A stable meshfree method for fully coupled flow-deformation analysis of saturated porous media

A fully coupled meshfree algorithm is proposed for numerical analysis of Biot’s formulation. Spatial discretization of the governing equations is presented using the Radial Point Interpolation Method (RPIM). Temporal discretization is achieved based on a novel three-point approximation technique with a variable time step, which has second order accuracy and avoids spurious ripple effects observed in the conventional two-point Crank Nicolson technique. Application of the model is demonstrated using several numerical examples with analytical or semi-analytical solutions. It is shown that the model proposed is effective in simulating the coupled flow deformation behaviour in fluid saturated porous media with good accuracy and stability irrespective of the magnitude of the time step adopted.     

求解渗流问题的高精度差分格式

在求解渗流问题的传统差分格式中,只有Crank Nicolson格式具有对时间t的二阶精度。本文在导数超收敛点概念的基础上,提出一种求解渗流问题的三阶精度差分格式,并将其与显式差分格式叠加形成组合差分格式以改善格式的稳定、收敛条件。算例计算结果表明,该组合格式具有精度高,稳定收敛限制宽松,易于编程等优点      

Crank-Nicolson差分格式及其稳定性研究

本文以自己独特的方式,构造了一维和二维抛物型方程的Ｃｒａｎｋ－Ｎｉｃｏｌｓｏｎ差分格式。本文不仅详细地给出了离散误差的表达式,而且论证了它们的稳定性。该差分格式具有精度高,稳定性好,计算量和存储量都比较小的特点,是一个很理想,便于应用的差分格式。     

Numerical performance of projection methods in finite element consolidation models

Projection, or conjugate gradient like, methods are becoming increasingly popular for the efficient solution of large sparse sets of unsymmetric indefinite equations arising from the numerical integration of (initial) boundary value problems. One such problem is soil consolidation coupling a flow and a structural model, typically solved by finite elements (FE) in space and a marching scheme in time (e.g. the Crank–Nicolson scheme). The attraction of a projection method stems from a number of factors, including the ease of implementation, the requirement of limited core memory and the low computational cost if a cheap and effective matrix preconditioner is available. In the present paper, biconjugate gradient stabilized (Bi‐ CGSTAB) is used to solve FE consolidation equations in 2‐D and 3‐D settings with variable time integration steps. Three different nodal orderings are selected along with the preconditioner ILUT based on incomplete triangular factorization and variable fill‐in. The overall cost of the solver is made up of the preconditioning cost plus the cost to converge which is in turn related to the number of iterations and the elementary operations required by each iteration. The results show that nodal ordering affects the perfor mance of Bi‐CGSTAB. For normally conditioned consolidation problems Bi‐CGSTAB with the best ILUT preconditioner may converge in a number of iterations up to two order of magnitude smaller than the size of the FE model and proves an accurate, cost‐effective and robust alternative to direct methods. Copyright © 2001 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.     

An alternative approach for quasi‐static large deformation analysis of saturated porous media using meshfree method

An alternative coupled large deformation formulation combined with a meshfree approach is proposed for flow–deformation analysis of saturated porous media. The formulation proposed is based on the Updated Lagrangian (UL) approach, except that the spatial derivatives are defined with respect to the configuration of the medium at the last time step rather than the configuration at the last iteration. In this way, the Cauchy stresses are calculated directly, rendering the second Piola–Kirchhoff stress tensor not necessary for the numerical solution of the equilibrium equations. Moreover, in contrast with the UL approach, the nodal shape function derivatives are calculated once in each time step and stored for use in subsequent iterations, which reduces the computational cost of the algorithm. Stress objectivity is satisfied using the Jaumann stress rate, and the spatial discretisation of the governing equations is achieved using the standard Galerkin method. The equations of equilibrium are satisfied directly, and the nonlinear parts of the system matrix are derived independent of the stresses of the medium resulting in a stable numerical algorithm. Temporal discretisation is effected based on a three‐point approximation technique that avoids spurious ripple effects and has second‐order accuracy. The radial point interpolation method is used to construct the shape functions. The application of the formulation and the significance of large deformation effects on the numerical results are demonstrated through several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

A Runge–Kutta,Taylor–Galerkin scheme for hyperbolic systems with source terms. Application to shock wave propagation in viscoplastic geomaterials

This paper presents an alternative formulation of Solid Dynamics problems based on (i) a mathematical model consisting of a system of hyperbolic PDEs where the source term is originated by the viscoplastic strain rate and (ii) a splitting scheme where the two‐step Taylor–Galerkin is used for the advective part of the PDE operator while the sources are integrated using a fourth‐order Runge–Kutta. Use of the splitting scheme results in a higher accuracy than that of the original two‐step Taylor–Galerkin. The scheme performs well when used with linear triangle or tetrahedra for (i) bending‐dominated situations (ii) localized failure under dynamic conditions and keeps the advantages of the two‐step Taylor–Galerkin concerning numerical dispersion and damping of short wavelengths. Copyright © 2006 John Wiley & Sons, Ltd.     

