Pseudospectral Approach Research Articles

Investigation of micropolar dusty fluid flow in a rotational frame with magnetic field: A meshless radial basis function pseudospectral approach

AbstractRotational flow has numerous applications across a wide range of fields, including industrial processes for mixing, pumping, and processing materials, geophysical fluid dynamics for studying the Earth's atmosphere and oceans, aerospace engineering for designing and testing aircraft and spacecraft, medical imaging for creating detailed images of the human body, and environmental engineering for studying the transport of pollutants in water and air. In rotational flow, pumping power is an important parameter that measures the amount of energy required to maintain the flow of the fluid in a rotating frame. It is significant because the presence of the Coriolis force and other rotational effects can substantially alter the fluid dynamics, making it difficult to maintain a steady flow. The present study deals with the flow of micropolar dusty fluid in a rotating frame under the influence of the magnetic field, applied transversely to the flow in a horizontal channel. The fluid is induced to flow between two parallel plates by the movement of the upper plate and a constant pressure gradient. A meshless radial basis function pseudospectral approach is employed to derive a set of partial differential equations that determine the primary and secondary velocity profiles of the fluid, as well as the particle and microrotation velocities and temperature distribution. The pumping power required to sustain the flow in the absence of a pressure gradient is also calculated based on these velocity profiles. The findings provide valuable insights into the behavior of micropolar dusty fluids in a rotational channel and can potentially inform the design of industrial processes involving such fluids.

A Pseudo-Spectral Method for Wall Shear Stress Estimation from Doppler Ultrasound Imaging in Coronary Arteries.

The Wall Shear Stress (WSS) is the component tangential to the boundary of the normal stress tensor in an incompressible fluid, and it has been recognized as a quantity of primary importance in predicting possible adverse events in cardiovascular diseases, in general, and in coronary diseases, in particular. The quantification of the WSS in patient-specific settings can be achieved by performing a Computational Fluid Dynamics (CFD) analysis based on patient geometry, or it can be retrieved by a numerical approximation based on blood flow velocity data, e.g., ultrasound (US) Doppler measurements. This paper presents a novel method for WSS quantification from 2D vector Doppler measurements. Images were obtained through unfocused plane waves and transverse oscillation to acquire both in-plane velocity components. These velocity components were processed using pseudo-spectral differentiation techniques based on Fourier approximations of the derivatives to compute the WSS. Our Pseudo-Spectral Method (PSM) is tested in two vessel phantoms, straight and stenotic, where a steady flow of 15mL/min is applied. The method is successfully validated against CFD simulations and compared against current techniques based on the assumption of a parabolic velocity profile. The PSM accurately detected Wall Shear Stress (WSS) variations in geometries differing from straight cylinders, and is less sensitive to measurement noise. In particular, when using synthetic data (noise free, e.g., generated by CFD) on cylindrical geometries, the Poiseuille-based methods and PSM have comparable accuracy; on the contrary, when using the data retrieved from US measures, the average error of the WSS obtained with the PSM turned out to be 3 to 9 times smaller than that obtained by state-of-the-art methods. The pseudo-spectral approach allows controlling the approximation errors in the presence of noisy data. This gives a more accurate alternative to the present standard and a less computationally expensive choice compared to CFD, which also requires high-quality data to reconstruct the vessel geometry.

Quantum chemical package Jaguar: A survey of recent developments and unique features.

This paper is dedicated to the quantum chemical package Jaguar, which is commercial software developed and distributed by Schrödinger, Inc. We discuss Jaguar's scientific features that are relevant to chemical research as well as describe those aspects of the program that are pertinent to the user interface, the organization of the computer code, and its maintenance and testing. Among the scientific topics that feature prominently in this paper are the quantum chemical methods grounded in the pseudospectral approach. A number of multistep workflows dependent on Jaguar are covered: prediction of protonation equilibria in aqueous solutions (particularly calculations of tautomeric stability and pKa), reactivity predictions based on automated transition state search, assembly of Boltzmann-averaged spectra such as vibrational and electronic circular dichroism, as well as nuclear magnetic resonance. Discussed also are quantum chemical calculations that are oriented toward materials science applications, in particular, prediction of properties of optoelectronic materials and organic semiconductors, and molecular catalyst design. The topic of treatment of conformations inevitably comes up in real world research projects and is considered as part of all the workflows mentioned above. In addition, we examine the role of machine learning methods in quantum chemical calculations performed by Jaguar, from auxiliary functions that return the approximate calculation runtime in a user interface, to prediction of actual molecular properties. The current work is second in a series of reviews of Jaguar, the first having been published more than ten years ago. Thus, this paper serves as a rare milestone on the path that is being traversed by Jaguar's development in more than thirty years of its existence.

