Application of the discontinuous spectral Galerkin method to groundwater flow

The discontinuous spectral Galerkin method uses a finite-element discretization of the groundwater flow domain with basis functions of arbitrary order in each element. The independent choice of the basis functions in each element permits discontinuities in transmissivity in the flow domain. This formulation is shown to be of high order accuracy and particularly suitable for accurately calculating the flow field in porous media. Simulations are presented in terms of streamlines in a bidimensional aquifer, and compared with the solution calculated with a standard finite-element method and a mixed finite-element method. Numerical simulations show that the discontinuous spectral Galerkin approximation is more efficient than the standard finite-element method (in computing fluxes and streamlines/pathlines) for a given accuracy, and it is more accurate on a given grid. On the other hand the mixed finite-element method ensures the continuity of the fluxes at the cell boundaries and it is particular efficient in representing complicated flow fields with few mesh points. Simulations show that the mixed finite-element method is superior to the discontinuous spectral Galerkin method producing accurate streamlines even if few computational nodes are used. The application of the discontinuous Galerkin method is thus of interest in groundwater problems only when high order and extremely accurate solutions are needed.     

基于全区视电阻率的线源FCSEM二维正演模拟

频率域可控源电磁是在大地电磁测深的基础上发展起来的一种人工源电磁测深法,其二维电磁响应的计算须采用数值模拟方法.本文以Matlab为程序编译工具,采用双二次插值的有限单元法,推导出相应的计算公式.为了模拟无穷远边界及满足计算机的内存需求,在保证计算精度的情况下设计了非均匀网格剖分.在程序编制中,只存储有限元系数矩阵的非零元素,大大减少了正演计算的时间.针对频率域可控源电磁法中卡尼亚电阻率在过渡区和近区畸变的问题,给出了全区视电阻率的迭代公式,并对典型的一维层状模型以及简单二维模型进行了计算.过渡区和近区数据经过校正后,可以正确反映出模型的地电特征,证明了线源下近区勘探的可能性.     

On the excitation of the earth's seismic normal modes

Summary The excitation of the earth's normal modes is formulated as an initial value problem. The static state of the earth, stressed from its hydrostatic reference situation, is considered as the initial state. The initial state is relaxed, at the time of the earthquake, by the removal of the forces maintaining the departure from hydrostatic equilibrium. Expressions are derived for the coefficients giving the relative excitation of the individual modes for the cases where these forces are compensating volume forces or compensating tractions on the faces of a dislocation. It is demonstrated that a point slip dislocation has a body force equivalent in the form of a double couple with a deviatoric moment tensor. However, for a source with volume change no moment tensor equivalent can be found. The volume change, apart from an elastic effect which can be represented by an isotropic moment tensor, has a direct gravitational effect on the excitation. This effect is due to a balanced force field consisting of a point force at the source and a continuous distribution of volume forces throughout the earth. The latter distribution, if not taken into account, may give rise to artificial phases in the frequency spectrum of the normal modes.     

A comparison of continuous mass‐lumped finite elements with finite differences for 3‐D wave propagation

The finite‐difference method on rectangular meshes is widely used for time‐domain modelling of the wave equation. It is relatively easy to implement high‐order spatial discretization schemes and parallelization. Also, the method is computationally efficient. However, the use of finite elements on tetrahedral unstructured meshes is more accurate in complex geometries near sharp interfaces. We compared the standard eighth‐order finite‐difference method to fourth‐order continuous mass‐lumped finite elements in terms of accuracy and computational cost. The results show that, for simple models like a cube with constant density and velocity, the finite‐difference method outperforms the finite‐element method by at least an order of magnitude. Outside the application area of rectangular meshes, i.e., for a model with interior complexity and topography well described by tetrahedra, however, finite‐element methods are about two orders of magnitude faster than finite‐difference methods, for a given accuracy.     

基于谱元法的频率域三维海洋可控源电磁正演模拟

高精度、快速有效的正演模拟算法是三维电磁正反演的前提.为了提高海洋电磁三维数值模拟的精度和效率,本文提出利用基于Gauss-Lobatto-Chebyshev（GLC）基函数的谱元法进行海洋可控源三维电磁正演模拟.谱元法结合有限元法和谱方法的优点.我们通过应用伽辽金加权残差法离散二次电场矢量亥姆赫兹方程,在单元内选择混合阶GLC多项式的张量积作为高阶矢量插值基函数,在求解大型稀疏线性方程组时利用直接求解器进行快速求解,从而实现了三维海洋可控源电磁快速高精度正演模拟.一维和三维模型正演结果验证了本文算法的有效性和准确性.典型模型的数值结果表明谱元法是一种有效的三维海洋可控源电磁正演数值方法,能在稀疏网格剖分情况下获得精确的海洋电磁正演模拟响应.     

