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 O(k²+kh²+h²) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

ASYMPTOTIC PROPERTIES OF ESTIMATORS IN PARTIALLY LINEAR SINGLE-INDEX MODEL FOR LONGITUDINAL DATA

In this article, a partially linear single-index model /or longitudinal data is investigated. The generalized penalized spline least squares estimates of the unknown parameters are suggested. All parameters can be estimated simultaneously by the proposed method while the feature of longitudinal data is considered. The existence, strong consistency and asymptotic normality of the estimators are proved under suitable conditions. A simulation study is conducted to investigate the finite sample performance of the proposed method. Our approach can also be used to study the pure single-index model for longitudinal data.

Displacement Response Reconstruction of Slender Flexible Structures Based on Cubic Spline Fitting Method

How to reconstruct a dynamic displacement of slender flexible structures is the key technology to develop smart structures and structural health monitoring(SHM), which are beneficial for controlling the structural vibration and protecting the structural safety. In this paper, the displacement reconstruction method based on cubic spline fitting is put forward to reconstruct the dynamic displacement of slender flexible structures without the knowledge of modeshapes and applied loading. The obtained strains and displacements are compared with the results calculated by ABAQUS to check the method's validity. It can be found that the proposed method can accurately identify the strains and displacement of slender flexible structures undergoing linear vibrations, nonlinear vibrations, and parametric vibrations. Under the concentrated force, the strains of slender flexible structures will change suddenly along the axial direction. With locally densified measurement points, the present reconstruction method still works well for the strain concentration problem.

NUAT T-splines of odd bi-degree and local refinement

This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.

Shifted Linear Interpolation Filter

Linear interpolation has been adapted in many signal and image processing applications due to its simple implementation and low computational cost. In standard linear interpolation the kernel is the second order B-spline. In this work we show that the interpolation error can be remarkably diminished by using the time-shifted B-spline as an interpolation kernel. We verify by experimental tests that the optimal shift is. In VLSI and microprocessor circuits the shifted linear interpolation (SLI) algorithm can be effectively implemented by the z-transform filter. The interpolation error of the SLI filter is comparable to the more elaborate higher order cubic convolution interpolation.

Function-on-Partially Linear Functional Additive Models

We consider a functional partially linear additive model that predicts a functional response by a scalar predictor and functional predictors. The B-spline and eigenbasis least squares estimator for both the parametric and the nonparametric components proposed. In the final of this paper, as a result, we got the variance decomposition of the model and establish the asymptotic convergence rate for estimator.

Computational Analysis for Solving the Linear Space-Fractional Telegraph Equation

Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.

THE SARD BEST APPROXIMATION OF LINEAR FUNCTIONAL

In numerical analysis, it is significant to approximate the linear functional Ef=sum from i=0 to m-1([integral from a to b(a1（x）f1（x）dx+ sum from f=0 to i1(b1f1（x1）)]) by a simpler linear functional Lf=sum from i=1 to m（a1f（x1）） In this paper, making use of natural Tchebysheff spline function, we give existence theorem and uniqueness theorem of L that is exact for the degree m to F; we also give three sufficient and necessary conditions in which L is the Sard best approximation to F.

Adaptive trajectory linearization control for hypersonic reentry vehicle

This paper presents an improved design for the hypersonic reentry vehicle(HRV) by the trajectory linearization control(TLC) technology for the design of HRV. The physics-based model fails to take into account the external disturbance in the flight envelope in which the stability and control derivatives prove to be nonlinear and time-varying, which is likely in turn to increase the difficulty in keeping the stability of the attitude control system. Therefore, it is of great significance to modulate the unsteady and nonlinear characteristic features of the system parameters so as to overcome the disadvantages of the conventional TLC technology that can only be valid and efficient in the cases when there may exist any minor uncertainties. It is just for this kind of necessity that we have developed a fuzzy-neural disturbance observer(FNDO) based on the B-spline to estimate such uncertainties and disturbances concerned by establishing a new dynamic system. The simulation results gained by using the aforementioned technology and the observer show that it is just due to the innovation of the adaptive trajectory linearization control(ATLC) system. Significant improvement has been realized in the performance and the robustness of the system in addition to its fault tolerance.