An approximate solution of the Blasius problem using spectral method

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

Pseudospectral analysis for multidimensional fractional Burgers equation based on Caputo fractional derivative

This study presents the Chebyshev pseudospectral approach in time and space to approximate a solution to the time-fractional multidimensional Burgers equation. The suggested approach utilizes Chebyshev–Gauss–Lobatto (CGL) points in both spatial and temporal directions. To figure out the fractional derivative matrix at CGL points, we use the Caputo fractional derivative formula. Further, the Chebyshev fractional derivative matrix is utilized to reduce the given problem in an algebraic system of equations. The numerical approach known as the Newton–Raphson is implemented to get the desired results for the system. Error analysis for the set of values of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ u $$\\end{document} is done for various model examples of fractional Burgers equations, where ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} represents the fractional order. The computed numerical results are in perfect agreement with the exact solutions.

A numerically robust and stable time–space pseudospectral approach for multidimensional generalized Burgers–Fisher equation

We study the pseudospectral discretization of a nonlinear multidimensional generalized Burgers–Fisher equation. The main objective of this study is to establish that the implementation of the pseudospectral method in time holds equal importance to its application in space when addressing nonlinearity. Typically, the literature recommends using a large number of grid points in the temporal direction to ensure satisfactory outcomes. However, the fact that we obtained excellent accuracy with a relatively small number of temporal grid points is a notable finding in this study. The Chebyshev–Gauss–Lobatto (CGL) points serve as the foundation for the recommended method. By applying the proposed method to the problem, a system of algebraic equations is obtained, which is subsequently solved using the Newton–Raphson technique. We have conducted stability analysis and error estimation for the proposed method. Different researchers’ considerations on test problems have been explored to illustrate the robustness and practicality of the approach presented. The results obtained using the proposed method demonstrate a high level of accuracy, surpassing the existing solutions by a significant margin.

Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems

This paper introduces a new, highly accurate model for periodic fractional optimal control problems (PFOCPs) based on Riemann–Liouville and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. This new model enables periodic solutions in fractional-order control models by preserving the periodicity of the fractional derivatives. Additionally, we develop a novel numerical solution method based on Fourier and Gegenbauer pseudospectral approaches. Our method simplifies PFOCPs into constrained nonlinear programming problems (NLPs), readily solvable by standard NLP solvers. Some key innovations of this study include: (i) The derivation of a simplified FD calculation formula to transform periodic FD calculations into evaluations of trigonometric Lagrange interpolating polynomial integrals, efficiently computed using Gegenbauer quadratures. (ii) The introduction of the αth-order fractional integration matrix based on Fourier and Gegenbauer pseudospectral approximations, which proves to be highly effective for computing periodic FDs. A priori error analysis is provided to predict the accuracy of the Fourier–Gegenbauer-based FD approximations. Benchmark PFOCP results validate the performance of the proposed pseudospectral method.

Numerical investigation of turbulence generation using Zakharov-like model equation

This study investigates the turbulence generation behavior with a Zakharov-like (ZL) equation in a fluid system. The model equation is derived using conservation equations (mass and momentum conservation), and the source of nonlinearity is the high amplitude of the acoustic wave. The Zakharov-like equation has been derived and then solved numerically, then turned into a modified nonlinear Schrödinger equation. Furthermore, modulation instability, or Benjamin–Feir instability, of the model equations, which leads to the emergence of Akhmediev breathers, is discussed. The numerical simulation uses a finite difference method for temporal evolution and a pseudo-spectral approach to determine spatial regimes. The outcomes indicate that the situation involving the nonlinear Schrödinger equation case displays a periodic pattern in space and time. The findings also demonstrate that the localization of structure and the Fermi, Pasta, and Ulam (FPU) recurrences are disrupted for the modified nonlinear Schrödinger equation and Zakharov-like equation cases. The energy spectrum exhibits a power law behavior that approximately follows k−1.65 in the ZL model equation case, and it is steeper than Kolmogorov's spectrum within the inertial sub-range.

