基于紧凑存储的任意四边形网格有限元法地震波场数值模拟

采用有限元法精确模拟复杂介质条件下的地震波场,一般采用三角单元,但在节点数相同的情况下,三角单元的计算精度不如矩形单元高,采用三角单元模拟复杂界面时,编制确定结构刚度矩阵非零元素位置的程序也较为麻烦。采用矩形单元离散含有倾斜或起伏界面的地质模型时,无法避免绕射噪声,若加密网格又会增加计算量。为此,本文基于任意四边形单元模拟声波的传播,在倾斜或起伏界面条件下,可以有效避免因离散引起的“阶梯状”界面,在不增加计算量以及内存占用的前提下,有效地消除离散绕射噪声。采用对角的集中质量矩阵代替一致质量矩阵,避免矩阵的求逆运算,从而提高显式有限元法的计算效率;对结构刚度矩阵采用紧凑存储格式,每一行需要存储的元素最多为5个,同时零元素不参与运算,既减少内存的占用,又极大地提高计算效率。      

Optimal implicit staggered‐grid finite‐difference schemes based on the sampling approximation method for seismic modelling

We propose new implicit staggered‐grid finite‐difference schemes with optimal coefficients based on the sampling approximation method to improve the numerical solution accuracy for seismic modelling. We first derive the optimized implicit staggered‐grid finite‐difference coefficients of arbitrary even‐order accuracy for the first‐order spatial derivatives using the plane‐wave theory and the direct sampling approximation method. Then, the implicit staggered‐grid finite‐difference coefficients based on sampling approximation, which can widen the range of wavenumber with great accuracy, are used to solve the first‐order spatial derivatives. By comparing the numerical dispersion of the implicit staggered‐grid finite‐difference schemes based on sampling approximation, Taylor series expansion, and least squares, we find that the optimal implicit staggered‐grid finite‐difference scheme based on sampling approximation achieves greater precision than that based on Taylor series expansion over a wider range of wavenumbers, although it has similar accuracy to that based on least squares. Finally, we apply the implicit staggered‐grid finite difference based on sampling approximation to numerical modelling. The modelling results demonstrate that the new optimal method can efficiently suppress numerical dispersion and lead to greater accuracy compared with the implicit staggered‐grid finite difference based on Taylor series expansion. In addition, the results also indicate the computational cost of the implicit staggered‐grid finite difference based on sampling approximation is almost the same as the implicit staggered‐grid finite difference based on Taylor series expansion.     

Transient analysis of wave propagation in non‐homogeneous elastic unbounded domains by using the scaled boundary finite‐element method

The scaled boundary finite‐element method has been developed for the dynamic analysis of unbounded domains. In this method only the boundary is discretized resulting in a reduction of the spatial dimension by one. Like the finite‐element method no fundamental solution is required. This paper extends the scaled boundary finite‐element method to simulate the transient response of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. To reduce the number of degrees of freedom and the computational cost, the technique of reduced set of base functions is applied. The scaled boundary finite‐element equation for an unbounded domain is reformulated in generalized coordinates. The resulting acceleration unit‐impulse response matrix is obtained and assembled with the equation of motion of standard finite elements. Numerical examples of non‐homogeneous isotropic and transversely isotropic unbounded domains demonstrate the accuracy of the scaled boundary finite‐element method. Copyright © 2006 John Wiley & Sons, Ltd.     

Three‐dimensional modelling of controlled‐source electromagnetic surveys using an edge finite‐element method with a direct solver

A fully three‐dimensional finite‐element algorithm has been developed for simulating controlled‐source electromagnetic surveys. To exploit the advantages of geometric flexibility, frequency‐domain Maxwell's equations of the secondary electric field were discretised using edge‐based finite elements while the primary field was calculated analytically for a horizontally layered‐earth model. The resulting system of equations for the secondary field was solved using a parallel version of direct solvers. The accuracy of the algorithm was successfully verified by comparisons with integral‐equations and iterative solutions, and the applicability to models containing large conductivity contrasts was verified against published data. The advantages of geometry‐conforming meshes have been demonstrated by comparing different mesh systems to simulate an inclined sheet model. A comparison of the performance between direct and iterative solvers demonstrated the superior efficiency of direct solvers, particularly for multisource problems.     

