Two-dimensional modelling of contaminant transport through saturated porous media using the radial point interpolation method (RPIM)

A new numerical tool is presented which models the two-dimensional contaminant transport through saturated porous media using a meshfree method called the radial point interpolation method (RPIM) with polynomial reproduction. In RPIM, an approximate solution is constructed entirely in terms of a set of nodes and no characterisation of the interrelationship of the nodes is needed. An advection-dispersion equation with sorption is considered to illustrate the applicability of the RPIM. The Galerkin weak form of the governing equation is formulated using two-dimensional meshfree shape functions constructed using thin plate spline radial basis functions. A computer program is developed for the implementation of the RPIM procedure. Three numerical examples are presented and the results are compared with those obtained from the analytical solution and finite element method. The experimental results are also used to validate the approach. The proposed RPIM has generated results with no oscillations and they are insensitive to Peclet constraints.     

三维径向点插值无网格法及其在流固耦合中的应用

径向基函数点插值无网格法(radial point interpolation method,RPIM)是一种新型的无网格法,其形函数具有插值特性,且形式简单,易于施加本质边界条件。文中介绍了径向基函数点插值无网格法的基本原理,推导了三维情况下点插值无网格法的基本公式。从变分原理出发,结合比奥固结理论,建立了流-固耦合的三维点插值无网格法基本方程和数值积分方法,并开发了相应计算程序。通过三维悬臂梁和单向固结问题的数值试验,验证了该方法对三维弹性问题和流-固耦合问题的适用性和有效性     

Modelling of contaminant transport through landfill liners using EFGM

Modelling of contaminant transport through landfill liners and natural soil deposits is an important area of research activity in geoenvironmental engineering. Conventional mesh‐based numerical methods depend on mesh/grid size and element connectivity and possess some difficulties when dealing with advection‐dominant transport problems. In the present investigation, an attempt has been made to provide a simple but sufficiently accurate methodology for numerical simulation of the two‐dimensional contaminant transport through the saturated homogeneous porous media and landfill liners using element‐free Galerkin method (EFGM). In the EFGM, an approximate solution is constructed entirely in terms of a set of nodes and no characterization of the interrelationship of the nodes is needed. The EFGM employs moving least‐square approximants to approximate the function and uses the Lagrange multiplier method for imposing essential boundary conditions. The results of the EFGM are validated using experimental results. Analytical and finite element solutions are also used to compare the results of the EFGM. In order to test the practical applicability and performance of the EFGM, three case studies of contaminant transport through the landfill liners are presented. A good agreement is obtained between the results of the EFGM and the field investigation data. Copyright © 2009 John Wiley & Sons, Ltd.     

基于单位分解的扩展径向点插值无网格法

针对线弹性断裂力学问题,提出扩展径向点插值无网格法（X-RPIM）。该方法基于单位分解思想,在传统径向点插值无网格法的位移模式中加入扩展项来描述裂纹两侧的不连续位移场和裂尖奇异场。由于其形函数具有Kronecker ? 函数性质,易于施加本质边界条件。详细描述了X-RPIM不连续位移模式的建立,支配方程的离散形式以及J积分计算混合模式裂纹的应力强度因子的实现过程,讨论了不同积分区域对应力强度因子的影响。数值算例分析证明了该方法在求解断裂问题时的可行性和有效性,同时说明扩展径向点插值无网格法在模拟裂纹扩展问题时具有良好的前景。     

重力异常二维正演中的无网格方法

无网格法是一类新型数值算法,具有精度高、高阶形函数构造与物性加载便利等特点,在计算力学领域应用广泛。将无网格方法（PIM、RPIM及EFGM）用于重力异常场二维正演计算:首先从重力异常二维变分问题出发,利用Galerkin法结合高斯积分公式推导了对应的无网格离散系统矩阵表达式;其次通过数值试验得出了RPIM-MQ、RPIM-exp及EFGM-exp形状参数的建议值,最后比较分析了最优形状参数下不同无网格法的计算效果。结果表明:无网格法适用于介质物性分布变化较大的重力异常二维正演,exp函数形状参数α_c最优取值区间为[1.5,1.7],β建议值为0.6,MQ函数q取值区间为-4.1~1.9;EFGM较PIM及RPIM具有更高的计算精度。      

广义节点无网格法在土力学固结分析中的应用

借鉴流形方法思想,引入广义节点的概念,对传统的无网格法进行了改进,建立了可具有任意高阶多项式插值函数的广义节点无网格方法.与传统无网格方法相比,广义节点无网格方法更具有一般性,当选取0阶广义节点位移插值函数时便可得到传统的无网格法.结合土工固结问题,通过推导建立了Blot固结方程的数值计算列式,对静态固结问题进行数值计算,通过对比分析验证了所建议方法的可行性.     

