To investigate the stability of a soil-rock-mixture (SRM) slope subjected to sequential excavation, two modifications are made to the numerical manifold method (NMM). One modification is the ...implementation of sequential excavation algorithms including an algorithm to find the excavated manifold elements and an algorithm named as “manifold element birth and death” to remove the excavated manifold elements in the excavation simulation. The other modification is the implementation of an improved shear strength reduction technique to evaluate the stability of a slope, as well as to obtain the factor of safety (FOS). In this technique, apart from the shear strength parameters, the Poisson's ratio υ is also adjusted for the purpose of eliminating the spurious plastic deformation that may happen in the deep areas of the slope. Two criterions including the NC criterion (Non-convergence criterion) and DPZ criterion (The criterion based on the distribution of plastic zones) are used to define the critical equilibrium state of the slope. With the improved NMM, three numerical examples including a homogeneous slope under one step of excavation, a slope under three steps of excavation and a SRM slope under two steps of excavation are solved. The simulation results show that: (1) the improved NMM is able to accurately simulate the excavation process of slopes, and predict the FOSs of slopes; (2) the FOSs based on the NC criterion are usually greater than or equal to those based on the DPZ criterion; (3) it is very difficult to form a slipping surface passing from the toe of the SRM slope to the top of the SRM slope; (4) rock blocks have some positive effects on the stability of a SRM slope.
•The NMM is used to simulate sequential excavation of a soil-rock-mixture (SRM) slope.•The sequential excavation algorithms are proposed to simulate the process of sequential excavation.•An improved shear strength reduction technique is adopted and implemented into the NMM.•Results obtained from NC criterion and DPZ criterion are compared.•The advantages of the NMM in discretization and accuracy are demonstrated.
In this study, the moving least squares based numerical manifold method (MLS-NMM) is firstly applied to discretize three-dimensional (3D) steady heat conduction problems of functionally graded ...materials (FGMs). In the 3D MLS-NMM, the influence domains of nodes in the moving least squares (MLS) are used as the mathematical patches to construct the mathematical cover (MC); while the shape functions of MLS-nodes as the weight functions subordinate to the MC. Compared with the traditional NMM using finite elements to form the MC, the cutting operations in generating the physical cover are unnecessary and the computation complexity is decreased significantly. Based on the Galerkin method, the discrete form of 3D heat conduction is derived. Finally, a series of numerical experiments concerning steady heat conduction problems are performed, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving steady heat conduction.
•Node influence domains in MLS are used to form mathematical cover in NMM.•The merits of combination of MLS and NMM are demonstrated.•The NMM model for analyzing the 3-dimensional heat conduction of FGMs is built.
•A new numerical manifold method is proposed based on the Hermitian interpolation and adoption of rectangular meshes.•C1 continuity is obtained in the proposed Hermitian numerical manifold method, ...and the convergence, accuracy and efficiency are considerably improved.•Strains at mathematical nodes inside the physical domain are directly available without extra postprocessing.•The linear dependence problems in most high-order numerical manifold methods are circumvented.
Numerous approaches have been proposed to enhance the accuracy and convergence of the numerical manifold method (NMM) in recent years, but most, if not all, of these approaches cannot ensure C1 continuity. Hermitian interpolation is an effective approach for obtaining high-order approximations. However, the requirement of rectangular meshes hinders the application of this approach in the finite element method. Taking advantage of the freedom in meshing in NMM, Hermitian interpolation is incorporated into NMM to obtain the C1 approximation. In contrast to the common high-order NMM, the Hermitian NMM (HNMM) improves the accuracy and convergence without causing the linear dependence problem. Moreover, the degrees of freedom (DOFs) of the mathematical nodes inside the physical domain have physical meanings, and the strains at nodes can be obtained directly without the need for extra postprocessing. The proposed HNMM is verified by solving numerous benchmark linear elastic problems, and the results are compared against those of linear and cubic Lagrangian NMMs. The numerical solutions for these examples confirm the remarkable superiority of the HNMM over the Lagrangian NMMs in terms of accuracy, convergence and efficiency.
