混沌时间序列预测的局域法在边坡变形分析中的应用

边坡作为一个复杂系统,其本身的各种参量是不确定的和随机的,在其演化过程中,表现出复杂的非线性行为,发生一系列的混沌现象。本文运用现代混沌理论,对边坡变形的预测问题进行探索性研究,把混沌时间序列理论引入到边坡工程研究中,对该理论的建立及预测方法进行系统地讨论,为该领域的研究提供完整的技术方法。通过对新滩滑坡的研究结果表明,混沌时间序列方法对混沌序列的预测较线性时间序列具有较高的精度。     

面向工程全局优化的混沌优化算法研究进展Research advances of chaos optimization algorithms for engineering global optimization

近年来,基于混沌的初值敏感性、伪随机性、遍历性以及自相似分形等非线性动力学特性所发展的混沌优化方法,是一种有潜力的工程全局优化新工具,已广泛应用于科学与工程技术的各学科领域。根据混沌优化方法的发展历程,以算法基本思想和工程应用研究状况为重点,评述了混沌神经网络优化方法、第一类混合混沌优化算法（基于混沌搜索）、第二类混合混沌优化算法（混沌序列代替随机序列）以及混沌分形优化四种主要混沌优化算法。混沌映射最早被引入神经网络,发展了混沌神经网络优化方法,可解决复杂的组合优化等全局优化问题。遗传算法及粒子群等启发式随机算法虽具全局搜索能力,但易出现早熟并陷入局部最优。然后,出现了混沌搜索的概念,研究者将其嵌入启发式算法建立了第一类混合混沌优化算法,可有效克服原启发式算法早熟收敛的缺点。随后,利用混沌映射产生的混沌序列代替启发式算法中的随机参数形成了第二类混合混沌优化算法。混合混沌优化算法有益于实现快速全局收敛和提高计算精度。最后,利用混沌分形特性,从分形理论出发提出一类新颖的混沌分形优化算法,可搜索到优化问题的所有全局最优解。此外,对混沌优化算法研究的几个发展方向进行了展望,诸如加强混沌优化算法的参数设计、处理大规模优化、多目标优化问题以及使用代理模型等。     

求解非线性方程组的混沌分形方法

混沌分形是动力系统普遍出现的一种现象,牛顿-拉夫森NR(Newton-Raphson)方法是重要的一维及多维迭代技术,其迭代本身对初始点非常敏感,该敏感区是牛顿-拉夫森法所构成的非线性离散动力系统Julia集,在Julia集中迭代函数会呈现出混沌分形现象,提出了一种寻找牛顿-拉夫森函数的Julia点的求解方法,利用非线性离散动力系统在其Julia集出现混沌分形现象的特点,提出了一种基于牛顿-拉夫森法的非线性方程组求解的新方法,计算实例表明了该方法的有效性和正确性.     

面向工程全局优化的混沌优化算法研究进展

近年来,基于混沌的初值敏感性、伪随机性、遍历性以及自相似分形等非线性动力学特性所发展的混沌优化方法,是一种有潜力的工程全局优化新工具,已广泛应用于科学与工程技术的各学科领域。根据混沌优化方法的发展历程,以算法基本思想和工程应用研究状况为重点,评述了混沌神经网络优化方法、第一类混合混沌优化算法(基于混沌搜索)、第二类混合混沌优化算法(混沌序列代替随机序列)以及混沌分形优化四种主要混沌优化算法。混沌映射最早被引入神经网络,发展了混沌神经网络优化方法,可解决复杂的组合优化等全局优化问题。遗传算法及粒子群等启发式随机算法虽具全局搜索能力,但易出现早熟并陷入局部最优。然后,出现了混沌搜索的概念,研究者将其嵌入启发式算法建立了第一类混合混沌优化算法,可有效克服原启发式算法早熟收敛的缺点。随后,利用混沌映射产生的混沌序列代替启发式算法中的随机参数形成了第二类混合混沌优化算法。混合混沌优化算法有益于实现快速全局收敛和提高计算精度。最后,利用混沌分形特性,从分形理论出发提出一类新颖的混沌分形优化算法,可搜索到优化问题的所有全局最优解。此外,对混沌优化算法研究的几个发展方向进行了展望,诸如加强混沌优化算法的参数设计、处理大规模优化、多目标优化问题以及使用代理模型等。     

