Numerical Simulation and Parameter Estimation of Fractional-Order Dynamic Epidemic Model for COVID-19

The outbreak of COVID-19 in 2019 resulted in numerous infections and deaths. In order to better study the transmission of COVID-19, this article adopts an improved fractional-order SIR model. Firstly, the properties of the model are studied, including the feasible domain and bounded solutions of the system. Secondly, the stability of the system is discussed, among other things. Then, the GMMP method is introduced to obtain numerical solutions for the COVID-19 system and combined with the improved MH-NMSS-PSO parameter estimation method to fit the real data of Delhi, India from April 1, 2020 to June 30, 2020. The results show that the fitting effect is quite ideal. Finally, long-term predictions were made on the number of infections. We accurately estimate that the peak number of infections in Delhi, India, can reach around 2.1 million. This paper also compares the fitting performance of the integer-order COVID-19 model and the fractional-order COVID-19 model using the real data from Delhi. The results indicate that the fractional-order model with different orders, as we proposed, performs the best.

Modeling and dynamics analysis of the fractional-order Buck-Boost converter in continuous conduction mode

In this paper, the fractional-order mathematical model and the fractional-order state-space averaging model of the Buck-Boost converter in continuous conduction mode （CCM） are established based on the fractional calculus and the Adomian decomposition method. Some dynamical properties of the current-mode controlled fractional-order Buck- Boost converter are analysed. The simulation is accomplished by using SIMULINK. Numerical simulations are presented to verify the analytical results and we find that bifurcation points will be moved backward as α and β vary. At the same time, the simulation results show that the converter goes through different routes to chaos.

Fractional-order visco-plastic constitutive model for uniaxial ratcheting behaviors

This paper proposes a novel unified visco-plastic constitutive model for uniaxial ratcheting behaviors. The cyclic deformation of the material presents remarkable time-dependence and history memory phenomena. The fractional(fractional-order)derivative is an efficient tool for modeling these phenomena. Therefore, we develop a cyclic fractional-order unified visco-plastic(FVP) constitutive model. Specifically, within the framework of the cyclic elasto-plastic theory, the fractional derivative is used to describe the accumulated plastic strain rate and nonlinear kinematic hardening rule based on the Ohno-Abdel-Karim model. Moreover, a new radial return method for the back stress is developed to describe the unclosed hysteresis loops of the stress-strain properly.The capacity of the FVP model used to predict the cyclic deformation of the SS304 stainless steel is verified through a comparison with the corresponding experimental data found in the literature(KANG, G. Z., KAN, Q. H., ZHANG, J., and SUN, Y. F. Timedependent ratcheting experiments of SS304 stainless steel. International Journal of Plasticity, 22(5), 858–894(2006)). The FVP model is shown to be successful in predicting the rate-dependent ratcheting behaviors of the SS304 stainless steel.

Modified Grey Model Predictor Design Using Optimal Fractional-order Accumulation Calculus

The major advantage of grey system theory is that both incomplete information and unclear problems can be processed precisely. Considering that the modeling of grey model(GM) depends on the preprocessing of the original data,the fractional-order accumulation calculus could be used to do preprocessing. In this paper, the residual sequence represented by Fourier series is used to ameliorate performance of the fractionalorder accumulation GM(1,1) and improve the accuracy of predictor. The state space model of optimally modified GM(1,1)predictor is given and genetic algorithm(GA) is used to find the smallest relative error during the modeling step. Furthermore,the fractional form of continuous GM(1,1) is given to enlarge the content of prediction model. The simulation results illustrated that the fractional-order calculus could be used to depict the GM precisely with more degrees of freedom. Meanwhile, the ranges of the parameters and model application could be enlarged with better performance. The method of modified GM predictor using optimal fractional-order accumulation calculus is expected to be widely used in data processing, model theory, prediction control and related fields.

