A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments. The author is grateful to the DFG (German Science Foundation—Deutsche Forschungsgemeinschaft) for the financial support under the grant number Di 430/4-2.     

A one-dimensional local discontinuous Galerkin Richards’ equation solution with dual-time stepping

We present a compact, high-order Richards’ equation solver using a local discontinuous Galerkin finite element method in space and a dual-time stepping method in time. Dual-time stepping methods convert a transient problem to a steady state problem, enabling direct evaluation of residual terms and resolve implicit equations in a step-wise manner keeping the method compact and amenable to parallel computing. Verification of our solver against an analytical solution shows high-order error convergence and demonstrates the solvers ability to maintain high accuracy using low spatial resolution; the method is robust and accurately resolves numerical solutions with time steps that are much larger than what is normally required for lower-order implicit schemes. Resilience of our solver (in terms of nonlinear convergence) is demonstrated in ponded infiltration into homogeneous and layered soils, for which HYDRUS-1D solutions are used as qualitative references to gauge performance of two slope limiting schemes.

     

Block fracturing analysis using nodal‐based discontinuous deformation analysis with the double minimization procedure

As a hybrid method, the nodal‐based discontinuous deformation analysis (NDDA) greatly improves the stress accuracy within each DDA block by coupling a well‐defined finite element mesh inside the DDA block; at the same time, the NDDA inherits the unique block kinematics of the standard DDA method. Each finite element mesh line inside the DDA block is treated as a potential crack, which enables the transformation of the block material from continuum to discontinuum through the tensile and shear fracturing mechanism. This paper introduces a double minimization procedure into the NDDA method to further improve the accuracy of the stresses evaluated at the finite element mesh lines and thus to obtain a more realistic fracture model. Three numerical examples are employed to demonstrate the improved stress accuracy by the implemented double minimization procedure and the accuracy and capability of the enhanced NDDA method in capturing brittle fracturing process. Copyright © 2013 John Wiley & Sons, Ltd.     

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.

     

A space-time adaptive method for reservoir flows: formulation and one-dimensional application

This paper presents a space-time adaptive framework for solving porous media flow problems, with specific application to reservoir simulation. A fully unstructured mesh discretization of space and time is used instead of a conventional time-marching approach. A space-time discontinuous Galerkin finite element method is employed to achieve a high-order discretization on the anisotropic, unstructured meshes. Anisotropic mesh adaptation is performed to reduce the error of a specified output of interest, by using a posteriori error estimates from the dual-weighted residual method to drive a metric-based mesh optimization algorithm. The space-time adaptive method is tested on a one-dimensional two-phase flow problem, and is found to be more efficient in terms of computational cost (degrees-of-freedom and total runtime) required to achieve a specified output error level, when compared to a conventional first-order time-marching finite volume method and the space-time discontinuous Galerkin method on structured meshes.     

模拟三维裂纹问题的扩展有限元法

扩展有限元法是一种在常规有限元框架内求解强和弱不连续问题的新型数值方法,其计算网格与不连续面相互独立,因此模拟移动不连续面时无需对网格进行重新剖分。给出了模拟三维裂纹问题的扩展有限元法。在常规有限元位移模式中,基于单位分解的思想加进一个阶跃函数和二维渐近裂尖位移场,反映裂纹处位移的不连续性。用两个水平集函数表示裂纹。采用线性互补法求解裂纹面非线性接触条件,不需要迭代,提高了计算效率。采用两点位移外推法计算裂纹前缘应力强度因子。给出了3个三维弹性静力问题算例,其结果显示了所提方法能获得高精度的应力强度因子,并能有效地处理裂纹面间的接触问题,同时表明扩展有限元结合线性互补法求解不连续问题具有较好的前景。     

Approximation of the Velocity by Coupling Discontinuous Galerkin and Mixed Finite Element Methods for Flow Problems

In this paper, we show how to couple the local discontinuous Galerkin method and the Raviart–Thomas mixed finite element method for elliptic equations modeling flow problems. We then show that the approximation of the velocity converges with the optimal order of k when we take the local discontinuous Galerkin that uses polynomials of degree k and the Raviart–Thomas space of polynomials of degree k?1.     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

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.     

