Numerical manifold method for thermo-mechanical coupling simulation of fractured rock mass

As a calculation method based on the Galerkin variation,the numerical manifold method(NMM)adopts a double covering system,which can easily deal with discontinuous deformation problems and has a high calculation accuracy.Aiming at the thermo-mechanical(TM)coupling problem of fractured rock masses,this study uses the NMM to simulate the processes of crack initiation and propagation in a rock mass under the influence of temperature field,deduces related system equations,and proposes a penalty function method to deal with boundary conditions.Numerical examples are employed to confirm the effectiveness and high accuracy of this method.By the thermal stress analysis of a thick-walled cylinder(TWC),the simulation of cracking in the TWC under heating and cooling conditions,and the simulation of thermal cracking of the SwedishÄspöPillar Stability Experiment(APSE)rock column,the thermal stress,and TM coupling are obtained.The numerical simulation results are in good agreement with the test data and other numerical results,thus verifying the effectiveness of the NMM in dealing with thermal stress and crack propagation problems of fractured rock masses.

Numerical investigation of geostress influence on the grouting reinforcement effectiveness of tunnel surrounding rock mass in fault fracture zones

Grouting is a widely used approach to reinforce broken surrounding rock mass during the construction of underground tunnels in fault fracture zones,and its reinforcement effectiveness is highly affected by geostress.In this study,a numerical manifold method(NMM)based simulator has been developed to examine the impact of geostress conditions on grouting reinforcement during tunnel excavation.To develop this simulator,a detection technique for identifying slurry migration channels and an improved fluid-solid coupling(FeS)framework,which considers the influence of fracture properties and geostress states,is developed and incorporated into a zero-thickness cohesive element(ZE)based NMM(Co-NMM)for simulating tunnel excavation.Additionally,to simulate coagulation of injected slurry,a bonding repair algorithm is further proposed based on the ZE model.To verify the accuracy of the proposed simulator,a series of simulations about slurry migration in single fractures and fracture networks are numerically reproduced,and the results align well with analytical and laboratory test results.Furthermore,these numerical results show that neglecting the influence of geostress condition can lead to a serious over-estimation of slurry migration range and reinforcement effectiveness.After validations,a series of simulations about tunnel grouting reinforcement and tunnel excavation in fault fracture zones with varying fracture densities under different geostress conditions are conducted.Based on these simula-tions,the influence of geostress conditions and the optimization of grouting schemes are discussed.

2D numerical manifold method based on quartic uniform B-spline interpolation and its application in thin plate bending

A new numerical manifold （NMM） method is derived on the basis of quartic uniform B-spline interpolation. The analysis shows that the new interpolation function possesses higher-order continuity and polynomial consistency compared with the conven- tional NMM. The stiffness matrix of the new element is well-conditioned. The proposed method is applied for the numerical example of thin plate bending. Based on the prin- ciple of minimum potential energy, the manifold matrices and equilibrium equation are deduced. Numerical results reveal that the NMM has high interpolation accuracy and rapid convergence for the global cover function and its higher-order partial derivatives.

A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.

An overview of particle-based numerical manifold method and its application to dynamic rock fracturing

This review summarizes the development of particle-based numerical manifold method(PNMM)and its applications to rock dynamics.The fundamental principle of numerical manifold method(NMM)is first briefly introduced.Then,the history of the newly developed PNMM is given.Basic idea of PNMM and its simulation procedure are presented.Considering that PNMM could be regarded as an NMM-based model,a comparison of PNMM and NMM is discussed from several points of view in this paper.Besides,accomplished applications of PNMM to the dynamic rock fracturing are also reviewed.Finally,some recommendations are provided for the future work of PNMM.

Numerical manifold method modeling of coupled processes in fractured geological media at multiple scales

