Cubic algebraic spline curves design

In this paper we propose a construction method of the planar cubic algebraic spline curve with endpoint interpolation conditions and a specific analysis of its properties. The piecewise cubic algebraic curve has G ² continuous contact with the control polygon at two endpoints and is G ² continuous between each segments of itself. The process of this method is simple and clear, and provides a new way of thinking to design implicit curves.     

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.     

基于Bezier曲面约束的多形态随机曲面生成方法

将随机图形与自由曲面的变形相结合,提出了随机曲面生成的均匀插值移位法．算法中,以Bezier曲面离散生成的de Casteljau算法为基础,利用不同性质的插值移位扰动量调控随机曲面的复杂多样性,可生成具有确定性的、随机性的、分形特征的和混合复杂性的四类不同形态特征的随机曲面．     

样条有限条塑性铰法分析板梁的极限荷载

用样条有限条塑性铰法分析了板梁的极限荷载。首先对条单元以样条位移函数表达的总势能进行求导而推导了位移-荷载关系式。然后用塑性铰法推导了单元的塑性刚度矩阵。因此该方法兼具二者优点:样条有限条法的位移量少和塑性铰法形成塑性刚度矩阵的便利。它还可以考虑梁的初始缺陷,如残余应力和初弯曲。通过与相关的试验数据比较,证明该方法有效与可靠。     

有理宏单元法求解泊松方程

基于有理超限插值,提出了一种在求解域边界布点的全域求解数值方法——有理宏单元法。推导出了三角形及四边形单元的有理混合函数,划分单元各边的节点并选定各边上的插值函数,建立了三角形及四边形母单元的形函数。利用等参变换,将求解域影射到相应的母单元上,得到了求解泊松方程边值问题的有理宏单元方程组。通过将求解域划分为一个或多个宏单元,有理宏单元法可对任意形状的二维区域求解。作有理宏单元法解泊松方程边值问题的算例,验证了本文方法的有效性。     

求解非线性偏微分方程的自适应小波精细积分法

以Burgers方程为例,提出了一种求解偏微分方程的自适应多层插值小波配置法,通过引入一种具有插值特性的拟Shannon小波并利用插值小波理论构造了多层自适应插值小波算子,从而在空间实现了偏微分方程的自适应离散.另外,精细时程积分方法和外推法的引入不但有助于提高求解速度和数值结果的精度,而且使时间积分步长的选取具有了自适应性.     

重叠网格方法的研究进展

从网格装配和插值计算两个主要方面对现有的重叠网格方法进行了综述。首先,从挖洞方法和建立嵌入网格关系环节的寻点技术出发归纳和介绍了网格装配方法;其次,介绍了数值迭代过程中的插值计算方法,并特别讨论了插值守恒性以及插值计算精度等问题;另外,对重叠网格方法的并行计算和应用成果也作了介绍;最后,通过总结认为重叠网格方法在改进网格装配方法、改善插值和并行计算效率等方面仍需进一步研究。     

区间五次Hermite样条多小波Euler-Bernoulli梁单元

首先采用区间五次Hermite样条函数,分别构造了三节点梁的边界和内部节点的多小波尺度函数,然后,基于节点尺度函数在区间内伸缩、平移的思想,构建了梁单元相互嵌套、逐级包含的多尺度位移近似空间序列;最后,采用最小势能原理,得到弯曲梁的平衡方程,从而构造了区间五次Hermite样条多小波Euler-Bernoulli梁单元.算例结果表明,该小波单元可通过改变尺度来重新划分网格,从而可自由调节单个小波单元的计算精度,其计算精度与在相同网格划分下采用任意多个传统三节点Hermite梁单元计算梁构件的完全一致.与其它小波单元相比较,该小波单元具有计算简单明了,物理意义明确,易于理解的特点.     

土体各向异性边界面模型研发的几个问题探讨

土体边界面模型通过采用塑性模量场插值,来反映土体变形的非线性,边界面模型适合对小应变情况下的土体非线性变形进行模拟. 在Wheeler 模型的基础上,结合边界面理论,构造一个土体各向异性边界面模型,模型可以反映天然土体的各向异性特性. 本文对该模型研发过程中遇到的几个问题进行理论探讨,为理解和掌握各向异性边界面理论,并进一步进行新型边界面模型的研发,提供了理论借鉴.     

Classification of Refinable Splines

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well-known B-splines, play a key role in computer aided geometric design. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete classification was obtained in [12]. However, refinable splines do not have to have integer dilations and integer translations. The classification of refinable splines with noninteger dilations and arbitrary translations is studied in this paper. We classify completely all refinable splines with integer translations and arbitrary dilations. Our study involves techniques from number theory and complex analysis.