Elements for the numerical analysis of wave motion in layered strata

A technique is developed for the numerical analysis of wave motion in layered strata. Semidiscrete particular solutions satisfying inhomogeneous boundary conditions are calculated by the finite element method. These solutions are combined with semidiscrete modes of an appropriate eigenvalue problem. The boundary conditions corresponding to rigid and rough footings on a layered stratum are treated in detail. Applications are considered which demonstrate the validity of the technique.     

Interaction of harmonic plane waves with a mobile elastic phase boundary in anti-plane shear

This paper is concerned with the effect of sustained infinitesimal harmonic plane wave excitation of a phase boundary in a non-linearly elastic material that is subject to anti-plane shear deformation. The phase boundary is capable of motion that is here described by a harmonic travelling waveform. The reflected wave is also a harmonic plane wave, however the transmitted wave may be either in the form of a harmonic plane wave or a harmonic surface wave. The phase boundary motion is determined on the basis of a standard kinetic relation that involves a single mobility parameter. This gives phase boundary motion that is synchronized with the incident wave for the case of a transmitted plane wave, but is not synchronized with the incident plane wave for the case of a transmitted surface wave. A certain fraction of the energy provided by the incident wave is dissipated by phase boundary motion in a fashion that can be explicitly quantified. Special incident angles can suppress the reflected wave, suppress the transmitted wave or cause the dissipation to vanish.     

半圆柱型沉积盆地对SH波散射的数值分析

该文实现了一种半无限域SH波散射问题的数值分析方法。采用传递矩阵法得到SH波斜入射时的自由场,将其作为输入;采用集中质量显式有限元方法计算区域内节点的位移;采用透射人工边界计算人工边界点的位移;通过编写的FORTRAN程序实现计算过程。运用该方法对均匀半空间内半圆柱型沉积盆地在SH波入射下的散射进行了分析,与Trifunac M D的解析解进行了对比,验证了该文方法的有效性,分析了不同入射角对地表位移和位移谱放大系数的影响。最后,对成层半空间内半圆柱型沉积盆地在SH波入射下的散射进行了分析。相对于解析方法而言,该方法可以考虑更为复杂地形情况。     

Implicit formulation with the boundary element method for nonlinear radiation of water waves

An accurate and efficient numerical method is presented for the two-dimensional nonlinear radiation problem of water waves. The wave motion that occurs on water due to an oscillating body is described under the assumption of ideal fluid flow. The governing Laplace equation is effectively solved by utilizing the GMRES (Generalized Minimal RESidual) algorithm for the boundary element method (BEM) with quadratic approximation. The intersection or corner singularity in the mixed Dirichlet–Neumann problem is resolved by introducing discontinuous elements. The fully implicit trapezoidal rule is used to update solutions at new time-steps, by considering stability and accuracy. Traveling waves generated by the oscillating body are absorbed downstream by the damping zone technique. To avoid the numerical instability caused by the local gathering of grid points, the re-gridding technique is employed, so that all the grids on the free surface may be re-distributed with an equal distance between them. The nonlinear radiation force is evaluated by means of the acceleration potential. For a mixed Dirichlet–Neumann problem in a computational domain with a wavy top boundary, the present BEM yields numerical solutions for the quadratic rate of convergence with respect to the number of boundary elements. It is also demonstrated that the present time-marching and radiation condition work successfully for nonlinear radiation problems of water waves. The results obtained from this study concur reasonably well with other numerical computations.     

