三维一致粘弹性人工边界及等效粘弹性边界单元

基于粘弹性人工边界推导了三维一致粘弹性人工边界单元的刚度及阻尼矩阵,利用单元矩阵等效原理采用普通有限单元构造了等效粘弹性边界单元来模拟三维粘弹性边界。均匀半空间算例与成层半空间算例证明三维粘弹性边界单元具有与集中粘弹性人工边界相近的精度,并且施加更为简便。     

无限域波动问题的有限元模型

本文基于半解析波动方程,推导了无限域粘弹性人工边界单元。在有限元波动模型中,统一了节点计算稳定性问题;提出了合理的人工边界反射系数公式,为有限元模型提供了理论基础。SH波数值分析表明本文有限元模型具有多向波动透射能力。     

采用粘弹性人工边界单元时显式算法稳定性分析

在土-结构地震反应或近场地震波动问题的分析中,常采用粘弹性人工边界单元将无限域问题转化为近场有限域问题进行计算。由于粘弹性人工边界单元的材料参数和单元尺寸与内部介质单元不同,采用显式时域逐步积分算法时,人工边界区与内部系统的数值稳定条件存在差异,但目前尚未有针对性的分析方法和研究成果,影响了显式数值稳定条件的确定和稳定积分时间步长的正确选取。针对二维粘弹性人工边界单元,该文提出一种分析显式时域逐步积分算法稳定性的方法:建立可代表人工边界区域特征的,包含人工边界单元的若干局部子系统,对各子系统的传递矩阵进行分析,给出采用显式时域逐步积分算法时各子系统的稳定条件解析解。通过对各子系统的稳定条件进行对比分析,获得了采用粘弹性人工边界单元时,显式时域逐步积分算法的统一稳定性条件。当内部介质区也满足该稳定条件时,这一条件成为使整体系统数值计算稳定的充分条件,可用于指导数值分析中离散时间步长的选取。     

地震作用下水-轴对称柱体相互作用的子结构分析方法

针对水-轴对称柱体动力相互作用问题,提出了一种地震作用下水-结构相互作用的时域子结构分析方法。基于三维不可压缩水体的波动方程和边界条件,利用分离变量法将其转换为环向解析、竖向和径向数值的二维模型;基于比例边界有限元推导了截断边界处无限域水体的动力刚度方程,并将水体内域有限元方程和人工边界处的动水压力进行耦合,从而得到结构表面的动水压力方程;将轴对称柱体结构的有限元方程与动水压力方程耦合,从而得到水-轴对称柱体结构系统的时域有限元方程;数值算例验证该文提出的水-轴对称动力相互作用的子结构方法,结果表明:该文方法具有很高的精度和计算效率。通过对水中轴对称结构地震响应和自振频率的分析表明:地震动水压力对结构自振频率和动力响应的影响随水深的增加而增大。     

求解粘弹性问题的时域自适应等几何比例边界有限元法

提出一种基于分段时域自适应算法和等几何分析的求解粘弹性问题的数值方法。利用时域分段展开,建立了递推格式的比例边界元求解方程,环向比例边界采用等几何技术离散,在继承常规比例边界有限元半解析、便于处理应力奇异性/无限域问题等优点的同时,可更准确地描述几何边界,由此进一步提高了计算精度;在时域,通过分段时域自适应计算,保证不同时间步长下的计算精度。通过数值算例,从计算精度、收敛性等方面,对所提方法的有效性进行了验证。     

粘弹性人工边界的虚位移原理

该文将结构及其近场地基作为动力平衡系统,将在人工边界上的波动分解为自由波和散射波,并将输入地震波动转化为作用于人工边界上的等效荷载以实现波动输入。基于以上假设通过分析结构及其近场地基系统的动力平衡关系和自由场的传播机制,给出了自由场的位移表达式、速度表达式,以及在人工边界上由自由场产生的等效荷载一般表达形式,最后建立了粘弹性人工边界统一的动力学积分弱解形式,同时基于有限元程序自动生成系统(FEPG)开发了粘弹性边界条件元件程序。经过计算验证:该文建立的具有粘弹性人工边界的动力学问题的积分弱解方程粘弹性边界条件元件程序可靠、正确。利用这些元件程序,在前处理中可像加位移或应力边界条件一样简便快捷地施加粘弹性边界条件。     

