Spline Interpolation Function Research Articles

On the Parameter-Uniform Convergence of Exponential Spline Interpolation in the Presence of a Boundary Layer

The paper is concerned with the problem of generalized spline interpolation of functions having large-gradient regions. Splines of the class C2, represented on each interval of the grid by the sum of a second-degree polynomial and a boundary layer function, are considered. The existence and uniqueness of the interpolation L-spline are proven, and asymptotically exact two-sided error estimates for the class of functions with an exponential boundary layer are obtained. It is established that the cubic and parabolic interpolation splines are limiting for the solution of the given problem. The results of numerical experiments are presented.

The approximation of generalized Log-aesthetic curves using Quintic Bezier curves

Generalized Log Aesthetic Curve segments (GLAC) are aesthetic curves that have monotonic curvature profile and hence they are considered fair. In the field of Computer-Aided Design (CAD), there exists a demand to construct fair curves for various design intent. However, we cannot implement GLAC in CAD system partly due to its transcendental form. A viable solution is to approximate GLACs using a quintic polynomial curve in the form of Bezier using curvature error measure. The problem of this approach is that it requires a formidable size of computations due to arc length reparametrization. In this paper, we introduce a new method of calculating curvature error measure using natural spline interpolation function to minimize computation effort while preserving the accuracy. The final section shows numerical examples depicting the proposed approximation of two types of the GLAC, which clearly indicate the efficiency of proposed method.

How should we interpret retrospective blood glucose measurements? Sampling and Interpolation

This study investigates blood glucose (BG) measurement interpolation techniques to represent intermediate BG dynamics, and the effect resampling of retrospective BG data has on key glycemic control (GC) performance results. Many GC protocols in the ICU have varying BG measurement intervals with gaps ranging from 0.5 to 4 hrs. Sparse data poses problems in model fitting techniques and GC performance comparisons, and thus interpolation is required to assume a continuous solution.Retrospective data from SPRINT in the Christchurch Hospital Intensive Care Unit (ICU) (2005-2007) was used to analyze various interpolation techniques. Piece-wise linear, spline and cubic interpolation functions, which force lines through data, as well as 1st and 2nd Order B-spline basis functions, used to identify the data, are investigated. Dense data was thinned to increase sparsity and obtain measurements (Hidden measurements) for comparison after interpolation. All of the piece-wise functions performed considerably better than the fitted interpolation functions. Linear piece-wise interpolation performed the best having a mean RMSE 0.39 mmol/L, within 2 standard deviations of the BG sensor error.The effect of minutely vs hourly sampling of the interpolated trace on key GC performance statistics was investigated using the retrospective data received from STAR GC in the Christchurch Hospital Intensive Care Unit (ICU), New Zealand (2011-2015). Minutely sampled BG resulted in significantly different key GC performance when compared to raw sparse BG measurements. Linear piece-wise interpolation provides the best estimate of intermediate BG dynamics and all analyses comparing GC protocol performance should use minutely linearly interpolated BG data.

Parabolic spline interpolation for functions with large gradient in the boundary layer

We consider the problem of Subbotin’s parabolic spline interpolation for functions with large gradient domains. In the case of the common piecewise uniform Shishkin’s mesh we obtain two-sided accuracy estimates for the class of functions with exponential boundary layer. The spline interpolation accuracy estimates are not uniform in a small parameter, while the error itself can grow unboundedly as the small parameter vanishes and the number N of nodes remains fixed. We include the results of some simulations.

On the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer

The problem of parabolic spline interpolation, according to Subbotin, of functions with large gradients in the boundary layer is considered. In the case of a uniform grid it is proved and in the case of the Shishkin mesh it is experimentally shown that with a parabolic spline interpolation of functions with large gradients in the exponential boundary layer, the error can unrestrictedly increase when the small parameter tends to zero and the number of grid nodes is fixed. An approximation process using parabolic splines with defect 1 is proposed; the error estimates are found to be uniform in the small parameter.

Solution identification and quantitative analysis of fiber-capacitive drop analyzer based on multivariate statistical methods.

A fiber-capacitive drop analyzer is an instrument which monitors a growing droplet to produce a capacitive opto-tensiotrace (COT). Each COT is an integration of fiber light intensity signals and capacitance signals and can reflect the unique physicochemical property of a liquid. In this study, we propose a solution analytical and concentration quantitative method based on multivariate statistical methods. Eight characteristic values are extracted from each COT. A series of COT characteristic values of training solutions at different concentrations compose a data library of this kind of solution. A two-stage linear discriminant analysis is applied to analyze different solution libraries and establish discriminant functions. Test solutions can be discriminated by these functions. After determining the variety of test solutions, Spearman correlation test and principal components analysis are used to filter and reduce dimensions of eight characteristic values, producing a new representative parameter. A cubic spline interpolation function is built between the parameters and concentrations, based on which we can calculate the concentration of the test solution. Methanol, ethanol, n-propanol, and saline solutions are taken as experimental subjects in this paper. For each solution, nine or ten different concentrations are chosen to be the standard library, and the other two concentrations compose the test group. By using the methods mentioned above, all eight test solutions are correctly identified and the average relative error of quantitative analysis is 1.11%. The method proposed is feasible which enlarges the applicable scope of recognizing liquids based on the COT and improves the concentration quantitative precision, as well.

