Chebyshev Polynomial Expansion Research Articles

Numerical solutions of integral and integro-differential equations using Chebyshev polynomials of the third kind

Our purpose in this study is to construct an algorithm based on the use of a finite expansion in Chebyshev polynomials of the third kind to solve singularly perturbed Volterra integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The convergence of the method is investigated. Finally, some numerical experiments, which confirm the theoretical results, are shown and comparisons with other methods in literature are given.

Quantum wavepacket method for state-to-state reactive cross sections in hyperspherical coordinates.

We present theory for calculating state-to-state differential cross sections (DCS) of triatomic reactive scattering in hyperspherical coordinates using a quantum wavepacket method. The adiabatically adjusting, principal axes hyperspherical coordinates proposed by Pack and Parker [J. Chem. Phys. 87, 3888 (1987)] are applied, which deal with all arrangement channels equivalently, allowing the analysis of the products in all three arrangement channels with one main propagation. The propagated wavepacket is analyzed by projecting it onto the product ro-vibrational states at a fixed, asymptotic radius, R, of the corresponding Jacobi coordinates; thus, the channel-specified S-matrix elements can be calculated by matching the projections with the boundary conditions in the Jacobi coordinates. For numerical demonstrations, state-to-state DCS of the H + HD ( 0 = 0, j 0 = 0) reaction and state-to-state reaction probabilities of the O + O2 ( 0 = 0, j 0 = 0) reaction and the F + HCl ( 0 = 0, j 0 = 0) → HF + Cl reaction for zero total angular momentum are presented. The second order split operator method and the Chebyshev polynomial expansion method were applied to propagate the wavefunction. The relative numerical efficiencies for calculating the state-to-state information of triatomic reactive scattering using the hyperspherical coordinate and the reactant Jacobi coordinate are discussed.

High-Performance Simulation for Quantum Charge Transport in Large Scale Materials -Application to Carbon Nanotubes and Organic Semiconductors-