带窄缝多箱体系统波浪荷载特性分析

针对波浪与带有窄缝多固定直角箱体结构作用产生的流体共振问题,建立了非线性波浪荷载分析二维时域模型。该模型采用域内源造波技术产生入射波浪,自由水面满足完全非线性运动学和动力学边界条件,窄缝内自由水面引入人工阻尼来等效由于涡旋运动和流动分离引起的粘性耗散,建立边界积分方程并采用高阶边界元离散求解物面上速度势等未知量,进而利用加速度势的方法来求得速度势时间导数,并基于伯努利方程积分得到作用结构上的瞬时波浪荷载。通过模拟带两窄缝的三箱体所受水平力与垂向力,并与已发表结果对比验证了模型的准确性。同时通过大量的数值计算,分析了箱体数量对各箱体所受波浪荷载大小及变化规律的影响。     

Construction of stationary waves on a falling film

Waves on a film falling down a vertical wall exhibit many distinct features. They tend to be locally stationary over several wavelengths, viz. they travel with constant speeds and shapes over a long distance. In the limit of very long (solitary) waves, these stationary waves also exhibit two length scales with small and short capillary waves running ahead of a large tear-drop shaped hump. We present a spectral-element method for this difficult multi-scale free surface problem. A boundary layer approximation of the equation of motion allows a Fourier expansion in the streamwise direction in conjunction with a domain decomposition in the direction normal to the wall that eliminates numerical instability. This mixed method hence enjoys both the exponential convergence rate of a spectral technique and the numerical advantage provided by a compactly supported basis which yields sparse projected differential operators. All stationary wave families, parameterized by the wavelength, are then constructed using a Newton continuation scheme. The constructed waves are favorably compared to experimentally measured wave shapes.     

饱和多孔介质一维瞬态波动问题的解析分析

采用基于混合物理论的多孔介质模型,提出了饱和多孔介质一维动力响应的初边值问题。利用拉氏变换和卷积定理,分别得到了边界自由排水时在任意应力边界条件和任意位移边界条件下瞬态波动过程的解析表达。几种典型的数值算例同时给出了两类边界条件下瞬态波动过程中多孔固体的位移场、应力场和孔隙流体的速度场、压力场。结果表明,饱和多孔介质的波动过程是多孔固体和孔隙流体中以同一速度传播的两种波动的耦合过程,时效特性分析也揭示了饱和多孔介质固有的表观粘弹性性质。     

A boundary element model for near surface contact stresses of rough surfaces

This study proposes a boundary element based near surface elastic stress prediction model for rough surface point contacts. The contact body is considered as a half space, whose boundary is divided into the rough finite field and the smooth infinite field. The integral kernel singularities in the displacement boundary integral equation are eliminated through coordinate transformation and the approach of rigid body motion. For the integral of the nearly singular stress kernels, a numerical scheme that combines the distance transformation and a subdivision technique is devised to improve the numerical accuracy and efficiency. The computational results are compared to closed-form solutions to assess the model accuracy. Using a three-dimensional sinusoidal roughness profile, the near surface stress concentrations induced by surface asperities are demonstrated. It is shown the exclusion of roughness profile in stress computation leads to significant errors in both the locations and the magnitudes of near surface stress concentrations.     

饱和两相介质近场波动问题的一种时域全显式数值计算方法

基于u-p形式的饱和两相介质弹性波动方程,开展了饱和两相介质近场波动问题时域显式数值计算方法的研究。通过对波动方程中的质量矩阵和孔隙流体压缩矩阵进行对角化处理,消除了方程中的动力耦联,实现了波动方程的解耦。分别应用中心差分法和Newmark常平均加速度法求解固相位移和速度,基于向后差分法求解孔隙流体压力,推导得到了饱和两相介质动力响应的时域显式逐步积分的计算列式,建立了饱和两相介质近场波动问题的一种新的时域全显式数值计算方法。进行了该文方法中矩阵对角化合理性的验证。将该方法的数值解与相应的解析解进行对比,二者符合良好,验证了该方法的正确性。将该文建立的时域数值计算方法与透射人工边界方法相结合,应用于饱和两相介质的近场波动问题,进行了饱和土场地地震响应的计算研究,计算结果符合弹性波动理论的基本规律,表明该方法对于饱和两相介质近场波动问题时域计算求解的适用性。基于该方法中时域递推计算格式的传递矩阵,进行了该方法稳定性特性的研究。该文建立的数值计算方法具有时域全显式算法的基本特征。方法中对动力响应的全部分量均采用递推和迭代的模式进行求解,避免了求解耦联的动力方程组。该方法具有较高的计算效率,...     

