Use Of Chebyshev Polynomials Research Articles

Forced vibrations of an elastic triangular plate supported around its perimeter by a unilateral support via the Chebyshev polynomial expansion

The present study introduces static and dynamic analysis of an elastic triangular plate on unilateral edge supports, including forced vibrations due to loading and unloading excitation. The plate is assumed to be subjected to a uniformly distributed load and an eccentrically applied concentrated load. Furthermore, the loads are assumed to display dynamic variations. The governing equation of the problem is derived by considering the static and the dynamic responses of the plate by including inertia forces. The analytical solution is conducted by employing a series of Chebyshev polynomials for the admissible displacement functions and using Lagrange’s equations of motion. Having found the governing equation of the problem, an approximate numerical solution is accomplished using an iterative process due to the non-linear properties of the unilateral edge supports. The static behavior of the plate under a concentrated load is investigated in detail numerically by considering a wide range of parameters of the plate geometry and the stiffness of the support. In deriving the governing equation of the problem and in the numerical solution, special care is taken to use nondimensional parameters. This approach ensures that the analysis and conclusions remain valid across a wide range of parameter values. Dynamic treatment of the problem is carried out in the time domain by assuming a stepwise time variation of the concentrated load and by employing the constant acceleration procedure in the numerical solution of the system of governing differential equations derived from the equation of motion. Time variations of the displacements, the contact points, and the reactions of the plate’s support are presented in figures for various values of the parameters of the plate, as well as those of the supports, particularly focusing on the non-linearity of the problem due to the plate liftoff from the unilateral support. The effects of the parameters and the loading are investigated in detail. The results reveal that the unilateral property of the support stiffness significantly affects the static and dynamic behavior of the triangular plate. The analysis can be extended easily to cover a general type of loading as well. The main issues addressed in this study include: the analysis of a triangular plate with unilateral perimeter support that does not resist tensile forces; the use of Chebyshev polynomials as admissible functions in both directions; the consideration of static and dynamic loads; the determination of lifting areas of the support in both cases and the assessment of the global vertical force balance in the plate, including inertia forces in the dynamic case.

Large deformation assessment of three-tower suspension bridges: An enhanced analytical model and solution algorithm

An enhanced nonlinear approach is proposed in this study to address the limitations of traditional deflection theory in analyzing large deformations of three-tower suspension bridges. In this approach, the three-tower suspension bridge is simplified as two quasi-independent bridges for analysis, and detailed considerations are given to the vertical and longitudinal deformations of the main cable, which produce the expression for the relationship between the cable's internal force and deformations. By employing the complementary energy principle and variational calculus, the nonlinear differential equations describing bridge deformations are then established. To ensure the displacement continuity at the middle tower, this paper proposes the use of Chebyshev polynomials to approximate the solutions of the differential equations, and the weighted residual method is then used to transform these differential equations into algebraic equations. On this basis, the developed nonlinear adjustment optimization algorithm enables efficient solving of the nonlinear equation and thus determination of bridge deformation under various load conditions. The accuracy and effectiveness of the proposed scheme are validated through the analysis of two suspension bridges with equal and unequal spans. Finally, the extended applications of the proposed scheme on the other type of multi-span suspension bridges are introduced for further applications.

Thermal damage analysis in tissue caused by electromagnetic radiation using space–time collocation method

Over the past half-century, the usage of external heat sources in medical applications has increased substantially. Controlling heat damage is essential for ensuring the efficacy of the treatment. Living tissues are highly non-homogeneous; hence, it is important to take into account the effects of local non-equilibrium on their thermal behavior. In the present study, two- and three- space dimensional time-space fractional single-phase-lag (SPL) and dual-phase-lag (DPL) models for bio-heat transfer in tissue are considered to study the thermal damage and temperature in tissue caused by electromagnetic radiation as an external heat source. The considered mathematical models are more general and consider non-Fourier as well as non-local effects. We obtain the numerical solution for the models by combining Gaussian RBFs and shifted Chebyshev polynomials in the space and time directions, respectively. The RBFs depend on Euclidean distance, so they can easily be used in multidimensional space domain, and the use of Chebyshev polynomials gives spectral accuracy in time direction. It is also explored how different parameters, such as blood perfusion rate Wb, phase lags τq, τt, and fractional derivatives α, β, affect the temperature distribution and thermal damage in the tissue.

Are Chebyshev-based stability analysis and Urabe’s error bound useful features for Harmonic Balance?

Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors’ knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe’s error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.