采用粘弹性人工边界时显式算法稳定性的改善研究

在显式动力计算中引入粘弹性人工边界时,受人工边界刚度和阻尼等因素影响,整体模型的数值积分稳定性将变得更为严格,这在一定程度上限制了粘弹性人工边界在大规模显式动力计算中的应用。该文基于对采用粘弹性人工边界的显式时域逐步积分算法稳定性条件的分析及其影响因素的研究,提出通过对人工边界附加集中质量来改善其数值积分稳定性的方法,发展了稳定性更优的改进粘弹性人工边界。为确定合理的人工边界质量值,利用基于局部子系统的稳定性分析方法推导得到改进粘弹性人工边界的稳定性条件,通过比较分析给出人工边界质量参数的建议值。采用该建议值后,粘弹性人工边界区的数值积分稳定性条件优于内部计算域的稳定性条件,整体计算模型的稳定性由内部计算域控制,此时可以用常规的稳定性判别准则来确定临界时间积分步长。数值算例表明,该文提出的粘弹性人工边界数值积分稳定性改善方法在提高计算效率的同时保持原人工边界的计算精度,具有较强的实用性。     

基于不同算法和边界条件的土中爆炸数值计算

运用共用节点算法、接触算法和任意拉格朗日-欧拉(ALE)算法对土中爆炸问题进行了数值模拟,考虑了3种不同的人工边界条件,即粘性边界、三维一致粘弹性边界和远置边界,从模型建立、模拟结果和计算时间3个方面对不同算法和边界条件下爆炸成腔效果进行了对比分析。结果表明:不同流固耦合算法各有优劣,需同时兼顾计算成本和模拟效果等因素选取合适的算法,而等效粘弹性边界单元可在保证计算精度的前提下有效降低计算成本,更好地模拟爆炸冲击过程中无限地基能量辐射阻尼效应问题。     

一种高精度圆柱形人工边界条件: 水-柱体相互作用问题

针对水-柱体动力相互作用问题,提出一种用于模拟无限域水体的圆柱形高精度时域人工边界条件。首先,基于三维可压缩水体的波动方程和边界条件,采用分离变量法建立了时空全局的精确人工边界条件;然后,将其动力刚度表示为外域模型和波导模型人工边界条件动力刚度的嵌套形式;之后,应用时间局部化方法得到时间局部的高精度人工边界条件;最后,离散高精度人工边界条件,并将其与近场有限元方程耦合,形成一种能够采用显式时间积分方法求解的时间二阶常微分方程组。数值算例表明:提出的三维圆柱形高精度人工边界条件精确、高效、稳定。     

外源波动问题数值模拟的一种实现方式

将人工边界上的波动场分解为无局部场地效应影响的自由场与局部场地效应引起的散射场两部分,无限域地基中的波动传播对人工边界的影响通过将位移场、速度场转化为应力场施加到人工边界节点上来反映,其中散射波场的模拟采用了杜修力等提出的一种远场近似解。在此基础上,利用通用有限元软件实现了斜入射条件下瞬态平面地震波作用引起的局部场地效应的时域数值模拟,为计算二维、三维局部不规则、非均匀场地的地震局部场地效应以及土结动力相互作用等外源波动问题提供了一种有效、实用的实现方式。建议方法的优点是可直接借助通用有限元软件强大的求解器和前后处理功能来计算和分析近场波动反应,且实现简便。实际上,由于通用有限元软件强大的求解功能,可方便地用于局部场地存在不均匀、非线性的情况。     

Non‐reflecting artificial boundaries for transient scalar wave propagation in a two‐dimensional infinite homogeneous layer

This paper presents an exact non‐reflecting boundary condition for dealing with transient scalar wave propagation problems in a two‐dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x →∞ and another extends toward x→‐ ∞, the non‐reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time‐dependent partial differential equation with only one spatial variable, can be further changed into a linear first‐order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non‐reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non‐reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi‐infinite media. Copyright © 2003 John Wiley & Sons, Ltd.     