Dynamic FE simulation of four‐story steel frame modeled by solid elements and its validation using results of full‐scale shake‐table test

Dynamic finite element analyses of a four‐story steel building frame modeled as a fine mesh of solid elements are performed using E‐Simulator, which is a parallel finite element analysis software package for precisely simulating collapse behaviors of civil and building structures. E‐Simulator is under development at the National Research Institute for Earth Science and Disaster Prevention (NIED), Japan. A full‐scale shake‐table test for a four‐story frame was conducted using E‐Defense at NIED, which is the largest shaking table in the world. A mesh of the entire structure of a four‐story frame with approximately 19 million degrees of freedom is constructed using solid elements. The density of the mesh is determined by referring to the results of elastic–plastic buckling analyses of a column of the frame using meshes of different densities. Therefore, the analysis model of the frame is well verified. Seismic response analyses under 60, 100, and 115% excitations of the JR Takatori record of the 1995 Hyogoken‐Nanbu earthquake are performed. Note that the simulation does not reproduce the collapse under the 100% excitation of the Takatori record in the E‐Defense test. Therefore, simulations for the 115% case are also performed. The results obtained by E‐Simulator are compared with those obtained by the E‐Defense full‐scale test in order to validate the results obtained by E‐Simulator. The shear forces and interstory drift angles of the first story obtained by the simulation and the test are in good agreement. Both the response of the entire frame and the local deformation as a result of elastic–plastic buckling are simulated simultaneously using E‐Simulator. Copyright © 2014 John Wiley & Sons, Ltd.     

Two‐dimensional physically based finite element runoff model for small agricultural watersheds: I. Model development

Soil erosion by water is the root cause of ecological degradation in the Shiwalik foothills of Northern India. Simulation of runoff and its component processes is a pre‐requisite to develop the management strategies to tackle the problem, successfully. A two‐dimensional physically based distributed numerical model, ROMO2D has been developed to simulate runoff from small agricultural watersheds on an event basis. The model employs the 2‐D Richards equation with sink term to simulate infiltration and soil moisture dynamics in the vadoze zone under variable rainfall conditions, and 2‐D Saint‐Venant equations under the kinematic wave approximation along with Manning's equation as the stage‐discharge equation for runoff routing. The various flow‐governing equations have been solved numerically by employing a Galerkin finite element method for spatial discretization using quadrilateral elements and finite difference techniques for temporal solutions. The ROMO2D computer program has been developed as a class‐based program, coded in C + + in such a way that with minor modifications, the model can be used to simulate runoff on a continuous basis. The model writes output for a runoff hydrograph of each storm. Model development is described in this paper and the results of model testing and field application are to be presented in a subsequent paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Time‐harmonic response of non‐homogeneous elastic unbounded domains using the scaled boundary finite‐element method

The scaled boundary finite‐element method is extended to simulate time‐harmonic responses of non‐homogeneous unbounded domains with the elasticity modulus and mass density varying as power functions of spatial coordinates. The unbounded domains and the elasticity matrices are transformed to the scaled boundary coordinates. The scaled boundary finite‐element equation in displacement amplitudes are derived directly from the governing equations of elastodynamics. To enforce the radiation condition at infinity, an asymptotic expansion of the dynamic‐stiffness matrix for high frequency is developed. The dynamic‐stiffness matrix at lower frequency is obtained by numerical integration of ordinary differential equations. Only the boundary is discretized yielding a reduction of the spatial dimension by one. No fundamental solution is required. Material anisotropy is modelled without additional efforts. Examples of two‐ and three‐dimensional non‐homogeneous isotropic and transversely isotropic unbounded domains are presented. The results demonstrate the accuracy and simplicity of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

电导率任意各向异性海洋可控源电磁三维矢量有限元数值模拟

现有海洋可控源电磁三维数值模拟方法大多基于电导率各向同性介质理论,不能模拟海底地层电导率各向异性的实际情况.本文给出了电导率各向异性三维介质中电性源海洋可控源电磁二次电场的边值问题以及相应的变分问题,采用长方体单元对研究区域剖分,将场分量定义在剖分单元的边上,利用矢量有限单元法求解变分问题,实现了电导率任意各向异性海洋可控源电磁三维矢量有限元数值模拟.这个新的正演方法可以计算电导率任意各向异性三维地电模型的海洋可控源电磁响应,基于二次场矢量有限元法直接求解电磁场,避免了传统有限元方法可能遇到的伪解问题和难于处理电场法向分量不连续的问题,提高了数值模拟计算精度.一维电导率各向异性模型电磁场数值解与解析解吻合得相当好,无论在源附近还是远离源处相对误差均不超过1%.电导率各向异性二维模型的计算结果与已有文献采用的非结构有限元模拟结果十分吻合.三维地电模型数值模拟结果显示,电导率各向异性张量电导率主轴分量和欧拉角对不同装置海洋可控源电磁响应均有着明显的影响.     

