Multi-timescale Method Research Articles

Three-to-one internal resonances of stepped nanobeam of nonlinearity

Abstract In this study, vibrations of stepped nanobeams were investigated according to Eringen’s nonlocal elasticity theory. Multi-time scale method, which is one of the perturbation methods, has been applied to solve dimensionless state equations. The solution is considered in two steps. First-order terms obtained from the perturbation expansion formed the linear problem in the first step. In the second step, the solution of the second order of the perturbation expansion was made and nonlinear terms emerged as corrections to the linear problem from this solution. The main issue that the study wants to emphasize is the examination of the mechanical effects of the steps, which are discontinuities encountered at the nanoscale, on the system. For this purpose, while the findings of the research were obtained, various nonlocal parameter values were obtained to capture the nano-scale effect, and frequency-response and nonlinear frequency-amplitude curves corresponding to the 1st Mode values of the beam for different step ratios and step locations were obtained to capture the step effect. One of the important features of the nonlinear system is the formation of internal resonance between the modes of the system. How this situation affects the characteristics of the system has also been examined and results have been given by graphs. The obtained data show that taking into account the nanoscale step is essential for the accuracy and sensitivity of many nanostructures such as sensors, actuators, biostructures, switches, etc. that are likely to be produced at the nanoscale in practice.

Optimal Power Flow Scheduling Strategy for Multi-Microgrids with Multi-Time Scale Method

Abstract: The energy management of a multi-microgrid (MG) system is essential for its stable and economic operation. This study proposes an optimal power flow scheduling strategy for the energy management of multi-MG systems. At the multi-MG level, the global central controller (GCC) is responsible for managing the MGs. The GCC calculates the amount of power exchanged within the MGs by using a novel optimal energy allocation policy. Based on the energy supply and demand mismatch, MGs are classified as providers and consumers. The GCC collects information, then distributes energy among the consumers and divides benefits to the providers. Each consumer determines the price of the purchased energy from other microgrids based on a priority parameter, in which the local load demand and renewable energy penetration rate are considered as important factors. At the MG level, with the goal of minimising the operating cost of the MG, the energy is controlled from two time scales, namely day-ahead and intraday, to optimise the output power of generators and energy storage devices. Finally, a simulation of a multi-MG system with three MGs demonstrate the effectiveness of the proposed optimal method.

The General Analytic Expression of a Harvested Logistic Model with Slowly Varying Coefficients

The harvested logistic model with a slow variation in coefficients has been considered. Two cases, which depend on the harvest rate, were identified. The first one is when the harvest is subcritical, where the population evolves to an equilibrium. The other is supercritical harvesting, where the population decreases to zero at finite times. The single analytic approximate expression, which is capable of describing both harvesting cases, is readily and explicitly obtained using the multi-time scaling method together with the perturbation approach. This solution fits for a wide range of coefficient values. In addition, such an expression is validated by utilizing numerical computations, which are obtained by using the fourth-order Runge–Kutta technique. Finally, the comparison shows a very good agreement between the two methods.

Multi-Timescale Simulations of Temperature Elevation for Ultrasonic Welding of CFRP with Energy Director

Ultrasonic welding is an energy-efficient technology that enables quick bonding of thermoplastic composite materials under normal temperature and pressure conditions. Here, numerical multi-timescale simulation is proposed to understand the welding principle, using numerical simulations of ultrasonic welding. The simulation results are validated by comparing with temperature measurements in welding tests. In the multi-timescale simulations, microsecond-scale simulations are performed first. The ultrasonic wave is modeled as a vibration load, and the energy dissipation per vibration at 25, 75, 125, 175, 225, and 275∘C is analyzed. Then, the time derivative of the temperature rise is obtained. In the normal scale simulations, the ultrasonic wave and holding pressure are replaced by a constant load, and the entire process of ultrasonic welding is simulated. The slope of the temperature rise is fitted to the time derivative of the temperature rise obtained from the microsecond-scale simulations, using the material constant as a parameter. Explicit multi-timescale simulations were performed to investigate the relationship between stress concentration and temperature rise due to ED geometry. The result reveals similar temperature behavior to the experimental one, indicating the validity of the multi-timescale method. It suggests that viscoelastic energy dissipation and stress concentration are responsible for the temperature spike.

