Chebyshev Orthogonal Polynomials Research Articles

NUMERICAL SOLUTION OF NONLINEAR SCHRODINGER EQUATION USING COLLOCATION METHOD WITH CHEBYSHEV POLYNOMIALS

The nonlinear Schrodinger equation (NSE) is a partial differential equation (PDE) with numerous applications in quantum mechanics. Analytical methods for the solution of NSE are almost impossible due to their complexity. Thus, the need for a numerical scheme to seek the approximate solution of the NSE. Hence, this paper considered the numerical solution of the NSE via the spectral collocation method (SCM) with Chebyshev orthogonal polynomials of the first kind. The scheme was efficiently constructed to solve the NSE equation with the aid of MAPLE 18, and numerical evidence compared with Adomian Decomposition Method (ADM), Homotopy Analysis Transform Method (HATM) and Residual Power Series Method (RPSM) as available in the literature. The findings show that the SCM is an effective solver for NSE with rapid convergence to the exact solution as the parameter �varies.

Effect of spatial setting angle on vibration of elastically restrained rotating beams

For a fully articulated rotor system, there may exist a spatial setting angle and elastically restrained root for each individual blade in rotation procedure due to flapping, drag and pitch hinges. Most of previous studies on rotating blades or beams focused on pitch angle and clamped root. In this paper, a three-dimensional dynamic model is proposed for elastically restrained rotating beams with spatial setting angle. The elastic deformation of the beam is expressed by three scalar variables in three-dimensional space including stretch deformation instead of the conventional axial deformation. The longitudinal shrinkage is considered to capture centrifugal forces. The penalty method is adopted to deal with the elastic constraints, which greatly simplifies the construction of admissible functions. A unified formulation is derived by a variational principle combined with Chebyshev orthogonal polynomials. One of the advantages of the present method is that any linearly independent and complete basis functions can be used as admissible function regardless of boundary conditions, and another advantage is that the present method is capable of handling arbitrary boundary conditions by only adjusting the values of penalty factor. The convergence and accuracy of the present method are validated by comparison, and the effects of boundary conditions, angular velocity and spatial setting angle on vibration characteristics are discussed in detail. The results show that natural frequencies increase as the spring stiffness rises in a certain range and the spatial setting angle plays a more significant role on dynamic characteristics for rotating beams.

A Theory of Functional Connections-Based hp-Adaptive Mesh Refinement Algorithm for Solving Hypersensitive Two-Point Boundary-Value Problems

This manuscript introduces the first hp-adaptive mesh refinement algorithm for the Theory of Functional Connections (TFC) to solve hypersensitive two-point boundary-value problems (TPBVPs). The TFC is a mathematical framework that analytically satisfies linear constraints using an approximation method called a constrained expression. The constrained expression utilized in this work is composed of two parts. The first part consists of Chebyshev orthogonal polynomials, which conform to the solution of differentiation variables. The second part is a summation of products between switching and projection functionals, which satisfy the boundary constraints. The mesh refinement algorithm relies on the truncation error of the constrained expressions to determine the ideal number of basis functions within a segment’s polynomials. Whether to increase the number of basis functions in a segment or divide it is determined by the decay rate of the truncation error. The results show that the proposed algorithm is capable of solving hypersensitive TPBVPs more accurately than MATLAB R2021b’s bvp4c routine and is much better than the standard TFC method that uses global constrained expressions. The proposed algorithm’s main flaw is its long runtime due to the numerical approximation of the Jacobians.

Transverse vibration of plate with multiple curved through cracks

A modeling method for transverse vibration of cracked plate containing multiple internal curved through cracks is presented and experimental studies are performed to validate its effectiveness. The cracked plate is assumed to be divided into several rectangular and curved trapezoidal domains in the global coordinates system. For each sub-domain, the transverse displacement is expanded as Chebyshev orthogonal polynomials series in local coordinates system. As for the shared edges between neighboring two sub-domains, the local collocation method is adopted and a number of Chebyshev zero points are selected to ensure the displacement continuity condition. To verify the convergence and correctness of the presented model and solutions, the natural frequencies, mode shapes and dynamic responses for several cases are compared with reference results and experimental results, with good agreement observed. In addition, the influence of several significant parameters of cracks are investigated, including the number, location and geometrical shape of the cracks. It is shown that the crack inside the plate presents less influence than that at the boundary on the mode change of the plate. The investigations in the study implies that the present model method is effective and convenient for plate with multiple curved through cracks.

