Chebyshev Polynomial Expansion Research Articles

Vibroacoustic analysis of submerged fluid-filled cylindrical shell

This study presents a comprehensive Chebyshev-Fourier spectral method for analyzing the free and forced vibroacoustic characteristics of submerged fluid-filled cylindrical shells in heavy fluids. The vibroacoustic interaction system of submerged cylindrical shells consists of three discrete regions: The Sanders shell theory is applied for the shell structure, whereas the wave equation is used to determine the fluid inside. The external fluid pressure on the shell is described using the Helmholtz boundary integral equation. The governing equations for the shell's motion, the formulation of the external fluid pressure on the fluid-structure-interaction (FSI) surface using first-class Chebyshev polynomial and Fourier series expansions, and the internal sound pressure employing two-dimensional Chebyshev polynomial and Fourier series expansions, are seamlessly integrated with Gauss-Chebyshev-Lobatto sampling techniques. Artificial springs are implemented to replicate the boundary conditions of the cylindrical shell. The precision of the proposed methodology is corroborated by comparisons with existing literature and finite element method (FEM) results. Furthermore, the research examines how the properties of the acoustical medium on the vibroacoustic responses of the cylindrical shell. Results indicate that the presence of air inside the shell slightly influence on its acoustic radiation. However, replacing the internal medium with water significantly alters the shell's original radiation characteristics.

On the Thermally Induced Interfacial Behavior of Thin Two-Dimensional Hexagonal Quasicrystal Films with an Adhesive Layer

The mechanical response of a quasicrystal thin film is strongly affected by an adhesive layer along the interface. In this paper, a theoretical model is proposed to study a thin two-dimensional hexagonal quasicrystal film attached to a half-plane substrate with an adhesive layer, which undergoes a thermally induced deformation. A perfect non-slipping contact condition is assumed at the interface by adopting the membrane assumption. An analytical solution to the problem is obtained by constructing governing integral–differential equations for both single and multiple films in terms of interfacial shear stresses that are reduced to a linear algebraic system via the series expansion of Chebyshev polynomials. The solution is compared to that without adhesive layers, and the effects of the aspect ratio of films, material mismatch, and the adhesive layer, as well as the interaction between films, are discussed in detail. It is found that the adhesive layer can soften the localized stress concentration. This study is instructive to the accurate safety assessment and functional design of a quasicrystal film system.

A fast Chebyshev spectral approach for vibroacoustic behavior analysis of heavy fluid-loaded baffled rectangular plates with general boundary conditions

A computationally efficient Chebyshev spectral approach is proposed to solve the vibroacoustic response of heavy fluid-loaded baffled rectangular plates. The governing equations for the displacements of motion in rectangular plates and the sound pressure in the Helmholtz integral are formulated using high-order, first-class Chebyshev polynomial expansions. This formulation is combined with Gauss-Chebyshev-Lobatto sampling. The quadruple integral encountered in solving the work done by the heavy fluid on the plate is reformulated into the configuration of a tensor product. The approach significantly reduces the computation time for solving the acoustic equations, bringing the vibration response of fluid-loaded plates closer to that of vacuum plates and limiting it to just a few seconds. The elastic boundary conditions of the rectangular plates are simulated using linear and rotational springs. Predictions of vibroacoustic behavior, including plate velocity, sound pressure, sound power, and radiation efficiency, are validated against literature results. The accuracy and efficiency of the Chebyshev spectral approach for the vibroacoustic coupling systems are demonstrated by the presence of exemplary agreements. Furthermore, this study investigates the effect of boundary conditions, geometric characteristics, and damping variables on the vibroacoustic behavior of rectangular plates.

