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。     

Modeling discontinuous rock mass based on smoothed finite element method

This paper aims at developing a method for modeling rock mass with preexisting multiple discontinuities within the framework of the smoothed finite element method (SFEM). The discontinuity is simulated by an interface element with zero thickness, the stiffness matrix of which are derived explicitly based on the SFEM. An elastic damage constitutive relation with residual strength is introduced in order to describe the nonlinear mechanical behavior of the discontinuities. The computation codes of the present method were developed. The present method has been verified to be a sound approach for modeling discontinuous rock mass, inheriting the advantages of the SFEM.     

扩展有限元法的数值方面

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,其原理是在裂尖附近用一些奇异函数和沿裂纹面用阶跃函数加强传统有限元的基,以考虑跨过裂纹的位移场的不连续,该加强策略允许计算网格独立于不连续体几何。讨论了扩展有限元法的一些数值方面,主要包括：水平集法确定界面和加强节点与加强方式、裂尖加强范围的选择、J积分区域的确定和积分方案等。     

断裂问题的扩展有限元法研究

扩展有限元（extended finite element method,XFEM）是近年来发展起来的、在常规有限元框架内求解不连续问题的有效数值计算方法,其基于单位分解的思想,在常规有限元位移模式中加入能够反映裂纹面不连续性的跳跃函数及裂尖渐进位移场函数,避免了采用常规有限元计算断裂问题时需要对裂纹尖端重新加密网格造成的不便。在推导扩展有限元算法的基础上,分析了应力强度因子的J积分计算方法及积分区域的选取。采用XFEM对I型裂纹进行了计算,有限元网格独立于裂纹面,无需在裂纹尖端加密网格;分析了积分区域、网格密度对应力强度因子计算精度的影响,指出了计算应力强度因子的合适参数,验证了此方法的可靠性和准确性。     

Extended Finite Element Method for the Analysis of Discontinuities in Rock Masses

The strength and deformability of rock mass primarily depend on the condition of joints and their spacing and partially on the engineering properties of rock matrix. Till today, numerical analysis of discontinuities e.g. joint, fault, shear plane and others is conducted placing an interface element in between two adjacent rock matrix elements. However, the applicability of interface elements is limited in rock mechanics problems having multiple discontinuities due to its inherent numerical difficulties often leading to non-convergent solution. Recent developments in extended finite element method (XFEM) having strong discontinuity imbedded within a regular element provide an opportunity to analyze discrete discontinuities in rock masses without any numerical difficulties. This concept is based on partition of unity principle and can be used for cohesive rock joints. This paper summarizes the mathematical frameworks for the implementation of strong discontinuities in 3 and 6 nodded triangular elements and also provides numerical examples of the application of XFEM in one and two dimensional problems with single and multiple discontinuities.     

Modelling of coupled fluid‐mechanical problems in fractured geological media using enriched finite elements

Geological environments, such as petroleum reservoirs, normally exhibit physical discontinuities, for example, fractures and faults. Because of the reduced thickness of these discontinuities, finite element formulations with strong discontinuity have been applied to the numerical modelling of geological environments. Until now, two relevant characteristics of petroleum reservoirs have not been addressed by these formulations. The first is the pore pressure jump in the direction normal to a discontinuity in a fluid‐mechanical coupling condition, which is present primarily in sealing faults owing to the contrast of permeability with the porous medium. The absence of this jump can affect the prediction of the deformability of a physical discontinuity. Furthermore, reservoir models frequently use coarse meshes. Thus, the method used to evaluate the pore pressure in the discontinuity may exhibit a strong dependence relative to the mesh refinement. Based on these characteristics, in this study, a formulation of an enriched finite element for application to coupled fluid‐mechanical problems with pre‐existing physical discontinuities saturated by a single fluid is presented. The formulation employs discontinuous interpolation functions and enables the reproduction of jumps of displacement and pore pressure associated with a discontinuity inside the element without the need to discretise it. An approximation to estimate the pore pressure in the discontinuity was developed, one which seeks to minimise the influence of refinement. The element's response is verified by comparison with a one‐dimensional analytical solution and simple examples that are simulated using commercial software. Copyright © 2015 John Wiley & Sons, Ltd.     

