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.     

Scaled boundary finite‐element analysis of a non‐homogeneous axisymmetric domain subjected to general loading

The scaled boundary finite‐element method is derived for elastostatic problems involving an axisymmetric domain subjected to a general load, using a Fourier series to model the variation of displacement in the circumferential direction of the cylindrical co‐ordinate system. The method is particularly well suited to modelling unbounded problems, and the formulation allows a power‐law variation of Young's modulus with depth. The efficiency and accuracy of the method is demonstrated through a study showing the convergence of the computed solutions to analytical solutions for the vertical, horizontal, moment and torsion loading of a rigid circular footing on the surface of a homogeneous elastic half‐space. Computed solutions for the vertical and moment loading of a smooth rigid circular footing on a non‐homogeneous half‐space are compared to analytical ones, demonstrating the method's ability to accurately model a variation of Young's modulus with depth. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Granular element method for three‐dimensional discrete element calculations

This paper endows the recently‐proposed granular element method (GEM) with the ability to perform 3D discrete element calculations. By using non‐uniform rational B‐Splines to accurately represent complex grain geometries, we proposed an alternative approach to clustering‐based and polyhedra‐based discrete element methods whereby the need for complicated and ad hoc approaches to construct 3D grain geometries is entirely bypassed. We demonstrate the ability of GEM in capturing arbitrary‐shaped 3D grains with great ease, flexibility, and without excessive geometric information. Furthermore, the applicability of GEM is enhanced by its tight integration with existing non‐uniform rational B‐Splines modeling tools and ability to provide a seamless transition from binary images of real grain shapes (e.g., from 3D X‐ray CT) to modeling and discrete mechanics computations.Copyright © 2013 John Wiley & Sons, Ltd.     

A three‐phase thermo‐hydro‐mechanical finite element model for freezing soils

Artificial ground freezing (AGF) is a commonly used technique in geotechnical engineering for ground improvement such as ground water control and temporary excavation support during tunnel construction in soft soils. The main potential problem connected with this technique is that it may produce heave and settlement at the ground surface, which may cause damage to the surface infrastructure. Additionally, the freezing process and the energy needed to obtain a stable frozen ground may be significantly influenced by seepage flow. Evidently, safe design and execution of AGF require a reliable prediction of the coupled thermo‐hydro‐mechanical behavior of freezing soils. With the theory of poromechanics, a three‐phase finite element soil model is proposed, considering solid particles, liquid water, and crystal ice as separate phases and mixture temperature, liquid pressure, and solid displacement as the primary field variables. In addition to the volume expansion of water transforming into ice, the contribution of the micro‐cryo‐suction mechanism to the frost heave phenomenon is described in the model using the theory of premelting dynamics. Through fundamental physical laws and corresponding state relations, the model captures various couplings among the phase transition, the liquid transport within the pore space, and the accompanying mechanical deformation. The verification and validation of the model are accomplished by means of selected analyses. An application example is related to AGF during tunnel excavation, investigating the influence of seepage flow on the freezing process and the time required to establish a closed supporting frozen arch. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Cauchy problem of three‐dimensional critical slip surfaces of slopes

Using the continuum‐based finite element approach in conjunction with the strength reduction technique, one can easily find out the factor of safety of a slope with rather high precision, but will encounter some obstacles in the accurate location of the corresponding three‐dimensional critical slip surface (CSS). On the basis of the Mohr–Coulomb failure criterion and the stress field in the limit equilibrium state of the slope, it is deduced that the three‐dimensional CSS is the solution of the Cauchy problem of a quasi‐linear partial differential equation (PDE) of first order. By means of the method of characteristics for the problem, the three‐dimensional CSS that may take any shape can be determined without the specification of its geometry. Copyright © 2010 John Wiley & Sons, Ltd.     

Three‐dimensional nonlinear finite element analysis of pile groups in saturated porous media using a new transmitting boundary

The dynamic behaviour of pile groups subjected to an earthquake base shaking is analysed. An analysis is formulated in the time domain and the effects of material nonlinearity of soil, pile–soil–pile kinematic interaction and the superstructure–foundation inertial interaction on seismic response are investigated. Prediction of response of pile group–soil system during a large earthquake requires consideration of various aspects such as the nonlinear and elasto‐plastic behaviour of soil, pore water pressure generation in soil, radiation of energy away from the pile, etc. A fully explicit dynamic finite element scheme is developed for saturated porous media, based on the extension of the original formulation by Biot having solid displacement (u) and relative fluid displacement (w) as primary variables (u–w formulation). All linear relative fluid acceleration terms are included in this formulation. A new three‐dimensional transmitting boundary that was developed in cartesian co‐ordinate system for dynamic response analysis of fluid‐saturated porous media is implemented to avoid wave reflections towards the structure. In contrast to traditional methods, this boundary is able to absorb surface waves as well as body waves. The pile–soil interaction problem is analysed and it is shown that the results from the fully coupled procedure, using the advanced transmitting boundary, compare reasonably well with centrifuge data. Copyright © 2007 John Wiley & Sons, Ltd.     

