Long range numerical simulation of short waves as nonlinear solitary waves

A new numerical method has been developed to propagate short wave equation pulses over indefinite distances and through regions of varying index of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation that propagates basis functions according to the wave equation. These basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation. The method can also be used to solve for harmonic waves in the high frequency (Eikonal) limit, including multiple arrivals. The solution involves discretizing the wave equation on a uniform Eulerian grid and adding a simple nonlinear “Confinement” term. This term does not change the amplitude (integrated through each point on the pulse surface) or the propagation velocity, or arrival time, and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain 2–3 grid cells in thickness indefinitely with no numerical spreading. With the method, only a simple discretized equation is solved each time step at each grid node. The method can be contrasted to Lagrangian Ray Tracing: it is an Eulerian based method that captures the waves directly on the computational grid, where the basic objects are codimension 1 surfaces (in the fine grid limit), defined on a regular grid, rather than collections of markers. In this way, the complex logic of current ray tracing methods, which involves allocation of markers to each surface and interpolation as the markers separate, is avoided.     

Modeling nonlinear waves in a numerical wave tank with localized meshless RBF method

This paper presents the numerical simulation of free surface waves in a 2D domain which represents a wave tank, using a localized approach of the meshless radial basis function (RBF) method. Instead of global collocation, the local approach breaks down the problem domain into subdomains, leading to a sparse global system matrix which is particularly advantageous in tackling the time consuming simulation process. Mixed Eulerian–Lagrangian approach is adopted for free surface tracking and fourth order Adams–Bashforth–Moulton scheme (ABM4) is used for time stepping. Both linear and nonlinear Stokes waves are simulated and compared to analytical solutions.     

Topology optimization for transient wave propagation problems in one dimension

Structures exhibiting band gap properties, i.e., having frequency ranges for which the structure attenuates propagating waves, have applications in damping of acoustic and elastic wave propagation and in optical communication. A topology optimization method for synthesis of such structures, employing a time domain formulation, is developed. The method is extended to synthesis of pulse converting structures with possible applications in optical communication.     

Instability in supercritical nonlinear wave equations: Theoretical results and symplectic integration

Nonlinear wave evolutions involve a dynamical balance between linear dispersive spreading of the waves and nonlinear self-interaction of the waves. In sub-critical settings, the dispersive spreading is stronger and therefore solutions are expected to exist globally in time. We show that in the supercritical case, the nonlinear self-interaction of the waves is much stronger. This leads to some sort of instability of the waves. The proofs are based on the construction of high frequency approximate solutions. Preliminary numerical simulations that support these theoretical results are also reported.     

Nonlinear localization strategies for domain decomposition methods: Application to post-buckling analyses

In this paper, we explore the capabilities of some nonlinear strategies based on domain decomposition for nonlinear analyses, and more particularly for post-buckling analyses of large slender structures. After having recalled the classical Newton-Krylov-Schur methods, chosen here to serve as a reference, we propose two versions specifically developed to treat nonlinear phenomena at the most relevant scale through nonlinear localizations per substructure. All these different strategies lead to solving similar condensed problems on which we apply classical Domain Decomposition Methods. Performances are discussed and comparative results in terms of convergence are presented in the case of beam frames with large rotations and local buckling.     

Snake solitons in the cubic–quintic Ginzburg–Landau equation

Comprehensive numerical simulations of pulse solutions of the cubic–quintic Ginzburg–Landau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons [C.I. Christov, M.G. Velarde, Dissipative solitons, Physica D 86 (1995) 323] are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied.     

Wave propagation in delaminated beam

The wave propagation model investigated herein is based on the known fact that material discontinuities affect the propagation of elastic waves in solids. The change in certain material characteristics, such as a local change in stiffness or inertia caused by a crack or the presence of material damage, will affect the propagation of transmitted elastic waves and will modify the received signal. Wave frequencies associated with the highest detection sensitivity depend, among other things, on the type of structure, the type of material, and the type of damage. This paper presents a method of wave propagation, which can be further used to detect small delaminations in beam-like structures. The considered beam is modelled by spectral finite elements.     

The eXogenous Kalman Filter (XKF)

It is well known that the time-varying Kalman Filter (KF) is globally exponentially stable and optimal in the sense of minimum variance under some conditions. However, nonlinear approximations such as the extended KF linearises the system about the estimated state trajectories, leading in general to loss of both global stability and optimality. Nonlinear observers tend to have strong, often global, stability properties. They are, however, often designed without optimality objectives considering the presence of unknown measurement errors and process disturbances. We study the cascade of a global nonlinear observer with the linearised KF, where the estimate from the nonlinear observer is an exogenous signal only used for generating a linearised model to the KF. It is shown that the two-stage nonlinear estimator inherits the global stability property of the nonlinear observer, and simulations indicate that local optimality properties similar to a perfectly linearised KF can be achieved. This two-stage estimator is called an eXogeneous KF (XKF).     

基于非线性薛定谔方程的高分三号图像内波参数反演

高分三号(GF-3)SAR遥感卫星的发射为内波的深入研究提供了丰富图像资料,利用其高空间分辨率和大幅宽成像的优势,能够对南中国海东沙岛附近的内波开展振幅和波速的高精度反演。采用高阶完全非线性薛定谔方程描述内波,建立反演振幅和波速的模型。收集2017年5月至8月南海东沙岛附近的GF-3遥感图像,探究南海陆坡处内波的振幅和传播速度演变。理论模型计算出的结果与2013年该研究区域的实测数据对比,发现反演振幅大小基本与实测相近。内波自东向西传播过程中,水深变浅,非线性作用增强,频散作用减弱,振幅和传播速度都不断减小。这为定量研究内波的能量输送、耗散以及未来的预报工作提供依据。     

一类非线性离散系统模糊控制器的分析和设计

针对一类非线性离散不确定系统,在系统状态不可测的情况下,以T-S模型描述不同状态空间的局部动态区域,并通过中心平均反模糊化、乘积推理、单点模糊化方法得到全局模糊系统模型.基于李亚普诺夫理论和线性矩阵不等式,设计了一种基于观测器的鲁棒控制器,并对离散状态下的此类系统进行了稳定分析.最后通过M ATLAB仿真,证明了该方法的有效性.     

基于量子遗传算法的非线性无约束优化方法

量子遗传算法(QGA)是量子计算和遗传算法相结合的产物,量子遗传算法将量子比特和量子旋转门表示引入到遗传算法中,具有比遗传算法更好的搜索效率和收敛性。非线性无约束优化是典型的工程应用问题,而复杂非线性函数的优化结果往往不能令人满意,如陷入局部最优等。利用量子遗传算法强大的搜索能力,可以很好的解决复杂非线性函数的无约束优化问题,实验表明量子遗传算法在该类问题中的有效性和可行性。     

Nonlinear optimal control as quantum mechanical eigenvalue problems

A novel approach for approximating the nonlinear optimal feedback control of a system with a terminal cost is proposed. To lessen the difficulty due to nonlinearity, we try to treat the system in a framework of linear theories. For this, we assume a quantum mechanical linear wave associated with the system. Since the control system is constrained by state equations, we handle the system according to quantum mechanics of constrained dynamics. A Hamiltonian is represented as a linear operator acting on a function that describes behavior of waves. Subsequently, nonlinear feedback is calculated without any time integration in the backward direction. Using eigenvalues and eigenfunctions of the linear Hamiltonian operator, an optimal feedback law is given as a combination of analytic functions of time and state variables. We take as an example a system described by two scalar variables for state and control input. Simulation studies on the system by the eigenvalue analysis show that the proposed method reduces calculation time to nearly a tenth that of a numerical calculation of a Hamilton-Jacobi equation by a finite difference method.     

Stability and performance of a variable gain controller with application to a dvd storage drive

This paper deals with the control design for optical storage drives. A nonlinear design is suggested to overcome the tradeoff between disturbance rejection, in the sense of tracking error reduction during low-frequency shock and vibration, and playability, in the sense of sensor noise tracking during high-frequency disc surface defects. Based on a variable gain control design, it is shown that improvements in disturbance rejection can be obtained without unnecessarily affecting playability. With this design, additional control is applied beyond a pre-defined error level. Accordingly, it is shown that large vibrations induce additional control effort giving improved disturbance rejection while, at the same time, small vibrations hardly induce any additional control effort thus leaving the playability properties unaffected. The variable gain strategy is studied regarding closed-loop stability and regarding performance. Closed-loop stability is derived on the basis of absolute stability theory. Performance is quantified using a measure derived from the linear sensitivity function. Based on the amplitude of the linear sensitivity function, the maximum absolute values of the periodic nonlinear response subjected to harmonic excitation are computed within a frequency range of interest. Experiments are performed on an industrial setup to validate the numerical results, and to illustrate the applicability of the nonlinear control design in a dvd application.     

