Trigonometric Interpolation Research Articles

A Boolean sum interpolation for multivariate functions of bounded variation

This paper deals with the approximation error of trigonometric interpolation for multivariate functions of bounded variation in the sense of Hardy-Krause. We propose interpolation operators related to both the tensor product and sparse grids on the multivariate torus. For these interpolation processes, we investigate the corresponding error estimates in the Lp norm for the class of functions under consideration. In addition, we compare the accuracy with the cardinality of these grids in both approaches.

Stress Characteristics and Mechanical Behavior of Rock Masses with an Opening under Complex Deep Underground Stress Conditions

In this study, the impact of principal stress states on the stress characteristics and initial failure of the rock mass surrounding a three-center arch opening was investigated using complex variable function methods and Discrete Element Method (DEM) numerical modeling. First, the mapping function of the opening was determined using the trigonometric interpolation method, and the influence of the number of terms in the mapping function on its accuracy was revealed. Based on this, the far-field stress state of the underground rock mass was characterized by the ratio of the minimum to maximum principal stress (λ) and the angle (β) between the principal stress and the vertical direction. This stress state was then converted into normal and shear stresses. Using complex variable function theory, the stress characteristics at the boundary of the opening under different stress states were analyzed. Finally, DEM numerical modeling was employed to study the initial failure characteristics at the boundary of the opening and its relationship with the stress distribution. The results indicate that the lateral pressure coefficient significantly affects the stability of the opening by influencing stress concentration around the surrounding rock. Low lateral pressure coefficients lead to tensile stress concentration at the boundary perpendicular to the maximum principal stress. As the coefficient increases, tensile stress decreases, and compressive stress areas expand. While the principal stress direction has a minor effect on stress concentration, it notably impacts stress distribution at the boundary. When λ < 1.0 and β = 45°, stress distribution asymmetry is most pronounced, with the highest compressive stress. The early failure distribution aligns with stress concentration areas, validating the use of stress analysis in predicting opening stability and failure characteristics.

Application of Universal Fast Trigonometric Interpolation and Extrapolation for Determining the Launch Point Coordinates of a Flight Vehicle

Application of Universal Fast Trigonometric Interpolation and Extrapolation for Determining the Launch Point Coordinates of a Flight Vehicle

New pipe element based on Hermite-Jackson interpolation

A new pipe finite element is proposed for piping analysis within the framework of linear shell theory. The approach involves the use of a mixed interpolation of classical polynomial and trigonometric polynomial spaces, with trigonometric interpolation performed via Hermite-Jackson polynomials along the mid-surface section. Error estimates are provided and a convergence analysis is performed under specific assumptions on the regularity of the solution. The proposed element is validated through several numerical examples, which demonstrate its accuracy and efficiency in terms of computational cost. This method represents a promising approach for addressing the challenges of piping analysis.

Dose-effect of exercise intervention on heart rate variability of acclimatized young male lowlanders at 3,680 m.

This study investigated whether exercise could improve the reduced HRV in an environment of high altitude. A total of 97 young, healthy male lowlanders living at 3,680m for >1 year were recruited. They were randomized into four groups, of which three performed-low-, moderate-, and high-intensity (LI, MI, HI) aerobic exercise for 4weeks, respectively. The remaining was the control group (CG) receiving no intervention. For HI, compared to other groups, heart rate (p = 0.002) was significantly decreased, while standard deviation of RR intervals (p < 0.001), SD2 of Poincaré plot (p = 0.046) and the number of successive RR interval pairs that differ by > 50ms divided by total number of RR (p = 0.032), were significantly increased after intervention. For MI, significantly increase of trigonometric interpolation in NN interval (p = 0.016) was observed after exercise. Further, a decrease in systolic blood pressure (SBP) after high-intensity exercise was found significantly associated with an increase in SD2 (r = - 0.428, p = 0.042). These results indicated that there was a dose effect of different intensities of aerobic exercise on the HRV of acclimatized lowlanders. Moderate and high-intensity aerobic exercise would change the status of the autonomic nervous system (ANS) and decrease the blood pressure of acclimatized lowlanders exposed to high altitude.

