Numerical solution of a nonlinear reaction-diffusion equation

A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkinfinite element method,which has been proved to be2nd-order accurate in time and4th-orderin space.The comparison between the exact and numerical solutions of progressive wavesshows that this numerical scheme is quite accurate,stable and efficient.It is also shown thatany local disturbance will spread,have a full growth and finally form two progressive wavespropagating in both directions.The shape and the speed of the long term progressive wavesare determined by the system itself,and do not depend on the details of the initial values.     

Analysis of aerodynamic problems with geometrically unspecified boundaries using an enhanced Lagrangian method

This paper presents solutions for several 2-D aerodynamic problems with geometrically unspecified boundaries. These solutions are obtained with an enhanced Lagrangian method based on stream function and Lagrangian-distance coordinates, which includes special procedures to substantially improve the numerical resolution of the shock waves and for the numerical implementation of the aerodynamic conditions defining the geometrically unspecified solid or fluid boundaries. The method is first validated for flows with specified solid boundaries by comparison with exact analytical solutions and with previous computational results obtained by numerical methods using Eulerian formulations. In all cases, this enhanced Lagrangian method displayed a very good accuracy and computational efficiency and a sharp numerical resolution of shock waves. Then this method is used to obtain solutions for problems with geometrically unspecified boundaries, such as: (i) indirect problems of determining the geometrical shape of airfoils and nozzle walls for a specified pressure distribution; (ii) supersonic nozzle design problem for a specified uniform flow at the nozzle outlet based on reflection-suppression condition; (iii) analysis of flexible-membrane airfoils; and (iv) analysis of jet-flapped airfoils.     

Scattering of elastic waves by three-dimensional surface topographies

Diffraction of plane harmonic waves by three-dimensional surface irregularities is investigated through the use of an indirect boundary integral equation method. The irregularity of an arbitrary shape is embedded in an elastic half-space and subjected to incident P, SV, SH, and Rayleigh waves. The material of the half-space is assumed to be linear, weakly anelastic, homogeneous and isotropic.

The accuracy of the method is demonstrated through comparison of the results with existing axisymmetric solutions. Several numerical examples for non-axisymmetric canyons are presented. The resulting amplification patterns exhibit strong sensitivity on type and angle of the incident waves and on the location of the observation point. Systematic comparisons of three-dimensional and corresponding two-dimensional models demonstrate similarity of the amplification pattern. The amplification is larger in some three-dimensional models than in two-dimensional ones. Strong coupling between SH and P-SV modes is observed for off-azimuthal incident waves. This phenomenon is specially pronounced for incident SH waves and it is intrinsic to three-dimensional scattering.     

An elastodynamic computational time-reversal method for shape reconstruction of traction-free scatterers

This study proposes an elastodynamic time-reversal imaging method for the shape reconstruction of flaws with the traction-free boundary. The proposed method is a generalization of the synthetic aperture focusing technique (SAFT) implemented with the aid of time-reversal back-propagation computation. The reconstruction formula is derived from the boundary condition applied on the flaw boundary without using a diffraction theory yielding an inherently normalized objective function. No restriction is imposed on the wave source or the material properties as long as the wave medium is linearly elastic and lossless. There are two major advantages with the proposed method. First, it is free from ray tracing. The method can thus work with an arbitrary source and the wave modes including inhomogeneous waves. Second, signal conditioning is not required to focus a scattered field to the correct location in the reconstructed image. Numerical examples are presented to demonstrate the performance and benefits of the proposed method. In the numerical examples, the shape of cavities and cracks in an elastic layer is reconstructed from synthetic data. The results show that the method works well with the reverberating incident fields in which inhomogeneous and body waves coexist. Also demonstrated is that the curvature of the cavities is resolved well at the scale of the incident wave length. Through the numerical examples, the method is shown to be a versatile imaging method potentially useful for ultrasonic nondestructive characterization of small flaws.     