A quasistatic analysis of a plate on consolidating layered soils by analytical layer‐element/finite element method coupling

In this paper, a coupling method between finite element and analytical layer‐elements is utilized to analyze the time‐dependent behavior of a plate of any shape and finite rigidity resting on layered saturated soils. Based on the integral transform techniques together with the aid of an order reduction method, an analytical layer‐element solution is derived from the governing equations for three‐dimensional Biot consolidation with respect to a Cartesian coordinate system and then extended to be the fundamental solution for the layered saturated soil under a point load. The Mindlin plate is modeled by eight‐noded isoparametric elements. The governing equations of the interaction between soil and plate in the Laplace‐Fourier transformed domain are deduced by referring to the coupling theory of FEM/BEM, and the final solution is obtained by applying numerical inversion. Numerical examples concerned with the time‐dependent response of a plate are performed to demonstrate the influence of soil and plate properties on the interaction process. Copyright © 2014 John Wiley & Sons, Ltd.     

A high strain‐rate constitutive model for sand and its application in finite‐element analysis of tunnels subjected to blast

The paper presents a constitutive model for simulating the high strain‐rate behavior of sands. Based on the concepts of critical‐state soil mechanics, the bounding surface plasticity theory and the overstress theory of viscoplasticity, the constitutive model simulates the high strain‐rate behavior of sands under uniaxial, triaxial and multi‐axial loading conditions. The model parameters are determined for Ottawa and Fontainebleau sands, and the performance of the model under extreme transient loading conditions is demonstrated through simulations of split Hopkinson pressure bar tests up to a strain rate of 2000/s. The constitutive model is implemented in a finite‐element analysis software Abaqus to analyze underground tunnels in sandy soil subjected to internal blast loads. Parametric studies are conducted to examine the effect of relative density and type of sand and of the depth of tunnel on the variation of stresses and deformations in the soil adjacent to the tunnels. Copyright © 2012 John Wiley & Sons, Ltd.     

构造应力场边界载荷反演的有限元逆逼近法

构造应力场问题的边界条件、加载方式是地应力场分析的难题,由于构造运动未知和地质构造复杂,无法直接求解。边界力反演是进行构造应力场研究的重要手段。从弹性力学基本方程出发,针对油气储层地应力场反问题,结合对区域构造应力场认识,利用阻尼最小二乘法建立了应力场反演的约束模型,提出了一种用于反演构造应力场边界力的反分析方法,该方法根据部分实测点的应力值进行边界力反演,并与有限元方法和有限差分法相结合,以边界力为参量逼近目标函数。根据研究区少量地应力实测资料采用该方法进行最优化计算,使得计算应力场与实测应力达到最优拟合,以弥补目前的回归反演方法和边界载荷调整法的不足来提高计算精度。算例表明了该方法的有效性,可用于解决构造应力场边界力的反演问题。     

A three‐dimensional staggered finite element approach for random parametric modeling of thermo‐hygral coupled phenomena in porous media

The aim of this paper is to present a three‐dimensional (3D) finite element modeling of heat and mass transfer phenomena in partially saturated open porous media with random fields of material properties. Randomness leads to transfer processes within the porous medium that naturally need a full 3D modeling for any quantitative assessment of these processes. Nevertheless, the counterpart of 3D modeling is a significant increase in computations cost. Therefore, a staggered solution strategy is adopted which permits to solve the equations sequentially. This appropriate partitioning reduces the size of the discretized problem to be solved at each time step. It is based on a specific iterative algorithm to account for the interaction between all the transfer processes. Accordingly, a suitable linearization of mass convective boundary conditions, consistent with the staggered algorithm, is also derived. After some validation tests, the 3D numerical model is used for studying the drying process of a cementitious material with regard to its intrinsic permeability randomness. Copyright © 2011 John Wiley & Sons, Ltd.     