波动方程的应力-速度有限元解耦交叠格式

本基于有限差分交叠格式和解耦有限元方法的基本概念,以应力－速度为变量,提出了求解波动的应力－速度有限元解耦交叠格式,这一格式不仅时空解耦,而且为显式,它适合于线性及非线性波动问题的数值模拟,已有的应力－速度有限元交叠格式（即格子法）为本的特例。通过解析解数值检验表明,本建议的方法具有较高的精度,而格子法计算精度较低。     

Dynamic-stiffness matrix in time domain of unbounded medium by infinitesimal finite element cell method

To calculate the dynamic-stiffness matrix in the time domain (unit-impulse response functions) of the unbounded medium, the infinitesimal finite element cell method based solely on the finite element formulation and working exclusively in the time domain is developed. As in the cloning algorithm, the approach is based on similarity of the unbounded media corresponding to the interior and exterior boundaries of the infinitesimal finite element cell. The derivation can be performed exclusively in the time domain, or alternatively in the frequency domain. At each time station a linear system of equations is solved. The consistent-boundary method to analyse a layered medium in the frequency domain and the viscous-dashpot boundary method are special cases of the infinitesimal finite element cell method. The error is governed by the finite element discretization in the circumferential direction, as the width of the finite-element cell in the radial direction is infinitesimal. The infinitesimal finite element cell method is thus ‘exact in the finite-element sense’. This method leads to highly accurate results for a vast class of problems, ranging from a one-dimensional spherical cavity to a rectangular foundation embedded in a half-plane.     

浮放储罐三维地震反应有限元分析

针对立式储罐,考虑液固耦合效应、地基与储罐结构的相互作用,采用有限元分析方法,对储罐在三维地震荷载作用下动反应进行了数值分析。分析结果表明：储罐三维地震加速度反应较一维地震加速度反应增加、提离高度明显放大、储罐轴向应力增加、基底剪力与弯矩增大。     

基于TrueGrid的起伏界面精准剖分及地震动模拟

利用谱元法的规则六面体单元进行网格剖分时,界面起伏较大处会出现阶梯状网格而导致模拟时产生数值散射。为消除阶梯状网格对起伏界面地震动模拟计算的影响,本文基于TrueGrid软件编写了应用程序,提出了起伏界面处六面体单元网格剖分方式,通过该程序可快速建立起伏界面处均匀的六面体网格模型。本文采取了删除四纵列拐角单元、删除一纵列拐角单元以及构造单元过渡环3种剖分方式,解决两个方向上（x-z与y-z方向）单元二合一过渡后拐角处产生扭曲单元的问题。将阶梯状网格经二合一处理后变为斜面网格,并投影至起伏界面,使得网格完全贴合起伏界面,改善了用台阶状网格近似描绘起伏界面的问题。将3种模型通过谱元法进行数值模拟计算验证了该剖分方式的正确性,对比结果发现删除拐角处一纵列单元方式与设置过渡环的方式均可使用,删除四纵列拐角单元方式不推荐使用,本文提出的方案有助于提高谱元法处理起伏界面问题的灵活性。     

基于二次场的可控源电磁法三维有限元-无限元数值模拟

本文提出了一种新的混合有限元-无限元三维可控源电磁法(CSEM)问题快速高精度正演模拟算法.首先从电场双旋度方程出发,推导了水平电偶极子源的二次场边值问题,采用无限元代替截断边界条件和有限元离散内部计算区域的新策略,达到减小计算区域的目的,基于并行直接求解技术,实现多源CSEM问题的快速精确求解.其次,通过层状解析模型测试,一方面验证了新算法的正确性,另一方面通过与其他三种已知CSEM问题求解策略进行对比,表明了本文提出的基于二次场有限元-无限元算法具有离散区域小、求解速度快和计算精度高等优点.最后,通过3D模型计算,清晰直观地模拟了场源阴影效应,为野外数据的处理与解释提供指导.     