Mathematical Modeling of Elastically Deformed States of Thin Isotropic Plates Using Chebyshev Polynomials

Abstract. In this paper a method for solving an inhomogeneous biharmonic equation while modeling elastically deformed states of thin isotropic rectangular plates using a system of orthogonal Chebyshev polynomials of the first kind is proposed. The method is based on representation of a solution to the initial biharmonic equation as a finite sum of Chebyshev series by each independent variable in combination with matrix transformations and properties of Chebyshev polynomials. The problem is examined for the case when a transverse load acts on the plate, and the hinge fastening along the edges of the plate is taken as boundary conditions. Using the extremes and zeros of Chebyshev polynomials of the first kind as collocation points, the boundary value problem is reduced to a system of linear algebraic equations. Decomposition coefficients of desired function with respect to Chebyshev polynomials act as unknowns in this system. As the comparison showed, the results obtained by this method with a high degree of accuracy coincide with similar results derived using analytical approach that are given in the article. The paper also presents the results of calculations using the proposed method in the case when two opposite edges of the plate are pinched and two others are pivotally fixed. The comparison with similar results of modeling the stress-strain states of rectangular plates which are presented in the open sources is carried out.

Dynamic modeling and vibration analysis of bolted flange joint disk-drum structures: Theory and experiment

Bolted flange joint disk-drum structures (BFJDSs) are key components in aero-engines, in which fatigue damage often occurs due to structural vibration. However, no study has so far given an effective theoretical dynamic model of BFJDSs. In this study, an effective dynamic model of BFJDSs is proposed. The present model considers the flange effect, non-uniform contact pressure of bolted joint, and bolt mass. The theoretical expression for describing the stiffness variation of bolted joints is presented. During the modeling process, the energy functions of disk, drum, and flange are obtained according to the Kirchhoff plate, Sanders’ shell, and Euler-Bernoulli beam theories, respectively. The non-uniform artificial spring technique is adopted to simulate the pressure distribution of bolted joint. This method is capable of taking into account the contact pressure, which is closer to real contact situation. The displacement functions of disk and drum are expanded by using the Chebyshev orthogonal polynomials, and the motion equations are obtained by employing the Lagrange equation. Then, the experimental studies are performed on BFJDSs to illustrate the effectiveness of the theoretical model. Finally, the frequency veering and coupling vibration phenomena are revealed. The comparison studies indicate that the established theoretical model can well predict the vibration characteristics of BFJDSs under both bolt looseness and no-looseness conditions. The maximum error can be reduced from 46.81% to 3.61% by using the present dynamic model.

Research on Multi-Objective Optimization on Explosion-Suppression Structure-Nonmetallic Spherical Spacers

Intense burning phenomena (fire disasters) need to be prevented in the combustible gas utilization and transportation processes to ensure industrial safety. Nonmetallic spherical spacers (NSSs) have been investigated and applied in lots of explosive atmospheres to prevent explosion execution in a confined space. In this work, a novel fuzzy-based analytic hierarchy process (FAHP) is developed to take into account the uncertainty in decision-making and effectively solve the problem of factor weight allocation in multi-objective optimization. Optimal Latin Hypercube Design (Opt LHD), Chebyshev Orthogonal Polynomials (COP), and Adaptive Simulated Annealing (ASA) were combined. A multi-objective optimization method is proposed for the structural parameter optimization problem on NSSs in order to achieve conflicting multiple-objective optimization of low displacement rate and minimal deformation. That is to say, the small volume (low displacement rate) and high explosion-suppression performance (minimal deformation) of NSSs were optimized simultaneously. The results show that, compared with the original NSS model’s deformation (2.85 mm) and displacement rate (3.63%), the optimized NSSs with weight allocation had optimized the deformation by 12.98% and displacement rate by 6.1%. Compared with the optimized design model of NSSs without weight allocation with a deformation of 2.75 mm and a displacement rate of 3.48%, the deformation has been optimized by 9.82%, and the displacement rate has been optimized by 2.0%. It was verified that the proposed method is effective. At the same time, it was verified that the suppression effect of NSSs can be enhanced by changing the shape of the NSS spacer reasonably by experimental verification.

