基于显式时间积分的球颗粒DDA计算方法

非连续变形分析方法（DDA）是一种平行于有限元法的新型数值计算方法,该方法基于最小势能原理,把每个离散块体的变形、运动和块体之间的接触统一到平衡方程中进行隐式求解。然而,传统DDA方法在计算过程中需组装整体刚度矩阵并联立求解方程组,在用于大型岩土工程问题的三维数值模拟时占用内存较大、耗时较长、计算效率极低。因此,提出一种基于显式时间积分的三维球颗粒DDA方法。该方法在求解过程中不需要组装整体刚度矩阵,在求解加速度时,由于质量矩阵为对角矩阵,可存储为一维向量占用内存较少,且可分块逐自由度求解,效率较高,在接触判断上采用最大位移准则简化了接触算法,采用较小的时步,保证了计算的精确性;通过几个典型算例验证了该方法的准确性及计算效率。     

Modified predictor-corrector solution approach for efficient discontinuous deformation analysis of jointed rock masses

The high computational costs associated with the implicit formulation of discontinuous deformation analysis (DDA) have been one of the major obstacles for its implementation to engineering problems involving jointed rock masses with large numbers of blocks. In this paper, the Newmark-based predictor-corrector solution (NPC) approach was modified to improve the performance of the original DDA solution module in modeling discontinuous problems. The equation of motion for a discrete block system is first established with emphasis on the consideration of contact constraints. A family of modified Newmark-based predictor-corrector integration (MNPC) scheme is then proposed and implemented into a unified analysis framework. Comparisons are made between the proposed approach and the widely used constant acceleration (CA) integration approach and central difference (CD) approach, regarding the stability and numerical damping features for a single-degree-of-freedom model, where the implications of the proposed approach on open-close iteration are also discussed. The validity of the proposed approach is verified by several benchmarking examples, and it is then applied to two typical problems with different numbers of blocks. The results show that the original CA approach in DDA is efficient for the simulation of quasi-static deformation of jointed rock masses, while the proposed MNPC approach leads to improved computational efficiency for dynamic analysis of large-scale jointed rock masses. The MNPC approach therefore provides an additional option for efficient DDA of jointed rock masses.     

多场耦合作用下岩体裂隙扩展演化关键问题研究

裂隙岩体热-水-应力（THM）耦合是目前研究的热点和难点。首先总结了裂隙岩体多场耦合的机制、模型、方法及研究内容,并通过分析裂隙对THM耦合的重要控制作用,提出了在THM耦合中考虑裂隙网络扩展演化及模拟的关键问题,同时指出了研究的3个关键点：（1）建立考虑裂隙网络演化的耦合模型;（2）裂隙扩展的数值模拟方法;（3）THM耦合及岩体变形、失稳全过程的数值模拟算法。随后通过对模拟多场耦合和裂隙扩展数值方法的归类比较,重点论述了目前适用于模拟多场耦合下裂隙扩展模拟的各种数值方法（包括有限单元法、无单元法、单位分解法、离散单元法、岩石破裂过程分析方法和数值流形方法）的优缺点,并通过对比研究,推荐采用数值流形方法（NMM）来实现对关键问题的模拟研究。最后,对研究思路和难点进行了初步探讨。     

Development of an implicit material point method for geotechnical applications

An implicit material point method (MPM), a variant of the finite element method (FEM), is presented in this paper. The key feature of MPM is that the spatial discretisation uses a set of material points, which are allowed to move freely through the background mesh. All history-dependent variables are tracked on the material points and these material points are used as integration points similar to the Gaussian points. A mapping and re-mapping algorithm is employed, to allow the state variables and other information to be mapped back and forth between the material points and background mesh nodes during an analysis. In contrast to an explicit time integration scheme utilised by most researchers, an implicit time integration scheme has been utilised here. The advantages of such an approach are twofold: firstly, it addresses the limitation of the time step size inherent in explicit integration schemes, thereby potentially saving significant computational costs for certain types of problems; secondly, it enables an improved algorithm accuracy, which is important for some constitutive behaviours, such as elasto-plasticity. The main purpose of this paper is to provide a unified MPM framework, in which both quasi-static and dynamic analyses can be solved, and to demonstrate the model behaviour. The implementation closely follows standard FEM approaches, where possible, to allow easy conversion of other FEM codes. Newton’s method is used to solve the equation of motion for both cases, while the formation of the mass matrix and the required updating of the kinematic variables are unique to the dynamic analysis. Comparisons with an Updated Lagrangian FEM and an explicit MPM code are made with respect to the algorithmic accuracy and time step size in a couple of representative examples, which helps to illustrate the relative performance and advantages of the implicit MPM. A geotechnical application is then considered, illustrating the capabilities of the proposed method when applied in the geotechnical field.     