An explicit finite element－finite difference method for analyzing the effect of visco－elastic local topography on the earthquake motion

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Accuracy‐constrained optimisation methods for staggered‐grid elastic wave modelling

The classical finite‐difference methods for seismic wave modelling are very accurate at low wavenumbers but suffer from inaccuracies at high wavenumbers, particularly at Nyquist wavenumber. In contrast, the optimisation finite‐difference methods reduce inaccuracies at high wavenumbers but suffer from inaccuracies at low wavenumbers, particularly at zero wavenumber when the operator length is not long and the whole range of wavenumbers is considered. Inaccuracy at zero wavenumber means that the optimisation methods only have a zeroth‐order accuracy of truncation and thus are not rigorously convergent. To guarantee the rigorous convergence of the optimisation methods, we have developed accuracy‐constrained optimisation methods. Different‐order accuracy‐constrained optimisation methods are presented. These methods not only guarantee the rigorous convergence but also reduce inaccuracies at low wavenumbers. Accuracy‐constrained optimisation methods are applied to staggered‐grid elastic wave modelling.     

Correlation of dynamic characteristics of a super‐tall building from full‐scale measurements and numerical analysis with various finite element models

The Di Wang Tower located in Shenzhen has 79 storeys and is about 325 m high. Field measurements have been conducted to investigate the dynamic characteristics of the super‐tall building. In parallel with the field measurements, seven finite element models have been established to model the multi‐outrigger‐braced tall building and to analyse the effects of various arrangements of outrigger belts and vertical bracings on the dynamic characteristics and responses of the Di Wang Tower under the design wind load and earthquake action. The distributions of shear forces in vertical structural components along the building height are also presented and discussed. The results from detailed modelling of group shear walls with several types of finite elements are addressed and compared to investigate various modelling assumptions. Finally, the performance of the finite element models is evaluated by correlating the natural frequencies and mode shapes from the numerical analysis with the finite element models and the field measurements. The results generated from this study are expected to be of interest to professionals and researchers involved with the design of tall buildings. Copyright © 2004 John Wiley & Sons, Ltd.     

有限元模拟中边界条件对计算结果的影响

本文从总电位法和异常电位法两个方面,分别就直流电法有限元中所采用的三类边界条件对计算结果的影响进行了分析,为采用合适的边界条件及选取最佳边界距离来提高正演精度提供了基础.结果表明,混合边界条件精度最高,可大大缩小求解区域而不影响计算精度,其次是Dirichlet边界条件精度较高,但测点越靠边界误差会越大,必须取足够大的边界区域,齐次边界条件的误差比较大,但如果采用非二极装置,通过电位差计算得到的视电阻率,由于无穷远边界对电位差的影响基本消除,视电阻率计算误差与混合边界条件下的接近,在反演中,为了节省计算时间,经常使用齐次边界条件进行有限元正演.     

An efficient three‐dimensional solid finite element dynamic analysis of reinforced concrete structures

Most of the finite element analyses of reinforced concrete structures are restricted to two‐dimensional elements. Three‐dimensional solid elements have rarely been used although nearly all reinforced concrete structures are under a triaxial stress state. In this work, a three‐dimensional solid element based on a smeared fixed crack model that has been used in the past mainly for monotonic static loading analysis is extended to cater for dynamic analysis. The only material parameter that needs to be input for this model is the uniaxial compressive strength of concrete. Steel bars are modelled as uniaxial elements and an embedded formulation allows them to have any orientation inside the concrete elements. The proposed strategy for loading or unloading renders a numerical procedure which is stable and efficient. The whole process is applied to two RC frames and compared against existing experiments in the literature. Results show that the proposed approach may adequately be used to predict the dynamic response of a structure. Copyright © 2005 John Wiley & Sons, Ltd.     

Implementation of a second‐order paraxial boundary condition for a water‐saturated layered half‐space in plane strain

A half‐space finite element and a transmitting boundary are developed for a water‐saturated layered half‐space using a paraxial boundary condition. The exact dynamic stiffness of a half‐space in plane strain is derived and a second‐order paraxial approximation of the stiffness is obtained. A half‐space finite element and a transmitting boundary are then formulated. The development is verified by comparison of the dynamic stiffness of impermeable and permeable rigid strip foundations with other published results. The advantage of using the paraxial boundary condition in comparison with the rigid boundary condition is examined. It is shown that the paraxial boundary condition offers significant gain and the resulting half‐space finite element and transmitting boundary can represent the effects of a water‐saturated layered half‐space with good accuracy and efficiency. In addition, the numerical method described herein maintains the strengths and advantages of the finite element method and can be easily applied to demanding problems of soil–structure interaction in a water‐saturated layered half‐space. Copyright © 2010 John Wiley & Sons, Ltd.     

