Iteratively coupled solution strategies for a four‐field mixed finite element method for poroelasticity

In this paper, we consider numerical algorithms for modeling of the time‐dependent coupling between the fluid flow and deformation in elastic porous media. Here, we employ a four‐field formulation which uses the total stress, displacement, flux, and pressure as its primary variables and satisfies Darcy's law and linear elasticity in mixed weak form. We present four different iteratively coupled methods, known as drained, undrained, fixed‐strain, and fixed‐stress splits, in which the diffusion operator is separated from the elasticity operator and the two subproblems are solved in a staggered way while ensuring convergence of the solution at each time step. A‐priori convergence results for each iterative coupling which differs from those found when using a traditional two‐field or three‐field formulation are presented. We also present some numerical results to support the convergence estimates and to show the accuracy and efficiency of the algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

饱和多孔介质中水力-力学-传质耦合过程的混合有限元法

对饱和多孔介质提出了一个含溶混污染物输运（传质）过程的混合元方法,其中污染物输运过程数学模型包含了对流、机械逸散、分子弥散和吸附等机制。固相位移、应变和有效应力,孔隙水压力、压力空间梯度和Darcy速度,污染物浓度、浓度空间梯度和浓度流量在单元内均为独立变量分别插值。基于胡海昌-Washizu三变量广义变分原理,结合可以滤掉虚假振荡的特征线方法,推导出饱和土中水力-力学-传质耦合问题控制方程的单元弱形式,并导出了混合元计算公式。数值模拟证明了所提出的方法可以提供与传统4点积分方案同样精度,同时能够提高计算效率。     

A mixed finite element approach for viscoelastic wave propagation

In this paper, we are interested in the modeling of wave propagation in viscoelastic media. We present a family of models which generalize the Zeners model. We achieve its mathematical analysis: existence and uniqueness of solutions, energy decay and propagation with finite speed. For the numerical resolution, we extend a mixed finite element method proposed in [8]. This method combines mass lumping with a centered explicit scheme for time discretization. For the resulting scheme, we prove a discrete energy decay result and provide a sufficient stability condition. For the numerical simulation in open domains we adapt the perfectly matched layers techniques to viscoelastic waves [23]. Various numerical results are presented.     

Numerical approximation of water–air two-phase flow by the mixed finite element method

A new model for two-phase flow of water and air in soil is presented. This leads to a system of two mass balance equations and two equations representing conservation of momentum of fluid and gas, respectively. This paper is concerned with the verification of this model for the special case of a rigid soil skeleton by computational experiments. Its numerical treatment is based on the Raviart–Thomas mixed finite element method combined with an implicit Euler time discretization. The feasibility of the method is illustrated for some test examples of one- and two-dimensional two-phase flow problems.     

双重孔隙-裂隙岩体中洞室变形及强度各向异性的三维有限元分析

将确定双重孔隙-裂隙介质凝聚力及内摩擦角的方法与描述介质强度各向异性的微结构-无迹张量方法相结合,引入到三维有限元程序中。以简单的算例,通过解析解与数值解的比较,验证了所开发的有限元程序的可靠性。针对一个假定的位于被3组正交裂隙所切割的围岩中的矩形洞室,使用Mohr-Coulomb准则进行弹塑性数值模拟,就不同的工况分析了围岩中的位移、应力及塑性区的状态。计算结果显示：裂隙组的不同展布及组合使得岩体的变形及强度性质有着不同的各向异性,从而相应地导致围岩中的位移、应力及塑性区的分布与量值产生明显的差异。     

Stochastic finite element methods for the seismic response of soils

Some of the available stochastic finite element methods are adapted and evaluated for the analyses of response of soils with uncertain properties subjected to earthquake induced random ground motion. In this study, the dynamic response of a soil mass, with finite element discretization, is formulated in the frequency domain. The spectral density function of the response variables are obtained from which the evaluation of the root-mean-squared and the most probable extreme values of the response are made. The material non-linearities are incorporated by using strain compatible moduli and damping of soils using an equivalent linear model for stress–strain behaviour of soils and an iterative solution of the response. The spatial variability of the shear modulus is described through a random field model and the earthquake included motion is treated as a stochastic process. The available formulations of direct Monte-Carlo simulation, first-order perturbation method, a spectral decomposition method with Neumann expansion and a spectral decomposition method with Polynomial Chaos are used to develop stochastic finite element analyses of the seismic response of soils. The numerical results from these approaches are compared with respect to their accuracy and computational efficiency. © 1998 John Wiley & Sons Ltd.     

