Form Of Spline Research Articles

Shape-Preserving Techniques with Fractional Order C1 Linear Trigonometric Spline

Generating a smooth curve or surface is the prime goal of interpolation. If the given interplant reflects the exact shape of the data, it would be an added advantage. In this study, a general form of Fractional Order Linear Trigonometric Spline (FOLTS) is proposed with continuity. A method for modeling imperative curves has been developed with the goal of using it in a variety of engineering, scientific, and design fields. The primary goal of this research is to combine linear trigonometric spline with Caputo derivative in order to obtain better control over the curve in each sub-interval. The curve can be manipulated locally with the help of one degree of freedom involved in the form of fractional parameters. To describe shape-preserving interpolation applications, two more parameters namely “s” and “t” have been developed. These parameters ensure that the interpolated piecewise curve is able to satisfy shape-preserving properties. The study also compares FOLTS with quadratic Lagrange and cubic spline.

Spline Search for Slip Surfaces in 3D Slopes

A novel method involving the transformation of spline surfaces is introduced to search for the critical slip surface in a three-dimensional (3D) slope, which corresponds to the minimum limit equilibrium method factor of safety for overall slope stability. A slipping surface in a slope can be represented as the intersection of any continuous geometrical entity with the slope topography, over which the mass of sliding soil is discretized to solve for the factor of safety satisfying given equilibrium conditions. Traditionally, many researchers have searched for a critical ellipsoidal or spherical surface, or surfaces formed using other simple shapes. However, the critical slip surface in complicated cases, for example, a landslide, is seldom purely ellipsoidal or spherical, which leads to overestimations of the true factor of safety in a slope. To provide greater flexibility for transforming the shape of the slip surfaces during a global search, the geometry representing the slip surface is assumed to be in the form of a nonuniform rational basis spline (NURBS) surface in this paper. The proposed method involves varying the parameters of a parametric exponential function, which spawns control points within its domain to create NURBS surfaces. The parameters in the exponential function are varied to transform the slipping surface using a metaheuristic search algorithm, such as particle swarm optimization. A major advantage of the proposed method is that the final spline surface in the search is formulated such that it can then be locally optimized using surface altering optimization methods by adjusting the locations of its control points.

NEW INFORMATION OPERATORS IN PROBLEMS OF NUMERICAL INTEGRATION OF FUNCTIONS OF THREE VARIABLES

The rapid development and implementation of the latest information technologies in many fields of science and technology contribute to the rise of new methods in mathematical modeling of systems and processes. In particular, new mathematical theories have appeared that can be effectively used for building and improvement of existing mathematical models of various phenomena and objects. One of these theories is the theory of new information operators, authored by the Laureate of the State Prize of Ukraine in Science and Technology, doctor of physical and mathematical sciences, professor (думаю, это излишняя информация в аннотации) O. M. Lytvyn. New information operators have found their application in the digital processing of signals and images, namely in the numerical integration of rapidly oscillating functions of many variables. Cubature formulas were built that are optimal in the order of accuracy and use the value of the non-oscillating factor of the integrand function not only at points, but also on planes or lines. The article demonstrates the application of new information operators to the numerical integration of functions of many variables, namely, the issue of approximate calculation of the integral from functions of three variables is considered in the case when the information about the function is given by its traces on the lines. The cubature formula uses an interlineation operator built on the basis of an interflatation operator with auxiliary functions in the form of piecewise constant splines. An estimate of the approximation error of the proposed cubature formula on the class of differentiable functions was obtained. The results of the calculation experiment in the computer mathematics system Mathcad are presented. Numerical calculations confirm the theoretical statements of the study.

Investigating the Regional Concentrations-Response Relations between PM2.5 and daily mortality in Japan

