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.     

Coupling multipoint flux mixed finite element methodswith continuous Galerkin methods for poroelasticity

We study the numerical approximation on irregular domains with general grids of the system of poroelasticity, which describes fluid flow in deformable porous media. The flow equation is discretized by a multipoint flux mixed finite element method and the displacements are approximated by a continuous Galerkin finite element method. First-order convergence in space and time is established in appropriate norms for the pressure, velocity, and displacement. Numerical results are presented that illustrate the behavior of the method.     

Finite Element Solvers for Biot’s Poroelasticity Equations in Porous Media

We study and compare five different combinations of finite element spaces for approximating the coupled flow and solid deformation system, so-called Biot’s equations. The permeability and porosity fields are heterogeneous and depend on solid displacement and fluid pressure. We provide detailed comparisons among the continuous Galerkin, discontinuous Galerkin, enriched Galerkin, and two types of mixed finite element methods. Several advantages and disadvantages for each of the above techniques are investigated by comparing local mass conservation properties, the accuracy of the flux approximation, number of degrees of freedom (DOF), and wall and CPU times. Three-field formulation methods with fluid velocity as an additional primary variable generally require a larger number of DOF, longer wall and CPU times, and a greater number of iterations in the linear solver in order to converge. The two-field formulation, a combination of continuous and enriched Galerkin function space, requires the fewest DOF among the methods that conserve local mass. Moreover, our results illustrate that three out of the five methods conserve local mass and produce similar flux approximations when conductivity alteration is included. These comparisons of the key performance indicators of different combinations of finite element methods can be utilized to choose the preferred method based on the required accuracy and the available computational resources.

     

A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow

We consider conjunctive surface-subsurface flow modeling, where surface water flow is described by the shallow water equations and ground water flow by Richards’ equation for the vadose zone. Coupling between the models is based on the continuity of flux and water pressure. Numerical approximation of the coupled model using the framework of discontinuous Galerkin (DG) methods is formulated. In the subsurface, the local discontinuous Galerkin (LDG) method is used to approximate ground water velocity and hydraulic head; a DG method is also used to approximate surface water velocity and elevation. This approach allows for a weak coupling of the models and the use of different approximating spaces and/or meshes within each regime. A simplified LDG method based on continuous approximations to water head is also described. Numerical results that investigate physical and numerical aspects of surface–subsurface flow modeling are presented. This work was supported by National Science Foundation grant DMS-0411413.     

A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium

We develop and analyze a mixed finite element method for the solution of an elliptic system modeling a porous medium with large cavities, called vugs. It consists of a second-order elliptic (i.e., Darcy) equation on part of the domain coupled to a Stokes equation on the rest of the domain, and a slip boundary condition (due to Beavers–Joseph–Saffman) on the interface between them. The tangential velocity is not continuous on the interface. We consider a 2-D vuggy porous medium with many small cavities throughout its extent, so the interface is not isolated. We use a certain conforming Stokes element on rectangles, slightly modified near the interface to account for the tangential discontinuity. This gives a mixed finite element method for the entire Darcy–Stokes system with a regular sparsity pattern that is easy to implement, independent of the vug geometry, as long as it aligns with the grid. We prove optimal global first-order L ² convergence of the velocity and pressure, as well as the velocity gradient in the Stokes domain. Numerical results verify these rates of convergence and even suggest somewhat better convergence in certain situations. Finally, we present a lower dimensional space that uses Raviart–Thomas elements in the Darcy domain and uses our new modified elements near the interface in transition to the Stokes elements.     

A finite element procedure for calculation of strain from interpolated displacements

Second-order universal kriging is proposed as an accurate model for interpolating displacements measured in the field to nodal points of a superimposed finite element mesh. These interpolated displacements are used in a modified finite element procedure to calculate strain. This model is compared to a local trend model to judge superiority. Interpolation models are tested by randomly sampling displacements obtained in a finite element analysis, then applying interpolation in attempts to reconstruct the original results.     

