applications of the exact implicitization in practical fields. Similar to the cubic parametric curves, cubic algebraic curves become the most widely used algebraic curves. In order to solve it, we use a piecewise cubic algebraic curve to give a global G² continuity approximation to the original parametric curve.Lastly, we discuss several computation problems on piecewise algebraic varieties. As the zeros of multivariate splines, the piecewise algebraic variety is a generalization of the classical algebraic variety. The intersection of spline curves/surfaces becomes an important problem in CAGD. However, this problem boils down to the computation of piecewise algebraic varieties. Hence, it is important to study the piecewise algebraic variety. As to it, we mainly do the following two items of work. On one hand, we discuss the interval iterative algorithm for computing the piecewise algebraic variety. The approach presented here is primarily based on the introduction to a concept ofε-deviation solutions and the effective evaluation the bound on the value of the derivative of the function on a given region. On the other hand, we give the effective and fast algorithm of real root isolation for zero-dimensional piecewise algebraic variety. It is primarily based on the computation of algebraic variety on a given convex polyhedron and the real roots of the univariate interval polynomial.