Fourier Series Weight in Quantum Machine Learning

In this work, we aim to confirm the impact of the Fourier series on the quantum machine learning model. We will propose models, tests, and demonstrations to achieve this objective. We designed a quantum machine learning leveraged on the Hamiltonian encoding. With a subtle change, we performed the trigonometric interpolation, binary and multiclass classifier, and a quantum signal processing application. We also proposed a block diagram of determining approximately the Fourier coefficient based on quantum machine learning. We performed and tested all the proposed models using the Pennylane framework.

Improved hole and tube elements in BEM for elasticity problems

Structures with holes are prevalent in engineering applications. Analyzing the stress concentration effects caused by holes using the finite element method (FEM) or the boundary element method (BEM) is challenging and time-consuming. Typically, a large number of elements are necessary in the vicinity of holes to achieve accurate results. In this paper, a series of improved hole elements and tube elements are constructed for simulating the holes and cylinders, respectively. Two construction methods for hole elements, based on Lagrange and trigonometric interpolations, respectively, are introduced. The general formulas for evaluating singular integrals over the proposed elements are derived. Additionally, the adaptive element sub-division technique is applied to address the nearly singular integrals over these elements and account for the interactions of multiple holes. Five numerical examples are studied to demonstrate the accuracy and efficiency of the proposed elements. The results demonstrate that employing hole and tube elements can reduce the number of nodes while maintaining stress accuracy, in comparison to using several quadratic elements.

A barycentric trigonometric Hermite interpolant via an iterative approach

In this work we construct an Hermite interpolant starting from basis functions that satisfy a Lagrange property. In fact, we extend and generalise an iterative approach, introduced by Cirillo and Hormann (2018) for the Floater–Hormann family of interpolants. Secondly, we apply this scheme to produce an effective barycentric rational trigonometric Hermite interpolant at general ordered nodes using as basis functions the ones of the trigonometric interpolant introduced by Berrut (1988). For an easy computational construction, we calculate analytically the differentiation matrix. Finally, we conclude with various examples and a numerical study of the convergence at equidistant nodes and conformally mapped nodes.

Lasso trigonometric polynomial approximation for periodic function recovery in equidistant points

This paper introduces a fully discrete soft thresholding trigonometric polynomial approximation on [−π,π], named Lasso trigonometric interpolation. This approximation is an ℓ1-regularized discrete least squares approximation under the same conditions of classical trigonometric interpolation on an equidistant grid. Lasso trigonometric interpolation is a sparse scheme which is efficient in dealing with noisy data. We theoretically analyze Lasso trigonometric interpolation quality for continuous periodic function. The L2 error bound of Lasso trigonometric interpolation is less than that of classical trigonometric interpolation, which improved the robustness of trigonometric interpolation. The performance of Lasso trigonometric interpolation for several testing functions (sin wave, triangular wave, sawtooth wave, square wave), is illustrated with numerical examples, with or without the presence of data errors.

Application of the Method of Fast Expansions to Construction of a Trajectory of Movement of a Body with Variable Mass from Its Initial Position in a Gained Final Position in a Gravitational Field

An analytical solution of the problem of the movement of a spacecraft from the starting point to the final point in a certain time is given. First, the method of fast sine expansions is used. The space problem considered here is essentially non-linear, what necessitates the use of trigonometric interpolation methods, which surpass all known interpolations in accuracy and simplicity. In this case, the problem of calculating Fourier coefficients by integral formulas is replaced by the solution of an orthogonal interpolation system. In this regard, two cases are considered on the segment \(\left[ {0,a} \right]\): universal interpolation and trigonometric sine and cosine interpolations. A theorem on the rapid decrease of expansion coefficients is proved, and a compact formula for calculating the interpolation coefficients is obtained. A general theory of fast expansions is given. It is shown that in this case, the Fourier coefficients decrease significantly faster with the growth of the ordinal number compared to the Fourier coefficients in the classical case. This property makes it possible to significantly reduce the number of terms taken into account in the Fourier series, significantly increase the accuracy of calculations and reduce the amount of calculations on a computer. The analysis of the obtained solutions of the spacecraft motion problem is carried out and their comparison with the exact solution of the test problem is proposed. An approximate solution by the method of fast expansions can be taken as an exact one, since the input data of the problem used from reference books have a higher error.

