Nonlinear Numerical Simulations Research Articles

Tailored Dispersion and Nonlinear Effects in Flint Glass Honeycomb PCF for Optical Communication

This paper describes a highly nonlinear flint glass-based honeycomb photonic crystal fiber (FGH-PCF) with a wavelength of 1550 nm. The PCF’s distinctive honeycomb lattice structure, combined with the nonlinear capabilities of flint glass, enables a wide range of nonlinear optical applications. To adjust the PCF's dispersion and nonlinear effects, numerical simulations and optimization approaches were used. To achieve maximum performance, fabrication procedures were carefully regulated. Dispersion values of −436.6 ps/(nm.km) for x polarization and −448.1 ps/(nm.km) for y polarization were verified by experimental characterization. The PCF displayed low confinement losses of 2.289 dB/cm (x polarization) and 4.935 dB/cm (y polarization), as well as birefringence of 2.202×10-3. The PCF measured 558.8 and 547.9 W-1 km-1 for x and y polarization, respectively, indicating a high nonlinear coefficient. The highly nonlinear FGH-PCF shows promising potential for nonlinear optical applications such as four-wave mixing, supercontinuum generation, frequency conversion, and parametric amplification. This research paves the way for compact and efficient nonlinear devices in modern optical communication systems and other cutting-edge technologies.

Analytical resistance of the laterally offset rectangular hollow sections connections under concentrated force

The study aims to address the lack of design techniques for welded rectangular hollow sections connections made with offset, i.e., where the members of the branch adjacent to the connection are laterally shifted away from the chord centreline towards the chord web. To estimate the plastic resistance of the laterally offset truss-type RHS T-connection, a theoretical investigation was conducted. As a result, an analytical model has been developed and verified against the outcomes of a series of non-linear numerical simulations. During the construction of the model, specific attention was paid to the most general case of the branch location relative to the chord web, namely, the scheme of connection where adjacent vertical faces of the branch and chord members are in different planes is considered as prevailing. The proposed analytical model is an extension of a plastic mechanism model for traditional stepped rectangular hollow sections (RHS) T-connections. The model predicts the yield load and the post-yield response of the connections under concentrated load. The model presented in this paper is compared with other recently proposed methods of analytically determining limit loads of offset connections, and it provides reliable results in yield load computation. Moreover, the model yields sufficient accuracy in predicting the ultimate load for the offset connections with matched faces of the chord web and brace lateral surface.

Ultimate in-plane shear behaviour of clay brick masonry elements strengthened with TRM overlays

This paper studies the response of unreinforced masonry (URM) members made of hydraulic lime mortar and fired clay bricks, commonly found in heritage structures, strengthened with textile reinforced mortar (TRM) overlays. The investigation includes URM and TRM-strengthened diagonal compression tests on square panels, and relatively large-scale wall specimens subjected to combined gravity and lateral cyclic loads. Complementary compression, tension, and interface material tests are also carried out. The diagonal panel tests show that the TRM effectiveness depends in a non-proportional manner on the overlays, render thickness, and substrate strength. The enhancement in stiffness, strength, and ultimate shear strain, using one to four mesh layers on each side, is shown to vary in the range of 49–132%, 102–536%, and 300–556% respectively. It is shown that strut crushing typically governs the response of such low-strength URM masonry elements confined by TRM overlays. The cyclic tests on the comparatively larger walls show that the TRM is effective, shifting the response from URM diagonal tension to rocking, and enhancing the stiffness, strength, and ultimate drift capacity by more than 160%, 30%, and 130%, respectively. It is shown that analytical assessment methods for predicting the response of TRM-strengthened and URM members in terms of stiffness, strength and load-deformation can be reliably adapted. The cumulative contribution of the URM and TRM components, in conjunction with a suitable fibre textile strain, is also found to offer an improved prediction of the shear strength compared to codified procedures. The findings enable the evaluation and improvement of analytical models for determining the main inelastic response parameters of TRM-strengthened masonry and provide information for validating future detailed nonlinear numerical simulations.

