A comparison of piecewise cubic Hermite interpolating polynomials,cubic splines and piecewise linear functions for the approximation of projectile aerodynamics

Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

Numerical solution of Poisson equation with wavelet bases of Hermite cubic splines on the interval

A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher effciency and precision in solving the Poisson equation.

A polynomial smooth epsilon-support vector regression based on cubic spline interpolation

Regression analysis is often formulated as an optimization problem with squared loss functions.Facing the challenge of the selection of the proper function class with polynomial smooth techniques applied to support vector regression models,this study takes cubic spline interpolation to generate a new polynomial smooth function |x|_ε~2 in ε-insensitive support vector regression.Theoretical analysis shows that S_ε~2-function is better than p_ε~2-function in properties,and the approximation accuracy of the proposed smoothing function is two order higher than that of classical p_ε~2-function.The experimental data shows the efficiency of the new approach.

关于第三类半对数spline插值函数的误差估计

本文讨论一种属于c2[a,b]的spline插值函数的误差估计,根据误差估计理论知它与三次样条插值函数具有相同的误差阶。

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and numerical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set,and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the shortcoming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a C1-cubic Hermite interpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global C2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an alternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15(1)(2007), pp. 41-53].

Dubuc-Deslauriers细分格式生成函数递推公式

细分格式是一种在初始控制网格基础上,通过迭代局部加细并应用特定拓扑规则,逐步形成光滑曲线或曲面的迭代方法.m重2N点Dubuc-Deslauriers细分格式是一种广泛应用的插值型格式.当重数m或N较大时,由于涉及多个控制顶点,Dubuc-Deslauriers细分格式面临计算效率和稳定性的挑战.通过将一次加细操作分解为多次小范围操作,Dubuc-Deslauriers细分格式的递推公式形式有效提高了计算稳定性.给出了m重2N点Dubuc-Deslauriers细分格式递推公式的生成函数的表达式,并探讨了2重和3重情况的特殊形式.

基于改进RRT算法的移动机器人路径规划研究

快速扩展随机树算法(rapidly-exploring random trees,RRT)规划移动机器人路径时,存在搜索盲目性强、搜索时间长、收敛速度慢、路径冗余点多且不平滑等问题。鉴于此,提出一种改进的RRT路径规划算法。首先,针对传统RRT算法盲目搜索以及局部极值的问题,提出概率目标偏置与人工势场结合的采样策略,引导随机树的扩展;其次,针对随机树扩展的避障能力差的问题,提出基于安全距离的碰撞检测以及动态变步长扩展策略;最后,针对路径上冗余点多以及曲率不连续的问题,提出考虑安全距离的剪枝优化和三次B样条曲线对初始路径进行拟合优化。仿真结果表明,在不同地图的路径规划中,相比于传统RRT算法,增强了通过狭窄通道能力,优化了路径的平滑性,搜索时间、迭代次数、路径长度分别减少约70%、40%、15%;相比于RRT衍生算法RRT-Connect,搜索时间、路径长度分别减少约25%、10%。

膳食叶黄素和玉米黄质与老年人低认知功能的剂量-反应关系

目的 使用限制性立方样条回归模型探讨膳食叶黄素和玉米黄质摄入量与老年人低认知功能的剂量-反应关系。方法 基于美国国家健康与营养调查(National Health and Nutrition Examination Survey, NHANES)数据库,选取年龄≥60岁且具有完整认知功能测试的老年人群。应用Logistic回归模型和限制性立方样条回归模型评估膳食叶黄素和玉米黄质摄入量与老年人低认知功能的关联和剂量-反应关系。结果 共纳入2 695名研究对象,其中认知功能正常组1 629例,低认知功能组1 066例。与认知功能正常组比较,低认知功能组年龄在70~<80岁和≥80岁的人群所占比例明显较高,糖尿病及中风发生比例也明显增高,此外,膳食叶黄素和玉米黄质摄入量以及叶黄素和玉米黄质总摄入量明显降低,差异均有统计学意义(P<0.05)。限制性立方样条回归分析显示,以性别分层,随着叶黄素和玉米黄质总摄入量的持续增加,男性和女性认知功能下降风险均呈近似“L”形的非线性变化(P非线性<0.01),但是在同样“L”形趋势的情况下,叶黄素和玉米黄质总摄入量是男性认知功能下降的易感因素。结论 叶黄素和玉米黄质总摄入量与老年人认知功能下降风险呈近似“L”形的非线性剂量-反应关系。

基于样条权函数神经网络的人脸识别研究

文章对人脸识别技术进行了调查,对比分析了传统识别技术以及基于深度学习识别技术的不同,指出由于深度学习的网络结构较为复杂,在人脸识别过程中容易出现算法过拟合的现象。针对上述过拟合问题,文章对基于权函数神经网络的人脸识别算法进行了研究,此类神经网络结构简单,仅有输入、输出两层结构,将人脸图片作为输入信息,网络训练目标为计算三次样条权函数,仿真实验证明该类网络具有较好的识别准确度和识别效率。