基于二次场的海洋可控源电磁法三维有限元正演

根据库伦规范势的定义,推导出关于磁矢量势和电标量势的偏微分方程,为了克服由电流源引起的奇异性和数值模拟计算困难,将电磁总场分解为一次场和二次场,一次场由基于Schelkunoff势函数的一维正演算法得到,二次场由有限元法计算得到,实现了海洋可控源电磁法三维有限元正演算法。通过一维数值模拟实例,验证该算法的计算精度。然后,利用该算法对带海底地形的三层储层模型进行正演,分析了海底地形对海洋控源电磁场各分量产生的影响。      

基于非结构化网格的2.5-D直流电阻率自适应有限元数值模拟

2.5-D直流电阻率有限元数值模拟中,模型的剖分及加密主要通过手动实现.另外,采用的单元类型比较规则如矩形单元等,不易实现复杂模型的模拟.为解决上述问题,文中提出了一种自适应有限元算法.算法中采用稳健的后验误差估计来自动预测下一次网格的单元尺寸,直到设定的迭代条件满足为止.另外,采用非结构化三角形单元实现了任意复杂模型的灵活剖分.基于此,利用垂直接触面模型分析和对比了不同自适应策略的效率.通过对比发现,点源附近的单元得到了加密以消除源的奇异性.另外,对于任意一种策略,有限元结果均能最终收敛到精确解.最后,模拟了两个模型：2-D单个异常体模型和2-D地形模型.     

基于二次场电导率分块连续变化的三维可控源电磁有限元数值模拟

从电偶源三维地电断面可控源电磁法的二次电场边值问题及其变分问题出发,采用任意六面体单元对研究区域进行剖分,并且在单元分析中同时对电导率及二次电场进行三线性插值,实现电导率分块连续变化情况下,基于二次场的可控源电磁三维有限元数值模拟.这个新的可控源电磁三维正演方法可以模拟实际勘探中地下任意形状及电性参数连续变化的复杂模型.理论模型的计算结果表明,均匀大地计算的视电阻率误差和相位误差分别为0.002%和0.0005°.分层连续变化模型的有限元计算结果表明,其与对应的分层均匀模型解析结果有明显差异.三维异常体组合模型以及倾斜异常体等复杂模型的有限元计算结果也有效地反映了异常形态.     

斜腿夹角对V形墩连续刚构桥地震响应的影响研究

为研究斜腿夹角对V形墩连续刚构桥地震响应的影响及合理斜腿夹角角度,以一座典型V形墩预应力混凝土连续刚构桥为研究对象,采用有限元分析方法研究了斜腿夹角θ对桥梁内力及位移的影响,得出了θ对结构地震响应的影响规律和变化曲线。研究结果表明:随着斜腿夹角的增加,在纵向地震力作用下,墩底纵向弯矩逐渐减小,墩顶和主梁墩顶支撑处纵向弯矩逐渐增大;在横向地震力作用下,跨中横向弯矩逐渐减小,墩底横向弯矩逐渐增大,墩顶横向弯矩基本不变;在竖向地震力作用下,墩底和墩顶竖向弯矩逐渐增大,主梁支撑处竖向弯矩逐渐减小;斜腿夹角对纵向或横向地震力作用下结构位移影响不大,对竖向地震力作用下的位移影响较大。在满足静力设计的前提下,当两斜腿夹角为90°时,结构地震响应相对较小,受力合理性最优。研究成果可为该类桥梁的抗震设计与斜腿夹角角度选取提供参考和依据。     

A novel spherical polar finite element for the solution of the steady-state scalar wave equation in three-dimensions

A novel method has been proposed for the finite element solution of the steady-state scalar wave equation in three-dimensions. In this the governing equation and the prescribed boundary conditions in the physical space are transformed into a spherical polar space in which the radial direction is logarithmically condensed; the physical problem domain is also mapped into the new space. The transformed equation is then solved in the mapped domain using conventional finite elements. Because physical dimensions of the problem are logarithmically condensed in the proposed spherical polar space, the method is particularly suitable for solving truly three-dimensional problems in which the aspect ratio(s) is large or very large. A number of illustrative examples considered show that the proposed method is capable of a high degree of accuracy, achieved efficiently and economically. A hybrid scheme has also been proposed for dealing with awkward-shaped domains.     