Fractional-order Maxwell model of seabed mud and its effect on surface-wave damping

A fractional-order Maxwell model is used to describe the viscoelastic seabed mud. The experimental data of the real mud well fit the results of the fractional-order Maxwell model that has fewer parameters than the traditional model. The model is then used to investigate the effect of the mud on the surface-wave damping. The damping rate of a linear monochromatic wave is obtained. The elastic resonance of the mud layer is observed, which leads to the peaks in the damping rate. The damping rate is a sum of the modal damping rates, which indicates the wave damping induced by the mud motion of particular modes. The analysis shows that near the resonance, the total damping rate is dominated by the damping rate of the corresponding mode.

Intelligent Networks for Chaotic Fractional-Order Nonlinear Financial Model

The purpose of this paper is to present a numerical approach based on the artificial neural networks(ANNs)for solving a novel fractional chaotic financial model that represents the effect of memory and chaos in the presented system.The method is constructed with the combination of the ANNs along with the Levenberg-Marquardt backpropagation(LMB),named the ANNs-LMB.This technique is tested for solving the novel problem for three cases of the fractional-order values and the obtained results are compared with the reference solution.Fifteen numbers neurons have been used to solve the fractional-order chaotic financial model.The selection of the data to solve the fractional-order chaotic financial model are selected as 75%for training,10%for testing,and 15%for certification.The results indicate that the presented approximate solutions fit exactly with the reference solution and the method is effective and precise.The obtained results are testified to reduce the mean square error(MSE)for solving the fractional model and verified through the various measures including correlation,MSE,regression histogram of the errors,and state transition(ST).

On a new fractional-order Logistic model with feedback control

In this paper,we formulate and analyze a new fractional-order Logistic model with feedback control,which is different from a recognized mathematical model proposed in our very recent work.Asymptotic stability of the proposed model and its numerical solutions are studied rigorously.By using the Lyapunov direct method for fractional dynamical systems and a suitable Lyapunov function,we show that a unique positive equilibrium point of the new model is asymptotically stable.As an important consequence of this,we obtain a new mathematical model in which the feedback control variables only change the position of the unique positive equilibrium point of the original model but retain its asymptotic stability.Furthermore,we construct unconditionally positive nonstandard finite difference(NSFD)schemes for the proposed model using the Mickens’methodology.It is worth noting that the constructed NSFD schemes not only preserve the positivity but also provide reliable numerical solutions that correctly reflect the dynamics of the new fractional-order model.Finally,we report some numerical examples to support and illustrate the theoretical results.The results indicate that there is a good agreement between the theoretical results and numerical ones.

部分饱和岩石的黏弹性Chapman-Kelvin模型的正演模拟方法研究

在讨论地震波传播理论时,大部分情况下是把地震波看作弹性波.事实上,地下介质是非完全弹性介质,这使得传统的均匀、完全弹性介质理论受到了严重挑战.地震波在裂隙岩石中传播受裂隙系统和流体含量的影响较大,以往许多关于频变各向异性的理论大多局限于单相流体假设,但是几乎所有储层中通常被一种以上的流体部分饱和或完全饱和.本文在Chapman理论基础上,提出了基于部分饱和的黏弹性Chapman-Kelvin模型的正演模拟新方法,以提高对部分饱和岩石的黏弹性地震波频变各向异性的认识.该方法基于Chapman和Kelvin理论模型,计算了双相不混溶流体饱和裂隙岩石中的黏弹性波频变各向异性弹性系数的表达式,提出包含喷射流和斑块效应的统一地震波传播的黏弹性Chapman-Kelvin新模型.通过对新模型进行数值试验,讨论了无裂缝和存在裂缝两种情况下,含裂隙储层部分饱和岩石中耦合的喷射流和斑块效应对黏弹性介质频变各向异性的影响.试验结果反映出黏弹性介质下地震波频变各向异性的变化规律,验证了本文提出的新方法和新模型正确.本文将黏弹性介质各向异性与裂缝中的流体流动参数相联系,有利于提高对含裂缝储层部分饱和岩石的黏弹性波频变地震各向异性的认识,以及地震学与油藏工程的结合程度.

