Simulation of marine controlled source electromagnetic measurements using a parallel fourier <Emphasis Type="Italic">hp</Emphasis>-finite element method

We introduce a new numerical method to simulate geophysical marine controlled source electromagnetic (CSEM) measurements for the case of 2D structures and finite 3D sources of electromagnetic (EM) excitation. The method of solution is based on a spatial discretization that combines a 1D Fourier transform with a 2D self-adaptive, goal-oriented, hp-Finite element method. It enables fast and accurate simulations for a variety of important, challenging and practical cases of marine CSEM acquisition. Numerical results confirm the high accuracy of the method as well as some of the main physical properties of marine CSEM measurements such as high measurement sensitivity to oil-bearing layers in the subsurface. In our model, numerical results indicate that measurements could be affected by the finite oil-bearing layer by as much as 10⁴% (relative difference). While the emphasis of this paper is on EM simulations, the method can be used to simulate different physical phenomena such as seismic measurements.     

二维电阻率倾斜各向异性对海洋可控源电磁场响应的影响

地下介质的电阻率常常表现为各向异性,海底褶皱带、逆冲断层带和倾斜层状沉积序列等地质构造可能形成宏观电阻率倾斜各向异性。这里采用规则矩形网格剖分有限元法,实现了二维电阻率倾斜各向异性海洋可控源电磁(CSEM)正演算法,模拟了二维电阻率倾斜各向异性模型海洋可控源电磁场响应。模型计算结果表明,电阻率倾斜各向异性围岩对含有海底高阻薄层的海洋可控源电磁响应产生严重畸变影响。因此,在海洋电磁资料解释中,电阻率倾斜各向异性的影响应该得到重视,忽略该影响将可能会导致数据解释错误。     

Embedded beam element with interaction surface for lateral loading of piles

This paper presents a numerical formulation of a three dimensional embedded beam element for the modeling of piles, which incorporates an explicit interaction surface between soil and pile. The formulation is herein implemented for lateral loading of piles but is able to represent soil–pile interaction phenomena in a general manner for different types of loading conditions or ground movements. The model assumes perfect adherence between beam and soil along the interaction surface. The paper presents a comparison of the results obtained by means of the present formulation and by means of a previously formulated embedded pile element without interaction surface, as well as reference semi‐analytical solutions and a fully 3D finite element (FE) model. It is seen that the proposed embedded element provides a better convergence behavior than a previously formulated embedded element and is able to reproduce key features of a full 3D FE model. Copyright © 2015 John Wiley & Sons, Ltd.     

各向异性介质中的浅海海洋可控源电磁响应特征

由于受到空气波的影响,浅海海洋可控源电磁数据对海底储油层的反映较弱,如何对浅海数据进行处理和解释一直是海洋电磁理论研究的热点。随着近年来海洋电磁理论的不断完善,浅海数据已经可以被较好地处理与反演,但是其解释水平仍然受基本理论研究不足的制约。针对这一现状,本文开展了海底电各向异性对浅海数据影响规律的研究。具体方法为:利用欧拉旋转建立不同的海底电性各向异性模型,然后采用交错网格有限差分法计算浅海海洋可控源电磁响应,最后通过分析同线情况下电场E_x分量的振幅和相位曲线特征以及海底电场及电流密度分布规律,分析各向异性对浅海海洋可控源电磁响应影响的物理机制,并讨论浅海各向异性情况下海洋电磁对高阻储油层的识别能力。得出的结论为各向异性介质中的浅海海洋电磁响应特征与深海有较大区别,在进行数据的处理、反演和解释时应区别于深海情况。     

Beam–solid contact formulation for finite element analysis of pile–soil interaction with arbitrary discretization

This paper presents an embedded beam formulation for discretization independent finite element (FE) analyses of interactions between pile foundations or rock anchors and the surrounding soil in geotechnical and tunneling engineering. Piles are represented by means of finite beam elements embedded within FEs for the soil represented by 3D solid elements. The proposed formulation allows consideration of piles and pile groups with arbitrary orientation independently from the FE discretization of the surrounding soil. The interface behavior between piles and the surrounding soil is represented numerically by means of a contact formulation considering skin friction as well as pile tip resistance. The pile–soil interaction along the pile skin is considered by means of a 3D frictional point‐to‐point contact formulation using the integration points of the beam elements and reference points arbitrarily located within the solid elements as control points. The ability of the proposed embedded pile model to represent groups of piles objected to combined axial and shear loading and their interactions with the surrounding soil is demonstrated by selected benchmark examples. The pile model is applied to the numerical simulation of shield driven tunnel construction in the vicinity of an existing building resting upon pile foundation to demonstrate the performance of the proposed model in complex simulation environments. Copyright © 2014 John Wiley & Sons, Ltd.     