A multi-timescale estimator for battery state of charge and capacity dual estimation based on an online identified model

Reliable online estimation of state of charge (SOC) and capacity is critically important for the battery management system (BMS). This paper presents a multi-timescale method for dual estimation of SOC and capacity with an online identified battery model. The model parameter estimator and the dual estimator are fully decoupled and executed with different timescales to improve the model accuracy and stability. Specifically, the model parameters are online adapted with the vector-type recursive least squares (VRLS) to address the different variation rates of them. Based on the online adapted battery model, the Kalman filter (KF)-based SOC estimator and RLS-based capacity estimator are formulated and integrated in the form of dual estimation. Experimental results suggest that the proposed method estimates the model parameters, SOC, and capacity in real time with fast convergence and high accuracy. Experiments on both lithium-ion battery and vanadium redox flow battery (VRB) verify the generality of the proposed method on multiple battery chemistries. The proposed method is also compared with other existing methods on the computational cost to reveal its superiority for practical application.

Influence of frictional mechanism on chatter vibrations in the cutting process\u2013analytical approach

The paper examines a nonlinear one-degree-of-freedom model of the cutting process. The classical regenerative mechanism of chatter is enriched by an additional friction phenomenon which generates frictional chatter. Additionally, the nonlinear cubic stiffness of the tool is taken into account. The aim of the paper is to investigate interactions between regeneration and the frictional effect. The proposed model is solved by the multi-time scale method. The cutting process stability (trivial solution) is determined in order to produce stability lobe diagrams and determine the influence of friction on the process. Finally, the maps of chatter amplitudes are presented and new frictional stability lobe diagrams are proposed to analyse the influence of friction.

Multi-scale modeling of dynamics and ignition to flame transitions of high pressure stratified n-heptane/toluene mixtures

The transitions from ignition to flames as well as the combustion dynamics in stratified n-heptane and toluene mixtures are numerically modeled by a correlated dynamic adaptive chemistry method coupled with a hybrid multi-timescale method (HMTS/CO-DAC) in a one-dimensional constant volume chamber. The study attempts to answer how the kinetic difference between n-alkanes and aromatics leads to different ignition to flame transitions and knocking-like acoustic wave formation at low temperature and engine pressure conditions with fuel stratification. It is found that the low temperature chemistry (LTC) and fuel stratification of n-heptane leads to the formation of multiple ignition fronts. Four different combustion wave fronts, a low temperature ignition (LTI) front followed by a high temperature ignition (HTI) front, a premixed flame (PF) front, and a diffusion flame (DF) front, are demonstrated. The fast LTI and HTI wave front propagation leads to a shock-like strong acoustic wave propagation, thus strongly modifying the dynamics of the subsequent diffusion and premixed flame fronts. On the other hand, for the toluene mixture, due to the lack of LTC, only two combustion wave fronts are formed, a HTI front and a premixed flame front, exhibiting stable flow field and no formation of shock-like acoustic wave. The dynamics of transition from combustion to shock waves is further analyzed by using a modified Burgers’ equation. The analysis for n-heptane/air mixture indicates that both the onset of LTI and the strong dependency of HTI on the equivalence ratio can either promote or attenuate the transition from strong acoustic wave to shock wave. However, the toluene/air mixture exhibits no coupling with acoustic wave, suggesting that the rich LTC reactivity with fuel stratification, specific to the n-alkane chemistry, can lead to knocking and acoustic formation.

