Numerically Research Articles

Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

Accelerating aeroelastic UVLM simulations by inexact Newton algorithms

AbstractWe consider the aeroelastic simulation of flexible mechanical structures submerged in subsonic fluid flows at low Mach numbers. The nonlinear kinematics of flexible bodies are described in the total Lagrangian formulation and discretized by finite elements. The aerodynamic loads are computed using the unsteady vortex-lattice method wherein a free wake is tracked over time. Each implicit time step in the dynamic simulation then requires solving a nonlinear equation system in the structural variables with additional aerodynamic load terms. Our focus here is on the efficient numerical solution of this system by accelerating the Newton algorithm. The particular structure of the aeroelastic nonlinear system suggests the structural derivative as an approximation to the full derivative in the linear Newton system. We investigate and compare two promising algorithms based on this approximation, a quasi-Newton type algorithm and a novel inexact Newton algorithm. Numerical experiments are performed on a flexible plate and on a wind turbine. Our computational results show that the approximation can indeed accelerate the Newton algorithm substantially. Surprisingly, the theoretically preferable inexact Newton algorithm is much slower than the quasi-Newton algorithm, which motivates further research to speed up derivative evaluations.

Analysis of the free vibration of a beam on a deformable support using a generalised analytical continuum model

ABSTRACT This paper deals with the application of a generalised analytical continuum foundation model recently proposed by the senior author to the dynamic analysis of beams. Specifically, analytical and numerical solutions of the problem of the free vibration of a beam on an equivalent two-parameter model and a simplified Winkler-type model are presented. In the two-parameter representation, a novel approach is used in which the shear parameter is taken care of by moving it from the soil-beam interface to the beam proper in the form of a geometric stiffness so that it serves as an enhancement to the conventional elastic beam stiffness. By doing so, the familiar Winkler support is maintained while the shear interaction among the springs is realised at the same time. Additional end springs are introduced to account for the shear continuity extending beyond the ends. The analytical and numerical results are compared with those obtained from the finite element software, Abaqus, by means of which the calibration factor is determined. The model is validated using measured data from the literature. The results are also compared with other existing models. Both the single- and two-parameter variants of the model performed remarkably well, especially the two-parameter variant.

Stochastic response and stability analysis of nonlinear vehicle energy-regenerative suspension system with time delay

With the rapid development of new energy vehicles, a novel vehicle energy-regenerative suspension for simultaneous vibration suppression and energy harvesting has become one of the key development directions of the next generation of vehicle design. In order to improve the performance of suspension system, the time-delayed feedback control is applied to the novel suspension system subjected to different external excitation. At the same time, there are few studies related to the stochastic dynamics of nonlinear vehicle energy-regenerative suspension system with time delay. This paper studies stochastic response and stability of nonlinear vehicle energy-regenerative suspension with time-delayed feedback control under the excitation of Gaussian white noise. First of all, a stochastic dynamic model of the nonlinear vehicle energy-regenerative suspension with time delay system is established. Secondly, using the stochastic averaging method based on generalized harmonic function, the steady-state probability density function (PDF) of amplitude is obtained. The accuracy of the presented approach is verified through the comparison of analytical and numerical solutions. In addition, through combining the stochastic averaging method with the Khasminskii method, the top Lyapunov exponent (TLE) of the system is obtained, and then the influence of time delay and system parameters on the asymptotic Lyapunov stability is discussed. Finally, the effects of time delay and system parameters on system performance are also studied.

Drift approximation by the modified Boris algorithm of charged-particle dynamics in toroidal geometry

AbstractIn this paper, we study the dynamics of charged particles under a strong magnetic field in toroidal axi-symmetric geometry. Using modulated Fourier expansions of the exact and numerical solutions, the long-term drift motion of the exact solution in toroidal geometry is derived, and the error analysis of the large-stepsize modified Boris algorithm over long time is provided. Numerical experiments are conducted to illustrate the theoretical results.

