External Force Term Research Articles

Uniform measure attractors for nonautonomous stochastic tamed 3D Navier–Stokes equations with almost periodic forcing

This article provides a characterization of uniform measure attractors for non-autonomous stochastic tamed 3D Navier–Stokes equations, specifically with a almost-periodic external force term. The existing literature has primarily focused on measure attractors within an autonomous framework; however, when the forcing function is almost periodic, the solution process generated by the corresponding equations behaves as a non-autonomous Markov process, rendering the existing framework inapplicable. Therefore, it is crucial to investigate this within a new framework, which is the motivation behind the study of uniform measure attractors in this paper.

Study on the evolution of heterogeneous double-cavity induced by near-wall and the fluctuation characteristics of load field

It is well known that the collapse of heterogeneous multi-cavity near the wall will induce the fluctuation of the load field. To address this problem, the Lattice Boltzmann Method (LBM) is applied to model the three-phase coupling between gas-liquid-solid. The objective is to investigate the evolution of heterogeneous double bubbles and the spatial-temporal distribution characteristics of wall loads induced near the wall. In this study, the pseudopotential Multi-Relaxation-Time Lattice Boltzmann Model (MRT-LBM) and the Carnahan-Starling Equation of State (C-S-EOS) with an extended format for the external force term are used. The effects of the distance of the bubble to the wall, the pressure differences between the inside and outside of the bubble, and the relative size of the bubble on the dynamic evolution and the load distribution characteristics of heterogeneous multi-bubbles near the wall are investigated in order to determine the influence of these factors. Under a two-dimensional pressure field, the collapse process of double cavitation bubbles is visualized. Through the flow field, the morphological changes of the cavitation bubble collapse near the wall are also described. Various parameters are found to have an influence on the evolution of double cavitation bubbles near the wall and the resulting load field. The study employs the Lattice Boltzmann Method and the Potential Model for the analysis of the heterogeneous bubble collapses in the near wall region.

Improved State-Space Approach Based on Lumped Mass Matrix for Transient Analysis of Large-Scale Locally Nonlinear Structures

Due to the assumption of acceleration variation in traditional step-by-step integration methods such as Newmark, sufficiently small time steps are required to ensure numerical stability and accuracy in dynamic systems. In contrast, the state-space approach, based on piecewise interpolation of discrete load functions, does not rely on predetermined acceleration assumptions and has demonstrated high efficiency in terms of stability and accuracy. The original state-space method requires the calculation of the inverse of the structural mass in the transition matrix. However, when a lumped mass matrix is used, this computation renders the entire mass matrix singular, resulting in an invalid solution expression. To address this issue, this study proposes an improved state-space approach for the transient analysis of large-scale structural systems with local nonlinearities. In this approach, a nonlinear force corrector is introduced as an external force term applied to the linear elastic system to account for the nonlinear behavior of locally yielding components. Consequently, the original nonlinear dynamic system can be transformed into an equivalent linear elastic transient system. Furthermore, based on the lumped mass matrix, a first-order ordinary differential state-space equation for such an equivalent linear elastic transient system is derived. Simulation results from three transient system examples show that the state-space approach outperforms the Newmark method in terms of accuracy and stability for dynamic systems characterized by high frequency and low damping. The prediction results show that the state-space approach appears to be insignificantly affected by the choice of the consistent or lumped mass matrix. The numerical results show that the root-mean-square errors between the consistent and lumped matrices in the top displacement time histories of a 15-storey plane frame under various seismic intensities are all less than 1%, and in the base reaction time histories responses the discrepancies are only about 0.5%, indicating that the use of lumped mass matrices is quite reliable. When many nodes or degrees of freedom have no assigned mass, the dimensionality of the state-space equation can be significantly reduced using the lumped mass approach. Therefore, the simulation of large-scale systems can be simplified by employing the improved state-space approach with lumped mass matrices, yielding results nearly identical to those obtained using traditional methods. In conclusion, the improved state-space approach has great potential for the simulation of transient behavior in large-scale systems with local nonlinearities.

Chaotic vibration of a curved CNT conveying magnetic fluid in the thermo-magnetic field considering the surface effects

The nonlinear chaotic vibration of curved single-walled carbon nanotube (CSWCNT) conveying magnetic fluid is studied based on the nonlocal Euler-Bernoulli beam model. The governing equation of CSWCNT is presented considering the axial thermo-magnetic load and the surface effect. By employing the Galerkin decomposition approximation method along with the admissible beam shape function satisfying the cantilevered beam boundary conditions, the nonlinear partial differential equation of the system is reduced to a nonlinear ordinary differential equation and numeric integration procedures are used to solve it. By considering the non-dimensional damping, cubic and quadratic terms, and amplitude of external force as controlling parameters, the bifurcation diagram and the largest Lyapunov exponent are employed to distinguish the chaotic, periodic, and quasi-periodic critical parameters of the curved CNT dynamics and validate the predicted results. Also, the phase plane and Poincare map for these critical parameters are provided. The results show that decreasing values of damping coefficient and quadratic term leads to quasi-periodic or chaotic motion, while system has periodic behaviour for smaller values of the external force and nonlinear cubic term. Also, magnetic field intensity and length of the CSWCNT help preventing the chaotic motion. Furthermore, adverse effects of higher values of temperature change, flow velocity and its correction factor are seen based on the obtained results.

Integrability, bilinearization, Bäcklund transformations and solutions for a generalized variable-coefficient Gardner equation with an external-force term in a fluid or plasma

Integrability, bilinearization, Bäcklund transformations and solutions for a generalized variable-coefficient Gardner equation with an external-force term in a fluid or plasma

Existence and uniqueness of solution to unsteady Darcy-Brinkman problem with Korteweg stress for modelling miscible porous media flow

The work investigates a model that combines a convection-diffusion-reaction equation for solute concentration with an unsteady Darcy-Brinkman equation for the flow field, including the Korteweg stress. Additionally, the flow field experiences an external body force term while the permeability fluctuates with solute concentration. Such models are used to describe flows in porous mediums such as fractured karst reservoirs, mineral wool, industrial foam, coastal mud, etc. The system of equations has Neumann boundary conditions for the solute concentration and no-flow conditions for the velocity field, and the well-posedness of the model is discussed for a wide range of initial data. The proofing techniques remain applicable in establishing the well-posedness of non-reactive and homogeneous porous media flows under the specified simplifications.

Effects of electron-to-ion mass ratio in driving magnetic oscillations of magnetohydrodynamic plasmas and self-organized criticality

Evolutions of nonlinear magnetic fields have been shown to be governed by a set of coupled nonlinear equations of second order in magnetohydrodynamic (MHD) plasmas by Lee and Parks [Geophys. Res. Lett. 19, 637–640 (1992)]. We have considered the same set of coupled nonlinear equations for further analysis in this work by neglecting the presence of external forcing term in it. Different modes of oscillations of magnetic field have been found to exist in special limiting cases of this set of undriven second order coupled nonlinear equations having frequencies that are multiples of lower hybrid frequency. Numerical solutions of these coupled equations have been analysed revealing a quasi-periodic route to chaotic oscillations of the nonlinear magnetic fields as electron-to-ion mass ratio signifying presence of linear coupling effects is increased. Some signatures of the phenomenon of self-organized criticality (SOC) in typical quasi-periodic oscillations of magnetic field have also been noticed using Fourier analysis. The presence of long range correlations has been witnessed in quasi-periodic oscillations whereas both long range correlations and anticorrelations are found in chaotic oscillations using rescaled range analysis. Concluding remarks are provided in addition to various results and discussions.

An effective and robust lattice Boltzmann model guided by atlas for hippocampal subregions segmentation.

Given the varying vulnerability of the rostral and caudal regions of the hippocampus to neuropathology in the Alzheimer's disease (AD) continuum, accurately assessing structural changes in these subregions is crucial for early AD detection. The development of reliable and robust automatic segmentation methods for hippocampal subregions (HS) is of utmostimportance. Our aim is to propose and validate a HS segmentation model that is both training-free and highly generalizable. This method should exhibit comparable accuracy and efficiency to state-of-the-art techniques. The segmented HS can serve as a biomarker for studying the progression ofAD. We utilized the functional magnetic resonance imaging of the Brain's Integrated Registration and Segmentation Tool (FIRST) to segment the entire hippocampus. By intersecting the segmentation results with the Brainnetome (BN) atlas, we obtained coarse segmentation of the four HS regions. This coarse segmentation was then employed as a shape prior term in the lattice Boltzmann (LB) model, as well as for initializing contours. Additionally, image gradients and local gray levels were integrated into the external force terms of the LB model to refine the coarse segmentation results. We assessed the segmentation accuracy of the model using the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset and evaluated the potential of the segmentation results as AD biomarkers on both the ADNI and Xuanwudatasets. The median Dice similarity coefficients (DSC) for the left caudal, right caudal, left rostral, and right rostral hippocampus were 0.87, 0.88, 0.88, and 0.89, respectively. The proportion of segmentation results with a DSC exceeding 0.8 was 77%, 78%, 77%, and 94% for the respective regions. In terms of volume, the correlation coefficients between the segmentation results of the four HS regions and the gold standard were 0.95, 0.93, 0.96, and 0.96, respectively. Regarding asymmetry, the correlation coefficient between the segmentation result's right caudal minus left caudal and the corresponding gold standard was 0.91, while for right rostral minus left rostral, it was 0.93. Over time, we observed a decline in the volumes of the four HS regions and the total hippocampal volume of mild cognitive impairment (MCI) converters. Analysis of inter-group differences revealed that, except for the right rostral region in the ADNI dataset, the p-values for the four HS regions in the normal controls (NC), MCI, and AD groups from both datasets were all below 0.05. The right caudal hippocampal volume demonstrated correlation coefficients of 0.47 and 0.43 with the mini-mental state examination (MMSE) and Montreal cognitive assessment (MoCA), respectively. Similarly, the left rostral hippocampal volume showed correlation coefficients of 0.50 and 0.58 with MMSE and MoCA,respectively. Our framework allows for direct application to different brain magnetic resonance (MR) datasets without the need for training. It eliminates the requirement for complex image preprocessing steps while achieving segmentation accuracy comparable to deep learning (DL) methods even with small sample sizes. Compared to traditional active contour models (ACM) and atlas-based methods, our approach exhibits significant speed advantages. The segmented HS regions hold promise as potential biomarkers for studying the progression ofAD.

Modified hybrid B-spline estimation based on spatial regulator tensor network for burger equation with nonlinear fractional calculus

In this paper, a modified hybrid B-spline approximation based on spatial regulator tensor network (MHBA-SRTN) is proposed for solving the application and the feasibility problems of fractional calculus-based Burger equation. The main innovation points include: (1) a dual singular kernel based fractional derivative operator is employed on the Burger’s equation with external force term to analyze the solutions and properties under perturbation condition for further realism simulation; (2) a modified hybrid B-spline basis function and its corresponding topological tensor network structure are further proposed to solve the provided Burger’s equation with more favorable approximation accuracy and solution uniqueness; (3) the tensor network transformed the hybrid B-spline construction process into the tensor solving with the introduced spatially regulated tensor network decomposition for a more robust solution procedure; (4) an acceleration approach for decomposition process is introduced to significantly reduce its computational and storage complexity. Extensive experiments are also carried out to verify the performance of MHBA-SRTN.

Lattice Boltzmann modelling of multicomponent and multiphase flow with high density ratio

Thermodynamic consistency is a key factor in heat and mass transfer simulation. A modified lattice Boltzmann pseudopotential model with the ability to simulate multiphase and multicomponent flow with a large density and viscosity ratio is proposed. Relative to a model using the exact difference method, a better thermodynamic consistency is obtained through introducing an improved external force term, a multi-relaxation-time collision operator, and a scaling coefficient k of the equation of state. The coefficient of scale k = 0.1 is preferred due to its better thermodynamic consistency, lower spurious current and a smaller surface tension. The inter-particle interaction strength has little influence over the thermodynamic consistency due to its limited contributions to the surface tension. With the model, the bubble dissolution under pressure is investigated and Henry’s law is validated.

Well-posedness for heat conducting non-Newtonian micropolar fluid equations

<abstract><p>In this paper, we consider the first boundary value problem for a class of steady non-Newtonian micropolar fluid equations with heat convection in the three-dimensional smooth and bounded domain $ \Omega $. By using the fixed-point theorem and introducing a family of penalized problems, under the condition that the external force term and the vortex viscosity coefficient are appropriately small, the existence and uniqueness of strong solutions of the problem are obtained.</p></abstract>