基于单元劈裂法的岩体黏结型结构面数值模拟

节理岩体中富含结构面等不连续体,其中很多结构面之间具有一定黏结强度,在冲击荷载作用下会发生脱结（debond）行为,从而消耗岩体应变能。在该结构面之间的黏结性质在很大程度上决定着岩体的破坏行为,发生脱结后,结构面之间不再具有黏结强度,岩体的力学性能发生弱化。为了对该类结构面进行模拟,在改进的Xu-Needleman势函数基础上推导出结构面单元,并将其嵌入到单元劈裂法中,模拟结构面的开裂过程,当结构面完全脱结后结构面单元就转化为一般节理单元。相应地,在数值实现过程中只是将结构面单元替换为一般节理单元即可。该结构面单元与单元劈裂法相结合,能够有效地模拟节理岩体的破坏过程。     

岩质边坡复合型破坏机制的改进运动单元法研究

目前运动单元法的研究主要集中于土质边坡,未涉及到岩质边坡稳定性分析问题;而岩体中孕育有不同特性的结构面,控制着岩质边坡的力学行为。为求解结构面控制作用下岩质边坡“结构面滑移-岩桥剪断”复合型破坏问题,研究了塑性滑移区节点在岩桥内和结构面上的运动约束条件,推导了含结构面的运动单元计算公式,提出了改进运动单元法。通过经典算例的对比分析,验证了改进运动单元法计算结果的准确性。研究结果表明:岩桥位置、结构面贯通度和结构面倾角是控制岩质边坡力学行为的3个主要影响因素。岩桥越接近坡顶,改进运动单元法所得安全系数越大,而Jennings法无法反映岩桥位置的影响效应。高贯通度的结构面导致岩质边坡发生“结构面滑移-岩桥剪断”复合型破坏模式,安全系数较小;而低贯通度的结构面导致完整岩石发生破坏,安全系数较大。水平或陡倾角结构面导致滑裂面穿切结构面,安全系数较大;而对于其它倾角情况下的结构面,岩质边坡发生“结构面滑移-岩桥剪断”复合型破坏模式,安全系数较小。实例应用结果说明该方法可以有效评价岩质边坡的稳定状态,可在类似工程中应用推广。     

岩体裂隙模拟的扩展有限元法应用研究

岩体中普遍存在着断层﹑节理和裂隙等结构面,这些结构面的存在和发展对岩体的整体强度﹑变形及稳定性有极大的影响。因此,研究岩体中原生结构面的萌生﹑发展以及贯通演化过程对评估岩体工程安全性和可靠性具有非常重要的理论与现实意义。扩展有限元法（XFEM）作为一种求解不连续问题的有效数值方法,模拟裂隙时独立于网格,因此,在模拟岩体裂隙扩展﹑水力劈裂等方面具有独特优势。针对扩展有限元法的基本理论及其在岩体裂隙扩展模拟中的应用展开了研究,建立了扩展有限元法求解岩体裂隙摩擦接触、岩体裂隙破坏等问题的数值模型,并将计算模型应用于岩质边坡稳定性分析和重力坝坝基断裂破坏等工程问题。     

XFEM,strong discontinuities and second-order work in shear band modeling of saturated porous media

We investigate shear band initiation and propagation in fully saturated porous media by means of a combination of strong discontinuities (discontinuities in the displacement field) and XFEM. As a constitutive behavior of the solid phase, a Drucker–Prager model is used within a framework of non-associated plasticity to account for dilation of the sample. Strong discontinuities circumvent the difficulties which appear when trying to model shear band formation in the context of classical nonlinear continuum mechanics and when trying to resolve them with classical numerical methods like the finite element method. XFEM, on the other hand, is well suited to deal with problems where a discontinuity propagates, without the need of remeshing. The numerical results are confirmed by the application of Hill’s second-order work criterion which allows to evaluate the material point instability not only locally but also for the whole domain.     

Calculating the effective permeability of sandstone with multiscale lattice Boltzmann/finite element simulations

The lattice Boltzmann (LB) method is an efficient technique for simulating fluid flow through individual pores of complex porous media. The ease with which the LB method handles complex boundary conditions, combined with the algorithm’s inherent parallelism, makes it an elegant approach to solving flow problems at the sub-continuum scale. However, the realities of current computational resources can limit the size and resolution of these simulations. A major research focus is developing methodologies for upscaling microscale techniques for use in macroscale problems of engineering interest. In this paper, we propose a hybrid, multiscale framework for simulating diffusion through porous media. We use the finite element (FE) method to solve the continuum boundary-value problem at the macroscale. Each finite element is treated as a sub-cell and assigned permeabilities calculated from subcontinuum simulations using the LB method. This framework allows us to efficiently find a macroscale solution while still maintaining information about microscale heterogeneities. As input to these simulations, we use synchrotron-computed 3D microtomographic images of a sandstone, with sample resolution of 3.34 μm. We discuss the predictive ability of these simulations, as well as implementation issues. We also quantify the lower limit of the continuum (Darcy) scale, as well as identify the optimal representative elementary volume for the hybrid LB–FE simulations.     

