The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.     

浅海沉积层中界面波的传播以及掩埋目标的回波

根据地声界面波的相关理论,采用时域有限差分法建立浅海沉积层中界面波传播模型。以沉积层中掩埋小目标为例,计算界面波对空腔掩埋物和实心掩埋物的不同作用结果。根据运动方程和应力应变关系采用中心差分网格建立声场的有限差分模型,计算了声场瞬态情况和质点振动的时间序列结果,从不同角度比较分析了地声界面波对掩埋目标的作用。根据浅海沉积层的性质计算了界面波的传播损失和对掩埋物体的目标强度,由于界面波传播衰减主要由沉积层的能量耗散引起,其规律接近横波的传播衰减,在高频情况下界面波在沉积层中传播的距离不远。文章试图建立一种对浅海沉积层中掩埋小尺度目标的探测新方法。通过一些典型模型的分析,从理论角度证明地声界面波对目标探测的可行性。     

基于虚拟介质方法数值模拟应力波穿透水舱

将虚拟介质方法推广到应力波与流固界面的相互作用,基于修正虚拟流体方法(MGFM)思想提出一种适用于流固界面应力波折射问题的界面状态计算法则。以此为基础,用Zwas格式和WENO格式分别离散固体和流体控制方程,数值模拟应力波穿透水舱。数值测试表明:虚拟介质方法推广到模拟流固界面上的应力波时,数值误差具有很好的收敛性;该研究提出的计算法则具有一阶精度。一维数值结果与理论解析解能很好地吻合,验证了用虚拟介质方法模拟应力波在流固界面处反射与透射的可行性。应力波穿透水舱的数值结果表明:虚拟介质方法与任意拉格朗日-欧拉(ALE)方法得到的数值结果相近;水舱加压后,透射波的强度也相应的增加;不同的水舱压力下,应力波在流固界面处的折射规律相近。该结论对舰船的防护设计有实际意义。     

The Method of Fundamental Solutions for One-Dimensional Wave Equations

A meshless numerical algorithm is developed for the solutions of one-dimensional wave equations in this paper. The proposed numerical scheme is constructed by the Eulerian-Lagrangian method of fundamental solutions (ELMFS) together with the D'Alembert formulation. The D'Alembert formulation is used to avoid the difficulty to constitute the linear algebraic system by using the ELMFS in dealing with the initial conditions and time-evolution. Moreover the ELMFS based on the Eulerian-Lagrangian method (ELM) and the method of fundamental solutions (MFS) is a truly meshless and quadrature-free numerical method. In this proposed wave model, the one-dimensional wave equation is reduced to an implicit form of two advection equations by the D'Alembert formulation. Solutions of advection equations are then approximated by the ELMFS with exceptionally small diffusion effects. We will consider five numerical examples to test the capability of the wave model in finite and infinite domains. Namely, the traveling wave propagation, the time-space Cauchy problems and the problems of vibrating string, etc. Numerical validations of the robustness and the accuracy of the proposed method have demonstrated that the proposed meshless numerical model is a highly accurate and efficient scheme for solving one-dimensional wave equations.     

Numerical modeling of complex interactions between underwater shocks and composite structures

To study the complex interactions between underwater shocks and composite structures, a strongly coupled Eulerian–Lagrangian numerical solver is developed. The coupled numerical solver consists of an Eulerian fluid solver, a Lagrangian solid solver, a one-fluid cavitation model, and an interface capturing scheme. The interface capturing scheme features a fluid characteristics method and a modified ghost fluid method (MGFM). The MGFM is reformulated for fluid–solid coupling by treating simultaneously the fluid characteristics equation and the solid equation of motion to determine the interface variables, leading to a strongly coupled Eulerian–Lagrangian scheme. Various components of the numerical solver are first individually tested and validated. The strongly coupled solver is then applied to realistic shock-structure interaction problems involving composite structures. The accuracy of the coupled solver is demonstrated via comparison with numerical predictions and experimental observations available in literature. Finally, the validated coupled numerical solver is utilized to study the effectiveness of a proof-of-concept shock mitigation scheme.     

Numerical simulation of ultrasonic transmission through curved interface

The direct boundary integral equation formalism is used to model reflected and transmitted wave fields due to a bounded ultrasonic beam incident upon an arbitrarily curved interface separating a fluid and a solid media. The numerical procedure involves application of point collocation with quadratic isoparametric approximation that reduce the integral equations to a discrete set of linear algebraic equations. Numerical results are provided for plane, concave and convex interfaces for incident beam profiles which are Gaussian or approximately rectangular. The case of surface wave excitation on a plane interface is considered and features of their numerical solution are discussed. Constraints on the various parameters of the numerical model and their effects on the accuracy of the solution are investigated in detail.     