Development of an adaptive CTM–RPIM method for modeling large deformation problems in geotechnical engineering

Acta Geotechnica - In this paper, a meshfree method called adaptive CTM–RPIM is developed to model geotechnical problems with large deformation. The developed adaptive CTM–RPIM is a...     

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.     

非平衡-非线性吸附情况下填埋场污染物运移分析

采用Langmuir等温吸附线方程描述非线性吸附性能, 基于改进的混合元方法，通过数值计算与分析，探讨了非平衡吸附条件下污染物运移过程及其机理。计算结果表明: 当考虑非平衡、非线性吸附性能时, 污染物穿透曲线即浓度的时程变化曲线尖锐而狭窄、峰值点前移, “拖长尾”现象不明显, 由此说明，土颗粒对污染物的非平衡、非线性吸附使得污染物的穿透能力增强, 滞留能力下降。进一步的变动参数比较分析表明：Langmuir等温线方程中的参数B、压实粘土衬里的渗透性及地下水渗流速度对污染物运移过程具有显著的影响。     

Modélisation du transport réactif en milieux poreux non saturés avec une méthode ELLAM en maillage variable

Modelling contaminant transfer with biological/chemical/radioactive processes needs appropriate numerical methods able to reproduce sharp concentration fronts. In this work, we develop a new Eulerian–Lagrangian Localized Adjoint Method (ELLAM) for solving the reactive transport equation with non-constant coefficients. To avoid interpolation (leading to errors), we use a moving grid to define the solution and test functions. The method is used to simulate first the infiltration of solute into a column of unsaturated porous medium and second the multispecies transport. The developed ELLAM gives accurate results without non-physical oscillations or numerical diffusion, even when using large time steps. To cite this article: A. Younes, C. R. Geoscience 336 (2004).     

Boundary element analysis of contaminant transport in fractured porous media

A two-dimensional boundary integral method to analyse the flow of contaminant in fractured media having a two- or three-dimensional orthogonal fracture network is presented. The method assumes that the fractures provide the paths of least resistance for transport of contaminants while the matrix, because of its low permeability, acts as ‘storage blocks’ into which the contaminant diffuses. Laplace transform is used to eliminate the time variable in the governing equation in order to facilitate the formulation of a boundary integral equation in the Laplace transform space. Conventional boundary element techniques are applied to solve for the contaminant concentrations at specified locations in the spatial domain. The concentration in the time domain is then obtained by using an efficient inversion technique developed by Talbot. The method is able to analyse the behaviour of waste repositories which have diminishing concentration due to the mass transport of the contaminant into the surrounding fractured media.     

Numerical study of soil heterogeneity effects on contaminant transport in unsaturated soil (model development and validation)

The movement of chemicals through soil to groundwater is a major cause of degradation of water resources. In many cases, serious human and stock health implications are associated with this form of pollution. The study of the effects of different factors involved in transport phenomena can provide valuable information to find the best remediation approaches. Numerical models are increasingly being used for predicting or analyzing solute transport processes in soils and groundwater. This article presents the development of a stochastic finite element model for the simulation of contaminant transport through soils with the main focus being on the incorporation of the effects of soil heterogeneity in the model. The governing equations of contaminant transport are presented. The mathematical framework and the numerical implementation of the model are described. The comparison of the results obtained from the developed stochastic model with those obtained from a deterministic method and some experimental results shows that the stochastic model is capable of predicting the transport of solutes in unsaturated soil with higher accuracy than deterministic one. The importance of the consideration of the effects of soil heterogeneity on contaminant fate is highlighted through a sensitivity analysis regarding the variance of saturated hydraulic conductivity as an index of soil heterogeneity. Copyright © 2011 John Wiley & Sons, Ltd.     

One‐dimensional contaminant transport through a deforming porous medium: theory and a solution for a quasi‐steady‐state problem

Contaminant migration through soil is usually modelled mathematically using the dispersion–advection equation. This type of model finds application when planning the remediation of contaminated land, predicting the movement of polluted groundwater and designing engineered landfills. Usually the analysis assumes that the porous media through which the contaminant migrates is stationary. However, the construction of landfills on clay soils means that the soil beneath the landfill will undergo time‐dependent deformation as the soil consolidates. To date, there are no published data on the effect a deforming porous media may have on contaminant transport beneath a landfill; indeed, there appears to be no theory of contaminant migration through a deforming soil. In this paper, a one‐dimensional theory of contaminant migration through a saturated deforming porous media is developed based on a small and large strain analysis of a consolidating soil and conservation of contaminant mass. By selection of suitable parameters, the new transport equation reduces to the familiar one‐dimensional dispersion–advection equation for a saturated soil with linear, reversible, equilibrium controlled sorption of the contaminant onto the soil skeleton. Analytic solutions to a quasi‐steady‐state contaminant transport problem for a deforming media are presented, and a preliminary assessment made of the potential importance of soil deformation on the results of a contaminant migration analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