BACKGROUND: The concentration-response (CR) functions of PM2.5 are sensitive to various local factors, such as meteorological conditions, PM sources and components, and population sensitivity to PM. However, the shape of local CR functions remains unclear in many regions in Japan. This research investigates the association between PM2.5 and cause-, age-, and location-specific mortality to identify susceptible subpopulations and regions using a nationwide dataset. METHODS: A two-stage analysis was performed to quantify the associations of PM2.5 with mortality. In the first stage, regional differences in the effects of PM were investigated. A time-series quasi-Poisson model, combined with a distributed lag non-linear model and a flexible functional form of splines, was applied to evaluate the location-specific associations between PM2.5 and mortality. Various covariates were controlled, including weather variables, day of the week, and seasonal and long-term time trends. Stratified analyses were performed by age groups and death cause categories. We used a random-effects meta-analytic model to estimate the pooled cumulative association in the second stage. Potential reasons for the regional heterogeneity of PM2.5 effects were further explored through random-effects meta-regression analyses, separately carried out for multiple variables as meta-predictors. RESULTS: Geographic variations in the concentration of PM2.5 were observed in Japan. An increase in PM2.5 concentration was associated with an increase in excess risk of mortality, and the magnitude of the corresponding associations appeared to vary by region in Japan. CONCLUSIONS: The regional risks of premature mortality from exposure to PM2.5 were investigated. We identified vulnerable regions and susceptible subpopulations. The disparity in the association of mortality risks and PM2.5 was apparent between geographical regions, indicating the importance of developing area-specific strategies and local risk prevention plans that can serve as a basis for policymaking to reduce the corresponding health effects.

An Algorithm for Restoring a Function from Different Functionals for Predicting Rare Events in the Economy

This paper aims to restore some parameters of functionals using cubic splines to forecast rare events in finance and economics. The article considers the mathematical method for recovering an unknown function from many different functionals, such as the value of a function, the value of its first derivative, second derivative, as well as a definite integral over a certain interval. Moreover, all observations can occur with an error. Therefore, the author uses a method of recovering a function from different functionals observed with an error. The function is restored in the form of a cubic spline, which has a value-second derivative representation. The optimization problem consists in minimizing several sums of squares of the deviation at once, for ordinary values, for the first derivatives, for the second derivatives, for integrals, and for roughness penalty. For greater flexibility, weights have been introduced both for each group of observations and for each individual observation separately. The article shows in detail how the elements of each corresponding matrix are filled in. The appendix provides an implementation of the method as a FunctionalSmoothingSpline function in R language. Examples of using the method for the analysis and forecasting of rare (discrete) events in the economy are given. Formulas for calculating the cross-validation score CV (α) for the automatic procedure for determining the smoothing parameter α from the data in our problem of recovering a function by many functionals are shown. The paper concludes that the presented method makes it possible to analyze and predict rare events, which will allow you to prepare for such future events, get some benefit from this, or reduce possible risks or losses.

Concentration-Response Functions as an Essence of the Results from Lags.

Among various aspects of environmental epidemiology, one is to assess the relationships between ambient air pollution and health outcomes. The goal of this work is to estimate the associations in the form of the parametric concentration–response functions (C-RF). Various forms of the C-RFs are proposed in this short-term health effect study. Emergency department (ED) visits for all respiratory health problems are analyzed as an illustrative example. A case-crossover (CC) technique is applied as a study design. Daily cases are organized as daily counts by the same day of the week in one common month. A conditional Poisson regression is used in the constructed statistical models. Temperature and relative humidity are included in the statistical models in the form of natural splines. Ground-level ozone concentration is considered an exposure. Ozone concentration values are transformed and submitted to the statistical models. The parameters of the transformation are determined by using the goodness of fit criterion. Counts of ED visits are analyzed in relation to a sequence of lagged exposure to ozone. The C-RF shapes are constructed for each individual lag. In a final step, the set of the estimated C-RF shapes is used to create a pooled C-RF shape. The results are positive and statistically significant for nine lagged exposures, from 0 to 8 days. The following relative risks (RR) were estimated from the constructed C-RFs at 30 ppb concentration of ozone: RR = 1.0531 (95% confidence interval: 1.0231, 1.0718), 1.0462 (1.0253, 1.0677), and 1.0387, (1.0240, 1.0531), realizing the CC method, CC method + transformation, and CC method + flexible transformation, respectively. The pooled C-RF shape gives a summary of the associations between ED visits for respiratory conditions and ambient ozone. The estimated shapes indicate lower air health effects than the standard CC methods. Among three considered statistical models, the CC method + flexible transformation is the most appropriate to use according to the goodness of fit criterion.