Temperature response in skin tissue during hyperthermia based on three-phase-lag bioheat model using RBF meshfree method

Various non-Fourier heat conduction models have been proposed to overcome the paradox of the infinite propagation speed of heat signals in Fourier’s law. The three-phase-lag (TPL) heat model is a non-Fourier model based on the linearized theory of coupled thermoelasticity that combines heat flux, temperature gradient, and thermal displacement gradient. The present paper proposes a novel method with meshfree and spectral nature to solve the two-dimensional three-phase lag (TPL) bioheat model for tissue heating during hyperthermia. In the space domain, radial basis functions (RBFs) and, in the time direction, shifted Chebyshev polynomials are employed to solve the model. The use of Chebyshev polynomials in the time direction enables one to get the solution in the whole time domain simultaneously with fewer nodes. Further, there is no need for background meshes in space due to the meshless nature of RBFs, and the proposed method is equally applicable to irregular domains. The proposed method is validated with the existing semi-analytical solution of the TPL model in one spatial dimension and found to be in good agreement. The temperature profile and the impact of various parameters, such as phase lag times, blood perfusion rate, and heat source parameters on heat transfer in tissue, have also been discussed.

A high-dimensional model to study the self-excited oscillations of rotary drilling systems

This paper analyzes the self-excited axial and torsional vibrations of rotary drilling systems with a model that combines a multi-degrees of freedom representation of the drilling structure with a rate-independent bit-rock interface law. The state-dependent delay introduced by the bit-rock interaction is captured by a bit trajectory function, which leads to the formulation of a nonlinear system of coupled partial differential equation (PDE) and ordinary differential equations (ODEs). This paper describes an algorithm that utilizes the Chebyshev spectral method to convert the system of PDE–ODEs into a system of coupled ODEs. The use of Chebyshev polynomials rather than Fourier series leads to a computationally more efficient algorithm, in view of the non-periodic nature of the bit trajectory function. By linearizing the system of coupled PDE–ODEs, this algorithm is further extended to assess numerically the stability of the high-dimensional model. The predictions of the linear stability analysis are validated with published results as well as with time-domain simulations. It is shown, as anticipated by Liu et al. (2014), that the stable region in the space of the operating parameters shrinks with increasing spatial resolution of the high-dimensional model, suggesting that the effects of damping on the stability of drilling system is generally negligible in the range of practical drilling parameters. Extensive parametric studies have been carried out using the high-dimensional model to analyze the influence of various factors on the torsional stick–slip oscillations. The bit wearflats are shown to play an important role in delaying or even suppressing the occurrence of torsional stick–slip oscillations. Finally, the axial vibrations propagating along the drillstring are revealed in the form of traveling waves, whereas the torsional vibrations take the form of standing waves, whose frequency is the same as the torsional resonance of the drillstring excited by the bit-rock interaction.

A local updating algorithm for personalized PageRank via Chebyshev polynomials

The personalized PageRank algorithm is one of the most versatile tools for the analysis of networks. In spite of its ubiquity, maintaining personalized PageRank vectors when the underlying network constantly evolves is still a challenging task. To address this limitation, this work proposes a novel distributed algorithm to locally update personalized PageRank vectors when the graph topology changes. The proposed algorithm is based on the use of Chebyshev polynomials and a novel update equation that encompasses a large family of PageRank-based methods. In particular, the algorithm has the following advantages: (i) it has faster convergence speed than state-of-the-art alternatives for local personalized PageRank updating; and (ii) it can update the solution of recent extensions of personalized PageRank that rely on complex dynamical processes for which no updating algorithms have been developed. Experiments in a real-world temporal network of an autonomous system validate the effectiveness of the proposed algorithm.

Efficient Design of Scaled Rectangular (Saramäki) Window

A rectangular-window sidelobe magnitude reduction method by widening the main lobe width was proposed by Saramäki. A technique for trading off main lobe width against sidelobe magnitude for any arbitrary window was reported by Lim et al.; a fast convergence algorithm for its implementation was proposed by the same authors in another article, where the derivatives of the window function are expressed in Chebyshev polynomials which have high arithmetic complexity. All the coefficients of a rectangular-window are equal; this special property is exploited, in this paper, for deriving the window function’s derivatives without the use of Chebyshev polynomials resulting in a great reduction in the arithmetic complexity.

