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.     

可控源电磁三维频率域有限元模拟

本文采用电磁场的磁矢量位和电标量势,将Maxwell方程组化为位势的类似于Helmholtz型方程,并引入罚项及稳定化方法克服了电磁三维有限元算法中的伪解及数值不稳定性;采用人工边界把计算区域局域化,将均匀半空间中水平电偶极子源产生的位势值作为人工边界上的第一类边界条件以表示源的作用,减少了实际的计算区域.理论模型和复杂模型的计算结果均表明,可控源电磁三维有限元数值模拟给出了稳定、可靠的电磁场分布.     

基于电场矢量波动方程的三维可控源电磁法有限单元法数值模拟

从可控源电磁法的基本原理出发,推导了基于电场矢量波动方程的三维边值问题,利用广义变分原理,把边值问题转换为变分问题,并引入散度条件,避免了伪解的出现,使有限元计算在理论上更加完备.在准静态近似条件下,把水平电偶极子在空中和大地的远区电场闭合表达式作为有限元计算中的区域外边界条件,解决了边界条件加载的困难;把应用于地震模拟中的伪delta函数引入到可控源电磁法中的三维有限元模拟中,消除了源点的奇异性,提高了方程组的稳定性.通过对均匀大地和层状介质模型的模拟,检验了程序的正确性,并对典型的地质体模型进行了数值模拟,分析了其变化规律.     

基于电场总场矢量有限元法的接地长导线源三维正演

电磁场数值模拟的背景场/异常场算法是三维正演的有效策略之一,优点为采用解析法计算电磁场背景场代替场源项、克服了场源奇异性,缺点为不适用于发射源布置于起伏地表或背景模型复杂的情形.总场算法是直接对电磁场总场开展数值模拟,其难点是有效加载场源、保证近区与过渡区数值解精度.本文以水平电偶源形式分段加载接地长导线源,并以电场总场Helmholtz方程为矢量有限元法控制方程,实现了基于非结构化四面体网格剖分的接地长导线源频率域电磁法三维正演.通过与均匀全空间中水平电偶源产生的电场解析解对比,验证了本文算法的正确性,并分析了四面体外接圆半径与其最短棱边的最大比值和四面体二面角最小值对数值解精度的影响规律.通过与块状高导体地电模型的积分方程法、有限体积法和基于磁矢量势Helmholtz方程的有限元法数值解对比,进一步验证了本文算法正确性,同时说明了非结构化四面体网格能够更加精细地剖分电性异常体,利于获得精确数值解.     

Urban fluvial flood modelling using a two‐dimensional diffusion‐wave treatment,part 1: mesh resolution effects

High‐resolution data obtained from airborne remote sensing is increasing opportunities for representation of small‐scale structural elements (e.g. walls, buildings) in complex floodplain systems using two‐dimensional (2D) models of flood inundation. At the same time, 2D inundation models have been developed and shown to provide good predictions of flood inundation extent, with respect to both full solution of the depth‐averaged Navier–Stokes equations and simplified diffusion‐wave models. However, these models have yet to be applied extensively to urban areas. This paper applies a 2D raster‐based diffusion‐wave model to determine patterns of fluvial flood inundation in urban areas using high‐resolution topographic data and explores the effects of spatial resolution upon estimated inundation extent and flow routing process. Model response shows that even relatively small changes in model resolution have considerable effects on the predicted inundation extent and the timing of flood inundation. Timing sensitivity would be expected, given the relatively poor representation of inertial processes in a diffusion‐wave model. Sensitivity to inundation extent is more surprising, but is associated with: (1) the smoothing effect of mesh coarsening upon input topographical data; (2) poorer representation of both cell blockage and surface routing processes as the mesh is coarsened, where the flow routing is especially complex; and (3) the effects of (1) and (2) upon water levels and velocities, which in turn determine which parts of the floodplain the flow can actually travel to. It is shown that the combined effects of wetting and roughness parameters can compensate in part for a coarser mesh resolution. However, the coarser the resolution, the poorer the ability to control the inundation process, as these parameters not only affect the speed, but also the direction of wetting. Thus, high‐resolution data will need to be coupled to a more sophisticated representation of the inundation process in order to obtain effective predictions of flood inundation extent. This is explored in a companion paper. Copyright © 2005 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.     

