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.

Dynamic GM（1,1） Model Based on Cubic Spline for Electricity Consumption Prediction in Smart Grid

Electricity demand forecasting plays an important role in smart grid expansion planning.In this paper,we present a dynamic GM(1,1) model based on grey system theory and cubic spline function interpolation principle.Using piecewise polynomial interpolation thought,this model can dynamically predict the general trend of time series data.Combined with low-order polynomial,the cubic spline interpolation has smaller error,avoids the Runge phenomenon of high-order polynomial,and has better approximation effect.Meanwhile,prediction is implemented with the newest information according to the rolling and feedback mechanism and fluctuating error is controlled well to improve prediction accuracy in time-varying environment.Case study using the living electricity consumption data of Jiangsu province in 2008 is presented to demonstrate the effectiveness of the proposed model.

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.

A Finite Element Cable Model and Its Applications Based on the Cubic Spline Curve

For accurate prediction of the deformation of cable in the towed system, a new finite element model is presented that provides a representation of both the bending and torsional effects. In this paper, the cubic spline interpolation function is applied as the trial solution. By using a weighted residual approach, the discretized motion equations for the new finite element model are developed. The model is calculated with the computation program complier by Matlab. Several numerical examples are presented to illustrate the numerical schemes. The results of numerical simulation are stable and valid, and consistent with the mechanical properties of the cable. The model can be applied to kinematics analysis and the design of ocean cable, such as mooring lines, towing, and ROV umbilical cables.

Denoising of hyperspectral imagery by cubic smoothing spline in the wavelet domain

The acquired hyperspectral images （HSIs） are inherently attected by noise wlm Dano-varylng level, which cannot be removed easily by current approaches. In this study, a new denoising method is proposed for removing such kind of noise by smoothing spectral signals in the transformed multi- scale domain. Specifically, the proposed method includes three procedures： 1 ） applying a discrete wavelet transform （DWT） to each band; 2） performing cubic spline smoothing on each noisy coeffi- cient vector along the spectral axis; 3 ） reconstructing each band by an inverse DWT. In order to adapt to the band-varying noise statistics of HSIs, the noise covariance is estimated to control the smoothing degree at different spectra｜ positions. Generalized cross validation （GCV） is employed to choose the smoothing parameter during the optimization. The experimental results on simulated and real HSIs demonstrate that the proposed method can be well adapted to band-varying noise statistics of noisy HSIs and also can well preserve the spectral and spatial features.

Performance of global look-up table strategy in digital image correlation with cubic B-spline interpolation and bicubic interpolation

Global look-up table strategy proposed recently has been proven to be an efficient method to accelerate the interpolation, which is the most time-consuming part in the iterative sub-pixel digital image correlation （DIC） algorithms. In this paper, a global look-up table strategy with cubic B-spline interpolation is developed for the DIC method based on the inverse compositional Gauss-Newton （IC-GN） algorithm. The performance of this strategy, including accuracy, precision, and computation efficiency, is evaluated through a theoretical and experimental study, using the one with widely employed bicubic interpolation as a benchmark. The global look-up table strategy with cubic B-spline interpolation improves significantly the accuracy of the IC-GN algorithm-based DIC method compared with the one using the bicubic interpolation, at a trivial price of computation efficiency.

DEFICIENT CUBIC SPLINES WITH AVERAGE SLOPE MATCHING

We obtain a deficient cubic spline function which matches the functions with certain area matching oner a greater mesh intervals, and also provides a greater flexibility in replacing area matching as interpolation. We also study their convergence properties to the interpolating functions.

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 efficiency and precision in solving the Poisson equation.

Non-overshooting and Non-undershooting Cubic Spline Interpolation for Empirical Mode Decomposition

To suppress the overshoots and undershoots in the envelope fitting for empirical mode decomposition （EMD）, an alternative cubic spline interpolation method without overshooting and undershooting is proposed. On the basis of the derived slope constraints of knots of a non-overshooting and non-undershooting cubic interpolant, together with ＂not-a-knot＂ conditions the cubic spline interpolants are constructed by replacing the requirement for equal second order derivatives at every knot with Brodlie＇ s derivative formula. Analysis and simulation experiments show that this approach can effectively avoid generating new extrema, shifting or exaggerating the existing ones in a signal, and thus significantly improve the decomposition performance of EMD.

Fuzzy clustering of time series gene expression data with cubic-spline

Data clustering techniques have been applied to ex- tract information from gene expression data for two decades. A large volume of novel clustering algorithms have been developed and achieved great success. However, due to the diverse structures and intensive noise, there is no reliable clustering approach can be applied to all gene expression data. In this paper, we aim to the feature of high noise and propose a cubic smoothing spline fitted for the time course ex- pression profile, by which noise can be filtered and then groups genes into clusters by applying fuzzy c-means clustering on the resulting splines (FCMS). The discrete values of radius of curvature are used to compute the similarity between spline curves. Results on gene expression data show that the FCMS has better performance than the original fuzzy c-means on reliability and noise robustness.

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.

