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.

Galerkin-based quasi-smooth manifold element(QSME)method for anisotropic heat conduction problems in composites with complex geometry

The accurate and efficient analysis of anisotropic heat conduction problems in complex composites is crucial for structural design and performance evaluation. Traditional numerical methods, such as the finite element method(FEM), often face a trade-off between calculation accuracy and efficiency. In this paper, we propose a quasi-smooth manifold element(QSME) method to address this challenge, and provide the accurate and efficient analysis of two-dimensional(2D) anisotropic heat conduction problems in composites with complex geometry. The QSME approach achieves high calculation precision by a high-order local approximation that ensures the first-order derivative continuity.The results demonstrate that the QSME method is robust and stable, offering both high accuracy and efficiency in the heat conduction analysis. With the same degrees of freedom(DOFs), the QSME method can achieve at least an order of magnitude higher calculation accuracy than the traditional FEM. Additionally, under the same level of calculation error, the QSME method requires 10 times fewer DOFs than the traditional FEM. The versatility of the proposed QSME method extends beyond anisotropic heat conduction problems in complex composites. The proposed QSME method can also be applied to other problems, including fluid flows, mechanical analyses, and other multi-field coupled problems, providing accurate and efficient numerical simulations.

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.

Numerical Method Based on Compatible Manifold Element for Thin Plate Bending

The typical quadrangular and triangular elements for thin plate bending based on Kirchhoff assumptions are the non- conforming elements with low computational accuracy and limitative application range in fmite element method（FEM）. Some compatible elements can be developed by the means of supplementing correction functions, increasing nodes in element or on the boundaries, expanding nodal degrees of freedom（DOF）, etc, but these elements are inconvenient to apply in practice for the high calculation complexity. In this paper, in order to overcome the defects of thin plate bending finite element, numerical manifold method（NMM） was introduced to solve thin plate bending deformation problem. Rectangular mesh was adopted as mathematical mesh to form f＇mite element cover system, and then 16-cover manifold element was proposed. Numerical manifold formulas were constructed on the basis of minimum potential energy principle, displacement boundary conditions are implemented by penalty function method, and all the element matrixes were derived in details. The 16-cover element has a simple calculation process for employing only the transverse displacement cover DOFs as the basic unknown variables, and has been proved to meet the requirements of completeness and full compatibility. As an application, the presented 16-cover element has been used to analyze bending deformation of square thin plate under different loads and boundary conditions, and the results show that numerical manifold method with compatible element, compared with finite element method, can improve computational accuracy and convergence greatly.

Modeling rock failure using the numerical manifold method followed bythe discontinuous deformation analysis

A complete rock failure process usually involves opening/sliding of preexisting discontinuities as well as frac- turing in intact rock bridges to form persistent failure sur- faces and subsequent motions of the generated rock blocks. The recently developed numerical manifold method （NMM） has potential for modelling such a complete failure process. However, the NMM suffers one limitation, i.e., unexpected material domain area change occurs in rotation modelling. This problem can not be easily solved because the rigid body rotation is not represented explicitly in the NMM. The discontinuous deformation analysis （DDA） is specially de- veloped for modelling discrete block systems. The rotation- induced material area change in the DDA modelling can be avoided conveniently because the rigid body rotation is represented in an explicit form. In this paper, a transition technique is proposed and implemented to convert a NMMmodelling to a DDA modelling so as to simulate a complete rock failure process entirely by means of the two methods, in which the NMM is adopted to model the early fracturing as well as the transition from continua to discontinua, while the DDA is adopted to model the subsequent motion of the generated rock blocks. Such a numerical approach also im- proves the simulation efficiency greatly as compared with a complete NMM modelling approach. The fracturing of a rock slab with pre-existing non-persistent joints located on a slope crest and the induced rockfall process are simulated. The validity of the modelling transition from the NMM to the DDA is verified and the applicability of the proposed nu- merical approach is investigated.

A generalized cover renewal strategy for multiple crack propagation in two-dimensional numerical manifold method

Partition of unity based numerical manifold method can solve continuous and discontinuous problems in a unified framework with a two-cover system,i.e.,the mathematical cover and physical cover.However,renewal of the topology of the two-cover system poses a challenge for multiple crack propagation problems and there are few references.In this study,a robust and efficient strategy is proposed to update the cover system of the numerical manifold method in simulation of multiple crack propagation problems.The proposed algorithm updates the cover system with a bottom-up process:1)identification of fractured manifold elements according to the previous and latest crack tip position;and 2)local topological update of the manifold elements,physical patches,block boundary loops,and non-persistent joint loops according to the scenario classification of the propagating crack.The proposed crack tracking strategy and classification of the renewal cases promote a robust and efficient cover renewal algorithm for multiple crack propagation analysis.Three crack propagation examples show that the proposed algorithm performs well in updating the cover system.This cover renewal methodology can be extended for numerical manifold method with polygonal mathematical covers.

THE VARIATIONAL PRINCIPLES AND APPLICATION OF NONLINEAR NUMERICAL MANIFOLD METHOD

The physical-cover-oriented variational principle of nonlinear numerical manifold method (NNMM) for the analysis of plastical problems is put forward according to the displacement model and the characters of numerical manifold method (NMM). The theoretical calculating formulations and the controlling equation of NNMM are derived. As an example, the plate with a hole in the center is calculated and the results show that the solution precision and efficiency of NNMM are agreeable.

THE VARIATIONAL PRINCIPLE AND APPLICATION OF NUMERICAL MANIFOLD METHOD

The physical-cover-oriented variational principle of numerical manifold method (NMM) for the analysis of linear elastic static problems was put forward according to the displacement model and the characters of numerical manifold method. ne theoretical calculating formulations and the controlling equation of NMM were derived. As an example, the plate with a hole in the center is calculated and the results show that the solution precision and efficiency of NMM are agreeable.