MODELING SPATIAL AND TEMPORAL VARIATION OF A POLLUTANT IN WATER ENVIRONMENT USING THE CUBIC SPLINE SCHEME

The cubic spline integration scheme has been successfully applied for the first time to obtain solutions for the two-dimensional shallow water wave and transport equations. Its principal advantages are its high accuracy, low computational cost, and flexibility of application. The study quantitatively analyzed and predicted the variation in time and space of a solid pollutant in water environment. The infiltration, adsorption, solubility, advection, and diffusion related factors at different spatial and temporal scales were taken into consideration in the model. Experiments were conducted in this study to examine the capabilities of the cubic spline scheme. The correspondence between the numerical results and the experimental data was fair. The results were also compared with the numerical results obtained using finite difference schemes with good agreement.

基于改进A^(*)算法的煤矿救援机器人路径规划

路径规划是煤矿救援机器人研究的重要内容之一。针对灾后煤矿环境非结构化的特点,以及传统A^(*)算法规划的路径长度非最短、拐弯次数多和平滑度较差等问题,提出一种基于改进A^(*)算法的煤矿救援机器人路径规划方法。对真实环境中的地图信息进行二值化处理,构建栅格地图;判断当前点与目标点的相对位置,利用改进A^(*)算法进行路径规划,得到一条从当前点到目标点的路径;利用Douglas-Peucker(D-P)算法提取路径上的关键节点,采用三次样条插值函数对关键节点进行拟合,完成对路径的平滑处理。改进A^(*)算法将传统A^(*)算法的8邻域搜索扩展为有目的性的13邻域搜索,在进行路径搜索时,先对当前点和目标点的位置关系进行判断,从而减少路径节点,减小路径长度,提升路径平滑度。Matlab仿真结果表明:与8邻域A^(*)算法、24邻域A^(*)算法、48邻域A^(*)算法相比,改进A^(*)算法在路径长度、拐弯次数、平滑度等方面有一定优化,更适用于煤矿救援机器人路径规划;与Fuzzy算法相比,改进A^(*)算法路径规划所用时间更短,规划的路径长度更短,拐弯次数更少。

神经功能、营养状况及运动能力量表评分与卒中相关肺炎风险关联性研究

目的分析美国国立卫生研究院卒中量表(National Institutes of Health stroke scale,NIHSS)评分、老年营养风险指数(geriatric nutritional risk index,GNRI)、运动功能独立性评定(motor function independence measure,MFIM)评分与卒中相关性肺炎(stroke-related pneumonia,SAP)风险的关系。方法纳入2021年11月至2022年5月卒中入院的患者,收集入院时NIHSS、GNRI、MFIM评分,根据卒中发病后1周内是否发生肺炎分为SAP组和非SAP组。使用受试者工作特征(receiver operating characteristic,ROC)曲线分析各评分最佳截断点并将评分转换为分类变量,采用多因素logistic回归模型和限制性立方样条分析各评分与SAP之间的关系。结果研究共纳入318例卒中患者,SAP组86例,非SAP组232例。logistic回归结果显示,NIHSS评分(OR=32.783,95%CI:16.366~65.671,P<0.001)、MFIM评分(OR=0.052,95%CI:0.027~0.100,P<0.001)和GNRI评分(OR=0.262,95%CI:0.144~0.476,P<0.001)与SAP存在关联。限制性立方样条分析显示,NIHSS评分(P_(总趋势)<0.001,P_(非线性)=0.002)、GNRI评分(P_(总趋势)<0.001,P_(非线性)<0.001)与SAP风险之间存在非线性剂量-反应关系。结论NIHSS、MFIM、GNRI评分和卒中患者SAP发生风险相关,其中NIHSS和GNRI评分与其存在非线性关联。

Laplace方程柯西问题的B样条方法

Laplace方程柯西问题极其不适定,需要有效的数值算法进行求解,本文提出一种B样条方法求解此问题。首先在三次B样条函数生成的平移不变空间中给出柯西问题逼近解的表达形式;然后借助B样条基函数导数可用低阶样条基函数表示及方程的性质,写出问题的变分形式;接着,为了降低噪音的影响,提出Tikhonov正则化方法,以获得稳定的数值解;最后分别对矩形区域和含非光滑边界的区域进行数值实验,证明此方法的有效性。

基于改进RRT算法的移动机器人路径规划研究

路径规划是移动机器人开发的核心技术,而传统RRT算法所规划的路径距离长、拐点多,对RRT算法进行改进,引入A*算法思想,改进待扩展节点的选择。引入人工势力场思想,改进扩展方向的选择。对规划的路径消除冗余点,并采用三次样条函数进行平滑处理。将改进的RRT算法应用于虚拟简单环境地图、虚拟复杂环境地图中,结果表明改进RRT算法所规划的路径平顺性好、路径长度短且所用时间短。最后将改进的RRT算法应用于实际的移动机器人路径规划中,路径规划结果验证了该算法的可行性。

