An operator‐split ALE model for large deformation analysis of geomaterials

Analysis of large deformation of geomaterials subjected to time‐varying load poses a very difficult problem for the geotechnical profession. Conventional finite element schemes using the updated Lagrangian formulation may suffer from serious numerical difficulties when the deformation of geomaterials is significantly large such that the discretized elements are severely distorted. In this paper, an operator‐split arbitrary Lagrangian–Eulerian (ALE) finite element model is proposed for large deformation analysis of a soil mass subjected to either static or dynamic loading, where the soil is modelled as a saturated porous material with solid–fluid coupling and strong material non‐linearity. Each time step of the operator‐split ALE algorithm consists of a Lagrangian step and an Eulerian step. In the Lagrangian step, the equilibrium equation and continuity equation of the saturated soil are solved by the updated Lagrangian method. In the Eulerian step, mesh smoothing is performed for the deformed body and the state variables obtained in the updated Lagrangian step are then transferred to the new mesh system. The accuracy and efficiency of the proposed ALE method are verified by comparison of its results with the results produced by an analytical solution for one‐dimensional finite elastic consolidation of a soil column and with the results from the small strain finite element analysis and the updated Lagrangian analysis. Its performance is further illustrated by simulation of a complex problem involving the transient response of an embankment subjected to earthquake loading. Copyright © 2007 John Wiley & Sons, Ltd.     

Arbitrary Lagrangian–Eulerian method for dynamic analysis of geotechnical problems

In this paper an arbitrary Lagrangian–Eulerian (ALE) method to solve dynamic problems involving large deformation is presented. This ALE method is based upon the operator-split technique in which the material displacements and mesh displacements are uncoupled. A brief history of the ALE method is first presented and then special issues such as time-stepping, mesh refinement, energy absorbing boundaries, dynamic equilibrium checks and remapping of state variables are explained. The ALE method and the updated-lagrangian (UL) method are then used to analyse some geotechnical problems to examine the significance of inertia effects, large deformation and contact mechanics. The results show the efficiency of the ALE method for solving dynamic geotechnical problems involving large deformation.     

弹塑性土质边坡的ALE方法有限元分析

有限元法被广泛用于解决几何和材料非线性的问题,但标准的有限元方法难以有效解决某些材料的大变形问题和计算中的网格扭曲问题。任意拉格朗日-欧拉法（ALE法）吸取了拉格朗日和欧拉法的优点,并克服了两者的缺点,可用于解决仅用拉格朗日或欧拉有限元法所难以解决的问题。基于ALE有限元方法和弹塑性大变形基本原理,研究了岩土工程中土质边坡在自重作用下的稳定问题;计算结果不仅能直观地显示失稳时的大变形状态,并能确定较符合实际的临界滑移面形状;同时分析了含软弱夹层复杂土质边坡的稳定性。结果表明,ALE方法能有效分析土质边坡的稳定性问题,适用于岩土工程的弹塑性分析。     

Solution of quasi‐static large‐strain problems by the material point method

A procedure for solving quasi‐static large‐strain problems by the material point method is presented. Owing to the Lagrangian–Eulerian features of the method, problems associated with excessive mesh distortions that develop in the Lagrangian formulations of the finite element method are avoided. Three‐dimensional problems are solved utilizing 15‐noded prismatic and 10‐noded tetrahedral elements with quadratic interpolation functions as well as an implicit integration scheme. An algorithm for exploiting the numerical integration procedure on the computational mesh is proposed. Several numerical examples are shown. Copyright © 2010 John Wiley & Sons, Ltd.     

An ALE method for penetration into sand utilizing optimization-based mesh motion

The numerical simulation of penetration into sand is one of the most challenging problems in computational geomechanics. The paper presents an arbitrary Lagrangian–Eulerian (ALE) finite element method for plane and axisymmetric quasi-static penetration into sand which overcomes the problems associated with the classical approaches. An operator-split is applied which breaks up solution of the governing equations over a time step into a Lagrangian step, a mesh motion step, and a transport step. A unique feature of the ALE method is an advanced hypoplastic rate constitutive equation to realistically predict stress and density changes within the material even at large deformations. In addition, an efficient optimization-based algorithm has been implemented to smooth out the non-convexly distorted mesh regions that occur below a penetrator. Applications to shallow penetration and pile penetration are given which make use of the developments.     

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.     