气相爆轰波在障碍物上Mach反射的实验验证

本文公布了气相爆轰波沿收缩管道传播时发生Mach反射的实验证据。在爆轰波通过的管道中安装不同楔角的楔块,形成管道的收缩。爆轰波在通过楔块时会发生Mach反射。利用烟熏玻璃片记录到了爆轰波Mach反射时形成的三波点迹线及其两侧胞格尺寸和密度的变化。据我们掌握的资料,这是首次用胞格结构变化的记录证实,气相爆轰波与无化学反应的空气中的冲击波一样,在一定的入射条件下会发生Mach反射。这一实验结果可使我们更深入了解爆轰波的本质,也为数值模拟气相爆轰波在障碍物上Mach反射现象提供了可对比的依据。     

Effects of surface waves on hydrodynamic pressures on rigid dams with arbitrary upstream face

The effects of surface gravity waves on earthquake‐induced hydrodynamic pressures on rigid dams with nonvertical upstream face are examined, taking the compressibility and viscosity of water into account. A simple closed‐form solution is obtained by using the Trefftz numerical method. The boundary under study is reduced to the upstream face only and the degrees of freedom of the problem are restricted to the number of trial functions used for analysis. For harmonic base excitation, a number of numerical examples for various geometries of the upstream face are presented. The hydrodynamic pressure distribution, the resultant lateral force and its effective height of application are evaluated. The results are strongly influenced by the upstream shape and, in a lesser degree, by the consideration of surface waves. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical evaluation of passive control of shock wave/boundary layer interaction on NACA0012 airfoil using jagged wall

Shock formation due to flow compressibility and its interaction with boundary layers has adverse effects on aerodynamic characteristics, such as drag increase and flow separation. The objective of this paper is to appraise the prac-ticability of weakening shock waves and, hence, reducing the wave drag in transonic flight regime using a two-dimensional jagged wall and thereby to gain an appropriate jagged wall shape for future empirical study. Different shapes of the jagged wall, including rectangular, circular, and triangular shapes, were employed. The numerical method was validated by experimental and numerical studies involving transonic flow over the NACA0012 airfoil, and the results presented here closely match previous experimental and numerical results. The impact of parameters, including shape and the length-to-spacing ratio of a jagged wall, was studied on aerodynamic forces and flow field. The results revealed that applying a jagged wall method on the upper surface of an airfoil changes the shock structure significantly and disinte-grates it, which in turn leads to a decrease in wave drag. It was also found that the maximum drag coefficient decrease of around 17%occurs with a triangular shape, while the max-imum increase in aerodynamic efficiency (lift-to-drag ratio) of around 10%happens with a rectangular shape at an angle of attack of 2.26?.     

爆轰波碰撞聚能无网格MPM法数值模拟

爆轰聚能经常被应用于工程爆破切割,为了探索新的聚能爆破形式,本文应用无网格MPM法对圆柱筒装药和导爆索多点起爆形式的爆轰波碰撞聚能问题进行三维数值模拟,采用显式积分算法完成爆轰波形成和汇聚过程以及爆轰压力场分布的数值计算,并将数值模拟结果与实验结果做了对比分析。由此验证了无网格MPM法的准确性,也为炸药聚能爆破提供了新的思路和高效的操作方法。     

Diffraction of solitary water waves around three-dimensional floating bodies

By using the matched asymptotic expansion method and the idea of edge layer, a mathematic model for describing the interaction between weakly nonlinear shallow-water waves and three-dimensional floating bodies is formed in the paper. As a numerical example, the diffraction of a solitary wave around a vertically floating circular cylinder has been investigated and the results are presented. The present method can further be extended to the study of wave diffraction around floating bodies of general shape. The project is supported by the National Natural Science Foundation of China.     

Height of burst explosions: a comparative study of numerical and experimental results

In the current work, we use the Constant Volume model and the numerical method, Regularized Smoothed Particle Hydrodynamics (RSPH) to study propagation and reflection of blast waves from detonations of the high explosives C-4 and TNT. The results from simulations of free-field TNT explosions are compared to previously published data, and good agreement is found. Measurements from height of burst tests performed by the Norwegian Defence Estates Agency are used to compare against numerical simulations. The results for shock time of arrival and the pressure levels are well represented by the numerical results. The results are also found to be in good agreement with results from a commercially available code. The effect of allowing different ratios of specific heat capacities in the explosive products are studied. We also evaluate the effect of changing the charge shape and height of burst on the triple point trajectory.      

