Numerical integration of partial differential equations using cubic splines

This paper presents techniques for the numerical solution of partial differential equations using cubic spline collocation.

The main spline relations are presented and incorporated into solution procedures for partial differential equations. The computational algorithm in every case is a tridiagonal matrix system amenable to efficient inversion methods. Truncation errors and stability are briefly discussed. Finally, some examples of their application to parabolic and hyperbolic systems with mixed boundary conditions are presented.

The results obtained are encouraging and justify further research in this field.     

基于NPS的常微分方程OBVP的研究

根据样条函数中基函数的特殊选取,构造出了一类新的非多项式样条函数,用来求解常微分方程OBVP的边值问题.给出求解的递推公式,并进行了相应的误差分析,结果显示该方法的局部截断误差为O（h6）.     

Smooth cutting pattern generation technique for membrane structures using geodesic line on subplane and spline interpolation

Practical techniques for smooth geodesic patterning of membrane structures were investigated, For the geodesic search, adjustment of the subplane of the extracted elements series was proposed, and various spline approximation methods were used to flatten the strip for the generation of a smooth pattern. This search approach is very simple, and the geodesic line could be easily attained by the proposed method without the need for a difficult computation method. Smooth cutting patterning can also be generated by spline approximation without the noise in discrete nodal information. Additionally, the geodesic cutting pattern saved about 21% of the required area for the catenary model due to the reduction of the curvature of the planar pattern seam line.     

广义三次DP曲线

给出一组带有两个形状参数的三次多项式基函数,它是三次DP基函数的扩展;分析该基函数的性质;基于该组函数定义了广义三次DP曲线,它不仅具有与DP曲线类似的性质,还可通过改变参数几或产使其具有形状可调性。当这两段曲线相拼接时,在满足一定的条件下,曲线可达到G^2或C^2连续,从而为自由曲线设计提供了一种有效的方法。     

Study of Quintic Spline Interpolation and Generated Velocity Profile for High Speed Machining

Modern high speed machining (HSM) machine tools often operates at high speed and high feedrate with high ac- celerations,in order to deliver the rapid feed motion.This paper presents an interpolation algorithm to generate continuous quintic spline toolpaths,with a constant travel increment at each step,while the smoother accelerations and jerks of two-order curve are obtained.Then an approach for reducing the feedrate fluctuation in high speed spline interpolation is presented.The presented ap- proach has been validated to quickly,reliably and effective with the simulation.     

An investigation into design of fair surfaces over irregular domains using data-dependent triangulation

Design of fair surfaces over irregular domains is a fundamental problem in computer-aided geometric design (CAGD), and has applications in engineering sciences (in aircraft, automobile, ship science etc.). In the design of fair surfaces over irregular domains defined over scattered data, it was widely accepted till recently that the classical Delaunay triangulation be used because of its global optimum property. However, in recent times it has been shown that for continuous piecewise linear surfaces, improvements in the quality of fit can be achieved if the triangulation pattern is made dependent upon some topological or geometric property of the data set or is simply data dependent. The fair surface is desired because it ensures smooth and continuous surface planar cuts, and these in turn ensure smooth and easy production of the surface in CAD/CAM, and favourable resistance properties. In this paper, we discuss a method for construction of C¹ piecewise polynomial parametric fair surfaces which interpolate prescribed ℜ³ scattered data using spaces of parametric splines defined on H³ triangulation. We show that our method is more specific to the cases when the projection on a 2-D plane may consist of triangles of zero area, numerically stable and robust, and computationally inexpensive and fast. Numerical examples dealing with surfaces approximated on plates, and on ships have been presented.     

New approach in depositing thick, layered cubic boron nitride coatings by oxygen addition—structural and compositional analysis

Cubic boron nitride (c-BN) can be produced by PVD and PA-CVD techniques by intensive ion bombardment leading to highly stressed films limiting its use in industrial applications. Various attempts have been undertaken to reduce the compressive stress of c-BN thin films. A significant reduction in compressive stress and a substantially improved adhesion was achieved by a new coating concept consisting of a two-step adhesion-promoting base layer, a compositional-graded nucleation layer obtained by a stepwise decrease of the oxygen content in the Ar/N₂/O₂ atmosphere and a low-stressed c-BN:O top layer with controlled oxygen addition. The four-layer c-BN:O film with a thickness of 3 μm was deposited by unbalanced radio frequency magnetron sputtering of a hot-pressed hexagonal boron nitride target on silicon substrates. The adhesion layer was deposited in a mixed Ar/O₂ atmosphere of 0.26 Pa with a stepwise increased nitrogen gas flow and a subsequent increase of the ion energy by increasing the substrate bias from 0 to − 250 V. The c-BN nucleation was gradually initiated by decreasing the O₂ gas flow. The present study was focused on the investigation of the morphology, the microstructure on the nanoscale, and the bonding structure using scanning electron microscopy (SEM), Fourier-Transmission infra-red spectroscopy (FTIR), high-resolution transmission electron microscopy (HRTEM) and electron energy loss spectroscopy (EELS) employing analytical scanning transmission electron microscopy (ASTEM). The HRTEM images revealed a four-layer coating consisting of a gradual nucleation of t-BN, on which a gradual nucleation of c-BN was achieved by decreasing the oxygen gas flow.     

Solid-state interfacial reactions and compound morphology of cBN grain and surface Ti coating

In order to research thoroughly the mechanism of the solid-state interfacial reactions and the induced compound morphology of cubic boron nitride (cBN) abrasive grains and the surface Ti-coating layer, annealing experiments of Ti-coated cBN grains were conducted at different temperatures of 550-950 °C for dwell times of 60 min under high-vacuum conditions. The corresponding interfacial characteristics were investigated by differential thermal analysis, X-ray diffraction and scanning electron microscopy. The results show that the relevant interfacial reactions and compound morphology between cBN and Ti are significantly influenced by the heat-treatment temperature. No reaction occurs below 550 °C, and TiN is the sole reaction product during heating from 650-750 °C. Three kinds of compounds, TiN, TiB₂ and TiB, can exist in the interfacial region at 950 °C. Here, the favorable interfacial structures, cBN/TiB₂/TiB/(TiB+TiN)/TiN/Ti, are developed for the excellent mechanical and chemical transition effects between cBN grains and Ti coating. The thermodynamic analysis finally predicts that there is a reasonable probability for the formation of such a special interfacial transition layer.     

基于坐标法的复杂曲面轮廓度的误差评定

复杂曲面的面轮廓度是一个应用广泛而又难于测量和评定的问题.在面轮廓度误差评定过程中,首先将复杂曲面的理论模型离散化后用双三次样条函数进行精确描述,并用该拟合曲面作为误差评定的依据;其次对测量后所得的测头球心位置曲面进行等距平移以得到实测轮廓曲面;最后再采用坐标轮换法求取实测点在以双三次样条函数所表示的理论轮廓上的对应点,从而实现对复杂曲面轮廓度的评定.由于本文用双三次样条函数所表示的曲面作为复杂曲面的一般表示,因此该评定算法可实现测量及评定软件的通用性,并通过计算机仿真,证明该算法的正确性与可行性.     

基于测量数据优化修匀的特性曲线样条插值方法

针对传感器特性曲线拟合存在的一些问题,提出了测量数据优化修匀思想以有效消除测量数据的随机误差.应用数值处理软件Matlab结合实验数据研究了基于不同修匀方案的特性曲线的三次样条插值表示并提供了可视化的结果.经过对比不难发现,依据优化修匀数据的三次样条插值误差明显减小,可应用于任意类型传感器或其它仪器仪表的特性曲线的描述与分析.