波动数值模拟中的吸收边界条件

为了提高人工边界条件在波动输入边界和自由场边界的精度,该文扩展了Higdon一阶吸收边界条件,并编制了相应的有限元计算程序。该方法将输入波分量引入到Higdon吸收边界条件当中,利用最小二乘法,用吸收边界邻域内的应变场和速度场计算Higdon边界条件公式中的参数,实现了吸收边界条件的自动更新。并且,该文提出了既适用于波...     

离散高阶Higdon-like透射边界

基于比例边界有限元法(SBFEM)半离散思想和Higdon透射微分算子提出了一种用于模拟二维层状介质标量波传播的高效离散高阶Higdon-like透射边界。对无限介质边界进行迦辽金有限元离散后,描述标量波的偏微分方程转换为局部坐标系下半离散矩阵方程组;然后使用高阶Higdon透射算子和辅助变量,在时域内得到了一个阶数不超过2阶的离散高阶透射边界。透射边界是由一组常微分方程构成,可以采用通常的时步积分方法求解,它在截断边界上非局部,在时间域局部。算例表明:该文提出的透射边界的计算精度可以随着辅助变量的增加而提高,但计算量却呈线性化增加,因而计算效率较全局方法有了显著提高。另外,由于该文的边界条件是直接建立在离散节点上的,所以它很方便与近场有限单元法耦合。     

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。     

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。     

扭转振动数值分析的粘弹性传输边界

传输边界是动力问题有限元计算中常见的边界处理方式。该文针对扭转振动引起半无限域内柱面剪切波有限元分析的传输边界,通过两种近似推导,提出了两种粘弹性传输边界,并对其计算精度进行了计算分析。数值分析结果显示,两种粘弹性边界都可以较好地模拟扭转振动分析时地基的无限性。同时,对这里考虑的扭转振动来说,粘弹性边界条件中的弹簧刚度与实际静力刚度相等时,传输边界的精度更高。     

基于波场分离的不规则地形下地震波输入方法

不规则地形条件下斜入射地震波场求解难度较大，以往的方法在计算精度和适用范围方面仍有不足。该文结合解析推导和有限元模拟，提出了一种基于波场分离技术的不规则地形条件下地震波输入方法，将地震P波和SV波在不同边界下进行波场分离：垂直入射时在侧面边界上分离为自由波场和散射波场，底部边界上分离为入射波场和边界外行场；斜入射时将输入侧对面的边界改为分离成入射波场和边界外行场；并充分考虑局部地形条件的影响，还基于改进的波动方法以便捷地输入节点力。同时对比了多组不同地震入射角度下规则场地和不规则场地的振动反应。结果表明：该方法在多类地形条件下计算精度与效率均较高，适用范围广且易于推广至复杂场地条件，并发现地震波入射角度和局部场地条件对地表位移响应影响较大。该研究可为不规则地形条件下的振动响应分析提供有效的手段。     

谱元法与透射边界的配合使用及其稳定性研究

在无限域波动模拟中引入透射边界条件时,目前多将边界上的透射公式与内域的有限元法结合使用,其计算精度由有限元方法决定,而谱元法因结合有限元和频谱法的优势则比有限元空间域积分具有更高的计算精度。该文基于谱元法非等距网格划分特性,研究了内域的谱元法与边界上的透射公式结合的理论方法,给出了相应的透射公式使用方法,并基于建立的谱元法波动数值模型探讨了透射公式的稳定性问题。研究表明:空间域插值系数需控制在一个合理范围内,空间域插值方法相对于时间域插值方法更为稳定,高频失稳出现可能性相对较小;Gamma算子的使用可提高模拟的精度,采用Gamma算子后对于高阶透射公式仍可出现低频漂移现象,可结合降阶消漂的方式实现稳定精度高的透射边界应用。     

求解无限域动力刚度矩阵的双渐近算法

采用连分式算法可以有效地求解无限域动力刚度表示的比例边界有限元方程, 它具有收敛范围广、收敛速度快等优点. 该文在高频渐近连分式算法的基础上考虑了低频渐近, 发展了一种针对矢量波动方程的双渐近算法. 随着展开阶数的增加, 双渐近算法可以在全频域范围内快速逼近准确解. 引入了系数矩阵?X(i)来增强连分式算法的数值稳定性. 通过在高频极限、低频极限时满足动力刚度表示的比例边界有限元方程, 建立了递推关系以求得动力刚度矩阵. 通过二维半无限楔形体、三维均质弹性半空间数值算例表明, 双渐近算法比单渐近算法更稳定、优越.     

