Fracture is one of the most common failure modes of materials and components and greatly restricts engineering design. Understanding of the crack propagation and evolution of rock and other ...engineering materials is of great significance to engineering construction. For the current numerical methods there are more or less limitations when analyzing the evolution of cracks, such as the mesh dependence of the crack path, the difficulty to deal with crack bifurcation and merging by the classic fracture criterion. In recent years, the phase field method (PFM) has been widely used in simulating crack growth. A phase field numerical manifold method (PFNMM) makes use of the advantages of the phase field method in simulating crack propagation and those of the numerical manifold method (NMM), is proposed for crack growth in rock. The implementation details of the proposed numerical model are presented. Several benchmark examples, including notched semi-circular bend test and Brazilian disc test, are adopted to validate the proposed numerical approach. After that, the multi-crack propagation process with different rock bridge inclination angles under uniaxial compression is simulated, which is in good agreement with the results derived from laboratory and PFC. And the results indicate that the PFNMM has broad application prospects in simulating crack growth of rock.
•A 3D first-order NMM-based multiscale scheme is presented for hydro-mechanics of the heterogeneous porous media.•Micro-scale dynamics is fully considered based on the extended Hill-Mandel ...lemma.•Simple, closed and efficient algorithms are presented for extracting macro-scale quantities and Jacobian.
A two-scale computational homogenization model is presented for modeling three-dimensional (3D) heterogeneous poro-elastic media in the framework of the Numerical Manifold Method (NMM). The micro-dynamics is fully incorporated using the extended first-order Hill-Mandel principle for simulation of the hydro-dynamic response. The micro- and macro-scale Initial Boundary-Value Problems (IBVPs) are simultaneously solved with the NMM by exchanging quantities between different scales. The micro-scale IBVPs are solved under both Linear Boundary Conditions (LBCs) and Periodic Boundary Conditions (PBCs). The macro-scale IBVP is solved using the Newton's algorithm iteratively. For both micro-scale LBCs and PBCs, highly efficient algorithms for extracting macro-scale quantities and Jacobian matrix from the micro-scale are established merely by simple matrix manipulations of the micro-scale Jacobian matrix without solving any IBVPs. By conducting benchmark simulations of the heterogeneous porous media under uniform, partial and impact loads, the accuracy, stability and versatility of the presented multiscale model are verified.
The major difficulty in the analysis of unconfined flow in porous media is that the free surface is unknown a priori, where the nonlinearity is even stronger than the unsaturated seepage analysis. ...There is much space for both the adaptive mesh methods and the fixed mesh methods to improve. In this study, firstly two variational principles fitted to the numerical manifold method (NMM) are formulated, each of which enforces the boundary conditions and the material interface continuity conditions. In the setting of the NMM together with the moving least squares (MLS) interpolation, then the discretization models corresponding to the variational formulations are built, which are utilized to locate the free surface and scrutinize the computational results respectively. Meanwhile, a novel approach is developed to update the free surface in iteration. With high accuracy and numerical stability but no need to remesh, the proposed procedure is able to accommodate complicated dam configuration and strong non-homogeneity, where internal seepage faces may develop, a seldom touched problem in the literature.
In this paper, a mixed two-scale numerical manifold computational homogenization model is presented for dynamic analysis and wave propagation of the discontinuous heterogeneous porous media based on ...the first-order homogenization theory. Instead of the conventional version which neglects microscopic dynamics, the extended Hill–Mandel lemma is employed to incorporate the microscale dynamic effects. Microscale and macroscale Initial Boundary Value Problems (IBVPs) are solved simultaneously using the Numerical Manifold Method (NMM) with the information conveyed between different scales. The microscale IBVP is solved under Linear Boundary Conditions (LBCs) and Periodic Boundary Conditions (PBCs) that are defined with macroscale solid displacement, fluid pressure and their first-order gradients. The discontinuous macroscale IBVP is solved iteratively using the Newton method with the macroscale internal forces and Jacobian determined by solving the microscale IBVPs. A stick–slip contact model is implemented using an augmented Lagrange multiplier method to impose frictional contact conditions along the macroscale discontinuities. Through various numerical simulations, the presented two-scale NMM is shown to be able to effectively and accurately capture the fully dynamic and wave propagation responses of the discontinuous heterogeneous porous media under the fluid injection and impact loading condition.
•A two-scale numerical manifold computational homogenization model is presented.•Micro dynamics is considered using the extended Hill-Mandel lemma.•Macro contact is modeled with an augmented Lagrange multiplier method.•Wave propagation in discontinuous heterogeneous porous media is modeled.