溃坝水流数值模拟研究进展

溃坝问题在水利工程的设计管理中具有重要地位,也是广大学者长期以来一直关注和研究的课题。回顾和总结了国内外对溃坝水流演进问题的研究进展:介绍了溃坝水流的数学模型及解析解法存在的困难,进而讨论了数值解法的最新进展;论述了求解溃坝水流一维问题的有限差分法、近似黎曼解的Godunov格式法、Boltzmann法、KFVS法和二维问题的TVD格式法、间断有限元法、有限体积法、特征线法,并分析了各种方法的适用范围和优缺点,及讨论了限制函数的使用;介绍了利用自由水面追踪方法计算溃坝水流的研究进展,并根据目前存在的不足和实际工程的需要,提出了进一步研究的方向和发展趋势。     

Numerical Analysis of Space–Time Finite Element Formulations for Miscible Displacements

A finite element formulation is proposed to approximate a nonlinear system of partial differential equations, composed by an elliptic subsystem for the pressure–velocity and a transport equation (convection–diffusion) for the concentration, which models the incompressible miscible displacement of one fluid by another in a rigid porous media. The pressure is approximated by the classical Galerkin method and the velocity is calculated by a post-processing technique. Then, the concentration is obtained by a Galerkin/least-squares space–time (GLS/ST) finite element method. A numerical analysis is developed for the concentration approximation. Then, stability, convergence and numerical results are presented confirming the a priori error estimates.     

An efficient finite–discrete element method for quasi‐static nonlinear soil–structure interaction problems

An efficient finite–discrete element method applicable for the analysis of quasi‐static nonlinear soil–structure interaction problems involving large deformations in three‐dimensional space was presented in this paper. The present method differs from previous approaches in that the use of very fine mesh and small time steps was not needed to stabilize the calculation. The domain involving the large displacement was modeled using discrete elements, whereas the rest of the domain was modeled using finite elements. Forces acting on the discrete and finite elements were related by introducing interface elements at the boundary of the two domains. To improve the stability of the developed method, we used explicit time integration with different damping schemes applied to each domain to relax the system and to reach stability condition. With appropriate damping schemes, a relatively coarse finite element mesh can be used, resulting in significant savings in the computation time. The proposed algorithm was validated using three different benchmark problems, and the numerical results were compared with existing analytical and numerical solutions. The algorithm performance in solving practical soil–structure interaction problems was also investigated by simulating a large‐scale soft ground tunneling problem involving soil loss near an existing lining. Copyright © 2011 John Wiley & Sons, Ltd.     

Kalman filter finite element method applied to dynamic ground motion

The purpose of this paper is to investigate the estimation of dynamic elastic behavior of the ground using the Kalman filter finite element method. In the present paper, as the state equation, the balance of stress equation, the strain–displacement equation and the stress–strain equation are used. For temporal discretization, the Newmark ¼ method is employed, and for the spatial discretization the Galerkin method is applied. The Kalman filter finite element method is a combination of the Kalman filter and the finite element method. The present method is adaptable to estimations not only in time but also in space, as we have confirmed by its application to the Futatsuishi quarry site. The input data are the measured velocity, acceleration, etc., which may include mechanical noise. It has been shown in numerical studies that the estimated velocity, acceleration, etc., at any other spatial and temporal point can be obtained by removing the noise included in the observation. Copyright © 2008 John Wiley & Sons, Ltd.     

离散单元法研究进展及应用综述

岩体是一种高度非线性、非连续的介质,从现代工程规模来看,需要对其物理力学性质有一个比较好地认识。相比而言,数值模拟就是一个比较好的认识工具,因为其具有时间短、费用低、简单易行等优点。讨论了连续介质数值方法应用于非连续介质存在的问题,指出了非连续介质数值方法更加适合非连续介质。介绍了非连续介质数值方法中应用比较广泛、发展比较成熟的离散单元法的基本情况,比较详细地阐述了离散单元法中几个有代表性的问题,详细地介绍了离散单元法的研究进展,对其应用也做了相关的介绍,最后进行归纳总结,探讨了离散单元法今后发展的几个方向。     

Formulation of discontinuous deformation analysis (DDA) — an implicit discrete element model for block systems

Continuum-based numerical methods have played a leading role in the numerical solution of problems in rock mechanics and engineering geology. However, for fractured rocks, a continuum assumption often leads to difficult parameters to define and over-simplified geometry to be realistic. In such case, discrete representations of fractures and individual blocks must be adopted. In this paper, a newly emerged member in the family of discrete element methods (DEM), the discontinuous deformation analysis (DDA), is presented, including its variational principle, governing equations, solution techniques and contact representation and detection algorithms. Its relative advantages and shortcomings are compared with the explicit distinct element method and the finite element method. An example of the analysis of tunnel stability is provided to demonstrate the capability of this new method.     

