Nonlocal Damping Research Articles

On a beam model related to flight structures with nonlocal energy damping

<p style='text-indent:20px;'>This paper deals with new results on existence, uniqueness and stability for a class of nonlinear beams arising in connection with nonlocal dissipative models for flight structures with <i>energy damping</i> first proposed by Balakrishnan-Taylor [<xref ref-type="bibr" rid="b2">2</xref>]. More precisely, the following <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional model is addressed <p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1">\begin{document}$ u_{tt}-\kappa \Delta u+\Delta ^2u-\gamma\left[\int_{\Omega}\left(|\Delta u|^2+|u_t|^2\right)dx \right]^q\Delta u_t+f(u) = 0 \ in \ \Omega \times \mathbb{R}^+, $\end{document}</tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary, the coefficient of extensibility <inline-formula><tex-math id="M3">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> is nonnegative, the damping coefficient <inline-formula><tex-math id="M4">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> is positive and <inline-formula><tex-math id="M5">\begin{document}$ q\ge 1 $\end{document}</tex-math></inline-formula>. The nonlinear source <inline-formula><tex-math id="M6">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> can be seen as an external forcing term of lower order. Our main results feature global existence and uniqueness, polynomial stability and a non-exponential decay prospect.

On the Exponential Stability of a Stratified Flow to the 2D Ideal MHD Equations with Damping

We study the stability of a kind of stratified flow of the two-dimensional inviscid incompressible magnetohydrodynamics (MHD) equations with velocity damping. The exponential stability for the perturbation near certain stratified flow is investigated in a strip-type area $\mathbb{R}\times[0,1]$. Although the magnetic field potential is governed by a transport equation, by using the algebraic structure of the incompressible condition, it turns out that the linearized MHD equations around the given stratified flow retain a nonlocal damping mechanism. Carefully analyzing the nonlinear structure and introducing some suitable weighted energy norm, we get the exponential stability by combining the exponential decay in time in the lower order energy with that in the high order energy.

Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping

The paper investigates the attractors and their robustness for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. We prove that the related evolution process has a finite-dimensional pullback attractor \begin{document}$ \mathscr{A}_\kappa $\end{document} and a pullback exponential attractor \begin{document}$ \mathscr{M}^\kappa_{exp} $\end{document} for each extensibility parameter \begin{document}$ \kappa\in [0,1] $\end{document} , respectively, and both of them are stable on the perturbation \begin{document}$ \kappa $\end{document} . In particular, these stability holds for the global and exponential attractors when the non-autonomous dynamical system degenerates to an autonomous one, so the results of the paper deepen and extend those in recent literatures [ 22 , 33 ].

Nonlocal damping consideration for the computer modelling of linear and nonlinear systems vibrations under the stochastic loads

The paper is devoted to the analysis of the nonlocal damping consideration influence on the computer modelling results of linear and nonlinear systems’ vibrations subjected to stochastic stationary loads. The shallow arc dynamic behavior is examined. The problem is solved using the Galerkin method.

Asymptotic behavior for the quasi-geostrophic equations with fractional dissipation in [formula omitted

In this paper the surface quasi-geostrophic equations (QGE) with fractional dissipation in R2 are considered. Our aim is to study the long-time behavior of solutions of QGE in the subcritical case. To this end we investigate the global well-posedness and global attractor for QGE in Hs(R2) via commutator estimates for nonlinear terms, a new iterative technique for estimates of higher order derivatives and with the help of a nonlocal damping term. Besides, by using the fractional Lieb–Thirring inequality, estimates of the finite Hausdorff and fractal dimensions of the global attractor are found.

