Nonlinear Stability Research Articles

Nonlinear torsional stability and buckling of spiral stiffened sigmoid FG thin circular cylindrical shells

This research employs an analytical approach to explore nonlinear torsional buckling behaviors of spiral-stiffened sigmoid FG (SSSFG) thin circular cylindrical shells. The research encompasses two variations of SSSFG cylindrical shells: those with internal and external spiral stiffeners. Furthermore, this study explores effects of two distinct material distributions for SSSFG cylindrical shells, specifically metal-ceramic-metal and ceramic-metal-ceramic layers, both following a sigmoid-law distribution. Theoretical formulations are obtained using the smeared stiffeners technique and classical shell theory, incorporating geometrical nonlinearity in the von-Kármán sense. The discretized nonlinear governing equations are obtained through Galerkin's method, and the approximate three-term solution for deflection is developed. Hence, explicit formulations are presented to ascertain the critical torsional load and depict the post-buckling torsional load–deflection curves. The outcomes of this study are validated by comparing them with the related research results existing in the literature. Significantly, the research highlights the influence of diverse parameters and the efficacy of stiffeners in enhancing the stability of SSSFG cylindrical shells. Researchers and engineers in this field may utilize the findings of this research as reference points for their design and research involving SSSFG cylindrical shells.

Nonlinear Convective Stability of a Critical Pulled Front Undergoing a Turing Bifurcation at Its Back: A Case Study

Nonlinear Convective Stability of a Critical Pulled Front Undergoing a Turing Bifurcation at Its Back: A Case Study

Stability of viscoelastic film on a slippery inclined plane

The linear and weakly nonlinear stability of viscoelastic film flowing down a slippery inclined plane is investigated analytically. Under the assumption of the long wave approximation, the first-order Benny equation of Oldroyd-B fluid thin film with slip condition is obtained. Through the normal mode analysis, the neutral stability curve and the temporal growth rates are calculated to explore the linear stability of the film. Linear results show that the critical Reynolds number decreases with the increase in slip length and viscoelastic parameter and that the liquid film may exhibit pure elastic instability. For the nonlinear stability analysis, both hydrodynamic instability and elastic instability are discussed. The primary bifurcations in the phase plane are identified by calculating the Landau coefficient, i.e., the unconditional stable region, the supercritical region, the subcritical region, and the explosive region. The dependence of primary bifurcation regions upon the slip length and Deborah number are studied, and the results indicate that the slip boundary and viscoelasticity destabilizes the flow. According to the Ginzburg–Landau equation, the threshold amplitude of the nonlinear equilibrium solution is analyzed as well.

Flying qualities based time-varying stability augmentation system design for tiltrotor conversion control

Tiltrotors have three flight modes that pose control problems and quality defects during the conversion process. To address this, a novel flying qualities-based time-varying stability augmentation system is designed to achieve multi-mode, nonlinear, and time-varying stability. The system integrates a nonlinear time-varying control law with the flying qualities requirements for all three flight modes. It consists of an inner and outer loop control framework, where the control law in the inner loop is designed based on the Lyapunov theorem of stability. The reference models in the outer loop are derived from the flying qualities criteria to meet level one flying qualities requirements. To evaluate the conversion process, a time-varying flying qualities evaluation method is developed, which includes the conversion path, pilot model, and time-varying flying qualities index. The proposed time-varying stability augmentation control system is then tested through simulation during the conversion process. A pilot-aircraft closed-loop system is established for conducting experiments. Comparison between simulation results and pilot-in-loop experiment results demonstrates the effectiveness of the proposed control system. Furthermore, it proves that the evaluation method is suitable for analyzing time-varying systems. This research can be valuable in designing and evaluating stability augmentation controls for strongly time-varying systems.

Multiplexable all-optical nonlinear activator for optical computing

As an alternative solution to surpass electronic neural networks, optical neural networks (ONNs) offer significant advantages in terms of energy consumption and computing speed. Despite the optical hardware platform could provide an efficient approach to realizing neural network algorithms than traditional hardware, the lack of optical nonlinearity limits the development of ONNs. Here, we proposed and experimentally demonstrated an all-optical nonlinear activator based on the stimulated Brillouin scattering (SBS). Utilizing the exceptional carrier dynamics of SBS, our activator supports two types of nonlinear functions, saturable absorption and rectified linear unit (Relu) models. Moreover, the proposed activator exhibits large dynamic response bandwidth (∼11.24 GHz), low nonlinear threshold (∼2.29 mW), high stability, and wavelength division multiplexing identities. These features have potential advantages for the physical realization of optical nonlinearities. As a proof of concept, we verify the performance of the proposed activator as an ONN nonlinear mapping unit via numerical simulations. Simulation shows that our approach achieves comparable performance to the activation functions commonly used in computers. The proposed approach provides support for the realization of all-optical neural networks.