一种新的有限域动力刚度改进连分式求解算法

比例边界有限元法仅需离散边界,网格划分灵活,且易于采用高阶单元,是结构动力分析的理想方法。针对有限域动力问题,基于广义特征值分解对动力刚度表示的比例边界有限元方程进行模态变换。通过选取特定的因子矩阵,简化了改进连分式算法的求解流程,提出了一种新的有限域动力刚度改进连分式求解算法。在动力刚度连分式渐近解的基础上引入辅助变量,建立了有限域动力问题的运动方程,其系数矩阵对称稀疏,可以利用现有的有限元求解器求解。正八边形板和重力坝算例表明,新算法具有良好的数值稳定性和计算精度,适用于实际工程问题的动力响应分析。     

求解弹性波有限差分法中自由边界处理方法的对比

自由边界条件在计算方法中的数值表征是地震波模拟中的一个重要内容,表征的有效性直接关系到所得波场能否代表地表介质特性的真实响应。该文评估了交错网格有限差分法中5 种常用自由边界处理方法:直接法、应力镜像法、改进应力镜像法、横向各向同性介质替换法和声学边界替换法,并与有限元法模拟结果进行了对比,波形曲线直观比较及波幅比与相关系数定量比较显示横向各向同性介质替换法与有限法模拟结果一致性最好。进一步的层状介质模型弹性波数值模拟结果表明:横向各向同性介质替换法的精度和可靠性最高,能真实表征地表介质中的地震波传播。     

Accurate H-shaped absorbing boundary condition in frequency domain for scalar wave propagation in layered half-space

An accurate absorbing boundary condition (ABC) is developed in frequency domain for finite element analysis of scalar wave propagation in unbounded layered half-space. The proposed ABC is H-shaped line that consists of two parts: a new ABC at horizontal bottom boundary of finite domain to replace semiinfinite strip below horizontal boundary and between two vertical boundaries, and a general consistent ABC at vertical lateral boundary to replace semiinfinite layered half-space outside vertical boundary. The key point for constructing the ABC is that a new continued fraction (CF) is presented to expand dynamic stiffness of underlying half-space, and the CF-based stress-displacement relationship is then transformed into an auxiliary variable system with square of horizontal wavenumber. The ABC has only one undetermined real parameter that is the CF-order independent of frequency and incidence angle of propagating outgoing waves. The parameter can be chosen relatively small value to achieve an accurate ABC. Moreover, the ABC can couple seamlessly with finite element method of finite domain. The finite domain can be chosen very small size due to high accuracy of the ABC. Numerical examples are finally given to demonstrate the effectiveness of the ABC.     

地震作用下深厚土层-结构相互作用的高效分析方法

基于黏弹性边界和将场地反应转化为等效地震荷载的有限元直接法是目前进行地震作用下土-结构相互作用分析的常用时程分析方法之一。当土层厚度较深时,整个深厚土层-结构系统的有限元模型自由度数目较多,尤其对于三维问题,计算效率低。该文提出一种高效分析方法,即一维场地反应分析仍然针对整个深厚土层,在后续的土-结构相互作用分析中将土-结构计算模型的底面人工边界从深厚土层底面（基岩面）向上移动到接近结构的位置,通过缩减土-结构相互作用模型尺寸来提高计算效率。采用理论分析与数值算例,通过与采用整个深厚土层的传统土-结构相互作用分析结果对比,说明提出的高效分析方法能够满足精度要求,并且给出底面人工边界位置以及边界条件和地震动输入方式的建议。     

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.     

