Hydrodynamic pressure on arch dams—by a mapping finite element method

A novel Finite Element Method (FEM) is proposed for the analysis of the uncoupled hydrodynamic pressures generated on arch dams due to a steady-state ground acceleration. In this method the equation governing hydrodynamic pressures and also the prescribed boundary conditions are all transformed from the Cartesian space to a logarithmically condensed cylindrical polar space; in this process the physical configuration of the reservoir-dam is also mapped into an ‘image’ domain. The transformed governing equation is then solved in the image domain, subject to the transformed boundary conditions, using standard finite elements. Because physical dimensions are logarithmically condensed in the image space, the proposed method is particularly suitable for dealing with large or very large aspect-ratio reservoir-dam systems, economically and efficiently. The high degree of accuracy which the proposed method is capable of, and also the simple way in which it can be applied to complex reservoir-dam shapes, have been demonstrated by means of examples. The method has also been applied to study the uncoupled hydrodynamic pressures on the upstream face of a cylindrical arch dam, generated by a steady-state vertical ground acceleration.     

The w-plane finite element method for the solution of the steady-state scalar wave equation in two dimensions

A novel finite element method has been described in this paper for the solution of the steady-state scalar wave equation in two dimensions. In this method the physical domain of the problem is mapped into an image domain; the governing equation and the prescribed boundary conditions are also appropriately transformed. Because logarithmic mapping functions are used, the physical domain is logarithmically condensed in the image plane. The method is therefore particularly suitable for the efficient and economical solution of large or very large aspect ratio problems. The high degree of accuracy which the method is capable of is demonstrated by means of two typical examples. Possible extension of the method to complex problems including non-linearity, multiplicity, etc. is also discussed.     

A semi-analytical artificial boundary for time-dependent elastic wave propagation in two-dimensional homogeneous half space

This paper presents a time-dependent semi-analytical artificial boundary for numerically simulating elastic wave propagation problems in a two-dimensional homogeneous half space. A polygonal boundary is considered in the half space to truncate the semi-infinite domain, with an appropriate boundary condition imposed. Using the concept of the scaled boundary finite element method, the wave equation of the truncated semi-infinite domain is represented by the partial differential equation of non-constant coefficients. The resulting partial differential equation has only one spatial coordinate variable and time variable. Through introducing a few auxiliary functions at the truncated boundary, the resulting partial differential equations are further transformed into linear time-dependent equations. This allows an artificial boundary to be derived from the time-dependent equations. The proposed artificial boundary is local in time, global at the truncated boundary and semi-analytical in the finite element sense. Compared with the scaled boundary finite element method, the main advantage in using the proposed artificial boundary is that the requirement for solving a matrix form of Lyapunov equation to obtain the unit-impulse response matrix is avoided, so that computer efforts are significantly reduced. The related numerical results from some typical examples have demonstrated that the proposed artificial boundary is of high accuracy in dealing with time-dependent elastic wave propagation in two-dimensional homogeneous semi-infinite domains.     

基于感兴趣区域CT重建的运动估计

为了对感兴趣区域运动信息进行估计,首先扩展中心切片定理并推导微分约束条件,然后用有限Hilbert变换算法求出方位角与径向图像导函数,最后用改进的光流方程求解极坐标系下的运动场。与一般运动估计算法相比,该算法不使用图像域差分运算计算导函数,因此具有更高精度,并且极坐标系下的运动估计更加适合医学诊断需要。数值仿真也验证了该算法的可行性。     

Stability and identification for rational approximation of frequency response function of unbounded soil

Exact representation of unbounded soil contains the single output–single input relationship between force and displacement in the physical or transformed space. This relationship is a global convolution integral in the time domain. Rational approximation to its frequency response function (frequency‐domain convolution kernel) in the frequency domain, which is then realized into the time domain as a lumped‐parameter model or recursive formula, is an effective method to obtain the temporally local representation of unbounded soil. Stability and identification for the rational approximation are studied in this paper. A necessary and sufficient stability condition is presented based on the stability theory of linear system. A parameter identification method is further developed by directly solving a nonlinear least‐squares fitting problem using the hybrid genetic‐simplex optimization algorithm, in which the proposed stability condition as constraint is enforced by the penalty function method. The stability is thus guaranteed a priori. The infrequent and undesirable resonance phenomenon in stable system is also discussed. The proposed stability condition and identification method are verified by several dynamic soil–structure‐interaction examples. Copyright © 2009 John Wiley & Sons, Ltd.     