Cubic spline interpolation of functions with high gradients in boundary layers

The cubic spline interpolation of grid functions with high-gradient regions is considered. Uniform meshes are proved to be inefficient for this purpose. In the case of widely applied piecewise uniform Shishkin meshes, asymptotically sharp two-sided error estimates are obtained in the class of functions with an exponential boundary layer. It is proved that the error estimates of traditional spline interpolation are not uniform with respect to a small parameter, and the error can increase indefinitely as the small parameter tends to zero, while the number of nodes N is fixed. A modified cubic interpolation spline is proposed, for which O((ln N/N)4) error estimates that are uniform with respect to the small parameter are obtained.

Monitoring the Shape of Satellite Wing Frame Using FBG Sensors in High Electronic Noise, Vacuum, and −196 °C Environment

In a normal environment on Earth, there are several methods for obtaining the deflection curve of a satellite wing frame, such as the laser method and the electronic resistance strain gauge method. However, none of them are suitable for the outer space environment. Therefore, developing a reliable sensing solution for a high electronic noise, vacuum, and extremely low-temperature environment may be a very challenging task. In this paper, a new fiber Bragg grating (FBG) sensors method is proposed to meet this harsh environment need. First, hundreds of FBG sensors are pasted onto the surface of a satellite wing frame to detect the strain data. Second, the cubic spline interpolation function, quadratic integral function, and quantic polynomial function are used for the data processing to obtain the shape of the satellite wing frame. Finally, a satellite wing frame is placed in a heat sink to conduct a shape monitoring experiment. Furthermore, a verification experiment in a normal environment shows the reliability of the FBG sensors method. Because the FBG sensors and the supporting equipment are available in the market, this method is a feasible solution for shape monitoring in outer space.

Optimisation and fabrication by 3D printing of a ceiling antenna for communication applications

This paper presents a new ceiling antenna optimised by applying differential evolution (DE) and fabricated by 3D printing. The frequency band of the antenna ranges from 0.75 GHz to 3 GHz. The geometric structure refers to volcano smoke antenna, which contains a teardrop as the radiating element and a curved ground plane. Curves in structure are determined by cubic spline interpolation function. The antenna design problem is converted into a constrained optimisation problem (COP), which is solved by DE next. Some optimised antennas are found, and one of them is presented in this paper. The measured result basically meets the antenna requirement, which means ceiling antenna proposed in this paper could work for 2G, 3G and 4G LTE bands and also for WiFi frequency bands theoretically. For such this complex-shaped three-dimensional antenna, we choose 3D printing to fabricate. The measured result almost matches the simulation.

Area PEc Neurons Use a Multiphasic Pattern of Activity to Signal the Spatial Properties of Optic Flow.

The cortical representation of visual perception requires the integration of several-signal processing distributed across many cortical areas, but the neural substrates of such perception are largely unknown. The type of firing pattern exhibited by single neurons is an important indicator of dynamic circuitry within or across cortical areas. Neurons in area PEc are involved in the spatial mapping of the visual field; thus, we sought to analyze the firing pattern of activity of PEc optic flow neurons to shed some light on the cortical processing of visual signals. We quantified the firing activity of 152 optic flow neurons using a spline interpolation function, which allowed determining onset, end, and latency of each neuronal response. We found that many PEc neurons showed multiphasic activity, which is strictly related to the position of the eye and to the position of the focus of expansion (FOE) of the flow field. PEc neurons showed a multiphasic activity comprised of excitatory phases interspersed with inhibitory pauses. This phasic pattern seems to be a very efficient way to signal the spatial location of visual stimuli, given that the same neuron sends different firing patterns according to a specific combination of FOE/eye position.

On the parabolic spline-interpolation of functions with a boundary layer

A subject of the article is a parabolic spline-interpolation of functions having high gradient domains. Uniform grids’ inefficiency was shown. The piecewise-uniform grids, concentrated in the boundary layer were scrutinized. There were announced asymptotically exact estimates on a class of functions with an exponential boundary layer in consideration with the traditional parabolic spline interpolation. There were obtained results showing non-uniform in small parameter estimates. A probable divergence of interpolation processes was extrapolated. A modified parabolic spline with obtained uniform in small parameter interpolation error estimations was introduced. The paper contains results of numerical experiments, which strengthen theoretical estimates.