单自由度体系地震动力响应的混沌特性分析

引入非线性动力学理论和混沌时间序列分析方法考察地震动作用下单自由度体系动力响应的混沌特性。输入典型近断层地震动记录,定量计算了代表性周期的单自由度弹性和非弹性体系加速度响应时程的非线性特性参数。计算表明,这些加速度响应的关联维数为分数维,最大Lyapunov指数大于0;地震动激励下单自由度体系的地震动力响应具有混沌特性,不是完全的随机信号,为理解结构地震动力响应的不规则性与复杂性提供了新思路和新视角。     

基于深部监测的滑坡动态特征分析

通过引力作用下理想气体流体力学方程组的无量纲化, 以空间尺度因子代 换时间参数, 根据量纲理论\Pi定理, 在方程中用与尺度因子对应的统计量物理 为度量单位, 从理论上推导出流体力学微分方程组的分离变量形式, 获得一组具有 分形结构特征一阶微分方程组. 引力作用下理想气体统计特征参量相对于空间尺度因子的一 般函数形式具有广义分形的结构特征, 这个结果表明局域性流体力学微分方程能够作为统计分 形结构的动力学基础.     

基于混沌神经网络理论的城市地面沉降量预测模型

通过分析城市地面沉降量时间序列的非线性动力学系统,认为该时间序列具有混沌特性。在此基础上,通过相空间重构的方法建立了用于城市地面沉降量预测的混沌神经网络模型;并利用此模型对高桥地面沉降量进行了预测,并和实际监测沉降量进行了比较,最大绝对预测误差为1. 7,预测的平均误差为0. 0833,研究结果表明,应用混沌神经网络模型进行城市沉降预测是可行、精确的。     

Al-Cu合金Portevin-Le Chatelier效应的多重分形分析

区别于传统的非线性动力学的吸引子结构分析,本文采用多重分形分析的研究方法讨论不同加载应变率条件下Al-Cu合金Portevin-Le Chatelier效应的多重分形特征,以揭示控制不同带型的内在动力机制。结果表明不管是哪种带型,都在一定尺度上存在着多重分形结构;同时发现B带的应力信号的均匀性高于A、C带,这反映了随着加载应变率的降低,动力系统由自组织临界性经拟周期状态过渡到混沌状态。     

概率约束评估的功能度量法的混沌控制

功能度量法是基于可靠度的结构优化设计中评估概率约束的一种方法,其改进均值(AMV)迭代格式具有简洁、高效的优点,但对一些非线性功能函数搜索最小功能目标点时可能陷入周期振荡或混沌解,本文利用混沌反馈控制的稳定转换法对功能度量法的AMV迭代格式实施收敛控制.首先展示一些功能函数应用功能度量法AMV格式迭代计算产生了周期解和混沌解现象,并对迭代算法进行了混沌动力学分析.然后利用稳定转换法对功能度量法迭代失败的参数区间进行混沌控制,使嵌入周期和混沌轨道的不稳定不动点稳定化,获得了稳定收敛解,实现了迭代解的周期振荡、分岔和混沌控制.     

基于功能度量法的概率优化设计的收敛控制

概率结构优化设计(PSDO)中概率约束的评定可以采用最近提出的、被认为更高效、稳定的功能度量法(PMA). 改进均值(AMV)迭代格式经常在PMA中使用,但它对一些非线性功能函数或非正态随机变量,搜索最小功能目标点时可能陷入周期振荡或混沌解,从而使PSDO的两层次算法或序列近似规划算法优化计算失败. 利用混沌反馈控制的稳定转换法对功能度量法的AMV迭代格式实施了收敛控制,使嵌入周期和混沌轨道的不稳定不动点稳定化,获得稳定收敛解,从而使概率约束的评定能正常进行;再由两层次算法或序列近似规划算法进行结构优化设计. 算例结果表明了稳定转换法实施收敛控制的有效性,以及序列近似规划算法相对高效的优点.      