Artificial intelligence (AI)-assisted simulation-driven earthquake-resistant design framework: Taking a strong back system as an example

Traditional earthquake-resistant structural design considers only a limited number of factors, mainly elastic structural properties, to determine key design parameters. However, these parameters are often not optimal because they do not take into account the extensive plasticity expected in building structures during earthquakes. To address this issue, an artificial intelligence (AI)-assisted simulation-driven framework has been developed in this study. This framework can automatically output optimal design parameters while considering nonlinear structural response under strong earthquakes and a large number of input factors. The primary innovation of the proposed framework lies in the fusion and integration of nonlinear numerical simulation and AI tools for earthquake-resistant design of building structures, marking a promising trend in this field. The framework consists of two steps. In the first step, a database that consists of optimal design parameters and covers a wide range of design inputs will be created through numerical nonlinear response history analyses (NRHAs). In the second step, AI models will be created and trained based on the database to automatically output the optimal design parameters. To illustrate the basic components underlying the proposed framework, the determination of the height-wise distribution (denoted by Ψ) of the total design lateral force for a strong back system is taken as an example. A database of 1200 samples was created through NRHAs, and an artificial neural network (ANN) model was created, optimised, and trained. The developed ANN model yielded optimal Ψ with the majority of absolute errors within 1%, demonstrating the feasibility of the proposed AI-assisted simulation-driven earthquake-resistant design framework.

PDEM-based seismic performance evaluation of circular tunnels under stochastic earthquake excitation

ABSTRACT Tunnels are a vital component of urban infrastructure that must be robust against seismic hazards. Given the randomness of earthquake occurrence, the seismic response of tunnel structures mut be studied by stochastic analysis methods. To this end, this study proposes a probability density evolution method (PDEM)-based framework to investigate the seismic performance of a circular tunnel under stochastic earthquake excitation. First, a suite of nonstationary earthquake motions compatible with the seismic design code was derived using a stochastic earthquake model. Then, a series of nonlinear dynamic numerical simulations were conducted for a typical circular tunnel that considers the soil-structure interaction. Finally, using the tunnel inclination angle as the performance index, the probability density function of the structural response of the tunnel was solved using the PDEM to obtain the corresponding exceedance probabilities of the tunnel under various damage states. The results show that the PDEM-based framework can be applied to evaluate the seismic performance of circular tunnels and could serve as a reference on the seismic fragility of tunnels and underground structures.

Quantitative stability analysis of complex nonlinear hydraulic turbine regulation system based on accurate calculation

The current stability studies of the hydraulic turbine regulation system (HTRS) mostly adopt the linear hydro-turbine model ignoring its strong nonlinearity, leading to insufficient disclosure of the true characteristics of the HTRS and also causing great inconvenience to the controller parameter tuning. To address this issue, based on the Hopf bifurcation theory, bisection method, and stability criterion, this paper proposes an algorithm (HBBSC) for determining the controller parameter constraint considering the nonlinearity of the hydro-turbine. Firstly, the nonlinear model of the hydro-turbine is constructed based on the model reconstruction strategy (NNGW) combining the backpropagation neural network (BPNN) with the improved grey wolf optimization algorithm (IGWO) to obtain an accurate nonlinear HTRS numerical simulation platform under the power control mode (PCM) and frequency control mode (FCM). Then, the HBBSC-based quantitative calculation procedure of stability region constraint is introduced in detail. Further, in a case study of stability region calculation of HTRS, the HBBSC is applied to calculate the stability region constraint, and the HBBSC-based stability region is verified through a simulation platform. Finally, the stability region of complex HTRS under all operating conditions is calculated based on HBBSC. The results indicate that the HBBSC can replace the traditional methods for stability region calculation during stability analysis of HTRS, outperforming the latter in accuracy and reliability.

Numerical study on steel plate girders ‐ Intermediate transverse stiffeners axial force

AbstractTransverse stiffeners are used in steel plate girders with slender webs to provide lateral support to the web along the span and thus increase the plate girder shear resistance. Furthermore, transverse stiffeners anchor the tension field, resulting in axial compression in the stiffener. Despite this, it is widely assumed that the design axial force provided by the FprEN 1993‐1‐5 [1] for the transverse stiffeners is significantly greater than that observed in experimental testing and numerical simulations.The results of 550 nonlinear numerical simulations of plate girders loaded in shear up to failure are presented. Shear resistance and axial stiffener forces are evaluated while the effects of a wide parameter range as web aspect ratio and thickness, flange‐to‐web area ratio, and steel grade are investigated. The numerical results are compared to the European standard design values. Finally, a statistical analysis of the set of results is presented.The numerical analyses show that the axial forces installed in the intermediate transverse stiffeners are significantly lower than those specified by FprEN 1993‐1‐5 [1].

