Order Spline Research Articles

The Hermite-Birkhoff Problem and Local Spline Approximation

This paper discusses the use of local spline approximations to solve the Hermite-Birkhoff problem. The solution to a specific problem using polynomial and non-polynomial local splines of the third order of approximation is considered. Here we discuss the case when the values of the function 𝑢(𝑥) and its derivative 𝑢’(𝑥) are given at the nodes of the grid in an alternative way: … , 𝑢(𝑥𝑗), 𝑢′(𝑥𝑗+1), 𝑢(𝑥𝑗+2), … . Note that when using polynomial and non-polynomial spline approximations, it is possible to obtain acceptable solutions in several interesting cases that are impossible when we use the classical approach. In the case of using local basis splines, many previously unsolvable problems turn out to be solvable. The results of the numerical experiments are presented.

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

Обобщенная математическая модель радиосигнала в различных базисах

The purpose of this article is to develop a generalized mathematical model of a radio signal in various bases. The proposed model allows us to generalize the known models of radio signals in basis functions of sine/cosine, basis functions of spline characters, bases of spline-Vilenkin-Chrestenson functions, bases of spline-Pontryagin-Vilenkin-Chrestenson functions, etc., constructed on the basis of spline-algebraic harmonic methods analysis. For the generalized model, special cases are formulated for the considered spaces of basis functions. Block diagrams and timing diagrams are presented that explain the main stages of the formation and processing of radio signals in various bases. Using the simulation method in the Matlab environment, the noise immunity of the proposed radio signals was assessed. Analysis of the obtained results showed that radio signals generated on the basis of a generalized mathematical model in various noise immunity bases under AWGN conditions are comparable (and even have a slight advantage) with similar signals generated in the bases of classical exponential functions. An assessment of structural secrecy is presented. It is shown that the structural secrecy of radio signals generated in various bases is ensured by the formation of signals in a priori unknown bases, the parameters of which: group, ring, spline order and number of the basis function change in accordance with a given code word, also unknown to potential unauthorized processing systems. Structural secrecy manifests itself in the incorrect choice of basis when processing signal data for reception, which entails an expansion of the signal spectrum. As a result, incorrect estimation of signal parameters and incorrect setting of the demodulator occurs.

Pemodelan Regresi Semiparametrik B Spline (Studi Kasus: Pengaruh Harga Emas dan Minyak Mentah Dunia Terhadap Indeks Harga Saham Gabungan)

<p dir="ltr"><span>The increases of the world gold and crude oil prices have a big role as a main factor that effect composite stock price index, the effect can make investors to buy stock from Bursa Efek Indonesia. Regression semiparametric used in this research for a purpose to get combined parametric and nonparametric with B Spline approach. B Spline is a development of spline to overcome weaknesses in making singular matrix at a high order spline with many knot points and close together. Variable parametric component is composite stock price index with crude oil price, and variable nonparametric component is composite stock price index with gold price that got obtained from January 2015 until December 2022. The result from this research is best regression semiparametric B-Spline modelling can be obtained using some combination of order and knot points. The optimal point is obtained on 2nd order using 4 knot point (1.135;1.319,15;1.320,75;1.323,25) with a minimum GCV value is 100.227,8. The best measure of goodness with a coefficient of determination value (R-Square) obtained a value 78,8%, because the value is more than 67% make it as a strong model. MAPE value is 3,37% that has a value less than 10 %, make this model have a perfect forecasting ability.</span></p><p><span id="docs-internal-guid-09fba3ae-7fff-a25a-8197-7dd90ea8b1e1"><span>Keywords: </span><span>Gold; Crude Oil; Composite Stock Price Index; Semiparametric B Spline; GCV</span></span></p>

Determination of channel curvature in the map forecast model
 of river meandering

The article is a development of the author's previous work aimed at predicting channel displacements during the meandering of lowland rivers. The hypothesis about the instability of a straight river bed is accepted as the main hypothesis about the causes of meandering. To construct a meandering model, a statement is used according to which the river bed shifts from its current position along the normal to the dynamic axis of the flow in the direction from the center of its curvature in proportion to the value of the curvature of the channel in a given section, and the proportionality coefficient is unknown in advance and must be determined in the process of cartographic (hydrological) monitoring. Since the curvature of a river channel is ex-pressed through the second derivative of a curve that approximates the outline of the bank, the important question is how to accurately calculate this curvature from the coordinates of bank points. A way to do that is proposed in the article. This way is based on approximating the coast-line with a 2nd order spline in the form of a circle most closely adjacent to the coastline. Since hydrological monitoring makes it possible to find normal channel displacements over a certain period of time, then knowing the curvature of the coastline in all sections of the channel, it is possible to calculate the unknown coefficient of proportionality of this displacement to the cur-vature. The obtained values of the coefficient of proportionality are proposed to be used for pre-dictive mapping of channel dynamics for future time intervals.