Soil and rock mixture (SRM) is ubiquitous in nature and also an excellent engineering fill material. Accurate evaluation of mechanical characteristics of SRM is of vital importance in the analysis of ...stability and deformation. Unfortunately, the presence of mega blocks in SRM makes it impossible to grasp the real attributes of full graded SRM using indoor or in-situ experiments. Consequently, numerical methods, especially those three-dimensional methods, become crucial to the objective evaluation of SRM mechanical properties. However, even the most powerful and sophisticated mesh generators become too weak to produce even low-quality meshes of real SRM samples. Accordingly, the numerical manifold method (NMM) is an excellent tool for this purpose since it successfully throws off shackles of mesh generation. In this study, we first produce an SRM sample according to the given size distribution and content of rock blocks. Then, we employ a tetrahedron mesh to act as the mathematical cover of the SRM sample; the mesh does not match the SRM sample domain. Next, we utilize Boolean operations in Ansys to generate the physical cover of the SRM sample and the robustness of the program is hence ensured to the greatest extent. As a preliminary application, finally we demonstrate the permeability evaluation of SRM, indicating that even for those SRM samples that cannot be meshed with the most powerful mesh generator, the proposed procedure can easily arrive at their solutions.
•A procedure is proposed for modeling soil-rock mixtures (SRM) based on the size distribution and content of rock blocks in SRM.•A method is developed for generating physical covers based on finite element meshes that are used by the numerical manifold method.•The main obstacles of finite element method in solving SRM problems are pointed out.
•Accuracy decay due to strain smoothing for crack tip enrichment in numerical manifold method (NMM) is shown.•Hybrid edge based smoothed NMM is proposed for accurate resolution of crack ...problems.•Compatible strain in enriched zones and smoothed strain in other areas are merged.•Local approximation based mass lumping is used for explicit dynamic procedures.
Modeling and prediction of dynamic behaviors of cracked bodies are vital to the safety evaluation of structures. The numerical manifold method (NMM) has achieved considerable success in crack analysis as it provides a powerful representation of complex discontinuities. Strain smoothing has been incorporated into enriched methods such as NMM to improve their accuracy, efficiency, and tolerance to mesh distortion. However, conventional strain smoothing cannot reproduce the large spatial variance in the strain surrounding crack tips. Consequently, the accuracy is reduced when it is applied to crack-tip enrichment bases. High-fidelity representation of crack-tip fields can be achieved by crack-tip enrichment. Therefore, it is desirable to combine compatible strain for the enriched areas with edge-based smoothed strain for the remaining areas. When explicit time integration is employed, it is beneficial to use a lumped mass matrix because it provides a larger critical time step and a straightforward solution of the linear algebraic equation. A local-approximation-based mass lumping strategy is used to obtain a block-diagonal mass matrix with small blocks. Finally, the proposed methodology is applied to various static and dynamic crack analyses, and highly accurate and stable results are obtained.
•NMM is enhanced to simulate crack initiation, propagation and coalescence.•Four typical brittle cracking examples under dynamic loading are well simulated.•Mesh size, wave propagation, crack ...propagation velocity, etc. are analyzed.•Effects of pre-existing geometrical defects (hole and notch) are analyzed.
Crack development in brittle materials under dynamic loading is widely involved in engineering, in which crack initiation, propagation and coalescence are typical phenomena. The numerical manifold method (NMM) is a unified continuous-discontinuous numerical method employing two cover systems, namely, mathematical covers and physical covers, which encounters no difficulty in the numerical representation of continua and complex discontinuities within one framework. In the present work, NMM is developed for the simulation of crack initiation, propagation and coalescence problems in brittle materials under dynamic loading based on the tensile strength criterion and the Mohr-Coulomb strength criterion for tensile and shear cracking, respectively. Four typical examples including the splitting of a rock bar, the Kalthoff-Winkler experiment, the cracking in tensile loaded pre-notched rectangular plates and the double-hole blasting of rectangular plates are simulated. The numerically derived crack development results are compared with corresponding theoretical or experimental results. The mesh size sensitivity is discussed for the first two examples; the dynamic cracking mechanism in the rock bar example is investigated along with the stress wave propagation analysis; the influence of the initial crack and hole locations on the crack path in the tensile loaded pre-notched example as well as the effect of the guiding notch in the double-hole blasting example are studied. Results indicate that the crack initiation, propagation and coalescence in brittle materials under dynamic loading are quite satisfactorily reproduced by NMM.
Since the advent of finite element methods, the dynamic response analysis of solids and structures follows such a route without exception. Firstly the spatial discretization is carried out and the ...system of second order ordinary differential equations with the degrees of freedom as the unknown functions of time is derived, which is called the semi-discrete scheme. Then the temporal discretization is performed to the system of ordinary differential equations and the system of algebraic equations, referred to as the fully-discrete scheme, is obtained. This route has been working well for most problems, where, the meshes deform continuously and, in all the time steps, all the degrees of freedom are valid and the number of them keeps invariant. In the simulation of crack propagation, however, even the number of degrees of freedom varies with crack propagation and those degrees of freedom associated with crack tips become meaningless after the crack tips move away. While this causes no difficulties in linear static solutions, it is not readily handled in time-dependent solutions, leading to the transfer issue of degrees of freedom. Opposite to the conventional order of discretization, in this study the temporal discretization is put prior to the spatial discretization. In this way, all the degrees of freedom are valid only within the current time step. The transfer issue of degrees of freedom is accordingly resolved elegantly. The implementation of the proposed procedure is in the framework of the numerical manifold method, illustrated by some typical examples, where compressed and sheared cracks are involved with frictional contact.
•An improved NMM with multiple layers of mathematical cover systems is proposed for the soil-rock-mixture(SRM) slopes.•An improved shear strength reduction technique is adopted and implemented into ...the improved NMM.•The FOSs based on the NC criterion is greater than those based on the DPZ criterion for SRM slopes.•The advantages of the improved NMM in discretization and accuracy are demonstrated.
In order to carry out stability analysis of soil-rock-mixture(SRM) slopes, an improved numerical manifold method (NMM) with multiple layers of mathematical cover systems is proposed. For SRM problems, the computational cost of the improved NMM is smaller than that of the traditional NMM. Besides, an improved shear strength reduction technique(ISSRT) which can eliminate spurious plastic deformation of the slopes is implemented. Based on the improved NMM with multiple layers of mathematical cover systems, the stability of three slopes, namely, a homogeneous slope and two SRM slopes with different contents of rock blocks is analyzed. The numerical results about the three slopes indicate that: 1) the proposed numerical model can obtain the FOS of a slope with high accuracy; 2) the bigger the content of rock blocks, the larger the FOS of the SRM slope will be; 3) due to the existence of rock blocks, the failure mode of a SRM slope is different from that of a homogeneous soil slope.
Numerical manifold method (NMM) has shown its ability to solve continuous and discontinuous deformation problems in a unified framework. However, due to the complexity of geometry description and the ...absence of a reliable 3D contact algorithm, the development of 3D-NMM still has a long-time challenge. In this study, an open-source software named MEG3D to generate the numerical model for 3D-NMM is developed. The MEG3D is a fast, light, and user-friendly interactive software to identify geometry, generate joint networks, generate structured finite mathematical mesh, cut blocks, and generate numerical manifold elements (MEs). In this software, a new C++ programming strategy with high modularization and good portability, and a novel data storage format and data structure were used. Based on the OpenGL library, real-time visualization and interactive interfaces were built. Examples containing curve block cutting, discrete fracture networks (DFN) model, soil-rock mixture slopes, complex shaped geometry, etc. have indicated that the software is robust, efficient, and user-friendly. This software can also generate models for the discrete element method (DEM), discontinuous deformation analysis (DDA), etc. Therefore, the MEG3D can be used as a general pre-processing program for 3D-NMM and other block-based numerical methods.
•A mass lumping scheme suitable for the second-order numerical manifold method is proposed.•The second-order numerical manifold method is further applied for dynamic analysis of structures.•Accuracy ...of the proposed mass lumping scheme is demonstrated.
The numerical manifold method (NMM) has been employed to solve many types of engineering problems. To improve accuracy of the traditional NMM, a second-order NMM with six-node triangular mesh was recently proposed. This second-order NMM is further applied for structural dynamic problems in this study. In order to reduce time consumption in solving large scale simultaneous algebraic equations, a mass lumping scheme which is suitable for the second-order NMM is proposed. A series of numerical examples show that natural frequencies of structures assessed from the proposed lumped mass matrix (LMM) are more accurate than those from the consistent mass matrix (CMM). In addition, even in the implicit time integration scheme to predict dynamic responses of structures, the CMM can be replaced by the proposed LMM without losing accuracy.