Characterizing complex flows using adaptive sparse dynamic mode decomposition with error approximation

Dynamic mode decomposition (DMD) is a powerful data-driven method for analyzing the coherent structures of dynamical systems. This work develops an adaptive sparse DMD with error approximation for the sparse reconstruction of complex flow fields. First, we propose a new sparse DMD model by redefining the penalty function, where adaptive weights are assigned to penalize different DMD modes. With the adaptive weights, the sparse DMD model is more capable of extracting important DMD modes and discarding unimportant ones. Second, we develop a novel error prediction model for the proposed sparse DMD. The key idea is to construct a multiple regression model between the sparse model and its error by employing the partial least squares regression. The error of the sparse DMD model can be reduced by integrating the error prediction model. Finally, we assess the proposed method by means of test cases, including a nonlinear parameterized function, a cylinder bundle flow, and the transient state of square cylinder flow. The results show that the proposed method can be used to capture the dominant modes and substantially increase the accuracy of flow reconstruction.     

Entropy measures for biological signal analyses

Entropies are among the most popular and promising complexity measures for biological signal analyses. Various types of entropy measures exist, including Shannon entropy, Kolmogorov entropy, approximate entropy (ApEn), sample entropy (SampEn), multiscale entropy (MSE), and so on. A fundamental question is which entropy should be chosen for a specific biological application. To solve this issue, we focus on scaling laws of different entropy measures and introduce an ensemble forecasting framework to find the connections among them. One critical component of the ensemble forecasting framework is the scale-dependent Lyapunov exponent (SDLE), whose scaling behavior is found to be the richest among all the entropy measures. In fact, SDLE contains all the essential information of other entropy measures, and can act as a unifying multiscale complexity measure. Furthermore, SDLE has a unique scale separation property to aptly deal with nonstationarity and characterize high-dimensional and intermittent chaos. Therefore, SDLE can often be the first choice for exploratory studies in biology. The effectiveness of SDLE and the ensemble forecasting framework is illustrated by considering epileptic seizure detection from EEG.     

有限元分析中的最优离散化

系统的最优离散化具有很大的实用价值和理论意义。采用这种方法可以用最少的自由度得到最精确的数值解。本文给出了两种有代表性的方法,并收集了一些应用实例,其中包括笔者和刘永成、朱建中在Zienkiewicz指导下所作的一些工作。这种方法作为一种重分析方法不仅可用于线性问题,而且也可以用于具有表面非线性的弹性接触问题。      

A geometrically exact approach to the overall dynamics of elastic rotating blades—part 2: flapping nonlinear normal modes

The geometrically exact equations of motion about the prestressed state discussed in part 1 (i.e., the nonlinear equilibrium under centrifugal forces) are expanded in the Taylor series of the incremental displacements and rotations to obtain the third-order perturbed form. The expanded form is amenable to a perturbation treatment to unfold the nonlinear features of free undamped flapping dynamics. The method of multiple scales is thus applied directly to the partial-differential equations of motion to construct the backbone curves of the flapping modes and their nonlinear approximations when they are away from internal resonances with other modes. The effective nonlinearity coefficients of the lowest three flapping modes of elastic isotropic blades are investigated when the angular speed is changed from low- to high-speed regimes. The novelty of the current findings is in the fact that the nonlinearity of the flapping modes is shown to depend critically on the angular speed since it can switch from hardening to softening and vice versa at certain speeds. The asymptotic results are compared with previous literature results. Moreover, 2:1 internal resonances between flapping and axial modes are exhibited as singularities of the effective nonlinearity coefficients. These nonlinear interactions can entail fundamental changes in the blade local and global dynamics.     