非饱和土弹塑性模型参数的试验确定及有限元法

简要介绍广泛应用的非饱和土Alonso模型的原理与发展,并通过3组共7个试样的非饱和土三轴试验,得到模型的11个参数。在饱和土Cam本构关系的基础上,推导出一个基于非饱和土Alonso模型的应力-应变增量方程的计算公式,该方程与饱和土的本构方程形式相同。编制了能够考虑净应力和吸力二者对土体硬化规律影响的有限元程序,从而为非饱和土弹塑性计算提供一条途径,便于进一步应用于实际工程。     

Hybrid time integration and coupled solution methods for nonlinear finite element analysis of partially saturated deformable porous media at small strain

The goal of the paper is to determine the most efficient, yet accurate and stable, finite element nonlinear solution method for analysis of partially saturated deformable porous media at small strain. This involves a comparison between fully implicit, semi‐implicit, and explicit time integration schemes, with monolithically coupled and staggered‐coupled nonlinear solution methods and the hybrid combination thereof. The pore air pressure p_a is assumed atmospheric, that is, p_a=0 at reference pressure. The solid skeleton is assumed to be pressure‐sensitive nonlinear isotropic elastic. Coupled partially saturated ‘consolidation’ in the presence of surface infiltration and traction is simulated for a simple one‐dimensional uniaxial strain example and a more complicated plane strain slope example with gravity loading. Three mixed plane strain quadrilateral elements are considered: (i) Q4P4; (ii) stabilized Q4P4S; and (iii) Q9P4; “Q” refers to the number of solid skeleton displacement nodes, and “P” refers to the number of pore fluid pressure nodes. The verification of the implementation against an analytical solution for partially saturated pore water flow (no solid skeleton deformation) and comparison between the three time integration schemes (fully implicit, semi‐implicit, and explicit) are presented. It is observed that one of the staggered‐coupled semi‐implicit schemes (SIS(b)), combined with the fully implicit monolithically coupled scheme to resolve sharp transients, is the most efficient computationally. Copyright © 2015 John Wiley & Sons, Ltd.     

基于粒子群算法与混合罚函数法的有限元优化反演模型及应用

岩土工程优化反分析是一个典型的复杂非线性函数优化问题,采用全局优化算法是解决这个问题的理想途径。针对常规反演方法应用于岩土工程参数反演时搜索效率低的缺点,结合粒子群算法和遗传算法的特点,充分考虑二者的互补性,提出一种效率较高的全局优化算法,以测点的实测值与计算值建立一种新的评价函数,将多目标优化问题转化为单目标优化问题,用混合罚函数法将约束问题变为无约束问题,构建了一种新的目标函数,将有限元程序ABAQUS作为一个模块嵌入到优化算法程序中,编制了有限元优化反演分析程序。并给出了应用实例验证了该法的有效性和实用性,是一种可行的参数反演方法,可应用于实际工程中复杂岩土介质初始应力场反演、渗流场以及位移反分析     

A finite element method for modeling coupled flow and deformation in porous fractured media

Modeling the flow in highly fractured porous media by finite element method (FEM) has met two difficulties: mesh generation for fractured domains and a rigorous formulation of the flow problem accounting for fracture/matrix, fracture/fracture, and fracture/boundary fluid mass exchanges. Based on the recent theoretical progress for mass balance conditions in multifractured porous bodies, the governing equations for coupled flow and deformation in these bodies are first established in this paper. A weak formulation for this problem is then established allowing to build a FEM. Taking benefit from recent development of mesh‐generating tools for fractured media, this weak formulation has been implemented in a numerical code and applied to some typical problems of hydromechanical coupling in fractured porous media. It is shown that in this way, the FEM that has proved its efficiency to model hydromechanical phenomena in porous media is extended with all its performances (calculation time, couplings, and nonlinearities) to fractured porous media. Copyright © 2015 John Wiley & Sons, Ltd.     