带有时变杠杆效应的随机波动率模型参数估计及其应用

为了更好地捕捉金融时间序列中杠杆效应的时变非对称性,本文基于线性样条思想,提出一种带有时变杠杆效应的半参数随机波动率模型,并利用贝叶斯MCMC方法对所提模型中的参数进行了估计.模拟研究表明贝叶斯MCMC方法在所提模型的参数估计方面有着良好的有限样本表现.最后利用本文所提出的带有时变杠杆效应的半参数随机波动率模型对2000年1月4日至2020年8月18日的上证综合指数和深证成份指数日收益数据进行了实证分析,结果表明利用本文所提出的模型拟合这两组实例数据是合理的.

孕期血清铁蛋白动态变化及其与分娩结局的相关性

目的分析孕期血清铁蛋白(SF)水平的动态变化,探讨不同孕周SF水平与分娩结局的相关关系。方法选取2021年1月至2023年8月期间,在安徽医科大学第一附属医院东城院区规律产检且连续3次SF检测的孕妇665名作为研究对象。分析孕期SF的动态变化,用线性回归模型和限制性立方样条分析不同孕周SF与分娩结局间的相关性,并构建SF动态变化与分娩结局的群组轨迹模型。结果随着孕周增加,SF水平下降,铁缺乏率增加。线性相关性分析显示孕8~16周、孕16~24周SF与分娩结局相关性无统计学意义(P>0.05);孕24-32周SF水平与新生儿身长(β:−0.009,95%CI:−0.015~−0.003)、新生儿体质量(β:−3.331,95%CI:−5.201~−1.461)及分娩孕周(β:−0.013,95%CI:−0.019~−0.008)呈负相关。限制性立方样条曲线拟合显示,孕8~16周SF、孕16~24周SF与分娩结局相关性无统计学意义(P整体>0.05),孕24~32周SF与分娩孕周和新生儿身长呈线性剂量-反应关系(P整体<0.05,P非线性>0.05),与新生儿体质量呈非线性相关关系(P整体<0.05,P非线性<0.05)。群组轨迹模型显示,SF浓度下降较快组早产率高于SF浓度下降中等组和较慢组(P<0.01),SF不同变化轨迹类型与新生儿体重相关性无统计学意义。结论孕24~32周SF水平升高与分娩孕周、新生儿身长和体质量相关,孕期SF持续下降与早产发生风险增加有关。

部分线性变系数模型的贝叶斯分位数回归

针对部分线性变系数模型的参数估计问题,采用B样条方法逼近非参数部分的未知光滑函数,进而利用非对称拉普拉斯分布实现贝叶斯分位数回归,并基于Gibbs抽样推导出所有未知参数的后验分布,通过数值模拟比较分析了贝叶斯分位数回归与分位数回归参数估计的优劣,结果表明,在均方误差准则下,贝叶斯分位数回归的估计效果更优。最后,通过实例分析说明了所提方法的有效性。

基于复合分位数回归的部分线性模型平均估计

针对部分线性模型的参数与非参数估计问题,基于复合分位数回归提出了一种稳健的模型平均估计量.为了提高估计效率,采用B样条的方法拟合子模型中的非参数函数.数值模拟和仿真实验证明了所提出的估计量预测效果优良.

自动泊车系统规划与控制方法研究

为了对泊车过程中的车辆进行跟踪控制,基于车辆动力学模型,研究设计了一种线性二次型控制器,使用B-spline曲线规划了一条曲率连续的泊车路径,针对控制器设计过程中权重系数选择困难,引入模糊逻辑算法对控制器进行优化处理,节省了人工成本,然而模糊逻辑算法的计算是实时的,且不同速度下车辆状态不同,因此将基于Frobenius范数的更新策略加入程序计算中以降低计算量,利用MATLAB/Carsim联合仿真验证了改进后控制系统的运算时间降为0.2 ms,实时性提升92%,适应度提升30%.