Damping of linear spin-wave modes in magnetic nanostructures: Local, nonlocal, and coordinate-dependent damping

A general perturbation theory for the description of weak damping of linear spin-wave modes in magnetic nanostructures is developed. This perturbative approach allows one to account for the usual uniform Gilbert damping, as well as for the spatially nonuniform (coordinate-dependent) and nonlocal (magnetization-texture-dependent) Gilbert-like dissipation mechanisms. Using the derived general expression, it is possible to calculate the damping rate of a particular spin-wave mode if the frequency and the spatial profile of this mode, along with the relevant parameters of a magnetic material, are known. The examples demonstrating the applications of the developed general formalism include (i) generalization of the damping rate of a spin-wave mode propagating in a magnetic sample for the case of a nonuniform static magnetization or/and bias magnetic field, (ii) calculation of a damping rate of a gyrotropic mode in a vortex-state magnetic nanodot, (iii) evaluation of the spin diffusion influence on the damping rate of spin-wave modes in a conducting ferromagnet, and (iv) calculation of damping rates of spin-wave modes in a ferromagnetic film in the presence of a spin pumping into an adjacent nonmagnetic metal layer. The developed formalism is especially useful in micromagnetic simulations, as it allows one to calculate damping rates of spin-wave modes based on the numerical solution of a conservative eigenmode problem.

Application of local elasticity continuum damping models in nonlinear dynamic analysis

In order to have a more generic representation of the damping phenomenon in seismic analysis of structures, in this paper, the already existing nonlocal elasticity continuum damping models have been adapted and extended to the inelastic domain. Adaptations of two nonlocal elasticity based damping models, Russell’s spatial hysteresis model and the extended Sorrentino model, into the nonlinear dynamic analysis is presented. Galerkin based finite element schemes are developed and the numerical implementation of the models are outlined. The performances of the nonlocal elasticity-based damping models are illustrated by studying the nonlinear dynamic responses of a four-story Reinforced Concrete frame designed to the Eurocodes. The incremental dynamic analysis study presented illustrates the fact that the proposed models are devoid of the large spurious damping actions commonly exhibited by the popular classical Rayleigh damping model. It has also been shown that in a nonlinear dynamic analysis, the proposed adaptation of the nonlocal elasticity continuum damping models might be a more realistic alternative to the popular classical Rayleigh damping model. The proposed models may be easily implemented in an existing software framework capable of solving both ordinary and integro-differential equations without adding markedly to the computational effort.

Application of nonlocal elasticity continuum damping models in nonlinear dynamic analysis

Application of nonlocal elasticity continuum damping models in nonlinear dynamic analysis

Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms

Optical nonlocalities are elusive and hardly observable in traditional plasmonic materials like noble and alkali metals. Here we report experimental observation of viscoelastic nonlocalities in the infrared optical response of epsilon-near-zero nanofilms made of low-loss doped cadmium-oxide. The nonlocality is detectable thanks to the low damping rate of conduction electrons and the virtual absence of interband transitions at infrared wavelengths. We describe the motion of conduction electrons using a hydrodynamic model for a viscoelastic fluid, and find excellent agreement with experimental results. The electrons’ elasticity blue-shifts the infrared plasmonic resonance associated with the main epsilon-near-zero mode, and triggers the onset of higher-order resonances due to the excitation of electron-pressure modes above the bulk plasma frequency. We also provide evidence of the existence of nonlocal damping, i.e., viscosity, in the motion of optically-excited conduction electrons using a combination of spectroscopic ellipsometry data and predictions based on the viscoelastic hydrodynamic model.

Analytical Solutions for Multi-Term Time-Space Fractional Partial Differential Equations with Nonlocal Damping Terms

AbstractIn this paper, we consider the analytical solutions of multi-term time-space fractional partial differential equations with nonlocal damping terms for general mixed Robin boundary conditions on a finite domain. Firstly, method of reduction to integral equations is used to obtain the analytical solutions of multi-term time fractional differential equations with integral terms. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time-space fractional partial differential equations with nonlocal damping terms to the multi-term time fractional differential equations with integral terms. By applying the obtained analytical solutions to the resulting multi-term time fractional differential equations with integral terms, the desired analytical solutions of the multi-term time-space fractional partial differential equations with nonlocal damping terms are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