Towards Tolerance Specifications for the Elastic Buckling Design of Axially Loaded Cylinders

Abstract The quest for safe lower bounds to the elastic buckling of axially loaded circular cylindrical shells has exercised researchers for the past 100 years. Recent work bringing together the capabilities of nonlinear numerical simulation, interpreted within the context of extended linear classical theory, has come close to achieving this goal of defining safe lower bounds. This paper briefly summarizes some of the important predictions emerging from previous work and presents new simulation results that confirm these earlier predictions. In particular, we show that for a specified maximum amplitude of the most sensitive, eigenmode-based geometric imperfections, normalized with respect to the shell thickness, lower bounds to the buckling loads remain constant beyond a well-defined value of the Batdorf parameter. Furthermore, we demonstrate how this convenient means of presenting the imperfection-sensitive buckling loads can be reinterpreted to develop practical design curves which provide safe, but not overly conservative, design loads for monocoque cylinders with a given maximum permitted tolerance of geometric imperfection. Hence, once the allowable manufacturing tolerance is specified during design or is measured post-manufacturing, the greatest expected knockdown factor for a shell of any geometry is defined. With the recent research interest in localized imperfections, we also attempt to reconcile their relation to the more classical, periodic, and eigenmode-based imperfections. Overall, this paper provides analytical and computational arguments that motivate a shift in focus in defect-tolerant design of thin-walled cylinders—away from the knockdown experienced for a specific geometric imperfection and towards the worst possible knockdown expected for a specified manufacturing tolerance.

Robust gain-scheduled missile autopilot design based on multifrequency extended state observers

This paper describes the design and implementation of a three-axis acceleration control autopilot for an asymmetric tail-controlled, skid-to-turn tactical missile. In an earlier flight test, degraded autopilot performance was attributed to multiple disturbances and uncertainties and the presence of hidden coupling terms, giving rise to a miss distance of greater than 20 m. To address these issues, the missile dynamics are decomposed into the angular rate dynamics as fast and the acceleration dynamics as slow subsystem using the singular perturbation theory to analyze a multi-time-scale property. Multifrequency extended state observers are then incorporated into the gain scheduling technique to attenuate disturbances, thus enhancing the control performance significantly. In the proposed engineering/practical design framework for missile autopilot, simple, conventional, and explicit tuning rules are provided. And the proposed control scheme can achieve input-to-state stability across the entire flight envelope under unknown but bounded disturbances. The advantages of the method over existing benchmark approaches are shown through nonlinear numerical simulations. This is supported by evidence from a new flight test result with a miss distance of only 2 m.

A chemotaxis reaction–diffusion model for Multiple Sclerosis with Allee effect

In this paper, we study a modification of the mathematical model describing inflammation and demyelination patterns in the brain caused by Multiple Sclerosis proposed in Lombardo et al. (J Math Biol 75:373–417, 2017). In particular, we hypothesize a minimal amount of macrophages to be able to start and sustain the inflammatory response. Thus, the model function for macrophage activation includes an Allee effect. We investigate the emergence of Turing patterns by combining linearised and weakly nonlinear analysis, bifurcation diagrams and numerical simulations, focusing on the comparison with the previous model.

Dispersion analysis of numerical schemes using 2D compressible linearized Navier–Stokes equation for direct numerical simulation