Finite element analysis of land subsidence above depleted reservoirs with pore pressure gradient and total stress formulations

The solution of the poroelastic equations for predicting land subsidence above productive gas/oil fields may be addressed by the principle of virtual works using either the effective intergranular stress, with the pore pressure gradient regarded as a distributed body force, or the total stress incorporating the pore pressure. In the finite element (FE) method both approaches prove equivalent at the global assembled level. However, at the element level apparently the equivalence does not hold, and the strength source related to the pore pressure seems to generate different local forces on the element nodes. The two formulations are briefly reviewed and discussed for triangular and tetrahedral finite elements. They are shown to yield different results at the global level as well in a three‐dimensional axisymmetric porous medium if the FE integration is performed using the average element‐wise radius. A modification to both formulations is suggested which allows to correctly solve the problem of a finite reservoir with an infinite pressure gradient, i.e. with a pore pressure discontinuity on its boundary. Copyright © 2001 John Wiley & Sons, Ltd.     

Thermal reservoir modeling in petroleum geomechanics

Thermal oil recovery processes involve high pressures and temperatures, leading to large volume changes and induced stresses. These cannot be handled by traditional reservoir simulation because it does not consider coupled geomechanics effects. In this paper we present a fully coupled, thermal half‐space model using a hybrid DDFEM method. A finite element method (FEM) solution is adopted for the reservoir and the surrounding thermally affected zone, and a displacement discontinuity method is used for the surrounding elastic, non‐thermal zone. This approach analyzes stress, pressure, temperature and volume change in the reservoir; it also provides stresses and displacements around the reservoir (including transient ground surface movements) in a natural manner without introducing extra spatial discretization outside the FEM zone. To overcome spurious spatial temperature oscillations in the convection‐dominated thermal advection–diffusion problem, we place the transient problem into an advection–diffusion–reaction problem framework, which is then efficiently addressed by a stabilized finite element approach, the subgrid‐scale/gradient subgrid‐scale method. Copyright © 2008 John Wiley & Sons, Ltd.     

横观各向同性软土地基上大圆筒防波堤稳定性研究

对于横观各向同性软土地基上沉入式大圆筒防波堤结构提出了一种准三维上限极限分析方法,所假设的破坏机制为大圆筒结构绕筒体内中轴线上某点发生转动失稳,泥面处形成楔体破坏,而筒底部形成圆弧滑裂面。本方法可以考虑土的三轴压缩、拉伸强度与直剪强度的差异。利用ABAQUS分别进行平面应变以及三维有限元分析,软土采用Hill本构模型,所得到的破坏模式以及大圆筒结构水平承载力与上限极限分析方法吻合较好,同时可以得出考虑地基土各向异性的大圆筒结构承载力比不考虑时有较大降低。     

XFEM formulation with sub-interpolation,and equivalence to zero-thickness interface elements

This paper describes a particular formulation of the extended finite element method (XFEM) specifically conceived for application to existing discontinuities of fixed location, for instance, in geological media. The formulation is based on two nonstandard assumptions: (1) the use of sub-interpolation functions for each subdomain and (2) the use of fictitious displacement variables on the nodes across the discontinuity (instead of the more traditional jump variables). Thanks to the first of those assumptions, the proposed XFEM formulation may be shown to be equivalent to the standard finite element method with zero-thickness interface elements for the discontinuities (FEM+z). The said equivalence is theoretically proven for the case of quadrangular elements cut in two quadrangles by the discontinuity, and only approximate for other types of intersections of quadrangular or triangular elements, in which the XFEM formulation corresponds to a kinematically restricted version of the corresponding interface plus continuum scheme. The proposed XFEM formulation with sub-interpolation, also helps improving spurious oscillations of the results obtained with natural interpolation functions when the discontinuity runs skew to the mesh. A possible explanation for these oscillations is provided, which also explains the improvement observed with sub-interpolation. The paper also discusses the oscillations observed in the numerical results when some nodes are too close to the discontinuity and proposes the remedy of moving those nodes onto the discontinuity itself. All the aspects discussed are illustrated with some examples of application, the results of which are compared with closed-form analytical solutions or to existing XFEM results from the literature.     

Kinematic modelling of shear band localization using discrete finite elements

Modelling shear band is an important problem in analysing failure of earth structures in soil mechanics. Shear banding is the result of localization of deformation in soil masses. Most finite element schemes are unable to model discrete shear band formation and propagation due to the difficulties in modelling strain and displacement discontinuities. In this paper, a framework to generate shear band elements automatically and continuously is developed. The propagating shear band is modelled using discrete shear band elements by splitting the original finite element mesh. The location or orientation of the shear band is not predetermined in the original finite element mesh. Based on the elasto‐perfect plasticity with an associated flow rule, empirical bifurcation and location criteria are proposed which make band propagation as realistic as possible. Using the Mohr–Coulomb material model, various results from numerical simulations of biaxial tests and passive earth pressure problems have shown that the proposed framework is able to display actual patterns of shear banding in geomaterials. In the numerical examples, the occurrence of multiple shear bands in biaxial test and in the passive earth pressure problem is confirmed by field and laboratory observations. The effects of mesh density and mesh alignment on the shear band patterns and limit loads are also investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