A 3D composite finite element used to model a vertical drain and its surrounding remoulded soil

This work addresses an enrichment technique for the three-dimensional (3D) finite element (FE) analysis of a vertical drain foundation because (1) 1D and 2D simulations are insufficient to integrally describe the consolidation behaviour and (2) drains are small both in spacing and size, resulting in enormous computational costs for a traditional 3D FE analysis. Based on the idea of the semi-analytical finite element method (FEM), which combines general FEM with the high accuracy of a closed-form solution, a new spatial element that contains a drain well and its neighbouring smear zone is presented. This new combined element is depicted by eight global independent nodes and two local dependent nodes, and a classical analytical theory is introduced to set up the relationship between the two kinds of nodes. Because permeability diversity between the drain and the smear zone is considered, both the effects of smearing and well resistance are taken into account with the composite element method (CEM). A detailed derivation of the CEM is performed using the weighted residual method. The accuracy of the proposed method is validated with a totally penetrating, single-drain ground analysis for seven calculation conditions. Additionally, the proposed CEM saves 1/4–1/2 mesh elements and helps to avoid slender elements for the FEM analysis of the drained foundation.     

Fast 2.5D finite element simulations of borehole resistivity measurements

We develop a rapid 2.5-dimensional (2.5D) finite element method for simulation of borehole resistivity measurements in transversely isotropic (TI) media. The method combines arbitrary high-order \(H^{1}\)- and \(\mathbf {H}\)(curl)-conforming spatial discretizations. It solves problems where material properties remain constant along one spatial direction, over which we consider a Fourier series expansion and each Fourier mode is solved independently. We propose a novel a priori method to construct quasi-optimal discretizations in physical and Fourier space. This construction is based on examining the analytical (fundamental) solution of the 2.5D formulation over multiple homogeneous spaces and assuming that some of its properties still hold for the 2.5D problem over a spatially heterogeneous formation. In addition, a simple parallelization scheme over multiple measurement positions provides efficient scalability. Our method yields accurate borehole logging simulations for realistic synthetic examples, delivering simulations of borehole resistivity measurements at a rate faster than 0.05 s per measurement location along the well trajectory on a 96-core computer.     

Calculating the effective permeability of sandstone with multiscale lattice Boltzmann/finite element simulations

The lattice Boltzmann (LB) method is an efficient technique for simulating fluid flow through individual pores of complex porous media. The ease with which the LB method handles complex boundary conditions, combined with the algorithm’s inherent parallelism, makes it an elegant approach to solving flow problems at the sub-continuum scale. However, the realities of current computational resources can limit the size and resolution of these simulations. A major research focus is developing methodologies for upscaling microscale techniques for use in macroscale problems of engineering interest. In this paper, we propose a hybrid, multiscale framework for simulating diffusion through porous media. We use the finite element (FE) method to solve the continuum boundary-value problem at the macroscale. Each finite element is treated as a sub-cell and assigned permeabilities calculated from subcontinuum simulations using the LB method. This framework allows us to efficiently find a macroscale solution while still maintaining information about microscale heterogeneities. As input to these simulations, we use synchrotron-computed 3D microtomographic images of a sandstone, with sample resolution of 3.34 μm. We discuss the predictive ability of these simulations, as well as implementation issues. We also quantify the lower limit of the continuum (Darcy) scale, as well as identify the optimal representative elementary volume for the hybrid LB–FE simulations.     

Mixed unsplit-field perfectly matched layers for transient simulations of scalar waves in heterogeneous domains

We discuss a new formulation for transient scalar wave simulations in heterogeneous semi-infinite domains. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries and adopt perfectly matched layers (PMLs) as the boundary wave absorbers. Within this framework, we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs. We use, as typically done, complex-coordinate stretching transformations in the frequency domain, and recover the governing partial differential equations in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which, numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error.     

3D finite element modeling of sand particle fracture based on in situ X‐Ray synchrotron imaging

Compressive loading of granular materials causes inter‐particle forces to develop and evolve into force chains that propagate through the granular body. At high‐applied compressive stresses, inter‐particle forces will be large enough to cause particle fracture, affecting the constitutive behavior of granular materials. The first step to modeling particle fracture within force chains in granular mass is to understand and model the fracture of a single particle using actual three‐dimensional (3D) particle shape. In this paper, the fracture mode of individual silica sand particles was captured using 3D x‐ray radiography and Synchrotron Micro‐computed Tomography (SMT) during in situ compression experiments. The SMT images were used to reconstruct particle surfaces through image processing techniques. Particle surface was then imported into Abaqus finite element (FE) software where the experimental loading setup was modeled using the extended finite element method (XFEM) where particle fracture was compared to experimental fracture mode viewed in radiograph images that were acquired during experimental loading. Load‐displacement relationships of the FE analysis were also compared with experimental measurements. 3D FE modeling of particle fracture offers an excellent tool to map stress distribution and monitors crack initiation and propagation within individual sand particles. Copyright © 2015 John Wiley & Sons, Ltd.     