Understanding heavy mineral enrichment using a three‐dimensional numerical model

Layered deposits of relatively light and heavy minerals can be found in many aquatic environments. Quantification of the physical processes which lead to the fine‐scale layering of these deposits is often limited with flumes or in situ field experiments. Therefore, the following research questions were addressed: (i) how can selective grain entrainment be numerically simulated and quantified; (ii) how does a mixed bed turn into a fully layered bed; and (iii) is there any relation between heavy mineral content and bed stability? Herein, a three‐dimensional numerical model was used as an alternative measure to study the fine‐scale process of density segregation during transport. The three‐dimensional model simulates particle transport in water by combining a turbulence‐resolving large eddy simulation with a discrete element model prescribing the motion of individual grains. The granular bed of 0·004 m in height consisted of 200 000 spherical particles (D50 = 500 μ m). Five suites of experiments were designed in which the concentration ratio of heavy (5000 kg m⁻³) to light particles (i.e. 2560 kg m⁻³) was increased from 6%, 15%, 35%, 60% to 80%. All beds were tested for 10 sec at a predefined flow speed of 0·3 m sec⁻¹. Analysis of the particle behaviour in the interior of the beds showed that the lighter particles segregated from the heavy particles with increasing time. The latter accumulated at the bottom of the domain, forming a layer, whereas the lighter particles were transported over the layer forming sweeps. Particles below the heavy particle layer indicated that the layer was able to armour the particles below. Consequentially, enrichment of heavy minerals in a layer is controlled by the segregation of a heavy mineral fraction from the light counterpart, which enhances current understanding of heavy mineral placer formation.     

Semi‐analytical solution to one‐dimensional consolidation for unsaturated soils with semi‐permeable drainage boundary under time‐dependent loading

This paper presents semi‐analytical solutions to Fredlund and Hasan's one‐dimensional consolidation of unsaturated soils with semi‐permeable drainage boundary under time‐dependent loadings. Two variables are introduced to transform two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. The pore‐water pressure, pore‐air pressure and settlement are obtained in the Laplace domain. Crump's method is adopted to perform the inverse Laplace transform in order to obtain semi‐analytical solutions in time domain. It is shown that the present solutions are more general and have a good agreement with the existing solutions from literatures. Furthermore, the current solutions can also be degenerated into conventional solutions to one‐dimensional consolidation of unsaturated soils with homogeneous boundaries. Finally, several numerical examples are provided to illustrate consolidation behavior of unsaturated soils under four types of time‐dependent loadings, including instantaneous loading, ramp loading, exponential loading and sinusoidal loading. Parametric studies are illustrated by variations of pore‐air pressure, pore‐water pressure and settlement at different values of the ratio of air–water permeability coefficient, depth and loading parameters. Copyright © 2017 John Wiley & Sons, Ltd.     

结构-地基动力相互作用的时域模型

结构-地基动力相互作用是结构地震响应分析及安全评估的一个非常重要课题。基于比例边界有限元法,提出了一种新颖的结构-地基动力相互作用的时域模型,即采用比例边界有限元子域模拟近场有限域部分,采用高阶透射边界模拟远场无限域部分。通过采用连分式展开和引入辅助变量,有限域的动力方程采用高阶的静力刚度矩阵和质量矩阵表示。高阶透射边界精确满足无限远处的辐射边界条件,具有全局精确、时间局部和收敛速度快等优点。它是基于改进的连分式法求解无限域动力刚度矩阵而建立的,在时域里表示为一阶常微分方程组。通过联立有限域和无限域的运动方程,建立了结构-地基相互作用的标准动力学方程,采用Newmark法可直接求解。3个算例结果表明,该算法在时域里比黏弹性边界更精确、有效。     

Homogenization framework for three‐dimensional elastoplastic finite element analysis of a grouted pipe‐roofing reinforcement method for tunnelling

This paper deals with the grouted pipe‐roofing reinforcement method that is used in the construction of tunnels through weak grounds. This system consists on installing, prior to the excavation of a length of tunnel, an array of pipes forming a kind of ‘umbrella’ above the area to be excavated. In some cases, these pipes are later used to inject grout to strengthen the ground and ‘connect’ the pipes. This system has proven to be very efficient in reducing tunnel convergence and water inflow when tunnelling through weak grounds. However, due to the geometrical and mechanical complexity of the problem, existing finite element frameworks are inappropriate to simulate tunnelling using this method. In this paper, a mathematical framework based on a homogenization technique to simulate ‘grouted pipe‐roofing reinforced ground’ and its implementation into a 3‐D finite element programme that can consider stage construction situations are presented. The constitutive model developed allows considering the main design parameters of the problem and only requires geometrical and mechanical properties of the constituents. Additionally, the use of a homogenization approach implies that the generation of the finite element mesh can be easily produced and that re‐meshing is not required as basic geometrical parameters such as the orientation of the pipes are changed. The model developed is used to simulate tunnelling with the grouted pipe‐roofing reinforcement method. From the analyses, the effects of the main design parameters on the elastic and the elastoplastic analyses are considered. Copyright © 2004 John Wiley & Sons, Ltd.     