Carbon-related materials, such as carbon nanotubes (CNTs), graphenes, and organic semiconductors, have attracted much attention as new functional materials used in electronic devices. For example, the carbon nanotubes (CNTs) and graphenes are expected as high-mobility materials in comparison with silicon semiconductors. On the other hand, the mobility of organic semiconductors is much lower, but organic semiconductors have advantages as low-cost, light-weight, and printable materials. Recently, to improve the low mobility, synthesis of new organic molecules has been actively carried out. Prediction of the charge transport properties of various new materials is very important from a viewpoint of materials development. Furthermore, the systematic understanding of various transport phenomena is a challenging issue in a field of basic science. Therefore, we have developed a new numerical simulation methodology called the time-dependent wave-packet-diffusion (TD-WPD) method [1], in which we evaluate the mobility from the wavepacket dynamics using Kubo’s formula. To reduce the computational cost, we apply Chebyshev polynomial expansion to the time-evolution operator. As the result, we perform the order-N calculations, which enable to treat up to a hundred million atoms. This indicates that we can directly compute the transport property from an atomic point of view and compare the simulation results with experimental observations of samples which have micron-order length. The transport property around room temperature is dominated by the phonons (the molecular vibrations). The effects of phonons on electronic states are included as the time-dependent electron transfer integrals, since the transfer integrals are dynamically modulated by phonons. We obtain the motion of atoms using the classical molecular dynamics, then evaluate the transfer integrals using the density functional theory (DFT). As the result, we can investigate the transport properties of micron-scale material at room temperature from an atomistic viewpoint within an accuracy of DFT. As a demonstration, we apply the TD-WPD method to the metallic (5,5)-CNTs and investigate how the transport property at room temperature changes according to the CNT length. Figure 1(a) shows the calculated length dependence of the resistance. We can confirm the crossover from the ballistic transport region to the diffusive band transport region. When the CNT length becomes small to zero, the resistance becomes independent of the length and converges to the inverse of quantized conductance. This is a typical behavior of the quantum ballistic transport limit. On the other hand, the resistance monotonically increases with the length when the CNT becomes longer than 1 micron. This means that the classical diffusive transport is realized in the long CNTs. From our simulation, the mean free path is estimated as 515 nm. We can specify the quasi-ballistic region for the region where the resistance deviates from the diffusive and ballistic transport limits, which is 300 -- 1000 nm for the (5,5)-CNTs. The calculated results quantitatively agree with experimental data. Then, we investigate the temperature dependence of mobility for representative single-crystal organic semiconductors, pentacene and C8-BTBT. It is well known that the high-quality single-crystal pentacene has the temperature-independent mobility of 1 cm2/Vs around room temperature while the single-crystal C8-BTBT exhibits the diffusive band-like transport with power-law temperature dependence. The mobility of C8-BTBT is about 10 times larger than that of pentacene at room temperature. In Fig. 1(b), circles and squares represent the calculated mobilities as a function of temperature for single-crystal pentacene and C8-BTBT, respectively. The calculated temperature dependences are quantitatively in good agreement with the experimental values for the single crystal devices. The simulations give the theoretically predicted upper limit of intrinsic mobility. Even if the same organic semiconductors are used for devices, the measured mobilities have distributions to some extent depending on the employed device architecture and the crystal quality. Low-quality devices show the low mobility with thermally activated behavior. To take the difference of device quality into account, we introduce the carrier-trap potentials in the simulation. The calculated mobility for pentacene are plotted by triangles. We can confirm the magnitude of mobility dramatically decreases and the temperature dependence changes to thermally activated behavior characterized by the hopping transport nature. In our presentation, we will present our TD-WPD methodology and discussions about the above-mentioned results for CNTs and organic semiconductors in detail. We also show you how our simulation can provide a systematic understanding for different transport mechanisms, such as the ballistic transport, diffusive-band transport, and hopping transport. [1] H. Ishii, et al., Phys. Rev. B 82, 085435 (2010); Phys. Rev. B 85, 245206 (2012); Phys. Rev. B 90, 155458 (2014); Phys. Rev. B 95, 035433 (2017). Figure 1

An analytical solution to the aeroelastic response of a two-dimensional elastic plate in axial potential flow

This paper presents a novel analytical model that predicts the two-way coupled aeroelastic response of a linear elastic plate in axial potential flow, including the effects of plate curvature. The plate deforms in dynamic balance of the inertia, elastic, and aerodynamic forces. Analytical solutions are obtained by deriving the generalized aerodynamic force with respect to the beam eigenfunctions, which are expressed in a Chebyshev polynomial expansion. Exact expressions are derived for the generated lift, thrust and required input power. The derived solution agrees well with the results reported in the literature for plate flutter and flapping wings.

Ab Initio Simulation of Transport Properties in Rb–CH4 and Cs–CH4 Laser Media

The pair interaction potentials of the weakly bound Rb–CH4 and Cs–CH4 systems, which are active media of alkali metal vapor lasers with broadband diode or excimer laser pumping, were calculated by the ab initio method. The electronic problem was solved by the coupled-cluster method in the CCSD(T) version including the basis set superposition error and extrapolation to an infinite basis set. The obtained pointwise ab initio potentials were approximated by the analytical functions based on the orthogonal Chebyshev polynomial expansion with correct asymptotic behavior at the dissociation limit and then used within the framework of the molecular kinetic theory of rarefied gases to evaluate the reduced collision integrals and mutual diffusion coefficients.