Search strategies for shape regularized active contour

Nonlinear shape models have been shown to improve the robustness and flexibility of contour-based object segmentation when there are appearance ambiguities between the object and the background. In this paper, we focus on a new search strategy for the shape regularized active contour (ShRAC) model, which adopts existing nonlinear shape models to segment objects that are similar to a set of training shapes. The search for optimal contour is performed by a coarse-to-fine algorithm that iterates between combinatorial search and gradient-based local optimization. First, multi-solution dynamic programming (MSDP) is used to generate initial candidates by minimizing only the image energy. In the second step, a combination of image energy and shape energy is minimized starting from these initial candidates using a local optimization method and the best one is selected. To generate diverse initial candidates while reducing invalid shapes, we apply two pruning methods to the search space of MSDP. Our search strategy combines the advantages of global combinatorial search and local optimization, and has shown excellent robustness to local minima caused by distracting suboptimal solutions. Experimental results on segmentation of different anatomical structures using ShRAC, as well as preliminary results on human silhouette segmentation are provided.     

A dynamic fuzzy model for a drum-boiler-turbine system

A nonlinear dynamic fuzzy model for natural circulation drum-boiler-turbine is presented. The model is derived from Åström-Bell nonlinear dynamic system and describes the complicated dynamics of the physical plant. It is shown that the dynamic fuzzy model gives in some appropriate sense accurate global nonlinear prediction and at the same time that its local models are close approximations to the local linearizations of the nonlinear dynamic system. This closeness is illustrated by simulation in various conditions.     

Local feedback stabilization and bifurcation control, I. Hopf bifurcation

Local bifurcation control problems are defined and employed in the study of the local feedback stabilization problem for nonlinear systems in critical cases. Sufficient conditions are obtained for the local stabilizability of general nonlinear systems whose linearizations have a pair of simple, nonzero imaginary eigenvalues. The conditions show, in particular, that generically these nonlinear critical systems can be stabilized locally, even if the critical modes are uncontrollable. The analysis also yields a direct method for computing stabilizing feedback controls. Use is made of bifurcation formulae which require only a series expansion of the vector field.     

A new approach to supply chain network equilibrium models

We establish nonlinear complementarity formulations for the supply chain network equilibrium models. The formulations have simple structures and facilitate us to study qualitative properties of the models. In this setting, we obtain weaker conditions to guarantee the existence and uniqueness of the equilibrium pattern for a supply chain. A smoothing Newton algorithm that exploits the network structure is proposed for solving these models. Not only is the smoothing Newton algorithm proved to be globally convergent without requiring the assumptions of monotonicity and Lipschitz continuity, but also it can overcome the flaw that the performance of the modified projection method heavily depends on the choice of the predetermined step size. Numerical results indicate the advantages of the nonlinear complementarity formulation and the smoothing Newton algorithm.     

动力定位船舶纵向运动的反步法控制器设计

本文研究了动力定位系统的控制策略,针对船舶动力定位系统模型的非线性特性,考虑到风浪流等外部环境扰动对船舶引起的恒值干扰及其不确定性,基于带有未知定常扰动的三自由度非线性船舶水面运动数学模型,应用自适应反步法设计船舶动力定位控制律,将Lyapunov函数选取和控制器设计相结合的回归设计方法,并在MATLAB软件环境下通过仿真研究验证模型和控制策略的有效性。在计算机仿真研究中,以26:1比例建造的2800mm×762mm×498mm船模为设计实例。     

Criteria for global minimum of sum of squares in nonlinear regression

Demidenko [2000. Is this the least squares estimate? Biometrika 87, 437-452] has established the relationship between the curvature of nonlinear regression and the local convexity of a sum of squares: the Hessian matrix is positive definite if the sum of squares is less than the minimum squared radius of the full curvature. In this paper, we continue developing the criteria for the global minimum of the sum of squares in nonlinear regression. In particular, the concept of the local unimodality is introduced; a function is called locally unimodal on a level set if it has a unique local minimum in each component of that level set. We show that the level of the local unimodality of the sum of squares is equal to the minimum squared radius of the intrinsic curvature of the nonlinear regression function. A new class of unidirected nonlinear regression models is introduced with an interpretation in terms of differential geometry. The criteria are illustrated by several popular nonlinear regression models.     

Numerical study of transient nonlinear harbor resonance

It is generally accepted that nonlinear wave-wave interactions play an important role in harbor resonance. Nevertheless it is not clear how waves take part in those interactions. The aim of this paper is to investigate those processes for a rectangular harbor at transient phases. Long-period oscillations excited by bichromatic waves are simulated by the Boussinesq model. The simulations start from calm conditions for the purpose of studying the response process. The internal wavemaker stops working after th...