SFQEDtoolkit: A high-performance library for the accurate modeling of strong-field QED processes in PIC and Monte Carlo codes

Strong-field quantum electrodynamics (SFQED) processes are central in determining the dynamics of particles and plasmas in extreme electromagnetic fields such as those present in the vicinity of compact astrophysical objects or generated with ultraintense lasers. SFQEDtoolkit is an open source library designed to allow users for a straightforward implementation of SFQED processes in existing particle-in-cell (PIC) and Monte Carlo codes. Through advanced function approximation techniques, high-energy photon emission and electron-positron pair creation probability rates and energy distributions are calculated within the locally-constant-field approximation (LCFA) as well as with more advanced models [Phys. Rev. A 99, 022125 (2019)]. SFQEDtoolkit is designed to provide users with high-performance and high-accuracy, and neat examples showing its usage are provided. In the near future, SFQEDtoolkit will be enriched to model the angular distribution of the generated particles, i.e., beyond the commonly employed collinear emission approximation, as well as to model spin and polarization dependent SFQED processes. Notably, the generality and flexibility of the presented function approximation approach makes it suitable to be employed in other areas of physics, chemistry and computer science. Program summaryProgram Title: SFQEDtoolkit (version 1.0)CPC Library link to program files:https://doi.org/10.17632/j7mhj3cg3m.1Developer's repository link:https://github.com/QuantumPlasma/SFQEDtoolkitLicensing provisions: GPLv3Programming language: C++, FortranNature of problem: The accurate simulation of strong-field QED processes in the interaction of particles or plasmas with strong electromagnetic fields is critical for present and future experiments with ultraintense lasers as well as to unveil the dynamics around compact astrophysical objects. State-of-the-art PIC and Monte Carlo codes implement strong-field QED processes by means of lookup tables, where the approximated function is evaluated on a fixed set of points. This typically results in numerical artifacts such as stairlike structures in the approximated function, and in a reduced efficiency when large size lookup tables are necessary. An easily implementable open-source accurate and efficient simulation tool is required to address problems at the high-intensity frontier of plasma physics and astrophysics.Solution method: Strong-field QED process rates and particle distributions are calculated by leveraging advanced approximation methods based on variable and function transformation, asymptotic expansions as well as Chebyshev polynomial expansions. This provides a numerically-stable and efficient approach that allows us to overcome all limitations of existing methods based on standard lookup tables.

Chebyshev collocation method for solving second order ODEs using integration matrices

The spectral collocation method for solving two-point boundary value problems for second order differential equations is implemented, based on representing the solution as an expansion in Chebyshev polynomials. The approach allows a stable calculation of both the spectral representation of the solution and its pointwise representation on any required grid in the definition domain of the equation and additional conditions of the multipoint problem. For the effective construction of SLAE, the solution of which gives the desired coefficients, the Chebyshev matrices of spectral integration are actively used. The proposed algorithms have a high accuracy for moderate-dimension systems of linear algebraic equations. The matrix of the system remains well-conditioned and, with an increase in the number of collocation points, allows finding solutions with ever-increasing accuracy.

The prediction of projectile-target intersection for moving tank based on adaptive robust constraint-following control and interval uncertainty analysis

To improve the hit probability of tank at high speed, a prediction method of projectile-target intersection based on adaptive robust constraint-following control and interval uncertainty analysis is proposed. The method proposed provides a novel way to predict the impact point of projectile for moving tank. First, bidirectional stability constraints and stability constraint-following error are constructed using the Udwadia-Kalaba theory, and an adaptive robust constraint-following controller is designed considering uncertainties. Second, the exterior ballistic ordinary differential equation with uncertainties is integrated into the controller, and the pointing control of stability system is extended to the impact-point control of projectile. Third, based on the interval uncertainty analysis method combining Chebyshev polynomial expansion and affine arithmetic, a prediction method of projectile-target intersection is proposed. Finally, the co-simulation experiment is performed by establishing the multi-body system dynamic model of tank and mathematical model of control system. The results demonstrate that the prediction method of projectile-target intersection based on uncertainty analysis can effectively decrease the uncertainties of system, improve the prediction accuracy, and increase the hit probability. The adaptive robust constraint-following control can effectively restrain the uncertainties caused by road excitation and model error.

