Nonlinear Kelvin Research Articles

Uniform Stabilization for the Semi-linear Wave Equation with Nonlinear Kelvin–Voigt Damping

Uniform Stabilization for the Semi-linear Wave Equation with Nonlinear Kelvin–Voigt Damping

Resonant excitation of single Kelvin–Helmholtz high-order waves in a magnetized electron fluid vortex

Thanks to the isomorphism between the drift-Poisson and Euler equations, inviscid two-dimensional fluid experiments can be performed in magnetized, single-component plasmas in Penning–Malmberg traps. Within this analogy, a trapped electron plasma column is equivalent to a two-dimensional vortex. Here, we focus our attention on the generation of V-states, i.e. $l$ -fold symmetric rotating vorticity patches where the deformation with respect to the circular cross-section has reached the nonlinear regime. We detail a linear theoretical analysis and devise an experimental routine to generate V-states through the precise excitation of single Kelvin–Helmholtz perturbations in a magnetized electron plasma. This technique makes use of suitable multipolar rotating electric fields, which are shown to be able to select the desired wavemode. In particular, with rotating fields, a hardware limitation in the highest accessible mode is removed and nonlinear Kelvin–Helmholtz waves of generic order $l$ can be attained, which pave the way for further investigations on the evolution and stability properties of V-states. Systematic experimental results for the selective mode growth in the linear and nonlinear regimes up to saturation and collapse are discussed.

A GPU-Accelerated Method for 3D Nonlinear Kelvin Ship Wake Patterns Simulation

The study of ship waves is important for ship detection, coastal erosion and wave drag. This paper proposed a highly paralleled numerical computation method for efficiently simulating three-dimensional nonlinear kelvin waves. First, a numerical model for nonlinear ship waves is established based on potential flow theory, the Jacobian-free Newton–Krylov (JFNK) method and the boundary integral method. To reduce the amount of data stored in the JFNK method and improve the computational efficiency, a banded preconditioner method is then developed by formulating the optimal bandwidth selection rule. After that, a Graphics Process Unit (GPU)-based parallel computing framework is designed, and we used the Compute Unified Device Architecture (CUDA) language to develop a GPU solution. Finally, numerical simulations of 3D nonlinear ship waves under multiple scales are performed by using the GPU and CPU solvers. Simulation results show that the proposed GPU solver is more efficient than the CPU solver with the same accuracy. More than 66% GPU memory can be saved, and the computational speed can be accelerated up to 20 times. Hence, the computation time for Kelvin ship waves simulation can be significantly reduced by applying the GPU parallel numerical scheme, which lays a solid foundation for practical ocean engineering.

A true triaxial creep constitutive model of rock considering the coupled thermo-mechanical damage

The main emphasis in contemporary mineral resource extraction has shifted towards deep Earth resources to address economic development requirements. The influence of temperature has emerged as an undeniable factor as engineering projects delve deeper into the Earth. The definition of a damage variable, which considers the coupled thermo-mechanical damage, has been proposed within the framework of damage mechanics. This definition aims to accurately describe the different stages of rock creep behavior in high-temperature environments. Furthermore, a constitutive model has been developed to describe the creep behavior of rocks under the combined influence of temperature and true triaxial stress. This model considers the series connection of the elastic body, nonlinear Kelvin body, Heard body, and visco-elastic-plastic body. The model has been parameterized and validated using available creep test data, and the equation for the yield surface of rocks under true triaxial stress has been derived through data fitting. The findings suggest that the developed constitutive model is capable of accurately describing the properties of rock creep behavior at different stages in high-temperature conditions, thus confirming its rationality and precision.