Generalized contact model for polyhedra in three‐dimensional discontinuous deformation analysis

We present a generalized contact computation model for arbitrarily shaped polyhedra to simplify the contact analysis in discontinuous deformation analysis. A list of generalized contact constraints can be established for contacting polyhedra during contact detection. Each contact constraint contains information for 2 contact points, unique contact plane, and related contact modes (open, locked, or sliding). Computational aspects of the generalized contact model include identification of contact positions and contact modes, uniform penalty formulation of generalized contact constraint, and uniform updating of contact modes and contact planes in the open‐close iteration. Compared with previous strategies, the generalized contact computation model has a simpler data structure and fewer memory requirements. Meanwhile, it simplifies the penalty formulation and facilitates the open‐close iteration check while producing enough accuracy. Illustrative examples show the ability of the method to handle the full range of polyhedral shapes.     

A practical and efficient numerical scheme for the analysis of steady state unconfined seepage flows

The scaled boundary finite‐element method (SBFEM), a novel semi‐analytical technique, is applied to the analysis of the confined and unconfined seepage flow. This method combines the advantages of the finite‐element method and the boundary element method. In this method, only the boundary of the domain is discretized; no fundamental solution is required, and singularity problems can be modeled rigorously. Anisotropic and nonhomogeneous materials satisfying similarity are modeled without additional efforts. In this paper, SBFE equations and solution procedures for the analysis of seepage flow are outlined. The accuracy of the proposed method in modeling singularity problems is demonstrated by analyzing seepage flow under a concrete dam with a cutoff at heel. As only the boundary is discretized, the variable mesh technique is advisable for modeling unconfined seepage analyses. The accuracy, effectiveness, and efficiency of the method are demonstrated by modeling several unconfined seepage flow problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Coupled finite‐element simulation of injection well testing in unconsolidated oil sands reservoir

This paper presents a finite‐element (FE) model for simulating injection well testing in unconsolidated oil sands reservoir. In injection well testing, the bottom‐hole pressure (BHP) is monitored during the injection and shut‐in period. The flow characteristics of a reservoir can be determined from transient BHP data using conventional reservoir or well‐testing analysis. However, conventional reservoir or well‐testing analysis does not consider geomechanics coupling effects. This simplified assumption has limitations when applied to unconsolidated (uncemented) oil sands reservoirs because oil sands deform and dilate subjected to pressure variation. In addition, hydraulic fracturing may occur in unconsolidated oil sands when high water injection rate is used. This research is motivated in numerical modeling of injection well testing in unconsolidated oil sands reservoir considering the geomechanics coupling effects including hydraulic fracturing. To simulate the strong anisotropy in mechanical and hydraulic behaviour of unconsolidated oil sands induced by fluid injection in injection well testing, a nonlinear stress‐dependent poro‐elasto‐plastic constitutive model together with a strain‐induced anisotropic permeability model are formulated and implemented into a 3D FE simulator. The 3D FE model is used to history match the BHP response measured from an injection well in an oil sands reservoir. Copyright © 2013 John Wiley & Sons, Ltd.     

A high‐frequency open boundary for transient seepage analyses of semi‐infinite layers by extending the scaled boundary finite element method

A high‐frequency open boundary has been developed for the transient seepage analyses of semi‐infinite layers with a constant depth. The scaled boundary finite element equation of pore water pressure is formulated first in the frequency domain. With the eigenvalue problem, the equation can be decoupled into modal equations whose modal dynamic permeability equation can be determined. The continued fraction technique is adopted to formulate the continued fraction solution in the frequency domain. All constants in the solution are determined recursively at the high‐frequency limit. By introducing auxiliary variables and the continued fraction solution to the relationship between the prescribed seepage flow and the pore water pressure in the frequency domain, the open boundary condition is obtained. After transformed to the time domain, the open boundary condition is expressed as a system of fractional differential equations. No convolution integral is required. The accuracy of the analysis results increases with the increasing orders of continued fraction. Copyright © 2015 John Wiley & Sons, Ltd.