Non-linear analysis of geomechanical problems using coupled finite and infinite elements

In modeling of many geomechanics problems such as underground openings, soil-foundation structure interaction problems, and in wave propagation problems through semi-infinite soil medium the soil is represented as a region of either infinite or semi-infinite extent. Numerical modeling of such problems using conventional finite elements involves a truncation of the far field in which the infinite boundary is terminated at a finite distance. In these problems, appropriate boundary conditions are introduced to approximate the solution of the infinite or semi-infinite boundaries as closely as possible. However, the task of positioning the finite boundary in conventional finite element discretization and the definition of the boundary and its conditions is very delicate and depends on the modeller's skill and intuition. Moreover, such a choice is influenced by the size of the domain to be discretized. Consequently, the dimensions of the global matrices and the time required for solution of the problem will increase considerably and also selection of the arbitrary location of truncated boundary may lead to erroneous result. In order to over come these problems, mapped infinite elements have been developed by earlier researchers (Simoni and Schrefier, 1987). In the present work the applicability of infinite element technique is examined for different geomechanics problems. A computer program INFEMEP is developed based on the conventional finite element and mapped infinite element technique. It is then validated using selected problems such as strip footing and circular footing. CPU time taken to obtain solutions using finite element approach and infinite element approach was estimated and presented to show the capability of coupled modeling in improving the computational efficiency. Mesh configurations of different sizes were used to explore the enhancement of both computational economy and solution accuracy achieved by incorporation of infinite elements to solve elastic and elasto-plastic problems in semi-infinite/finite domain as applied to geotechnical engineering. © Rapid Science Ltd. 1998     

Parametric variational principle for elastic–plastic consolidation analysis of saturated porous media

Finite element procedures for numerical solution of various engineering problems are often based on variational formulations. In this paper, a parametric variational principle applicable to elastic-plastic coupled field problems in consolidation analysis of saturated porous media is presented. This principle can be used to solve problems where materials are inconsistent with Drucker's postulate of stability, such as in non-associated plasticity flow or softening problems. The finite element formulation was given, and it can be solved by either the conventional method or a parametric quadratic programming method.     

Pseudo-plane strain analysis of wave propagation problems arising from detonations of explosives in cylindrical boreholes

Wave propagation problems, such as blasting for excavation of a new tunnel oriented perpendicular to an existing tunnel, are truly three dimensional in nature. Dynamic finite element analysis with three-dimensional elements is, however, very expensive. The cheaper and simpler alternative would be to model the problem approximately in two dimensions. This paper shows that dynamic finite element analysis of such problems using conventional two-dimensional plane strain elements produces responses which are erroneously excessive. This is accredited to the inability to the inability of the two-dimensional elements to correctly model the rapid attenuation of the amplitudes of the outward propagating waves. To overcome this problem, a pseudo-plane strain concept is introduced and has been found to be a viable alternative. Numerical results are presented to demonstrate the application of the concept.     

An improved search strategy for the critical slip surface using finite element stress fields

The finite element method can be used to advantage in slope stability problems. This paper proposes a technique to search for the critical slip surface as well as to define and calculate the factor of safety for the slope, when the finite element method is used to model its formation. First, stresses are estimated at each Gaussian point from the finite element analysis. Then, the global stress smoothing method is applied to get a continuous stress field. Based on this stress field, the factor of safety is calculated for a specified slip surface by a stress integration scheme. An improved search strategy is proposed for a noncircular critical surface which starts with a search method for a circular critical surface. During the search process, points defining a trial slip surface can freely move in the finite element mesh subject to some kinematical constraints. This method can be applied to both the limit equilibrium method and the finite element method. Effects of the slope stress history and soil parameters on the resulting critical surface are investigated.     

求解非稳定对流占优水质模型的非标准Galerkin有限元法

在求解非稳定地下水溶质运移模型时,若对流项占优,则模型表现出双曲方程的特性。针对这种特性,采用非标准Galerkin有限元方法进行求解是解决这类问题的有效途径。分别采用Wavelet-Galer-kin有限元方法、迎风有限元方法和特征有限元方法对强对流溶质运移模型进行了求解,并将其结果与标准Galerkin有限元和解析解进行对比。结果表明:标准Galerkin有限元方法会产生强烈的数值振荡;Wavelet-Galerkin有限元方法的时空定位效果好;迎风有限元方法能够有效降低数值振荡现象,但迎风因子对解的影响较大,而且会带来时间延迟;特征有限元方法能够提高解的精度,故可以认为特征有限元方法是求解强对流地下水溶质运移模型的首选方法。     

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.     

Use of coupled finite element analysis in unsaturated soil problems

This paper presents a variation of Biot's consolidation theory for analysing problems involving unsaturated soils, and implemented using the finite element method. The numerical method is applied to a few geotechnical problems as examples and the results obtained are compared to some published data. The illustrative examples show how the numerical method can be used to analyse seepage and consolidation problems associated with unsaturated soils and demonstrate the flexibility and applicability of the presented method. Copyright © 2000 John Wiley & Sons, Ltd.     