Frictionless contact formulation for dynamic analysis of nonlinear saturated porous media based on the mortar method

A finite element algorithm for frictionless contact problems in a two‐phase saturated porous medium, considering finite deformation and inertia effects, has been formulated and implemented in a finite element programme. The mechanical behaviour of the saturated porous medium is predicted using mixture theory, which models the dynamic advection of fluids through a fully saturated porous solid matrix. The resulting mixed formulation predicts all field variables including the solid displacement, pore fluid pressure and Darcy velocity of the pore fluid. The contact constraints arising from the requirement for continuity of the contact traction, as well as the fluid flow across the contact interface, are enforced using a penalty approach that is regularised with an augmented Lagrangian method. The contact formulation is based on a mortar segment‐to‐segment scheme that allows the interpolation functions of the contact elements to be of order N. The main thrust of this paper is therefore how to deal with contact interfaces in problems that involve both dynamics and consolidation and possibly large deformations of porous media. The numerical algorithm is first verified using several illustrative examples. This algorithm is then employed to solve a pipe‐seabed interaction problem, involving large deformations and dynamic effects, and the results of the analysis are also compared with those obtained using a node‐to‐segment contact algorithm. The results of this study indicate that the proposed method is able to solve the highly nonlinear problem of dynamic soil–structure interaction when coupled with pore water pressures and Darcy velocity. Copyright © 2015 John Wiley & Sons, Ltd.     

Study of the free surface flow of water–kaolinite mixture by moving particle semi‐implicit (MPS) method

The flow of water–kaolinite mixtures exhibits a non‐Newtonian nature that differs from the flow of Newtonian fluid. The varying viscosities and shear history of non‐Newtonian fluid flows necessitate the use of a rheology model in moving particle semi‐implicit (MPS) for the numerical studies. On the other hand, the Lagrangian method has the advantage of handling free surface flows with large deformation and fragmentation. This study proposes a mesh‐free Lagrangian method, namely, the MPS method, together with a simple rheology model to investigate the non‐Newtonian free surface flows. The rheological parameters required in the rheology model are determined based upon experiments. The proposed method is applied to a water–kaolinite mixture collapse problem and is proved to be capable of reproducing the significant flow features observed in laboratory experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

High‐order ADE scheme for solving the fluid diffusion equation in non‐uniform grids and its application in coupled hydro‐mechanical simulation

To improve the stability and efficiency of explicit technique, one proposed method is to use an unconditionally stable alternating direction explicit (ADE) scheme. However, the standard ADE scheme is only moderately accurate and restricted to uniform grids. This paper derives a novel high‐order ADE scheme capable of solving the fluid diffusion equation in non‐uniform grids. The new scheme is derived by performing a fourth‐order finite difference approximation to the spatial derivatives of the diffusion equation in non‐uniform grid. The implicit Crank‐Nicolson technique is then applied to the resulting approximation, and the subsequent equation is split into two alternating direction sweeps, giving rise to a new high‐order ADE scheme. Because the new scheme can be potentially applied in coupled hydro‐mechanical (H‐M) simulation, the pore pressure solutions from the new scheme are then sequentially coupled with an existing geomechanical simulator in the computer program Fast Lagrangian Analysis of Continua. This coupling procedure is called the sequentially explicit coupling technique based on the fourth‐order ADE scheme (SEA‐4). Verifications of well‐known consolidation problems showed that the new ADE scheme and SEA‐4 can reduce computer runtime by 46% to 75% to that of Fast Lagrangian Analysis of Continua's basic scheme. At the same time, the techniques still maintained average percentage error of 1.6% to 3.5% for pore pressure and 0.2% to 1.5% for displacement solutions and were still accurate under typical grid non‐uniformities. This result suggests that the new high‐order ADE scheme can provide an efficient explicit technique for solving the flow equation of a coupled H‐M problem, which will be beneficial for large‐scale and long‐term H‐M problems in geoengineering.     

Gibson大变形固结理论的一种连续介质力学表述

为了克服Gibson大变形固结理论的一维局限性,提出了一个新的连续介质力学模型,其中,采用第2类Piola- Kirchhoff应力和Green应变分别替代Lagrange应力和孔隙比。该模型可以考虑土体压缩性和渗透性的非线性变化以及几何非线性,并具有易于推广到多维模型的优势。结合试验资料,探讨了新模型的固结系数和对流系数的变化规律,结果表明,对流系数随着应变增加而减小,反映了大变形固结过程中土体自重效应在不断减弱。     

边坡稳定分析的自适应性局部网格加密解法

常规的有限元强度折减法在确定边坡塑性区时,不能直观地反映出塑性区的形状和滑裂面位置,计算结果精度低。在塑性区范围内进行网格局部加密,不仅使安全系数的计算精度大幅度提高,而且可求出滑裂面的形状和位置。针对求解塑性区搜索滑裂面这一具体问题,提出塑性应变界限值网格加密准则,采用自适应性网格局部加密技术求解边坡的滑裂面和稳定安全系数。数值计算结果表明,所提出的加密准则能正确地确定滑裂面的形状和位置,计算所得稳定安全系数和理论稳定安全系数仅相差0.2 %。     

