水平集和无网格耦合法在裂纹扩展中的应用

提出了一种模拟裂纹扩展的水平集和无网格耦合方法。由于水平集和无网格方法都是建立在离散节点上,因而可以很自然地实现耦合。在该方法中,两个在裂尖处相互正交的水平集不仅用于描述裂纹的几何形态和裂尖位置,而且用于建立无网格伽辽金法（简称EFGM）不连续近似函数中的Heaviside跳跃项和裂尖处的Westergaard扩展项。当裂纹扩展时,则由水平集更新算法确定新裂纹的位置。水平集和无网格耦合法无需使用可视法、衍射法或透明法,克服了这些方法在裂尖处人为引入的不连续且能很好地再生 奇异场;而且节点影响域不受裂纹线切割的影响,在计算中往往使用较小的影响域,保持了整体刚度矩阵的带状、稀疏性;另外,水平集简化了扩展节点的选取和附加函数的建立,其更新过程无需求解演化方程,实现简单且易于编程。数值算例表明本文方法具有较高的计算精度,其模拟的裂纹扩展路径与试验结果吻合得很好,从而验证了本文方法的正确性和可行性。     

模拟三维裂纹问题的多尺度扩展有限元法

扩展有限元法模拟裂纹时独立于网格,因此该方法是目前求解裂纹问题最有效的数值方法。为了在计算代价不大的情况,实现大型结构分析中考虑小裂纹或提高裂纹附近精度,在裂纹附近一般采用小尺度单元,其他区域采用大尺度单元。提出了分析三维裂纹问题的多尺度扩展有限元法,在需要的地方采用小尺度单元。基于点插值构造了六面体任意节点单元。所有尺度单元都采用8节点六面体单元,这样六面体任意节点单元可方便有效地连接不同尺度单元。采用互作用积分法计算三维应力强度因子。边裂纹和中心圆裂纹算例分析结果表明,该方法是正确和有效的。     

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。     

改进的数值流形法在水力劈裂中的应用

数值流形法在非连续变形分析领域具有独特优势。结合裂纹尖端场函数的基本概念,分析了水力劈裂破坏问题,模拟了水力劈裂破坏过程,避免了扩展有限元中的阶跃函数和水平集概念。为了避免裂纹尖端在单元内部不同位置而产生误差,对裂纹尖端附近一定范围内的每一个物理覆盖附加奇异覆盖函数。选取一个算例比较分析了内水压力对应力强度因子的影响,当考虑裂纹面内压时,定量分析比较了各因素对应力强度因子的影响大小,并应用于分支裂纹水力劈裂破坏。计算结果表明,改进后的计算结果与解析解相吻合。未考虑裂纹面内压,误差往往较大。考虑裂纹面内压后,随着裂纹长度的增加,误差逐渐减小;随着网格密度的增加,误差也逐渐减小。分支裂纹的渐进破坏结果表明该改进方法的可行性,具有较大的实际应用价值。     

求解潜水含水层剖面二维稳定流问题探讨

本文在对现有求解潜水含水层剖面二维流的多种方法及其特点进行分析的基础上,探讨新的求解方法.提出的方法只需进行一次网格剖分,在网格节点编号保持不变的情况下,把潜水面与单元相交的形状概化为五边形、任意四边形和三角形,按五边形、任意四边形和三角形子程序准确地求出潜水面穿越的单元的渗透矩阵.同时不用计算完全在潜水面之上的单元的渗透矩阵.本文给出具体的思路、算法步骤,计算潜水面的位置,通过比较数值解与解析解,两者虽然存在一定的误差,但是该方法总体结果是比较令人满意的.     

旅行时线性插值射线追踪提高计算精度和效率的改进方法

旅行时线性插值（LTI）射线追踪算法是基于线性假设的,在向前处理过程中仅用按行或按列扫描的方法来计算节点旅行时没有考虑逆向传播射线,导致其计算精度与网格剖分大小有关,在处理复杂介质时会使得追踪出来的射线路径不一定满足最短旅行时。因此,笔者提出了两点改进措施：在向前处理时需采用全方位循环的方法来计算节点最小旅行时;在网格边界加入次生节点。模型试算结果表明：采用全方位循环的LTI法考虑了来自各个方向的射线,可提高其对复杂模型的适应能力;在节点间距相同的情况下,网格边界插入次生节点的LTI法较传统的LTI法计算精度至少可以提高一个数量级,同时,计算速度也更快;随着节点间距剖分的越精细,计算耗时下降也越明显,计算速度较传统的方法可提高n~10n倍。     