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete-in-time case

In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a discrete-in-time setting. Of particular interest is the case when the lowest-order Raviart–Thomas approximating space or cell-centered finite differences are used in the mixed formulation and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.     

A finite element algorithm for parameter identification of material models for fluid saturated porous media

In this contribution an algorithm for parameter identification of geometrically linear Terzaghi–Biot‐type fluid‐saturated porous media is proposed, in which non‐uniform distributions of the state variables such as stresses, strains and fluid pore pressure are taken into account. To this end a least‐squares functional consisting of experimental data and simulated data is minimized, whereby the latter are obtained with the finite element method. This strategy allows parameter identification based on in situ experiments. In order to improve the efficiency of the minimization process, a gradient‐based optimization algorithm is applied, and therefore the corresponding sensitivity analysis for the coupled two‐phase problem is described in a systematic manner. For illustrative purpose, the performance of the algorithm is demonstrated for a slope stability problem, in which a quadratic Drucker–Prager plasticity model for the solid and a linear Darcy law for the fluid are combined. Copyright © 2001 John Wiley & Sons, Ltd.     

Design of raft–pile foundation using combined optimization and finite element approach

This paper describes the application of a structural optimization approach combined with the finite element method for the optimal design of a raft–pile foundation system. The analysis takes into account the non-linear behaviour of the soil medium and the piles. For the optimization process, the sensitivity analysis is carried out using the approximate semi-analytical method while the constraint approximation is obtained from the combination of extended Bi-point and Lagrangian polynomial approximation methods. The objective function of the problem is the cost of the foundation. The design variables are the raft thickness, cross-section, length and number of piles. The maximum displacement and differential displacement are selected as the constraints. The proposed method is shown to achieve an optimum design of raft–pile foundation efficiently and accurately. Copyright © 1999 John Wiley & Sons, Ltd.     

土石坝拟静力抗震稳定分析的强度折减有限元法

基于拟静力抗震设计概念,提出利用强度折减有限单元法分析土石坝的抗震稳定性,给出了两种确定地震惯性力的方法：（1）依据《水工建筑物抗震设计规范》[1],并结合有关土石坝动态分布系数计算了沿坝高分布的地震惯性力;（2）直接利用土石坝有限元地震动力反应分析得到的单元节点加速度反应,依据建议的方法确定坝体各单元节点的地震惯性力。将上述计算确定的地震惯性力与其他形式的外荷载共同作用到土石坝上,采用强度折减有限元法确定土石坝坝体的拟静力抗震安全系数。对于稳定渗流期,水位降落期等不同工况,或需要考虑振动孔隙水压力作用的饱和无黏性土填筑坝等不同计算条件,给出了使用折减强度有限元法分析坝体抗震稳定性的实现途径和方法。研究表明,有限元法对边界条件、复杂断面条件和材料分区及荷载组合均具有较强的适应能力,因此,使用有限元法分析土石坝抗震稳定性具有显著的优越性。     

三维边坡稳定分析的有限元弹塑性迭代解法

针对已知滑动面的三维边坡稳定分析问题,在塑性力学上限定理的基础上,提出了求解安全系数的三维有限元弹塑性迭代解法。对于滑动面的非线性特性,采用常规的矩形（平面）或立方体（空间）单元来描述,应力则符合相关联流动法则的Mohr-Coulomb强度准则。通过对弹塑性增量理论中流动法则的分析,证明了当结构达到极限状态时,滑动面内切向应力的方向与滑体的主滑方向一致,说明最终以滑动面内的切向应力作为滑体滑动力的有效性。通过迭代的方式,逐步降低滑动面材料的抗剪参数,使边坡达到极限状态,不仅快速求得了滑动面的安全系数,且能够得到边坡在接近临界失稳状态时滑动面内切向应力的分布情况和边坡的变形规律,为边坡采取加固措施提供了参考依据。最后,以椭球面滑动和楔形滑动2个经典算例和工程实例,验证了该方法的有效性和正确性。     

三维油藏中应力与渗流的摄动-有限元分析

提出了一个较为实际的石油油藏的三维力学模型。以已有的三维介质中应力与渗流耦合问题的变分原理为基础,用摄动法证明了光油藏厚度变化为小量时,三维问题或简化为平面应变问题,并可采用不同厚度的平面应变单元进行有限元分析,所得到的解即是该三维问题的零阶摄动解。给出了摄动法的推导和有限元格式,计算了一个水驱法进行油田二次开发的算例并给出了计算结果。     