Multi-timescale analysis of fatigue crack growth on interfaces via cohesive-zone models

The paper describes a new non-linear finite-element formulation to analyse fatigue debonding or delamination, along predefined interfaces, which is multi-scale in time. At the small timescale level, cyclic loading and the related oscillating response are considered in an explicit way, whereas at the large timescale level, both the real loading actions and the related response in terms of displacement and stress fields are replaced with ‘minimum’ and ‘maximum’ functions over the time of the analysis, which also implies doubling the degrees of freedom of the finite-element model. A cohesive-zone model capable of simulating sub-critical damage growth and hysteretic local response is used on the interface. With a conventional cycle-by-cycle incremental procedure, the analysis would require a number of increments significantly higher than the number of cycles, and would be therefore unfeasible for most industrial applications. Instead, with the developed multi-timescale method, the cycle-by-cycle time integrat...

Accelerating cyclic plasticity simulations using an adaptive wavelet transformation based multitime scaling method

SUMMARYCyclic finite element simulations of complex materials, for example, polycrystalline metals, are widely used to study fatigue failure due to plasticity and damage. Typically, this requires the simulation of a large number of cycles to failure for accurate determination of evolving deformation variables. Modeling cyclic deformation using conventional methods of time integration in semidiscretization techniques can however be computationally challenging. Single time scale integration methods typically follow the high frequency characteristics and discretize each cycle into a number of time steps over which integration is performed. To overcome this computational challenge, the wavelet transformation‐based multitime scale (WATMUS) method proposed in an earlier work by the authors is advanced and validated in this paper to perform accelerated finite element simulations of materials undergoing rate‐dependent plasticity for large number of cycles. Specifically, the WATMUS algorithm is integrated with crystal plasticity finite element method to perform accelerated simulations of polycrystalline alloys. The WATMUS methodology is also endowed with adaptive capabilities to optimally construct the wavelet basis functions and determine coarse‐scale cycle steps. Accuracy and efficiency of the WATMUS methodology is conclusively demonstrated by comparing the results with cyclic single‐time scale crystal plasticity finite element simulations performed on image‐based microstructure of titanium alloys. Copyright © 2012 John Wiley & Sons, Ltd.

Wavelet transformation based multi-time scale crystal plasticity FEM for cyclic deformation in titanium alloys under dwell load

Titanium alloys are used in high end applications due to their desirable properties. However under dwell fatigue loading, these alloys exhibit premature crack initiation and failure when compared to normal cyclic loading. This early crack initiation can be attributed to the inhomogenous plastic deformation occurring in these alloys during the hold period of dwell cyclic loading and is strongly influenced by the underlying micro-structure. This necessitates the consideration of micro-structural features and their influence on the response, for accurate life prediction in these alloys. Crystal plasticity based finite element simulations capture the micro-structural influence on the response and can prove effective for accurate prediction of crack initiation in these alloys. However 70–80% of total life in Ti-alloys based components is spent in crack initiation and this may involve crystal plasticity based finite element simulations for large number of cycles. Such simulations using conventional single scale time integration schemes can be computationally intractable. A multi-time scale method using wavelet based decomposition is thus developed in this work for accurate and computationally efficient dwell fatigue simulations of these alloys for large number of cycles.

Multi-timescale modeling of ignition and flame regimes of n-heptane-air mixtures near spark assisted homogeneous charge compression ignition conditions

The flame regimes of ignition and flame propagation as well as transitions between different flame regimes of n-heptane-air mixtures in a one-dimensional, cylindrical, spark assisted homogeneously charged compression ignition (HCCI) reactor are numerically modeled using a multi-timescale method with reduced kinetic mechanism. It is found that the initial mixture temperature and pressure have a dramatic impact on flame dynamics. Depending on the initial temperature gradient, there exist at least six different combustion regimes, an initial single flame front propagation regime, a coupled low temperature and high temperature double-flame regime, a decoupled low temperature and high temperature double-flame regime, a low temperature ignition regime, a single high temperature flame regime, and a hot ignition regime. The results show that the low temperature and high temperature flames have distinct kinetic and transport properties as well as flame speeds, and are strongly influenced by the low temperature chemistry. The pressure and heat release rates are affected by the appearance of different flame regimes and the transitions between them. Furthermore, it is found that the critical temperature gradient for ignition and acoustic wave coupling becomes singular at the negative temperature coefficient (NTC) region. The results show that both the NTC effect and the acoustic wave propagation in a closed reactor have a dramatic impact on the ignition front and acoustic interaction.