Induced aseismic slip and the onset of seismicity in displaced faults

Abstract We address aseismic fault slip and the onset of seismicity resulting from depletion-induced or injection-induced stresses in reservoirs with pre-existing vertical or inclined faults. Building on classic results, we discuss semi-analytical modelling techniques for fault slip including dislocation theory, Cauchy-type singular integral equations and the use of Chebyshev polynomials for their solution and an eigenvalue-based stability analysis. We consider slip patch development during depletion for faults with zero, constant static and slip-weakening friction, and our results confirm earlier findings based on numerical simulation, in particular the aseismic growth of two slip patches that may subsequently merge and/or become unstable resulting in nucleation of seismic slip. New findings include improved approximate expressions for the induced seismic moment per unit strike length and a description of the effect of coupling between the slip patches which affects both forward simulation and eigenvalue computation for high values of the ratio of fault throw to reservoir height. Our implementation based on analytical inversion and semi-analytical integration with Chebyshev polynomials is more efficient and more robust than our numerical integration approach. It is not yet well suited for Monte Carlo simulation, which typically requires sub-second simulation times, but with some further development that option seems to be within reach. Moreover, our results offer a possibility for embedded fault modelling in large-scale numerical simulation tools.

Efficient, High-Fidelity Modeling and Simulation of Global Navigation Satellite System Constellations

This paper presents a method for modeling and simulation of Global Navigation Satellite System (GNSS) constellations by the use of Chebyshev polynomials fit to publicly available precision data. The Global Positioning System (GPS) is used to demonstrate the method, but the method applies to all GNSS with precise ephemeris data including Galileo and GLONASS. The method facilitates highly accurate satellite orbit and clock modeling, including relativistic effects, with mean fit differences of the order in position, in velocity, and microseconds in clock offset. The method also improves simulation run time by using a fast technique for evaluating the Chebyshev polynomials. Runtime comparisons demonstrate the polynomial method reduces runtime by up to 72% of more traditional methods. The speed of the method makes it well suited for practical, accurate Monte Carlo simulation techniques. The method can be used in simulations for the verification and validation of GNSS receivers embedded in autonomous aerospace navigation systems, for uncertainty analysis, and for other purposes. Details for validating the precision data are given. Pseudocode algorithms are provided for fitting and evaluating Chebyshev polynomials. Also provided are high-level concepts of one approach for integrating the method into larger, multisystem aerospace simulations, including the effects of signal propagation delay and Earth blockage.

Приближенное решение гиперсингулярного интегрального уравнения I рода, ограниченное на обоих концах отрезка интегрирования, с применением рядов Чебышева

A hypersingular integral equation on the interval of integration is considered. The hypersingular integral is understood in the sense of Hadamard, that is, in the finite part. The class of such equations is widely used in problems of mathematical physics, in technology, and most importantly: in recent years, they are one of the main devices for modeling problems in electrodynamics. With the use of Chebyshev polynomials of the second kind, the unknown function, the right-hand side and the kernel are replaced in the equation. The expansion coefficients of these functions are calculated using quadrature formulas of the highest algebraic degree of accuracy, i.e., Gauss quadrature formulas. Thus, the equation is discretized. The result is an infinite system of linear algebraic equations for the expansion coefficients of the unknown function. The fact that the hypersingular integral equation in the case under consideration has a unique solution in the class of sufficiently smooth functions is taken into account. The constructed computational scheme is substantiated using the general theory of functional analysis. The calculation error is estimated under certain conditions relative to the right-hand side and the kernel of the equation. The described method for solving the hypersingular integral equation is illustrated by test examples that show the high efficiency of the method.

Free-boundary MRxMHD equilibrium calculations using the stepped-pressure equilibrium code

The stepped-pressure equilibrium code (SPEC) (Hudson et al 2012 Phys. Plasmas 19, 112 502) is extended to enable free-boundary multi-region relaxed magnetohydrodynamic (MRxMHD) equilibrium calculations. The vacuum field surrounding the plasma inside an arbitrary ‘computational boundary’, , is computed, and the virtual-casing principle is used iteratively to compute the normal field on so that the equilibrium is consistent with an externally produced magnetic field. Recent modifications to SPEC are described, such as the use of Chebyshev polynomials to describe the radial dependence of the magnetic vector potential, and a variety of free-boundary verification calculations are presented.