Properties of a diffuse interface model based on a porous medium theory for solid–liquid dissolution problems

In this paper, a local non-equilibrium diffuse interface model is introduced for describing solid–liquid dissolution problems. The model is developed based on the analysis of Golfier et al. (J Fluid Mech 457:213–254, 2002) upon the dissolution of a porous domain, with the additional requirement that density variations with the mass fraction are taken into account. The control equations are generated by the upscaling of the balance equations for a solid–liquid dissolution using a volume averaging theory. This results into a diffuse interface model (DIM) that does not require an explicit treatment of the dissolving interface, e.g., the use of arbitrary Lagrangian–Eulerian (ALE) methods, for instance. Test cases were performed to study the features and influences of the effective coefficients inside the DIM. In particular, an optimum expression for the solid–liquid exchange coefficient is obtained from a comparison with the referenced solution by ALE simulations. Finally, a Ra–Pe diagram illustrates the interaction of natural convection and forced convection in the dissolution problem.     

Natural convection of compressible and incompressible gases in undeformable porous media under cold climate conditions

A numerical model for convective heat and mass transport of compressible or incompressible gas flows with soil-water phase change is presented. In general, the gaseous phase is considered as compressible and the model accounts for adiabatic processes of compression heating and expansion cooling. The inherently compressible gaseous phase may nevertheless be considered as incompressible by adopting the Oberbeck–Boussinesq approximations. The numerical method used to solve the equations that describe natural convection is based on a Galerkin finite element formulation with adaptive mesh refinement and dynamic time step control. As most existing numerical studies have focused on the behavior of incompressible fluids, model substantiation examines the influence of fluid compressibility on two-widely used benchmarks of steady-state convective heat and mass transport. The relative importance of the effect of pressure-compressibility cooling is shown to increase as the thermal gradient approaches the magnitude of the adiabatic gradient. From these results, it may be concluded that pore-air compressibility cannot be neglected in medium to large-sized enclosures at small temperature differentials. After demonstrating its ability to solve fairly complex transient problems, the model is used to further our understanding of the thermal behavior of the toe drain at the LA2-BSU dam in the province of Quebec, Canada.     

A new mixed finite element method for poro‐elasticity

Development of robust numerical solutions for poro‐elasticity is an important and timely issue in modern computational geomechanics. Recently, research in this area has seen a surge in activity, not only because of increased interest in coupled problems relevant to the petroleum industry, but also due to emerging applications of poro‐elasticity for modelling problems in biomedical engineering and materials science. In this paper, an original mixed least‐squares method for solving Biot consolidation problems is developed. The solution is obtained via 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 involves four separate categories of unknowns: displacements, stresses, fluid pressures and velocities. Each of these unknowns is approximated by linear continuous functions. The mathematical formulation is implemented in an original computer program, written from scratch and using object‐oriented logic. The performance of the method is tested on one‐ and two‐dimensional classical problems in poro‐elasticity. The numerical experiments suggest the same rates of convergence for all four types of variables, when the same interpolation spaces are used. The continuous linear triangles show the same rates of convergence for both compressible and entirely incompressible elastic solids. This mixed formulation results in non‐oscillating fluid pressures over entire domain for different moments of time. The method appears to be naturally stable, without any need of additional stabilization terms with mesh‐dependent parameters. Copyright © 2007 John Wiley & Sons, Ltd.     

Simulation of long‐term consolidation behavior of soft sensitive clay using an elasto‐viscoplastic constitutive model

The phenomenon of excess pore water pressure increase or stagnation and continuing large ground deformation in soft sensitive clay following the completion of construction of embankment is simulated for a case study at Saint Alban, Quebec, Canada. The present model employs an updated Lagrangian finite element framework and is combined with an automatic time increment selection scheme. The simulation based on an elasto‐viscoplastic constitutive model considers soil‐structure degradation effect. It is shown that without consideration for the microstructural degradation effect, it is not possible to reproduce the field responses of soft sensitive clay even during the construction of the embankment. When the soil‐structure degradation effect is considered, the present model can offer reasonably accurate prediction for the consolidation behavior of soft sensitive clay, including the so‐called anomalous pore water pressure generation and continuing large deformation even after the end of construction, which has been posing numerous uncertainties on the long‐term performance of earth structures. Copyright © 2013 John Wiley & Sons, Ltd.     