Nonlinear Stability of Curved Multiphase Composite Panels: Influence of Agglomeration in Randomly Distributed Carbon Nanotubes with Nonuniform In-Plane Loads

Nonlinear Stability of Curved Multiphase Composite Panels: Influence of Agglomeration in Randomly Distributed Carbon Nanotubes with Nonuniform In-Plane Loads

A computer-vision-based deep learning model of smoke diffusion

Smoke is a lethal hazard in fire accidents and the biggest obstacle to emergency rescue. Predicting its diffusion range is important in fire command and emergency rescue. This article utilizes computer-vision-based deep learning methods to learn the characteristics of smoke diffusion and achieve real-time prediction of smoke-temperature fields. Firstly, a tunnel fire was simulated to verify the reliability of the deep learning model in predicting smoke-temperature fields. Thirty-six sets of fire simulation conditions were established with various values of the fire source location and heat release rate (HRR) and longitudinal ventilation airflow velocity. This created a database of 17,280 smoke-temperature fields. Based on windless scenario with the fire source located in the middle of the tunnel, it was determined that the SiLU activation function provides the best nonlinear modelling ability and stability in training neural networks. Based on the smoke-temperature-field database, a U-Net network was constructed to predict the smoke diffusion process of the tunnel fire. Under different scenarios, the U-Net model achieved 90 % accuracy in smoke-temperature-field prediction. The proposed U-Net model was then used to predict smoke dispersion in a petroleum storage depot scenario. The results show that this approach can help to delineate safe areas during fire emergencies.

Nonlinear stability of entropy waves for the Euler equations

Nonlinear stability of entropy waves for the Euler equations

Effects of throughflow and gravity modulation on triple-diffusive convection in a sparsely packed porous medium

This article investigates the impact of four different gravity modulations on triple-diffusive convection in a porous layer with throughflow. The study conducts a weakly nonlinear stability analysis. Additionally, cubic Ginzburg–Landau equation is derived to observe the effects of throughflow, gravity modulation, and concentration. The solution of Ginzburg–Landau equation is used to calculate heat transfer and mass transport in terms of the Nusselt and Sherwood numbers, respectively. Furthermore, the mean Nusselt and Sherwood numbers are calculated. The study investigates the influence of Lewis numbers, solutal Rayleigh numbers, and Peclet numbers on the stability and instability of the system graphically for the four types of gravity modulations. The results show that the Sawtooth modulation destabilizes the system, while the square waveform modulation stabilizes it. The mean Nusselt number decreases with both the first Lewis number Le 1 and the second Lewis number Le 2, as well as the mean Sherwood number, which decreases with the first solutal Rayleigh number Rs 1 and the second solutal Rayleigh number Rs 2. This indicates system stability. Furthermore, it is noted that the heat and mass transport for triangular modulation fall between those of sawtooth modulation and square modulation.

Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

The stability characteristics of a thin film of viscoelastic (Walters’ B′ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number kc which include the viscoelastic parameter Γ, angle of inclination of the plane θ, Reynolds number Re and Weber number We. The weakly nonlinear stability analysis that is based on the second Landau constant J2, reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is 3(1+3Γ)Re−3cotθ−4ReWek2 = 0. It is found that all the four distinct flow zones of this problem exist in Re-k-, θ-k- and Γ-k-plane after the critical value of Rec,θc and Γc, respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of Γ irrespective of the values of Re and θ, as for example, a solution of polyisobutylene in cetane, compared with the viscous (Γ=0) film flow case. Finally, we scrutinize the effect of Γ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.

Multi-objective shape-section optimization of free-form latticed shells using the RBF-NSGA-II algorithm

In this study, a novel RBF-NSGA-II optimization framework based on the response surface methodology (RSM) and the constrained NSGA-II algorithm is proposed to simultaneously improve the static, nonlinear stability and economic performance of free-form latticed shells. The bending stiffness of joints is innovatively considered one of the design variables for shape-member section coupled optimization. The augmented radius basis functions (RBFs) are adopted to develop the metamodels of both the objectives and constraints. The improved minimum distance selection method (TMDSM) is used to select the optimal solution from the Pareto optimal solution set. Moreover, three alternative solutions are presented as supplements to meet various engineering requirements. The proposed RBF-NSGA-II algorithm is validated to have high accuracy for multi-objective optimization problems with high nonlinearity. Compared with those of the initial structures, the strain energy and steel consumption of the optimal structure decreased by 82.23 % and 17.97 %, respectively, while the buckling load considering both the geometric nonlinearity and material nonlinearity increased by 1.2 times. The involvement of the joint stiffness is demonstrated to have a great influence on the multi-objective optimization results. It is necessary to consider the joint stiffness as one of the design variables during shape-member section coupled optimization.