A mixed mode rock fracture model for the prediction of crack path

Crack propagation in rocks is simulated by using a displacement substitution method based on a mixed mode fracture criterion. The main advantage of this model is that it can distinguish between mode I and mode II stress intensity factors simultaneously. A typical finite element program is used to compute displacements adjacent to the crack tip. The maximum circumferential tensile stress is adopted as a ‘yield surface’ for the calculation of the load factor in each crack increment. Pure mode I and mixed mode examples have been analysed to validate the capability of the model. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

A Timoshenko finite element straight beam with internal degrees of freedom

We present hereafter the formulation of a Timoshenko finite element straight beam with internal degrees of freedom, suitable for nonlinear material problems in geomechanics (e.g., beam type structures and deep pile foundations). Cubic shape functions are used for the transverse displacements and quadratic for the rotations. The element is free of shear locking, and we prove that one element is able to predict the exact tip displacements for any complex distributed loadings and any suitable boundary conditions. After the presentation of the virtual power and the weak form formulations, the construction of the elementary stiffness matrix is detailed. The analytical results of the static condensation method are provided. It is also proven that the element introduced by Friedman and Kosmatka in 1 , with shape functions depending on material properties, is derived from the new beam element. Validation is provided using linear and material nonlinear applications (reinforced concrete column under cyclic loading) in the context of a multifiber beam formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

A quadratic variation indirect boundary element method for traction boundary-value problems of two-dimensional elastostatics

Boundary integral equations for traction boundary-value problems of two-dimensional elastostatics are derived by the indirect boundary element method. Quadratic variation functions for the representation of geometry, fictitious forces and displacements over each boundary element are described. A system of equations approximating to the boundary integral equations is obtained by a Galerkin formulation in which the integral equation is written at Gauss integration points of elements. The method of computation of the Cauchy principal value is described. Examples of application to the analysis of stress and displacement around underground excavations demonstrate the accuracy and efficiency of the formulation.     

Three‐dimensional modeling of problems in poro‐elasticity via a mixed least‐squares method using linear tetrahedral elements

In a previous publication we developed a new mixed least‐squares method for poro‐elasticity. The approximate solution was obtained via a minimization of a least‐squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involved four independent types of variables: displacements, stresses, pore pressures and velocities. All of them were approximated by linear continuous triangles. Encouraged by the computational results, obtained from the two‐dimensional implementation of the method, we extended our formulation to three dimensions. In this paper we present numerical examples for the performance of continuous linear tetrahedra within the context of the mixed least‐squares method. The initial results suggest that the method works well in the nearly and entirely incompressible limits for elasticity. For poro‐elasticity, the obtained pore pressures are stable without exhibiting the oscillations, which are observed when the standard Galerkin formulation is used. Copyright © 2010 John Wiley & Sons, Ltd.     

Coupling of finite and hierarchical infinite elements: application to a non‐homogeneous cross‐anisotropic half‐space subjected to a non‐uniform circular loading

A method is presented for coupling cubic‐order quadrilateral finite elements with the finite side of a new coordinate ascent hierarchical infinite element. At a common side shared by a hierarchical infinite element and an arbitrary number of finite elements, the displacements are minimized in the least square sense with respect to the degrees‐of‐freedom of the finite elements. This leads to a set of equations that relate the degrees‐of‐freedom of the finite and hierarchical infinite elements on the shared side. The method is applied to a non‐homogeneous cross‐anisotropic half‐space subjected to a non‐uniform circular loading with Young's and shear moduli varying with depth according to the power law. A constant mesh constructed from coupled finite and hierarchical infinite elements is used and convergence is sought simply by increasing the degree of the interpolating polynomial. The displacements and stresses produced by conical and parabolic circular loads applied on the surface are obtained. The efficiency of the proposed method is demonstrated through convergence and comparison studies. New results produced by a frusto‐conical circular load applied on the surface of a half‐space made up of heavily consolidated London clay are provided. The non‐homogeneity parameter and degree of anisotropy are shown to influence the soil response. Copyright © 2012 John Wiley & Sons, Ltd.     

Simulation of arch dam – foundation interaction with a new friction interface element

A new model for three-dimensional non-linear contact problems with irreversible friction is presented here for the interaction between the rock foundation and an arch dam structure. Based on the finite element method and load incremental theory, a constraint contact element with displacements and contact stresses as node parameters is developed. In this approach, four contact conditions are considered, i.e. fixed, slip, free and mixed. This model can simulate frictional slippage, decoupling and re-bonding of two bodies initially mating at a common interface or with any initial gaps. Furthermore boundary conditions for this element are discussed and treatment measures proposed. This method is shown to be effective and to have the advantage of being easily implemented into standard finite element programs. Solutions are obtained for a centrally loaded, simply supported composite beam and for an end-loaded elastica with initial gaps in regional contact with a rigid surface. The results obtained for these examples are compared to the plane stress solutions by contact friction analysis. As an application example, Quanshui arch dam located in Ruyuan County of Guangdong Province in southern China is simulated with the new element.     