Calculation of the Energy Coefficients of Reflection and Transmission for the Layered Periodic Media

Currently, much attention is paid to the study of photonic crystals – materials with an ordered structure characterized by a strictly periodic change in the refractive index at scales comparable to the wavelengths of radiation in visible and near infrared ranges. This is a dynamically developing direction of modern materials science. It is connected with the possibility of creating LEDs with high efficiency, new types of lasers with low threshold generation, light waveguides, optical switches, filters, as well as digital computing devices based on Photonics. The aim of this work is to calculate the reflection and transmission of a polarized light wave from a layered system of nanostructures that form a periodic medium. The calculation is carried out by two methods: the method of characteristic matrices and the method based on the use of Chebyshev polynomials. The authors have created a basic component of the computer program for calculating the reflection coefficient and the transmittance of layered nanostructures. The paper calculates the spectra of reflection and transmission coefficients and presents the analysis of the results obtained. The basic element is chosen as a basic nanostructure: a layer of magnesium oxide MgO 100 nm thick, a diamond layer 160 nm thick, an arsenic layer AsBr3 tribromide 80 nm thick, a silicon layer 120 nm thick. The authors compare the two methods used: the results are almost the same, which makes it possible in practice for such structures to use a simpler method for the computational procedure based on Chebyshev polynomials.

Design and analysis of novel Chebyshev neural adaptive backstepping controller for boost converter fed PMDC motor

An adaptive backstepping Chebyshev neural network controller (ABCNNC) is proposed for the boost converter fed PMDC motor to track the angular velocity. The computational complexity of the neural network is avoided by the use of Chebyshev polynomials as the basis function. The online weight update of the Chebyshev neural network (CNN) is designed for the closed loop system based on the Lyapunov stability analysis to obtain the asymptotically stable system. A detailed analysis of the steady state and transient performance is performed and results are compared with that of conventional PI controller and radial basis function neural network controller (RBFNNC). To ensure the robustness of the proposed ABCNNC, it is being analysed for a wide range of variations in load torque and the set point changes and it is validated by comparing with the conventional PI control approach and RBFNNC. Comparison of results validates that the proposed ABCNNC shows the enhanced transient and steady state responses for the uncertainties caused by disturbances, than conventional PI controller and RBFNNC.

On the use of Chebyshev polynomials in the Rayleigh-Ritz method for vibration and buckling analyses of circular cylindrical three-dimensional graphene foam shells

This study aims to perform free vibration and buckling analyses of circular cylindrical shells made of three-dimensional (3D) graphene foam. Three kinds of porosity distribution are considered including two kinds of symmetrical distribution and uniform distribution. The Kirchhoff-Love shell theory is utilized to establish the mathematical model of 3D graphene foam (3D-GF) shells. Natural frequencies and buckling loads of the shells under various boundary conditions are obtained via the Rayleigh-Ritz method in conjunction with Chebyshev polynomials. Results show that the mechanical characteristics of the 3D-GF shells are significantly affected by the porosity coefficient and porosity distribution. Moreover, the effect of porosity coefficient is closely related to boundary conditions of the shells. It is also shown that as the porosity coefficient increases, the porosity distribution effect on the mechanical characteristics of 3D-GF shells becomes increasingly significant.

Stable method for solving the Cauchy problem with the use of Chebyshev polynomials

PurposeThe purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ1) and find proper method for the problem regularization.Design/methodology/approachThe solution of the direct and the inverse (Cauchy-type) problems for the Laplace’s equation is presented in the paper. The form of the solution is noted as the linear combination of the Chebyshev polynomials. The collocation method in which collocation points had been determined based on the Chebyshev nodes was applied. To solve the Cauchy problem, the minimum of functional describing differences between the target and the calculated values of temperature and the heat flux on a section of the domain’s boundary was sought. Various types of the inverse problem regularization, based on Tikhonov and Tikhonov–Philips regularizations, were analysed. Regularization parameter α was chosen with the use of the Morozov discrepancy principle.FindingsCalculations were performed for random disturbances to the heat flux density of up to 0.01, 0.02 and 0.05 of the target value. The quality of obtained results was next estimated by means of the norm. Effect of the disturbance to the heat flux density and the type of regularization on the sought temperature distribution on the boundary Γ1 was investigated. Presented methods of regularization are considerably less sensitive to disturbances to measurement data than to Tikhonov regularization.Practical implicationsDiscussed in this paper is an example of solution of the Cauchy problem for the Laplace’s equation in the rectangular domain that can be applied for determination of the temperature distribution on the boundary of the heated element where it is impossible to measure temperature or the measurement is subject to a great error, for instance on the inner wall of the boiler. Authors investigated numerical examples for functions with singularities outside the domain, for which values of gradients change significantly within the calculation domain what corresponds to significant changes in temperature on the wall of the boiler during the fuel combustion.Originality/valueIn this paper, a new method for solving the Cauchy problem for the Laplace’s equation is described. To solve this problem, the Chebyshev polynomials and nodes were used. Various types of regularization of this problem were considered.