A Symmetrical Streamline Stabilization scheme for high advective transport

An overview of numerical techniques and previous investigations related to the solution of advection‐dominated transport processes is presented. In addition a new Symmetrical Streamline Stabilization (S³) scheme is introduced. The basis of the technique is to treat the transport equation in two steps. In the first step the dispersion part is approximated by a standard Galerkin approach, while in the second step the advection is approximated by a least‐squares method. The two parts are reassembled, resulting in one system of equations. The resulting coefficients' matrix is symmetric. Only half of a sparse matrix needs to be stored. Robust iterative algorithms for symmetrical systems of equations such as the preconditioned conjugate gradient method (PCG) can be successfully used. The new method leads to an implicit introduction of an ‘artificial diffusion’ term. Solute transport with high Peclet and Courant numbers does not lead to oscillations due to an inherent upwind damping. The upwind effect acts only in flow direction. The efficiency of the new formulation in terms of accuracy and computation time is shown in comparison with the Galerkin approach for mesh parallel and mesh oblique high advective solute transport. Copyright © 2000 John Wiley & Sons, Ltd.     

An efficient quasi-Newton approximation-based SORM to estimate the reliability of geotechnical problems

The first order reliability method (FORM) is efficient, but it has limited accuracy; the second order reliability method (SORM) provides greater accuracy, but with additional computational effort. In this study, a new method which integrates two quasi-Newton approximation algorithms is proposed to efficiently estimate the second order reliability of geotechnical problems with reasonable accuracy. In particular, the Hasofer–Lind–Rackwitz–Fiessler–Broyden–Fletcher–Goldfarb–Shanno (HLRF–BFGS) algorithm is applied to identify the design point on the limit state function (LSF), and consequently to compute the first order reliability index; whereas the Symmetric Rank-one (SR1) algorithm is nested within the HLRF–BFGS algorithm to compute good approximations, yet with a reduced computational effort, of the Hessian matrix required to compute second order reliabilities. Three typical geotechnical problems are employed to demonstrate the ability of the suggested procedure, and advantages of the proposed approach with respect to conventional alternatives are discussed. Results show that the proposed method is able to achieve the accuracy of conventional SORM, but with a reduced computational cost that is equal to the computational cost of HLRF–BFGS-based FORM.     

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.     

固结问题有限元法时步自适应研究

关于提高有限元法解的精度和稳定性问题已受到不少学者的重视。基于Biot固结理论,为提高有限元法计算的精度和稳定性,研究了固结过程中的时步自适应。采用基本原理推导出时步控制公式,从而全面控制应力和孔隙压力场。研究结果有助于准确模拟应力应变的发展,对粘土心墙土石坝等建筑物的设计及施工有重要意义。     

非饱和土体降雨入渗的全过程数值模拟

采用Crank-Nicolson型差分格式建立了非饱和土降雨入渗的精细化数值模型,此模型对时间和空间均具有二阶精度。每个时间步长的计算采用迭代过程来提高计算精度,利用此计算模型对某一具体的降雨过程中土体含水量的变化进行了数值模拟计算。     

A continuum‐discrete model using Darcy's law: formulation and verification

This paper presents a numerical scheme for fluid‐particle coupled discrete element method (DEM), which is based on poro‐elasticity. The motion of the particles is resolved by means of DEM. While within the proposition of Darcian regime, the fluid is assumed as a continuum phase on a Eulerian mesh, and the continuity equation on the fluid mesh for a compressible fluid is solved using the FEM. Analytical solutions of traditional soil mechanics examples, such as the isotropic compression and one‐dimensional upward seepage flow, were used to validate the proposed algorithm quantitatively. The numerical results showed very good agreement with the analytical solutions, which show the correctness of this algorithm. Sensitivity studies on the effect of some influential factors of the coupling scheme such as pore fluid bulk modulus, volumetric strain calculation, and fluid mesh size were performed to display the accuracy, efficiency, and robustness of the numerical algorithm. It is revealed that the pore fluid bulk modulus is a critical parameter that can affect the accuracy of the results. Because of the iterative coupling scheme of these algorithms, high value of fluid bulk modulus can result in instability and consequently reduction in the maximum possible time‐step. Furthermore, the increase of the fluid mesh size reduces the accuracy of the calculated pore pressure. This study enhances our current understanding of the capacity of fluid‐particle coupled DEM to simulate the mechanical behavior of saturated granular materials. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Explicit integration of a porosity‐dependent hydro‐mechanical model for unsaturated soils