Investigation of The Effect of Design Variables on Slip Assembly: Spline Module and Slip Length

The driveshaft which transmits the power from the engine to the wheels in the motor vehicles is one of the vital elements in the driveline. Driveshafts have two basic motions depending on the road conditions, angular and axial movements. Angular movement is provided by the universal joints while the axial movement is provided by the slip assembly which involves a group of sliding components. Therefore, the slip assembly is an important part of the drive shaft. Basically, a slip assembly consists of two parts which are connected to each other by means of their spline forms. The parts can move back and forth longitudinally while they transmit the torque thanks to the connection interface of their spline forms. And so, they can slide and adjust the drive shaft lengths due to the position of the axles under road conditions.
 During the motor vehicle movement, slip assembly are subjected to torsion like the other components on the driveshaft. In this context, spline size and magnitude of the length compensation are highly important in the design process of the slip assembly. In this study, the effect of the spline size and the slip length have been investigated for the yoke shaft design, by using analytical and numerical methods in terms of shear stress. It has been observed that the analytical and the numerical methods give the similar results in shear stress on the pitch diameter of the spline. Thus, the analytical method can be preferred instead of the finite element analysis (FEA), especially considering that the FEA is a time consuming method compared to the analytical method in the design process.

Determination of parameters for the formation process of rare events in the economy for their subsequent forecasting

The article presents a method for determining unknown parameters of the process that forms rare events in the economy.The idea behind the study of rare events in economy is to consider these events not just from a statistical point of view, but from the point of view of the processes that form these events, moreover, as well as a process can be an arbitrary algorithm. Such an event generation process will use parameters, which can be static or dynamic. For example, if we consider the consumption process, which forms discrete purchases of an uncontrolled customer, then such parameters can be the maximum stock and a dynamically changing consumption rate. In general, the process can be arbitrary and possess various parameters. The task is to determine these parameters of an unknown process obtaining only a sample of rare events. The idea of the method is to minimize the loss function, which is determined based on the differences between the events generated during the operation of the process and events from the initial sample of observations. Each event, in addition to the time of occurrence, is also characterized by additional information, for example, the purchase volume. We are trying to find out such parameters of such a process that would allow us to get a very similar sample of events. The dynamic parameters of the process are set in the form of cubic splines of a special structure. For an unambiguous determination of each dynamic parameter, a roughness penalty of the corresponding splines is introduced into the objective function. An example of a process and its structure of parameters to be determined is shown. Optimization is performed numerically, based on the Nelder–Mead algorithm, which runs on a grid to determine the global optimum. The process parameters are determined in steps, at the beginning just to get a few events, then the next events. That allows one large optimization task to be divided into a sequence of simple tasks, this significantly reduces the overall complexity. An assumption is described that must be fulfilled for such a technique to be valid. An example of determining unknown parameters is considered on the example of the consumption process. After determining the process parameters, one can proceed to extrapolation of parameters and forecast future events.

ИДЕНТИФИКАЦИЯ УЯЗВИМОСТИ СОСТОЯНИЙ ПРИРОДНО-ТЕХНИЧЕСКИХ ОБЪЕКТОВ НА ОСНОВЕ ЛИНЕЙНОЙ СПЛАЙН-ИНТЕРПОЛЯЦИИ ИНТЕНСИВНОСТИ ИНТЕРФЕЙСНОГО ТРАФИКА

The method of application of spline interpolation in solving the problems of identification of abnormal states (A-events) in information data flows and classification of the specified events in the control of natural and technical objects (PTO) is considered. The approach is based on the representation of the intensity of interface traffic by piecewise linear splines and implemented using a modeling stand. At the first stage, descriptions are generated and formed in the form of linear splines representing the states of controlled objects, one of which is subject to external disturbance. At the second stage, the generated descriptions of splines are used to assess discrepancies between the studied distributions and the influence of a number of factors on the reliability of decisions made using probabilistic modeling methods in the Anylogic environment.