An Electromagnetic Load Identification Method Based on the Polynomial Structure Selection Technique

Electromagnetic loads can effectively monitor motor health and improve motor design. Considering the weak correlation of the modal shape and Chebyshev orthogonal polynomial in the space-time independent electromagnetic load identification method, a proposed method combining the polynomial structure selection technique together with limited measured displacement responses is presented, in which an error reduction ratio is used to pick out the significant mode shape matrix and the Chebyshev orthogonal polynomial. The time-history function of the electromagnetic load is reconstructed by combining the significant mode shape matrix and the identified concentrated load through modal transformation, and the corresponding spatial distribution function is fitted by the significant Chebyshev orthogonal polynomial. Eventually, a comparative numerical study considering the selection of significant components and measurement noise is carried out to prove the effectiveness of the presented method.

Спектральные методы полиномиальной интерполяции и аппроксимации

Classical problem of interpolation and approximation of functions with polynomials is considered here as a special case of spectral representation of functions. We developed this approach earlier for Legendre and Chebyshev orthogonal polynomials. Here we use Newton's fundamental polynomials as basis functions. We demonstrate that the spectral approach has computational advantages over the divided differences method. In a number of problems, Newton and Hermite interpolations are indistinguishable in our approach and computed by the same formulas. Also, computational algorithms that we constructed earlier with the use of orthogonal polynomials are adapted without modifications for use of Newton and Hermite polynomials.

An effective model for bolted flange joints and its application in vibrations of bolted flange joint multiple-plate structures: Theory with experiment verification

Bolted flange joints are widely used in various engineering structures to combine two or more substructures together. However, the understanding of mechanical mechanism of bolted flange joints is still insufficient due to the lack of efficient mathematical models. This paper proposes an effective mathematical model for bolted flange joints, which is then employed to conduct vibration analysis on bolted flange joint multiple-plate structures. During the modeling process, the flange is modeled by the Kirchhoff plate theory, and the bolted joints are modeled by the two-dimensional face constraint in bolted joint affected region (BJAR). The artificial spring technique is adopted to simulate the face constraint. The friction contact model and equivalent linearization technique are adopted to analyze the stick and slip states of bolted joints. By adopting the Chebyshev orthogonal polynomials as admissible functions, governing equations are derived. Then, the Rayleigh-Ritz method is applied to study the free vibration characteristics, and the Newmark-beta method is employed to study the vibration response under base excitation. The experimental studies are performed on the bolted flange joint two-plate structure (BFJTPS) to illustrate the effectiveness of the proposed theoretical model. The results indicate that the proposed mathematical model can well predict the vibration characteristics of multiple-plate structures under both bolt looseness and no-looseness conditions. The soft nonlinearity phenomenon under bolt looseness condition can be successfully revealed by the model.

A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells

In this research study, we aim to approximate a solution for the mathematical model of the Human Immunodeficiency Virus (HIV) infection of CD4+T-cells. An operational matrix method based on Chebyshev orthogonal polynomials has been adapted to obtain numerical solutions for the model of HIV infection of CD4+T-cells. The proposed numerical scheme is built on a system of a nonlinear algebraic equation, including coefficients of a finite Chebyshev series that are represented the approximate solutions of the model. Results are compared to existing methods to verify the accuracy of the numerical scheme.

Higher Order Orthogonal Polynomials as Activation Functions in Artificial Neural Networks

Activation functions are used in Artificial Neural Networks to provide non-linearity to the system. Several different activation functions in use are very well known by almost any AI practitioner however this is not the case for polynomial activation functions. Increasing attention to these valuable mathematical functions can encourage more research and help to fill the gap. During this work, Chebyshev and Hermite orthogonal polynomials were used as activation functions. Calculations were conducted on 3 different datasets with different hyperparameters. According to the results, calculations done by Chebyshev activation functions take less time, but Chebyshev can be more fragile depending on the solved problem. On the other hand, Hermite shows a more robust and generalized behavior, it is less dependent on the problem type, and it improves by necessary adjustments.