Finite difference simulation of natural convection of two-phase hybrid nanofluid along a vertical heated wavy surface

This study employs finite difference modelling to investigate the natural convection of a two-phase hybrid nanofluid consisting of Al 2 O 3 and Cu nanoparticles dispersed in water along a vertically heated wavy surface. The hybrid nanofluid has unique features that influence convective heat transfer. Numerical solutions analyze fluid flow patterns and temperature distributions, providing insights into heat exchange and thermal management. The research highlights the importance of hybrid nanofluids, notably those containing Al 2 O 3 and Cu in a water base, in improving overall heat transfer efficiency. The mathematical model accounts for laminar and incompressible fluid flow with a Prandtl number of Pr = 6.2, Lewis number Le = 10, and maximum 10 % concentration of hybrid nanoparticles. After transforming the governing equations into a non-dimensional form, the implicit finite difference method is used to solve them. The solution employs the in-house FORTRAN 90 code and compares its outcomes with the benchmark results. Various parameters, such as the Schmidt number (Sc = 1 to 10), volume fraction of nanoparticles ( ϕ = 0.0 to 0.1), wavy amplitude (A = 0.0 to 0.3), and N BT = (0.05 to 0.2), are investigated regarding temperature, velocity, local skin friction coefficient ( C f ) , local Nusselt number (Nu), streamlines, and isotherms. Elevated volume fraction generally decreases skin friction, increases temperature, and may reduce velocity. Meanwhile, higher skin friction, temperature, velocity, and local Nusselt numbers often correlate with elevated Schmidt numbers. Changes in amplitude and N BT affect skin friction, temperature, velocity, and local Nusselt numbers, providing information on the dynamics of fluid systems. For example, when the volume fraction (ϕ) increases from 0 to 0.1 at Y = 2, the temperature rises by 75 % , while the velocity gradually decreases by 11.11 % . This is in contrast to findings according to when the N BT rises from 0.05 to 0.2 at Y = 1. In this case, the temperature drops by 18.75 % , followed by a 9.68 % fall in fluid velocity. Understanding these interactions enhances comprehension of heat transfer and fluid dynamics. By investigating these interactions, we not only improve our understanding of heat transfer and fluid dynamics, but also open the door to new strategies for improving thermal systems and manufacturing procedures.

Uniqueness and numerical inversion in bioluminescence tomography with time-dependent boundary measurement

In the paper, an inverse source problem in bioluminescence tomography (BLT) is investigated. BLT is a method of light imaging and offers many advantages such as sensitivity, cost-effectiveness, high signal-to-noise ratio and non-destructivity. It thus has promising prospects for many applications such as cancer diagnosis, drug discovery and development as well as gene therapies. In the literature, BLT is extensively studied based on the (stationary) diffusion approximation (DA) equation, where the distribution of peak sources is reconstructed and no solution uniqueness is guaranteed without proper a priori information. In this work, motivated by solution uniqueness, a novel dynamic coupled DA model is proposed. Theoretical analysis including the well-posedness of the forward problem and the solution uniqueness of the inverse problem are given. Based on the new model, iterative inversion algorithms under the framework of regularizing schemes are introduced and applied to reconstruct the smooth and non-smooth sources. We discretize the regularization functional with the finite element method and give the convergence rate of numerical solutions. Several numerical examples are implemented to validate the effectiveness of the new model and the proposed algorithms.

Dynamics of Hepatitis B Virus Transmission with a Lévy Process and Vaccination Effects

This work proposes a novel stochastic model describing the propagation dynamics of the hepatitis B virus. The model takes into account numerous disease characteristics, and environmental disturbances were collected using Lévy jumps and the conventional Brownian motions. Initially, the deterministic model is developed, and the asymptotic behavior of the model’s solution near the equilibria is examined. The deterministic model is transformed into a stochastic model while retaining the Lévy jumps and conventional Brownian motions. Under specific assumptions, the stochastic system is shown to have a unique solution. The study further investigates the conditions that ensure the extinction and persistence of the infection. The numerical solutions to both stochastic and deterministic systems were obtained using the well-known Milstein and RK4 techniques, and the analytical findings are theoretically confirmed. The simulation suggests that the noise intensities have a direct relationship with the amplitudes of the stochastic curves around the equilibria of the deterministic system. Smaller values of the intensities imply negligible fluctuations of trajectories around the equilibria and, hence, better describe the extinction and persistence of the infection. It has also been found that both Brownian motions and the Lévy jump had a significant influence on the oscillations of these curves. A discussion of the findings of the study reveals other important aspects as well as some future research guidelines. In short, this study proposes a novel stochastic model to describe the propagation dynamics of the hepatitis B virus.