КОМПЕНСАЦИЯ ВЛИЯНИЯ ТЕМПЕРАТУРЫ В АВИАЦИОННЫХ ПЬЕЗОРЕЗИСТИВНЫХ ДАТЧИКАХ ДАВЛЕНИЯ

Currently, piezoresistive pressure sensors (PDS) find expanded applications in various microelectronicdevices used in aviation technology. The behavior of the electrical signal of suchPDS mainly depends on the ambient temperature. It is known that the temperature drift of the PDSoutput signal is influenced by various factors: the temperature effect, the dependence of the resistanceof the sensitive element on the concentration of impurities, the dependence of the Young'smodulus of the sensor membrane and substrate materials on temperature, etc. It was found that thepreviously developed analytical calibration model of the sensor output signal, which takes intoaccount the models describing individual temperature effects, does not allow the pressure to bemeasured with the required accuracy in the temperature range characteristic of aviation equipmentfrom –60 C to 140 C. Therefore, conventional polynomial mathematical models are used todescribe the dependence of the PDS output signal on the measured pressure and temperature.The work uses a traditional approach, when the dependence of the output voltage on pressure is represented using a polynomial of relatively low order, and the dependences of the coefficients ofthis polynomial on temperature are also specified by the corresponding polynomials. Unfortunately,the temperature dependences of the coefficients are adequately described only by high-orderpolynomials (at least 7), which complicates the model identification procedure and leads to computationerrors. Therefore, the authors proposed to look for the dependence of the coefficients ontemperature in the form of cubic splines. The paper describes in detail the identification techniqueof the polynomial model under consideration and obtains expressions for correcting the PDS readingswhen measuring pressure in wide temperature ranges. In order to experimentally confirm theefficiency of the proposed method, an intelligent industrial automated system for the calibration oftraffic rules, described in the work, was used. It is shown how it can be used to take experimentaldata to calibrate the sensor readings over a wide temperature range, and the procedure for identifyingthe mathematical model of the pressure sensor required to minimize the cost of its certificationis described. The results of experimental studies of specific pressure sensors used in aviationtechnology are presented.

ESTIMASI MODEL REGRESI SEMIPARAMETRIK SPLINE TRUNCATED MENGGUNAKAN METODE MAXIMUM LIKELIHOOD ESTIMATION (MLE)

Regression modeling with a semiparametric approach is a combination of two approaches, namely the parametric regression approach and the nonparametric regression approach. The semiparametric regression model can be used if the response variable has a known relationship pattern with one or more of the predictor variables used, but with the other predictor variables the relationship pattern cannot be known with certainty. The purpose of this research is to examine the estimation form of the semiparametric spline truncated regression model. Suppose that random error is assumed to be independent, identical, and normally distributed with zero mean and variance , then using this assumption, we can estimate the semiparametric spline truncated regression model using the Maximum Likelihood Estimation (MLE) method. Based on the results, the estimation results of the semiparametric spline truncated regression model were obtained p=(inv(M'M)) M'y

Аппроксимация физического сплайна с большими прогибами

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

Аппроксимация физического сплайна с большими прогибами

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

Soft computing analysis of thermohydraulic enhancement using twisted tapes in a flat-plate solar collector: Sensitivity analysis and multi-objective optimization

Flat-plate solar collectors are a cost-effective technology for water heating in residential buildings. Comprehensive sensitivity analysis and multi-objective optimization of five crucial factors in a flat-plate solar collector equipped with twisted tapes are investigated. The studied factors are the Nusselt number (Nu), friction factor (f), solar collector efficiency (η), thermal performance factor (ξ), and the temperature difference of the heat transfer fluid across the device (ΔT). The simulation of functions is carried out by four soft computing methods, such as the linear and cubic form of multivariate adaptive regression splines (MARS), group method of data handling (GMDH), and multivariate polynomial regression (MPR). The linear form of the MARS algorithm shows the highest accuracy in predicting Nu, f, and η by R2 = 0.99986, 0.9984, and 0.98821, respectively. The Reynolds number (Re) is from 4000 to 14,000, and the twisted tape ratio (Y) is between 0 and 15. The simulated annealing optimization is employed to find the global optima. According to the study, Nu, η, ξ, and ΔT reach their maximum when f is minimum, which occurs for Y = 5 and Re in the 13,200 to 13,400 range for different global solar radiation levels. The maximum temperature difference happens at noon, about 6.54 K at the Re = 5362.5 and Y = 5 with the collector surface area by 2 m2. When the appropriate working conditions are met so that these factors are simultaneously optimal, ΔT, hence the efficiency, can be increased.