Surface TM wave propagating at the boundary of isotropic collisionless plasma moving at a relativistic velocity

The dispersion properties of a surface TM wave propagating at the boundary of isotropic collision-less plasma moving at a relativistic velocity are described. The phenomenon of aberration, whereby the wave normal of the surface wave deviates from the boundary plane and rotates in the direction of motion, takes place only in the laboratory frame. The subrelativistic motion of the plasma is manifested by a decrease in the cutoff frequency of the surface TM wave and by the appearance of a long-wavelength limit in the TM wave spectrum, which shifts toward shorter wavelengths with increasing plasma velocity, thus reflecting the indissoluble unity of space and time.     

Wave generation in an elastic half-space by a normal point load moving uniformly over the free surface

The motion is determined of an elastic, homogeneous and isotropic half-space, excited by a normal point load traveling uniformly over the free surface. Using the differential transform technique [9], exact, closed expressions in terms of algebraic functions are found for the displacements at any point of the half-space. The displacements appear as a superposition of a primary wave, i.e. The wave that would exist if the medium were unbounded, and a secondary wave accounts for the presence of the boundary. The primary wave has previously been given by Eason, Fulton and Sneddon.     

Active Control of Laminar Boundary Layer Using Various Wall Motions

In this study, three types of surface motion of wall motion actuators were proposed and used to control the Tollmien-Schlichting (T-S) wave in the laminar boundary layer of a plate. These three types of motion are standing transverse wave (with out-of-plane displacement), traveling transverse wave (with out-of-plane displacement) and standing longitudinal wave (with in-plane displacement). The length of a wall motion actuator was set to 1, 2 or 3 cycles of waveform. Numerical simulation was performed on the generation of T-S wave and its suppression with the three types of surface motion of the wall motion actuator and the dependence of control effect on the amplitude and the phase of the surface motion, and the number of waveform was investigated.     

出平面线源荷载对半空间中半圆形凸起的圆柱形弹性夹杂的散射

采用复变函数法和Green函数法,求解出平面线源荷载对半空间中半圆形凸起的圆柱形弹性夹杂的散射。首先,给出在含有半圆形凸起的圆柱形弹性夹杂的弹性半空间中,水平表面上任意一点承受时间谐和的出平面线源荷载作用时的位移函数,取该位移函数作为Green函数;然后,采用分区的思想,分别构造出夹杂内的驻波和夹杂外的散射波,满足"公共边界"处位移和应力的连续性条件,建立起求解该问题的无穷代数方程组;最后,给出了动应力集中系数和水平地面位移幅值的数值结果,并进行了讨论。     

A noncollinear TM surface wave propagating at the moving plasma-vacuum interface

The conditions of existence and the dispersion properties of a noncollinear TM electromagnetic surface wave propagating at the moving interface between a low-temperature isotropic collisionless plasma and vacuum are studied. It is found that the boundary motion shifts the surface wave normal out of the interface and inclines it toward the direction of motion. The possibility of using the moving interface for a substantial transformation of the TM surface wave spectrum in the cutoff frequency region is pointed out.     

Boundary element analysis of wave-current interaction around a large structure

A numerical approach for wave-current interaction around a large structure is investigated, based on potential flow theory, linear waves and small current velocity approximation. The velocity potential in a wave-current coexisting field is separated into a steady current potential and an unsteady wave potential. The boundary element method was then employed to compute the unsteady wave potential with effects of both a uniform current and a large body taken into consideration. It is demonstrated that the steady current potential can be expressed as the sum of a uniform current and a steady disturbance due to the presence of the object. The variation of current velocity in the vicinity of the object is then calculated by using a surface vorticity boundary integral meethod. Boundary element analysis is also used for the numerical solutions of the surface vorticity method. Substituting both unsteady wave potential and current velocity into the first-order dynamic surface boundary condition, the water surface elevation around a large structure in a wave-current coexisting field can then be obtained. Comparisons of numerical predictions with experimental results ar also made; qualitative good agreements are obtained.     