A three‐dimensional transmitting boundary formulated in Cartesian co‐ordinate system for the dynamics of non‐axisymmetric foundations

A three‐dimensional transmitting boundary is formulated in the Cartesian co‐ordinate system. It is developed for the dynamic soil–structure interaction problems of arbitrary shape foundations in laterally heterogeneous strata overlying rigid bedrock. Dynamics of a rectangular rigid surface foundation on a homogeneous stratum is analysed by a hybrid approach in which the finite region including foundation is modelled by the conventional finite element method and the surrounding infinite region by the newly developed transmitting boundary. To demonstrate its strength, the present method is applied to a rectangular foundation in a horizontally heterogeneous ground consisting of two distinct regions divided by and welded along a vertical plane. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

三角网格有限元法声波与弹性波模拟频散分析

本文对声波与弹性波方程进行有限元法离散,构造有限元法频散关系的一般特征值问题,分析了时间离散格式为中心差分的三角网格有限元法声波与弹性波模拟的频散特性. 比较了三种质量矩阵即分布式质量矩阵、集中质量矩阵和混合质量矩阵对有限元法频散的影响;选取四种典型三角网格,分析了混合质量矩阵有限元（MFEM）频散的方向各向异性;数值频散、方向各向异性随插值阶数的增加逐渐减弱,当空间为三阶插值时,频散主要表现为随采样率的变化而几乎无明显方向各向异性, 其频散幅值也较小. 控制其他影响因素不变的情况下,研究了不同波速比介质中弹性波的数值频散. 最后给出了三角网格MFEM的数值耗散性.     

Comparison of free-surface and rigid-lid finite element models of barotropic instabilities

The main goal of this work is to appraise the finite element method in the way it represents barotropic instabilities. To that end, three different formulations are employed. The free-surface formulation solves the primitive shallow-water equations and is of predominant use for ocean modeling. The vorticity–stream function and velocity–pressure formulations resort to the rigid-lid approximation and are presented because theoretical results are based on the same approximation. The growth rates for all three formulations are compared for hyperbolic tangent and piecewise linear shear flows. Structured and unstructured meshes are utilized. The investigation is also extended to time scales that allow for instability meanders to unfold, permitting the formation of eddies. We find that all three finite element formulations accurately represent barotropic instablities. In particular, convergence of growth rates toward theoretical ones is observed in all cases. It is also shown that the use of unstructured meshes allows for decreasing the computational cost while achieving greater accuracy. Overall, we find that the finite element method for free-surface models is effective at representing barotropic instabilities when it is combined with an appropriate advection scheme and, most importantly, adapted meshes.     

An Hermitian finite element solution of the two-dimensional saturated-unsaturated flow equation

This paper describes a Galerkin-type finite element solution of the two-dimensional saturated-unsaturated flow equation. The numerical solution uses an incomplete (reduced) set of Hermitian cubic basis functions and is formulated in terms of normal and tangential coordinates. The formulation leads to continuous pressure gradients across interelement boundaries for a number of well-defined element configurations, such as for rectangular and circular elements. Other elements generally lead to discontinuous gradients; however, the gradients remain uniquely defined at the nodes. The method avoids calculation of second-order derivatives, yet retains many of the advantages associated with Hermitian elements. A nine-point Lobatto-type integration scheme is used to evaluate all local element integrals. This alternative scheme produces about the same accuracy as the usual 9- or 16-point Gaussian quadrature schemes, but is computationally more efficient.     

稀疏存储的显式有限元三角网格地震波数值模拟及其PML吸收边界条件

有限元法是复杂介质地震模拟的有力工具,它能比较客观地反映地震波的传播,比较细致地再现地震图像.但是,为了获得较精确的结果,有限元法模拟地震波的传播需要的网格点数多,具有计算量大和消耗内存多的缺点.针对上述缺点,本文对刚度矩阵采用压缩存储行(CSR)格式,以减少计算量并节省内存;采用集中质量矩阵得到对角的质量矩阵以提高有限元法(显式有限元)的计算效率;时间离散采用保能量的Newmark算法以提高有限元法的计算精度;采用变分形式(弱形式)的PML吸收边界条件对人工截断边界进行处理.通过与高精度的数值方法--谱元法的数值试验的对比表明,上述方法的引入可使有限元法在计算精度和计算效率方面均可取得比较显著的改进.为了获得相当的计算精度,相比于7阶谱元法,显式有限元法需要更精细的网格.然而,显式有限元法的计算速度比前者快近2倍,而内存需求仅为谱元法的1/4~1/6.