Three‐dimensional vibration control of high‐tech facilities against earthquakes and microvibration using hybrid platform

High‐tech equipments engaged in the production of ultra‐precision products have very stringent vibration criteria for their functionality in normal operation conditions and their safety during an earthquake. Most previous investigations were based on simplified planar models of building structures, despite the fact that real ground motions and structures are always three‐dimensional. This paper hence presents a three‐dimensional analytical study of a hybrid platform on which high‐tech equipments are mounted for their vibration mitigation. The design methodology of the hybrid platform proposed in this study is based on dual‐level performance objectives for high‐tech equipments: safety against seismic hazard and functionality against traffic‐induced microvibration. The passive devices (represented by springs and viscous dampers) and the active actuators are designed, respectively, to meet vibration criteria corresponding to safety level and functionality level. A prototype three‐story building with high‐tech equipments installed on the second floor is selected in the case study to evaluate the effectiveness of the hybrid platform. The optimal location of the platform on the second building floor is determined during the design procedure in terms of the minimal H ₂ cost function of absolute velocity response. The simulation of the coupled actuator‐platform‐building system subjected to three‐dimensional ground motions indicates that the optimally designed hybrid platform can well achieve the dual target performance and effectively mitigate vibration at both ground motion levels. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparing the performance of a 2‐D finite element and a 2‐D finite volume model of floodplain inundation using airborne SAR imagery

The performances of a finite volume model (SFV) and finite element model (TELEMAC‐2D) in reproducing inundation on a 16 km reach of the river Severn, United Kingdom, are compared. Predicted inundation extents are compared with 4 airborne synthetic aperture radar images of a major flood event in November 2000, and these are used to calibrate 2 values of Manning's n for the channel and floodplain. The four images are shown to have different capacities to constrain roughness parameters, with the image acquired at low flow rate doing better in determining these parameters than the image acquired at approximately peak flow. This is assigned to the valley filling nature of the flood and the associated insensitivity of flood extent to changes in water level. The level of skill demonstrated by the models, when compared with inundation derived using a horizontal water free surface, also increases as flow rate drops. The two models show markedly different behaviours to the calibration process, with TELEMAC showing less sensitivity and lower optimum values for Manning's n than SFV. When the models are used in predictive mode, calibrated against one image and predicting another, SFV performs better than TELEMAC. Copyright © 2007 John Wiley & Sons, Ltd.     

2.5D direct‐current resistivity forward modelling and inversion by finite‐element–infinite‐element coupled method

To reduce the numerical errors arising from the improper enforcement of the artificial boundary conditions on the distant surface that encloses the underground part of the subsurface, we present a finite‐element–infinite‐element coupled method to significantly reduce the computation time and memory cost in the 2.5D direct‐current resistivity inversion. We first present the boundary value problem of the secondary potential. Then, a new type of infinite element is analysed and applied to replace the conventionally used mixed boundary condition on the distant boundary. In the internal domain, a standard finite‐element method is used to derive the final system of linear equations. With a novel shape function for infinite elements at the subsurface boundary, the final system matrix is sparse, symmetric, and independent of source electrodes. Through lower upper decomposition, the multi‐pole potentials can be swiftly obtained by simple back‐substitutions. We embed the newly developed forward solution to the inversion procedure. To compute the sensitivity matrix, we adopt the efficient adjoint equation approach to further reduce the computation cost. Finally, several synthetic examples are tested to show the efficiency of inversion.     

A comparison of one‐ and two‐dimensional approaches to modelling flood inundation over complex upland floodplains

A much understudied aspect of flood inundation is examined, i.e. upland environments with topographically complex floodplains. Although the presence of high‐resolution topographic data (e.g. lidar) has improved the quality of river flood inundation predictions, the optimum dimensionality of hydraulic models for this purpose has yet to be fully evaluated for situations of both topographic and topological (i.e. the connectivity of floodplain features) complexity. In this paper, we present the comparison of three treatments of upland flood inundation using: (a) a one‐dimensional (1D) model (HEC‐RAS v. 3·1·2) with the domain defined as series of extended cross‐sections; (b) the same 1D model, but with the floodplain defined by a series of storage cells, hydraulically connected to the main river channel and other storage cells on the floodplain according to floodplain topological characteristics; (c) a two‐dimensional (2D) diffusion wave treatment, again with explicit representation of floodplain structural features. The necessary topographic and topological data were derived using lidar and Ordnance Survey Landline data. The three models were tested on a 6 km upland reach of the River Wharfe, UK. The models were assessed by comparison with measured inundation extent. The results showed that both the extended cross‐section and the storage cell 1D modes were conceptually problematic. They also resulted in poorer model predictions, requiring incorrect parameterization of the main river to floodplain flux in order to approach anything like the level of agreement observed when the 2D diffusion wave treatment was assessed. We conclude that a coupled 1D–2D treatment is likely to provide the best modelling approach, with currently available technology, for complex floodplain configurations. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional hydrodynamic and salinity transport modelling of Danshuei River estuarine system and adjacent coastal sea,Taiwan