GRAPES模式标准初始化方案设计与实现

文章介绍了GRAPES模式标准初始化系统的方案设计及程序结构,对3种不同方案产生的初始场进行了分析。结果表明:水平风场、温度场、位温场对插值方法的精度并不是很敏感,而高度场以及气压Exner函数则对垂直插值方法的精度要求很高;在3种模式变量产生方案中,方案一的效果较差,但它计算速度快,在地形梯度不大的地方可以选择使用;方案二与方案三的效果则比较好,用户可根据自己的需求任意选择。GRAPES模式标准初始化系统在完成其主要功能的同时,通过灵活的程序设计,对标准初始化系统的区域、分辨率、垂直层次、水平插值方法、垂直插值方法等的任意选取,使得标准初始化系统可以很容易满足模式初值的各种需求,为模式开发研究实验提供了各种便利。

用双三次样条函数和GPS资料反演现今中国大陆构造形变场

将中国大陆现今构造变动视为一种连续的地壳变形 ,利用双三次样条函数模拟了近期 GPS测定的大陆内部及周边地区 4 12个测站速率 ,反演大陆地区自洽的构造变动速度场和应变率场。模拟结果显示 :印度板块与欧亚板块的碰撞、挤压是构成中国大陆内部岩石层水平形变的主要驱动力。印度板块在东喜马拉雅构造结深深插入青藏高原 ,造成地壳大规模的缩短和抬升。青藏高原东南部的喜马拉雅带、拉萨和羌塘地块以及青藏高原东南边的川滇地区 ,内部构造活动强烈 ,其内部的构造变形包含地壳碎片的冲断、褶皱和侧向逃逸。大陆地壳 (或岩石圈 )的增厚 ,尤其是喜马拉雅山脉南北向的快速缩短和青藏高原东西向的缓慢拉张 ,大约吸收了印欧板块会聚量的 85 % ,西藏中东地区东西向的拉张速率达到了 (16± 2 .0 ) mm/ a,且顺时针方向扭转明显。印度板块相对欧亚板块运动的欧拉极为 (2 9.7°N,19.3°E,0 .392°/ Ma) ;华南地块相对于欧亚大陆向东 (10 2°± 7.4°)南的运动速率是 (11± 1.5 4 )mm/ a,华南块体相对欧亚板块运动的欧拉极为 (6 2 .2 5°N,12 6 .5 6°E,0 .14 1°/ Ma) ;塔里木地块相对较稳定 ,其西部运动速度高于东部运动速度 ,作顺时针方向旋转。总体上讲 ,中国大陆运动方向为北偏东呈辐射状 ,从西部近南北方向的?

基于三次样条函数的激光雷达数据可视化插值法

传统的可视化软件与方法不能够准确地反映受大气湍流以及大气边界层影响的大气参数的细节信息,难以实现连续性的大气参数演变过程。提出了一种基于三次样条函数的激光雷达数据可视化插值法。分析了激光雷达数据的离散型特点,确定影响大气激光雷达数据可视化图形的关键因素,利用一定大气范围内的观测数据相互影响的关系,结合三次样条插值函数对插值数据进行修正并拟合观测数据的整体变化趋势。实验结果表明,该方法能够提高离散激光雷达数据间插值数据的准确性与可视化图形的平滑性,为激光雷达数据分析提供了有力手段。

一种加三项余弦窗的加窗插值FFT算法

提出了一种基于三项余弦窗exact Blackman窗的插值FFT算法。讨论了exact Blackman窗的频率响应,详细推导了基于exact Blackman窗的插值FFT算法的计算公式,并采用三次样条插值函数计算频率修正系数和复振幅的修正系数。仿真计算结果表明,利用三次样条函数计算的谐波幅值误差小于0.1%,频率误差小于0.01Hz,相位误差小于5%。新的插值FFT算法能够有效地提高电力系统谐波的测量精度,与其他四项余弦窗加窗插值FFT算法相比,具有较小的运算量和较好的实时性。

比赛重要性及比赛进程与顶级乒乓球运动员心理状态的关系

显示基于比分变化的函数拟合和插值可以较好地揭示乒乓球运动员赛中的心理状态。为了解比赛重要性与我国顶级乒乓球运动员赛中心理状态的关系,以近3年的比赛比分数据为基础绘制了4名主力运动员的比分变化曲线,结果显示,乒乓球重要比赛(3大赛事)对不同主力运动员的心理状态有不同的影响。在此基础上,能够结合比赛进程的、针对性的心理训练方案有助于提高运动员参加重大比赛时应对心理压力的能力。