基于Kelvin模型的圆形巷道位移反分析的唯一性

讨论了黏弹性岩体中圆形巷道位移反分析的唯一性问题。以Kelvin模型为研究对象,在圆形巷道位移解析解的基础上,利用参数可辨识条件,对岩体参数进行了位移反分析唯一性的研究。结果表明:无论测量出多少个测点的位移,都不能唯一地反分析出所有岩体参数,至少要已知2个参数,才有可能唯一地反分析出其余参数;随着已知参数的增多,各参数的可辨识性逐渐增强;竖向地应力与水平地应力是否相等,且二者作为已知条件时其大小是否为0对反分析结果有较为明显的影响;适当减少待反演参数的数量对提升反分析结果的唯一性有显著影响;所有参数中地应力具有最好的可辨识性,其次是弹性模量,泊松比及黏滞系数最差。

A modified Kelvin impact model for pounding simulation of base-isolated building with adjacent structures

Base isolation can effectively reduce the seismic forces on a superstructure, particularly in lowto medium-rise buildings. However, under strong near-fault ground motions, pounding may occur at the isolation level between the baseisolated building （BIB） and its surrounding retaining walls. To effectively investigate the behavior of the BIB pounding with adjacent structures, after assessing some commonly used impact models, a modified Kelvin impact model is proposed in this paper. Relevant parameters in the modified Kelvin model are theoretically derived and numerically verified through a simple pounding case. At the same time, inelasticity of the isolated superstructure is introduced in order to accurately evaluate the potential damage to the superstructure caused by the pounding of the BIB with adjacent structures. The reliability of the modified Kelvin impact model is validated through numerical comparisons with other impact models. However, the difference between the numerical results from the various impact analytical models is not significant. Many numerical simulations of BIBs are conducted to investigate the influence of various design parameters and conditions on the peak inter-story drifts and floor accelerations during pounding. It is shown that pounding can substantially increase floor accelerations, especially at the ground floor where impacts occur. Higher modes of vibration are excited during poundings, increasing the inter-story drifts instead of keeping a nearly rigid-body motion of the superstructure. Furthermore, higher ductility demands can be imposed on lower floors of the superstructure. Moreover, impact stiffness seems to play a significant role in the acceleration response at the isolation level and the inter-story drifts of lower floors of the superstructure. Finally, the numerical results show that excessive flexibility of the isolation system used to minimize the floor accelerations may cause the BIB to be more susceptible to pounding under a limited seismic gap.

基于黏弹性Chapman-Kelvin模型的裂缝储层频变AVAZ反演研究

为了在地震资料和裂缝储层特征之间建立联系,对裂缝储层采用了等效介质模型.而传统的等效介质模型未充分考虑非完全弹性介质理论和基于频变各向异性理论的双相或多相流体假设,也不能对实际裂缝储层中的地震波频散和衰减现象提供准确合理的解释,并且储层参数的反演研究对裂缝储层的定性预测和定量描述举足轻重.为此,本文首先根据所提出的黏弹性Chapman-Kelvin动态等效介质模型,该模型考虑了耦合的双相流体假设、黏弹性理论、喷射流以及斑块效应,并在此基础上分析了裂缝储层参数(主要为裂缝密度、裂缝长度、孔隙度和含水饱和度)对地震波频变特征的影响.然后基于黏弹性Chapman-Kelvin模型与Schoenberg和Protazio概括的Zoeppritz方程所计算出的频变反射系数,分析了反射PP波和PS波的频变AVAZ(Amplitude Versus Angle and Azimuth)特性和PP波频变反射系数与裂缝储层参数的关系.同时考虑到发生地震频散时,反射系数和频率产生关系,构建了在角度、方位和时间域内的新型正演方程.最后,基于PP波频变反射系数对裂缝密度、裂缝长度、孔隙度和含水饱和度的变化有较好的敏感性特点,进行了两种反演方法研究:其一是基于贝叶斯理论直接反演方法,其中以对数绝对范数作为似然函数和高斯分布,即L2范数度量作为先验约束;其二是基于频变反射系数的MCMC(Markov Chain Monte Carlo)随机反演方法.通过频变AVAZ进行裂缝储层参数的反演研究,结果表明,MCMC随机反演方法在缺失先验的储层参数信息时,反演结果的不确定性较强.当存在有效且足够的先验信息时,反演结果的可靠性进一步提升.而基于贝叶斯理论直接反演方法,则显示出基于频变AVAZ来区别大尺度裂缝和微尺度裂隙的潜力.