A path flux analysis method for the reduction of detailed chemical kinetic mechanisms

A direct path flux analysis (PFA) method for kinetic mechanism reduction is proposed and validated by using high temperature ignition, perfect stirred reactors, and steady and unsteady flame propagations of n-heptane and n-decane/air mixtures. The formation and consumption fluxes of each species at multiple reaction path generations are analyzed and used to identify the important reaction pathways and the associated species. The formation and consumption path fluxes used in this method retain flux conservation information and are used to define the path indexes for the first and the second generation reaction paths related to a targeted species. Based on the indexes of each reaction path for the first and second generations, different sized reduced chemical mechanisms which contain different number of species are generated. The reduced mechanisms of n-heptane and n-decane obtained by using the present method are compared to those generated by the direct relation graph (DRG) method. The reaction path analysis for n-decane is conducted to demonstrate the validity of the present method. The comparisons of the ignition delay times, flame propagation speeds, flame structures, and unsteady spherical flame propagation processes showed that with either the same or significantly less number of species, the reduced mechanisms generated by the present PFA are more accurate than that of DRG in a broad range of initial pressures and temperatures. The method is also integrated with the dynamic multi-timescale method and a further increase of computation efficiency is achieved.

Wavelet transformation based multi-time scaling method for crystal plasticity FE simulations under cyclic loading

Microstructure based mechanistic calculations, coupled with physically motivated crack initiation criterion, can provide effective means to predict fatigue cracking in polycrystalline materials. However the accommodation of large number of cycles to failure, as observed in the experiments, could be computationally exhaustive to simulate using conventional single time scale finite element analysis. To meet this challenging requirement, a novel wavelet transformation based multi-time scaling algorithm is proposed for accelerated crystal plasticity finite element simulations in this paper. An advantage over other conventional methods that fail because of assumptions of periodicity etc., is that no assumption of scale separation is needed with this method. The wavelet decomposition naturally retains the high frequency response through the wavelet basis functions and transforms the low frequency material response into a “cycle scale” problem with monotonic evolution. The method significantly enhances the computational efficiency in comparison with conventional single time scale integration methods. Adaptivity conditions are also developed for this algorithm to improve accuracy and efficiency. Numerical examples for validating the multi-scaling algorithm are executed for a one dimensional viscoplastic problem and a 3D crystal plasticity model of polycrystalline Ti alloy under the cyclic loading conditions.

A dynamic multi-timescale method for combustion modeling with detailed and reduced chemical kinetic mechanisms

A new on-grid dynamic multi-timescale (MTS) method is presented to increase significantly the computation efficiency involving multi-physical and chemical processes using detailed and reduced kinetic mechanisms. The methodology of the MTS method using the instantaneous timescales of different species is introduced. The definition of the characteristic time for species is examined and compared with that of the computational singular perturbation (CSP) and frozen reaction rate methods by using a simple reaction system. A hybrid multi-timescale (HMTS) algorithm is constructed by integrating the MTS method with an implicit Euler scheme, respectively, for species with and without the requirement of accurate time histories at sub-base timescales. The efficiency and the robustness of the MTS and HMTS methods are demonstrated by comparing with the Euler and VODE solvers for homogenous ignition and unsteady flame propagation of hydrogen, methane, and n-decane–air mixtures. The results show that both MTS and HMTS reproduce well the species and temperature histories and are able to decrease computation time by about one-order with the same kinetic mechanism. Compared to MTS, HMTS has slightly better computation efficiency but scarifies the stability at large base time steps. The results also show that with the increase of mechanism size and the decrease of time step, the computation efficiency of multi-timescale method increases compared to the VODE solver. In addition, it is shown that the integration of the multi-timescale method with the path flux analysis based mechanism reduction approach can further increase the computation efficiency. Unsteady simulations of outwardly propagating spherical n-decane–air premixed flames demonstrate that the multi-timescale method is rigorous for direct numerical simulations with both detailed and reduced chemistry and can dramatically improve the computation efficiency.