最小二乘配点法解一维非稳定流问题

最小二乘配点无网格法是一种新型高效的无网格法。该方法除节点外又在研究域内引入辅助点,近似函数仍然只通过节点构造,微分方程在所有节点和辅助点上满足。用最小二乘配点法计算河间承压非稳定流问题,算例表明,最小二乘配点法比有限差分法计算精度高,稳定性好。     

杂交边界点方法求解动态断裂力学问题

结合杂交边界点法和双互易法则,推导出求解动力问题的纯边界类型无网格方法--双互易杂交边界点方法,并将该方法用于求解含中心裂纹的方板受瞬态载荷作用的问题。该方法将问题的解分为通解和特解两部分,通解使用杂交边界点法求解,特解则利用局部径向基函数近似,域内布点仅仅为了径向基插值,因此仍然是一种纯边界类型的无网格方法。同时,将移动最小二乘近似中的基函数扩充,使该方法能更好地模拟裂纹尖端应力场的奇异性,具有后处理简单、精度高的优点。数值算例表明了该方法的稳定性和有效性。     

Certified solutions for hydraulic structures using the node‐based smoothed point interpolation method (NS‐PIM)

A meshfree node‐based smoothed point interpolation method (NS‐PIM), which has been recently developed for solid mechanics problems, is applied to obtain certified solutions with bounds for hydraulic structure designs. In this approach, shape functions for displacements are constructed using the point interpolation method (PIM), and the shape functions possess the Kronecker delta property and permit the straightforward enforcement of essential boundary conditions. The generalized smoothed Galerkin weak form is then applied to construct discretized system equations using the node‐based smoothed strains. As a very novel and important property, the approach can obtain the upper bound solution in energy norm for hydraulic structures. A 2D gravity dam problem and a 3D arch dam problem are solved, respectively, using the NS‐PIM and the simulation results of NS‐PIM are found to be the upper bounds. Together with standard fully compatible FEM results as a lower bound, we have successfully determined the solution bounds to certify the accuracy of numerical solutions. This confirms that the NS‐PIM is very useful for producing certified solutions for the analysis of huge hydraulic structures. Copyright © 2009 John Wiley & Sons, Ltd.     

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

A new method to determine semi-analytical solutions of one-dimensional contaminant transport problem with nonlinear sorption is described. This method is based on operator splitting approach where the convective transport is solved exactly and the diffusive transport by finite volume method. The exact solutions for all sorption isotherms of Freundlich and Langmuir type are presented for the case of piecewise constant initial profile and zero diffusion. Very precise numerical results for transport with small diffusion can be obtained even for larger time steps (e.g., when the Courant-Friedrichs-Lewy (CFL) condition failed).     

非饱和渗流Richards方程数值求解的欠松弛方法

非饱和土渗流理论是岩土工程问题的基础理论,在土石坝渗流、污染物传输、冻土渗流相变和边坡稳定分析等领域有着广泛的应用。非饱和土渗流Richards方程的数值求解过程中,某些参数如水力传导系数计算不当可能引起非线性方法,如Picard方法或Newton方法的迭代收敛震荡,从而导致非线性迭代方法收敛缓慢和精度降低。为了消除或降低迭代收敛震荡对求解精度和计算性能的影响,目前主要采用欠松弛方法。通过一维入渗算例和二维非均质土坝渗流算例演示已有欠松弛方法的局限性,进而提出新的短项混合欠松弛法,并对其实用性和可靠性进行验证。     

大变形黏土防渗层中的污染物迁移和转化规律研究

国内湖泊疏浚污染底泥堆场一般以较厚的黏土层作为主要防渗层,由于在上覆底泥作用下黏土层会发生较大的固结变形,因此,在研究黏土防渗层中的污染物运移和转化规律时,应该考虑土体变形的影响。基于Gibson一维大变形固结理论和饱和多孔介质中的污染物对流扩散方程,建立了二者耦合的可变形多孔介质中污染物的运移和转化模型,其中首次考虑了土体自重和生物降解作用的影响。利用所建立模型的数值解,研究了在可变形黏土防渗层中的污染物运移和转化规律,同时分析了模型中不同项和主要参数的作用和影响。研究结果表明,土体大变形对黏土防渗层中污染物的运移有着较复杂的影响,一方面土体变形会加速污染物的运移;另一方面土体固结带来的渗透性减小会增加污染物的穿透时间,二者的不同作用取决于众多的影响因素,如土层厚度和吸附作用等。研究结果对于评估天然黏土防渗层对污染物的阻隔作用有重要的指导意义。