The dispersion analysis of numerical schemes for finite difference computations based on 2D compressible linearized Navier–Stokes equation (LNSE) is presented here. The analysis presents a more holistic view of the applicability of the schemes while solving compressible NSE, rather than analyzing model 1D or 2D convection or convection–diffusion equations. A theoretical dispersion analysis of model 2D−LNSE is performed first which categorizes it into vortical, entropic and two acoustic modes. The variation of the theoretical dispersion relation for each of the modes with wavenumber, Mach number and Reynolds number is analyzed. It is noted that the entropic mode is the most diffusive among all the modes. Two acoustic modes are less diffusive than the vortical mode up to a certain absolute wavenumber Kb depending upon the Mach number and Reynolds number. It is also shown that while vortical and entropic modes are non-dispersive in nature, the two acoustic modes are dispersive only up to K=Kb. Next, numerical dispersion and diffusion corresponding to each of the modes are analyzed for second order central difference (CD2) and sixth order central compact difference scheme proposed in Lele (J. Comput. Phys., Vol-103(1), 1992) for discretization of convective and diffusive derivatives using fourth-order Runge–Kutta (RK4) time integration method. Results show the existence of numerical unstable zone and regions where spurious upstream propagating waves are noted for each of the modes in the spectral plane. From these and the relative size of the dispersion relation preserving zones for the modes, applicability of the numerical schemes can be assessed. Analysis of the variation of the numerical unstable zone with Mach number, orientation of the mean flow and the unit CFL number Nc=Δt/Δx for each of the modes are presented. It is noted that entropic mode displays the lowest critical CFL number of all the four modes. Lastly, linearized and nonlinear numerical simulations for the convection of acoustic and vortical pulse with varying spectral content are presented. The linearized simulations are also compared with corresponding exact solutions. Obtained results indicate a direct correspondence of the stability analysis performed based on 2D−LNSE even for nonlinear simulations which result in additional intra modal energy transfer and interactions.

Improving the Estimation of the Occurrence Time of an Impending Major Earthquake Using the Entropy Change of Seismicity in Natural Time Analysis

This article is focused on a new procedure concerning a more accurate identification of the occurrence time of an impending major earthquake (EQ). Specifically, we first recapitulate that, as was recently shown [P. Varotsos et al., Communications in Nonlinear Science and Numerical Simulation 125 (2023) 107370], natural time analysis of seismicity supplemented with the non-additive Tsallis entropy Sq leads to a shortening of the time window of an impending major EQ. This has been shown for the Tohoku mega-EQ of magnitude M9 that occurred in Japan on 11 March 2011, which is the largest event ever recorded in Japan. Here, we also show that such a shortening of the time window of an impending mainshock can be achieved for major, but smaller EQs, of the order of M8 and M7. In particular, the following EQs are treated: the Chiapas M8.2 EQ, which is Mexico’s largest EQ for more than a century that took place on 7 September 2017 near the coast of Chiapas state in Mexico, the 19 September 2017 M7.1 EQ that occurred within the Mexican flat slab, and the M7.1 Ridgecrest EQ on 6 July 2019 in California.

Seismic response of soil-shield tunnel systems in sandwiched liquefiable soil deposits

Long-extended underground structures such as shield-driven tunnels may be constructed through sandwiched liquefiable soil profiles, which can compromise their integrity during strong earthquakes. In this study, the influence of sandwiched liquefiable soil layer on the seismic response of shield tunnels under ground motions with different frequency contents is evaluated through nonlinear numerical simulations. Six soil-shield tunnel finite element models with different sandwiched liquefiable soil profiles are developed using the open-source software framework OpenSees. The solid–fluid fully coupled dynamic effective stress analysis method is adopted for the nonlinear seismic analysis of liquefiable soil. The numerical results demonstrate that the frequency contents of earthquake motion and the spatial location of the sandwiched liquefiable soil profile have great influence on the dynamic responses of the segmental tunnel. In all cases studied, the bottom half of the tunnel embedded in the liquefiable soil layer experienced the most unfavorable condition for the seismic performance. The deformations and internal forces of the segmental joints around the tunnel foot are remarkably higher than at other joints under the same ground motion intensity. Moreover, the Kobe motion with the low-frequency content would significantly amplify the seismic responses of both the site and the segmental tunnel compared to the Chi-Chi motion with the high-frequency content in different sandwiched liquefiable soil deposits. It is also found that the racking ratio of the non-liquefiable soil layer is greater than that of the liquefiable soil layer, which means that the surrounding soil with large shear deformation pushes the underground structure leading to larger adverse lateral deformation of the structure.