Implicit integration algorithm for Hoek-Brown elastic-plastic model

A realistic strength criterion often used to describe the yielding behaviour of a jointed rock mass at a continuum level is the well-known Hoek and Brown criterion. This paper is concerned with a 3-D stress generalization of the Hoek-Brown failure criterion by means of an elliptical functional which leads to a smooth deviatoric trace in the stress space. For its incorporation into a finite element analysis involving plasticity calculations, the formulation of an implicit stress integration algorithm is presented. The key computational methodology alludes to the notion of consistent tangent modulus and implicit return mapping schemes (radial and closest point return) for stress integration in a finite element analysis. Within the context of non-linear elastoplastic analysis, it is found that formulation of such consistent modulus and success into achieving numerical efficiency are closely intertwined. Indeed, the procedure results into accurate and rapid convergence of the displacement finite element scheme during the search for equilibrium. This means that considerable savings in computational time can be achieved for large scale problems. Numerical examples which focus on the Hoek-Brown plasticity model are presented in order to fully appreciate the robustness of the algorithm, and hence the viability of such method to practical problems.     

岩体水力学基础(五)——岩体渗流场与应力场耦合的裂隙网络模型

岩体是由裂隙和其间的岩块组成。当岩块处于近乎隔水状态时，岩体中的渗流可看作为裂隙网络流。本文通过分析岩体裂隙网络渗流场、应力场以及它们之间的相互力学关系，提出了岩体渗流场与应力场耦合的裂隙网络模型及数值计算方法。     

裂隙岩体不稳定温度场的复合单元算法研究

基于复合单元法,结合三维热传导-对流方程和“充填模型”,提出了裂隙岩体不稳定温度场的复合单元模型。该模型前处理简便快捷,计算网格生成时无需考虑裂隙的存在,网格剖分不受限制,随后利用复合单元前处理程序,依据裂隙的位置和方位将其自动离散在单元内。对常规热传导-对流方程进行自伴随性调整,应用变分原理,推导出裂隙岩体不稳定温度场的复合单元算法,该算法可分别计算出岩块子单元和裂隙的温度值,且可真实反映裂隙中水流与相邻岩块间的热能量交换规律。将复合单元数值模型计算的不稳定温度场结果与相应的实测数据进行对比分析可知,数值计算结果与实测数据基本一致,验证了裂隙岩体不稳定温度场复合单元算法的可靠性与有效性。算例分析表明,裂隙中水流与相邻岩块间有明显的热传导和热对流作用。     

修正剑桥模型的隐式积分算法在ABAQUS中的数值实施

利用大型有限元软件ABAQUS所提供的用户材料子程序UMAT接口,针对修正的剑桥本构模型开发了隐式积分算法,并且与自动选择时间步长的增量有限元方程迭代解法相结合,对正常固结土与超固结土的三轴排水与不排水试验进行了数值模拟。结果表明,所发展的隐式本构积分算法与时间步长自动选择方法具有较好的稳定性和较高的计算精度,能够得到比较合理的数值分析结果。     