卫星测高混合边值问题的球谐级数解法

研究了球界面下卫星测高问题的解法,利用有限逼近方法得到了下列结论：若陆地部分是球冠,则卫星测高问题的解可以转换成关于球谐级数位系数的线性方程组,并且位系数的阶和次是以分离形式出现的,从而确保该解法具有实用意义. 利用重力场360阶模型进行模拟计算的结果表明：该解法得到的位系数的相对精度达到了10－１１. 同时证明了常用的Stokes问题、Dirichlet问题、Neumann问题可以看成卫星测高问题的特殊情况.     

Reflection of equatorial Kelvin waves at eastern ocean boundaries Part I: hypothetical boundaries

A baroclinic shallow-water model is developed to investigate the effect of the orientation of the eastern ocean boundary on the behavior of equatorial Kelvin waves. The model is formulated in a spherical polar coordinate system and includes dissipation and non-linear terms, effects which have not been previously included in analytical approaches to the problem. Both equatorial and middle latitude response are considered given the large latitudinal extent used in the model. Baroclinic equatorial Kelvin waves of intraseasonal, seasonal and annual periods are introduced into the domain as pulses of finite width. Their subsequent reflection, transmission and dissipation are investigated. It is found that dissipation is very important for the transmission of wave energy along the boundary and for reflections from the boundary. The dissipation was found to be dependent not only on the presence of the coastal Kelvin waves in the domain, but also on the period of these coastal waves. In particular the dissipation increases with wave period. It is also shown that the equatorial β-plane approximation can allow an anomalous generation of Rossby waves at higher latitudes. Nonlinearities generally have a small effect on the solutions, within the confines of this model.     

深海热液硫化物矿体3D瞬变电磁正演

深海热液硫化物矿体瞬变电磁的正演是考虑深海环境的全空间条件下三维体的涡流电磁响应.采用全空间矢量有限元法模拟计算深海热液硫化物矿的三维瞬变电磁响应,对硫化物矿体采用矩形单元模型剖分,应用Galerkin法推导有限元方程,先计算频率域响应,再通过Fourier反变换将其转换至时间域,得出深海热液硫化物矿矿体的瞬变电磁响应.并用双半空间模型的解析解检验了全空间矢量有限元法模拟计算算法和程序的正确性,最后按照等比例缩小电磁物理实验原则,比对数值计算和物理实验结果论证了全空间3D模型数值的正确性.结果表明：对于海水、矿体以及围岩复杂电磁边界,应用全空间矢量有限元法模拟计算深海热液硫化物矿瞬变电磁响应异常与物理模拟结果一致,而且计算方法简单精确,异常幅值明显,边界清晰.     

基于Lorenz规范的空间-波数域大地电磁三维正演

大地电磁勘探方法由于其成本低、施工简单、探测深度广等优点,广泛应用于矿产资源普查、油气勘探和深部构造研究等领域.如何提高大规模三维大地电磁数值模拟的精度和效率一直是研究热点.本文基于空间-波数域方法,实现了基于Lorenz规范的空间-波数域三维大地电磁数值模拟.基于二次场计算原理,引入Lorenz规范,将Maxwell方程组转化为关于二次场矢量位的亥姆霍兹方程;利用水平方向二维傅里叶变换,将空间域三维偏微分方程转换为多个波数下相互独立的常微分方程,方程采用二次插值有限单元法计算,得到定带宽线性方程组,方程计算量小、并行性好,采用追赶法求解,提高了算法效率;引入压缩算子,用迭代法逐次逼近真实解.充分利用了空间-波数域方法数值精度高、内存需求少、效率高的特点.设计棱柱体模型验证了算法的正确性、分析了算法的收敛性,说明算法对不同频率、不同电导率对比度模型均具有很好的适应性.利用Dublin（DTM1）模型进行三维大地电磁数值模拟,结果表明：在满足精度要求的前提下,空间-波数域算法比空间域算法占用内存少、耗时短;相比基于Coulomb规范的空间-波数域算法,基于Lorenz规范的空间-波数域方法耗时更短、占用内存更少,效率提高至少3倍以上,体现了新方法的优势.

     

各向异性介质空间-波数混合域直流电阻率法三维数值模拟研究

将各向异性介质分成各向同性背景介质和各向异性异常介质,并提出一种空间-波数混合域方法实现了各向异性介质下直流电阻率法的三维数值模拟.不同于传统直流电阻率数值模拟方法,本文算法直接对空间域异常电位满足的偏微分方程沿水平方向进行二维傅里叶变换,使水平方向转换成波数域,保留垂直方向为空间域,可根据地下介质电流密度变化的快慢灵活剖分.这样可把空间域异常电位满足的三维偏微分方程转化成不同波数满足的一维常微分方程,把一个大规模三维数值模拟问题分解为多个一维数值模拟问题,利用一维有限单元法求解方程组,并通过采用压缩算子迭代计算,最终获得较为精确的数值解.与自适应有限单元法对比验证了本文算法的正确性;测试了算法的收敛性,结果表明在满足精度要求的情况下,算法的收敛性只与异常体和围岩之间的电导率差异相关,而与异常体大小和埋深无关;分析了算法计算效率,结果表明算法的计算效率与剖分网格节点成线性关系,算法可在微型计算机中较快计算出剖分节点总数超过千万的各向异性模型的结果;设计简单观测系统并验证其具有反映地下各向异性结构特性的能力;最后模拟异常体沿着不同方向旋转不同角度时的响应特征,对比分析可知异常体为各向异性情况下,沿不同方向旋转相同角度或沿同一个方向旋转不同角度,在同一个位置上测量得到的视电阻率有差异.