抛物形缓坡方程的数值研究

对2种典型的抛物形缓坡方程进行比较系统的数值研究。通过对4种典型地形上的波浪变形的数值模拟，详细讨论了网格节点数对数值解精度的影响、模型对初始入射角的敏感程度、非线性项对数值结果的影响等。研究结果可为实际应用抛物形缓坡方程研究大区域复杂地形上的波浪传播提供一定的理论指导。     

Finite element simulation and visualization of leaky Rayleigh wavesfor ultrasonic NDE

The generation and propagation properties of transient leaky Rayleigh waves are characterized by a two-dimensional finite element model. The displacement vector is used as the primary variable for the solid medium and a potential scalar, which is a replacement for the pressure, is taken as the fundamental variable for the fluid medium. The coupled system of finite element equations are solved in the time domain by direct integration through the central difference scheme. Three configurations are considered: the conversion of a Rayleigh surface wave into a leaky Rayleigh wave, a focused beam probing a fluid/solid interface at the Rayleigh angle, and the interaction of a defocused wave with the interface. The wave velocity in the fluid (water) is lower than the Rayleigh wave velocity in the solid (aluminum). The wave propagation profile in each case is predicted by the model. The finite element model proves to be an effective tool for surface acoustic device design and ultrasonic NDE     

弱粘接界面固-固界面波特性及与界面粘接性质关系数值研究

根据两固体粘接结构在不同粘接强度下的弹簧模型边界条件,通过傅里叶积分变换方法进行波动方程求解,理论分析和数值计算了相近横波速度的两种固体间界面波的频散及衰减特性.计算结果表明,当切向弹簧劲度系数从滑移粘接界面向完好粘接界面逐渐变化时,界面波的频散特性随之变化.在此基础上进一步计算了不同界面粘接条件下法向线源脉冲激发的界...     

A Petrov–Galerkin finite element scheme for the regularized long wave equation

A Petrov-Galerkin finite element method (FEM) for the regularized long wave (RLW) equation is proposed. Finite elements are used in both the space and the time domains. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weight functions. An implicit, conditionally stable, one-step predictor–corrector time integration scheme results. The accuracy and stability are investigated by means of local expansion by Taylor series and the resulting equivalent differential equation. An analysis based on a linear Fourier series solution and the Von Neumanns stability criterion is also performed. Based on the order of the analytical approximations and of the domain discretization it is concluded that the scheme is of third order in the nonlinear version and of fourth order in the linear version. Three numerical experiments of wave propagation are presented and their results compared with similar ones in the literature: solitary wave propagation, undular bore propagation, and cnoidal wave propagation. It is concluded that the present scheme possesses superior conservation and accuracy properties.This work has been partially supported by the Fundação para a Ciência e Tecnologia, under project POCTI/ECM/41800/2001.     

用改进的Boussinesq方程模拟潜堤上的波浪变形

研究了一种改进的Boussinesq方程,采用预报校正格式对该方程进行了数值离散,并对淹没潜堤上的波浪变形进行了数值模拟,从数值模拟结果和实测值的比较结果来看,该方程能较好地模拟波浪在潜堤上传播时波面的变形过程,可以用于实际中的波浪问题计算。这种改进的Boussinesq方程本身及其求解方法需做进一步的完善。研究结果为实际应用Boussinesq方程来研究复杂地形上的波浪传播提供一定的理论指导。     

基于精确描述的超声散射声场的建立

从超声波在连续介质中的传播特性出发,在建立声波在均匀介质中传播的波动方程的基础上,对声波在非均匀介质中传播的问题进行了研究。利用格林函数,建立了关于精确散射场和全场的波动方程。为了便于利用计算机等处理器对方程进行求解,将矩量法引入到方程的离散化问题中,从而为波动方程的求解提供了理论基础。试验验证结果表明：基于离散化的散射场方程和全场动方程,在一定对比度的条件下,采用迭代方法能够实现超声散射层析成像。     

求解带有不连续波数的二维变系数 Helmholtz方程的一种高精度紧致差分方法

很多实际物理问题都可以由带有不连续波数的变系数 Helmholtz 方程进行数值模拟。Helmholtz 方程的数值方法研究是热点问题之一,具有重要的理论和实际意义。由于波数的不连续性,使用传统的有限差分方法求解带有不连续波数的 Helmholtz 方程时通常无法达到原有差分格式的精度。结合浸入界面方法的思想,对带有不连续波数的二维变系数 Helmholtz 方程构造了一类新的四阶紧致有限差分格式,数值实验验证了新方法的可靠性和有效性。     

A new application of an improved DRBEM model for water wave propagation over a frictional uneven bottom

This paper presents a numerical scheme to approximate water wave diffraction, refraction and frictional dissipation over an axi-symmetric pit. Based on an improved extended mild-slope equation (EMSE) including bottom friction effect, as the elliptic governing differential equation, dual reciprocity boundary element method (DRBEM) is employed to model water wave propagation over an axi-symmetric pit. To the authors' knowledge, this is the first application of DRBEM for water wave scattering over a pit. In order to promote accuracy of the model, not only effects of the bottom curvature and the slope-squared terms which are neglected in the mild-slope equation (MSE), are considered, but also effect of the bottom friction is measured by the improved EMSE. Numerical results are compared with existing analytical or numerical solutions or with experimental data by several examples. Through these numerical experiments reliability and efficiency of present DRBEM model for determining the total wave field over an uneven bottom is approved.     