A polynomial‐based method for topology optimization of phononic crystals with unknown‐but‐bounded parameters

SummaryThe design and analysis of phononic crystals (PnCs) are generally based on the deterministic models without considering the effects of uncertainties. However, uncertainties that existed in PnCs may have a nontrivial impact on their band structure characteristics. In this paper, a sparse point sampling–based Chebyshev polynomial expansion (SPSCPE) method is proposed to estimate the extreme bounds of the band structures of PnCs. In the SPSCPE, the interval model is introduced to handle the unknown‐but‐bounded parameters. Then, the sparse point sampling scheme and the finite element method are used to calculate the coefficients of the Chebyshev polynomial expansion. After that, the SPSCPE method is applied for the band structure analysis of PnCs. Meanwhile, the checkerboard and hinge phenomena are eliminated by the hybrid discretization model. In the end, the genetic algorithm is introduced for the topology optimization of PnCs with unknown‐but‐bounded parameters. The specific frequency constraint is considered. Two numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Band structure analysis of phononic crystals based on the Chebyshev interval method.

Due to environmental changes and manufacturing errors, uncertainties inherently exist in phononic crystals, especially in the material properties and geometric parameters. To handle the uncertainties with limited information, two different methods are proposed for band structure analysis of phononic crystals with uncertainties. One is the Monte Carlo method (MCM). The main shortcoming of it is the high computational cost. The other is the simplex Chebyshev polynomial expansion (SPCE) method. In addition, the computational efficiency of the SPCE method is much higher. In the SPCE method, the simplex Chebyshev is employed to estimate the band structures of phononic crystals. Meanwhile, the simplified incremental sampling strategy is introduced for the simplex Chebyshev to retain calculation accuracy and improve computational efficiency simultaneously. In the Chebyshev surrogate model, the samples yielded with the MCM are used to calculate the interval ranges of the band structures in phononic crystals. Three numerical examples, including a two-dimensional (2D) Helmholtz resonator phononic crystal, a 2D solid-solid phononic crystal, and a three-dimensional phononic crystal, are introduced to verify the effectiveness and efficiency of the proposed method.

Analytical Solution for Optimal Control by the Second Kind Chebyshev Polynomials Expansion

Second kind Chebyshev polynomials are modified set of defined Chebyshev polynomials by a slightly different generating function. This paper presents new and efficient algorithm for achieving an analytical approximate solution to optimal control problems. The proposed solution is based on state parameterization, such that the state variable is approximated by the second kind Chebyshev polynomials with unknown coefficients. At first, the equation of motion, boundary conditions and performance index are changed into some algebraic equations. This task converts the optimal control problem into an optimization problem, which can then be solved easily. The presented technique approximates the control and state variables as a function of time. After optimizing, the system is converted into a feedback mode for having the closed loop profits. The results proved the algorithm convergence. Finally by analyzing two numerical examples, the reliability and effectiveness of the proposed method by comparing two different methods is demonstrated.

Marangoni convection in water–alumina nanofluids: Dependence on the nanoparticle size

A linear stability analysis for the onset of Marangoni convection in a horizontal layer of distilled water/alumina nanofluid heated from below is investigated using a model which incorporates the effects of Brownian and thermophoreticdiffusions. The analysis uses realistic constitutive models for thermal conductivity and dynamic viscosity which include a dependence on average size of nanoparticles in addition to the volume fraction of nanoparticles and temperature. The lower boundary of the layer is a rigid surface at fixed temperature while the upper boundary behaves as a non-deformable free surface cooled by convection to an exterior region at a fixed temperature. Both boundaries are assumed to be impenetrable to nanoparticles so that their distribution is determined from a conservation condition. The numerical results are obtained using the method of expansion of Chebyshev polynomials. Stability boundary curves are investigated for nanoparticle sizes ranging from 15 nm to 100 nm. Nanofluids with nanoparticle of smaller average size are found to be more stable than those with larger sized nanoparticles.

Identification of time-varying neural dynamics from spike train data using multiwavelet basis functions

BackgroundTracking the changes of neural dynamics based on neuronal spiking activities is a critical step to understand the neurobiological basis of learning from behaving animals. These dynamical neurobiological processes associated with learning are also time-varying, which makes the modeling problem challenging. New methodWe developed a novel multiwavelet-based time-varying generalized Laguerre–Volterra (TVGLV) modeling framework to study the time-varying neural dynamical systems using natural spike train data. By projecting the time-varying parameters in the TVGLV model onto a finite sequence of multiwavelet basis functions, the time-varying identification problem is converted into a time invariant linear-in-the-parameters one. An effective forward orthogonal regression (FOR) algorithm aided by mutual information (MI) criterion is then applied for the selection of significant model regressors or terms and the refinement of model structure. A generalized linear model fit approach is finally employed for parameter estimation from spike train data. ResultsThe proposed multiwavelet-based TVGLV approach is used to identify both synthetic input-output spike trains and spontaneous retinal spike train recordings. The proposed method gives excellent the performance of tracking either sharply or slowly changing parameters with high sensitivity and accuracy regardless of the a priori knowledge of spike trains, which these results indicate that the proposed method is shown to deal well with spike train data. Comparison with existing methodsThe proposed multiwavelet-based TVGLV approach was compared with several state-of-art parametric estimation methods like the steepest descent point process filter (SDPPF) or Chebyshev polynomial expansion method. The conventional SDPPF algorithm, or SDPPF with B-splines wavelet expansion method was shown to have the poor performance of tracking the time-varying system changes with the synthetic spike train data due to the slow convergence of the adaptive filter methods. Although the Chebyshev polynomial basis function method gave the good parametric estimation results, it requires prior parameter estimation. It was shown that the proposed multiwavelet-based TVGLV method can track the time-varying parameter changes rapidly and accurately. ConclusionsThe multiwavelet-based TVGLV modeling framework developed in this paper can not only provide a computational modeling scheme for investigating such nonstationary properties, track more general forms of changes in time-varying neural dynamics, and but also may potentially be applied to investigate the spatial-temporal information underlying biomedical spiking signals.

Comparison of Chebyshev and Legendre Polynomial Expansion of Phase Function of Cloud and Aerosol Particles

Chebyshev and Legendre polynomial expansion is used to reconstruct the Henyey-Greenstein phase function and the phase functions of spherical and nonspherical particles. The result of Legendre polynomial expansion is better than that of Chebyshev polynomial for around 0-degree forward angle, while Chebyshev polynomial expansion produces more accurate results in most regions of the phase function. For large particles like ice crystals, the relative errors of Chebyshev polynomial can be two orders of magnitude less than those of Legendre polynomial.

Chebyshev-polynomial expansion of the localization length of Hermitian and non-Hermitian random chains.

We study Chebyshev-polynomial expansion of the inverse localization length of Hermitian and non-Hermitian random chains as a function of energy. For Hermitian models, the expansion produces this energy-dependent function numerically in one run of the algorithm. This is in strong contrast to the standard transfer-matrix method, which produces the inverse localization length for a fixed energy in each run. For non-Hermitian models, as in the transfer-matrix method, our algorithm computes the inverse localization length for a fixed (complex) energy. We also find a formula of the Chebyshev-polynomial expansion of the density of states of non-Hermitian models. As explained in detail, our algorithm for non-Hermitian models may be the only available efficient algorithm for finding the density of states of models with interactions.

Numerical calculation of the Casimir-Polder interaction between a graphene sheet with vacancies and an atom

In this work the Casimir{Polder interaction energy between a rubidium atom and a disordered graphene sheet is investigated beyond the Dirac cone approximation by means of accurate real-space calculations. As a model of defected graphene, we consider a tight-binding model of \Pi-electrons on a honeycomb lattice with a small concentration of point defects. The optical response of the graphene sheet is evaluated with full spectral resolution by means of exact Chebyshev polynomial expansions of the Kubo formula in large lattices with in excess of ten million atoms. At low temperatures, the optical response of defected graphene is found to display two qualitatively distinct behavior with a clear transition around non-zero Fermi energy, \mu~\mu*. In the vicinity of the Dirac point, the imaginary part of optical conductivity is negative for low frequencies while the real part is strongly suppressed. On the other hand, for high doping, it has the same features found in the Drude model within the Dirac cone approximation, namely, a Drude peak at small frequencies and a change of sign in the imaginary part above the interband threshold \omega > 2\mu. These characteristics translate into a non-monotonic behavior of the Casimir{Polder interaction energy with very small variation with doping in the vicinity of the neutrality point while having the same form of the interaction calculated with Drude's model at high electronic density.

Accelerating Real-Time Time-Dependent Density Functional Theory with a Nonrecursive Chebyshev Expansion of the Quantum Propagator.

Solutions of the real-time time-dependent density functional theory (RT-TDDFT) equations provide an affordable route to understanding the electronic dynamics that underpins many spectroscopic techniques. From the solutions of the RT-TDDFT equations, it is possible to extract optical absorption and circular dichroism spectra, as well as descriptions of charge transfer and charge transport dynamics. In order to apply RT-TDDFT to increasingly large systems, it is necessary to develop methods to overcome computational bottlenecks. One current bottleneck is the [Formula: see text] cost required to form the time propagator for the RT-TDDFT equations, because of the full matrix diagonalization that is required at each time step. Here, we present a (semi)diagonalization-free formation of the propagator based on a nonrecursive Chebyshev polynomial expansion. The Chebyshev expansion relies only on matrix multiply operations which have lower computational cost and are furthermore extremely parallelizable. We demonstrate the accuracy and stability of the Chebyshev approach, and then discuss the favorable scaling of the method, compared to traditional approaches based on matrix diagonalization. The Chebyshev expansion method should enable the application of RT-TDDFT methods to large systems such as nanocrystals and biomolecules.

One-step wave extrapolation matrix method for reverse time migration

We have developed a new method for solving the acoustic full-wave equation, which we call the one-step wave extrapolation (OSE) matrix method. In our method, the wave equation is redefined by introducing a complex (analytic) wavefield and reformulating the traditional acoustic full-wave equation as a first-order partial differential equation in time. Afterward, the analytical wavefield is separated to its real and imaginary parts, and the resulting first-order coupled set of equations is solved by the Tal-Ezer’s technique, which consists of using the Chebyshev polynomial expansion to approximate the matrix exponential operator. The matrix is antisymmetrical with a square-root pseudodifferential operator, which is computed using the Fourier method. In this way, the implementation of the proposed method is straightforward and if the appropriate number of Chebyshev polynomial expansion terms is chosen, the proposed numerical algorithm is unconditionally stable and propagates seismic waves free of numerical dispersion for any seismic velocity variation in a recursive manner. Moreover, in our method, the number of Fourier transforms is explicitly determined and it is a function of the maximum eigenvalue of the matrix operator and time-step size. A numerical modeling example is shown to demonstrate that the proposed method has the capability to extrapolate waves using a time stepping up to Nyquist limit. We have also developed a reverse time migration example with illumination compensation. The migration results based on the OSE method demonstrate the capability of this new method to image complex structures in the presence of strong velocity contrasts.

Dynamic response analysis of structure under time-variant interval process model

Due to the aggressiveness of the environmental factor, the variation of the dynamic load, the degeneration of the material property and the wear of the machine surface, parameters related with the structure are distinctly time-variant. Typical model for time-variant uncertainties is the random process model which is constructed on the basis of a large number of samples. In this work, we propose a time-variant interval process model which can be effectively used to deal with time-variant uncertainties with limit information. And then two methods are presented for the dynamic response analysis of the structure under the time-variant interval process model. The first one is the direct Monte Carlo method (DMCM) whose computational burden is relative high. The second one is the Monte Carlo method based on the Chebyshev polynomial expansion (MCM-CPE) whose computational efficiency is high. In MCM-CPE, the dynamic response of the structure is approximated by the Chebyshev polynomials which can be efficiently calculated, and then the variational range of the dynamic response is estimated according to the samples yielded by the Monte Carlo method. To solve the dependency phenomenon of the interval operation, the affine arithmetic is integrated into the Chebyshev polynomial expansion. The computational effectiveness and efficiency of MCM-CPE is verified by two numerical examples, including a spring-mass-damper system and a shell structure.