Minimal chaotic models from the Volterra gyrostat

Low-order models obtained through Galerkin projection of several physically important systems (e.g., Rayleigh–Bénard convection, mid-latitude quasi-geostrophic dynamics, and vorticity dynamics) appear in the form of coupled gyrostats. Forced dissipative chaos is an important phenomenon in these models, and this paper introduces and identifies “minimal chaotic models” (MCMs), in the sense of having the fewest external forcing and linear dissipation terms, for the class of models arising from an underlying gyrostat core. The identification of MCMs reveals common conditions for chaos across a wide variety of physical systems. It is shown here that a critical distinction is whether the gyrostat core (without forcing or dissipation) conserves energy, depending on whether the sum of the quadratic coefficients is zero. The paper demonstrates that, for the energy-conserving condition of the gyrostat core, the requirement of a characteristic pair of fixed points that repel the chaotic flow dictates placement of forcing and dissipation in the minimal chaotic models. In contrast if the core does not conserve energy, the forcing can be arranged in additional ways for chaos to appear in the subclasses where linear feedbacks render fewer invariants in the gyrostat core. In all cases, the linear mode must experience dissipation for chaos to arise. The Volterra gyrostat presents a clear example where the arrangement of fixed points circumscribes more complex dynamics.

Attractors' Analysis and Bifurcation Diagrams for an Impacting Inverted Pendulum in the Presence of a Two-Terms Harmonic Excitation

Abstract In this work, we examine the nonlinear dynamics of an inverted pendulum between lateral rebounding barriers. We continue the numerical investigation started in (Demeio et al., 2006, “Response Scenario and Non-Smooth Features in the Nonlinear Dynamics of an Impacting Inverted Pendulum,” ASME J. Comput. Nonlin. Dyn., 1(1), pp. 56–64) by adding the contribution of the second harmonic in the external forcing term. We investigate the behavior of the periodic attractors by bifurcation diagrams with respect to each amplitude and by behavior charts of single attractors in the amplitude parameters plane for fixed frequency. We study the effects of the second harmonic term on the existence domain of each attractor, on local bifurcations, and on the changes in the basins of attraction. The behavior of some robust chaotic attractor is also considered. In the evolution of the periodic attractors we have observed that the addition of the second harmonic generates a rich variety of behaviors, such as loss of stability and formation of isolas of periodic orbits. In the case of chaotic attractors, we have studied one attractor at high frequency, ω = 18, and one at low frequency, ω = 3. In the high frequency case we detect a transition from a scattered to a confined attractor, whereas at the lower frequency the chaotic attractor is present over a wide range of the second harmonic's amplitude. Finally, we extend the investigation of the chaotic attractors by bifurcation diagrams with respect to the frequency.

Propagation of ion-acoustic wave and its fractal representations in spin polarized electron plasma

Propagation of small-amplitude quantum ion-acoustic waves and its fractal representations is investigated in an electron-ion quantum plasma with separated spin electrons in the framework of the KdV and EKdV equations derived using reductive perturbation technique. These two equations are transformed into planar dynamical systems by using suitable traveling wave transformation. Qualitatively different phase portraits of these two systems are drawn with their respective Sagdeev’s pseudopotential curves and surface plots. Distinct orbits in the phase portraits give rise to distinct wave solutions. Periodic and superperiodic wave solutions are investigated numerically and solitary wave solutions are derived analytically. The effect of parameters such as Mach number, direction cosine, spin polarization and frequency ratio on these wave solutions are presented. For instance, it is seen that the spin density polarization ratio has an impressive effect on the amplitude of the wave solution, while the frequency ratio has no effect on its amplitude. Also, we have provided a physical explanation for our finding. Further, the dynamical features of the original system and reconstructed system using fractal interpolation function are investigated under the external forcing term. Chaotic and quasiperiodic phenomena are observed for different initial conditions. The results may help to better understand the degenerate electron gas that exists in dense astrophysical bodies.

Robustness of 3D Navier–Stokes System with Increasing Damping

The principal objective of the paper is the study of the three-dimensional Navier–Stokes system with non-autonomous perturbation force term and increasing damping term, which often appears in the fluid system within saturated porous media and other complex media. With some suitable assumptions on the system parameters and external force term, based on the known result on global well-posedness, the existence of pullback attractors is educed, and the system robustness is shown via the upper semicontinuity of system attractors as the perturbation parameter approaches a certain value.