Finite element analysis of wave motion

To estimate wave motion, the finite element method is presented based on the linear interpolation function and two-step explicit numerical integration in time. For the determination of free surface position, the two-step scheme based on the Eulerian technique is usefully employed. Travel and run-up of solitary wave have been analysed and compared with the analytical solution. Both results are reasonably well in agreement. This method is applied to estimate the wave force on the practical breakwater to show the validity of the method.     

An implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

In this work, a new comprehensive method has been developed which enables the solution of large, non‐linear motions of rigid bodies in a fluid with a free surface. The application of the modern Eulerian–Lagrangian approach has been translated into an implicit time‐integration formulation, a development which enables the use of larger time steps (where accuracy requirements allow it). Novel features of this project include: (1) an implicit formulation of the rigid‐body motion in a fluid with a free surface valid for both two or three dimensions and several moving bodies; (2) a complete formulation and solution of the initial conditions; (3) a fully consistent (exact) linearization for free surface flows valid for any boundary elements such that optimal convergence properties are obtained when using a Newton–Raphson solver. The proposed framework has been completed with details on implementation issues referring mainly to the computation of the complete initial conditions and the consistent linearization of the formulation for free surface flows. The second part of the paper demonstrates the mathematical and numerical formulation through numerical results simulating large free surface flows and non‐linear fluid structure interaction. The implicit formulation using a fully consistent linearization based on the boundary element method and the generalized trapezoidal rule has been applied to the solution of free surface flows for the evolution of a triangular wave, the generation of tsunamis and the propagation of a wave up to overturning. Fluid–structure interaction examples include the free and forced motion of a circular cylinder and the sway, heave and roll motion of a U‐shaped body in a tank with a flap wave generator. The presented examples demonstrate the applicability and performance of the implicit scheme with consistent linearization. Copyright © 2001 John Wiley & Sons. Ltd.     

A numerical model for determining the motion of a bubble close to a fixed rigid structure in a fluid

This paper is concerned with the problem of modelling the motion of a bubble close to a rigid structure in an infinite fluid. It is well known that the boundary integral method is a powerful technique for modelling the motion of a single bubble in a fluid. In this paper we shall present a modified boundary integral method for modelling the motion of a bubble close to a fixed finite rigid structure, and discuss a numerical scheme for solving the resulting integral equation for three-dimensional problems. Finally, we illustrate our method with some typical numerical results.     

Transmitting boundary conditions for 1D peridynamics

The peridynamic theory reformulates the equations of continuum mechanics in terms of integro‐differential equations instead of partial differential equations. It is not straightforward to apply the available artificial boundary conditions for continua to peridynamic modeling. We therefore develop peridynamic transmitting boundary conditions (PTBCs) for 1D wave propagation. Differently from the previous method where the matching boundary condition is constructed for only one boundary material point, the PTBCs are established by considering the interaction and exchange of information between a group of boundary material points and another group of inner material points. The motion of the boundary material points is recursively constructed in terms of their locations and is determined through matching the peridynamic dispersion relation. The effectiveness of the PTBCs is examined by reflection analyses, numerical tests, and numerical convergent conditions. Furthermore, two‐way interfacial conditions are proposed. The PTBCs are then applied to simulations of wave propagation in a bar with a defect, a composite bar with interfaces, and a domain with a seismic source. All the analyses and applications demonstrate that the PTBCs can effectively remove undesired numerical reflections at artificial boundaries. The methodology may be applied to modeling of wave propagation by other nonlocal theories. Copyright © 2016 John Wiley & Sons, Ltd.