The greatest challenges of rigorously modeling coupled hydro-mechanical(HM)processes in fractured geological media at different scales are associated with computational geometry.These challenges include dynamic shearing and opening of intersecting fractures at discrete fracture scales as a result of coupled processes,and contact alteration along rough fracture surfaces that triggers structural and physical changes of fractures at micro-asperity scale.In this paper,these challenges are tackled by developing a comprehensive modeling approach for coupled processes in fractured geological media based on numerical manifold method(NMM)at multiple scales.Based on their distinct geometric features,fractures are categorized into three different scales:dominant fracture,discrete fracture,and discontinuum asperity scales.Here the scale is relative,that of the fracture relative to that of the research interest or domain.Different geometric representations of fractures at different scales are used,and different governing equations and constitutive relationships are applied.For dominant fractures,a finite thickness zone model is developed to treat a fracture as a porous nonlinear domain.Nonlinear fracture mechanical behavior is accurately modeled with an implicit approach based on strain energy.For discrete fractures,a zero-dimensional model was developed for analyzing fluid flow and mechanics in fractures that are geometrically treated as boundaries of the rock matrix.With the zero-dimensional model,these fractures can be modeled with arbitrary orientations and intersections.They can be fluid conduits or seals,and can be open,bonded or sliding.For the discontinuum asperity scale,the geometry of rough fracture surfaces is explicitly represented and contacts involving dynamic alteration of contacts among asperities are rigorously calculated.Using this approach,fracture alteration caused by deformation,re-arrangement and sliding of rough surfaces can be captured.Our comprehensive model is able to handle the computational challenges with accurate representation of intersections and shearing of fractures at the discrete fracture scale and rigorously treats contacts along rough fracture surfaces at the discontinuum asperity scale.With future development of three-dimensional(3D)geometric representation of discrete fracture networks in porous rock and contacts among multi-body systems,this model is promising as a basis of 3D fully coupled analysis of fractures at multiple scales,for advancing understanding and optimizing energy recovery and storage in fractured geological media.

Searching for critical slip surfaces of slopes using stress fields by numerical manifold method

This study first reviews the numerical manifold method(NMM)which possesses some advantages over the traditional limit equilibrium methods(LEMs)in calculating the factors of safety(Fs)of the slopes.Then,with regard to a trial slip surface(TSS),associated stress fields reproduced by NMM as well as the enhanced limit equilibrium method are combined to compute Fs.In order to search for the potential critical slip surface(CSS),the MAX-MIN ant colony optimization algorithm(MMACOA),one of the best performing algorithms for some optimization problems,is adopted.Procedures to obtain Fs in conjunction with the potential CSS are described.Finally,the proposed numerical model and traditional methods are compared with stability analysis of three typical slopes.The numerical results show that Fs and CSSs of the slopes can be accurately calculated with the proposed model.

基于NMM法的水泥环过套管电阻率测井影响及校正研究

通过对套管井漏失电流的检测,过套管电阻率测井能够有效测量金属套管外地层电阻率,在剩余油气资源评价和油藏监测等方面具有独特优势。套管和地层之间的水泥环对漏失电流的大小及测量结果有一定影响。采用NMM(数值模式匹配法)对水泥环的影响进行分析,数值模拟结果表明当地层与水泥环电阻率比值小于1时,测井响应值偏大,而且水泥环对测量的影响随该比值的减小和水泥环厚度的增加而增大。在此基础上对校正方法进行研究,提出校正的条件及采用校正图版进行水泥环校正的方法步骤。

岩石力学与工程问题的DDA和NMM模拟研究进展

全面回顾了非连续变形分析(DDA)方法与数值流形方法(NMM)在岩石力学与工程问题模拟领域的研究与应用进展。首先论述了相关理论与算法的研究现状,包括DDA中旋转虚假体积膨胀及能量损耗问题的解决,开裂破坏模拟的子块体单元DDA方法和节点DDA方法,以及颗粒DDA方法的发展等;也包括对NMM开裂破坏模拟能力及双重覆盖系统的研究,无网格NMM及粒子NMM的发展等;其中还涉及单纯形积分、高阶方法、材料本构模型、显式与并行计算的研究等;此外还对接触相关问题的研究进行了专门论述。随后,对DDA与NMM在岩体边坡失稳与落石、地下岩体结构的变形与破坏、应力波传播与岩石爆破、岩体结构地震破坏与失稳及岩体锚固工程领域的代表性研究与应用工作进行了详细论述。在此基础上,凝练和论述了DDA和NMM研究与应用面临的主要问题与挑战,包括计算控制参数、计算效率、多物理场、三维方法、程序的商业化开发等,并给出了相应的对策建议。

Modeling complex crack problems using the three-node triangular element fitted to numerical manifold method with continuous nodal stress