无压渗流问题分析的多节点有限元方法

采用基于平均值插值的多节点有限元方法分析有自由面渗流问题。在自由面附近采用多节点单元逼近自由面,利用平均值插值建立多节点单元的形函数,在远离自由面的区域采用四边形/三角形单元剖分。给定一个初始自由面位置,通过对渗流控制方程的多节点有限元求解,根据自由面上节点水头值判断自由面节点的调整方向和大小,最终迭代求出自由面的位置。土坝渗流问题的数值分析表明了所提方法的有效性和足够的计算精度。     

可变网格的有限单元分析在边坡稳定计算中的应用

本文在非线性有限单元法的基础上，引进八节点的狭长单元来描述边坡中的滑动面，同时对于有限元单元网格可调节点坐标进行变更形成新的网格，于是利用原有基本网格架，通过多次有限元分析求出边坡中可能滑动面的稳定系数曲线Ｋ＝ｆ（），并进而确定最危险的滑动面，在此基础上开展一系列稳定分析工作。     

基于水平集法的非均质岩石建模及水力压裂传播特性研究

基于水平集法的基本思想,讨论了含有不同类型包裹体分布的岩石的二维和三维有限元建模方法。对于二维有限元模型的建立,考虑了以椭圆形为例的规则包裹体的周期分布、位置及包裹体大小均随机变化的多种情况。建议了包裹体和基岩之间界面的材料特性过渡处理方法。同时给出了含有非规则形状包裹体的建模方法。对于三维有限元模型的建立,则考虑了以任意大小椭球体为例的包裹体分布情况。该种建模方法的优点是对于不同的含有任意分布的包裹体的岩石试件,均可以采用相同的有限元网格,即,材料特性的变化不受有限元网格的制约。该法缺点是增加了计算资源。最后结合基于弥散裂缝模型的水力压裂数值计算方法,模拟了含有不同包裹体分布的岩石试件的水力压裂传播特点。     

无网格法中节点非均匀自动布置及背景网格生成

针对形状复杂的区域,提出了一种基于节点间距函数的非均匀节点自动布置和背景网格的生成算法。该算法对不规则区域适应能力强,节点可任意加密,无需人工干预,且有实现简单、耗时少等特点。这些自动布置的离散节点可直接用于无网格数值计算。用所布置的节点进一步生成背景网格简单易行且高效,生成的背景网格还可直接用来进行后置处理。     

模拟岩石压剪状态下主次裂纹萌生开裂的扩展有限元法

压剪应力状态下,岩石类材料中常见两类裂纹：翼型张拉裂纹和次生压剪裂纹。基于扩展有限元方法（XFEM）,提出了模拟压剪裂纹面作用机制的扩展有限元位移增强方案,并给出了扩展有限元法分叉裂纹处理方法,分别用最大周向拉应力准则和Mohr-Coulomb准则判断张拉裂纹和压剪裂纹的萌生和扩展。基于Matlab平台编写了数值计算程序Betaxfem 2D,通过两个算例对所提方案进行了验证,所得结果与有限元法（FEM）计算结果吻合很好。模拟了单轴压缩载荷下含预制闭合裂纹试件的裂纹分叉、扩展过程,与试验结果的对比表明,所提方案可以模拟和预测岩石类材料张拉、压剪交互分叉裂纹的萌生和扩展行为。     

多裂纹扩展的扩展有限元法分析

建立了求解多裂纹扩展的扩展有限元法。引入裂纹交叉汇合加强函数以分析多裂纹交叉汇合过程;在裂纹附近区域使用广义形函数,并引入线增函数消除混合单元,可有效地提高裂纹附近的精度;用砂浆法（线段-线段接触法）结合增广型Lagrange乘子法处理裂纹段的接触条件,可以精确地模拟裂纹面约束,并方便地求解控制方程。算例分析了两方面内容：（1）计算交叉裂纹体的应力强度因子,结果表明提出的方法精度高;（2）模拟多裂纹扩展及交叉汇合过程,模拟的裂纹扩展路径与试验结果吻合得较好,表明了方法的可靠性。     

A 3D composite finite element used to model a vertical drain and its surrounding remoulded soil

This work addresses an enrichment technique for the three-dimensional (3D) finite element (FE) analysis of a vertical drain foundation because (1) 1D and 2D simulations are insufficient to integrally describe the consolidation behaviour and (2) drains are small both in spacing and size, resulting in enormous computational costs for a traditional 3D FE analysis. Based on the idea of the semi-analytical finite element method (FEM), which combines general FEM with the high accuracy of a closed-form solution, a new spatial element that contains a drain well and its neighbouring smear zone is presented. This new combined element is depicted by eight global independent nodes and two local dependent nodes, and a classical analytical theory is introduced to set up the relationship between the two kinds of nodes. Because permeability diversity between the drain and the smear zone is considered, both the effects of smearing and well resistance are taken into account with the composite element method (CEM). A detailed derivation of the CEM is performed using the weighted residual method. The accuracy of the proposed method is validated with a totally penetrating, single-drain ground analysis for seven calculation conditions. Additionally, the proposed CEM saves 1/4–1/2 mesh elements and helps to avoid slender elements for the FEM analysis of the drained foundation.     