Nonlocal Gilbert damping tensor within the torque-torque correlation model

An essential property of magnetic devices is the relaxation rate in magnetic switching which depends strongly on the damping in the magnetisation dynamics. It was recently measured that damping depends on the magnetic texture and, consequently, is a non-local quantity. The damping enters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter $\alpha$, that does not, in a straight forward formulation, account for non-locality. Efforts were spent recently to obtain Gilbert damping from first principles for magnons of wave vector $\mathbf{q}$. However, to the best of our knowledge, there is no report about real space non-local Gilbert damping $\alpha_{ij}$. Here, a torque-torque correlation model based on a tight binding approach is applied to the bulk elemental itinerant magnets and it predicts significant off-site Gilbert damping contributions, that could be also negative. Supported by atomistic magnetisation dynamics simulations we reveal the importance of the non-local Gilbert damping in atomistic magnetisation dynamics. This study gives a deeper understanding of the dynamics of the magnetic moments and dissipation processes in real magnetic materials. Ways of manipulating non-local damping are explored, either by temperature, material's doping or strain.

On the origin of nonlocal damping in plasmonic monomers and dimers

The origin and importance of nonlocal damping is discussed through simulations with the generalized nonlocal optical response (GNOR) theory, in conjunction with time-dependent density functional theory (TDDFT) calculations and equivalent circuit modeling, for some of the most typical plasmonic architectures: metal–dielectric interfaces, metal–dielectric–metal gaps, spherical nanoparticles and nanoparticle dimers. It is shown that diffusive damping, as introduced by the convective–diffusive GNOR theory, describes well the enhanced losses and plasmon broadening predicted by ab initio calculations in few-nm particles or few-to-sub-nm gaps. Through the evaluation of a local effective dielectric function, it is shown that absorptive losses appear dominantly close to the metal surface, in agreement with TDDFT and the mechanism of Landau damping due to generation of electron–hole pairs near the interface. Diffusive nonlocal theories provide therefore an efficient means to tackle plasmon damping when electron tunneling can be safely disregarded, without the need to resort to more accurate, but time-consuming fully quantum-mechanical studies.

How nonlocal damping reduces plasmon-enhanced fluorescence in ultranarrow gaps

The nonclassical modification of plasmon-assisted fluorescence enhancement is theoretically explored by placing two-level dipole emitters at the narrow gaps encountered in canonical plasmonic architectures, namely dimers and trimers of different metallic nanoparticles. Through detailed simulations, in comparison with appropriate analytical modelling, it is shown that within classical electrodynamics, and for the reduced separations explored here, fluorescence enhancement factors of the order of $10^{5}$ can be achieved, with a divergent behaviour as the particle touching regime is approached. This remarkable prediction is mainly governed by the dramatic increase in excitation rate triggered by the corresponding field enhancement inside the gaps. Nevertheless, once nonclassical corrections are included, the amplification factors decrease by up to two orders of magnitude and a saturation regime for narrower gaps is reached. These nonclassical limitations are demonstrated by simulations based on the generalised nonlocal optical response theory, which accounts in an efficient way not only for nonlocal screening, but also for the enhanced Landau damping near the metal surface. A simple strategy to introduce nonlocal corrections to the analytic solutions is also proposed. It is therefore shown that the nonlocal optical response of the metal imposes more realistic, finite upper bounds to the enhancement feasible with ultrasmall plasmonic cavities, thus providing a theoretical description closer to state of the art experiments.

On the Asymptotic Stability of Stationary Solutions of the Inviscid Incompressible Porous Medium Equation

We study the stability of stationary solutions of the two dimensional inviscid incompressible porous medium equation (IPM). We show that solutions which are near certain stable stationary solutions must converge as t → ∞ to a stationary solution of the IPM equation. It turns out that linearizing the IPM equation about certain stable stationary solutions gives a non-local partial damping mechanism. On the torus, the linearized problem has a very large set of stationary (undamped) modes. This makes the problem of long-time behavior more difficult since there is the possibility of a cascading non-linear growth along the stationary modes of the linearized problem. We solve this by, more or less, doing a second linearization around the undamped modes, exploiting a special non-linear structure, and showing that the stationary modes can be controlled.