Frictional slip plane growth by localization detection and the extended finite element method (XFEM)

This paper is concerned with developing a numerical tool for detecting instabilities in elasto‐plastic solids (with an emphasis on soils) and inserting a discontinuity at these instabilities allowing the boundary value problem to proceed beyond these instabilities. This consists of implementing an algorithm for detection of strong discontinuities within a finite element (FE) framework. These discontinuities are then inserted into the FE problem through the use of a displacement field enrichment technique called the extended finite element method (XFEM). The newly formed discontinuities are governed by a Mohr–Coulomb frictional law that is enforced by a penalty method. This implementation within an FE framework is then tested on a compressive soil block and a soil slope where the discontinuity is inserted and grown according to the localization detection. Copyright © 2010 John Wiley & Sons, Ltd.     

Three dimensional FEM of moving coordinates for the analysis of transient vibrations due to moving loads

This paper is concerned with the transient vibration analysis of railway-ground system under fast moving loads. A 3D finite element method in a convected coordinate system moving with the load is formulated, together with viscous-elastic transmitting boundary conditions in order to limit the finite element mesh. A method is proposed to introduce Rayleigh type material damping in the finite element formulation in the moving coordinate system, while measures have also been taken to improve the numerical stability of the solution procedure. The performance of the transmitting boundary and the entire solution procedure are assessed via comparison with the ordinary finite element solution of some relatively simple problems and through a comparison with field measurements. The reasonable agreement found from these comparisons demonstrates the validity of the proposed method.     

A stabilized extended finite element framework for hydraulic fracturing simulations

We present a stabilized extended finite element formulation to simulate the hydraulic fracturing process in an elasto‐plastic medium. The fracture propagation process is governed by a cohesive fracture model, where a trilinear traction‐separation law is used to describe normal contact, cohesion and strength softening on the fracture face. Fluid flow inside the fracture channel is governed by the lubrication equation, and the flow rate is related to the fluid pressure gradient by the ‘cubic’ law. Fluid leak off happens only in the normal direction and is assumed to be governed by the Carter's leak‐off model. We propose a ‘local’ U‐P (displacement‐pressure) formulation to discretize the fluid‐solid coupled system, where volume shape functions are used to interpolate the fluid pressure field on the fracture face. The ‘local’ U‐P approach is compatible with the extended finite element framework, and a separate mesh is not required to describe the fluid flow. The coupled system of equations is solved iteratively by the standard Newton‐Raphson method. We identify instability issues associated with the fluid flow inside the fracture channel, and use the polynomial pressure projection method to reduce the pressure oscillations resulting from the instability. Numerical examples demonstrate that the proposed framework is effective in modeling 3D hydraulic fracture propagation. Copyright © 2016 John Wiley & Sons, Ltd.     

The Andra Couplex 1 Test Case: Comparisons Between Finite-Element, Mixed Hybrid Finite Element and Finite Volume Element Discretizations

We present here results for the Andra Couplex 1 test case, obtained with the code Cast3m. This code is developped at the CEA (Commissariat l'nergie atomique) and is used mainly to solve problems of solid mechanics, fluid mechanics and heat transfers. Different types of discretization are available, among them finite element, finite volume and mixed hybrid finite element method. Cast3m is also a componant of the platteform Alliances (co-developped by Andra, CEA), which will be used by Andra for the safety calculation of an underground waste disposal in year 2004. We solve the Darcy equation for the water flow and a convection–diffusion transport equation for the Iodine 129 which escapes from a repository cave into the water. The water flow is calculated with a MHFE discretization. It is shown that this method provides sharp results even on relatively coarse grids. The convection–diffusion transport equation is discretized with FE (Finite Element), MHFE (Mixed Hybrid Finite Element) and FV (Finite Volume) methods. In our comparison, we point out the differences of these methods in term of accuracy, respect of the maximum principle and calculations cost. Neither the finite element nor the mixed hybrid finite element approach respects the maximum principle. This results in the presence of negative concentrations near the repository cave, whereas FV calculations respect the monotonicity. We show that mass lumping techniques suppress this problem but with strong restrictions on the grid. FE and MHFE approaches are more accurate than FV for the diffusion equation, but the overall results are equivalent since the advective terms are dominant in the far field and are discretized with centered schemes. We conclude by studying the influence of the grid: a very fine grid near the repository solves almost all the problems of monotonicity, without employing mass lumping techniques. We also observed a very important increase of the accuracy on a structured grid made up of rectangles.     