Three- and four-wave resonances in the nonlinear quadratic Kelvin lattice

In this paper we investigate analytically and numerically the nonlinear Kelvin lattice, namely a chain of masses and nonlinear springs, as in the α-Fermi-Pasta–Ulam-Tsingou (FPUT) chain, where, in addition, each mass is connected to a nonlinear resonator, i.e., a second mass free to oscillate. Both nonlinearities are quadratic in the equations of motion. This setup represents the simplest prototype of nonlinear wave propagation on a nonlinear metamaterial. In the linear case, we diagonalize the system, and the two branches of the dispersion relation can be found. Using this result, we derive in the nonlinear case the equations of motion for the normal variables in Fourier space, obtaining a system governed by triad interactions among the two branches of the dispersion relation. We find that the transfer of energy between these two branches is ruled by three- and four-wave resonant interactions. We perform numerical simulations of the primitive equations of motion and highlight the role of resonances as an efficient mechanism for transferring energy. Moreover, as predicted by the theory, we provide direct evidence that four-wave resonances appear on a time scale that is longer than the time scale for three-wave resonances. We also assess the recurrence behaviour (usual in the FPUT system) for the nonlinear Kelvin lattice, and we show that, while recurrence is observed if all the energy is placed, at time t=0, in the lowest mode of the acoustical branch, a non-recurrent behaviour is observed if the initial energy is located in the optical branch.

A nonlinear rheological shear constitutive model of bolted joints considering initial damage and damage evolution

Understanding the shear mechanics mechanism of bolted joints is of great significance for predicting and preventing geological disasters. Most current studies seldom consider the rheological effects of bolted joints. In this paper, a comprehensive rheological constitutive model is proposed, accounting for initial damage and damage evolution across different rheological stages and bolt characteristics. The model incorporates an elastoplastic Hooke body for instantaneous deformation, parametric nonlinear Kelvin and viscous models for attenuation and steady creep stages, and a visco-plastic model based on time-dependent shear strength for accelerated creep stage. Additionally, a bolt-rock cooperative deformation model is introduced, considering the evolution of the bolt's elastic modulus. The resulting elasto-viscoplastic constitutive model effectively describes the shear rheological behavior of bolted joints, with its validity and superiority demonstrated through comparisons with shear creep tests and the Maxwell model. This research aims to provide valuable theoretical guidance for the construction and reinforcement of rock mass engineering projects.

Model of Angular Object Positioning Process by a System of Oblique Friction Force Fields on Parallel Planes

Abstract The primary contribution of the paper is a proposal of a method to minimize the angular positioning error in the process of transferring cuboidal objects between oblique friction force fields generated by two conveyors located on parallel planes. The research included infeed conveyors with two variants of inputs: angular and straight. It was assumed that the object, while moving between oblique conveyors at different heights, performs a 3D movement. The object is treated as a rigid body with a soft base, edges, and corners that can be subjected to significant local deformations. A modified nonlinear Kelvin model was used to describe the normal reaction forces at the contact points of the object with the bearing surfaces of the conveyors, and the modified static Bengisu–Akay friction model represents the tangential forces. Research shows that the use of a slight offset between the bearing surfaces of the conveyors and the highest possible proportion between the motion velocities of the infeed and outfeed conveyors have positive effect on improving the angular precision of the positioned objects. This conclusion applies to both variants of the infeed conveyor inputs. The results of the research have practical application in the design of high-performance conveyor transport systems, used in the processes of picking cuboidal objects, requiring a strictly defined angular orientation.

Linear and Nonlinear Kelvin–Helmholtz Instability and Magnetohydrodynamic Wave Emission in Sheared Astrophysical Plasma Flows

The evolution of the Kelvin–Helmholtz instability (KHI) and magnetohydrodynamic (MHD) wave emission is investigated at shear-flow boundaries of magnetized plasmas. While MHD wave emission has been suggested to be only possible during the nonlinear stages, we find that there is also significant wave emission during the KHI’s linear stages. These emitted MHD waves may have stronger impacts than KHI surface waves since they can act to transport energy away from the local region of the shear flow. The removal of energy from the shear-flow region, instead of just the local redistribution considered in previous studies, and its propagation away from the interface could have major implications for the evolution of astrophysical objects characterized by fast plasma flow shears.

A Chemical Damage Creep Model of Rock Considering the Influence of Triaxial Stress.