Artificial Intelligence to Design a Mask Insensible to the Distance From the Camera to the Scene Objects

The sharpness of an image depends on the spatial frequency response of the photographic imaging system and the sensor characteristics. In conventional digital cameras, only those objects within a distance range are in focus, while other objects are captured with different amounts of blurring depending on their distance to the focal plane. This can be desired for some applications; however, this can also be undesired because some objects in the scene may be blurred and impossible to recover. In the field of augmented reality, simulating this natural effect of showing sharp objects in combination with blurred objects increases the visual realism of augmented video. In order to simulate this effect, it is very important to capture all objects in the scene with high quality so that it could be possible to dynamically blur different objects in the scene at runtime. In this paper, we present an algorithm to find a set of possible complex-amplitude transmittance masks capable of considerably reducing the impact of focus errors in the scene objects. Computer simulations are used to compare the masks found in this paper with a classic mask in the state of the art. The main contribution of this paper is the use of Chebyshev polynomials to model an optical mask, and then, use artificial intelligence to establish some properties of this mask, such as the depth of field, the resolution, and the amount of gathered light.

Algorithms For Positive Polynomial Approximation

We propose several algorithms for positive polynomial approximation. The main tool is a novel iterative method to compute non negative interpolation polynomials at any order, which is shown to converge under conditions that make it suitable for the numerical approximation of positive functions. Our method is based on the special representations of non negative polynomials provided by the Lukacs Theorem, and a key point is the use of Chebyshev polynomials for the initial step of the iterations. Numerical results illustrate the convergence properties of the proposed algorithms, and they are completed with a first application of this technique to the positive discretization of the advection equation.

A novel low complexity downlink linear precoding algorithm for massive MIMO systems

The tremendous advancements in 4G and beyond 4G wireless standards requires a high demand for the increased data rate, high spectral efficiency and minimal power requirement. Massive MIMO is a key technology that can be used to attain all the above requirements at the cost of increased complexity. The performance of this massive MIMO system can be optimized by simple linear precoding techniques, the complexity of which lies on the inversion of large size matrix. In this paper, we propose a large scale low complexity matrix inversion algorithm which is highly suitable for parallel architecture. The proposed algorithm makes use of Chebyshev polynomial and Weyl’s inequality and named as Weyl’s Chebyshev acceleration (WCA) algorithm. This algorithm is further simplified by exploiting the diagonal dominance property of the positive definite Hermitian Gram matrix that is to be inverted. The specialty of this algorithm is, it is inner product free which makes this highly suitable for parallel computing environment and thus the algorithm becomes speedy. The performance of the proposed algorithm is evaluated in urban micro cell scenario and is proven to be more efficient in terms of BER performance and complexity. Also the proposed WCA based precoding algorithm achieves approximately 50% of complexity saving in the total flop count in comparison with the existing algorithms for the micro cell scenario. The BER performance reaches the near optimal results of ZF algorithm as SNR increases.

Cosmographic analysis with Chebyshev polynomials

The limits of standard cosmography are here revised addressing the problem of error propagation during statistical analyses. To do so, we propose the use of Chebyshev polynomials to parameterize cosmic distances. In particular, we demonstrate that building up rational Chebyshev polynomials significantly reduces error propagations with respect to standard Taylor series. This technique provides unbiased estimations of the cosmographic parameters and performs significatively better than previous numerical approximations. To figure this out, we compare rational Chebyshev polynomials with Pad\'e series. In addition, we theoretically evaluate the convergence radius of (1,1) Chebyshev rational polynomial and we compare it with the convergence radii of Taylor and Pad\'e approximations. We thus focus on regions in which convergence of Chebyshev rational functions is better than standard approaches. With this recipe, as high-redshift data are employed, rational Chebyshev polynomials remain highly stable and enable one to derive highly accurate analytical approximations of Hubble's rate in terms of the cosmographic series. Finally, we check our theoretical predictions by setting bounds on cosmographic parameters through Monte Carlo integration techniques, based on the Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic data, using the JLA supernovae sample and the most recent versions of Hubble parameter and baryon acoustic oscillation measurements. We find that cosmography with Taylor series fails to be predictive with the aforementioned data sets, while turns out to be much more stable using the Chebyshev approach.