岩石剪切断裂面接触算法的开发与应用

为了克服现有数值计算方法在计算复杂断裂面接触滑移过程中效率低的问题，通过吸收边界元的建模思想，提出了新的数值计算方法。在该算法中，所有网格均位于岩块的边界，利用显示差分方法计算得到岩块之间的相互作用力以及岩块的位移量。利用“小球在抛物面上滑移”、“哑铃在不同倾角的斜面上”两个模拟试验对文中的算法进行验证，同时利用文中算法与有限元算法对直剪试验进行模拟。结果表明：算法能够准确描述物体之间的相互接触，准确计算得到接触块体之间的法向位移。算法能够准确判断出接触物体的“滑移状态”和“稳定状态”，摩擦力的数值计算结果与解析解的误差小于10?10。在计算断裂面接触和剪应力变化规律时，算法与有限元算法的计算结果一致，但文中计算方法的效率显著提高。     

岩石弹塑性损伤本构模型建立及在隧道工程中的应用

在实际隧道施工过程中,隧道开挖引起地下岩体应力重分布使得围岩的微裂纹扩展损伤,并伴随有塑性流动变形。在地下水环境中对于孔隙和微裂隙围岩介质受到应力作用时,在内部将产生高孔隙水压力影响岩石的力学性质,也改变了围岩的破坏模式。为了研究损伤引起的刚度退化和塑性导致的流动两种破坏机制的耦合作用,从弹塑性力学和损伤理论的角度出发,同时引入修正有效应力原理来考虑孔隙水压力的作用,建立基于Drucker-Prager屈服准则的弹塑性损伤本构模型;针对该本构模型推导了孔隙水压力作用下弹塑性损伤本构模型的数值积分算法-隐式返回映射算法,分别对预测应力返回到屈服面的光滑圆锥面或尖点奇异处两种可能的情况给出了详细的描述,隐式返回映射算法具有稳定性和准确性的特点;大多数弹塑性损伤模型中涉及参数多且不易确定的问题,采用反分析方法获得损伤参数,解决了损伤参数不易确定的难题;采用面向对象的编程方法,使用C++语言编制了弹塑性损伤本构求解程序,并对所建立的弹塑性损伤模型和所编程序进行了试验和数值两个方面的验证;最后将其在吉林抚松隧道工程中进行应用,模拟了塑性区和损伤区的发展变化。研究结果表明：所建立的弹塑性损伤本构模型能够较好地描述岩石的力学性能、塑性和损伤变化趋势,所编程序能够进行实际工程问题的模拟,对现场施工给予一定的指导。     

Demonstration and application of NMM‐based fractured porous medium flow model

In natural rock masses, the shapes of three‐dimensional (3‐D) blocks cut by arbitrary fracture networks may be very complex. Owing to the geometric complexity and difficulty of mesh discretization of 3‐D blocks and fracture facets, explicit consideration of fracture networks in flow analysis of fractured porous medium (FPM) is very challenging. Using the numerical manifold method based on independent covers (NMMIC), an FPM flow model was proposed that can deal with very complex 3‐D fracture networks. In this paper, the convergence of NMMIC was first demonstrated. The theoretical basis of the arbitrary refinement of computational meshes was proven. Moreover, three peculiarities of NMMIC meshes, that is, arbitrary shape, arbitrary connection, and arbitrary refinement of independent covers, were concluded. Finally, some two‐dimensional (2‐D) tunnel flow examples were analyzed and the numerical results were compared with the analytical results. 3‐D examples with complex fracture distributions were also analyzed. In addition, the computational scale of the developed program was tested by increasing the number of computational elements. The results show that our model can accurately analyze the groundwater flow of rocks surrounding tunnels with complex fracture distributions.     

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。     

Evaluation of numerical stress-point algorithms on elastic–plastic models for unsaturated soils with hardening dependent on the degree of saturation

Constitutive models of unsaturated soils, and in particular those based on constitutive variables which include both degree of saturation and suction, are characterised by strong non linearities due to hydromechanical coupling. In this paper, a refined Runge–Kutta–Dormand–Prince explicit algorithm and a fully implicit Euler scheme are compared for the integration of the latter class of models. The explicit and implicit procedures have been tested along different hydromechanical paths, involving various hydraulic and mechanical external control conditions. Accuracy and efficiency of the algorithms have been investigated. The results confirm that substepping is mandatory for the explicit algorithm to converge regardless the initial step size and to remain sufficiently accurate. The value of the incremental hydromechanical work per unit volume was calculated during the explicit integration procedure. The numerical results show that the maximum size of the substep which can be adopted to meet a given tolerance depends on the gradient of the incremental work per unit volume. Therefore, the latter appears a good candidate to identify problematic integration steps in terms of convergence. Accuracy of the implicit algorithm also depends on the chosen step size, although the algorithm proved to be convergent in all the paths analysed.     