Relationships among some locally conservative discretization methods which handle discontinuous coefficients

This paper presents the relationships between some numerical methods suitable for a heterogeneous elliptic equation with application to reservoir simulation. The methods discussed are the classical mixed finite element method (MFEM), the control-volume mixed finite element method (CVMFEM), the support operators method (SOM), the enhanced cell-centered finite difference method (ECCFDM), and the multi-point flux-approximation (MPFA) control-volume method. These methods are all locally mass conservative, and handle general irregular grids with anisotropic and heterogeneous discontinuous permeability. In addition to this, the methods have a common weak continuity in the pressure across the edges, which in some cases corresponds to Lagrange multipliers. It seems that this last property is an essential common quality for these methods. T.F. Russell: Partially supported by the National Science Foundation Grant Nos. DMS-0084438 and DMS-0222300.     

Poroelastic two‐phase material modeling: theoretical formulation and embedded finite element method implementation

This paper presents the formulation of FEMs for the numerical modeling of a poroelastic two‐phase (aggregates/mixture phase) solid. The displacement and pressure fields are decomposed, following the Enhanced Assumed Strain (EAS) method, into a regular part and an enhanced part. This leads to discontinuous strain and pressure gradient fields allowing to capture the jump in mechanical and hydrical properties passing through the interface between the aggregates and the mixture phase. All these enhanced fields are treated in the context of the embedded FEM through a local enhancement of the finite element interpolations as these jumps appear. The local character of these interpolations leads after a static condensation of the enhanced fields to a problem exhibiting the same structure as common poroelastic finite element models but incorporating now the mechanical and hydrical properties of a two‐phase solid. Copyright © 2015 John Wiley & Sons, Ltd.     

Computational simulation for the morphological evolution of nonaqueous phase liquid dissolution fronts in two-dimensional fluid-saturated porous media

This paper deals with the computational aspects of nonaqueous phase liquid (NAPL) dissolution front instability in two-dimensional fluid-saturated porous media of finite domains. After the governing equations of an NAPL dissolution system are briefly described, a combination of the finite element and finite difference methods is proposed to solve these equations. In the proposed numerical procedure, the finite difference method is used to discretize time, while the finite element method is used to discretize space. Two benchmark problems, for which either analytical results or previous solutions are available, are used to verify the proposed numerical procedure. The related simulation results from these two benchmark problems have demonstrated that the proposed numerical procedure is useful and applicable for simulating the morphological evolution of NAPL dissolution fronts in two-dimensional fluid-saturated porous media of finite domains. As an application, the proposed numerical procedure has been used to simulate morphological evolution processes for three kinds of NAPL dissolution fronts in supercritical NAPL dissolution systems. It has been recognized that: (1) if the Zhao number of an NAPL dissolution system is in the lower range of the supercritical Zhao numbers, the fundamental mode is predominant; (2) if the Zhao number is in the middle range of the supercritical Zhao numbers, the (normal) fingering mode is the predominant pattern of the NAPL dissolution front; and (3) if the Zhao number is in the higher range of the supercritical Zhao numbers, the fractal mode is predominant for the NAPL dissolution front.     

砂土液化变形的有限元-无网格耦合方法

通过构造Biot固结理论u-p方程的无网格伽辽金-有限元耦合方法,对砂土液化变形问题进行了数值模拟。对于饱和砂土,采用Oka等提出的弹塑性本构模型,同时采用更新的Lagrange计算格式推导了控制方程。耦合方法能够发挥有限元和无网格各自的优点,既避免了由于单元变形扭曲而引起的计算中断,也可节约计算时间,算例验证了该方法在地震液化问题中的有效性。     

Adaptive integration of constitutive rate equations

In finite element calculations the constitutive model plays a key role. The evaluation of the stress response of the constitutive relation for a given strain increment, which is a time integration in the case of models of the rate type, is a typical sub task in such calculations. Adaptive behaviour of the time integration is essential to assure numerical stability and to control the accuracy of the solution. An adaptive second order semi-implicit method is developed in this paper. Its numerical behaviour is compared with an adaptive second order explicit scheme. The two proposed methods control the local error and guarantee numerical stability of the time integration. We include several numerical geotechnical element tests using hypoplasticity with intergranular strain. The element tests simulate the behaviour of a finite element method based on the displacement formulation.