Generalized fairing algorithm of parametric cubic splines

Kjellander has reported an algorithm for fairing uniform parametric cubic splines. Poliakoff extended Kjellander’s algorithm to non-uniform case. However, they merely changed the bad point’s position, and neglected the smoothing of tangent at bad point. In this paper, we present a fairing algorithm that both changed point’s position and its corresponding tangent vector. The new algorithm possesses the minimum property of energy. We also proved Poliakoff’s fairing algorithm is a deduction of our fairing algorithm. Several fairing examples are given in this paper.

AN INTERPOLATION METHOD BY BIVARIATE CUBIC SPLINES WITH C^2 -JOIN ON TYPE-II TRIANGULARS AT A RECTANGULAR DOMAIN

In this paper, an interpolating method for bivariate cubic splines with C2-join on type-II triangular at a rectangular domain is given, and the approximation degree, inter- polating existence and uniqueness of the cubic splines are studied.

Cubic Spline Interpolation on a Class of Triangulations

In this paper, we consider spaces of cubic C^1-spline on a class of triangulations. By using the inductive algorithm, the posed Lagrange interpolation sets are constructed for cubic spline space. It is shown that the class of triangulations considered in this paper are nonsingular for S1/3 spaces. Moreover, the dimensions of those spaces exactly equal to L. L. Schuraaker＇s low bounds of the dimensions. At the end of this paper, we present an approach to construct triangulations from any scattered planar points, which ensures that the obtained triangulations for S1/3 space are nonsingular.

A simple piecewise cubic spline method for approximation of highly nonlinear data

Approximation methods are used in the analysis and prediction of data, especially laboratory data, in engineering projects. These methods are usually linear and are obtained by least-square-error approaches. There are many problems in which linear models cannot be applied. Because of that there are logarithmic, exponential and polynomial curve-fitting models. These nonlinear models have a limited application in engineering problems. The variation of most data is such that the nonlinearity cannot be approximated by the above approaches. These methods are also not applicable when there is a large amount of data. For these reasons, a method of piecewise cubic spline approximation has been developed. The model presented here is capable of following the local nonuniformity of data in order to obtain a good fit of a curve to the data. There is C1 continuity at the limits of the piecewise elements. The model is tested and examined with four problems here. The results show that the model can approximate highly nonlinear data efficiently.

Improvement of Orbit Prediction Algorithm for Spacecraft Through Simplified Precession-Nutation Model Using Cubic Spline Interpolation Method

For the on-orbit flight missions,the model of orbit prediction is critical for the tasks with high accuracy requirement and limited computing resources of spacecraft.The precession-nutation model,as the main part of extended orbit prediction,affects the efficiency and accuracy of on-board operation.In this paper,the previous research about the conversion between the Geocentric Celestial Reference System and International Terrestrial Reference System is briefly summarized,and a practical concise precession-nutation model is proposed for coordinate transformation computation based on Celestial Intermediate Pole(CIP).The idea that simplifying the CIP-based model with interpolation method is driven by characteristics of precession-nutation parameters changing with time.A cubic spline interpolation algorithm is applied to obtain the required CIP coordinates and Celestial Intermediate Origin locator.The complete precession nutation model containing more than 4000 parameters is simplified to the calculation of a cubic polynomial,which greatly reduces the computational load.In addition,for evaluating the actual performance,an orbit propagator is built with the proposed simplified precession-nutationmodel.Compared with the orbit prediction results obtained by the truncated series of IAU2000/2006 precession-nutation model,the simplified precession-nutation model with cubic spline interpolation can significantly improve the accuracy of orbit prediction,which implicates great practical application value in further on-orbit missions of spacecraft.

Stability Analysis of Spatial Cubic Spline Geometric Nonlinear Beam Element Considering the Second-Order Effect

To analyze the stability problem of spatial beam structure more accurately, a spatial cubic spline geometric nonlinear beam dement was proposed considering the seeond-order effect. The deformation field was built with cubic spline function, and its curvature degree of freedom （DOF） was eliminated by static condensation method. Then we got the geometric nonlinear stiffness matrix of the new spatial two.node Euler-Bernouili beam dement. Several examples proved calculation accuracy of the critical load by meshing a bar to one element using the method of this paper was equivalent to mesh a bar to 3 or 4 traditional nonlinear beam dements.

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 ｜×｜ε^ 2, in g-insensitive support vector regression. Theoretical analysis shows that Sε^2 -function is better than pε^2 -function in properties, and the approximation accuracy of the proposed smoothing function is two order higher than that of classical pε^2 -function. The experimental data shows the efficiency of the new approach.

Combining Cubic Spline Interpolation and Fast Fourier Transform to Extend Measuring Range of Reflectometry

The reflectometry is a common method used to measure the thickness of thin films. Using a conventional method,its measurable range is limited due to the low resolution of the current spectrometer embedded in the reflectometer.We present a simple method, using cubic spline interpolation to resample the spectrum with a high resolution,to extend the measurable transparent film thickness. A large measuring range up to 385 m in optical thickness is achieved with the commonly used system. The numerical calculation and experimental results demonstrate that using the FFT method combined with cubic spline interpolation resampling in reflectrometry, a simple,easy-to-operate, economic measuring system can be achieved with high measuring accuracy and replicability.