A three-node triangular element fitted to numerical manifold method with continuous nodal stress, called Trig_3-CNS(NMM)element, was recently proposed for linear elastic continuous problems and linear elastic simple crack problems. The Trig_3-CNS(NMM) element can be considered as a development of both the Trig_3-CNS element and the numerical manifold method(NMM).Inheriting all the advantages of Trig_3-CNS element, calculations using Trig_3-CNS(NMM) element can obtain higher accuracy than Trig_3 element without extra degrees of freedom(DOFs) and yield continuous nodal stress without stress smoothing. Inheriting all the advantages of NMM, Trig_3-CNS(NMM) element can conveniently treat crack problems without deploying conforming mathematical mesh. In this paper,complex problems such as a crucifix crack and a star-shaped crack with many branches are studied to exhibit the advantageous features of the Trig_3-CNS(NMM) element. Numerical results show that the Trig_3-CNS(NMM) element is prominent in modeling complex crack problems.

Numerical manifold method based on isogeometric analysis

In this study, numerical manifold method(NMM) coupled with non-uniform rational B-splines(NURBS) and T-splines in the context of isogeometric analysis is proposed to allow for the treatments of complex geometries and local refinement. Computational formula for a 9-node NMM based on quadratic B-splines is derived. In order to exactly represent some common free-form shapes such as circles, arcs, and ellipsoids, quadratic non-uniform rational B-splines(NURBS) are introduced into NMM. The coordinate transformation based on the basis function of NURBS is established to enable exact integration for the manifold elements containing those shapes. For the case of crack propagation problems where singular fields around crack tips exist, local refinement technique by the application of T-spline discretizations is incorporated into NMM, which facilitates a truly local refinement without extending the entire row of control points. A local refinement strategy for the 4-node mathematical cover mesh based on T-splines and Lagrange interpolation polynomial is proposed. Results from numerical examples show that the 9-node NMM based on NURBS has higher accuracies. The coordinate transformation based on the NURBS basis function improves the accuracy of NMM by exact integration. The local mesh refinement using T-splines reduces the number of degrees of freedom while maintaining calculation accuracy at the same time.

A high order numerical manifold method and its application to linear elastic continuous and fracture problems

The numerical manifold method(NMM) is a partition of unity(PU) based method. For the purpose of obtaining better accuracy with the same mesh, high order global approximation can be adopted by increasing the order of local approximations(LAs). This,however, will cause the "linear dependence"(LD) issue, where the global matrix is rank deficient even after sufficient constraints are enforced. In this paper, through quadrilateral mesh to form the mathematical cover, a high order numerical manifold method called Quad4-COLS(NMM) is developed, where the constrained and orthonormalized least-squares method(CO-LS) is used to construct the LAs. The developed Quad4-COLS(NMM) does not need extra nodes or DOFs to construct high order global approximations, while is free from the LD issue. Nine numerical tests including five tests for linear elastic continuous problems and four tests for linear elastic fracture problems are carried out to validate the accuracy and robustness of the proposed Quad4-COLS(NMM).

An h-adaptive numerical manifold method for solid mechanics problems

The numerical manifold method(NMM) features its dual cover systems, namely the mathematical cover and physical cover,which provide a unified framework for mechanics problems involving continuum and discontinuum deformation. Uniform finite element meshes can be and are usually used to construct the mathematical cover. Though this strategy can handle different kinds of problems in a unified way, it is not economical for cases with steep deformation gradients or singularities. In this paper, using the recovery-based error estimator, an h-adaptive NMM on quadtree meshes is proposed to deal with such cases. The quadtree meshes serve as the mathematical meshes, on which the local refinement is carried out. When the quadtree meshes are refined,the corresponding mathematical cover, physical cover and manifold elements are updated accordingly. To handle the hanging nodes in the quadtree meshes, we resort to mean value coordinates. Comparing to the refinement based on manifold elements,the proposed strategy guarantees consistent structured meshes throughout the adaptive process, thus retaining the unique feature of original NMM. In contrast with polygonal finite element method, an advantage of the proposed method is that the meshes do not need to conform to the crack face and material boundary, which means the adaptive NMM is very suitable for problems with complex geometric boundary. Several representative mechanics problems, including crack problems, are analyzed to investigate the effectiveness of the proposed method. It is demonstrated that the proposed adaptive NMM has higher accuracy and better performance comparing to uniform refinement strategy.