Strain parameters of discontinuities in rock for finite element calculation

The bulk strain induced by excavation of rooms for storage of highly radioactive waste is usually unimportant, but the change in aperture and shear displacement of permeable fractures may be of practical importance. Hence, the normal and shear of such discontinuities have to be considered and are commonly predicted by using special deformation moduli termed “normal” and “shear” stiffnesses in numerical calculations. For the use of some finite element methods, these have to be converted to compression and shear moduli ,which, in turn, requires that the thickness of the discontinuity is known. The normal and shear strain can be expressed in terms of soil mechanical parameters, yielding the compression modulus M and the shear modulus G, with the required form for finite element calculations. By definition they are functions of the normal and shear stiffness K_n and K_s. It is concluded that calculation of the normal and shear strain of discontinuities with clastic fillings by the use of finite element methods and deformation moduli derived from stiffness numbers is very uncertain except when the geometry of the weaknesses can be accurately defined.     

Instability of wave propagation in saturated poroelastoplastic media

In the present work, stationary discontinuities and fluttery instabilities of wave propagation in saturated poro‐elastoplastic media are analysed in the frame of Biot theory. The generalized Biot formulations are particularly employed for simulating non‐linear coupled hydro‐mechanical behaviour of the media. Inertial coupling effect between the solid and the fluid phases of the media is also taken into account. The non‐associated Drucker–Prager criterion to describe non‐linear constitutive behaviour of pressure dependent elasto‐plasticity for the solid skeleton of the media is particularly considered. With omission of compressibility of solid grains and the pore fluid, the critical conditions of stationary discontinuities and flutter instabilities occurring in wave propagation are given in explicit forms. It is shown that when the stationary discontinuity is triggered at the surface of discontinuity there still may exist real wave speeds. The wave speeds across the stationary discontinuity surface entirely cease to be real only in non‐associated plasticity, certain ranges of value of Poisson's ratio and when compression stress normal to the surface of discontinuity dominates the stress state at the surface. It is also indicated that the fluttery instabilities, under which some wave speeds cease to be real even in strain hardening stage, may occur prior to stationary discontinuities only for non‐associated plasticity under certain conditions. These conditions are: (1) both the porosity and the Poisson's ratio possess relatively low values and (2) the deviatoric part of the effective stress normal to the surface of discontinuity is compressive. A region in the porosity–Poisson's ratio plot, in which fluttery instabilities are possible to occur, is given. Copyright © 2002 John Wiley & Sons, Ltd.     

裂隙岩体渗流场的无单元法模拟及渗流特性研究

裂隙岩体因含有发育程度不同的裂隙、节理和断层等不连续面,致其渗透性具有各向异性、不连续性等特点,因此传统的有限元法对分布密集的裂隙岩体渗流场求解有一定的难度。本文提出了采用无单元Glaerkin法求解有自由面裂隙渗流问题,并推导了无单元法求解渗流场的基本方程和积分格式,给出了应用罚函数法处理渗流边界条件和自由面处理方法。采用IDL语言编制了二维无单元法计算软件LIDAREFM。文中以北京怀柔桥梓镇某裂隙岩体边坡渗流场计算为例,研究了复杂裂隙共同作用下渗流场特性和自由面分布,讨论了不同开度、不同连通程度的裂隙对渗流场的影响。研究结果表明:无单元法可以较好地解决有密集裂隙的岩体渗流场的求解问题,实现了裂隙处结点任意加密以及积分网格的独立布置,避免了对有自由面和裂隙穿越的子域的重新处理,简化了渗流问题的求解过程。     

Land subsidence due to gas/oil production in inhomogeneous transversally anisotropic half-space by a boundary element method

In a previous paper¹ the authors have developed and implemented a new boundary element (BE) model to simulate and predict land subsidence occurring over three-dimensional gas/oil fields in a homogeneous and isotropic half-space. The approach relies on Betti's reciprocal theorem and makes use of the classical fundamental solution of Boussinesq in the framework of the theory of linear poroelasticity. The BE method is here extended to inhomogeneous, transversally anisotropic soils by the aid of a two-dimensional finite element (FE) model which provides a fundamental numerical solution for the actual multi-layer setting of the subsurface system. The new FE–BE approach is then used to simulate the subsidence caused by gas production over the deep reservoir of Campo Ravenna Terra, Ravenna (Italy) from 1950 to 1980. The results compare very favourably with the outcome from a full more expensive three-dimensional FE model of the same occurrence.