Analysis on abnormal low-frequency vertical vibration of medium–low-speed maglev vehicle

A newly-developed medium–low-speed maglev test vehicle was used to obtain the vertical dynamic response signals of the vehicle-line coupled system when the test vehicle was standing still on four different types of lines through field tests. It was found that abnormal low-frequency vertical vibration occurred on the car body, so that an excellent ride index of the test vehicle cannot be ensured. As the stability of the vehicle at standstill is the prerequisite to the stability of the vehicle running at a certain speed, it is necessary to investigate the causes underlying the abnormal vibration and propose corrective measures through theoretical studies in combination with the measured results. Therefore, this paper established a coupled vehicle-track system dynamics model considering active feedback control, and investigated the stability and bifurcation behavior of the coupled system through nonlinear dynamics theory and numerical simulation. Field test results show that: the vertical ride index of the test vehicle was rated as “Unqualified” when it was standing still; the vertical vibration eigenfrequency of the car body is basically the same (between 2.10 and 2.20 Hz) when the test vehicle was standing still on the four different lines, indicating that the abnormal vibration was basically independent of the line type; the vertical dynamic response of each measurement point on the electromagnet, F-rail and bridge was normal. The theoretical and simulation studies show that: the numerical simulation of the theoretical model can effectively reproduce the abnormal vibration; the malfunction in the nonlinear characteristics of the air spring vertical damping (i.e. the damping malfunction) is the main reason for this vibration, and the abnormal vibration corresponded to the supercritical Hopf bifurcation dynamics behavior in nonlinear dynamics; after the air spring was restored to normal, the equilibrium point of the coupled system was stable under the influence of different initial disturbances. This study may provide an important reference for reducing the abnormal low-frequency vertical vibration of the test vehicle at standstill.

From thin plates to Ahmed bodies: linear and weakly nonlinear stability of rectangular prisms

We study the stability of laminar wakes past three-dimensional rectangular prisms. The width-to-height ratio is set to $W/H=1.2$ , while the length-to-height ratio $1/6< L/H<3$ covers a wide range of geometries from thin plates to elongated Ahmed bodies. First, global linear stability analysis yields a series of pitchfork and Hopf bifurcations: (i) at lower Reynolds numbers $Re$ , two stationary modes, $A$ and $B$ , become unstable, breaking the top/bottom and left/right planar symmetries, respectively; (ii) at larger $Re$ , two oscillatory modes become unstable and, again, each mode breaks one of the two symmetries. The critical $Re$ values of these four modes increase with $L/H$ , reproducing qualitatively the trend of stationary and oscillatory bifurcations in axisymmetric wakes (e.g. thin disk, sphere and bullet-shaped bodies). Next, a weakly nonlinear analysis based on the two stationary modes $A$ and $B$ yields coupled amplitude equations. For Ahmed bodies, as $Re$ increases, state $(A,0)$ appears first, followed by state $(0,B)$ . While there is a range of bistability of those two states, only $(0,B)$ remains stable at larger $Re$ , similar to the static wake deflection (across the larger base dimension) observed in the turbulent regime. The bifurcation sequence, including bistability and hysteresis, is validated with fully nonlinear direct numerical simulations, and is shown to be robust to variations in $W$ and $L$ in the range of common Ahmed bodies.

Diagnosing Hydro-Mechanical Effects in Subsurface Fluid Flow Through Fractures

Hydro-mechanically induced transient changes in fracture volume elude an analysis of pressure and flow rate transients by conventional diffusion-based models. We used a previously developed fully coupled, inherently non-linear numerical simulation model to demonstrate that harmonic hydraulic excitation of fractures leads to systematic overtones in the response spectrum that can thus be used as a diagnostic criterion for hydro-mechanical interaction. The examination of response spectra, obtained from harmonic testing at four different field sites, for the occurrence of overtones confirmed their potential for the hydro-mechanical characterization of tested reservoirs. A non-dimensional analysis identified relative aperture change as the critical system parameter.

Numerical Analysis on Effects of Soil Improvement on Pile Forces on Existing High-Rise Building

Nowadays, seismic codes are regularly updated with new knowledge and a better understanding of the earthquake phenomenon. With these updates, existing buildings require a reevaluation of their stability and a process of reinforcement and/or retrofitting. This study investigated the effects of two types of ground improvement which use cement-mixing soil surrounding the foundation structure to reduce and redistribute forces acting on piles. This is especially important when the reevaluation of high-rise buildings leads to increased forces in the piles. Typically, buildings are designed while assuming fixed base boundary conditions at the foundation level, without considering soil–pile–structure interaction (SPSI). SPSI significantly influences the response of high-rise buildings supported by soft soil. Increasing the lateral resistance of the surrounding soil can reduce the influence of SPSI. In this study, a detailed dynamic numerical analysis was used to investigate the dynamic response of an SPSI system of a high-rise building under seismic load. A dynamic analysis was conducted on a modified layout of a real building, using real-time earthquake motion. The finite element program DIANA FEA was used to perform nonlinear 3D FEM numerical simulations, taking into account the essential SPSI phenomena, gap-slip between the piles and the soil, and free-field boundary conditions. A comparison of the data suggests that the bending moment and shear forces in the piles are reduced in magnitude and evenly distributed along the upper part of the pile, which reduces the stress concentration of the bending moment and shear forces at the contact between the piles and the pile cap.