Reliability assessment of rainfall-induced slope stability using Chebyshev–Galerkin–KL expansion and Bayesian approach

Soil spatial variability has essential influence on the reliability of geotechnical structures. Karhunen–Loève (KL) series expansion is an effective approach to characterize such features of soil properties. Acquiring solution for Fredholm integral equation of the second type is a necessary prerequisite; however, the corresponding analytical expressions are only available for limited circumstances. To overcome this challenge, a newly proposed method called Chebyshev–Galerkin–KL expansion was developed to discretize the random fields of soil parameters, from which the approximated eigenvalues and eigenfunctions can be obtained using the Chebyshev orthogonal polynomials of the second kind combined with Galerkin technique. Application of the proposed approach is illustrated through reliability analysis of an unsaturated slope example under different rainfall patterns, where the uncertainty in selection of a “best” soil-water characteristic curve (SWCC) model and statistical uncertainties in SWCC model parameters are taken into account. Results show that the developed approach is feasible to generate random fields with sufficient accuracy. Under a constant rainfall duration, the Advanced pattern may lead to shallow landslide with the highest probability, followed by Intermediate and Delayed. It should be noted that Bayesian inference and determination of optimal SWCC model should be carried out prior to reliability analysis. Otherwise, the landslide risk level would be exaggerated.

A New Approach to the Evaluation of the Lightning Horizontal Electric Field Over a Lossy Ground Based on the Time-Domain Cooray–Rubinstein Formula Fitted by Orthogonal Polynomials

The calculation of lightning return stroke electromagnetic field is the basis for evaluating transmission line lightning-induced voltages. The Cooray–Rubinstein formula in the time domain is widely used to evaluate the horizontal electric field of lightning return stroke on a lossy ground, but the evaluation of the formula requires high computation cost. Therefore, this study presents a new approach to evaluate the Cooray–Rubinstein formula based on orthogonal polynomial fitting. This method converts the complex convolution in the Cooray–Rubinstein formula into the sum of multiple simple convolutions by using an effective truncation scheme of Legendre orthogonal polynomial and the second Chebyshev orthogonal polynomial. The objective is to realize an efficient evaluation method to the horizontal electric field of the lightning return stroke. On the premise of ensuring the calculation accuracy, the effectiveness and correctness of the proposed method are verified through numerical experiments. Moreover, the application of the time-domain Cooray–Rubinstein formula based on two orthogonal polynomial fitting methods is discussed. Compared with the existing numerical methods for solving the Cooray–Rubinstein formula, the proposed method is easy to implement and efficient, which provides a new idea for evaluating the horizontal electric field of lightning return stroke above the lossy ground in the time domain.

Research on High-frequency stock price prediction based on Chebyshev-Stacking and Weighted LSTM neural network [1

To solve the problem of high-frequency stock price prediction, this paper proposed a prediction model based on Chebyshev-Stacking and a weighted LSTM neural network. The proposed method extracts the function characteristic information of the high-frequency stock price series through Chebyshev orthogonal polynomial basis expansion. Considering that the potential model structure between each component of the function feature vector and the residual sequence predicted by the LSTM neural network is unknown and there is a certain noise, this paper used the Stacking framework to enhance the data and weighed the bias and variance of the prediction model. In addition, since the number of predictor variable periods of the LSTM neural network is a hyperparameter, the model averaging method based on distance covariance is used for optimization. The results of actual data analysis show that the proposed method is significantly better than the original LSTM neural network in terms of mean square error, absolute error, and relative error. By selecting the different number of training sets, the robustness of the improved model is verified. Finally, the proposed method can also be used in practical applications such as daily average temperature prediction, missile trajectory prediction, and real-time monitoring of atmospheric environment quality.

Spectral treatment for the fractional-order wave equation using shifted Chebyshev orthogonal polynomials

This paper will examine the approximate solution for the fractional-order wave equation. The method used for this purpose fundamentally is based on the second kind of Chebyshev polynomials besides the Chebyshev collocation mechanism in addition to the forward finite difference scheme. The fractional-order terms are expressed through Caputo’s fractional-order derivative definition. With these tools, the problem will be converted into an algebraic system of equations that are solved numerically. The accuracy and applicability of the proposed technique are demonstrated through some given numerical applications.