Arch dam–fluid interactions: By fem–bem and substructure concept

Arch dams can be conveniently analysed by the finite element method. For dam–fluid interaction problems, the fluid domain may be more conveniently handled by the boundary element method as a substructure first before connecting to the dam substructure. The added-mass matrix calculated from the fluid domain is symmetrized and lumped first so that the banded and symmetrical characteristics of the finite element method are retained. In the boundary element formulation, a mirror image method and quadratic elements are used for computational efficiency and accuracy. The strong singular terms are handled by using a solution which satisfies the governing equation and the free surface boundary condition. Infinite boundary conditions at the upstream of the reservoir can be reasonably approximated from the fundamental solution with accurate results, if the interior pressure distribution in the fluid domain is neglected. Numerical solutions on hydrodynamic pressure distribution and the natural frequencies of the dam–reservoir system with various water levels are obtained and compared with available analytical and experiment results.     

Transient analysis of three-dimensional dynamic interaction between multilayered soil and rigid foundation

The study of dynamic soil-structure interaction is significant to civil engineering applications, such as machine foundation vibration, traffic-induced vibration, and seismic dynamic response. The scaled boundary finite element method (SBFEM) is a semi-analytical algorithm, which is used to solve the dynamic response of a three-dimensional infinite soil. It can automatically satisfy the radiation boundary condition at infinity. Based on the dynamic stiffness matrix equation obtained by the modified SBFEM, a continued fraction algorithm is proposed to solve the dynamic stiffness matrix of layered soil in the frequency-domain. Then, the SBFEM was coupled with the finite element method (FEM) at the interface to solve the dynamic stiffness matrices of the rigid surface/buried foundation. Finally, the mixed-variable algorithm was used to solve the three-dimensional transient dynamic response of the foundation in the time domain. Numerical examples were performed to verify the accuracy of the proposed algorithm in solving the dynamic stiffness matrix of the infinite domain in the frequency domain and the dynamic transient displacement response of the foundation in the time domain. Compared with the previous numerical integration technique, the dynamic stiffness matrix in the frequency domain calculated by using the proposed algorithm has higher accuracy and higher efficiency.     

The scaled boundary finite element method in structural dynamics

The scaled boundary finite element method is extended to solve problems of structural dynamics. The dynamic stiffness matrix of a bounded (finite) domain is obtained as a continued fraction solution for the scaled boundary finite element equation. The inertial effect at high frequencies is modeled by high‐order terms of the continued fraction without introducing an internal mesh. By using this solution and introducing auxiliary variables, the equation of motion of the bounded domain is expressed in high‐order static stiffness and mass matrices. Standard procedures in structural dynamics can be applied to perform modal analyses and transient response analyses directly in the time domain. Numerical examples for modal and direct time‐domain analyses are presented. Rapid convergence is observed as the order of continued fraction increases. A guideline for selecting the order of continued fraction is proposed and validated. High computational efficiency is demonstrated for problems with stress singularity. Copyright © 2008 John Wiley & Sons, Ltd.     

A continued‐fraction‐based high‐order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry

A high‐order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued‐fraction solution of the dynamic‐stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued‐fraction solution and introducing auxiliary variables, a high‐order local transmitting boundary is formulated as an equation of motion with symmetric and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high‐order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

波动谱元模拟中透射边界稳定性分析

无限域波动数值模拟中,人工边界的稳定性是获得可靠模拟结果的前提。具有高阶精度的谱元法和透射边界两者结合的数值模拟方案显示出较好的模拟精度和数值稳定性,然而,仍然存在数值失稳现象,其失稳机理和稳定条件尚不明确,相应的理论分析极为欠缺。该文针对透射边界在高阶谱元法中的稳定性,依据高阶谱单元中非等间距节点的周期延拓特点,通过构建内域和边界数值格式的向量形式来分析人工边界反射系数。进而保证边界对谱元法中存在的真实模态和虚假模态的反射系数均小于等于1,从而得到透射边界的稳定条件,其表现为无量纲边界参数和谱元参数之间的关系,其含义为透射边界人工波速与介质物理波速的比值限定在一定范围内。同时揭示了透射边界引发高频失稳的机理,即边界对谱元法中虚假模态的反复反射放大所致。最后采用数值实验验证了透射边界稳定条件。     

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

A new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ?? which encloses a finite computational domain Ω. On ?? a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ??, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C⁰ finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.