Cubic and Hermite splines for VTK

A cubic spline is a spline where each curve is defined by a third-order polynomial, while a Hermite spline has each polynomial specified in Hermite form, being computed using tangent information as well as the position of the points. We propose two new classes for VTK, vtkCubicSpline and vtkHermiteSpline, which compute interpolating splines using a Cubic and a Hermite Spline Interpolation function, respectively. We also propose two new auxiliary classes, vtkParametricCubicSpline and vtkParametricHermiteSpline, that create parametric functions for the 1D interpolating aforementioned splines.

The Cubic α-Catmull-Rom Spline

By extending the definition interval of the standard cubic Catmull-Rom spline basis functions from [0,1] to [0,α], a class of cubic Catmull-Rom spline basis functions with a shape parameter α, named cubic α-Catmull-Rom spline basis functions, is constructed. Then, the corresponding cubic α-Catmull-Rom spline curves are generated based on the introduced basis functions. The cubic α-Catmull-Rom spline curves not only have the same properties as the standard cubic Catmull-Rom spline curves, but also can be adjusted by altering the value of the shape parameter α even if the control points are fixed. Furthermore, the cubic α-Catmull-Rom spline interpolation function is discussed, and a method for determining the optimal interpolation function is presented.

Material optimization of functionally graded heterogeneous cylinder for wave propagation

In this paper, optimization of volume fraction distribution in a thick hollow heterogeneous cylinder subjected to impulsive internal pressure is considered. Dynamic behavior and wave propagation are considered in radial and axial directions. Volume fractions of constituent materials on a finite number of design points are taken as design variables and the volume fractions at any arbitrary point in the cylinder are obtained via cubic spline interpolation functions. The objectives are to minimize the amplitude of stress waves propagating through the structure during a specified time interval, while the total mass of the structure is also minimized. Minimizing the displacement amplitude of the outer surface of the cylinder will also be considered as another objective function. Multi-objective Genetic Algorithm jointed with interior penalty-function method for implementing constraints is effectively employed to find the global solution of the optimization problem. Obtained results indicates that by using the mentioned objective functions, considerably more efficient usage of materials can be achieved compared with the common power law volume fraction distribution. Based on our results, the proposed methodology provides a framework for designing functionally graded structures with optimum material tailoring.

A motion control method for face hobbing on CNC hypoid generator

In this paper, the relative motion relation was derivated among the virtual cradle, generating gear, cutter and workpiece on the CNC hypoid generator. A directly axis coordinate solving method of CNC hypoid generator was proposed by taking cutter axis vector as control objectives to transform the attitude of whole cradle-type generator. The method didn't solve complex nonlinear equations. We also provide a motion control solution for face-hobbing on the CNC hypoid generator using Siemens 840Dsl NC system.In this method, the online real-time programming is achieved, because of the index motion and generating motion of machining process are separated, and spline interpolation and electronic gearbox function are adopted. The tooth form equation which is determined by the motion control model of CNC hypoid generator is derived. The calculation example of a gear sets shows that the theoretical tooth form of the epicycloid hypoid gear machined by such motion control method is consistent with the standard tooth form computed by Gleason CAGE Software. With NC machining simulation software VERICUT, virtual NC hypoid gear cutting machine is established, which realizes face hobbing simulation of such gear sets. At last, a face hobbing experiment is conducted on YK2235 CNC hypoid generator manufactured by Tianjin No. 1 Machine Tool Works, and such face hobbing motion control model is verified as correct through tooth form measurement and rolling test.

Optimal material tailoring of 2D heterogeneous cylinder for a prescribed temperature field in transient heat conduction

In this paper, optimization of volume fraction distribution in a finite length thick hollow cylinder made of heterogeneous material is considered. The cylinder has two-directional functionally graded constituent and subjected to transient thermal loading. Two-dimensional heat conduction in the cylinder is considered and variation of temperature with time as well as temperature distribution through the cylinder is investigated. A graded finite element method combined with direct time integration method is used to model the structure and solve time-dependent equations. Volume fractions of constituent materials on a finite number of design points are taken as design variables and the volume fractions at any arbitrary point in the cylinder are obtained via cubic spline interpolation functions. The objective is to minimize the difference between the actual distribution of the temperature through the structure and a desired target distribution after a prescribed time while the total mass of the structure is also minimized. Multi-objective Genetic Algorithm jointed with interior penalty-function method for implementing constraints is effectively employed to find the global solution of the optimization problem. Obtained results indicate that by using the uniform distribution of temperature and minimum mass as objective functions, considerably more effective usage of materials can be achieved compared with the common power law volume fraction distribution. The proposed methodology provides a framework for designing functionally graded structures with optimum material tailoring for a prescribed field variable problem.

