Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry

The boundary knot method (BKM) of very recent origin is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the numerical discretization of general partial differential equation systems. Unlike the method of fundamental solutions, the use of non‐singular general solution in the BKM avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain. The purpose of this paper is to extend the BKM to solve 2D Helmholtz and convection–diffusion problems under rather complicated irregular geometry. The method is also first applied to 3D problems. Numerical experiments validate that the BKM can produce highly accurate solutions using a relatively small number of knots. For inhomogeneous cases, some inner knots are found necessary to guarantee accuracy and stability. The stability and convergence of the BKM are numerically illustrated and the completeness issue is also discussed. Copyright © 2003 John Wiley & Sons, Ltd.     

Real-time accurate hand path tracking and joint trajectory planning for industrial robots (I)

Previously, researchers raised the accuracy for a robot’s hand to track a specified path in Cartesian space mainly through increasing the number of knots on the path and the number of the path’s segments, which results in the heavier online computational burden for the robot controller. Aiming at overcoming this drawback, the authors propose a new kind of real-time accurate hand path tracking and joint trajectory planning method. Through selecting some extra knots on the specified hand path by a certain rule and introducing a sinusoidal function to the joint displacement equation of each segment, this method can greatly raise the path tracking accuracy of robot’s hand and does not change the number of the path’s segments. It also does not increase markedly the computational burden of robot controller. The result of simulation indicates that this method is very effective, and has important value in increasing the application of industrial robots. Foundation item: Foundation of the Robotics Laboratory, Chinese Academy of Sciences (No. RL200002) Biography of the first author: TAN Guan-zheng, Dr., professor, born in Oct. 1962, majoring in artificial intelligence, robotics and automation.     

基于边界节点法的MATLAB 工具箱开发

0引言 Helmholtz方程作为简化的波动方程,长期以来备受关注.求解Helmholtz方程的数值方法主要有有限差分法、有限元法和边界元法等.这些方法主要基于网格近似或者需要背景网格积分,在处理高波数或高频率Helmholtz方程及高维问题时存在网格划分困难和内存消耗大等缺点.基于点的近似思想,无网格方法在一定程度上减少对网格的依赖,甚至不需要网格。     

纽结理论与三叶结分子合成研究进展

拓扑学中的纽结理论在化学和生物学中有非常重要的应用,首先介绍了纽结理论在化学中的应用发展,然后对三叶结分子合成方法的发展及其可能的反应机理进行了阐述。     

Almost strictly total positivity of NUAT B-spline basis

Total positivity of spline basis has been well known in the theory of computer aided geometry design, which is highly related with good shape preserving property. Almost strictly total positivity is stronger, which could determine the positive minors while the other is zero. In this paper, a geometrical approach is proposed to prove the collection matrices of NUAT B-spline basis are almost strictly totally positive. In this paper, knot insertion algorithm combined with coefficient variation of NUAT B-spline function, we put forward an intuitive and geometrical method.     

基于Matlab的径向基函数数值计算软件包

为推动基于无网格方法的计算软件的发展,介绍基于Matlab自主开发的径向基函数（Radial Basis Function,RBF）数值计算软件包,阐述软件的理论基础、设计思路以及该软件包的功能和特点,并结合边界节点法（Boundary Knot Method,BKM）的数值实例给出软件的使用过程．该软件包可以根据不同的数学物理模型选择合适的数值算法来求解多种实际物理问题,也可对不同数值算法得到的结果进行比较．最后,总结应用Matlab进行数值计算软件开发的优缺点．     

NURBS边界曲面直接生成法

由于非均匀有理B样条(NURBS)曲面的复杂性,传统NURBS边界曲面的生成是先构造孔斯曲面,再由孔斯曲面向NURBS曲面转换得到,其操作过程比较烦琐。针对此问题,提出了NURBS边界曲面直接生成算法,该算法根据给定的四条NURBS边界曲线,结合孔斯曲面生成方法直接插值生成NURBS曲面,从而避免了通过孔斯曲面向NURBS曲面转换所带来的计算代价,因此同传统方法相比,具有较低的计算代价。实验结果表明:该算法简化了曲面生成步骤,减少了曲面转换过程的计算量,生成的曲面边界信息明确,且连续性好。     

应用NURBS曲面磨光多面体

本文应用ＮＵＲＢＳ曲面磨光多面体,产生了处处Ｃ′连续的过渡曲面．多面体的磨光分为边的磨光和顶点的磨光两种情形,边的磨光相对较容易,而顶点的磨光则很困难．本文所采用的应用ＮＵＲＢＳ曲面磨光多面体的顶点和边的方法,不仅可以统一实现二者的磨光操作,而且方法简单且统一,产生了Ｃ′连续的过渡面．较之以前的方法,首先,利用ＮＵＲＢＳ曲面可以精确地描述对边磨光所用的柱面（等半径或非等半径）,其次,在对顶点的磨光中,同以往的方法不同,将与该顶点相邻的边的过渡曲面相互分离,并首次引入了“补面”的概念,使得对该点所产生的过渡曲面处处Ｃ′连续．本算法首先构造用以磨光多面体顶点和边的ＮＵＲＢＳ曲面的边界曲线网络图产生边界曲线的控制点及其权值（ＮＵＲＢＳ表示）,然后依据连续性准则,产生ＮＵＲＢＳ曲面的控制信息．     

Symbolic computation with finite biquandles

A method of computing a basis for the second Yang–Baxter cohomology of a finite biquandle with coefficients in Q$\mathbb{Q}$ and _Zp${\mathbb{Z}}_p$ from a matrix presentation of the finite biquandle is described. We also describe a method for computing the Yang–Baxter cocycle invariants of an oriented knot or link represented as a signed Gauss code. We provide a URL for our Maple implementations of these algorithms.     

纽结带形建筑表皮的拓扑学分析

引进拓扑学的纽结理论来指导分析纽结带形表皮的设计,首先从分析纽结带形表皮的拓扑学特征入手,提出了纽结类带形表皮和链环类带形表皮设计的造型规律,同时利用琼斯多项式来阐述纽结带形表皮的拓扑同痕不变量,并分别对纽结带形表皮的复杂度,以及纽结带形表皮间的环绕、串联和拼合进行阐述,力求利用拓扑学中纽结理论解决一些建筑纽结带形表皮的设计中所遇到的问题。