Propagation of small waves in inextensible strings

The model of an inextensible uniform string subject to constant gravitation is used to study the propagation of transversal waves in one-dimensional continua. Perturbation analysis of the equations of motion yields as a result the local representation of small waves in terms of a normalized Riemann function. By means of the latter, shape and speed of propagating waves may be discussed. A refined analysis confirms that on first order, small waves travel along characteristics of the unperturbed equilibrium configuration. An explicit power law for the waves’ amplitudes is given, and the findings are supported by the numerical results.     

功能梯度圆柱板中的周向衰逝导波

与传播波模态不同,衰逝波模态波数为纯虚数或复数,它们随传播距离呈指数或阻尼式衰减。复数根描述的衰逝波对结构缺陷形状和尺寸的导波检测具有重要作用,但其求解却是非常困难的,通常要借助于各种迭代技术。本文提出一种计算求解衰逝波问题的改进的Legendre正交多项式方法,该方法可将复杂的变系数微分方程组计算转换为特征值求解问题,无需迭代便能计算得到包含实波数域、虚波数域和复波数域的完整的频散曲线。通过具体算例验证了提出方法的正确性。应用提出方法计算了不同梯度圆柱板中的衰逝导波,绘制了三维频散曲线,研究了不同径厚比和梯度场对衰逝导波频散的影响,分析了衰逝导波的位移和应力分布,讨论了衰逝导波的传播特性。     

Connection machine simulation of ultrasonic wave propagation in materials III: The three-dimensional case

A formalism for the simulation by parallel processing of the propagation of ultrasonic pulses of arbitrary shape (i.e. not restricted to continuous waves) in complex media is presented. Recursive relationships yielding the time evolution of the displacement field are derived for both homogeneous and heterogeneous materials. In the latter case, formulas for cross-points at the intersection of up to eight different materials are obtained. A few examples of numerical results obtained with the proposed method are shown to demonstrate its applicability and efficiency. The problem of visualization of the results is also briefly discussed.     

曲面曲率对Rayleigh波传播特性的影响

对任意形状的均匀各向同性线弹性曲面物体,用 WKB～（1）方法求解了沿曲面传播的Rayleigh表面波的运动微分方程,同时考虑了波传播方向及其垂直方向曲面曲率对波的穿透性的影, 所获波动方程的势函数解答表明,在一般情况下垂直波传播方向的曲面曲率对波的穿透深度的影响是不容忽视的.进而以同种介质平面表面情况下的Rayleigh面波的传播特性为基准,给出了曲面曲率引起波数或波速变化的解析表达式.通过理论分析和数值算例,描述了曲面上Rayleigh面波传播行为的一些基本特征.     

流固耦合地震波动问题的显式谱元模拟方法

流固耦合地震波动问题主要研究由流体和固体构成的复杂系统中地震波传播特性及其规律. 传统模拟方法中一般以声波方程、弹性波方程的数值解分别描述理想流体和弹性固体中的波动, 并实时地处理两种不同性质介质之间的相互耦合作用, 数值格式复杂且限制数值模拟精度与计算效率. 本文采用谱元法结合多次透射公式人工边界条件实现了一种流固耦合地震波动问题的高阶显式数值计算方法. 该方法利用了流固耦合问题统一计算框架,可将饱和多孔介质的Biot波动方程分别退化为理想流体的声波方程和弹性固体的弹性波方程. 通过P波垂直入射的水平成层理想流体-饱和多孔介质-弹性固体场地模型、P波斜入射的不规则层状界面以及任意形状界面的理想流体-饱和多孔介质-弹性固体场地模型等三个算例, 与传递函数法解析解以及集中质量有限元法计算结果进行对比分析, 证明了本文方法的正确性与有效性. 数值模拟结果表明, 本文方法相较传统有限元法可以少得多的节点数量获得更高的数值精度, 并且在较宽的频率范围内都能可靠地模拟出流固耦合系统的动力响应, 充分体现出本文方法兼顾高精度、计算效率和复杂场地建模灵活的特点.      