Stability Calculation of the Plane Steel Frame Structures Using Tangent Modulus Theory

This paper considers the phenomenon of the instability of steel plane frame structures in the elastic–plastic domain. The numerical analysis uses finite element method, where the corresponding stiffness matrices are based upon the trigonometric and hyperbolic interpolation functions of normal forces. The tangent modulus concept is applied when buckling occurs in the plastic range. Thus, the stiffness matrices for nonlinear material behavior are also derived. The procedure for determining effective length factors of compressed columns, which implies a global stability analysis of entire frames, is established too. It means that the proposed approach is based on the assumption that the collapse of the structure occurs when the most loaded columns reach their stability limit. So, the presented procedure enables the calculation of the real critical load of plane frame structures in the elastic–plastic domain and determines the corresponding effective length factors.

Shape control tools for periodic Bézier curves

Bézier curves are an essential tool for curve design. Due to their properties, common operations such as translation, rotation, or scaling can be applied to the curve by simply modifying the control polygon of the curve. More flexibility, and thus more diverse types of curves, can be achieved by associating a weight with each control point, that is, by considering rational Bézier curves. As shown by Ramanantoanina and Hormann (2021), additional and more direct control over the curve shape can be achieved by exploiting the correspondence between the rational Bézier and the interpolating barycentric form and by exploring the effect of changing the degrees of freedom of the latter (interpolation points, weights, and nodes). In this paper, we explore similar editing possibilities for closed curves, in particular for the rational extension of the periodic Bézier curves that were introduced by Sánchez-Reyes (2009). We show how to convert back and forth between the periodic rational Bézier and the interpolating trigonometric barycentric form, derive a necessary condition to avoid poles of a trigonometric rational interpolant, and devise a general framework to perform degree elevation of periodic rational Bézier curves. We further discuss the editing possibilities given by the trigonometric barycentric form.

Rational quadratic trigonometric spline fractal interpolation functions with variable scalings

Rational quadratic trigonometric spline fractal interpolation functions with variable scalings

Investigation of the Errors of Fast Trigonometric Interpolation in Solving the Problem of Stresses in a Bar

Investigation of the Errors of Fast Trigonometric Interpolation in Solving the Problem of Stresses in a Bar

Trigonometric Hermite interpolation method for Fredholm linear integral equations

Abstract This paper presents a new trigonometric composite Hermite interpolation method for solving Fredholm linear integral equations. This operator approximates locally both the function and its derivative, which is known on the subdivision nodes. Then we derive a class of quadrature rules with endpoint corrections based on integrating the composite Hermite interpolant. We also provide error estimation and numerical examples to illustrate that this new operator can provide highly accurate results.

On trigonometric interpolation in an even number of points

In contrast to odd-length trigonometric interpolants, even-length trigonometric interpolants need not be unique; this is apparent from the representation of the interpolant in the (real or complex) Fourier basis, which possesses an extra degree of freedom in the choice of the highest-order basis function in the even case. One can eliminate this degree of freedom by imposing a constraint, but then the interpolant may cease to exist for certain choices of the interpolation points. On the other hand, the Lagrange representation developed by Gauss always produces an interpolant despite having no free parameters. We discuss the choice Gauss's formula makes for the extra degree of freedom and show that, when the points are equispaced, its choice is optimal in the sense that it minimizes both the standard and 2-norm Lebesgue constants for the interpolation problem. For non-equispaced points, we give numerical evidence that Gauss's formula is no longer optimal and consider interpolants of minimal 2-norm instead. We show how to modify Gauss's formula to produce a minimal-norm interpolant and that, if the points are equispaced, no modification is necessary.