解非均质各向异性地下水渗流模型的改进型有限差分法

常用的求解地下水渗流模型有限差分法,三角形单元Δikj内水文地质参数相同,以相同参数的三角形单元进行参数分区,这种方法,对某些情形可以达到解决地下水渗流场模拟和预报问题,但对非均质各向异性有一定的局限性。本文阐述了以三角形单元的棱边控制面积作为参数分区的最小单元的参数分区方法,建立了单元非等参有限差分方程,给出了实际应用例子。该方法可更准确刻画非均质各向异性问题,同时兼容以往的差分方程,可退化成一般有限差分格式。     

Transient seepage analysis in zoned anisotropic soils based on the scaled boundary finite‐element method

The scaled boundary finite‐element method, a semi‐analytical computational scheme primarily developed for dynamic stiffness of unbounded domains, is applied to the analysis of unsteady seepage flow problems. This method is based on the finite‐element technology and gains the advantages of the boundary element method as well. Only boundary of the domain is discretized, no fundamental solution is required and singularity problems can be modeled rigorously. Anisotropic and non‐homogeneous materials satisfying similarity are modeled with no additional efforts. In this study, firstly, formulation of the method for the transient seepage flow problems is derived followed by its solution procedures. The accuracy, simplicity and applicability of the method are demonstrated via four numerical examples of transient seepage flow – three of them are available in the literature. Homogenous, non‐homogenous, isotropic and anisotropic material properties are considered to show the versatility of the technique. Excellent agreement with the finite‐element method is observed. The method out‐performs the finite‐element method in modeling singularity points. Copyright © 2014 John Wiley & Sons, Ltd.     

利用算子分裂迎风均衡格式解对流为主溶质运移问题

水污染模拟问题是水流问题与溶质运移问题的耦合问题.各种常见的数值解法在以对流为主溶质运移问题的求解中都会遇到困难,如用有限单元法或有限差分法时,会产生数值弥散与过量这两类误差.引入算子分裂迎风均衡格式法求解对流为主的水污染模拟问题,较好地克服了数值弥散和数值解出现振荡问题,该格式具有良好的稳定性、单调性及守恒性特点.     

基于稳定性耦合分析法的余王扁边坡稳定性分析

边坡稳定性分析的极限平衡法与有限元法(FEM)的耦合分析法,首先利用有限元法分析获得边坡岩土体的整体应力场,在此基础上利用极限平衡法进行边坡的稳定系数求解。该方法既反映了边坡的稳定和变形之间的关系,又克服了极限平衡法与有限元法的不足,使二者的优点相互补充,获得的稳定系数基于极限平衡理论体系,可以同传统的稳定系数评价体系接轨。以西安市雁塔区余王扁削坡后边坡为例,用稳定性耦合分析法对其进行了稳定性分析,并把分析结果与各种极限平衡法的计算结果进行了比较,证明了耦合分析方法的可靠性和可行性。      

一种适用于岩土力学无界域的有限元与边界元耦合法的具体实现

针对有限元与边界元耦合法在岩土力学无异域中应用的具体实现问题进行了研究，提出了在微机上实现的具体措施和需建立的程序模块，将有限元与边界元耦合法计算结果、有限元计算结果与理论解析解进行了比较。     

Semi‐analytical elastostatic analysis of unbounded two‐dimensional domains

Unbounded plane stress and plane strain domains subjected to static loading undergo infinite displacements, even when the zero displacement boundary condition at infinity is enforced. However, the stress and strain fields are well behaved, and are of practical interest. This causes significant difficulty when analysis is attempted using displacement‐based numerical methods, such as the finite‐element method. To circumvent this difficulty problems of this nature are often changed subtly before analysis to limit the displacements to finite values. Such a process is unsatisfactory, as it distorts the solution in some way, and may lead to a stiffness matrix that is nearly singular. In this paper, the semi‐analytical scaled boundary finite‐element method is extended to permit the analysis of such problems without requiring any modification of the problem itself. This is possible because the governing differential equations are solved analytically in the radial direction. The displacement solutions so obtained include an infinite component, but relative motion between any two points in the unbounded domain can be computed accurately. No small arbitrary constants are introduced, no arbitrary truncation of the domain is performed, and no ill‐conditioned matrices are inverted. Copyright © 2002 John Wiley & Sons, Ltd.     