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。

用模式匹配算法研究层状各向异性倾斜地层中多分量感应测井响应

利用数值模式匹配技术,研究并建立层状各向异性倾斜地层中多分量感应测井响应的快速数值模拟算法.首先将位于井轴上的三个相互正交磁偶极子转化成关于极角θ的三个谐变分量的叠加,以便将三个正交磁偶极子电磁场的正演问题完全简化成三个谐变分量电磁场的轴对称定解问题,并给出电磁场各个谐变分量在井轴上满足的边界条件,保证电磁场在井轴附近仍然可解.然后利用数值模式匹配技术建立电磁场各个谐变分量的正演过程,得到层状各向异性地层中磁流源并矢Green函数的半解析表达式,给出计算层状各向异性倾斜地层中多分量感应测井响应的具体方法,最后通过数值计算结果证明该算法的有效性并考察几种不同情况下多分量感应测井响应特征.

岩石冲击破坏的数值流形方法模拟(英文)

岩石冲击破坏是一种典型的动力学问题。为了更好地利用数值流形方法对动力学问题进行分析,本文在对原数值流形方法中的动力学问题求解思想进行分析的基础上,采用动力有限元方法中的Newmark法对该算法进行了改进。改进后的数值流形方法与原来相比具有3个明显的优势:①当选择合适的参数后,该方法能够保证解的无条件收敛;②可以采用比原算法大的多的时间步长;③充分地考虑了动力学问题中的阻尼效应。最后通过一个算例说明了改进后的数值流形方法能够很好地模拟岩石在冲击载荷作用下破坏的全过程,克服了有限元法不能模拟岩石破坏后块体运动情况的不足。

部分重叠覆盖的数值流形方法初步研究

数值流形方法(简称流形法)采用基于完全重叠覆盖的有限元网格作为数学网格,物理网格与数学网格的不匹配导致结构一些关键部位的计算精度下降。针对此问题,首次提出部分重叠覆盖的流形法,采用以独立覆盖为主的分析方式,独立覆盖之间仅用较小的部分重叠区域保持连续性,从而将现有的基于完全重叠覆盖的流形法扩展到一般意义上的流形研究。对矩形部分重叠覆盖进行初步研究,通过单个矩形数学网格各结点之间的自由度约束方式,很方便地将完全重叠的覆盖形式及流形法公式转化为部分重叠的覆盖形式及公式。算例分析初步验证了这种新型流形法的有效性。

基于独立覆盖的流形法的收敛性及覆盖网格特性

针对前期提出的基于部分重叠覆盖的数值流形方法,将其内涵范围缩小,仅研究其中的一种情况——基于独立覆盖的数值流形方法。从完备性和协调性2个方面讨论该方法的收敛性,特别强调其收敛性是基于各个独立覆盖的逼近而建立起来的,独立覆盖之间条形连接区域的尺寸要取小,并由此推断及用实例说明,覆盖网格可以具备"3个任意"的优良特性——任意形状、任意连接以及由此而来的可任意加密的能力,从而有望使数值计算的前处理工作大为简化。

数值流形方法中的网格重分技术及其应用

研究了网格重新划分的判断准则及新网格系统的自动生成 ,采用Sherpard形函数提出了新旧网格信息转换的参数传递方法 ,给出了一种通用的 ,适合二维刚塑性分析的网格重划分方法。数值算例表明 ,该方法具有高的精度和计算效率。

非轴对称条件下用三维模式匹配法计算电阻率测井响应

采用模式匹配法研究了非轴对称条件下普通电阻率测井响应.根据非轴对称的地层模型,建立了新型的坐标系,确定在井轴与地层法线所形成的XOY平面上用数值方法,与此面垂直的方向上用解析方法.在XOY平面上的数值分析中,构建了广义特征值问题;在Z轴方向的解析部分中,人为地划分了一些介质层,并根据电磁场在层界面的连续性条件,推导出了各层之间场量的传递关系,从而摒弃了传统上二维模式匹配法中的复杂的反射阵和透射阵理论,完成了三维模式匹配理论.多种地层模型的数值结果对比表明,该方法在轴对称条件下的计算结果和二维模式匹配法的计算结果具有很好的一致性.此外,应用此算法还研究了倾斜地层等各种非轴对称的地层模型电阻率测井响应,系统地考察了井斜角对普通电阻率测井响应的影响.