多元样条与分片代数簇计算的若干研究
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摘要
本文主要对多元样条与分片代数簇计算展开若干研究。一方面,我们对多元样条函数在数值分析中的若干应用进行了讨论,主要包括二元样条在有限元,二维奇异积分的计算,以及近似隐式化中的应用。另一方面,我们讨论了分片代数簇计算的某些问题,主要包括零维分片代数簇的区间迭代算法和实根分离算法。主要工作如下:
首先,我们讨论了二元B-样条有限元方法。在有限元方法中,多元样条函数主要用来构造各种类型的模型函数。本章我们主要构造了一类均匀(非均匀)2-型三角剖分下的二元样条基函数组用以插值于边界函数,并且结合均匀(非均匀)2-型三角剖分下带齐次边界条件的样条空间S_2~(1,0)(△_(mn)~(2)),利用有限元方法来求解椭圆型偏微分方程;而后,我们研究了二元样条函数在二维奇异积分中的应用。近年来国内外许多学者都采用多元样条函数来求解数值积分。特别地,拟插值算子常被应用于各种奇异积分,包括Cauchy主值积分和有限部分积分以及振荡积分的计算上。它们在奇异积分方程的求解中有着重要的应用。由于2-型三角剖分下的二元B-样条基函数具有结构简单,对称性好,并且具有良好逼近性质的拟插值算子,因此它在实际应用中有广泛的应用。本章我们利用均匀2-型三角剖分下样条空间S_4~(2,3)(△_(mn)~(2))上的两类具有良好逼近性质的拟插值算子,构造了具有更高精度的数值积分公式并且将其应用到Hadamard有限部分和的计算上。
然后,我们讨论了三次代数样条在参数曲线近似隐式化中的应用。由于参数曲线曲面的精确隐式化不一定可以实现。即使可以实现,许多情形下我们也不必这么做。这主要由于精确隐式化的计算很复杂并且系数很大,以及隐式曲线曲面具有不希望的自交奇异点和多余分支。这就引起了计算的不稳定性和几何造型中拓扑结构的不一致性,从而大大限制了隐式化在实际问题中的运用。与三次参数曲线类似,三次代数曲线成为人们广泛研究和应用的代数曲线。因此,我们构造了整体G~2-连续的三次代数样条来逼近参数曲线以实现近似隐式化。
最后,我们讨论了分片代数簇计算中的某些问题。分片代数簇作为多元样条组的公共零点集合,是经典代数簇的推广,丰富和发展。它不仅和许多实际问题如多元样条插值,代数簇的光滑拼接,CAD和CAGD等有关,而且还为研究经典代数几何提供了理论依据。CAGD中大量的曲线曲面类型是样条曲线曲面。因此给出样条曲线曲面的求交算法是CAGD中的重要问题之一,而这一问题的本质可以归结为分片代数簇的计算。因此,研究分片代数簇的计算是十分重要的。针对分片代数簇的计算,我们做了以下两方面的工作:一方面,我们讨论了任意剖分下分片代数簇的区间迭代解法,主要将代数簇的区间迭代算法有效应用到分片代数簇的计算上。该算法主要通过引入ε-偏差解和给出区域内极值的有效估计来实现。另一方面,我们给出了零维分片代数簇实根简单而有效的分离算法。该算法主要基于凸多面体内代数簇的计算和一元区间多项式实根的计算来实现。
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 S_2~(1,0)(△_(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 S_4~(2,3)(△_(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 G~2 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.
首先,我们讨论了二元B-样条有限元方法。在有限元方法中,多元样条函数主要用来构造各种类型的模型函数。本章我们主要构造了一类均匀(非均匀)2-型三角剖分下的二元样条基函数组用以插值于边界函数,并且结合均匀(非均匀)2-型三角剖分下带齐次边界条件的样条空间S_2~(1,0)(△_(mn)~(2)),利用有限元方法来求解椭圆型偏微分方程;而后,我们研究了二元样条函数在二维奇异积分中的应用。近年来国内外许多学者都采用多元样条函数来求解数值积分。特别地,拟插值算子常被应用于各种奇异积分,包括Cauchy主值积分和有限部分积分以及振荡积分的计算上。它们在奇异积分方程的求解中有着重要的应用。由于2-型三角剖分下的二元B-样条基函数具有结构简单,对称性好,并且具有良好逼近性质的拟插值算子,因此它在实际应用中有广泛的应用。本章我们利用均匀2-型三角剖分下样条空间S_4~(2,3)(△_(mn)~(2))上的两类具有良好逼近性质的拟插值算子,构造了具有更高精度的数值积分公式并且将其应用到Hadamard有限部分和的计算上。
然后,我们讨论了三次代数样条在参数曲线近似隐式化中的应用。由于参数曲线曲面的精确隐式化不一定可以实现。即使可以实现,许多情形下我们也不必这么做。这主要由于精确隐式化的计算很复杂并且系数很大,以及隐式曲线曲面具有不希望的自交奇异点和多余分支。这就引起了计算的不稳定性和几何造型中拓扑结构的不一致性,从而大大限制了隐式化在实际问题中的运用。与三次参数曲线类似,三次代数曲线成为人们广泛研究和应用的代数曲线。因此,我们构造了整体G~2-连续的三次代数样条来逼近参数曲线以实现近似隐式化。
最后,我们讨论了分片代数簇计算中的某些问题。分片代数簇作为多元样条组的公共零点集合,是经典代数簇的推广,丰富和发展。它不仅和许多实际问题如多元样条插值,代数簇的光滑拼接,CAD和CAGD等有关,而且还为研究经典代数几何提供了理论依据。CAGD中大量的曲线曲面类型是样条曲线曲面。因此给出样条曲线曲面的求交算法是CAGD中的重要问题之一,而这一问题的本质可以归结为分片代数簇的计算。因此,研究分片代数簇的计算是十分重要的。针对分片代数簇的计算,我们做了以下两方面的工作:一方面,我们讨论了任意剖分下分片代数簇的区间迭代解法,主要将代数簇的区间迭代算法有效应用到分片代数簇的计算上。该算法主要通过引入ε-偏差解和给出区域内极值的有效估计来实现。另一方面,我们给出了零维分片代数簇实根简单而有效的分离算法。该算法主要基于凸多面体内代数簇的计算和一元区间多项式实根的计算来实现。
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 S_2~(1,0)(△_(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 S_4~(2,3)(△_(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 G~2 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.
引文
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