Influence of Discrete Longitudinal Ribs on Harmonic Waves in Cylindrical Shells

The influence of discrete longitudinal ribs on harmonic waves in cylindrical shells is studied by numerical examples. It is shown that the presence of discrete ribs is manifested as an increased number and changed shape of the dispersion curves     

波浪作用下直立结构物附近强湍动掺气流体运动的数值模拟

波浪破碎卷入气体易对建筑物受力产生压力振荡, 了解波浪作用下建筑物附近掺气水流的运动特性是精确计算建筑物受力的前提. 基于OpenFOAM开源程序包和修正速度入口造波方法建立三维数值波浪水槽, 模型采用S-A IDDES湍流模型进行湍流封闭, 并采用修正的VOF 方法捕捉自由液面, 数值模拟了规则波在1:10的光滑斜坡上与直立结构物的相互作用过程, 重点分析了结构物附近的水动力和掺气水流运动特性. 结果表明, 建立的数值模型能精确地捕捉波浪作用下直立结构物附近的自由液面的变化以及气泡输运过程, 较好地描述气体卷入所形成的气腔形态以及多气腔之间的融合、分裂等过程; 波浪与直立结构物相互作用产生强湍动掺气水流, 其运动过程十分复杂; 掺气流体输运过程中水气界面周围一直伴随着涡的存在, 其中, 气泡的分裂与周围正负涡量剪切作用密切相关, 且其输运轨迹主要受周围流场的影响; 研究揭示了结构物附近湍动能与掺气特性的关系, 发现波浪作用下直立结构物附近湍动能的分布与掺气水流特征参数(气泡数量、空隙率)整体呈现一定的线性关系.      

Finite element simulations of fully non‐linear interaction between vertical cylinders and steep waves. Part 2: numerical results and validation

In Ma, Wu, Eatock Taylor [Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure. International Journal for Numerical Methods in Fluids 2001], designated Part 1 hereafter, we have developed the methodology and solution procedure for simulating the three‐dimensional interaction between fixed bodies and steep waves based on a finite element method (FEM). This paper provides extensive numerical results and validation. The effectiveness of the radiation condition is investigated by comparing the results from short and long tanks; the accuracy of the computed data is confirmed through comparison with analytical solutions. The adopted mathematical model is also validated by comparing the obtained numerical results with experimental data. Various test cases, including non‐linear bichromatic and irregular waves and the interactions between waves and one or two cylinders, are analysed. Copyright © 2001 John Wiley & Sons, Ltd.     

细长体后部非定常超空泡研究

采用积分方程方法,研究了轴对称细长体后部非定常超空泡问题.应用时间有限差分离散化方法,对积分方程进行了求解.以细长锥体空化器为例,文中分别给出了当锥角和空化数改变(简称扰动)时,空泡长度和形状的变化规律.当流场周期扰动时,分析计算了超空泡的尺度变化.分别采用本文方法和理论公式,对空泡长度与空化数的关系曲线进行了对比.数值结果表明,扰动周期越短,空泡长度的变化越小;在相同的扰动频率下,空泡越长,时间滞后越长;空泡长度相同时,扰动频率越高,时间滞后越长.在高频脉冲扰动下,有脉冲波形沿着空泡表面传播,其传播速度为来流速度.在周期小扰动情况下,扰动波形沿着空泡表面传播,传播速度也是来流速度.本文得到的数值结果为水下航行体空化器的分析和设计提供参考作用.     

Nonlinear wave propagation and reflection — Comparing the numerics with the analytics

The accuracy of numerical methods needs always a special attention. In this paper, analytical and numerical methods have been compared to describe the initial stage of nonlinear propagation and reflection of longitudinal ultrasonic waves. The perturbation method has been used to derive the analytical solution and the finite difference scheme to find the numerical solution for multiple free-boundary reflections of a harmonic burst at ultrasonic frequencies. The comparison of results at relatively small nonlinearities reveals a good qualitative and quantitative agreement between the analytical and numerical solutions. The method for determining analytically the exact region of interaction for counter-propagating waves is outlined in detail. At higher frequencies and larger nonlinear effects some quantitative differences between analytical and numerical results appear. The results are applicable in modelling nonlinear wave motion, including NDT and nonlinear one-dimensional vibrations.