Chebyshev Approximation for Nanofluid flow of Non-Isothermal Channel Flow under Constant Heat Flux

This paper investigates the Nano-fluid for a non-isothermal channel flow under the effect of a constant pressure gradient acting along the channel axis. Two-dimensional, non-isothermal, steady flow of an incompressible fluid in a channel is taken into consideration. Upper and lower walls of the channel are kept at the same constant heat flux. To consider the effect of conductivity and viscosity, Maxwell and Brinkman’s models are used respectively. The effects of volume fraction, pressure gradient and Reynolds numbers on velocity and temperature profiles are discussed for the Nano-fluid Alumina. Water is used as a base fluid. The comparisons of the flow characteristics, including the distributions of velocity, temperature and volumetric flow rate of aluminum oxide are also given in the paper. Shear stress distribution along the channel axis and pressure gradient for different volume fraction are presented as well. Discretization is performed using a Pseudospectral technique based on Chebyshev polynomial expansions. The resulting nonlinear, coupled boundary value problem is solved iteratively using Chebyshev pseudospectral method.

On computation of Hough functions

Abstract. Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for the Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact MATLAB code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wave numbers.

Numerical solutions of quasi-two-dimensional models for laminar water hammer problems

ABSTRACTNumerical solutions of two quasi-two-dimensional models for water hammer problems under laminar flow conditions are proposed. The solutions are based on modified Chebyshev polynomial expansion of radial distribution of velocities. The collocation method is used to solve for the expansion coefficients. Spatial variation in the axial direction and time advancement are treated using the method of characteristics. One model includes the radial velocity components in the continuity equation while the other is focused on the cross-sectional mean velocity without the radial component. The effect of including the radial velocity component in the flow system is evaluated. The Chebyshev polynomials constitute an orthogonal basis for approximating numerical solutions and show suitable behaviour of the expansion coefficients.

Multiplons in the two-hole excitation spectra of the one-dimensional Hubbard model

Using the density-matrix renormalization group in combination with the Chebyshev polynomial expansion technique, we study the two-hole excitation spectrum of the one-dimensional Hubbard model in the entire filling range from the completely occupied band (n = 2) down to half-filling (n = 1). For strong interactions, the spectra reveal multiplon physics, i.e., relevant final states are characterized by two (doublon), three (triplon), four (quadruplon) and more holes, potentially forming stable compound objects or resonances with finite lifetime. These give rise to several satellites in the spectra with largely different spectral weights as well as to different two-hole, doublon–hole, two-doublon etc continua. The complex multiplon phenomenology is analyzed by interpreting not only local and k-resolved two-hole spectra but also three- and four-hole spectra for the Hubbard model and by referring to effective low-energy models. In addition, a filter-operator technique is presented and applied which allows to extract specific information on the final states at a given excitation energy. While multiplons composed of an odd number of holes do neither form stable compounds nor well-defined resonances unless a nearest-neighbor density interaction V is added to the Hamiltonian, the doublon and the quadruplon are well-defined resonances. The k-resolved four-hole spectrum at n = 2 represents an interesting special case where a completely stable quadruplon turns into a resonance by merging with the doublon–doublon continuum at a critical wave vector. For all fillings with , the doublon lifetime is strongly k-dependent and is even infinite at the Brillouin zone edges as demonstrated by k-resolved two-hole spectra. This can be traced back to the ‘hidden’ charge-SU(2) symmetry of the model which is explicitly broken off half-filling and gives rise to a massive collective excitation, even for arbitrary higher-dimensional but bipartite lattices.