     

双轴各向异性介质中回线源瞬变电磁三维拟态有限体积正演算法

采用模拟离散的有限体积法实现了双轴各向异性地层回线源瞬变电磁三维正演.首先引入内积定义,采用自然边界条件,将瞬变电磁法的控制方程转化为弱形式表示.将计算区域划分为一系列的控制体积单元,采用交错网格对控制方程进行模拟有限体积空间离散,包括旋度算子离散和空间内积离散.基于斯托克斯定理的旋度积分定义公式实现旋度算子离散.中点平均实现电导率双轴各向异性的空间内积离散,从而得到离散化的控制方程.时间步迭代采用无条件稳定的欧拉后向差分格式.并通过均匀全空间中稳定电流回线源的磁场解析表达式得到回线源初始时刻的电磁场分布.为了同时保证计算精度和效率,本文采用分段等间隔的时间步迭代,利用直接法求解器PARDISO实现其快速求解.最后通过对比层状模型和各向异性半空间模型的正演计算结果,验证了本文算法的计算精度和计算效率;计算三维双轴各向异性模型的正演响应可知,水平方向电导率变化对电磁响应产生显著影响,而垂直方向的电导率变化对电磁响应几乎没有影响.产生这一现象的主要原因是回线源产生的感应电流主要是水平方向的,因此响应主要受到水平方向电导率的影响,垂直方向的电导率影响很小.

     

基于矢量位和标量位的空间波数混合域电磁三维正演模拟

高效、高精度电磁三维数值模拟是制约大规模电磁数据精细化三维反演成像、人机交互定量解释的核心问题.针对一问题,本文提出一种基于矢量位和标量位的空间波数混合域电磁场三维数值模拟方法.该方法利用沿水平方向的二维傅里叶变换将空间域矢量位和标量位耦合偏微分方程组转换为波数之间相互独立的常微分耦合方程组,将一个大规模三维问题分解为多个一维小问题,具有高度并行性,由此大大减少了计算量和存储量;保留垂向为空间域,浅层网格剖分适当加密,深层网格剖分适当稀疏,有效兼顾了计算精度与计算效率;采用有限单元法求解不同波数的常微分方程,充分利用追赶法求解定带宽线性方程组的高效性进一步提高数值模拟效率.在模型算例中,设计棱柱体模型验证了本文方法的正确性、计算精度和计算效率.数值试验结果表明本文方法具有数值精度高、并行度高、占用内存小、计算效率高的特性,比传统有限单元法三维数值模拟方法计算效率高1~2个数量级,且网格剖分规模越大,该方法计算效率优势越明显.

     

Analytical layer-element solution to axisymmetric dynamic response of transversely isotropic multilayered half-space

Starting with the governing equations of motion and the constitutive equations of transversely isotropic elastic body, and based on the corresponding algebraic operations and the Hankel transform, the analytical layer-elements of a finite layer and a half-space are obtained in the transformed domain. According to the continuity conditions between adjacent layers, the global stiffness matrix equation is obtained by assembling the analytical layer-element of each single layer. The solutions in the transformed domain are acquired by introducing the boundary conditions into the global stiffness matrix equation, and thus, the corresponding solutions in frequency domain are achieved by taking the inversion of Hankel transform. Finally, some numerical examples are given to illustrate the accuracy of the proposed method, and to study the influence of properties and the frequency of excitation on the dynamic response of the medium.     

A generalized transformation approach for simulating steady-state variably-saturated subsurface flow

A numerical method is proposed to accurately and efficiently compute a direct steady-state solution of the nonlinear Richards equation. In the proposed method, the Kirchhoff integral transformation and a complementary transformation are applied to the governing equation in order to separate the nonlinear hyperbolic characteristic from the linear parabolic part. The separation allows the transformed governing equation to be applied to partially- to fully-saturated systems with arbitrary constitutive relations between primary (pressure head) and secondary variables (relative permeability). The transformed governing equation is then discretized with control volume finite difference/finite element approximations, followed by inverse transformation. The approach is compared to analytical and other numerical approaches for variably-saturated flow in 1-D and 3-D domains. The results clearly demonstrate that the approach is not only more computationally efficient but also more accurate than traditional numerical solutions. The approach is also applied to an example flow problem involving a regional-scale variably-saturated heterogeneous system, where the vadose zone is up to 1 km thick. The performance, stability, and effectiveness of the transform approach is exemplified for this complex heterogeneous example, which is typical of many problems encountered in the field. It is shown that computational performance can be enhanced by several orders of magnitude with the described integral transformation approach.     