Efficient Chebyshev polynomial approach to quantum conductance calculations: Application to twisted bilayer graphene

In recent years, Chebyshev polynomial expansions of tight-binding Green's functions have been successfully applied to the study of a wide range of spectral and transport properties of materials. However, the application of the Chebyshev approach to the study of quantum transport properties of noninteracting mesoscopic systems with leads has been hampered by the lack of a suitable Chebyshev expansion of Landaeur's formula or one of its equivalent formulations in terms of Green's functions in Keldysh's perturbation theory. Here, we tackle this issue by means of a hybrid approach that combines the efficiency of Chebyshev expansions with the convenience of complex absorbing potentials to calculate the conductance of two-terminal devices in a computationally expedient and accurate fashion. The versatility of the approach is demonstrated for mesoscopic twisted bilayer graphene (TBG) devices with up to $2.3\times10^6$ atomic sites. Our results highlight the importance of moir\'e effects, interlayer scattering events and twist-angle disorder in determining the conductance curves in devices with a small twist angle near the TBG magic angle $\theta_m \approx 1.1^\circ$.

Investigation of eigenfunctions of perturbation of the radial component of flow velocity

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of a generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The influence of taking into account the exponential dependence of fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of incompressible fluid flow in an annular channel is investigated. The eigenvalue spectra were obtained numerically. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity disturbances, their possible growth or decay over time. Graphs of the eigenfunctions of the generalized flow stability equation in an annular channel have been constructed. It is shown that the structure of the spectra largely depends on both the properties of the liquid, determined by the functional dependence of viscosity, and on the geometry of the channel. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It has been established that at small values of the thermoviscosity parameter the spectrum is comparable to the spectrum for isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are a greater number of points at which the problem has non-zero amplitudes of transverse velocity disturbances. It is worth noting that the diversity of eigenvalue spectra corresponds to the diversity of eigenfunctions, which have a nontrivial distribution of the oscillation amplitude over the cross section in each case. Smooth curves are obtained for a narrow channel, as for the case of a flat channel. However, as the ratio of the channel radii increases, “jumps” appear. Also, note that the eigenfunctions do not have the property of symmetry; this follows from the fact that the velocity profile in the unperturbed state also does not have symmetry. The maximum values of the eigenfunctions are shifted to the right from the center of the channel, which corresponds to the fact that disturbances arise and grow intensively near the hot wall.

T-stress evaluation of cracks by means of continuous dislocations model in two-dimensional elasticity

Numerical procedure for analyzing T-stresses of cracks in two dimensional elasticity was developed using the method of continuously distributed dislocations model. Boundary conditions on the crack surfaces are reduced into singular integral equations in which dislocation densities are unknown. The weight function of the dislocation density is approximated as an expansion of Chebyshev polynomials of the first kind with finite number of terms and the coefficient of each term is determined by satisfying the boundary conditions at suitably selected collocation points. Some crack problems were solved by the proposed method in order to demonstrate rapid convergence of the calculated T-stresses with the increase of the number of the collocation points.

Грубо-сеточные поправки в вычислении действия матричной экспоненты подпространством Крылова

A coarse grid correction (CGC) approach is proposed to enhance the efficiency of the matrix exponential and phi matrix function evaluations. The approach is intended for iterative methods computing the matrix-vector products with these functions. It is based on splitting the vector by which the matrix function is multiplied into a smooth part and a remaining part. The smooth part is then handled on a coarser grid, whereas the computations on the original grid are carried out with a relaxed stopping criterion tolerance. Estimates on the error are derived for the two-grid and multigrid variants of the proposed CGC algorithm. Numerical experiments demonstrate the efficiency of the algorithm, when employed in combination with Krylov subspace and Chebyshev polynomial expansion methods.