Design Method and Teeth Contact Simulation of PEEK Involute Spline Couplings.

In order to design an involute spline made of PEEK (polyetheretherketone)-based material with better performance and improve the design rules of involute splines, initially, an involute splines design theory for PEEK (polyetheretherketone)-based materials is presented, which combines international standards (ISO 4156, 1. 2. 3, 2005), American standards (ANSI B92.2M. 1989), and traditional empirical formulas. Second, using the involute splines calibration method in international standards as a guide, we developed the involute splines calibration method for PEEK-based materials by estimating the impact of energy consumption caused by viscoelasticity on temperature field calibration. Next, the contact characteristics of the designed spline were analyzed using ABAQUS2022 software to confirm the accuracy and reliability of the design and calibration methods. Finally, finite element simulation was used to analyze the influence of different pressure angles, moduli, combined lengths, and other parameters on the contact characteristics of the spline in order to realize the optimal design of PEEK-based material involute splines, to offer a theoretical foundation and improved design methodology for cylindrical straight-tooth involute splines.

Application of Splines of the Seventh Order Approximation to the Solution of the Fredholm Integral Equations with Weekly Singularity

We consider the construction of a numerical solution to the Fredholm integral equation of the second kind with weekly singularity using polynomial spline approximations of the seventh order of approximation. The support of the basis spline of the seventh order of approximation occupies seven grid intervals. In the beginning, in the middle, and at the end of the integration interval, we apply various modifications of the basis splines of the seventh order of approximation. We use the Gaussian-type quadrature formulas to calculate the integrals with a weakly singularity. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the required error of approximation. Numerical examples of the application of spline approximations of the seventh order to solve integral equations are given.

Associations between body mass index and all-cause mortality: A systematic review and meta-analysis.

Fasting insulin and c-reactive protein confound the association between mortality and body mass index. An increase in fat mass may mediate the associations between hyperinsulinemia, hyperinflammation, and mortality. The objective of this study was to describe the "average" associations between body mass index and the risk of mortality and to explore how adjusting for fasting insulin and markers of inflammation might modify the association of BMI with mortality. MEDLINE and EMBASE were searched for studies published in 2020. Studies with adult participants where BMI and vital status was assessed were included. BMI was required to be categorized into groups or parametrized as non-first order polynomials or splines. All-cause mortality was regressed against mean BMI squared within seven broad clinical populations. Study was modeled as a random intercept. β coefficients and 95% confidence intervals are reported along with estimates of mortality risk by BMIs of 20, 30, and 40 kg/m2 . Bubble plots with regression lines are drawn, showing the associations between mortality and BMI. Splines results were summarized. There were 154 included studies with 6,685,979 participants. Only five (3.2%) studies adjusted for a marker of inflammation, and no studies adjusted for fasting insulin. There were significant associations between higher BMIs and lower mortality risk in cardiovascular (unadjusted β -0.829 [95% CI -1.313, -0.345] and adjusted β -0.746 [95% CI -1.471, -0.021]), Covid-19 (unadjusted β -0.333 [95% CI -0.650, -0.015]), critically ill (adjusted β -0.550 [95% CI -1.091, -0.010]), and surgical (unadjusted β -0.415 [95% CI -0.824, -0.006]) populations. The associations for general, cancer, and non-communicable disease populations were not significant. Heterogeneity was very large (I2 ≥ 97%). The role of obesity as a driver of excess mortality should be critically re-examined, in parallel with increased efforts to determine the harms of hyperinsulinemia and chronic inflammation.

An assessment of the total Lagrangian material point method: Comparison to conventional MPM, higher order basis, and treatment of near-incompressibility

The total Lagrangian material point method (TLMPM) is a recent modification to the conventional material point method (MPM). It poses the weak form in the undeformed configuration, and has shown great potential in solid mechanics applications because it seamlessly eliminates the so-called cell-crossing instability and overcomes numerical fracture. Despite its promise, it has not been widely adopted by the MPM community yet, and only a handful of research groups have employed it. At the same time, due to TLMPM’s recent appearance, there are open research questions, such as the subject of near-incompressibility that has not been thoroughly studied within its context. In this paper we attempt to strengthen TLMPM’s position in the field by providing an independent assessment of the method in a set of benchmark problems. Furthermore, we evaluate the need of using higher order spline shape functions, which have been shown to be crucial for the success of the conventional MPM. Finally, we present a simple projection technique for addressing solid mechanics problems that fall within the near-incompressible regime. The numerical results show that TLMPM is indeed a strong alternative to conventional MPM, whereas our near-incompressibility approach produces solutions that are free of volumetric locking and pressure oscillations, while introducing very minor modifications to existing codes.