顺层岩质边坡渐进破坏及失稳机理的数值模拟研究

提出了利用Lagrangian乘子网格来模拟顺层边坡层间结构面接触行为的方法,并推导了考虑接触约束的Lagrangian控制方程的弱形式及其有限元离散方程。Lagrangian乘子网格容许界面出现较大的切向滑移,较好地解决了接触问题中的高度非线性和响应不平滑性。应用以上方法对沪－蓉－西高速公路某顺层岩质高边坡进行了数值模拟分析,得到了临滑状态下的位移场、应力应变场、结构面上的接触状态、层间滑移距离以及接触摩擦力的分布规律。数值模拟结果表明,顺层岩质边坡的破坏是一个渐进的过程,由最初的层间错动,发展为顺层蠕滑,最后在坡底剪出而破坏;其破坏型式由最初的顺层滑动逐步向溃曲破坏转化。     

A framework for automatic modeling of underground excavations and optimizing three‐dimensional boundary and finite element meshes derived from them—framework

Many problems in mining and civil engineering require using numerical stress analysis methods to repeatedly solve large models. Widespread acceptance of tunneling methods, such as New Austrian Tunneling Method, which depend heavily on numerical stress analysis tools and the fact that the effects of excavation at the face of a tunnel are distinctively three–dimensional (3D), necessitates the use of 3D numerical analysis for these problems. Stress analysis of a practical mining problem can be very lengthy, and the processing time can be measured in days or weeks at times. A framework is developed to facilitate efficient modeling of underground excavations and to create an optimal 3D mesh by reducing the number of surface and volume elements while keeping the result of stress analysis accurate enough at the region of interest, where a solution is sought. Fewer surface and volume elements mean fewer degrees of freedom in the numerical model, which directly translates into savings in computational time and resources. The mesh refinement algorithm is driven by a set of criteria that are functions of distance and visibility of points from the region of interest, and the framework can be easily extended by adding new types of criteria. This paper defines the framework, whereas a second companion paper will investigate its efficiency, accuracy and application to a number of practical mining problems. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Application of a Coupled Eulerian–Lagrangian approach on geomechanical problems involving large deformations

Geotechnical boundary value problems involving large deformations are often difficult to solve using the classical finite element method. Large mesh distortions and contact problems can occur due to the large deformations such that a convergent solution cannot be achieved. Since Abaqus, Version 6.8, a new Coupled Eulerian–Lagrangian (CEL) approach has been developed to overcome the difficulties with regard to finite element method and large deformation analyses. This new method is investigated regarding its capabilities. First, a benchmark test, a strip footing problem is investigated and compared to analytical solutions and results of comparable finite element analyses. This benchmark test shows that CEL is well suited to deal with problems which cannot be fully solved using FEM. In further applications the CEL approach is applied to more complex geotechnical boundary value problems. First, the installation of a pile into subsoil is simulated. The pile is jacked into the ground and the results received from these analyses are compared to results of classical finite element simulations. A second case study is the simulation of a ship running aground at an embankment. The results of the CEL simulation are compared to in situ measurement data. Finally, the capabilities of the new CEL approach are evaluated regarding its robustness and efficiency.     

Large deformation dynamic analysis of saturated porous media with applications to penetration problems

This paper outlines the development as well as implementation of a numerical procedure for coupled finite element analysis of dynamic problems in geomechanics, particularly those involving large deformations and soil-structure interaction. The procedure is based on Biot’s theory for the dynamic behaviour of saturated porous media. The nonlinear behaviour of the solid phase of the soil is represented by either the Mohr Coulomb or Modified Cam Clay material model. The interface between soil and structure is modelled by the so-called node-to-segment contact method. The contact algorithm uses a penalty approach to enforce constraints and to prevent rigid body interpenetration. Moreover, the contact algorithm utilises a smooth discretisation of the contact surfaces to decrease numerical oscillations. An Arbitrary Lagrangian–Eulerian (ALE) scheme preserves the quality and topology of the finite element mesh throughout the numerical simulation. The generalised-α method is used to integrate the governing equations of motion in the time domain. Some aspects of the numerical procedure are validated by solving two benchmark problems. Subsequently, dynamic soil behaviour including the development of excess pore-water pressure due to the fast installation of a single pile and the penetration of a free falling torpedo anchor are studied. The numerical results indicate the robustness and applicability of the proposed method. Typical distributions of the predicted excess pore-water pressures generated due to the dynamic penetration of an object into a saturated soil are presented, revealing higher magnitudes of pore pressure at the face of the penetrometer and lower values along the shaft. A smooth discretisation of the contact interface between soil and structure is found to be a crucial factor to avoid severe oscillations in the predicted dynamic response of the soil.