The smooth extension embedding method with Chebyshev polynomials

AbstractWe propose an implementation of the smooth extension embedding method (SEEM), first described by Agress and Guidotti's study, in the setting of Chebyshev polynomials. SEEM is a hybrid fictitious domain/collocation method which solves general boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, SEEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non‐periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve elliptic and parabolic problems in complex geometries, with non‐constant coefficients, and for general boundary conditions. We consider applications to two and three dimensional boundary value problems as well as an initial boundary value problem via a genuinely space–time discretization.

Uncertainty analysis of quasi-zero stiffness metastructure for vibration isolation performance

Quasi-zero stiffness (QZS) metamaterials and metastructures have great advantages of being highly integrable and lightweight for vibration isolation in aerospace and aviation applications. However, the geometric uncertainty introduced from additive manufacturing (AM) significantly affects the metamaterial/metastructure’s vibration isolation performance and therefore, needs to be evaluated accurately and efficiently in the design process. In this study, a high-order sparse Chebyshev polynomial expansion (HOSPSCPE) method is first utilized to quantify the influence of AM-induced geometric uncertainty in the QZS microstructure. Excellent accuracy and much higher efficiency (about 470 times faster) of the proposed method are observed when compared to the widely used Monte Carlo method (MCM). Uncertainty analyses are then conducted for vibration isolation performance of the QZS metastructures and band gap properties of the QZS locally resonant metamaterials, respectively. The numerical results demonstrate that the geometric uncertainty analysis can provide useful guidance and recommendations for the manufacturing-influenced design of QZS metastructures and metamaterials.

Spectral density reconstruction with Chebyshev polynomials

Accurate calculations of the spectral density in a strongly correlated quantum many-body system are of fundamental importance to study its dynamics in the linear response regime. Typical examples are the calculation of inclusive and semiexclusive scattering cross sectionsin atomic nuclei and transport properties of nuclear and neutron star matter. Integral transform techniques play an important role in accessing the spectral density in a variety of nuclear systems. However, their accuracy is in practice limited by the need to perform a numerical inversion which is often ill-conditioned. In the present work we extend a recently proposed quantum algorithm which circumvents this problem. We show how to perform controllable reconstructions of the spectral density over a finite energy resolution with rigorous error estimates. An appropriate expansion in Chebyshev polynomials allows for efficient simulations also on classical computers. We apply our idea to obtain the local density of states for graphene in a magnetic field as a proof of principle. This paves the way for future applications in nuclear and condensed matter physics.

Real-space density kernel method for Kohn–Sham density functional theory calculations at high temperature

Kohn-Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. We present a density matrix based method for Kohn-Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann-Feynman forces, and Hellmann-Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw-Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.

Systematic Analysis Method for Nonlinear Response Tensors

We propose a systematic analysis method for identifying essential parameters in various linear and nonlinear response tensors without which they vanish. By using the Keldysh formalism and the Chebyshev polynomial expansion method, the response tensors are decomposed into the model-independent and dependent parts, in which the latter is utilized to extract the essential parameters. An application of the method is demonstrated by analyzing the nonlinear Hall effect in the ferroelectric SnTe monolayer for example. It is shown that in this example the second-neighbor hopping is essential for the nonlinear Hall effect whereas the spin–orbit coupling is unnecessary. Moreover, by analyzing terms contributing to the essential parameters in the lowest order, the appearance of the nonlinear Hall effect can be interpreted by the subsequent two processes: the orbital magneto-current effect and the linear anomalous Hall effect by the induced orbital magnetization. In this way, the present method provides a microscopic picture of responses. By combining with computational analysis, it stimulates further discoveries of anomalous responses by filling in a missing link among hidden degrees of freedom in a wide variety of materials.

Investigation of the spectral characteristics of a thermoviscous fluid flow in an annular channel

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of the generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The effect of taking into account the exponential dependence of the fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of an incompressible fluid flow in a flat channel for various wall temperatures is studied. Spectral patterns of eigenvalues of the generalized equation are constructed. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity perturbations, their possible growth or decay with time. It is shown that the structure of the spectra largely depends both on the properties of the liquid, determined by the index of the functional dependence of viscosity, and on the geometry of the channel. It has been established that for small values of the thermoviscosity parameter, the spectrum is comparable to the spectrum for an isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has nonzero amplitudes of transverse velocity perturbations. The stability of a thermoviscous fluid flow depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It is shown that, at fixed values of the Reynolds number and wave number, the flow can become unstable with an increase in the thermoviscosity parameter.

Hermite-Chebyshev pseudospectral method for inhomogeneous superconducting strip problems and magnetic flux pump modeling

The numerical simulation of superconducting devices is a powerful tool for understanding the principles of how they work and improving their design. Usually, these simulations are based on a finite element method but, recently, a different approach, based on the spectral technique, has been presented for very efficient solution of several applied superconductivity problems described by 1D integro-differential equations or a system of such equations. Here, we propose a new pseudospectral method for 2D magnetization and transport current superconducting strip problems with an arbitrary current–voltage relation, spatially inhomogeneous strips and strips in a nonuniform applied field. The method is based on bivariate expansions in Chebyshev polynomials and Hermite functions. It can be used for numerical modeling of magnetic flux pumps of different types and investigating AC losses in coated conductors with local defects. Using a realistic 2D version of the superconducting dynamo benchmark problem as an example, we show that our new method is a competitive alternative to finite element methods.

Entropy generation on flow and heat transfer of a reactive MHD Sisko fluid through inclined walls with porous medium

The present research focuses on flow, heat transfer, and entropy production properties in the context of their applications. The use of non-Newtonian materials in biomedical rheological models has garnered considerable attention. Having such practical and potential applications, the current study explores the incompressible, gravity-driven flow of a combustible Sisko liquid down inclined heated permeable walls embedded in a porous medium. The flow is said to be fully developed and modified Darcy’s law is employed to express the porous medium. Such a flow is critical to simulate free-surface flows in trapezoidal reservoirs or channels, and around curved or inclined hulls. The spectral collocation technique with expansions in Chebyshev polynomials was instrumental in achieving numerical solutions. Furthermore, a parametric study is done to explore the behavior of flow, thermal, entropy generation, and Bejan number profiles. The significant outcomes of the current study are that the rise in values of material parameter decreases the velocity, thermal, and entropy generation profiles but increases the Bejan number profile. The upsurge in values of magnetic parameter decreases the velocity and entropy generation profiles but increases the thermal and Bejan number profile. The rise in values of the Frank-Kameneskii parameter increases the Bejan number, thermal, and entropy generation profiles.

Dynamical screening in the description of plasma transport properties with arbitrary degeneracy

AbstractFor the first time, the contribution of electron–electron scattering to the electrical conductivity of plasma is calculated with simultaneous consideration of degeneracy and dynamic screening. A procedure for calculating electron–electron correlation functions used in the linear response theory in the formulation of Zubarev is developed. The Chebyshev polynomial expansion of the Fermi–Dirac distribution function is applied. Calculations were carried out in the random‐phase approximation for the dielectric function.

Modeling wind speed time series by Chebyshev polynomial expansion method

ABSTRACT This paper sets out to simulate non-Gaussian wind speed time series. Given a set of historical wind speed observations, if the cumulative distribution function is not explicitly known, a probability distribution based on Fourier series is developed to recover the analytical expression of the cumulative distribution function. The marginal transformation is then applied to map wind speed observations to the standard normal space, where the autocorrelation function is represented by a weighted sum of Chebyshev polynomials. Finally, an explicit formula is derived to generate samples of wind speed time series. Testing on three sets of historical wind speed observations, it shows that the proposed method can well match the cumulative distribution function and autocorrelation function of wind speed time series, the absolute errors between the fitted cumulative distribution function and empirical cumulative distribution function are less than ; the autocorrelation function of generated wind speed time series is in a good agreement with that of historical wind speed observations, the differences are less than .