UPPER SEMICONTINUITY OF UNIFORM RANDOM ATTRACTORS FOR DELAY PARABOLIC EQUATION

This paper concentrates on the upper semicontinuity of uniform random attractors for a class of delay parabolic equations with additive noise and nonautonomous external force terms. Firstly, through the uniform estimation of the solution, it is proved that the solution of the equation has a closed uniform pullback absorbing set with respect to the symbolic space. Then, by Arzela-Ascoli theorem, we prove uniformly pullback compactness of solutions as well as the existence and uniqueness of uniform random attractors. Finally, we prove the upper semicontinuity of the uniform random attractors when time delay approaches to zero.

Bifurcations in a forced Wilson-Cowan neuron pair

We investigate bifurcations of periodic solutions observed in the forced Wilson-Cowan neuron pair by both the brute-force computation and the shooting method. By superimposing the results given by both methods, a detailed topological classification of periodic solutions is achieved that includes tori and chaos attractors in the parameter space is achieved. We thoroughly explore the parameter space composed of threshold values, amplitude, and angular velocity of an external forcing term. Many bifurcation curves that are invisible when using brute-force method are solved by the shooting method. We find out a typical bifurcation structure including Arnold tongue in the angular velocity and the amplitude of the external force parameter plane, and confirm its fractal structure. In addition, the emergence of periodic bursting responses depending on these patterns is explained.

A new linear stability analysis approach for microchannel flow based on the Boltzmann Bhatnagar–Gross–Krook equation

Microchannels are important components of microelectromechanical systems (MEMSs) that encounter rarefaction effects due to their small-scale characteristics. The influence of rarefaction effects on the flow stability of microchannels should be investigated to improve MEMS performance. Based on kinetic theory, a linear stability analysis approach for low-speed rarefied flows was developed by using the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation with an external force term. This approach was applied to study the linear temporal stability of microchannel flows. A slip flow model was introduced for comparison. The corresponding eigenvalue problem was solved with a Chebyshev collocation method. This novel approach yielded a critical Reynolds number of 5778. Analysis of the validity and accuracy of the slip flow model shows that although this model cannot capture the Knudsen layer structure, this approach effectively improves the prediction accuracy of the growth rate of the least stable mode. However, the prediction accuracy gradually decreases with increasing Knudsen number. Compared with the stability results obtained from the BGK equation, the Navier–Stokes equations-based stability analysis method always underestimates the disturbance growth rate, regardless of whether a slip flow model is used. The stability analysis results show that rarefaction effects stabilize the flow. The degree of rarefaction does not affect the trends of growth rate and phase velocity with wavenumber, nor does it affect the shape of the velocity eigenfunctions. For a rarefied case, increasing the Mach number has a destabilizing effect on low-speed microchannel flows.

A lattice Boltzmann method for two-phase nanofluid under variable non-uniform magnetic fields

In this study, a new lattice Boltzmann scheme is developed for the two-phase CuO–H2O nanomagnetic fluid (ferrofluid) under a non-uniform variable magnetic field. It introduces the second-order external force term including both MHD (magnetohydrodynamic) and FHD (ferrohydrodynamic) into the lattice Boltzmann equation. The square cavity and a heat source inside the circular cavity with natural convections of nanofluid are investigated, respectively. The effects of Rayleigh number (Ra), the volume fraction of nanoparticles (φ), Hartmann number (Ha) generated by MHD, and magnetic number (MnF) generated by FHD on the nanofluid flow and heat transfer properties, as well as the total entropy generation (Stot) have been examined. The two-phase lattice Boltzmann model has demonstrated that it is more accurate in predicting the heat transfer of nanofluid than the single-phase model. Consequently, the results calculated by the single-phase and the two-phase methods show the opposite trends. It indicates that nanoparticles could enhance heat transfer with maximum values of 1.78% or deteriorate heat transfer with maximum values of 14.84%. The results of the circular cavity show that Ha could diminish the flow intensity, whereas MnF could enhance it. The average Nusselt number (Nuave) on the heat source decreases with the augments of Ha and MnF but increases with Ra. An optimal volume fraction φ = 1% for heat transfer is obtained except for Ra = 104. Stot achieves the maximum value at Ha = 40 when Ra = 105. It increases with a rise of Ra but reduces with an increment of φ.

A general numerical scheme for the optimal control of fractional Birkhoffian systems

This paper gives a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d’Alembert principle which includes the control as an external force term. Following the strategy of variational integrators, the fractional Pfaff–Birkhoff–d’Alembert principle is directly discretized to develop the equivalent discrete fractional forced Birkhoff equations that served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by existing algorithms. An illustrative example is given to show the efficiency and simplicity of the proposed method.