爆炸冲击波在复杂坑道内传播规律的数值研究

研究二维爆炸流场中空气冲击波在复杂坑道内传播的机制.对多物质流采用Euler方法和NND格式,对爆轰产物采用JWL状态方程;自行编制二维爆炸模拟计算程序.试验结果与数值模拟基本相符,反映了爆炸冲击波在坑道内传播的详细过程.结果表明:采用穿廊式出入口,可以削弱进入坑道内55%的空气冲击波强度;采用穿廊加扩散室可以削弱进入坑道75%的冲击波强度.     

A SPACE–TIME FINITE ELEMENT METHOD FOR THE EXTERIOR STRUCTURAL ACOUSTICS PROBLEM: TIME-DEPENDENT RADIATION BOUNDARY CONDITIONS IN TWO SPACE DIMENSIONS

A time-discontinuous Galerkin space–time finite element method is formulated for the exterior structural acoustics problem in two space dimensions. The problem is posed over a bounded computational domain with local time-dependent radiation (absorbing) boundary conditions applied to the fluid truncation boundary. Absorbing boundary conditions are incorporated as ‘natural’ boundary conditions in the space–time variational equation, i.e. they are enforced weakly in both space and time. Following Bayliss and Turkel, time-dependent radiation boundary conditions for the two-dimensional wave equation are developed from an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a non-dimensional wavenumber. In this paper, we undertake a brief development of the time-dependent radiation boundary conditions, establishing their relationship to the exact impedance (Dirichlet-to-Neumann map) for the acoustic fluid, and characterize their accuracy when implemented in our space–time finite element formulation for transient structural acoustics. Stability estimates are reported together with an analysis of the positive form of the matrix problem emanating from the space–time variational equations for the coupled fluid-structure system. Several numerical simulations of transient radiation and scattering in two space dimensions are presented to demonstrate the effectiveness of the space–time method.     

固体中二维纵波非线性方程的摄动解

针对二维纵波非线性方程的求解问题进行了研究.从二维纵波非线性方程出发,围绕声马赫数对该方程的解进行二阶摄动展开,求解出二维非线性纵波方程的二阶摄动解.根据计算结果定义二次谐波沿x轴的传播分量和沿y轴的传播分量与入射基波的振幅比为非线性系数,并对基波和二次谐波的传播特性进行详细讨论,探究了二维纵波在固体中传播时其入射波振...     

位于弹性半空间上的理想流体层动力反应--平面SV波入射

根据弹性固体与理想流体动力学方程,导出了固-液介质交界面上波的透射与反射系数的理论公式,分析了平面SV波从弹性半空间入射到与理想流体层的交界面时,弹性半空间的弹性模量和密度,平面SV波的入射角和频率,以及流体的体积模量和密度对理想流体层动压力的影响。     

Rayleigh waves at the boundary surface of modified couple stress generalized thermoelastic with mass diffusion

In this problem, we have studied propagation of Rayleigh waves in an homogeneous isotropic modified couple stress generalized thermoelastic with mass diffusion solid half space in the context of Lord–Shulman (L-S), Green–Lindsay (G-L) theories of thermoelasticity. Secular equations are derived mathematically by using appropriate boundary conditions. The values of determinant of secular equation, Rayleigh wave velocity and attenuation coefficient with respect to angular velocity for different values of wave number and relaxation times in the absence and presence of mass diffusion, are computed numerically. The numerical simulated results are depicted graphically for copper material.     

Boundary element method for band gap calculations of two-dimensional solid phononic crystals

A boundary element method (BEM) is developed to calculate the elastic band gaps of two-dimensional (2D) phononic crystals which are composed of square or triangular lattices of solid cylinders in a solid matrix. In a unit cell, the boundary integral equations of the matrix and the scatterer are derived, the former of which involves integrals over the boundary of the scatterer and the periodic boundary of the matrix, while the latter only involves the boundary of the scatterer. Constant boundary elements are adopted to discretize the boundary integral equations. Substituting the periodic boundary conditions and the interface conditions, a linear eigenvalue equation dependent on the Bloch wave vector is derived. Some numerical examples are illustrated to discuss the accuracy, efficiency, convergence and the computing speed of the presented method.