Material distribution optimization of 2D heterogeneous cylinder under thermo-mechanical loading

In this paper optimization of volume fraction distribution in a thick hollow cylinder with finite length made of two-dimensional functionally graded material (2D-FGM) and subjected to steady state thermal and mechanical loadings is considered. The finite element method with graded material properties within each element (graded finite elements) is used to model the structure. Volume fractions of constituent materials on a finite number of design points are taken as design variables and the volume fractions at any arbitrary point in the cylinder are obtained via cubic spline interpolation functions. The objective function selected as having the normalized effective stress equal to one at all points that leads to a uniform stress distribution in the structure. Genetic Algorithm jointed with interior penalty-function method for implementing constraints is effectively employed to find the global solution of the optimization problem. Obtained results indicates that by using the uniform distribution of normalized effective stress as objective function, considerably more efficient usage of materials can be achieved compared with the power law volume fraction distribution. Also considering uniform distribution of safety factor as design criteria instead of minimizing peak effective stress affects remarkably the optimum volume fractions.

Application of Moments in Qualitative Analysis on Borehole Trajectory Curving

There are many calculation methods of horizontal borehole trajectory, such as cubic spline interpolation function which is based on three-moment equation. By calculating the moments of deviation and azimuth from cubic spline interpolation function respectively, it is easy obtain functions of deviation and azimuth change rate and average borehole curvature to analysis on curving of borehole trajectory. So it is one of the most common methods. However, the moments are always be neglected. Moments of deviation and azimuth, which could be used to qualitatively estimate borehole trajectory curving, are first proposed tentatively in this paper. Synthetically employing other useful parameters of estimation borehole trajectory curving, tens of high-angle directional and horizontal wells’ drilling data is computed and analyzed. Contrast to the variation characteristics of deviation and azimuth change rate, the deviation and azimuth moments have the similar change tendency respectively. The results of this study signify that the proposed parameters could provide some reference for assessment on borehole trajectory curving.

Climate Change Effects on Long-term World-crop Production: Incorporating a Crop Model into Long-term Yield Estimates

Annual crop yield forecasts are necessary for analysis because evaluating climate–change impacts on world food markets requires supply–response functions, including output prices of the prior year. This research was undertaken to develop yield–response functions of the world food model to evaluate climate–change effects by incorporating a crop model into the yield–trend function. Yield–trend functions of rice, wheat, maize, and soybeans were obtained by estimating logistic functions or linear functions with a logarithmic time–trend term and climate variables. Furthermore, temperature and solar–radiation elasticities of yields were calculated using a crop model of the FAO and IIASA. The functions of the maximum rate of gross biomass production and the maximum net rate of CO2 exchange of leaves in the crop model were modified by introducing cubic spline interpolation and logistic functions. Smoothing these two functions alleviates drastic changes, but reveals small changes in the elasticities of crop yields compared to the kinked functions and these more realistic elasticities can improve the evaluation accuracy of climate–change impacts on crop supply and demand. These variable elasticities of temperature and solar–radiation were inserted into the yield–trend functions, whereupon the global effects of changes in climate variables, including rainfall, were analyzed. The changes in yields obtained using climate variables of two of the four RCP scenarios were compared with the baseline, for which climate variables were fixed. Results of trend analyses show that yields of rice, wheat, maize, and soybeans under RCP8.5 are lower than those under RCP2.6, except for wheat in China. Results of geographical analysis show that climate change can be expected to affect wheat and maize productions in low–latitude countries. Furthermore, results suggest that climate change will depress rice production in sub–Saharan African countries in the 2040s.

A Variational Principle Approach for Vibration of Non-Uniform Nanocantilever Using Nonlocal Elasticity Theory

In this article free vibration of a nanocantilever with nonuniform cross section is studied using nonlocal elasticity within the scope of continuum mechanics. Based on an exact variational principle approach, an asymptotic partial differential equation of infinite order is derived along with the corresponding boundary conditions. These equations involve essential higher-order differential terms which, if neglected, have previously led to some rather intriguing observations and conclusions. A reduced sixth-order differential equation is, then, solved for a nanocantilever by applying finite element method using quintic spline interpolation functions. The finite element model developed will be of practical use and reference to physicists and engineers, alike, in the analysis and design of more complicated nanostructures.The paper also resolves some of the strange observations from similar studies like: (i) the non-existence of real eigenvalues (for the first and second modes) for nonlocal parameter (e0a/L)>0.62; and (ii) the existence of a height ratio where frequency becomes independent of size effects, this ratio was defined as the ‘critical height ratio’. It is clear from this study that these observations were a result of using a governing equation and boundary conditions which were not exact thus leading to erroneous observations and conclusions.