Exponential stability for the wave equation with degenerate nonlocal weak damping

A damped nonlinear wave equation with a degenerate and nonlocal damping term is considered. Well-posedness results are discussed, as well as the exponential stability of the solutions. The degeneracy of the damping term is the novelty of this stability approach.

Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping&lt;sup&gt;*&lt;/sup&gt;

In this paper we consider new results on well-posedness and long-time dynamics for a class of extensible beam/plate models whose dissipative effect is given by the product of two nonlinear terms. The addressed model contains a nonlocal nonlinear damping term which generalizes some classes of dissipations usually given in the literature, namely, the linear, the nonlinear and the nonlocal frictional ones. A first mathematical analysis of such damping term is presented and represents the main novelty in our approach.

Maximizing modal damping in layered structures via multi-objective topology optimization

This paper presents a study on maximizing single and multiple modal damping ratios (MDR) of a structure by finding the optimal layouts of damping and/or base materials using an extended moving iso-surface threshold (MIST) topology optimization. Firstly, by using the Lagrange’s equation, a general formulation of loss factor is derived and then used to extract MDR for classical and non-classical damping. Secondly, in the extended MIST, new general formulations for the physical response functions of individual mode are derived for the classical non-local damping and the non-classical damping, in which for non-classical damping modal strain energy (MSE) is used to estimate MDR and derive physical response function. Thirdly, to maximize multiple MDRs simultaneously or overcome the mode switching issue, a multi-objective optimization strategy is developed; and a concurrent design optimization of both base and damping layer is proposed to maximize MDR. Finally several numerical examples are presented to validate and illustrate the efficiency of the present extended MIST approach.

Influence of nonlocal damping on the field-driven domain wall motion

We derive the complete expression of nonlocal damping in noncollinear magnetization due to the nonuniform spin current pumped by precessional magnetization and incorporate it into a generalized Thiele equation to study its effects on the dynamics of the transverse and vortex domain walls (DWs) in ferromagnetic nanowires. We demonstrate that the transverse component of nonlocal damping slows down the field-driven DW propagation and increases the Walker breakdown field whereas it is neglected in many previous works in literature. The experimentally measured DW mobility variation with the damping tuned by doping with heavy rare-earth elements that had discrepancy from micromagnetic simulation are now well understood with the nonlocal damping. Our results suggest that the nonlocal damping should be properly included as a prerequisite for quantitative studies of current-induced torques in noncollinear magnetization.

Experimental evaluation of viscous damping coefficient in the fractional underdamped oscillator

In this article, we obtain the viscous damping coefficient β theoretically and experimentally in the spring–mass–viscodamper system. The calculation is performed to obtain the quasi-period τ. The influence of the viscosity of the fluid and the damping coefficient is analyzed using three fluids, water, edible oil, and gasoline engine oil SAE 10W-40. These processes exhibit temporal fractality and non-local behaviors. Our general non-local damping model incorporates fractional derivatives of Caputo type in the range [Formula: see text], and the viscous damping coefficients are determined in terms of the inverse Mittag-Leffler function. The classical models are recovered when the order of the fractional derivatives is equal to 1.

Nonlocal damping of helimagnets in one-dimensional interacting electron systems

We investigate the magnetization relaxation of a one-dimensional helimagnetic system coupled to interacting itinerant electrons. The relaxation is assumed to result from the emission of plasmons, the elementary excitations of the one-dimensional interacting electron system, caused by slow changes of the magnetization profile. This dissipation mechanism leads to a highly nonlocal form of magnetization damping that is strongly dependent on the electron-electron interaction. Forward scattering processes lead to a spatially constant damping kernel, while backscattering processes produce a spatially oscillating contribution. Due to the nonlocal damping, the thermal fluctuations become spatially correlated over the entire system. We estimate the characteristic magnetization relaxation times for magnetic quantum wires and nuclear helimagnets.