An adaptive substepping explicit integration scheme is developed for a porosity‐dependent hydro‐mechanical model for unsaturated soils. The model is referred to as the modified σ –Θ model in this paper, which features the employment of the subloading surface plasticity and the stress–saturation approach. On numerical aspects, convex/nonconvex subloading surfaces in the σ –Θ space may result in incorrect loading–unloading decisions during the integration. A new loading–unloading decision method is developed here to solve the problem and then embedded into the explicit integration scheme for the modified σ –Θ model. In addition, to enhance the accuracy of the explicit integration, local errors from both hydraulic and mechanical components are included in the error control for each substep. A drift correction method is also developed to ensure the state point lies on the subloading surface in the σ –Θ space within a set error level. The performance of the loading–unloading decision method for the modified σ –Θ model is discussed through comparing it with the conventional loading–unloading decision method. The importance of involving the hydraulic component in the error control is also demonstrated. The accuracy and efficiency of the proposed adaptive substepping explicit integration scheme for the modified p–Θ model are also studied via several numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Application of the numerical manifold method for stress wave propagation across rock masses

In this paper, the numerical manifold method (NMM) is extended to study wave propagation across rock masses. First, improvements to the system equations, contact treatment, and boundary conditions of the NMM are performed, where new system equations are derived based on the Newmark assumption of the space–time relationship, the edge‐to‐edge contact treatment is further developed for the NMM to handle stress wave propagation across discontinuities, and the viscous non‐reflection boundary condition is derived based on the energy minimisation principle. After the modification, numerical comparisons between the original and improved NMM are presented. The results show that the original system equations result in artificial numerical damping, which can be overcome by the Newmark system equations. Meanwhile, the original contact scheme suffers some calculation problems when modelling stress wave propagation across a discontinuity, which can be solved by the proposed edge‐to‐edge contact scheme. Subsequently, the influence of the mesh size and time step on the improved NMM for stress wave propagation is studied. Finally, 2D wave propagation is modelled, and the model's results are in good agreement with the analytical solution. Copyright © 2013 John Wiley & Sons, Ltd.     

A time‐discontinuous Galerkin method for the dynamical analysis of porous media

We present a time‐discontinuous Galerkin method (DGT) for the dynamic analysis of fully saturated porous media. The numerical method consists of a finite element discretization in space and time. The discrete basis functions are continuous in space and discontinuous in time. The continuity across the time interval is weakly enforced by a flux function. Two applications and several numerical investigations confirm the quality of the proposed space–time finite element scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

Evaluating the performance of an explicit dynamic relaxation technique in analyzing non-linear geotechnical engineering problems

Explicit dynamic relaxation is an efficient tool that has been used to solve problems involving highly non-linear differential equations. The key feature of this method is the ability to use explicit dynamic algorithms in solving static problems. Few attempts have been made to date to apply this technique in conventional geotechnical engineering. In this study, an algorithm that incorporates the application of a stiffness dependent time step scheme is proposed. The algorithm has been successfully used to solve 2D and 3D non-linear geotechnical engineering problems. To calibrate the developed algorithm, numerical simulations have been conducted for a strip and square footings supported by Mohr–Coulomb material. Performance of four different types of brick elements used in collapse load calculation is examined in terms of convergence speed and accuracy. In addition, the role of employing adaptive time steps in reducing the number of iterations needed for convergence is also evaluated.     

Accelerating strategies to the numerical simulation of large‐scale models for sequential excavation

In this paper, a novel combination of well‐established numerical procedures is explored in order to accelerate the simulation of sequential excavation. Usually, large‐scale models are used to represent these problems. Due to the high number of equations involved, the solver algorithm represents the critical aspect which makes the simulation very time consuming. The mutable nature of the excavation models makes this problem even more pronounced. To accomplish the representation of geometrical and mechanical aspects in an efficient and simple manner, the proposed solution employs the boundary element method with a multiple‐region strategy. Together with this representational system, a segmented storage scheme and a time‐ordered tracking of the changes form an adequate basis for the usage of fast updating methods instead of frontal solvers. The present development employs the Sherman–Morrison–Woodbury method to speed up the calculation due to sequential changes. The efficiency of the proposed framework is illustrated through the simulation of test examples of 2D and 3D models. Copyright © 2006 John Wiley & Sons, Ltd.     

Mixed finite element discretization and Newton iteration for a reactive contaminant transport model with nonequilibrium sorption: convergence analysis and error estimates

We present a numerical scheme for reactive contaminant transport with nonequilibrium sorption in porous media. The mass conservative scheme is based on Euler implicit, mixed finite elements, and Newton method. We consider the case of a Freundlich-type sorption. In this case, the sorption isotherm is not Lipschitz but just Hölder continuous. To deal with this, we perform a regularization step. The convergence of the scheme is analyzed. An explicit order of convergence depending only on the regularization parameter, the time step, and the mesh size is derived. We give also a sufficient condition for the quadratic convergence of the Newton method. Finally, relevant numerical results are presented.