Shaking table test and numerical analyses of a full scale three-leaf masonry wall

This paper discusses the main results of a full-scale shaking table test campaign carried out under the auspices of the EU funded research project SERA, whose objective is to investigate the seismic performance of three-leaf masonry walls with weak lime-mortar joints. These masonry walls are widely found in seismic prone regions in the Mediterranean area, thus assessing their behaviour under dynamic actions is an important pre-requisite for the seismic vulnerability evaluation of a plethora of historical centres. The first part of the paper presents a preliminary study on the mechanical properties of the wall component materials that was carried out through an ad-hoc experimental campaign. The outcomes are of particular interest for the characterization of the mortar and of the infill materials, that were designed to reproduce the low strength that is typically found in old masonry buildings. The design of the masonry wall that was tested and the test set-up are presented next. The applied loading protocol consisted of the horizontal component of a ground motion record that is repeatedly applied to the shaking table with increasing intensity. Finally, the main results of the experimental test are discussed. The damage patterns, drift ratios and base shear are presented for the ground motion sequence. The results are also discussed through a dynamic capacity curve that shows the attainment of different limit states with increasing ground motion intensity. A set of nonlinear numerical simulations, both static and dynamic, using a 3D FE model of the wall verify the experimental study as they report good agreement with the experimental tests and exhibit stable numerical behaviour.

Evaporation-driven gravitational instability in the liquid layer of a polymer solution: Theoretical and numerical studies

The drying of the thin liquid film of polymer solution is a complex process accompanying instabilities induced by concentration gradients that develop during the evaporation. This study investigates evaporation-driven gravitational instabilities (Rayleigh–Taylor instability) in the thin liquid film of a polymer solution based on theoretical and numerical analyses, specifically focusing on the onset of instability at the early stage of the drying process. Evaporation-driven convection is taken into account by employing a dynamic moving boundary condition and concentration-dependent material functions (dynamic viscosity and diffusivity). A theoretical analysis predicts the onset of gravitational instability (τc), showing good agreement with the results of a numerical simulation analysis (τd), suggesting scaling relationships between the onset of instability and the initial concentration (Ci), τc∼Ci−1/2, for a slow evaporation rate and dilute concentration range, which rapidly increases with an increase in the concentration due to an instability-retardation effect. A nonlinear two-dimensional numerical simulation visualizes the development of a polymer-dense layer, pluming, and a convection–diffusion flow throughout the liquid film under transient evaporation. The concentration-gradient-driven convection–diffusion flow enhances the gravitational instability and pluming further at higher frequencies. Meanwhile, for a polymer solution with high viscosity, viscosity thickening delays the onset of instability, retarding the development of instability. This study provides a fundamental understanding of the mechanism behind how convective instability develops in a drying polymer solution during evaporation and information on how to develop pluming of a polymer-dense skin layer at early stages of drying of a polymer solution film. The findings suggest that the proper control of hydrodynamic instability induced by a concentration gradient in a thin polymer solution film is important to prevent or enhance the formation of a convection pattern in the thin polymer film.

Splitting of doubly quantized vortices in holographic superfluid of finite temperature

The temperature effect on the linear instability and the splitting process of a doubly quantized vortex is studied. Using the linear perturbation theory to calculate out the quasi-normal modes of the doubly quantized vortex, we find that the imaginary part of the unstable mode increases with the temperature till some turning temperature, after which the imaginary part of the unstable mode decreases with the temperature. On the other hand, by the fully non-linear numerical simulations, we also examine the real time splitting process of the doubly quantized vortex, where not only do the split singly quantized vortex pair depart from each other, but also revolve around each other. In particular, the characteristic time scale for the splitting process is identified and its temperature dependence is found to be in good agreement with the linear instability analysis in the sense that the larger the imaginary part of the unstable mode is, the longer the splitting time is. Such a temperature effect is expected to be verified in the cold atom experiments in the near future.