A three‐dimensional, time‐dependent hydrodynamic and salinity model was applied to the Danshuei River estuarine system and adjacent coastal sea in Taiwan. The model forcing functions consist of tidal elevations along the open boundary and freshwater flows from the main stem and tributaries in the Danshuei River system. The bottom roughness height was calibrated and verified with model simulation of barotropic flow, and the turbulent diffusivities were calibrated through comparison of time‐series of salinity distributions. The overall model verification was achieved with comparisons of residual current and salinity distribution. The model simulation results are in qualitative agreement with the available field data. The model was then used to investigate the tidal current, residual current, and salinity patterns under the low freshwater flow condition in the modelling domain. The results reveal that the extensive intrusion of saline water imposes a significant baroclinic forcing and induces a strong residual circulation in the estuary. The downriver net velocity in the upper layer increases seaward despite the enlargement of the river cross‐section in that direction. Strong residual circulation can be found near the Kuan‐Du station. This may be the result of the deep bathymetric features there. Copyright © 2007 John Wiley & Sons, Ltd.     

Inversion of inline and broadside marine controlled‐source electromagnetic data with constraints derived from seismic data

We present a structural smoothing regularization scheme in the context of inversion of marine controlled‐source electromagnetic data. The regularizing hypothesis is that the electrical parameters have a structure similar to that of the elastic parameters observed from seismic data. The regularization is split into three steps. First, we ensure that our inversion grid conforms with the geometry derived from seismic. Second, we use a seismic stratigraphic attribute to define a spatially varying regularization strength. Third, we use an indexing strategy on the inversion grid to define smoothing along the seismic geometry. Enforcing such regularization in the inversion will encourage an inversion result that is more intuitive for the interpreter to deal with. However, the interpreter should also be aware of the bias introduced by using seismic data for regularization. We illustrate the method using one synthetic example and one field data example. The results show how the regularization works and that it clearly enforces the structure derived from seismic data. From the field data example we find that the inversion result improves when the structural smoothing regularization is employed. Including the broadside data improves the inversion results even more, due to a better balancing between the sensitivities for the horizontal and vertical resistivities.     

Direct finite element method for nonlinear earthquake analysis of 3‐dimensional semi‐unbounded dam–water–foundation rock systems

A direct finite element method for nonlinear earthquake analysis of 2‐dimensional dam–water–foundation rock systems has recently been presented. The analysis procedure uses standard viscous‐damper absorbing boundaries to model the semi‐unbounded foundation‐rock and fluid domains and specifies the seismic input as effective earthquake forces at these boundaries. Presented in this paper is a generalization of the direct finite element method with viscous‐damper boundaries to 3‐dimensional dam–water–foundation rock systems. Step‐by‐step procedures for determining the effective earthquake forces starting from a ground motion specified at a control point on the foundation‐rock surface is developed, and several numerical examples are computed and compared with independent benchmark solutions to demonstrate the effectiveness of the analysis procedure for modeling 3‐dimensional systems.     

耦合有限单元法扩边的直流电阻率勘探无单元Galerkin法正演

本文分析了目前直流电阻率正演模拟中的无单元Galerkin法（EFGM）和有限单元法（FEM）的优缺点,针对采用第一类边界条件需要足够大的计算域时EFGM计算成本高的问题,在计算域外围区域采用FEM扩边,提出了直流电阻率的无单元Galerkin-有限单元耦合法（EFG-FE）.采用具有Kronecker delta函数性质的径向基点插值法（RPIM）构造EFGM形函数,在外围区域将EFGM与FEM直接耦合,无需其他处理手段,消除了传统EFGM与FEM耦合中存在的界面耦合困难.EFG-FE将模型计算域分割为EFGM区域和FEM区域,模型核心区域采用EFGM计算,发挥EFGM灵活性、适应性强和高精度的优点,使得模型建立简单方便,对任意复杂地电模型适应性强,同时获得高精度模拟结果.在模型计算域外围采用快速扩展的FEM单元网格进行剖分,利用其数值稳定性和高效性,使用少量FEM节点和单元网格将计算域大范围扩大满足第一类边界条件,同时不大幅增加计算成本,进而提高计算效率.最后,通过不同正演方法的模型算例的模拟结果对比,验证了本文提出的EFG-FE有效可行,其模拟结果具有很高的模拟精度,且相比于采用第三类边界条件的EFGM提高了计算效率,具有更好的模拟性能.

     

Urban fluvial flood modelling using a two‐dimensional diffusion‐wave treatment,part 2: development of a sub‐grid‐scale treatment

This paper develops and tests a sub‐grid‐scale wetting and drying correction for use with two‐dimensional diffusion‐wave models of urban flood inundation. The method recognizes explicitly that representations of sub‐grid‐scale topography using roughness parameters will provide an inadequate representation of the effects of structural elements on the floodplain (e.g. buildings, walls), as such elements not only act as momentum sinks, but also have mass blockage effects. The latter may dominate, especially in structurally complex urban areas. The approach developed uses high‐resolution topographic data to develop explicit parameterization of sub‐grid‐scale topographic variability to represent both the volume of a grid cell that can be occupied by the flow and the effect of that variability upon the timing and direction of the lateral fluxes. This approach is found to give significantly better prediction of fluvial flood inundation in urban areas than traditional calibration of sub‐grid‐scale effects using Manning's n. In particular, it simultaneously reduces the need to use exceptionally high values of n to represent the effects of using a coarser mesh process representation and increases the sensitivity of model predictions to variation in n. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Controlled source electromagnetic three‐dimensional grid‐modelling based on a complex resistivity structure of the seafloor: effects of acquisition parameters and geometry of multi‐layered resistors

A comprehensive controlled source electromagnetic (CSEM) modelling study, based on complex resistivity structures in a deep marine geological setting, is conducted. The study demonstrates the effects of acquisition parameters and multi‐layered resistors on CSEM responses. Three‐dimensional (3D) finite difference time domain (FDTD) grid‐modelling is used for CSEM sensitivity analysis. Interpolation of normalized CSEM responses provides attributes representing relative sensitivity of the modelled structures. Modelling results show that fine grid, 1 × 1 km receiver spacing, provides good correlations between CSEM responses and the modelled structures, irrespective of source orientation. The resolution of CSEM attributes decreases for receiver spacing >2 × 2 km, when using only in‐line data. Broadside data in the grid geometry increase data density by 100 – approximately 200% by filling in in‐line responses and improve the resolution of CSEM attributes. Optimized source orientation (i.e., oblique to the strike of an elongated resistor) improves the structural definition of the CSEM anomalies for coarse‐grid geometries (receiver spacing ≥3 × 3 km). The study also shows that a multi‐resistor anomaly is not simply the summation but a cumulative response with mutual interference between constituent resistors. The combined response of constituent resistors is approximately 50% higher than the cumulative response of the multi‐resistor for 0.5 Hz at 4000 m offset. A gradual inverse variation of offset and frequency allows differentiation of CSEM anomalies for multi‐layered resistors. Similar frequency‐offset variations for laterally persistent high‐resistivity facies show visual continuity with varying geometric expressions. 3D grid‐modelling is an effective and adequate tool for CSEM survey design and sensitivity analysis.     

RC pile–soil interaction analysis using a 3D‐finite element method with fibre theory‐based beam elements

To evaluate the overall response of a structural system including its foundation and surrounding soil, an equivalent finite element model with reduced degrees of freedom using fibre theory‐based beam element was proposed. The proposed model was based on investigations of the subgrade soil reaction of a single‐layer model, and was verified for the cyclic behaviour of a laterally loaded single RC pile in terms of the load–displacement relationship, pile deformation, and soil pressures on the pile surface. Also investigated was the effect of the interfacial element between pile and soil on the behaviour of the laterally loaded pile. Copyright © 2006 John Wiley & Sons, Ltd.