A Stable High-Order Perturbation of Surfaces/Asymptotic Waveform Evaluation Method for the Numerical Solution of Grating Scattering Problems

A Stable High-Order Perturbation of Surfaces/Asymptotic Waveform Evaluation Method for the Numerical Solution of Grating Scattering Problems

Microscopic structure of electromagnetic whistler wave damping by kinetic mechanisms in hot magnetized Vlasov plasmas

Electromagnetic transverse perturbations propagating parallel to the external magnetic field in a warm electron plasma, specifically the warm electron whistler-mode waves, are simulated in Maxwellian as well as κ distributed (with energetic tail) electrons. The Vlasov-Maxwell phase-space continuum simulations are applied to the stable and unstable (i.e. isotropic and anisotropic) VDFs. The variation of real frequency from both numerical solution of dispersion relation and simulations show limited sensitivity to electron temperature in low wave-number regime as compared to high wave number regime, however the opposite holds for the imaginary frequency or the decay rate. The analytically predicted reduction in the decay rate of the whistler-mode with increasing electron temperature is recovered by the Vlasov-Maxwell simulations. The phase-space portraits of the these cases show that after the linear damping phase of the evolution, the particles are trapped in the wave magnetic field leading to the wave amplitudes oscillating about a mean value which follow the theoretical analysis. Palmadesso and Schmidt (1971) Physics of Fluids 14, 1411.

Desvendando a influência das heterogeneidades e da extração no fluxo de água subterrânea e no transporte de soluto em um aquífero de carbonato fraturado, na Sicília, Itália

AbstractThe migration of a groundwater contaminant plume consisting of light nonaqueous phase liquids (LNAPLs) along the Ionian coastline of Sicily, Italy, has been reported to follow a trajectory that is inconsistent with the regional hydraulic gradient. The influence of several faults affecting the fractured carbonate bedrock aquifer, and groundwater abstraction from a well, were hypothesized to be responsible for the anomalous trajectory of the contaminant plume. A conceptual hydrogeological model was developed for the study area that incorporated structural information derived from geophysical surveys and the mapping of fractures in bedrock outcrops. This conceptual model was incorporated into numerical groundwater flow and solute transport models to simulate the groundwater transport of the light nonaqueous phase liquids. Four model scenarios representing different levels of complexity were tested to assess the relative influence of geological heterogeneity and groundwater abstraction on the migration of the contaminant plume. Results show that underground major discontinuity systems, invoking the presence of the faults in the model domain accounted for the observed migration of the contaminant plume, act as conduits for groundwater flow. Conversely, groundwater abstraction from a well was found to result in relatively minor, localized impacts to the migration of the contamination plume. This study demonstrates the importance of incorporating geological heterogeneity into groundwater modelling and environmental risk assessments associated with the storage of LNAPLs.

Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition

The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed based on the artificial boundary method (ABM), addressing issues related to the semi-infinite boundary. We adopt the finite difference method (FDM) for deriving the numerical solution by employing the L1 scheme to approximate the fractional derivative. To confirm the precision of this method, a source term is added to establish an exact solution for verification purposes. A comparative evaluation of the ABC versus the direct truncated boundary condition (DTBC) is conducted, with their effectiveness and soundness being visually scrutinized and assessed. This study investigates the impact of the motion of plates at different fluid flow velocities, focusing on the effects of dynamic elements influencing flow mechanisms and velocity. This research’s primary conclusion is that a higher fractional parameter correlates with the fluid flow. As relaxation parameters decrease, the delay effect intensifies and the fluid velocity decreases.

Can physics-informed neural networks beat the finite element method?

Abstract Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen–Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.