An automatic time increment selection scheme for simulation of elasto-viscoplastic consolidation of clayey soils

An automatic time increment selection scheme for numerical analysis of long-term response of geomaterials is presented. The scheme is simple, rational and stable. Governed by a simple empirical expression, it can adaptively adjust the time increments depending on the strain rate-dependent temporal history of the material response. The proposed expression requires only a few parameters whose selection is a trivial task since they have a small effect on accuracy but have a significant effect on computational efficiency. This generalization has been made possible because of the enforcement of certain predefined control criteria to avoid extreme conditions. If any of the control criteria is satisfied, the computation is restarted by going a few time steps back to ensure the smoothness of the computed responses and time increments are again continuously adjusted through the governing equation provided. Performance of the automatic time increment selection scheme is investigated through finite element analyses of the long-term consolidation response of clay under different geotechnical profiles and loading conditions. Both elastic and elasto-viscoplastic constitutive relations are considered, including the consideration of the destructuration effects of geomaterials. Numerical results show that the performance of the automatic time increment selection scheme is reasonably excellent. While offering reasonable accuracy of the numerical solution, it can ensure temporal stability at optimal computational efficiency. In addition to the Euler implicit method, the automatic time increment selection scheme also performs well even when the explicit fourth-order Runge–Kutta method is employed for the integration of time derivatives.     

An unconditionally stable explicit and precise multiple timescale finite element modeling scheme for the fully coupled hydro-mechanical analysis of saturated poroelastic media

An unconditionally stable, fully explicit and highly precise multiple timescale finite element modeling scheme is described for a fully coupled hydro-mechanical (FCHM) analysis of saturated poroelastic media. The finite element method (FEM) is used for the discretization of the FCHM differential equation in the space domain. Direct integration is performed based on the precise time step integration method (PTSIM) for the time derivatives. Two configurations for the proposed scheme are constructed (abbreviated as PTSIM-f1 and -f2, respectively). The stability and convergence of the PTSIM-f1 and -f2 are proved using a matrix-based spectral analysis in the time domain. It is demonstrated that the explicit scheme proposed in this paper is unconditionally stable and independent of the time-step size. The algorithmic error estimation results indicate that the numerical modeling performed using PTSIM-f1 and -f2 in the time domain match the computer precision. Theoretically, the algorithmic error is caused by only the mesh discretization. Therefore, the proposed modeling scheme is a semi-analytical scheme. The applicability and accuracy of the proposed scheme are examined using sample calculations. By comparing with the analytical solutions, it is indicated that the modeling results have significant advantages over the standard FEM in terms of precision and computational efficiency for large timescales.     

Explicit Fracture Network Modelling: From Multiple Point Statistics to Dynamic Simulation

This paper addresses the problem of explicit fractured media modelling in an operational case. On one side, realistic fracture models are mainly used for research purposes in order to investigate better the flow behaviour impacted by the complex multi-scale fracture network. Often, a very fine grid and hence an increased computation time are needed. On the other hand, an operational fractured reservoir is still generally modelled using an implicit fracture media representation. The upscaled petrophysical properties and dual media are defined on a coarse grid to limit the computational time of dynamic simulation. The challenge of this work is to demonstrate that an explicit fracture modelling is not reserved only for the research domain, but can be applied to an operational case study. The static model is constructed using a multiple point statistics approach in order to represent complex interaction patterns of fractures and faults observed at the analogue outcrop. The dynamic behaviour is simulated based on this spatial fracture network representation.     