两条B样条曲线求交的高效计算方法

曲线曲面间求交计算在CG和CAD中有着广泛的应用.牛顿法等迭代法计算效率高但需要良好的初始值;裁剪法具有良好的鲁棒性但计算效率不理想,尤其是对于相切情况的求交问题.为此,提出一种计算2条B样条曲线交点的混合方法.首先提出一种高效的线性复杂度裁剪方法,用于获得良好的初始值;然后提出一种与导数无关且效率更高的改进的割线法,用于验证贯穿性相交情况;最后提出一个相切情况下收敛阶为2的迭代公式,其性能远优于现有的牛顿法和裁剪法.理论上,混合方法若与根隔离法相结合,可以应用于更多类型曲线间的求交问题.数值实验结果表明,与现有的同类方法相比,在贯穿情况下,所提方法的计算效率提高约10%,在相切情况下则提高约100%~300%.

部分线性变系数模型的贝叶斯复合分位数回归

部分线性变系数模型由参数和非参数2部分组成,具有适应范围广和解释性强双重优点。针对该模型的参数估计问题,采用B样条方法逼近非参数部分的未知光滑函数,进而利用复合非对称拉普拉斯分布实现贝叶斯复合分位数回归,并基于Gibbs抽样算法推导出所有未知参数的后验分布,以获取参数的估计值。通过数值模拟对贝叶斯复合分位数回归与贝叶斯分位数回归、贝叶斯线性回归参数估计效果进行比较分析,结果显示:当误差服从非正态分布时,在均方误差准则下,贝叶斯复合分位数回归估计表现更优。基于上述3种方法对实例数据进行预测分析,结果表明:在平均绝对偏差和均方误差预测意义下,基于贝叶斯复合分位数回归的预测效果更好。

某试验场强化路三维虚拟路面重构与对比分析

对某试验场强化路面中线轨迹的测试数据,进行样条插值,还原路面轨迹,对测试的路面纵向5路高程数据,根据路面横向几何特征进行横向插值:横向几何特征一致的路面,采用线性插值;横向几何特征随机变化的路面,采用样条插值;横向几何特征不一致,但存在一定规律的路面,按其规律插值。最终依据相应的插值方法编程实现三维虚拟路面重构。进一步对重构的RDF和CRG格式路面进行对比的结果表明:CRG格式更适用于三维虚拟路面重构和动态载荷预测的动力学分析。路面特征越复杂,CRG路面在计算效率上相对RDF路面的优势更为突显。

五孔探针实验数据处理的线性插值法

在使用五孔探针校准数据时通常采用样条插值或最小二乘法拟合,相同的探针校准曲线用这两种方法分别插值得到的结果有较大的差异。本文通过对两种方法理论上分析,提出了一种实用性更强的线性插值法。对于特性良好的探针,该方法的精度接近样条插值;对于特性较差的探针,能够在不改变探针特性曲线的基础上保证插值精度。相对于样条插值和最小二乘法拟合,该方法有着更为广泛的适应性。

应用样条插值提高GRAPES模式物理过程反馈精度

GRAPES(Global/Regional Assimilation and PrEdiction System)模式动力框架中垂直方向变量的跳层设置采用Charney-Phillips分布,在整层上进行位温、水物质的计算,物理过程中在半层上对其进行处理。这样在GRAPES模式中,进入物理过程之前和物理过程计算完毕之后,都要采用线性插值进行整层和半层之间物理量的转换。由于线性插值精度欠佳,为提高上述反馈过程的精度,并保证水物质的正定性。该研究引入样条插值,并在水物质的插值过程中进行保单调处理,有效减小了位温场、水物质场的预报偏差,并提升了模式的综合预报性能。