Comparison of an Adaptive Wavelet Method and Nonlinearly Filtered Pseudospectral Methods for Two-Dimensional Turbulence

An adaptive wavelet method for solving the two-dimensional Navier–Stokes equations is compared with nonlinear Fourier filtering and nonlinear wavelet filtering of the pseudospectral method at each time step. The methods are each applied to a highly nonlinear flow typical of two-dimensional turbulence, the merger of two positive vortices pushed together by a weaker negative vortex, and the results are compared with a reference classical pseudospectral method. Nonlinear Fourier filtering uses 1.7 times fewer active modes than the reference simulation at the time of merger (when the flow is most complicated) and retains the overall dynamics and structure of the flow. However, it induces spurious oscillations in the background. Nonlinear wavelet filtering simulation uses 9.2 times fewer modes than the reference simulation at the time of merger, and reduces the errors in the solution. The adaptive wavelet simulation replicates precisely the dynamics and spatial structure of the reference simulation while retaining the high compression rate of the nonlinear wavelet filtering simulation. In addition we observe that the number of active wavelet modes remains quasi-constant during the whole merging process, independent of the strength of the vorticity gradients. On the contrary, the number of active Fourier modes is multiplied by 5 when the vorticity gradients are strongest. The increased accuracy of the adaptive wavelet simulation is due to the security zone added around the active coefficients and to the compression of the nonlinear term of the Navier–Stokes equations in the wavelet basis. These results suggest that nonlinear Fourier filtering of a classical pseudospectral method cannot produce significant improvement, but that the adaptive wavelet method combines a consistently high compression rate with high accuracy. Received 22 April 1997 and accepted 11 August 1997     

增维精细积分法的适用范围研究

对线性定常结构动力系统提出的增维精细积分法,能够将非齐次动力方程转化为齐次动力方程,不用对状态矩阵求逆就能方便高效地求解出结构的动力响应。本文在仔细分析增维精细积分法性质的基础上,提出了其适用条件,进一步拓宽了其应用范围,并给出了将荷载项展开成傅里叶级数时,相应增维精细积分法的表达式。同时,在一个时间步长内,通过对非齐次项作线性化假设,成功地将增维精细积分法应用到了非线性动力分析领域。本文方法计算格式统一,易于编程,具有很高的计算效率。数值算例证明了本文方法的有效性。     

弹性矩形板非线性振动的多模态解

本文将非线性振动矩形板的振型函数展开为梁函数和Ｂ样条函的乘积形式。由哈密顿原理导出了系统的运动微分方程，得到了以多个线性模态表示的大振幅振动板的位移和非线性频率比。计算结果表明：该法具有很高的计算速度和精度。     

Nonlinear dynamical analysis of noisy time series

Empirical time series in the life sciences are often nonstationary and have small signal-to-noise ratios, making it difficult to accurately detect and characterize dynamical structure. The usual response to high noise is averaging, but time domain averaging is inappropriate, especially when the dynamics are nonlinear. We review alternative delay-space averaging methods based on the topology and short-term predictability of nonlinear dynamics and illustrate their application using the TISEAN software (Hegger, Kantz & Schreiber, 1999). The methods were applied to a Lorenz series, which resembles the dynamics found by Kelly, Heathcote, Heath and Longstaff (2001) in series of decision times. The Lorenz series was corrupted with up to 80% additive Gaussian noise, a lower signal-to-noise ratio than has been used in any previous test of these methods, but consistent with Kelly et al.'s data. Prediction methods performed the best for detecting nonstationarity and nonlinear dynamics, and optimal predictability provided an objective criterion for setting the parameters required by the analyses. Local linear filtering methods performed best for characterization, producing informative plots that revealed the nature of the underlying dynamics. These results suggest that a methodology based on delay-space averaging and prediction could be useful with noisy empirical data series.     

Analytical treatment on magnetohydrodynamic (MHD) flow and heat transfer due to a stretching hollow cylinder

This paper studied on magnetohydrodynamics flow and heat transfer outside a stretching cylinder. Momentum and energy equations are reduced using similarity transformation and converted into a system of ordinary differential equations which are solved analytically by the homotopy analysis method. The effects of the parameters involved, namely the magnetic parameter (M), Prandtl number (Pr) and Reynolds number (Re) on the velocity and temperature fields are investigated. The obtained results are valid for the whole solutions' domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd.