Based on the energy equilibrium, the computational formula of critical buckling length of multi-layer rock slope is derived. Considering interlayer and cross joints, the numerical manifold method is ...used to simulate the buckling evolution process of Malvern hills slope in New Zealand, and the theoretical calculation and numerical simulation results are compared with the field measured data. The results show that numerical manifold method can accurately simulate slope buckling failure process by preforming interlayer and cross joints. The process of slope buckling deformation and instability failure can be divided into interlayer dislocation-slight bending, slope toe traction-sharp uplift and accelerated sliding-landslide formation. Under the long-term action of self-weight, the evolution of slope buckling from formation to failure mainly includes three stages: initial bending, sharp bending and landslide formation. The angle between cross joint and slope normal is defined as β. Among the four kinds of cross joints with the angle β of 0°, 15°, 30° and 45°, the slope with 45° cross joint is most prone to slipping and bending deformation, the degree of buckling is the largest, and the number of time steps of slipping and bending is the least. When β is in the range of 30°−45°, the numerical simulation results are in good agreement with the reality.
Dynamics of the widespread saturated and semi-saturated soil–rock mixtures (SRMs) are of importance in practical engineering. In this paper, a numerical manifold model is presented for ...hydro-mechanical simulation of the saturated and semi-saturated SRMs. In the case of the semi-saturated SRMs, the model is based on the generalized Biot theory involving immiscible two-phase wetting and non-wetting fluids in deformable porous media. The wetting, non-wetting fluid pressure and skeleton displacement are chosen to be primary variables which are related to the wetting and non-wetting saturation and permeability by experimental relationships. For the mechanical problem, the material interfaces between soil and rock are deemed discontinuous by imposing a stick–slip contact constraint using an augmented Lagrange multiplier approach. For the hydraulic problem, the material interfaces are continuous for the fluid pressure and flux fields. Within the framework of the numerical manifold method (NMM), the discretized model including the interfacial discontinuities always can be established using the triangular mesh and the lumped mass representation is always available to increase computational efficiency. Besides the benchmark problems of saturated and semi-saturated porous media, two examples of soil–rock foundation and slope are performed to demonstrate the versatility and robustness of the model and to investigate the hydro-mechanical responses of the porous SRMs.
•A numerical manifold model is proposed for the semi-saturated porous SRMs.•The displacement discontinuity is modeled using a stick–slip contact model.•The effect of rock blocks on the dynamic responses of SRM is investigated.
In this paper, a high-efficiency structural topology optimization framework based on combination of numerical manifold method (NMM) and parameterized level set method (PLSM) is established. The NMM ...uses two cover systems to discretize the model, which can accurately represent the complex boundary of the design domain. A new numerical manifold element has been derived for the application of NMM in PLSM-based optimization. To obtain enhanced accuracy for structural analysis, a multilevel subdivision technique for numerical manifold element generation is presented, and the related numerical integration scheme coupled with the interpolation point selection strategy is provided. For the proposed topology optimization, the new update method of element stiffness matrix is exclusively formulated as well as the calculation method of volume fraction. Some representative structural optimization problems demonstrate that the proposed structural topology optimization method is very effective for two-dimensional (2D) and three-dimensional (3D) structural topology optimization problems.
In this paper, a novel quasi-smooth tetrahedral numerical manifold method (NMM) and its two-dimensional (2D) counterpart are proposed. A new topology optimization method is established by combining ...the quasi-smooth manifold element (QSME) with the parameterized level set method (PLSM). The QSME introduces an innovative displacement function characterized by high accuracy and high-order continuity, effectively addressing the “linear dependence” (LD) issue inherent in traditional high-order NMM. To integrate QSME and PLSM, the corresponding optimization formulations and sensitivity analyses are provided. In order to fully utilize advantages of this novel quasi-smooth NMM and the PLSM, an element subdivision technique based on model recognition is proposed to accurately capture the physical boundaries. Additionally, a volume fraction update method based on element refinement is proposed. Taking advantage of the characteristics of the PLSM, a structure visualization method based on the sign distance function is developed to accurately describe curve boundary. This method allows for precise visualization of optimized structures. This study verifies high efficiency of the QSME-based PLSM for minimum compliance topology optimization problems in both 2D and 3D structures. Some representative structural optimization examples are tested to demonstrate effectiveness of the proposed method in both 2D and 3D problems, especially in complex design domain.
Due to rock masses' nonlinear failure property, it is inappropriate to investigate the stability of rock slopes using the traditional SRM (strength reduction method) which is based on the linear MC ...(Mohr-Coulomb) failure criterion. To conduct 3D analysis (three dimensional) of rock slopes, we propose a 3D-NSRNMM (3D nonlinear strength reduction numerical manifold method) that is based on the nonlinear GHB (Generalized Hoek-Brown) failure criterion. To effectively implement the proposed 3D-NSRNMM, two methods are adopted to convert the GHB parameters into the average and instantaneous equivalent MC parameters. With the proposed 3D-NSRNMM, the influences of different types of equivalent MC parameters, and boundary conditions on rock slopes' stability are investigated. The numerical results assessed from the proposed 3D-NSRNMM indicate that: 1) boundary conditions will significantly influence the safety factor and failure mode of a rock slope obtained from 3D analysis; 2) the safety factor from two-dimensional analysis is more conservative compared with 3D analysis; 3) Furthermore, safety factors based on the instantaneous equivalent MC parameters are very close to those based on the average equivalent MC parameters, but 3D rock slopes' failure modes based on the two different types of equivalent MC parameters are a little different from each other.
•A 3D-NSRNMM that is on the basis of the nonlinear GHB criterion is proposed.•Two methods are adopted to convert GHB parameters into equivalent MC parameters.•Stability analyses of 3D rock slopes are investigated using the 3D-NSRNMM.•Boundary conditions significantly influence the safety factor and failure mode of 3D rock slopes.
By introducing the concept of mathematical cover and physical cover, the numerical manifold method (NMM) is able to solve continuous and discontinuous problems in a unified way. In this paper, the ...NMM is developed to analyze three dimensional (3D) fracture propagation. The maximum tensile stress criterion is implemented to determine whether the fracture will propagate and the direction of fracture propagation. Three benchmark problems are analyzed to validate the present algorithm and program. The numerical results replicate available experimental results and existing numerical results. The present algorithm and 3D NMM code are promising for 3D fracture propagation. They deserve to be further developed for the analysis of rock mechanic problems in which the initiation and propagation of multiple fractures, tensile and shear fractures, and fracture propagation under compressive loading are taken into account.
•Formulations of the Quad4-CNS (NMM) element for crack problems are presented.•The derivatives of Quad4-CNS (NMM) shape function are continuous at nodes.•Quad4-CNS (NMM) element is able to achieve ...continuous nodal stress without any smoothing operation.•Quad4-CNS (NMM) element is very suitable for solving crack propagation problems.•Comparing to extended quadrilateral element (XQ4), Quad4-CNS (NMM) element can obtain better results.
Formulations of a four-node quadrilateral element fitted to numerical manifold method (NMM) with continuous nodal stress called Quad4-CNS (NMM) element for two-dimensional (2D) crack analysis are presented. This Quad4-CNS (NMM) element can be considered as a development of the recently published four-node quadrilateral element with continuous nodal stress (Quad4-CNS). In contrast to the four-node iso-parametric quadrilateral element (Quad4), the Quad4-CNS element has higher order of global approximations, much better accuracy and continuous nodal stress. Moreover, it is free from the “linear dependence” which otherwise cripples many of the partition of unity (PU) based methods with high order global approximations. Due to the adoption of two cover systems, namely, the mathematical cover and physical cover, the NMM is capable of solving continuous and discontinuous problems in a unified way. The purpose of this paper is to synergize the advantages of both the Quad4-CNS element and the NMM to precisely model linear elastic fracture problems. A number of numerical examples indicate the accuracy and robustness of the present Quad4-CNS (NMM) element.
Micromechanical modeling of geomaterials is challenging because of the complex geometry of discontinuities and potentially large number of deformable material bodies that contact each other ...dynamically. In this study, we have developed a numerical approach for micromechanical analysis of deformable geomaterials with dynamic contacts. In our approach, we detect contacts among multiple blocks with arbitrary shapes, enforce different contact constraints for three different contact states of separated, bonded, and sliding, and iterate within each time step to ensure convergence of contact states. With these features, we are able to simulate the dynamic contact evolution at the microscale for realistic geomaterials having arbitrary shapes of grains and interfaces. We demonstrate the capability with several examples, including a rough fracture with different geometric surface asperity characteristics, settling of clay aggregates, compaction of a loosely packed sand, and failure of an intact marble sample. With our model, we are able to accurately analyze (1) large displacements and/or deformation, (2) the process of high stress accumulated at contact areas, (3) the failure of a mineral cemented rock samples under high stress, and (4) post-failure fragmentation. The analysis highlights the importance of accurately capturing (1) the sequential evolution of geomaterials responding to stress as motion, deformation, and high stress; (2) large geometric features outside the norms (such as large asperities and sharp corners) as such features can dominate the micromechanical behavior; and (3) different mechanical behavior between loosely packed and tightly packed granular systems.
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