Verification for numerical modelling of jointed rock mass using thin layer elements

Thin layer finite elements are employed to model the response of rock joints in finite element analyses of rock mechanics problems. The case of a circular opening embedded in a rock mass is investigated. Results are obtained for both solid and jointed rock masses, and the effects of the presence of joints on the stress and strain field are investigated. Results from the finite element analysis are verified with respect to measurements from physical modelling. It is shown that the comparison between measured and computed results is satisfactory and that the presence of joints has a pronounced effect on the stress and strain field as compared to that realized in the absence of any rock joints. This study provides valuable data which contribute to the verification of thin layer finite elements for modelling the response of rock joints.     

A comparison between finite element and finite difference explicit formulations for triangular and quadrilateral plane-strain elements

For non-linear dynamic problems, it has been recognized that an explicit time-integration method of approach is a very efficient way of solving the dynamic equations of motion. The numerical formulation and computation for such problems fall into the two general categories of finite elements and finite differences. Over the years, there have been many arguments between schools which adopt the finite element approach and those which adopt the finite difference approach. At one extreme, arguments areconcerned with the superiority of each approach and at the other end of the spectrum the arguments are about which approach is a subset of the other. The most common of these arguments are concerned with efficiency and accuracy. This publication addresses the accuracy issue with specific reference to explicit calculations in which the analysis domain is discretized into triangular or quadrilateral plane-strain elements. It concludes that if the same basic assumptions are made in the two approaches, they, will give identical answers for problems in this category.     

On implementation of bounding surface plasticity models with no overshooting effect in solving boundary value problems

This paper presents a new scheme that can be used to overcome the overshooting effect, one of the well known problems occurs during application of bounding surface plasticity models in numerical analysis of boundary value problems. The scheme is based on definition of clouds of loading surfaces with a specific margin of strains within which unloading does not accompanied kinematic hardening. The basic concept of the scheme is introduced and the methodology and the relevant step by step algorithm to implement this scheme are presented. This scheme has been incorporated in the UNSW bounding surface model and implemented in a finite difference code and used to simulate cyclic triaxial tests as well as complicated monotonic and dynamic boundary value problems. The satisfactory performance of the scheme is demonstrated and its efficiency is discussed thorough these simulations.     

Finite element methods for variable density flow and solute transport

Saltwater intrusion into coastal freshwater aquifers is an ongoing problem that will continue to impact coastal freshwater resources as coastal populations increase. To effectively model saltwater intrusion, the impacts of increased salt content on fluid density must be accounted for to properly model saltwater/freshwater transition zones and sharp interfaces. We present a model for variable density fluid flow and solute transport where a conforming finite element method discretization with a locally conservative velocity post-processing method is used for the flow model and the transport equation is discretized using a variational multiscale stabilized conforming finite element method. This formulation provides a consistent velocity and performs well even in advection-dominated problems that can occur in saltwater intrusion modeling. The physical model is presented as well as the formulation of the numerical model and solution methods. The model is tested against several 2-D and 3-D numerical and experimental benchmark problems, and the results are presented to verify the code.     

Numerical limit analysis of three-dimensional slope stability problems in catchment areas

Risk analysis of existing slopes in catchment areas requires quantification of their stability. This quantification becomes particularly difficult when dealing with larger areas under 3D conditions and including saturated and unsaturated water flow. This paper proposes the use of an effective numerical procedure to solve three-dimensional slope stability problems in large areas subjected to pore pressure effects. This numerical approach, numerical limit analysis, utilizes the finite element method and mathematical programming techniques. Mathematical programming is needed because the basic plasticity theorems for limit analysis can be cast as optimization problems. The generated optimization problem is formulated under a second-order cone programming framework, which is known to solve large-scale problems with great computational efficiency. The main objective of this work was to determine the slope safety factor and the collapse mechanism of soils governed by the Drucker–Prager yield criterion for large-scale 3D problems including pore pressure effects. This approach is applied to an experimental catchment in the Oregon Coast Range that failed after an intense rainfall. The results were compared with a previous stability analysis of the area available in the literature that used a novel 3D limit equilibrium method.     

Implicit and explicit procedures for the yield vertex non-coaxial theory

The yield vertex non-coaxial model is different from classical elastoplastic models, in that there is an additional plastic strain rate tangential to yield surfaces, as well as the plastic strain rate normal to yield surfaces, when orientations of principal stress change. This feature raises concerns on its finite element implementations. In nonlinear finite element numerical iterations, a large tangential plastic strain rate is likely to make the trial total strain rate direct inside a yield surface, which entails convergence difficulty. Some modifications are introduced on the non-coaxial model itself to make numerical convergence easier in the work published in Yang and Yu (2010) [20]. This paper is an extension of the previous work. Instead of modifying the non-coaxial model itself, this paper concerns the use of finite element explicit procedure, which is suitable for highly discontinuous problems. The simulations of shallow foundation load-settlement responses indicate that the finite element explicit procedure, assisted with a robust and explicit automatic substepping integration scheme of the non-coaxial model, does not encounter numerical difficulty. In addition, the overall trends of implicit and explicit simulations are similar.     

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。