Fractures have attracted the attention of computational scientists for several decades. The modeling and simulation of fractures have been a major motivation for developing enriched finite element ...methods (FEMs), such as the numerical manifold method (NMM). However, ill-conditioning has always haunted NMM and other enriched FEMs when they are utilized for linear elastic fracture problems. Generally, ill-conditioning for a fracture problem is caused by two main issues: the arbitrary cut of the mesh by the fracture path and linear dependence related to the crack-tip enrichments. It is significantly challenging to overcome these two types of ill-conditioning using a single technique. In this study, we employ a preconditioner based on global normalization and local Gram–Schmidt orthogonalization of bases to eliminate these two ill-conditioning issues in NMM entirely and simultaneously. Various numerical examples have demonstrated that the proposed preconditioning strategy is highly effective in reducing the condition number and iteration counts of a iterative solver. It is highly robust, stable, and efficient and can be incorporated into enriched FEM programs to significantly facilitate the analyses of linear elastic fractures.
•Ill-conditioning due to mesh cut and crack-tip enrichment is explored.•Symmetric preconditioning overcomes ill-conditioning.•Stability and efficiency are significantly enhanced for linear elastic fractures.
A high-order numerical manifold method (HONMM) is developed using four-node quadrilateral (QUAD4) element to enhance accuracy and computational efficiency in solid mechanical problems. In the current ...study, the high-order global approximations are built by increasing the order of local approximations without facing the linear dependence (LD) problem, which is one of the main issues in the partition of unity (PU)-based methods with high-order approximations. To remove the LD problem, a new and simple scheme is proposed. Four problems are utilized to evaluate the efficiency of the QUAD4-based HONMM (QHONMM). A cantilever beam example is analyzed to compare different orders of QHONMM. Also, a block example with different loads and boundary conditions is used to investigate the sensitivity of the QHONMM results. Moreover, to demonstrate the capabilities of the QHONMM in dynamic analysis, free and forced vibrations of a simple beam under distributed and moving loads are analyzed. The results showed that using QUAD4 elements, the proposed QHONMM performs better than the conventional NMM in terms of accuracy and computational costs.
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.
A three-dimensional (3D) numerical manifold formulation with continuous nodal gradients (H8-CNS) is presented for dynamic analysis of saturated elasto-plastic porous media based on the three-variable ...(u−w−p) formulation. Using meshes of 8-node hexahedra as mathematical patches, the skeleton displacement (u) and fluid velocity (w) interpolations are constructed with a constrained and ortho-normalized least-squares (CO-LS) scheme and a piecewise constant interpolation for fluid pressure field is employed. One feature of H8-CNS is to overcome locking in both limiting cases of low permeability and rigid skeleton. Another feature of H8-CNS is to achieve higher accuracy in stress results for porous media problems involving both elastic and elasto-plastic skeleton behaviors. H8-CNS is also able to fully capture dynamic responses of porous media under high-frequency loading. The accuracy and stability of H8-CNS are verified by performing a variety of numerical simulations. Time integration of H8-CNS is shown to be stable and accurate using the energy balance condition.
•A 3D NMM H8-CNS is given for dynamics of elasto-plastic porous media.•For elastic/plastic analysis, H8-CNS gives more accurate stress than mixed FEM.•H8-CNS is immune from locking.
Grouting is a commonly used technique in rock engineering to enhance the joint strength and improve the stability of surrounding rock. Grout penetration characteristic is controlled by grouting ...parameters and has a significant role on practice. In the study, a numerical manifold method (NMM) for grout penetration process simulation in fractured rock mass is firstly proposed. The fluid flow behaviour of the grout is assumed to be a Bingham fluid and control equations are established using discrete fracture network model. The global discretization equation, element sub-matrixes and NMM simulation algorithm for grouting are presented. Then, numerical tests for grouting process in a single fracture and a regular fracture network are conducted firstly to verify the proposed NMM grouting model by comparing with analytical solutions and experimental results. Furthermore, the effects of mesh size, fracture and slurry parameters on the grouting performance are systematically investigated using a random fracture network example. The numerical results indicate that the grouted zone and propagation depth decrease as the mesh size of numerical model and yield strength increases, while it increases as initial fracture aperture and grouting pressure increases.
•A novel local refinement strategy based on the numerical manifold method is developed.•Variable-midside-node elements are introduced to link different-sized elements.•The refinement regions transfer ...automatically with the crack growth.•Numerical integration in the transition element is achieved by a special algorithm.
Uniform finite element meshes are usually used to discrete problem domain in the numerical manifold method. This strategy provides high interpolation accuracy but is ineffective when dealing with cases involving steep deformation gradients or singularities. The major objective of this research is to develop a local multilevel mesh refinement strategy on the basis of the numerical manifold method, in which the determination of mathematical elements to be refined and the multilevel refinement to the target region can be performed automatically. To accurately capture the singularity while saving computational cost during crack growth, a follow-up refinement scheme is implemented to realize real-time refinement for the crack tip region. The variable-midside-node elements with conforming shape functions are integrated into the present formulation to solve the mismatching problem induced by different different-sized elements in an effective way. A special subdivision algorithm is proposed for appropriately and accurately treating numerical integration of manifold elements in the transition elements. The numerical performance of the proposed method is first validated by a linear elastic example. Next, six fracture problems involving multiple and branched cracks are simulated. The obtained results indicate high accuracy, low computational cost, and good performance of the proposed method in fracture analysis.
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