基于Kelvin-Voigt模型的黏弹性岩体位移反分析唯一性研究

在岩土工程中,准确的岩体力学参数对于工程项目的建设和维护极为重要,相较于现场原位试验、实验室试验、数值模拟等获取参数的方法,岩体位移反分析方法具有方便操作、条件获取难度小、数据准确等特点,而反分析结果的唯一性是位移反分析能否成功的重要影响因素。为了研究位移反分析结果的唯一性问题,使用Laplace变换和蠕变柔量的方法,推导了在圆形巷道中符合Kelvin-Voigt模型的黏弹性岩体的位移解析解,并运用参数可辨识准则研究了已知参数对黏弹性岩体反分析唯一性的影响。研究结果表明,无论在任何条件下,都无法唯一地反演出所有6个参数,在至少有2个参数是已知的情况下,才可能唯一地反演出其他参数;地应力分量被唯一反演的概率最大,弹性模量、泊松比和黏滞系数根据控制条件的不同,其可辨识性存在差异,并随着已知参数数量的增加,参数可辨识性也随之增强;地应力是否为0对反演结果的唯一性有着显著的影响。工程算例表明,研究结果对于岩体参数的反分析计算具有重要的指导意义。

Establishment of a Fractional Order COVID-19 Model and Its Feasibility Analysis

This paper investigates an improved SIR model for COVID-19 based on the Caputo fractional derivative. Firstly, the properties of the model are studied, including the feasible domain and bounded solutions of the system. Secondly, the stability of the system is discussed, among other things. Then, the GMMP method is introduced to obtain numerical solutions for the COVID-19 system. Numerical simulations were conducted using MATLAB, and the results indicate that our model is valuable for studying virus transmission.

Kelvin尾迹SAR图像仿真研究

一方面利用Kelvin尾迹水动力模型,通过对同一吃水深度(4 m)、不同船速(1-20 m/s)下的Kelvin尾迹波的仿真,提出了此模型适用的船只航行速度范围以及随着船速的变化,Kelvin尾迹波的变化情况。另一方面结合海表面波模型和雷达后向散射模型,比较同一船速、不同吃水深度(4 m、6 m和8 m)下的SAR仿真图像。结果表明,仅考虑Kelvin尾迹水动力模型时,当吃水深度不变时,在一定速度范围内,随船速的增大,Kelvin尾迹的横波波长逐渐增大,尖波变的逐渐清晰,超出此范围,Kelvin尾迹水动力模型就不再适用了;当同时考虑3种模型时,不同吃水深度下,在一定速度范围内,同一船速所仿真出的Kelvin尾迹SAR图像是不一样的,随着吃水深度的加深,Kelvin尾迹横波也随之清晰,超出此速度范围则无法看到横波。

修正广义Kelvin蠕变固结模型研究

软土变形的时效特性产生于蠕变和固结,但蠕变和固结基于不同的变形机制,因此研究蠕变固结耦合效应的计算模型对于软土工程来说是十分重要的。采用将非线性本构模型和Biot固结模型耦合的方法,建立修正的广义Kelvin蠕变固结模型,即非线性弹-黏弹固结模型。首先,根据软土室内试验成果分析变形机制和应变随应力和时间的变化规律,找到函数拟合关系;其次,将Duncan模型和修正的广义Kelvin模型相结合,建立描述上述函数拟合关系的本构模型。然后,将该本构模型代入Biot固结理论的平衡方程,并提出基于非线性变形特征的蠕变固结模型。该模型共有14个模型参数,包括8个Duncan参数,均可通过常规三轴试验确定。通过一个实际工程的计算,完成该模型的计算流程,并对计算结果进行分析和比较,验证所建模型的有效性,探讨软土蠕变对地基中孔隙水压力及应力水平的影响,在此基础上获得一些有益的结论。研究结果表明,采用修正的广义Kelvin蠕变固结模型进行软土变形特性分析是一种合理的方法,具有模型机制明确、参数确定方法简单的优点。另外,虽然蠕变和固结都会给工程带来工后沉降问题,但两者对工程的影响是不一样的,改善地基的排水条件可弱化蠕变效应并提高工程的安全度。