Subdiffusive processes in BWRs

The objective of this work is the analysis of subdiffusive processes in BWR reactors. Nuclear reactors are highly heterogeneous systems where the phenomena at the reactor scale are not possible to describe with constitutive standard laws (normal diffusion) of energy, momentum, and mass. However, there are methods to achieve it such as the volumetric average to upscale the reactor and incorporate up-scaled constitutive laws. However, it can also be proposed constitutive laws of fractional order to consider anomalous diffusion. For the neutronic processes, the neutron diffusion theory is non-appropriated to the reactor scale, because this theory is applied to the homogeneous reactor. Given the heterogeneity of these systems in describing the dynamic behaviour of BWRs with point models, it represents a challenge to the incorporation of the heterogeneous effects whose main characteristic is that the transport phenomena are subdiffusive with relaxation times (i.e., non-instantaneous) by nature. The point models called reducer-order models are important for stability analysis and to develop control strategies. In linear or nonlinear stability analysis and system control, they are carried out with reduced order models that are zero dimension, i.e., point models that depend exclusively on time. This is because existing methodologies propose mathematical expressions in the time or frequency domain for stability analysis or to establish control strategies. The challenge is for ROM models to be able to consider subdiffusive effects on neutron motion. This is achieved by various routes from the P1 theory that includes the temporal term with relaxation effects (relaxation time) or by proposing a more general constitutive equation of fractional order for the neutron current vector, as is presented in this work. Moreover, the Fractional Order Models behavior must be as a transport theory based model, as well as in those nuclear and thermohydraulic codes which are very useful but complex, for which is sometimes it is impossible to obtain analytical expressions. To incorporate these effects in this work we consider point models of fractional order, which represent the subdiffusive phenomenon when the fractional order (known as anomalous diffusion coefficient) is greater than 0 and less than 1.

Nonlinear Stability and Asymptotic Behavior of Periodic Wave Trains in Reaction–Diffusion Systems Against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations

We present a nonlinear stability theory for periodic wave trains in reaction–diffusion systems, which relies on pure L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the semigroup generated by the linearization about the wave train and introduce a spatio-temporal phase modulation to capture the most critical dynamics, which is governed by a viscous Burgers’ equation. We then aim to close a nonlinear stability argument by iterative estimates on the corresponding Duhamel formulation, where, hampered by the lack of localization, we must rely on diffusive smoothing to render decay of the semigroup. However, this decay is not strong enough to control all terms in the Duhamel formulation. We address this difficulty by applying the Cole–Hopf transform to eliminate the critical Burgers’-type nonlinearities. Ultimately, we establish nonlinear stability of diffusively spectrally stable wave trains against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations. Moreover, we show that the perturbed solution converges to a modulated wave train, whose phase and wavenumber are approximated by solutions to the associated viscous Hamilton–Jacobi and Burgers’ equation, respectively.

Nonlinear stability of ES-FG porous sandwich cylindrical shells subjected to axial compression in thermal environment

Stability analysis of cylindrical shells subjected to axial compression is a common problem in engineering applications. The phenomenon of instability can easily appear with thin cylindrical shells. Therefore, to increase the load-carrying capacity of this type of structure, the cylinders are increased in thickness by adding a porous core layer and reinforcing orthogonal stiffeners. In this study, the nonlinear buckling and postbuckling behavior of functionally graded (FG) porous sandwich cylindrical shells with orthogonal eccentrically stiffeners (ES) are investigated via analytical approach. The FG porous sandwich cylindrical shells are rested on Pasternak elastic foundations and subjected to axial compression load in a thermal environment. The reinforcement of the shells is achieved through closely spaced stringers and rings, with the material properties of both the shell and stiffeners graded continuously in the thickness direction. The governing equations are derived based on the Donnell’s shell theory with von Karman geometrical nonlinearity and the smeared stiffeners technique. The Galerkin procedure is employed to ascertain the critical load and post-buckling response, employing a three-term deflection solution. The study analyzed the effects of parameters such as porosity coefficient, material, temperature, dimensional parameters, stiffener, and foundation characteristics.