On the displacement control performance of base-isolated structures incorporating nonlinear negative stiffness device

Low-frequency dominant far-field harmonic-like long period ground motions (FGMs) excitations have the potential to amplify the seismic response of base-isolated structures (BIS) and may result in isolator displacements exceeding their maximum limits. The application of negative stiffness devices (NSD) in isolated structures primarily focuses on SDOF systems, which fails to capture the displacement response of the superstructure on the isolation story. To address this issue, this study proposes the application of NSD to a 2DOF system in a BIS, with root mean square value as the metric for displacement transmissibility. Firstly, the displacement transmissibility of base-isolated structures with negative stiffness devices (BIS-NSD) and BIS is derived using the triple-harmonic balance (T-HB) method and the single-harmonic balance (S-HB) method, respectively. Subsequently, the applicability of the T-HB method and the S-HB method in obtaining analytical solution for the displacement transmissibility of BIS-NSD and BIS is revealed and validated through numerical solution. Parameters are selected based on fixed-point theorem for comparative analysis of the displacement and acceleration transmissibility between BIS-NSD and BIS. Finally, a comparative analysis between BIS-NSD and BIS under FGMs confirms the application advantages of NSD. The results indicate that compared to BIS, BIS-NSD significantly reduces the displacement transmissibility of the superstructure and the isolation story in the low-frequency range. As the primary resonance frequency decreases, NSD widens the vibration bandwidth of BIS. BIS-NSD effectively mitigates the increase in displacement caused by low-frequency ground motions. The acceleration transmissibility demonstrates that, without sacrificing the acceleration of the superstructure, it effectively controls the displacement of the isolation story.

Well-Balanced conservative central upwind scheme for solving the dam-break flow problem over erodible bed

This work deals with the numerical solution of dam-break flow over an erodible bed. The mathematical model is a combination of the shallow water, the transport diffusion and the bed morphology change equations. The system is solved by a well-Balanced central upwind scheme with conservative property. Several tests are illustrated in order to validate the accuracy and the performance of the model. A comparison of central upwind scheme and Roe scheme is presented.

A numerical code for the analysis of magnetic white-dwarf spectra that includes field effects on the chemical equilibrium

We present a new magnetic-atmosphere model code for obtaining synthetic spectral fluxes of hydrogen-rich magnetic white dwarfs. To date, observed spectra have been analyzed with models that neglect the magnetic field's effects on the atomic populations. In this work, we incorporate state-of-art theory in the evaluation of numerical densities of atoms, free electrons, and ions in local thermodynamical equilibrium under the action of a magnetic field. The energy distribution of atoms is rigorously evaluated for arbitrary field strength. This energy pattern includes going from tightly bound states to metastable or truly bound, highly excited states embedded in the continuum, that is, over the first Landau level. Finite nuclear mass effects and the coupling between the internal atomic structure and the motion of the atom across the magnetic field are also considered. Synthetic fluxes are generated with integrations of numerical solutions of polarized radiative transfer over the visible stellar disk using a spherical $t$-design method. The atmosphere code is tested with observations from the Sloan Digital Sky Survey for a group of known magnetic white dwarfs. Physical stellar parameters are obtained from least-squares fits to the observed energy distribution and compared with results of previous works. We show that the use of zerofield ionization equilibrium in spectral analyses can lead to underestimated effective temperatures for highly magnetic white dwarfs.

Analysis of nonlinear problems using the Dual Mesh Control Domain Method with arbitrary meshes

The Dual Mesh Control Domain Method (DMCDM), proposed and developed by the second author, is a recent advancement in the field of numerical methods. This method represents a blend of the Finite Element Method (FEM) and the Finite Volume Method (FVM), each having its primary domains of popularity. The DMCDM incorporates the concepts of interpolation, duality, and numerical integration from the former and the idea of satisfying the integral (global) form of the differential (or balance) equations from the latter. In this paper, we present a fresh and systematic approach to solving two-dimensional nonlinear problems, including advection–diffusion problems, with arbitrary primal meshes. We develop an approach where element-level equations are derived and assembled in a manner similar to that of the FEM, as opposed to using nodal equations, as has been the convention in all of the previous works involving the DMCDM. Additionally, we extend the previous works to include higher-order interpolation, specifically quadratic interpolation. In short, we provide formulations for four-node bilinear and nine-node biquadratic elements of quadrilateral shape, enabling the application of the DMCDM to arbitrary primal and dual meshes. Finally, we explore the Newton iteration method for linearizing the nonlinear system of equations that arises from the DMCDM discretization. We present numerical results and comparisons against available exact and numerical solutions to verify the accuracy and robustness of the DMCDM.