Localization of beam generated whistler wave and turbulence generation in reconnection region of magnetopause

Whistler waves have been studied for many years in relation to turbulence and particle heating, and observations show that they are crucial to magnetic reconnection. Recent research has revealed a close relationship between magnetic reconnection and turbulence. The current work investigates the whistler turbulence caused by the energetic electron beam in the magnetic reconnection sites of magnetopause and also due to dynamic evolution of magnetic islands. For this, we develop a model based upon the two-fluid approximation to study whistler dynamics, propagating in the medium with the pre-existing chain of magnetic islands and under the influence of background density perturbation originating from ponderomotive nonlinearity of wave. Dynamics of nonlinear whistler have been solved with pseudo-spectral approach and a finite difference method with a modified predictor–corrector method and a Runge Kutta method for the semianalytical model. In the current research, we study how the nonlinear whistler wave contributes to the significant space phenomenon, i.e., turbulence, localization, and magnetic reconnection. We have also investigated the formation of a current sheet in a magnetopause region of the order of few-electron inertial length. We analyzed the power spectrum at the magnetopause when the system reached a quasi-steady condition. Our new approach to study whistler turbulence by an energetic electron beam at the magnetic reconnection sites has extensive applications to space plasmas, shedding a new light on the study of magnetic reconnection in nature.

A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations

<p>Nonlinear optimal control problems governed by variable-order fractional integro-differential equations constitute an important subgroup of optimal control problems. This group of problems is often difficult or impossible to solve analytically because of the variable-order fractional derivatives and fractional integrals. In this article, we utilized the expansion of Lagrange polynomials in terms of Chebyshev polynomials and the power series of Chebyshev polynomials to find an approximate solution with high accuracy. Subsequently, by employing collocation points, the problem was transformed into a nonlinear programming problem. In addition, variable-order fractional derivatives in the Caputo sense were represented by a new operational matrix, and an operational matrix represented fractional integrals. As a result, the mentioned integro-differential optimal control problem becomes a nonlinear programming problem that can be easily solved with the repetitive optimization method. In the end, the proposed method is illustrated by numerical examples that demonstrate its efficiency and accuracy.</p>

Asynchronous quantum Kármán vortex street in two-component Bose-Einstein condensate with PT symmetric potential

The dynamics of a miscible two-component Bose-Einstein condensate (BEC) with PT (parity-time) symmetric potential are investigated numerically. The dynamical behaviors of the system is described by Gross-Pitaevskii (GP) equations under the mean-field theory. Firstly, the ground state of the system is obtained by the imaginary-time propagation method. Then dynamical behaviors are numerically simulated by the time-splitting Fourier pseudo-spectral approach under periodic boundary conditions. By adjusting the width and velocity of the obstacle potential, various patterns such as no vortex, oblique drifting vortex dipole, V-shaped vortex pairs, irregular quantum turbulence and combined modes are studied. It is noted that the shedding vortex pairs in components 1 and 2 are staggered, which is called “the asynchronous quantum Kármán vortex street”. Here, the ratio of the distance between two vortex pairs in one row to the distance between vortex rows is approximately 0.18, which is less than the stability criterion 0.28 of classical fluid. We calculated the drag force acting on the obstacle potential during generation of the asynchronous quantum Kármán vortex street. It is found that periodical oscillation of the drag force is generated via drifting up or down of the vortex pairs. Meanwhile, we analyzed the influence of the imaginary part of the PT symmetric potential with gain-loss for wake. The trajectory and frequency of the vortex are changed, due to the imaginary part breaks the local symmetry of the system. In addition, the imaginary part affects the stability of the asynchronous quantum Kármán vortex street. Lots of numerical simulations are carried out to determine the parameter regions of various vortex shedding modes. We also proposed an experimental protocol to realize the asynchronous quantum Kármán vortex street in the miscible two-component BEC with PT symmetric potential.