Simulations of hydro-fracking in rock mass at meso-scale using fully coupled DEM/CFD approach

The paper deals with two-dimensional (2D) numerical modelling of hydro-fracking (hydraulic fracturing) in rocks at the meso-scale. A numerical model was developed to characterize the properties of fluid-driven fractures in rocks by combining the discrete element method (DEM) with computational fluid dynamics (CFD). The mechanical behaviour of the rock matrix was simulated with DEM and the behaviour of the fracturing fluid flow in newly developed and pre-existing fractures with CFD. The changes in the void geometry in the rock matrix were taken into account. The initial 2D hydro-fracking simulation tests were carried out for a rock segment under biaxial compression with one injection slot in order to validate the numerical model. The qualitative effect of several parameters on the propagation of a hydraulic fracture was studied: initial porosity of the rock matrix, dynamic viscosity of the fracking fluid, rock strength and pre-existing fracture. The characteristic features of a fractured rock mass due to a high-pressure injection of fluid were realistically modelled by the proposed coupled approach.

     

基于统计强度理论的破裂岩体本构模型研究

借助损伤力学思想,基于统计强度理论,提出一种适用于深部工程破裂区破裂岩体的本构模型建立方法,并通过室内试验和数值试验进行了验证。将破裂岩体划分为无数微元立方体,微元立方体的强度与岩石破裂程度有关,且各立方体强度随机分布,故可用强度反映破裂岩体的破裂程度,据此提出一种破裂岩体本构模型建立方法。其中,根据岩石破裂面间的摩擦力做功等于材料破裂后释放的应变能,得到从力学角度定义的岩石破裂程度变量;另外,假定微元立方体强度分布服从Weibull分布,应力水平满足Hoek-Brown准则。利用泥质砂岩破裂岩体典型三轴试验结果,建立泥质砂岩破裂岩体本构模型,并进行了验证,结果表明模型计算曲线与试验曲线吻合度较好。利用离散元软件PFC进行了补充数值试验验证研究,证明了泥质砂岩破裂岩体理论模型的良好计算效果,进而证明提出的本构模型建立方法的可行性。     

Implicit and explicit integration schemes in the anisotropic bounding surface plasticity model for cyclic behaviours of saturated clay

Two integration algorithms, namely the implicit return mapping and explicit sub-stepping schemes, are adopted in the anisotropic bounding surface plasticity model for cyclic behaviours of saturated clay and are implemented into finite element code. The model is a representative of a series of bounding surface models that have typical characteristics, including isotropic and kinematic hardening rules and a rotational bounding surface to capture complex but important cyclic behaviours of soils, such as cyclic shakedown and degradation. However, there is no explicit current yield surface in the model to which the conventional implicit algorithm returns the stress state back or the sub-stepping integration corrects the drift of the stress state. Hence, necessary modifications have been made for both of the integration schemes. First, the image stress point is mapped or corrected to the bounding surface instead of mapping back or correcting the stress state to the yield surface. Second, the unloading–loading criterion is checked to determine the image stress point rather than checking the yield criterion after giving the trial stress state in a conventional way. Comparative studies on the accuracy, stability and efficiency of the two integration schemes are conducted not only at the element level but also in solving boundary value problems of monotonic and cyclic bearing behaviours of rigid footings on saturated clay. For smaller strain increments, there is no significant difference in the accuracy between the two integration schemes, but the explicit integration shows a higher efficiency and accuracy. For relatively larger increments, the implicit return mapping algorithm presents good accuracy and more robustness, while the sub-stepping algorithm shows deteriorating accuracy and suffers the convergence problem. With the tolerance used in the present model, the bearing capacity of the rigid footing predicted by the return mapping algorithm is closer to the available analytical and numerical solutions, while the bearing capacity predicted by the sub-stepping algorithm shows a marginal increase.     

岩体材料物理网格对流形元覆盖系统形成的影响

流形元的数学覆盖与物理网格相对独立,使流形元覆盖系统的形成比有限元单元网格的形成有更大的灵活性和方便性,但在形成流形元覆盖系统时对物理网格也有一定的依赖。笔者讨论了使用规则网格作为数学覆盖时,岩体材料普遍存在的裂隙和岩层界面这两种物理网格对流形元覆盖系统形成的影响,指出数学覆盖的构造要与岩体裂隙和岩层界面相适应,并提出了相应的流形元覆盖系统的形成方法,有利于流形元法应用于大型岩体工程问题。