Roller vibroprotection mechanism with a quasi-zero stiffness section

A new numerical method for designing the profile of the support surface for moving the vibration protection mechanism pressure roller for the operator’s seat in road and construction machines has been developed. The roller vibration protection mechanism of the type under consideration provides for the required force characteristic with a quasi-zero stiffness section of a given long length. This advantage of the roller vibration protection mechanism makes it possible to recommend it for application in vibroprotection systems for road and construction machinery operators. Difficulties in creating such mechanisms are caused by the nonlinear nature of the system of algebra-differential equations that describe the relationship between the static force characteristic and the geometry of the mechanism support profile. That is why it is difficult to obtain a solution in the analytical form of a support surface profile and requires a numerical solution. As an example of the efficiency of the proposed method, we obtained a support surface profile for a roller vibration protection mechanism with a given force characteristic in the form of a three-segment spline with 0.1 meters length quasi-zero stiffness section. The developed method lets us use the specified parameter values as the initial data. In the Simscape Multibody application of the MATLAB system, a simulation model of a dynamic vibration protection mechanism system was developed. It allows simulating movements and studying the modes of this mechanism at the design stage. We give an example of the time dependencies of the mechanism points coordinates obtained using a simulation dynamic model for damped free vibrations.

Dynamic spline bas-relief modeling with isogeometric collocation method

In this paper, we propose a novel framework for dynamic spline bas-relief modeling based on isogeometric collocation method. In the proposed method, the bas-relief model is separated into two parts: the base surface and the relief surface; and both of them are represented in spline form. To simplify the construction, the spline bases for representing the base surface and relief surface are in the same spline space. Then the bas-relief surface can be added to the deformed base surface in normal direction directly, which can preserve the shape of bas-relief automatically. In our framework, efficient isogeometric collocation method is used to solve the PDE problems involved in the spline bas-relief construction. In general, the proposed method is efficient, easy to implement, and can produce high-quality bas-relief models. The proposed method has been tested with a set of binary images to show its effectiveness, especially for the Chinese calligraphy.

Reverse-engineering and modeling the 3D passive and active responses of skeletal muscle using a data-driven, non-parametric, spline-based procedure

In this work we present a non-parametric data-driven approach to reverse-engineer and model the 3D passive and active responses of skeletal muscle, applied to tibialis anterior muscle of Wistar rats. We assume a Hill-type additive relation for the stored energy into passive and active contributions. The terms of the stored energy have no upfront assumed shape, nor material parameters. These terms are determined directly from experimental data in spline form solving numerically the functional equations of the tests from which experimental data is available. To characterize typical longitudinal-to-transverse behavior in rodent's muscle, experiments from Morrow et al. (J. Mech. Beh. Biomed. Mater. 2010; 3: 124–129) are employed. Then, the passive and active behaviors of Wistar rats are determined from the experiments of Calvo et al. (J. Bomech. 2010; 43:318–325) and Ramirez et al. (J. Theor. Biol. 2010; 267:546–553). The twitch shape is not assumed, but reverse-engineered from experimental data. The influence of the strain and the stimulus voltage and frequency in the active response, are also modeled. A convenient stimulus power-related variable is proposed to comprise both voltage and frequency dependencies in the active response. Then, the behavior of the resulting muscle model depends only on the muscle strain maintained during isometric tests in the muscle and the stimulus power variable, along the time from initiation of the tetanus state.

Convergence analysis and parity conservation of a new form of a quadratic explicit spline with applications to integral equations

In this study, a new form of a quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve integral equations.

Valve Stiction Model Estimation in Closed-loop Operation

The estimation of valve stiction model is studied. In industrial applications, valve outputs are often not available, the stiction nonlinear block appears before a linear dynamic block which is operation in a closed-loop system. By parameterizing the valve stiction model as a form of cubic splines, an identification method is proposed using a relaxation iteration scheme. Parameter estimation for the linear part is accomplished through a two-stage procedure. Firstly, an unbiased estimation is obtained by the high-order ARX (AutoRegressive exogenous) model. Then the ARX model is reduced to a Box-Jenkins model. The consistency of the method is established. Simulation data sets and real operation data sets are used to illustrate the method.

Elastic guide rail oscillation due to moving concentrated load

This article illustrates the solution of a differential equation describing oscillations of an elastic tensioned guide rail, which consist of string bundle enclosed in an elastic cylindrical shell, while concentrated load, simulated by a material point, moves along it. The oscillatory system is considered in such way that the guide rail supports freely. The existing external and internal forces of resistance to movement of the guide rail are also taken into account. Initial and boundary conditions are zero. In article «A string bend under a moving load», published in the journal «Vestnik BGU. Seriya 1, Fizika. Matematika. Informatika» (2004, No. 1), the deflection of a flexible guide rail under load was obtained by solving an equation with deviating argument. In this article, an algorithm is constructed for finding deflection of an elastic tensioned guide rail in the form of a cubic splines. All the results of calculations are presented in a dimensionless form.