ALLIANCE: Spectral solver for kinetic plasma turbulence

The ALLIANCE (ALLIANCE - spectrAL soLver for kInetic plAsma turbuleNCE) code is developed to solve a new set of four-dimensional electromagnetic drift-kinetic equations in slab geometry [1]. The nonlinear equations are useful for the study of magnetized plasma systems at scales comparable to, or larger than the ion gyroradius. In particular, it is suited for the study the kinetic turbulent cascade in astrophysical plasma, while preserving finite Larmor radius effects at the fluid-kinetic transition. The equations solved are in spectral Fourier-Laguerre-Hermite form, a pseudo-spectral approach is used for the nonlinear terms, and the code is parallelised over multiple directions. After a presentation of the code, validation runs are shown, and benchmarks for serial and parallel computations are presented.

An efficient energy-stable pseudospectral method for simulating vortex dynamics of the Ginzburg–Landau–Schrödinger equation

In this article, we concentrate on designing an efficient energy-stable pseudospectral method for the vortex dynamics of the Ginzburg–Landau–Schrödinger equation (GLSE) on bounded domain under homogeneous Neumann boundary condition. Based on a Lagrange multiplier (LagM) approach, we firstly introduce an equivalent system, named LagM–GLSE. Secondly, we discretize the LagM–GLSE by Crank–Nicolson Leap-Frog scheme in temporal direction. While in spatial direction, we utilize a cosine pseudospectral approach. Thus, the resulting scheme has a spectral/second-order convergence rate in space/time. The discretized scheme is efficient and easy to implement. Moreover, we prove that the LagM based scheme can dissipate/conserve the original total energy. Finally, ample numerical results are provided to show the efficiency and accuracy of our scheme. We also apply the lagM based scheme to simulate an interesting vortex dynamics with different parameters for the GLSE.

Data-informed statistical finite element analysis of rail buckling

In this paper, the statistical finite element method is developed further to synthesize distributed rail response data with nonlinear finite element model predictions within and outside the measured loading range. In the data-generating model of the statistical finite element method, the distributed sensing data is decomposed into a finite element model component, a model-reality mismatch component, and a noise component. Each component is considered uncertain and is represented as a Gaussian random vector with a corresponding prior density. The finite element prior density is updated using the Bayesian statistical framework in light of the distributed fiber optic sensing data. The calculated posterior density enables one to infer the true structural response. The required finite element prior density is determined by solving a conventional stochastic forward problem using a polynomial chaos expansion of random fields and a non-intrusive pseudo-spectral projection approach. In this paper, a lab test involving a section of a rail (i.e., a beam-column structural member) instrumented with distributed fiber optic sensors and subjected to axial load causing progressive lateral buckling is considered to demonstrate the use of the statistical finite element method to improve rail response prediction. The results show improved prediction of true rail strain in linear and nonlinear regimes compared with a pure finite element model-based prediction.

Transport Features on Bidirectional Nanofluid Flow with Convective Heating and Variable Darcy Regime

The performance of Au and nanoparticles in Casson flow propelled by the rotational effect subject to convective heating and variable Darcy phenomenon is presented. As a means to contribute significantly to the biomedical industry for proper prediction and treatment of diseases like cancer, stenosis, et cetera, the Casson fluid model and magnetohydrodynamic effect are employed in this study to investigate the bidirectional-rotating boundary layer flow of blood since Casson rheology best describe the mammalian blood flow dynamics and that blood is electrically conducting in nature. The enormous performance of nanoparticles made them extremely useful; hence, they gained recognition over the conventional refrigerants and were widely used by scientists, modelers, and researchers for controlling and treating diseases via drug targeting (chemotherapy), etc. In the three-dimensional plane, the assumed electrically conducting fluid conveying nanoparticles is properly mixed through the Coriolis force, thus rotating with angular velocity across the variable porous medium. A numerical tool, Chebyshev Collocation Method (CCM) with pseudo-spectral approach, is deployed on the resulting transformed ODEs. Profile distributions of respective boundary layers to distinct responses of model parameters, including the Eckman, Darcy, and Biot numbers on Au and nanoparticles, are analyzed and discussed. The dominance of nanoparticles on both velocities is revealed, while Au nanoparticles demonstrated a higher thermal performance rate. A rise in the Darcy parameter has a diametrically opposed behavior on the primary and secondary velocities.