Impact of electro-osmotic, activation energy and chemical reaction on Sisko fluid over Darcy–Forchheimer porous stretching cylinder

This study provides numerical solution for two-dimensional electro-osmotically motivated electro-magneto-hydrodynamic Sisko fluid through an elongating porous cylinder. The electro-osmotic, activation energy with chemical reaction are also interpreted for this flow model. The numerical equations that govern this flow problem are renewed into dimensionless form by applying appropriate transformations exploiting Runge–Kutta–Fehlberg fifth order through shooting technique method. The after-effects are illustrated as graphs for the concerned physical parameters and their impact on convection and conduction is also studied. The velocity rises for electro-osmotic parameter whereas drops for magnetic and porous parameter. The temperature shows an increasing profile for the curvature, electric and thermal conductivity parameter while decreases for Prandtl number. The concentration of nanoparticles in the fluid boosts for activation energy but deflates for curvature parameter. The present findings appear to be in good accord when compared to earlier published studies. Numerous opportunities and applications are presented by applying electro-osmotic forces to non-Newtonian fluid flow, especially in the fields of nanotechnology, electro-kinetics and micro-fluids. The current study can be applied to design effective electro-magnetic devices, particularly in certain thermal transport characteristic regime. The main findings demonstrate the great utility of electro-osmosis in micro-fluidic devices, chemical analysis, soil analysis and cement slurries for managing flow and heat transmission.

Unraveling the transformative impact of ternary hybrid nanoparticles on overlapped stenosis with electroosmotic vascular flow kinetics and heat transfer

This study examines how ternary hybrid nanoparticles affect electroosmotic vascular flow kinetics and heat transfer. Through a meticulous exploration of their intricate interplay, this research unveils unprecedented insights into their transformative impact. By comprehensively analyzing the dynamics of vascular flow and thermal behavior under the influence of ternary hybrid nanoparticles, novel advancements are revealed. The assessment is innovative because it incorporates electroosmotic force on blood flow that contains three different nanoparticles (Ti O2, Al2O3 and Si O2). To evaluate the numerical solution, an unraveled approach using the finite element method is employed, ensuring both stability and convergence of the solution. The computed numerical results are presented in graphs and tables, showcasing the relationship between key factors. The comparative analysis uncovers the unparalleled performance and remarkable efficacy of these nanoparticles in enhancing the electroosmotic vascular flow and optimizing heat transfer. The electric field due to the electroosmosis flow interacts with flow pattern and influence the potential flow and vortex formation. This research presents a paradigm shift in the understanding of biomedical engineering and fluid dynamics, offering promising prospects for revolutionizing healthcare technologies and achieving unprecedented levels of thermal management efficiency across diverse applications.

Viscous dissipation and Joule heating in case of variable electrical conductivity Carreau–Yasuda nanofluid flow in a complex wavy asymmetric channel through porous media

This paper focuses on flow structures and thermal fields of the Carreau–Yasuda (CY) nanofluid model through a two-dimensional, wavy, complicated vertical asymmetrical conduit filled with porous elements. Formulations of the viscous dissipation in the case of CY nanofluids are derived and nonlinear radiation flux as well as joule heating are examined. Buongiorno’s nanofluid approach, which involves Brownian motion and thermophoresis aspects is considered. The electrical conductivity of the suspension is considered as a variable where it depends upon the ambient temperature and concentration distributions and the Joule heating impacts are not neglected. The approach of solving the problem is contingent upon converting the system to dimensionless form then the lubrication approach with low magnetic Reynold numbers is applied. Numerical solutions are found for the resultant system, and wide ranges are considered for Weissenberg number We, non-Newtonian parameter n and Darcy number [Formula: see text], namely, [Formula: see text], [Formula: see text] and [Formula: see text], respectively. The major results indicated that gradients of the pressure are higher in case of shear thickening [Formula: see text] comparing to in the instance of shear thinning [Formula: see text]. Also, the velocity is enhanced, close to the channel’s lowest portion, as the Weissenberg number is growing. The variable electrical conductivity gives a higher mass transfer rate compared to the constant property.