An On-Grid Dynamic Multi-Timescale Method with Path Flux Analysis for Multi-Physics Detailed Modeling of Combustion

Combustion is a kinetic and multi-physical-chemical process involving many time and length scales from atomic excitation to turbulent mixing. This paper presents an overview of the recent progress of numerical modeling using detailed and reduced chemical kinetic mechanisms for multi-scale combustion problems. A particular focus of this review is to introduce two new methods, an on-grid dynamic multi-time scale (MTS) method and a path flux analysis (PFA) to increase the computation efficiency involving multi-physical chemical processes using large kinetic mechanisms. Firstly, the methodology of the on-grid dynamic MTS method using the instantaneous time scales of different reaction groups is introduced. The definition of species timescales and the algorithm to generate species groups are presented. Secondly, a concept of hybrid multi-time scale (HMTS) method to selectively remove time histories of uninterested fast modes is introduced to increase the algorithm flexibility and efficiency. The efficiency and the robustness of the MTS and HMTS methods are demonstrated by comparing with the Euler and ODE solvers for ignition and flame propagation of hydrogen, methane, and n-decane-air mixtures. Thirdly, the PFA method to generate a comprehensive reduced mechanism by using the reaction path fluxes in multiple generations is summarized. The results are compared with that of the direct relation graph (DRG) method for n-decane and n-heptane-air ignition delay time. The PFA method shows a significant improvement in the accuracy of mechanism reduction over the DRG method in a broad range of species number and physical properties. Finally, the PFA method is integrated with MTS and HMTS methods to further increase the computation efficiency of combustion with large mechanisms. The new algorithm is used to predict unsteady flame propagation of n-decane-air mixtures in a spherical chamber. Excellent computation accuracy and efficiency are demonstrated.

A Multi-Time Scale Method in Almost-Periodic Stability Problems

This paper considers the problem of stability of a class of almost-periodic ordinary differential equations. Necessary and sufficient stability conditions are derived by use of a method of “multiple time scales” suggested by similar techniques in perturbation theory. The conditions obtained involve a spectral analysis of a compact operator determined by the original system coefficients, and in principle are computable to any desired degree of accuracy by finite dimensional methods.

Stability, Instability, Invertibility and Causality

1. Introduction. A number of interesting stability criteria for feedback systems have recently appeared in the control theory literature. The procedures used in proving these criteria can roughly be divided into three classes; the first based on Popov-like methods, the second using Lyapunov theory with Lyapunov functions derived from spectral factorizations or Riccati-type algebraic matrix equations, and the third treating the stability problem from a functional analysis point of view. Each of these methods has relative merits, e.g., the Lyapunov methods seem to be the only ones which allow us to obtain an estimate of the domain of attraction in the case of nonglobal stability. However, the method based on functional analysis appears to be the more satisfactory one, in view of the essential simplicity, of the intuitive nature of the results (loop gain less than one, passivity conditions), and of the fact that it unifies the various criteria (as, e.g., the circle criterion and the Popov criterion). It therefore deserves more investigation and exposition than it has thus far been given. A peculiarity of this method, as presently employed, is that most of the analysis and estimates have to be made on extended spaces which, although derived from normed spaces, are themselves not normed. This entails in general rather cumbersome mathematical manipulations. One however suspects the