In order to accurately describe the characteristics of each stage of rock creep behavior under the combined action of acid environment and true triaxial stress, based on damage mechanics, chemical damage is connected with elastic modulus; thus, the damage relations considering creep stress damage and chemical damage are obtained. The elastic body, nonlinear Kelvin body, linear Kelvin body, and viscoelastic-plastic body (Mogi-Coulomb) are connected in series, and the actual situation under the action of true triaxial stress is considered at the same time. Therefore, a damage creep constitutive model considering the coupling of rock acid corrosion and true triaxial stress is established. The parameters of the deduced model are identified and verified with the existing experimental research results. The yield surface equation of rock under true triaxial stress is obtained by data fitting, and the influence of intermediate principal stress on the creep model is discussed. The derived constitutive model can accurately describe the characteristics of each stage of true triaxial creep behavior of rock under acid environment.

Simulation of Es Layer Modulated by Nonlinear Kelvin–Helmholtz Instability

AbstractIn this study, a two‐dimensional numerical model of the nonlinear Kelvin–Helmholtz (KH) instability is used to investigate the nonlinear evolution processes of the ion layer, which is associated with the middle and low latitude E region irregularities. Based on simulation results, the two‐stage process in the evolution of the nonlinear KH instability is found in KH vortex morphology. The simulated nonlinear KH instability process presents multiple striation structures, including horizontal structures of the primary ion layer vortex with a size of a few kilometers and the secondary KH vortex with a size of 2 km. The polarization electric field is induced in the horizontal structure of the sporadic E (Es) layer associated with the vortex structure of neutral wind. The polarization electric field presents a similar tendency to the quasi‐periodic vortex structure. The horizontal distorted Es layer structure and anomalously large polarization electric field can provide the initial condition and lead to plasma instability, which can further develop to meter‐scale irregularities of QP echoes.

Weakly nonlinear incompressible Kelvin–Helmholtz instability in plane geometry

A weakly nonlinear (WN) theoretical model for the two-dimensional incompressible Kelvin–Helmholtz instability (KHI) is proposed. Its solution form is the complete expansion in real space. The transition from linear to nonlinear growth is analytically studied via third-order solutions of plane KHI initiated by a single-mode surface perturbation. The difference between the WN growth of the Rayleigh–Taylor instability (RTI) in plane geometry and the WN growth of the KHI in plane geometry is discussed. It is found that there are resonance solutions in the higher harmonics of KHI but not in RTI. The vertex of spikes and bubbles is deflected because of the shearing effect in KHI compared with that in RTI. The spike velocity increases with the Atwood number before a particular time and then inversely after that time. There is no such reversal in RTI. However, the bubble velocity with the Atwood number of KHI has the same evolution rule as that of RTI. In addition to the influence of initial perturbation on the nonlinear saturation amplitude, the influence of Atwood number on the nonlinear saturation amplitude of the fundamental mode is obtained. Finally, compared with the numerical simulations, at the same initial conditions, the analytical solutions up to the third harmonics are well consistent with the simulated datum in the linear and weakly nonlinear stages for a widely varied Atwood number.

Domains of Magnetic Pressure Balance in Parker Solar Probe Observations of the Solar Wind

Abstract The Parker Solar Probe (PSP) spacecraft is performing the first in situ exploration of the solar wind within 0.2 au of the Sun. Initial observations confirmed the Alfvénic nature of aligned fluctuations of the magnetic field B and velocity V in solar wind plasma close to the Sun, in domains of nearly constant magnetic field magnitude ∣ B ∣, i.e., approximate magnetic pressure balance. Such domains are interrupted by particularly strong fluctuations, including but not limited to radial field (polarity) reversals, known as switchbacks. It has been proposed that nonlinear Kelvin–Helmholtz instabilities form near magnetic boundaries in the nascent solar wind leading to extensive shear-driven dynamics, strong turbulent fluctuations including switchbacks, and mixing layers that involve domains of approximate magnetic pressure balance. In this work we identify and analyze various aspects of such domains using data from the first five PSP solar encounters. The filling fraction of domains, a measure of Alfvénicity, varies from median values of 90% within 0.2 au to 38% outside 0.9 au, with strong fluctuations. We find an inverse association between the mean domain duration and plasma β. We examine whether the mean domain duration is also related to the crossing time of spatial structures frozen into the solar wind flow for extreme cases of the aspect ratio. Our results are inconsistent with long, thin domains aligned along the radial or Parker spiral direction, and compatible with isotropic domains, which is consistent with prior observations of isotropic density fluctuations or flocculae in the solar wind.