Fully coupled hydraulic fracture simulation using the improved element partition method

The improved element partition method (IEPM) is a newly developed fracture simulation approach. IEPM allows a fracture to run across an element without introducing extra degrees of freedom. It can also simulate any number of fractures in a prescribed mesh without remeshing. In this study, the IEPM is extended to hydraulic fracture simulation. First, the seepage and volumetric storage matrix of a cracked element are derived using virtual nodes (the intersection points of a crack with element edges). Subsequently, the fully coupled hydromechanical equation is derived for this cracked element. To eliminate the extra degrees of freedom (virtual nodal quantities), the water pressure and displacement of the virtual nodes are associated with their adjacent nodes through least squares interpolation. Finally, the fully coupled equation in terms of nodal quantities is obtained. The verification cases validate the method. By using this method, the field-scale hydraulic fracturing process is well simulated. The proposed approach is simple and efficient for field-scale hydraulic fracture simulation.     

Enriched finite element procedures for analyzing decoupled bolts installed in rock mass

Numerical procedures are developed to analyze interaction between fully grouted bolts and rock mass using ‘enriched finite element method (EFEM)’. A solid element intersected by a rock bolt along any arbitrary direction is termed as ‘enriched’ element. The nodes of an enriched element have additional degrees of freedom for determining displacements, stresses developed in the bolt rod. The stiffness of the enriched element is formulated based on properties of rock mass, bolt rod and grout, orientation of the bolt and borehole diameter. Decoupling at grout–bolt interface and elasto‐plastic behavior of rock mass have also been incorporated into the EFEM procedures. The results of this method are compared with analytical pull‐out test results presented by Li and Stillborg (Int. J. Rock Mech. Min. Sci. 1999; 36 :1013–1029). In addition, a numerical example of a bolted tunnel is provided to demonstrate the efficacy of the proposed method for practical applications. Copyright © 2010 John Wiley & Sons, Ltd.     

A fracture mapping and extended finite element scheme for coupled deformation and fluid flow in fractured porous media

This paper presents a fracture mapping (FM) approach combined with the extended finite element method (XFEM) to simulate coupled deformation and fluid flow in fractured porous media. Specifically, the method accurately represents the impact of discrete fractures on flow and deformation, although the individual fractures are not part of the finite element mesh. A key feature of FM‐XFEM is its ability to model discontinuities in the domain independently of the computational mesh. The proposed FM approach is a continuum‐based approach that is used to model the flow interaction between the porous matrix and existing fractures via a transfer function. Fracture geometry is defined using the level set method. Therefore, in contrast to the discrete fracture flow model, the fracture representation is not meshed along with the computational domain. Consequently, the method is able to determine the influence of fractures on fluid flow within a fractured domain without the complexity of meshing the fractures within the domain. The XFEM component of the scheme addresses the discontinuous displacement field within elements that are intersected by existing fractures. In XFEM, enrichment functions are added to the standard finite element approximation to adequately resolve discontinuous fields within the simulation domain. Numerical tests illustrate the ability of the method to adequately describe the displacement and fluid pressure fields within a fractured domain at significantly less computational expense than explicitly resolving the fracture within the finite element mesh. Copyright © 2013 John Wiley & Sons, Ltd.     

水力劈裂对岩体中自然裂纹的影响研究

建立了求解自然裂纹和水力裂纹扩展的扩展有限元法,对裂纹附近区域的节点采用广义形函数,并采用线增函数消除混合单元,以提高裂纹附近的精度。引入水力劈裂的非耦合模型,即假设裂纹中的水压力为均布力;用砂浆法（线段-线段接触法）结合增广型拉格朗日乘子法处理受压裂纹段的接触条件。并通过算例分析了以下内容：计算了受压裂纹和裂纹面分布均布水压力的水力裂纹的应力强度因子,并与解析解进行了比较,结果表明,提出的方法具有很高的精度;模拟了水力裂纹对自然裂纹面的影响,并分析了自然裂纹面上的接触力和接触状态。     

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。