A new approximation in determination of vertical displacement behavior of a concrete-faced rockfill dam

A new simplified method based on one-dimensional displacement theory and 2-D finite element (FE) analysis was developed to predict the vertical displacement behavior of a concrete-faced rockfill dam. The FE analyses were carried out at the end of construction (EOC) and the end of first filling of reservoir. The proposed method was calibrated by using continuously monitored vertical displacement of the dam’s body to determine the mobilized modulus of elasticity of the rockfills at the EOC. The prediction capability of the method was demonstrated using field measurements against the findings from the 2-D FE analysis simulating characteristics of construction stages of the dam. The validity of the method was also examined on another membrane-faced rockfill dam by comparing the geodetic measurements of vertical displacement measurements of the dam’s body with the calculated vertical displacements from 2-D FE analysis at the EOC.     

三维有限元位移场插值问题的研究和应用

对三维有限元位移场的插值问题进行了研究,提出基于Delaunay三角剖分的二次细分网格法。该法首先建立2个模型--整体模型和局部二次细分模型,对整体模型进行有限元计算和反分析得到初始地应力场和位移场;然后,基于Delaunay三角剖分,对整体模型的三维有限元位移场进行插值,插值结果作为局部模型的边界位移,局部二次细分模型再在此基础上进行有限元分析。最后,将该方法应用于工程实践,证明能提高计算效率,是切实可行的。     

A stable space-time FE method for the shallow water equations

We consider the finite element (FE) approximation of the two dimensional shallow water equations (SWE) by considering discretizations in which both space and time are established using a stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to establish stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We establish a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptive mesh refinements on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to some commonly applied benchmarks for the SWE.

     

Coupled hydromechanical‐fracture simulations of nonplanar three‐dimensional hydraulic fracture propagation

This paper presents an algorithm and a fully coupled hydromechanical‐fracture formulation for the simulation of three‐dimensional nonplanar hydraulic fracture propagation. The propagation algorithm automatically estimates the magnitude of time steps such that a regularized form of Irwin's criterion is satisfied along the predicted 3‐D fracture front at every fracture propagation step. A generalized finite element method is used for the discretization of elasticity equations governing the deformation of the rock, and a finite element method is adopted for the solution of the fluid flow equation on the basis of Poiseuille's cubic law. Adaptive mesh refinement is used for discretization error control, leading to significantly fewer degrees of freedom than available nonadaptive methods. An efficient computational scheme to handle nonlinear time‐dependent problems with adaptive mesh refinement is presented. Explicit fracture surface representations are used to avoid mapping of 3‐D solutions between generalized finite element method meshes. Examples demonstrating the accuracy, robustness, and computational efficiency of the proposed formulation, regularized Irwin's criterion, and propagation algorithm are presented.     

Viscoelastic functionally graded finite element method with recursive time integration and applications to flexible pavements

The finite‐element (FE) method is used for modeling geotechnical and pavement structures exhibiting significant non‐homogeneity. Property gradients generated due to non‐homogeneous distributions of moisture is one such example for geotechnical materials. Aging and temperature‐induced property gradients are common sources of non‐homogeneity for asphalt pavements. Investigation of time‐dependent behavior combined with functionally graded property gradation can be accomplished by means of the non‐homogeneous viscoelastic analysis procedure. This paper describes the development of a generalized isoparametric FE formulation to capture property gradients within elements, and a recursive formulation for solution of hereditary integral equations. The formulation is verified by comparison with analytical and numerical solutions. Two application examples are presented: the first describes stationary crack‐tip fields for viscoelastic functionally graded materials, and the second example demonstrates the application of the proposed procedures for efficient and accurate simulations of interfaces between layers of flexible pavement. Copyright © 2011 John Wiley & Sons, Ltd.     

基于有限元计算的边坡三维滑裂面搜索

提出一种基于有限元计算的边坡三维滑裂面搜索的方法,根据有限元计算的应力结果,计算可能滑裂面的安全系数,然后用遗传算法搜索出最危险滑裂面,并得出相应的安全系数作为边坡的安全度评价指标。该方法的优越性是只需进行一次有限元计算,不需要假定滑裂面,可以用于具有复杂地貌和地质构造的三维边坡。通过算例,验证该方法的可行性和程序的正确性,以大岗山水电站库区边坡工程实例验证该方法的实用性。