Analysis of the positioning process of objects on an oblique plane with barriers

The paper presents a model of the positioning process of cuboidal objects on an oblique plane with barriers, caused by the interaction of inertia, gravity and contact forces. In this model, the contact area of the object base with the bearing planes is represented by evenly spaced discrete elements and the corners and edges of the object identify the contact with the positioning barriers. The object performs free spatial movement and may be in simultaneous contact with many bearing surfaces with different frictional properties. The static Bengisu-Akay friction model represents tangent contact forces, and the nonlinear Kelvin collision model describes normal forces. Due to high sliding velocities, the Stribeck's effect in the friction model does not have a significant impact on the course of the positioning process, hence it was assumed that the static and kinetic friction coefficients are equal to each other. Moreover, on the basis of the conducted analyses of the results of numerical and experimental tests, a geometrical configuration of the positioning barriers was proposed, allowing the object to be intercepted on an oblique plane and placed on the outfeed conveyor.

Study on Nonlinear Damage Creep Model for Rocks under Cyclic Loading and Unloading

To determine the nonlinear creep characteristics of rocks under cyclic loading and unloading conditions, a nonlinear Kelvin model and damage viscoplastic model are proposed. The models are connected in series with a linear elastic body to establish a nonlinear damage creep model. The differential damage constitutive equations of the proposed creep model under one‐dimensional and three‐dimensional stress states are derived based on the creep mechanics and elasticity theory. The damage and unloading creep equations are then obtained based on the superposition principle, and a simple and feasible method for determining the model parameters is determined. Finally, the step cyclic loading and unloading creep test data for lherzolite and limestone are used to verify the rationality and feasibility of the nonlinear damage creep model. The results show that the theoretical creep curves of the nonlinear damage creep model are consistent with the experimental curves which indicates that the proposed model can not only determine the creep properties of lherzolite and limestone under cyclic loading and unloading but also determine the nonlinear characteristics of rocks in the transient and steady‐state creep stages and particularly within the accelerating creep stage.

Model of positioning objects by the system of oblique friction force fields on horizontal and vertically offset planes

The paper presents a model of the process of angular positioning of cuboidal objects using a system of two oblique friction force fields produced by the conveyor belts located on parallel planes a and b, offset vertically by value H. It has been assumed that, while moving from friction force field a to lower placed field b, the object performs a 3D motion and comes into contact with the friction force fields at the corners, edges or the base surface. The object is treated as a rigid body with a soft surface of the bottom, which can be subjected to significant local deformations when interacting with obstacles. Two continuous collision models were used to describe the normal reaction forces at the contact points of the object with the bearing surfaces of the conveyors: the nonlinear MSC Adams model and the modified nonlinear Kelvin model. In addition, three friction models were included, i.e. two static Bengisu-Akay and MSC Adams models and one dynamic LuGre model. Numerical and experimental research demonstrate that the use of friction force fields with vertical offset of a short distance (e.g. H = 0.005 m) enhances the rotation angle Φ minimization of the object positioned.