Numerical simulation of modified nonlinear Schrodinger equation and turbulence generation

This article presents a numerical model to study wave turbulence in fluids. The model equation is derived by incorporating energy conservation (along with the usual fluid equations of a compressible flow), and the source of nonlinearity is the rise in temperature due to the acoustic wave's high amplitude. The nonlinear Schrödinger (NLS) and modified nonlinear Schrödinger (MNLS) equations have been derived and then solved numerically. A numerical simulation of the MNLS equation is used for investigating the turbulence generation and a semi-analytical method to understand the physics of localized structures of nonlinear waves. The numerical simulation is based on a pseudo-spectral approach to resolve spatial regimes, a finite difference method for temporal evolution. The results show a periodic pattern viz. Fermi–Pasta–Ulam (FPU) recurrence for NLS, while turbulence generation breaks down the FPU recurrence in MNLS. The turbulent power spectrum in the inertial sub-range approximately follows the Kolmogorov–Zakharov scaling (∼k−1.2).

Damping of three-dimensional waves on coating films dragged by moving substrates

Paints and coatings often feature interfacial defects due to disturbances during the deposition process, which, if they persist until solidification, worsens the product quality. In this article, we investigate the stability of a thin liquid film dragged by a vertical substrate moving against gravity, a fundamental flow configuration in various coating processes. The receptivity of the liquid film to three-dimensional disturbances is analyzed with Direct Numerical Simulations (DNS) and an in-house Integral Boundary Layer (IBL) film model. The latter was used for linear stability analysis and nonlinear wave propagation analysis. The numerical implementation of the IBL film model combines a finite volume formulation with a pseudo-spectral approach for the capillary terms that allows one to investigate non-periodic surface tension-dominated flows. The numerical model was successfully validated with DNS computations. The combination of these numerical tools allows one to describe the mechanisms of capillary and nonlinear damping and identify the instability threshold of the coating processes. The results show that transverse modulations can be beneficial for damping two-dimensional waves within the range of operational conditions considered in this study, which are relevant to air-knife and slot-die coating.

Thermal conductivity reconstruction method with application in a face milling operation

PurposeThis paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.Design/methodology/approachThe discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.FindingsThe paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.Originality/valueThe paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

Fourier Neural Operator Network for Fast Photoacoustic Wave Simulations

Simulation tools for photoacoustic wave propagation have played a key role in advancing photoacoustic imaging by providing quantitative and qualitative insights into parameters affecting image quality. Classical methods for numerically solving the photoacoustic wave equation rely on a fine discretization of space and can become computationally expensive for large computational grids. In this work, we applied Fourier Neural Operator (FNO) networks as a fast data-driven deep learning method for solving the 2D photoacoustic wave equation in a homogeneous medium. Comparisons between the FNO network and pseudo-spectral time domain approach were made for the forward and adjoint simulations. Results demonstrate that the FNO network generated comparable simulations with small errors and was orders of magnitude faster than the pseudo-spectral time domain methods (~26× faster on a 64 × 64 computational grid and ~15× faster on a 128 × 128 computational grid). Moreover, the FNO network was generalizable to the unseen out-of-domain test set with a root-mean-square error of 9.5 × 10−3 in Shepp–Logan, 1.5 × 10−2 in synthetic vasculature, 1.1 × 10−2 in tumor and 1.9 × 10−2 in Mason-M phantoms on a 64 × 64 computational grid and a root mean squared of 6.9 ± 5.5 × 10−3 in the AWA2 dataset on a 128 × 128 computational grid.

On the best volume fraction distributions for functionally graded cylinders, spheres and disks – A pseudospectral approach

The paper is devoted to the optimization of axisymmetric structures made of functionally graded materials and subject to mechanical and thermal loads. The novelty of the results is that the volume fraction distribution is not limited to a power-law variation, as in most of the works available in the literature, but can be any (piecewise continuous) function. This approach leads to an intrinsic tailoring approach, in the sense it occurs without prefixing the spatial distribution of effective mechanical properties a priori, and therefore exploiting at best the inhomogeneity of functionally graded material. After recalling the governing equations and showing some recent results concerning candidate solutions for the optimal volume fraction distribution in some particular cases, several instances of the optimization problems aiming at minimizing occurring maximum stresses are formulated. We show that all these formulations can be treated within the same numerical approach based on the so-called pseudospectral methods. In the last part of the paper we describe how these methods have been effectively applied to the considered problems and we discuss the yielded solutions comparing them, where possible, with power-law solutions.