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,是进行饱和两相介质近场波动问题时域计算求解的一种有效的算法。     

一种高阶精度人工边界条件:出平面外域波动问题

针对无限外域中的出平面波动问题,提出一种用于近场波动有限元分析的高阶精度人工边界条件。首先,采用变量分离法求解远场初边值问题,建立了时空全局的精确动力刚度人工边界条件;然后,发展了一种由有理函数近似和辅助变量实现构成的时间局部化方法,并将其应用于动力刚度人工边界条件,得到时间局部的高阶精度人工边界条件;最后,沿人工边界离散高阶精度人工边界条件,并将其与近场集中质量有限元方程耦合,形成对称的时间二阶常微分方程组,采用一种新的显式时间积分方法进行求解。数值算例表明:提出的高阶精度人工边界条件精确、高效、稳定并且容易在现有的有限元代码中实现。     

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.     

A damping boundary condition for coupled atomistic–continuum simulations

Many atomistic–continuum coupling techniques employ an overlapping subdomain to suppress spurious wave reflections. In this paper, we propose the imposition of a new damping condition on the overlapping subdomain to enhance the capability of such methods in eliminating spurious wave reflections. In this technique, the total displacements of the atoms in the overlapping subdomain are decomposed into fine and coarse scales. The fine scale displacements represent the oscillations which cannot be resolved by the continuum mesh and must be eliminated to avoid the artificial reflections. This is achieved by modifying the equations of motion of the fine scale displacements to include a damping term. The flexibility of the proposed technique is verified by applying it to the bridging scale method and bridging domain method. Numerical simulations of one- and two-dimensional problems demonstrate the effectiveness of the technique in enhancing the elimination of the spurious wave reflections in coupled atomistic–continuum techniques.     

波浪作用下粘弹性海床动力响应的数值分析

基于广义Biot理论,采用Kelvin-Voigt模型描述海床土骨架的应力应变时间的本构关系,从Galerkin加权余量法出发建立以土骨架位移u和孔隙流体全位移U表达的u~U形式的有限元边值方程,采用Newmark逐步积分法求解时域内动力方程。数值计算表明土骨架和孔隙流体的加速度对粘弹性海床动力响应的影响极小。当土的粘滞系数的数值小于其弹性剪切模量的大小时,体粘滞系数和偏粘滞系数的变化对超静孔压幅值的影响较小,但对有效应力幅值的影响相对显著。对于粘滞系数较大的海床,波浪荷载可能导致其变形在很短的时间内单调迅速增长至破坏。     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Note on sway, roll and yaw motions of a ship with forward speed: analytical study

Summary The paper deals with the prediction of coupled sway-roll-yaw motions of a ship with constant forward speed, subject to regular wave force. The governing equations derived in time domain after balancing the hydrodynamic and exciting forces are solved analytically while considering the motion variables, wave frequency and external force are complex quantities. Numerical experiments were carried out for a Panamax container ship under the action of a sinusoidal wave of periodicity 11.2 sec with the variations of speed and wave height. The effect of various parameters on motion responses and system stability is investigated.     

Perfectly matched layers for transient elastodynamics of unbounded domains

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non‐tangential angles‐of‐incidence and of all non‐zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337–1375], the authors presented, inter alia, time‐harmonic governing equations of PMLs for anti‐plane and for plane‐strain motion of (visco‐) elastic media. This paper presents (a) corresponding time‐domain, displacement‐based governing equations of these PMLs and (b) displacement‐based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti‐plane PML is found to be symmetric, whereas that of the plane‐strain PML is not. Numerical results are presented for the anti‐plane motion of a semi‐infinite layer on a rigid base, and for the classical soil–structure interaction problems of a rigid strip‐footing on (i) a half‐plane, (ii) a layer on a half‐plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd.