A new strategy for 3D non-persistent crack propagation by the numerical manifold method (NMM) with tetrahedral meshes is proposed and corresponding computer programs are coded by C++ language. The Mohr-Coulomb failure criterion with a tensile cut-off (MCCT) is selected as the crack initiation and failure criterion for 3D crack propagation. The triangle crack tracking method is firstly used to obtain initial new crack surfaces, which are reconstructed by the cutting surface method (CFM) for tetrahedral meshes to produce the final new crack surfaces. Seven possible types of cutting surfaces that may be generated during the surface-block cutting process are discussed, and then corresponding optimal processing algorithms (OPAs) for new divisions are presented. Considering the convenience of the forthcoming contact analysis, all crack related surfaces (CRFs) for a boundary inconsistent manifold element (BIE) containing a crack are required to be further triangulated. Four numerical examples for 3D non-persistent crack propagation with tetrahedral meshes are illustrated to show the validity and applicability of the present 3D NMM algorithm and program. The numerical results are in good agreement with existing test or numerical results, thus it indicates that the new strategy for 3D non-persistent crack propagation with tetrahedral meshes is feasible and effective. 

A new strategy for 3D non-persistent crack propagation by the numerical manifold method with tetrahedral meshes

GMLS-based numerical manifold method in mechanical analysis of thin plates with complicated shape or cutouts

To simulate the moving boundary problems, the moving least square–based numerical manifold method, abbreviated as MLS-based NMM, was proposed. The MLS-based NMM has been applied successfully to open crack problems, which exhibits the high accuracy and strong robustness. In this study, we extend the MLS-based NMM to simulate the cracked rock considering the contact of the crack surface. Simultaneously, in order to simulate the progressive failure of the cracked rock, an improved strength-based criterion is proposed. The criterion is based on the Mohr–Coulomb criterion and maximum tensile stress criterion. Because rock can be regarded as a quasi-brittle material, a characteristic distance is used to calculate the crack tip stress and correct the crack propagation direction which avoids the phenomenon of “Zig-zag” for the crack propagation path based on the fracture mechanics criterion. The proposed strength-based criterion can acquire the crack tip stress and propagation direction and also realize the automatic determination of the crack propagation length in each step of the crack growth. A Brazilian disc problem and a rectangular plate problem are adopted to verify the numerical model. At last, the numerical model is applied to study the progressive failure process of the rock slope. The results indicate that the proposed method can deal with the crack propagation in the rock and the opening/sliding of rock blocks along discontinuities in a natural way.

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MLS-Based Numerical Manifold Method for Modeling the Cracked Rock Considering the Contact of the Crack Surface

The contact is a typical non-linear problem that exists in various projects. For traditional three-node triangular mesh and four-node quadrilateral mesh, the accuracy and convergence of the calculation results are affected by the quality of the mesh. The test space and trial space in the virtual element method (VEM) do not need to be accurately calculated, avoiding mesh dependence. In this paper, the formulation of linear elasticity and the formulation of the frictionless node-to-segment (NTS) contact model via VEM are shown. There are four numerical simulations. The sensitivity of the virtual element method to mesh distortion is studied in the first numerical simulation. The exactness and convergence of the algorithm are investigated by the second numerical example. The second numerical example simultaneously explores the penalty factor’s effect on the results. The third example investigated the impact of mesh shape and number of Voronoi mesh elements on the results by comparing normal contact stresses. The fourth numerical example studies the application of the method to engineering. Those numerical examples show that the virtual element method is insensitive to mesh distortion and could solve the joint contact in engineering.

The Virtual Element Method for the Dam Foundation With Joint

In many structural applications, the transition region, where the solids connect to the shells, is of considerable interest. Due to the complex behavior of this area, accurate modeling is necessary in practical problems. A numerical manifold method (NMM) based on the least square point interpolation method (LSPIM) free from linear dependence, is proposed to simulate this region. First, the basic formulations of local approximation based on LSPIM and the method to construct composite shape function are briefly introduced. This new composite shape function possesses the Kronecker-delta, inter-element compatibility, and higher-order completeness properties. Then, the formulations of the hexahedral element, 3D shell and 3D transition elements are derived based on the proposed method. Several examples are carried out to validate the accuracy and robustness of the suggested method for simulation of transition regions. 

The LSPIM-based numerical manifold method (NMM) for modeling transition elements

The meshless numerical manifold method (MNMM) inherits two covers of the numerical manifold method. A mathematical cover is composed of nodes' influence domains and a physical cover consists of physical patches, which are produced through cutting mathematical cover by physical boundaries. Because two covers are adopted, MNMM can naturally and uniquely solve both the continuous and discontinuous problems under Galerkin's variational framework. However, Galerkin's meshless method needs background integration grids to realize solving, which often does not match the nodes' influence domains, so the accuracy of numerical integration is reduced. Consider that MNMM allows even distribution of nodes and the physical cover contains the characteristics of the boundary and nodes' influence domains, the study presents a new numerical integration strategy to ensure that the background integration grids match the nodes' influence domains. The method can be applied to continuous and discontinuous problems, and is proved to be equivalent to the influence domain integration. At the same time, a reasonable arrangement of mathematical nodes is made to assure the background integration grid that it is accordant and straightforward. In this way, the number of physical patches of each integral point is the same, which improves the accuracy of interpolation calculation. The effectiveness of the proposed method is verified by numerical examples of both continuous and discontinuous problems.

A numerical integration strategy of meshless numerical manifold method based on physical cover and applications to linear elastic fractures

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A numerical integration strategy of meshless numerical manifold method based on physical cover and applications to linear elastic fractures