弱形式时域完美匹配层

应用高精度人工边界条件可有效提升近场波动数值模拟计算效率.完美匹配层是吸收层形式高精度人工边界条件,匹配层内场方程和界面条件通常分别采用复坐标延伸技术变换强形式无限域内波动方程和界面条件得到,亦曾将无限域界面条件当作匹配层界面条件.场方程和界面条件构建过程相互独立,可能出现匹配不合理而引发数值失稳、计算精度低下等问题.本文提出采用复坐标延伸技术变换弱形式无限域波动方程以构建完美匹配层的方法.弱形式波动方程耦合了波动方程及界面条件,进而规避了变换后所得场方程与界面条件之间的匹配不合理问题.新方法可直接建立弱形式匹配层,在此基础上亦可给出强形式匹配层.弱形式便于有限元离散,强形式便于有限差分离散.基于弱形式完美匹配层,结合勒让德谱元建立了弹性介质近场波动谱元模拟方案.利用算例验证了新方案的精度及数值稳定性.本文工作可直接推广至多相耦合介质近场波动数值模拟.

     

复阻尼多自由度系统动力分析的模态叠加法

工程实际中,复阻尼多自由度系统的瞬态响应过去一直是通过频域方法求解的。频域方法的一个突出问题是求传递函数矩阵的计算工作量过大。本文给出一种有效而实用的时域解法———实模态叠加法,此外还介绍了复模态叠加法。实模态叠加法是基于一个事实,即n维复向量空间中的复向量可以在n维实向量空间中的一组线性无关的实向量构成的基下表出;另外,就此方法还讨论了复阻尼多自由度系统初始运动条件的给出和转换问题。复模态叠加法则是通过变量替换的方法,变其中的复特征值问题为形式上的一个实特征值问题来解决的。     

均质地基中桩-桩位移相互作用系数的有限元分析

在实际工程分析中用有限元法模拟无限域需要考虑很大一部分桩周土体来保证计算的精度,从而导致计算量的增大,同时对计算机的要求也很高。特别是对于大规模的群桩问题,有限元的计算工作量使一般的计算机无法满足其要求,这就限制了其在实际工程中的应用。本文基于叠加原理用有限元法计算桩-桩位移相互作用系数,不仅能减少群桩的计算量,又能提高计算精度。     

A method for nonlinear dynamic response analysis of semi-infinite foundation soil

This paper presents development of a special finite difference method for the nonlinear dynamic response analysis of semi-infinite foundation soil. Semi-infinite domain is mapped into the finite domain using special mapping. For the region of engineering interest, mapping is isometrical, and for far field, shrink mapping which transforms an infinite interval into a finite interval is adopted. Using linear and nonlinear constitutive models, the responses of semi-infinite foundation soil are computed using a proposed method with a small mesh model and an extensive mesh model. Surface loadings or incident earthquake waves are applied to the models in the computations. Good agreements were obtained among the theoretical and computed results of the two models and the effectiveness of the proposed method was demonstrated.     

用边界单元法模拟二维地形对大地电磁场的影响

本文用边界单元法模拟二维地形对大地电磁场的影响.首先用格林公式将二维大地电磁的边值问题转变成积分方程,然后用边界单元法解积分方程,得到地形上的大地电磁场和它的法向导数,由此可计算电阻率.与有限单元法相比,本方法剖分后的地形与实际地形的拟合程度高,向计算机输入原始数据的工作简单,可以在微机上计算地形引起的视电阻率.本方法适合于在野外生产现场进行大地电磁法的地形改正.计算表明,地形对H_x型波的影响比对E_x型波严重得多;随着周期的增长,地形对H_x型波的影响变得稳定.     

地磁学中的球面各向同性随机矢量场

从地磁场随机模拟的需要出发，研究球面各向同性随机矢量场．首先指出，与三维空间中的随机矢量场不同，在球上均匀性蕴含于各向同性；三维均匀各向同性随机矢量场一定是球上各向同性随机矢量场，反之不真．其次，以地核磁场的随机模型为例，给出了球面各向同性模型的空间两点关联张量的并矢表示，并将其在不变坐标系分解；利用核幔边界地球主磁场的球谐功率谱的幂级数拟合模型，估算空间两点关联张量．因此，即使根据地球主磁场观测认定地核场具有球面各向同性，亦不能推定地核内部磁流体运动是三维均匀各向同性的．