A fracture mapping and extended finite element scheme for coupled deformation and fluid flow in fractured porous media

This paper presents a fracture mapping (FM) approach combined with the extended finite element method (XFEM) to simulate coupled deformation and fluid flow in fractured porous media. Specifically, the method accurately represents the impact of discrete fractures on flow and deformation, although the individual fractures are not part of the finite element mesh. A key feature of FM‐XFEM is its ability to model discontinuities in the domain independently of the computational mesh. The proposed FM approach is a continuum‐based approach that is used to model the flow interaction between the porous matrix and existing fractures via a transfer function. Fracture geometry is defined using the level set method. Therefore, in contrast to the discrete fracture flow model, the fracture representation is not meshed along with the computational domain. Consequently, the method is able to determine the influence of fractures on fluid flow within a fractured domain without the complexity of meshing the fractures within the domain. The XFEM component of the scheme addresses the discontinuous displacement field within elements that are intersected by existing fractures. In XFEM, enrichment functions are added to the standard finite element approximation to adequately resolve discontinuous fields within the simulation domain. Numerical tests illustrate the ability of the method to adequately describe the displacement and fluid pressure fields within a fractured domain at significantly less computational expense than explicitly resolving the fracture within the finite element mesh. Copyright © 2013 John Wiley & Sons, Ltd.     

碎石垫层强度与变形特性试验研究和有限元分析

碎石垫层已广泛应用于土木工程各个领域,但碎石垫层的强度与变形特性尚不十分清楚,需要进一步研究。通过对5种不同厚度碎石垫层在高应力下的室内模型试验和有限元分析,揭示了不同厚度碎石垫层的强度特性和变形特征。通过有限元分析与试验结果的对比分析,得出了不同厚度的碎石垫层荷载与沉降变形、竖向应力与侧向约束力都近似呈线性变化。碎石垫层的厚度对沉降大小有一定影响,在其他条件相同的情况下,碎石垫层越厚,沉降越大。在一定荷载作用下,碎石垫层在较短的时间内自身压缩变形已基本趋于稳定,而后随时间的增加,其压缩变形量很小。提出了一种测定碎石类土侧压力系数、泊松比及变形模量的试验测定方法,以期为测定其他类别土的侧压力系数、泊松比及变形模量提供参考。     

考虑张剪破坏的边坡下限原理有限元法

在工程中发生的很多边坡破坏模式不仅表现为剪切破坏,还表现为在边坡后缘产生一定深度的张拉破坏。这种既考虑剪切破坏又考虑张拉破坏的屈服准则,是经典的Mohr-Coulomb（简称M-C）屈服准则难以模拟的。从空间方位离散的角度出发,对经典的M-C屈服面线性化的方式进行改造,建立了基于方位离散的线性化屈服准则,同时在屈服准则中考虑了张拉破坏准则,并引入伪黏聚力,保证在每一个离散方位平面上既不违背屈服准则,又不违背张拉破坏准则,建立了同时考虑张拉及剪切破坏的下限原理有限元法。算例表明了该方法的有效性,同时也证明了仅考虑剪切破坏时,会过高地估计边坡的安全系数。     

热-水-应力-迁移耦合条件下双重孔隙-裂隙介质的抗剪强度及有限元分析

考虑裂隙的连通率、间距、孔隙基质和裂隙材料在表征单元（REV）中的体积分数,并假定双重孔隙-裂隙介质的等效内摩擦角保持常数,而等效黏聚力是固有黏聚力、等效塑性应变、基质吸力、溶质浓度及温度的函数,提出了一种在热-水-应力-迁移耦合条件下确定表征单元内任一平面上等效的黏聚力及内摩擦角的方法。针对一个假定的位于非饱和双重孔隙-裂隙岩体中的高放废物地质处置模型进行了数值模拟及分析。结果表明,基质吸力对于等效黏聚力的增强作用大于等效塑性应变和溶质浓度的减弱作用,使得等效黏聚力得到了提高,故减少了围岩中的塑性区;由此岩体应力、孔（裂）隙水压力及流速、孔（裂）隙溶质浓度的分布及量值也发生相应的改变。