The effects of vertical mixing on nonlinear Kelvin waves

Abstract

Extracting non-linear viscoelastic material properties from violently-collapsing cavitation bubbles

Determination of high strain-rate constitutive material parameters from highly compliant, non-linear materials including hydrogels and soft tissues remains a significant challenge. To address this challenge we recently developed a laser-based Inertial Microcavitation Rheometry (IMR) technique, capable of constitutively characterizing soft materials at ultra-high strain-rates (O(103)∼O(108)s−1). This technique utilizes spatially-focused, and temporally-resolved, single cavitation bubbles in conjunction with a theoretical framework for determining the non-linear viscoelastic material properties of the surrounding soft material.Here, by advancing our spatiotemporal imaging capability and tracking the motion of native material features close to the inertially cavitating bubble, we find significant deviation from the IMR predicted material displacements at a critical material Mach number of 0.08, marking the transition to violent collapse. Interestingly, this critical Mach number appears to be almost independent of the material properties of the surrounding hydrogel. We show that through proper temporal resolution of the cavitation dynamics, all of the highly nonlinear viscoelastic properties of the surrounding material can still be uniquely determined from just the first expansion and collapse cycle alone, as long as the local Mach number is below the critical threshold. Furthermore, we show that the inclusion of a higher order strain stiffening term on the rate-dependent non-linear spring can lead to better estimation of the material properties and full recovery of the equilibrium shear modulus, which was not possible with previous nonlinear neo-Hookean Kelvin–Voigt model formulations. Thus, the developments presented here significantly expand the applicability and robustness of IMR for accurately determining the rate-dependent, finite deformation constitutive behavior of soft materials.

Dynamics of Two Linearly Elastic Bodies Connected by a Heavy Thin Soft Viscoelastic Layer

In this paper, we extend the asymptotic analysis in (Licht et al. in J. Math. Pures Appl. 99:685–703, 2013) performed, in the framework of small strains, on a structure consisting of two linearly elastic bodies connected by a thin soft nonlinear Kelvin–Voigt viscoelastic adhesive layer to the case in which the total mass of the layer remains strictly positive as its thickness tends to zero. We obtain convergence results by means of a nonlinear version of Trotter’s theory of approximation of semigroups acting on variable Hilbert spaces. Differently from the limit models derived in (Licht et al. in J. Math. Pures Appl. 99:685–703, 2013), in the present analysis the dynamic effects on the surface to which the layer shrinks do not disappear. Thus, the limiting behavior of the remaining bodies is described not only in terms of their displacements on the contact surface, but also by an additional variable that keeps track of the dynamics in the adhesive layer.

A Long‐Lived Sharp Disruption on the Lower Clouds of Venus

AbstractPlanetary‐scale waves are thought to play a role in powering the yet unexplained atmospheric superrotation of Venus. Puzzlingly, while Kelvin, Rossby, and stationary waves manifest at the upper clouds (65–70 km), no planetary‐scale waves or stationary patterns have been reported in the intervening level of the lower clouds (48–55 km), although the latter are probably Lee waves. Using observations by the Akatsuki orbiter and ground‐based telescopes, we show that the lower clouds follow a regular cycle punctuated between 30°N and 40°S by a sharp discontinuity or disruption with potential implications to Venus's general circulation and thermal structure. This disruption exhibits a westward rotation period of ∼4.9 days faster than winds at this level (∼6‐day period), alters clouds' properties and aerosols, and remains coherent during weeks. Past observations reveal its recurrent nature since at least 1983, and numerical simulations show that a nonlinear Kelvin wave reproduces many of its properties.

Shape morphing in molecular cloud-cloud collision affected by coherent instabilities

This paper presents the results of numerical simulation of Molecular Cloud-Cloud Collision in the under the head-on and glancing impacts. The calculations are performed according to different colliding scenarios between two dissimilar (by size, displacement and inner distribution of matter) spherical clouds. The aim is to study the matter compression in clumps and filaments formed, the generation of vortex coherent structures in new formations imbalanced after the collision and to analyze the influence of emergent instabilities on clouds morphing. In-house Eulerian code is used to perform a numerical experiment on high-performance computers. Nebula shaping and the compression oscillations in a shock core depend on initial impulse of colliding molecular clouds. The article gives a morphological analysis of gas redistribution by density under local compression in a bow-shock layer and filaments which are generated. The mechanism of perturbations to matter turbulization, fragmentation and disruption of new formations is revealed. Strong shock compression pulsations generated in the gas core and waved rarefactions in the interstellar medium are interpreted as possible Nonlinear Thin Shell Instability and Kelvin–Helmholtz Instability aftereffect.