A Multiple Scale Approach to Nonlinear Vibrations and Stability of Doubly Curved Shells/Panels Under External and Parametric Excitations

This study focuses on nonlinear vibrations and stability of doubly curved shells/panels. The thin-walled shell structure is assumed to be acted upon by external loads as well as parametric excitation. Using the first-order shear deformation Fülgge theory assumptions, governing equations of motion are derived for a general doubly curved geometry. Nonlinear components which are compatible with the Von-Karman kinematic assumptions are retained in the formulation and the rest of nonlinear components are neglected due to the fact that strains are assumed to be small and rotations to be moderate. The formulation can be readily reduced to any geometry including cylindrical shells and spherical panels, as benchmark examples in this study. Employing the Galerkin discretization approach and applying a modification to the Volmir assumption, a reduced order form of the original system is determined, which is shown to be a great representative of the original system with much more computational advantages. Using the method of multiple scales, the reduced order nonlinear governing equations are solved to discover the primary resonance, forced vibration characteristics and stability margins of thin-walled shells/panels under the action of parametric excitation. Several interesting findings on nonlinear behavior of cylindrical shells and spherical panels (as benchmark geometries) are presented while giving a demonstration of the accuracy and computational advantages of the proposed model.

Higher order mKdV breathers: nonlinear stability

We are interested in stability results for breather solutions of the 5th, 7th and 9th order mKdV equations. We show that these higher order mKdV breathers are stable in $H^2(\R)$, in the same way as \emph{classical} mKdV breathers. We also show that breather solutions of the 5th, 7th and 9th order mKdV equations satisfy the same stationary fourth order nonlinear elliptic equation as the mKdV breather, independently of the order, 5th, 7th or 9th, considered.

Transition Threshold for the 3D Couette Flow in a Finite Channel

In this paper, we study nonlinear stability of the 3D plane Couette flow ( y , 0 , 0 ) (y,0,0) at high Reynolds number R e {Re} in a finite channel T × [ − 1 , 1 ] × T \mathbb {T}\times [-1,1]\times \mathbb {T} . It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity v 0 v_0 satisfies ‖ v 0 − ( y , 0 , 0 ) ‖ H 2 ≤ c 0 R e − 1 \|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1} for some c 0 > 0 c_0>0 independent of R e Re , then the solution of the 3D Navier-Stokes equations is global in time and does not transit away from the Couette flow in the L ∞ L^\infty sense, and rapidly converges to a streak solution for t ≫ R e 1 3 t\gg Re^{\frac 13} due to the mixing-enhanced dissipation effect. This result confirms the threshold result obtained by Chapman via an asymptotic analysis(JFM 2002). The most key ingredient of the proof is the resolvent estimates for the full linearized 3D Navier-Stokes system around the flow ( V ( y , z ) , 0 , 0 ) (V(y,z), 0,0) , where V ( y , z ) V(y,z) is a small perturbation(but independent of R e Re ) of y y .

On the existence and stability of stationary solutions for the compressible micropolar fluids

We consider the existence, uniqueness and nonlinear stability of stationary solutions for the three-dimensional compressible micropolar fluids with the external force of general form. More precisely, with the weighted L2 and L∞ estimates on solutions to the linearized problem, we show the existence and uniqueness of stationary solutions in some suitable function space by the contraction mapping principle. The stability of the steady flow is studied by an elementary energy method.

Stability of penetrative convective currents in local thermal non-equilibrium

The aim of this paper is to investigate the onset of penetrative convection in a Darcy–Brinkman porous medium under the hypothesis of local thermal non-equilibrium. For the problem at stake, the strong form of the principle of exchange of stabilities has been proved, i.e. convective motions can occur only through secondary stationary motions. We perform linear and nonlinear stability analyses of the basic state, with particular regard to the behaviour of stability thresholds with respect to the relevant physical parameters characterizing the problem. The Chebyshev- τ method and the shooting method are employed and accurately implemented to solve the differential eigenvalue problems arising from linear and nonlinear analyses to determine critical Rayleigh numbers. Via numerical simulations, the stabilizing effect of the upper bounding plane temperature, of the Darcy number and of the interaction coefficient for the heat exchange, is demonstrated.

Novel convergence to forced traveling waves for a nonlocal dispersal population model in a shifting environment

Some new types of the nonlinear stability of forced traveling waves for a nonlocal dispersal population model in a shifting environment are given in this paper. Firstly, when the speed of the shifting habitat edge c is larger than the spreading speed of the species c⁎, by applying the comparison principle, the Fourier transform and the L2-weighted energy estimate, we show that all forced traveling waves with the speed c>c⁎ are exponentially stable in the form of e−μt, for some μ>0. In order to extend the stability results to c=c⁎, we take another weight function and construct the different weighted Sobolev space. Then by using the Fourier transform and the L1-weighted energy estimate, we prove that all forced traveling waves with the speed c>c⁎ are exponentially stable in the form of t−12e−νt, for some ν>0, while all forced traveling waves with the speed c=c⁎ are algebraically stable in the form of t−12, which are quite different from the previous stability results.