Контактная задача об осесимметричном кручении упругого слоя посредством цилиндрического штампа

The paper studies the axis-symmetric contact problem between an elastic layer and a rigid cylindrical circular stamp under torque. The stamp adheres to the upper boundary of the layer whereas the lower boundary of the layer is rigidly fastened. With the application of Hankel integral transform solving the problem reduces to solving the first kind Fredholm integral equation (IE) with symmetrical kernel, represented as a sum of its principal part, Weber-Sonin integral, and the regular kernel. It is estimated that once its height attains a certain level, the layer actually deforms as a semi-space. In the process, through Abel IE method, the solution of the well-known Reissner-Sagoci problem is obtained once again and the original first kind Fredholm IE is reduced to the second kind Fredholm IE. Concurrently, using the collocation method combined with Gauss type quadrature formulas for integral estimation, the original IE reduces to a finite system of linear algebraic equations. To obtain this quadrature formula, properties of Gegenbauer and Chebyshev orthogonal polynomials are used. In the enough wide range of change of characteristic elastic and geometrical parameters of the problem numerical analysis is performed and patterns of changes of tangential contact stresses under the stamp as well as the angle of twist of the stamp are identified.

Bayesian-based method for the simultaneous identification of structural damage and moving force

The input force is usually difficult to measure directly in the damage identification due to the system uncertainty and noise uncertainty. A Bayesian method for the simultaneous identification of damage and moving force is proposed. The moving force on the structure are transformed into equivalent nodal forces through the element function, and these equivalent nodal forces are expanded by using Chebyshev orthogonal polynomials. The posterior probability density function of the damage parameter and the force orthogonal parameter under Bayesian theory is deduced in time-domain. The prior information of the Chebyshev orthogonal polynomials adopts the uniform prior, and the damage parameters adopt the Gaussian prior in accordance with the characteristics of each parameter. The Laplace approximation method for time domain data is used to determine the hyperparameters of elemental damage and moving force. The first-order and second-order sensitivities of the dynamic response with damage and force parameters are derived to update the hyperparameters and to accelerate gradient-based system identification. The uncertainty of the identified results of damage parameters is quantified by using an approximate Gaussian distribution. Two numerical cases of a simply supported beam and the Tanjiang Bridge on the Shen-Mao Railway show that the proposed method can simultaneously identify the elemental damage and moving forces with uncertainty quantified.

Nonlocal anisotropic elastic shell model for vibrations of double-walled carbon nanotubes under nonlinear van der Waals interaction forces

In this paper, a novel nonlocal anisotropic elastic shell model is developed to investigate the nonlinear vibrations of double-walled carbon nanotubes (DWCNTs) in the framework of Sanders–Koiter shell theory. Van der Waals interaction forces between the two concentric single-walled carbon nanotubes (SWCNTs) composing a DWCNT are modelled via Lennard-Jones potential and He’s formulation. In the linear vibration analysis, the displacement field of each SWCNT is expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction, and Rayleigh–Ritz method is considered to get approximate natural frequencies and modal shapes. In the nonlinear vibration analysis, the three displacements of each SWCNT are re-expanded by means of the approximate eigenfunctions derived in the linear analysis, and an energy approach based on Lagrange equations is adopted to obtain a set of nonlinear ordinary differential equations of motion, which is then solved numerically. Molecular dynamics simulations are performed in order to calibrate the proper value of nonlocal parameter to be inserted in the constitutive equations of the proposed elastic continuum model. A simplified linear distribution of van der Waals interaction forces is initially adopted to analyse the nonlinear vibrations of DWCNTs, obtaining a hardening nonlinear behaviour. By considering a more realistic nonlinear distribution of van der Waals interaction forces, a stronger hardening nonlinear behaviour is found.

Connectivity of rough contacts in plane and axially symmetric cases

It is customary to assume that a contact of rough elastic solids is always multiply connected, i.e. have regions in it where contact surfaces are apart of each other and contact pressure is zero. The issue of the connectivity in rough elastic contacts has both theoretical and practical interest. In this paper we extend the earlier conducted analysis of rough contacts in plane formulation on the case of axially symmetric rough elastic contacts and compare our findings. The main goal of the paper is to obtain an analytical solution of axially symmetric rough elastic contact and analyze its properties such as contact connectivity and contact pressure smoothness compared to the smoothness of the surface roughness profile. This goal is achieved by using solution expansions in Legendre orthogonal polynomials. In case of a plane contact of rough elastic solids it has been achieved using the solution expansions in Chebyshev orthogonal polynomials. In both axially symmetric and plane cases of rough elastic contacts the ranges of contact parameters for which contacts are singly connected have been determined.