Urn模型的B网及其应用

Urn模型的离散概率分布被用于构造逼近格式,具有Bernstein多项式的许多特性。本文用B网技术对Urn模型概率分布给出新的解释,指出了Urn模型B网的诸多性质,导出分布函数曲线的导数公式,构造了Urn分布的对偶基及与此相关的拟插值算子。     

A cubic quadrilateral spline element with concave shapes

Basic requirement for applying isoparametric element is that the element has to be convex and no violent distortion is allowed. In this paper, a cubic quadrilateral spline element with 12 nodes has been developed using the triangular area coordinates and the B-net method, which can exactly model the cubic field for quadrilateral element with both convex and concave shapes. Neither mapping nor coordinate transformation is required and the spline element can obtain high accuracy solutions and insensitive to mesh distortions.     

多元弱样条空间和最小确定集

根据研究多元弱样条函数的B—网方法,给出了某些多元弱样条函数空间的最小确定集的构造方法,并从而求出了它们的维数．本文还讨论了对偶基的局部支集性质。     

Bivariate polynomial and continued fraction interpolation over ortho-triples

By means of the barycentric coordinates expression of the interpolating polynomial over each ortho-triple, some properties are obtained. Moreover, the explicit coefficients in terms of B-net for one ortho-triple, and two ortho-triples are worked out, respectively. Thus the computation of multiple integrals can be converted into the sum of the coefficients in terms of the B-net over triangular domain much effectively and conveniently. Based on a new symmetrical algorithm of partial inverse differences, a novel continued fractions interpolation scheme is presented over arbitrary ortho-triples in R², which is a bivariate osculatory interpolation formula with one-order partial derivatives at all corner points in the ortho-triples. Furthermore, its characterization theorem is presented by three-term recurrence relations. The new scheme is advantageous over the polynomial one with some numerical examples.     

基于面积坐标与B网方法的四边形样条单元

传统等参元方法中, S型等参元完备阶较低,对网格畸变敏感, L型等参元具有高阶完备性但需要使用内部节点. 另外,由于引入等参变换, 采用数值积分可能导致总刚度矩阵出现奇异性.利用三角形面积坐标与B网方法建立了一类平面四边形的样条单元函数,它们的特点是满足协调条件, 克服网格畸变敏感性.其中8节点和12节点单元分别为2次和3次样条函数,对直角坐标分别具有二阶和三阶完备性, 高于相同节点的S型等参元.通过算例测试了这些样条单元, 并与等参元和其它四边形单元比较,数值结果显示了它们的高精度和有效性.      

非均匀二型三角剖分二元二次样条的数值积分公式

给出了构成矩形域的4个三角形子区域的二元样条拟插值算子的等价形式,对这4个三角形子区域分别建立了数值积分公式,相加后得到一般矩形域上的数值积分公式,同时给出了构造数值积分公式所需的结点处的函数值与相应的求积系数。进一步,利用算子范数、连续模及拟插值算子的保多项式性,针对具有不同连续性的被积函数,得到了相应的数值积分公式的求积余项。研究表明,提出的数值积分公式不仅具有较高的计算精度,而且计算量约为二元张量积型求积公式的1/5。数值算例进一步说明了数值积分公式的有效性。     

二元弱样条函数空间的维数

基于B-网方法,在一定条件下构造了由一列星形域构成的三角剖分 |VI|∪i=1star(vi)相应定点剖分Ⅰ1|VI|∪i=1 star(vi)下2μ次μ阶光滑二元弱样条函数空间Wμ2μ(Ⅰ1|VI|∪i=1 star(vi)的一个最小决定集,据此给出了该空间的维数.作为两个应用的实例,我们给出了非均匀(Ⅰ)型三角剖分△(1)mn及非均匀(Ⅱ)型三角剖分△(2)mn相应的定点剖分下二元弱样条函数空间Wμ2μ(I1△(1)mn)和Wμ2μ(I1△(2)mn)的维数.     

采用三角形面积坐标的四边形17节点样条单元

利用二元4次样条插值基和三角形面积坐标构造17节点四边形单元．这个新单元具有4次完备阶,通过一些算例测试表明了该单元有较高精度并对网格畸变不敏感．     

A 3D pyramid spline element

In this paper,a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method,which achieves the second order completeness in Cartesian coordinates.Some appropriate examples were employed to evaluate the performance of the proposed element.The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted,and it is comparable to the Lagrange element Q27.It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.     

The mixed directional difference–summation algorithm for generating the Bézier net of a trivariate four-direction Box-spline

Trivariate Box-splines lack an efficient and general exact evaluation technique. This paper presents one possible and underexploited approach to solving this problem. The algorithm we propose is based on mixed directional differences and summations for computing the Bézier net coefficients of all trivariate four-direction Box-splines of any degree over tetrahedral tessellations of the domain. A Matlab package, called MDDS, for computing the Bézier net both in the trivariate and bivariate cases, is also provided.