The Application of Splines of the Seventh Order Approximation to the Solution of Integral Fredholm Equations

There are various numerical methods for solving integral equations. Among the new numerical methods, methods based on splines and spline wavelets should be noted. Local interpolation splines of a low order of approximation have proved themselves well in solving differential and integral equations. In this paper, we consider the construction of a numerical solution to the Fredholm integral equation of the second kind using spline approximations of the seventh order of approximation. The support of the basis spline of the seventh order of approximation occupies seven grid intervals. We apply various modifications of the basis splines of the seventh order of approximation at the beginning, the middle, and at the end of the integration interval. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the required error of approximation. Numerical examples of the application of spline approximations of the seventh order for solving integral equations are given.

Adapting Cubic Hermite Splines to the Presence of Singularities Through Correction Terms

Hermite interpolation is classically used to reconstruct smooth data when the function and its first order derivatives are available at certain nodes. If first order derivatives are not available, it is easy to set a system of equations imposing some regularity conditions at the data nodes in order to obtain them. This process leads to the construction of a Hermite spline. The problem of the described Hermite splines is that the accuracy is lost if the data contains singularities. The consequence is the appearance of oscillations, if there is a jump discontinuity in the function, that globally affects the accuracy of the spline, or the smearing of singularities, if the discontinuities are in the derivatives of the function. This paper is devoted to the construction and analysis of a new technique that allows for the computation of accurate first order derivatives of a function close to singularities using a Hermite spline. The idea is to correct the system of equations of the spline in order to attain the desired accuracy even close to the singularities. Once we have computed the first order derivatives with enough accuracy, a correction term is added to the Hermite spline in the intervals that contain a singularity. The aim is to reconstruct piecewise smooth functions with O(h^4) accuracy even close to the singularities. The process of adaption will require some knowledge about the position of the singularity and the jumps of the function and some of its derivatives at the singularity. The whole process can be used as a post-processing, where a correction term is added to the classical cubic Hermite spline. Proofs for the accuracy and regularity of the corrected spline and its derivatives are given. We also analyse the mechanism that eliminates the Gibbs phenomenon close to jump discontinuities in the function. The numerical experiments presented confirm the theoretical results obtained.

ADI+MDA orthogonal spline collocation for the pressure Poisson reformulation of the Navier–Stokes equation in two space variables

A numerical method for solving a pressure Poisson reformulation of the Navier–Stokes equation in two space variables is presented. The method discretizes in space using orthogonal spline collocation with splines of order r. The velocity terms are obtained through an alternating direction implicit extrapolated Crank –Nicolson scheme applied to a Burgers’ type equation and the pressure term is found by applying a matrix decomposition algorithm to a Poisson equation satisfying non-homogeneous Neumann boundary conditions at each time level. Numerical results suggest that the scheme exhibits convergence rates of order r in space in the H1 norm and semi-norm for the velocity and pressure terms, respectively, and is order 2 in time. Finally, the scheme is applied to the lid-driven cavity problem and is compared to standard benchmark values.

Design and optimization of complex mechanism flip shaping subsystem based on genetic algorithm and rigid-flexible coupled dynamic model.

In this paper, the cam connecting rod system of the high-speed group vertical machine flipping shaping mechanism is the research object. In order to solve the key problem that the flipping shaping mechanism cannot accurately complete the action when the vibration of the mechanism is large. In this paper, the finite element method is used to construct the dynamic model of the connecting rod subsystem of the flipped shaping mechanism. And The dynamic model of cam roller subsystem is established by centralized parameter method. Based on the MATLAB Genetic Algorithm toolbox and using Newmark's method, the dynamic equations of the flipped plastic mechanism system are solved. The optimal parameters of the connecting rod of the mechanism, the cam profile curve and the swing power and swing torque of the mechanism at different speeds are analyzed. The results show that the speed and convex contour line are important factors affecting the performance of the mechanism. And the pendulum force (swing torque) is the main cause of the vibration of the mechanism on the frame. Therefore, the mechanism pendulum dynamic and the swing moment are selected as the objective functions of the optimization model. By selecting the node parameters of the sixth order spline motion law and the cross-section parameters of the connecting rod as the design variables. The cam linkage system is optimally designed to obtain the optimal value. Finally, the optimal design of the flipped shaping mechanism was analyzed and compared with the original mechanism.