THEORETICAL STUDY OF THREE-DIMENSIONAL NUMERICAL MANIFOLD METHOD

The three-dimensional numerical manifold method（NMM） is studied on the basis of two-dimensional numerical manifold method. The three-dimensional cover displacement function is studied. The mechanical analysis and Hammer integral method of three-dimensional numerical manifold method are put forward. The stiffness matrix of three-dimensional manifold element is derived and the dissection rules are given. The theoretical system and the numerical realizing method of three-dimensional numerical manifold method are systematically studied. As an example, the cantilever with load on the end is calculated, and the results show that the precision and efficiency are agreeable.

Incompatible numerical manifold method for fracture problems

The incompatible numerical manifold method （INMM） is based on the finite cover approximation theory, which provides a unified framework for problems dealing with continuum and discontinuities. The incompatible numerical manifold method employs two cover systems as follows. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the weighted functions, and the physical cover system describes geometry of the domain and the discontinuous surfaces therein. In INMM, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the process of displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the INMM is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack. To improve the precision of the INMM, the analytical solution is used at the tip of a crack, and thus the cover displacement functions are extended with higher precision and computational efficiency. Some numerical examples are given.

Numerical Manifold Method and Its Application in the Study of Crustal Movements in the Sichuan-Yunnan Area

The numerical manifold method （NMM） can calculate the movements and deformations of structures or materials. Both the finite element method （FEM） for continua and the discontinuous deformation analysis （DDA） for block systems are special cases of NMM. NMM has separate mathematical covers and physical meshes： the mathematical covers define only fine or rough approximations; as the real material boundary, the physical mesh defines the integration fields. The mathematical covers are triangle units; the physical mesh includes the fault boundaries, joints, blocks and interfaces of different crust zones on the basis of a geological tectonic background. Aiming at the complex problem of continuous and discontinuous deformation across the Chinese continent, the numerical manifold method （NMM） is brought in to study crustal movement of the Stchuan-Yunnan area. Based on the GPS velocity field in the Sichuan-Yunnan area, a crustal strain and stress field is simulated and analyzed. Moreover, results show that the NMM is a more suitable method than DDA in simulating the movement of the Sichuan-Yunnan area. Finally, a kind of mechanism of crustal motion in the Sichuan-Yunnan area is discussed in the paper.

Numerical simulation of crack propagation by manifold method

In this paper, by means of the maximum circle tensile stress on curve of constant ω and stress intensity factors by a path independent contour integral method, trajectories of maxed mode crack propagation are simulated through numerical manifold method. The crack propagation is traced dynamically by modifying the neighboring connection between the crack-top and nodes within elements in the calculating process. This method has the advantages such as less modified area, easiness of programming, high realizability and so on. Then a single sharp nicked specimen is used to verified the numerical result. It is shown that the provided method is reasonable and effective.

Modeling wave propagation across rock masses using an enriched 3D numerical manifold method

The three-dimensional numerical manifold method(3DNMM) method is further enriched to simulate wave propagation across homogeneous/jointed rock masses. For the purpose of minimizing negative effects from artificial boundaries, a viscous nonreflecting boundary, which can effectively absorb the energy of a wave, is firstly adopted to enrich 3DNMM. Then, to simulate the elastic recovery property of an infinite problem domain, a viscoelastic boundary, which is developed from the viscous nonreflecting boundary, is further adopted to enrich 3DNMM. Finally, to eliminate the noise caused by scattered waves, a force input method which can input the incident wave correctly is incorporated into 3DNMM. Five typical numerical tests on P/S-wave propagation across jointed/homogeneous rock masses are conducted to validate the enriched 3DNMM. Numerical results indicate that wave propagation problems within homogeneous and jointed rock masses can be correctly and reliably modeled with the enriched 3DNMM.

A stable one-point quadrature rule for three-dimensional numerical manifold method

We present a numerically stable one-point quadrature rule for the stiffness matrix and mass matrix of the three-dimensional numerical manifold method(3D NMM).The rule simplifies the integration over irregularly shaped manifold elements and overcomes locking issues,and it does not cause spurious modes in modal analysis.The essential idea is to transfer the integral over a manifold element to a few moments to the element center,thereby deriving a one-point integration rule by the moments and making modifications to avoid locking issues.For the stiffness matrix,after the virtual work is decomposed into moments,higher-order moments are modified to overcome locking issues in nearly incompressible and bending-dominated conditions.For the mass matrix,the consistent and lumped types are derived by moments.In particular,the lumped type has the clear advantage of simplicity.The proposed method is naturally suitable for 3D NMM meshes automatically generated from a regular grid.Numerical tests justify the accuracy improvements and the stability of the proposed procedure.

Three-dimensional numerical manifold method for heat conduction problems with a simplex integral on the boundary

The three-dimensional numerical manifold method(3D-NMM),which is based on the derivation of Galerkin's variation,is a powerful calculation tool that uses two cover systems.The 3D-NMM can be used to handle continue-discontinue problems and extend to THM coupling.In this study,we extended the 3D-NMM to simulate both steady-state and transient heat conduction problems.The modelling was carried out using the raster methods(RSM).For the system equation,a variational method was employed to drive the discrete equations,and the crucial boundary conditions were solved using the penalty method.To solve the boundary integral problem,the face integral of scalar fields and two-dimensional simplex integration were used to accurately describe the integral on polygonal boundaries.Several numerical examples were used to verify the results of 3D steady-state and transient heat-conduction problems.The numerical results indicated that the 3D-NMM is effective for handling 3D both steadystate and transient heat conduction problems with high solution accuracy.