地震时桥梁碰撞分析的等效Kelvin撞击模型

建立地震时桥梁碰撞分析的Kelvin撞击模型的参数确定方法。基于Hertz接触理论,考虑波动效应,按照最大撞击力与最大撞击变形的比值,确定Kelvin撞击模型的碰撞刚度;数值分析了影响Kelvin撞击模型阻尼系数取值的邻梁碰撞恢复系数。结果表明:Kelvin撞击模型的碰撞刚度随Hertz接触刚度、撞击速度以及短梁与长梁的长度比的增大而增大;波动效应对碰撞刚度的影响明显,不能忽略;邻梁碰撞恢复系数随撞击速度增大而减小,随短梁与长梁的长度比减小而减小;确定Kelvin撞击模型的碰撞刚度和邻梁碰撞恢复系数对城市桥梁的合理取值范围应分别为3×105kN/m―6×105kN/m和0.7―0.95。

不同条件下海面Kelvin尾迹的红外仿真

针对舰船在海上航行时形成的尾迹包含多类信息,利用Kelvin尾迹模型得到舰船速度为5m/s、10m/s和15m/s的尾迹最大波高分别为0.5m、1.5m和2.5m。通过海浪PiersonMoscowitz谱模型描述海面风速分别为5m/s、8m/s和10m/s下的复杂海面背景模型。基于海面背景和尾迹区域的几何差异,建立了Kelvin尾迹的红外发射模型,得到不同海面风速、不同舰船航速和不同探测天顶角下的红外特征仿真图像。仿真结果表明,红外图像的灰度极大值位于尾迹波峰处,海面风速由5m/s增加到10m/s时,尾迹区域与海面背景平均灰度差值由100逐渐减小,直至无差异。相同海面风速下,舰船速度由5m/s增加到15m/s,尾迹波长由10m增加到40m,红外特征明显。改变探测器天顶角分别为0°和30°时,红外图像差异较小,当天顶角增大到60°时,尾迹区域灰度值接近海面背景,差值小于30,尾迹的辨识难度增加。

Kelvin模型阻尼层合板的振动分析

基于Kelvin模型粘弹性材料本构关系导出了阻尼层合板的动力学微分方程组,给出了四边简支阻尼层合板的固有频率和损耗因子的解析解。与文献结果比较表明,将Kelvin模型应用于粘弹结构的动力特性问题求解,计算模型简便,且计算结果比常复数模型更为精确。分析了阻尼层参数变化对结构振动特性的影响。分析结果表明:增加阻尼层厚度,可以有效提高结构损耗因子;增加阻尼层材料的剪切模量,结构损耗因子增大到一定值后又逐渐减小,在减振设计中阻尼层的模量存在最佳值。

机织建筑膜材料的广义Kelvin-Voigt蠕变模型

对平纹组织为基布的PVC涂层膜材料进行了经纬两方向各自5种不同应力条件下的单轴向拉伸蠕变试验,给出了广义Kelvin-Voigt模型的具体蠕变方程,并利用广义三参数、五参数与七参数Kelvin-Voigt的蠕变方程对实验数据进行了分析.结果显示:随着模型参数个数的增加,预测效果也呈现出递增趋势;当模型参数为7个时,各应力拟合的确定系数均在99.9%以上,表明七参数广义Kelvin-Voigt蠕变模型已可以很好地预测膜材料的蠕变特性.

确定Kelvin模型粘弹性材料参数的一种实验方法

从Kelvin模型和Euler-bernoulli梁理论出发,利用弹性-粘弹性相应原理和弹性材料动力学理论得出粘弹性梁的动力学方程。利用中心差分法分析了粘弹性材料悬臂梁的动态函数。根据振动特性,将材料参数与梁应变响应周期及振幅衰减相联系;利用这种联系,提出了测定Kelvin模型粘弹性材料参数的一种简单实验方法;分别取不同长度的橡胶悬臂梁进行了相应的实验,测出了材料的剪切模量和粘性系数;分析实验结果,得出一些有益的结论。