Solution of Integral Equations Using Local Splines of the Second Order

Splines are an important mathematical tool in Applied and Theoretical Mechanics. Several Problems in Mechanics are modeled with Differential Equations the solution of which demands Finite Elements and Splines. In this paper, we consider the construction of computational schemes for the numerical solution of integral equations of the second kind with a weak singularity. To construct the numerical schemes, local polynomial quadratic spline approximations and second-order nonpolynomial spline approximations are used. The results of the numerical experiments are given. This methodology has many applications in problems in Applied and Theoretical Mechanics

High order spline finite element method for the fourth-order parabolic equations

In this paper, we concern with the development of error estimates for high-order spline finite element approximation to a class of fourth-order parabolic equations. The L2 estimates for semi-discrete scheme are derived by constructing two auxiliary problems (one is a steady-state problem, the other is similar to the primal problem) and by using the structure of solutions of the second-order ordinary differential equation and the similarity theory of matrices, and are optimal in terms of the regularity of the exact solution. The H2 energy estimates for spline-element central difference approximation are established under a condition of stability (for explicit scheme), and are optimal for space variable in H2-norm and for time variable in H1,∞-norm. Numerical examples are presented to validate the theory.

Nonlenear Integro-differential Equations and Splines of the Fifth Order of Approximation

In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation, which contains the first derivative of the function. Our method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations. The method based on the application of the local polynomial splines of the fifth order of approximation is proposed. Theorems about the errors of the approximation of a function and its first derivative by these splines are given. With the help of the proposed splines, the function and the derivative are replaced by the corresponding approximation. Note that at the beginning, in the middle and at the end of the interval of the definition of the integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding formulas for numerical differentiation. which we also apply for the solution of integro-differential equations. The formulas for approximation of the function and its derivative are presented. The results of the numerical solution of several integro-differential equations are presented. The proposed method is shown that it can be applied to solve integro-differential equations containing the second derivative of the solution.

3D time-dependent scattering about complex shapes using high order difference potentials

We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches.The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary.The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.

Interpolant-based demosaicing routines for dual-mode visible/near-infrared imaging systems.

Dual-mode visible/near-infrared imaging systems, including a bioinspired six-channel design and more conventional four-channel implementations, have transitioned from a niche in surveillance to general use in machine vision. However, the demosaicing routines that transform the raw images from these sensors into processed images that can be consumed by humans or computers rely on assumptions that may not be appropriate when the two portions of the spectrum contribute different information about a scene. A solution can be found in a family of demosaicing routines that utilize interpolating polynomials and splines of different dimensionalities and orders to process images with minimal assumptions.

Use of mathematical modeling for solving tasks of optimal control of an electric drive

The object of study in this article is the control processes of electromechanical systems with an electric drive. The subject matter is a mathematical model in the problem of optimal control of the following electric drive. This article is devoted to the mathematical modeling of the control processes of electromechanical systems with an electric drive. In this paper, we use the approximation of vector-valued functions of one variable by splines of the first degree to solve the problem of rotation of the motor shaft in electromechanical system. As you know, it is not always possible to find exact solutions in optimal control problems. In this regard, there is a need to obtain approximate solutions for optimal control problems. Therefore, an urgent task is the development of new methods for the approximate solution to the problems of optimal control of the electric drive, which provide higher approximation accuracy. The following results are presented: Graphs of phase coordinates and optimal control of approximation by splines of the first order are obtained. The results obtained were compared with the exact solution. The given examples illustrate the high accuracy of approximate solutions to the control problem obtained using the proposed method. Conclusions. The scientific novelty of the results obtained lies in the fact that a method is proposed for an approximate solution to the problem of optimal control of a servo drive using spline functions, in which unknown control parameters are found simultaneously with unknown parameters of phase coordinates, which gives the best approximation to the exact solution.

On a Phase Transition in General Order Spline Regression

In the Gaussian sequence model Y = θ0+ε in Rn, we study the fundamental limit of statistical estimation when the signal θ0 belongs to a class Θn(d, d0, k) of (generalized) splines with free knots located at equally spaced design points. Here d is the degree of the spline, d0 is the order of differentiability at each inner knot, and k is the maximal number of pieces. We show that, given any integer d≥0 and d0 ∈ {-1, 0, ..., d-1}, the minimax rate of estimation over Θn(d, d0, k) exhibits the following phase transition: inf~θ supθ∈n(d,d0,k)Eθ‖~θ - θ‖2 ≍ d {k log log(16n/k), 2 ≤ k ≤ k0, k log(en/k), k ≥ k0 + 1. The transition boundary k0, which takes the form ⌊(d + 1)/(d - d0)⌋ + 1, demonstrates the critical role of the regularity parameter d0 in the separation between a faster log log(16n) and a slower log(en) rate. We further show that, once encouraging an additional ‘d-monotonicity’ shape constraint (including monotonicity for d = 0 and convexity for d = 1), the above phase transition is removed and the faster k log log(16n/k) rate can be achieved for all k. These results provide theoretical support for developing ℓ0-penalized (shape-constrained) spline regression procedures as useful alternatives to ℓ1- and ℓ2-penalized ones.