Research Note: Sparsity‐Aware Multiple Microseismic Event Localization Blind to the Source Time‐Function

We consider the problem of simultaneously estimating three parameters of multiple microseimic events, i.e., the hypocenter, moment tensor, and origin time. This problem is of great interest because its solution could provide a better understanding of reservoir behavior and can help to optimize the hydraulic fracturing process. The existing approaches employing spatial source sparsity have advantages over traditional full‐wave inversion‐based schemes; however, their validity and accuracy depend on the knowledge of the source time‐function, which is lacking in practical applications. This becomes even more challenging when multiple microseimic sources appear simultaneously. To cope with this shortcoming, we propose to approach the problem from a frequency‐domain perspective and develop a novel sparsity‐aware framework that is blind to the source time‐function. Through our simulation results with synthetic data, we illustrate that our proposed approach can handle multiple microseismic sources and can estimate their hypocenters with an acceptable accuracy. The results also show that our approach can estimate the normalized amplitude of the moment tensors as a by‐product, which can provide worthwhile information about the nature of the sources.     

反应位移法抗震设计中地基弹簧系数取值的探讨

随着城市轨道交通工程抗震性能越来越受到关注,反应位移法作为一种简捷的抗震设计方法被大量应用于地下结构抗震设计中。地基弹簧系数的确定方法是反应位移法抗震设计中最为重要的因素之一,本文选取试验法、李英民法、MIDAS法和有限元法,结合北京典型地层,对比分析这四种方法地基弹簧系数取值对地下结构的弯矩、轴力和剪力的影响规律。结果表明:(1)在反应位移法计算中,地基弹簧系数取值对结构内力响应具有重要影响,对结构轴力和剪力的影响较大,而对于弯矩的影响相对较小;(2)地基弹簧系数获取方法优选试验法,在无地勘资料情况下,经验公式和有限元法可作为备选方法;(3)综合考虑四种不同地基弹簧系数确定方法对结构内力影响程度,为保证结构设计的安全性,可对试验法下结构内力进行系数修正,修正系数建议值为1.10。     

SOURCE ARRAY SCALING FOR WAVELET DECONVOLUTION*

A seismic source array is normally composed of elements spaced at distances less than a wavelength while the overall dimensions of the array are normally of the order of a wavelength. Consequently, unpredictable interaction effects occur between element and the shape of the far field wavelet, which is azimuth-dependent, can only be determined by measurements in the far field. Since such measurements are very often impossible to make, the shape of the wavelet—particularly its phase spectrum—is unknown. A theoretical design method for overcoming this problem is presented using two scaled arrays. The far field source wavelets from the source arrays have the same azimuth dependence at scaled frequencies, and the far field wavelets along any azimuth are related by a simple scaling law. Two independent seismograms are generated by the two scaled arrays for each pair of source-receiver locations, the source wavelets being related by the scaling law. The technique thus permits the far field waveform of an array to be determined in situations where it is impossible to measure it. Furthermore it permits the array design criteria to be changed: instead of sacrificing useful signal energy for the sake of the phase spectrum, the array may be designed to produce a wavelet with desired amplitude characteristics, without much regard for phase.     

A flux-limiting wetting–drying method for finite-element shallow-water models,with application to the Scheldt Estuary

We present a flux-limiting wetting–drying approach for finite-element discretizations of the shallow-water equations using discontinuous linear elements for the elevation. The key ingredient of the method is the use of limiters for generalized nodal fluxes. This method is implemented into the Second-generation Louvain-la-Neuve Ice-ocean Model (SLIM), and is verified against standard test cases. The method is further applied to the wetting and drying of sand banks in the Scheldt Estuary, which is located in northern Belgium and the southern Netherlands. The results obtained for both the benchmarks and the realistic problem illustrate the accuracy of the method in describing the hydrodynamics in the vicinity of dry areas. In particular, the method strictly conserves mass, and there is no transport through dry areas.     

三维地形频率域人工源电磁场的边界元模拟方法

提出了一种用边界元法计算频率域人工源三维地形电磁场的数值模拟方法.首先用矢量积分理论和电磁场边界条件，将上半空间（空气）和下半空间（地下介质）两个区域电磁场边值问题变为仅对地形界面的两个矢量面积分方程.然后将对地形界面的积分剖分为一系列的三角单元积分.在三角单元积分中，假设单元中电磁场为无限大气空间电磁场与地形影响的叠加,并假设地形影响为常项,这样既保证了计算精度又使得计算方法简便.通过分解和计算，每一个矢量面积分方程分解为对应三个坐标方向的三个常量线性方程，这些线性方程组成了对角占优的线性方程组，可用SSOR方法求解.文中给出了垂直磁偶源的垂直磁场地形影响的例子.