增量位移反分析在水电地下洞室工程中的应用

反分析是确定计算模型参数的有效方法.通常多采用量测所得全量位移进行反演计算.但地下工程中许多量测数据为增量位移,且实际施工过程可以通过建立分步开挖的有限元模型来模拟.据此,结合某水电站地下洞室工程中地下厂房的开挖,建立了模拟动态施工的有限元模型,利用某一开挖步施工前后量测值之差,采用增量位移优化反分析方法对洞室附近初始地应力场及围岩弹性模量进行了反演.计算所得增量位移与实测值符合较好,表明了这种方法的可行性.同时,根据分析结果对该方法进行了评价.     

Upwind‐mixed methods for transport equations

In this paper, we continue our analysis of upwind‐mixed methods for advection–diffusion equations, which have been developed and analyzed by the first author over the past several years. In previous work, our analysis has been limited to low order approximating spaces, positive definite diffusion coefficients and Dirichlet boundary conditions. In this paper, we extend our results to higher order approximating spaces, possibly zero diffusion, and more physically realistic boundary conditions. Moreover, unlike previous papers, we avoid the use of Gronwall's Inequality, which can result in extremely large constants in the stability and error bounds. Numerical results are presented for constant, linear and quadratic approximating spaces. This revised version was published online in July 2006 with corrections to the Cover Date.     

A multidirectional semi‐analytical method for analysis of laterally loaded pile groups in multi‐layered elastic strata

A semi‐analytical method for calculating the response of single piles and pile groups subjected to lateral loading is developed in this paper. Displacements anywhere in the soil domain are tied to the displacements of the piles through decay functions. The principle of virtual work and the calculus of variations are used to derive the governing differential equations that describe the response of the piles and soil. The eigenvalue method and the finite difference technique are used to solve the system of coupled differential equations for the piles and soil, respectively. The proposed method takes into account the soil surface displacement along and perpendicular to the loading direction and produces displacement fields that are very close to those produced by the finite element method but at lower computational effort. Compared with the previous method that considered only the soil displacement along the loading direction, accounting for the multi‐directional soil displacement field produces responses for the piles and soil that are closer to those approximated by the finite element method. Copyright © 2016 John Wiley & Sons, Ltd.     

Comparison of continuous and discontinuous constitutive models to simulate concrete behaviour under mixed‐mode failure conditions

The paper presents detailed FE simulation results of concrete elements under mixed‐mode failure conditions according to the so‐called shear‐tension test by Nooru‐Mohamed, characterized by curved cracks. A continuous and discontinuous numerical two‐dimensional approach was used. In order to describe the concrete's behaviour within continuum mechanics, two different constitutive models were used. First, an elasto‐plastic model with isotropic hardening and softening was assumed. In a compression regime, a Drucker–Prager criterion with a non‐associated flow rule was used. In turn, in a tensile regime, a Rankine criterion with an associated flow rule was adopted. Second, an isotropic damage constitutive model was applied with a single scalar damage parameter and different definitions of the equivalent strain. Both constitutive laws were enriched by a characteristic length of micro‐structure to capture properly strain localization. As an alternative approach, the extended finite element method was used. Our results were compared with the experimental ones and with results of other FE simulations reported in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

海底缓坡场地地震侧移数值分析方法

地震动使海底倾斜土层软化、液化并产生永久变形和位移。基于有限元理论,提出一种海底缓坡场地地震引起水平侧移的数值计算方法,将波浪荷载简化为恒定压力荷载和初始孔压,采用二维有效应力动力有限元分析方法进行液化分析,同时由模量软化理论得到土层在地震动各时段的模量,通过非线性静力方法计算软化、液化引起的水平侧移。由算例分析了土层坡度、液化层及上覆非液化层厚度、波浪荷载等因素对侧移的影响,通过对比分析表明了该方法的有效性,可为近海工程场地地震地质灾害评价提供参考数据。     

A FIC‐based stabilized mixed finite element method with equal order interpolation for solid–pore fluid interaction problems

A new mixed displacement‐pressure element for solving solid–pore fluid interaction problems is presented. In the resulting coupled system of equations, the balance of momentum equation remains unaltered, while the mass balance equation for the pore fluid is stabilized with the inclusion of higher‐order terms multiplied by arbitrary dimensions in space, following the finite calculus (FIC) procedure. The stabilized FIC‐FEM formulation can be applied to any kind of interpolation for the displacements and the pressure, but in this work, we have used linear elements of equal order interpolation for both set of unknowns. Examples in 2D and 3D are presented to illustrate the accuracy of the stabilized formulation for solid–pore fluid interaction problems. Copyright © 2016 John Wiley & Sons, Ltd.