A boundary integral equation method for the fluid-solid interaction problem

&lt;abstract&gt;&lt;p&gt;In this paper, a boundary integral equation method is proposed for the fluid-solid interaction scattering problem, and a high-precision numerical method is developed. More specifically, by introducing the Helmholtz decomposition, the corresponding problem is transformed into a coupled boundary value problem for the Helmholtz equation. Based on the integral equation method, the coupled value problem is reduced to a system of three coupled hypersingular integral equations. Semi-discrete and fully-discrete collocation methods are proposed for the singular integral equations. The presented method is based on trigonometric interpolation and discretized singular operators applied to differentiated interpolation. The convergence of the method is verified by a numerical experiment.&lt;/p&gt;&lt;/abstract&gt;

Investigation of the Errors of Fast Trigonometric Interpolation in Solving the Problem of Stresses in a Bar

Using the fast trigonometric interpolation, the problem on stresses in a rectangular bar is solved. The obtained approximate analytical solution is compared with the exact one. During this analysis the relative error of the displacement components, the stress tensor components, the discrepancy between the Lame equilibrium equations, and the discrepancy of the boundary conditions are investigated. It has been established that when using the second-order boundary function in fast expansions and a small number of terms in the Fourier series (from two to six), the maximum relative error δmath max of the displacement components and the stress tensor components is less than one percent. With an increase in the order of the boundary function and/or the number of terms N in the Fourier series, δmath decreases rapidly. Increasing the order of the boundary function is a more effective way to reduce the calculation error of δmath max than increasing the number of terms in the Fourier series. When studying the intensity of stresses σ in a bar with different overall dimensions of a rectangular cross-section, but with the same area of all sections, it has turned out that the smallest value of σmath all cross-sections is observed for a bar with a square section.

Cubic Trigonometric Hermite Interpolation Curve: Construction, Properties, and Shape Optimization

Cubic Hermite interpolation curve plays a very important role in interpolation curves modeling, but it has three shortcomings including low continuity, difficult shape adjustment, and the inability to accurately represent some common engineering curves. We construct a cubic trigonometric Hermite interpolation curve to make up the three shortcomings of cubic Hermite interpolation curve once and for all. The cubic trigonometric Hermite interpolation curve not only inherits the features of cubic Hermite interpolation curve but also achieves C2 continuity, has local and global adjustability, and can accurately represent elliptical arc, circular arc, quadratic parabolic arc, cubic parabolic arc, and astroid arc that often appear in engineering. In addition, we give the schemes for optimizing the shape of the cubic trigonometric Hermite interpolation curve based on internal energy minimization. The schemes include optimizing the shape of planar curve and spatial curve. Some modeling examples show that the proposed schemes are effective and the cubic trigonometric Hermite interpolation curve is more practical than cubic Hermite interpolation curve.

On the theory of interpolation of functions on sets of matrices with the Hadamard multiplication

This article is devoted to the problem of interpolation of functions defined on sets of matrices with multiplication in the sense of Hadamard and is mainly an overview. It contains some known information about the Hadamard matrix multiplication and its properties. For functions defined on sets of square and rectangular matrices, various interpolation polynomials of the Lagrange type, containing both the operation of matrix multiplication in the Hadamard sense and the usual matrix product, are given. In the case of analytic functions defined on sets of square matrices with the Hadamard multiplication, some analogues of the Lagrange type trigonometric interpolation formulas are considered. Matrix analogues of splines and the Cauchy integral are given on sets of matrices with the Hadamard multiplication. Some of its applications in the theory of interpolation are considered. Theorems on the convergence of some Lagrange interpolation processes for analytic functions defined on a set of matrices with multiplication in the Hadamard sense are proved. The results obtained are based on the application of some well-known provisions of the theory of interpolation of scalar functions. Data presentation is illustrated by a number of examples.