•Formulations of a high-order numerical manifold method (HONMM) for crack problems are presented.•The derivatives of HONMM shape function are continuous among internal element edges and nodes.•The ...proposed HONMM is very suitable for solving crack propagation problems.•High accuracy for crack analysis can be obtained from the HONMM.
Recent attempts to solve solid mechanical problems using the numerical manifold method (NMM) are very fruitful. In the present work, a high-order numerical manifold method (HONMM) which is able to obtain continuous stress/strain field is proposed. By employing the same discretized model as the traditional NMM (TNMM), the proposed HONMM can yield much better accuracy without increasing the number of degrees of freedom (DOFs), and obtain continuous stress/strain field without recourse any stress smoothing operation in the post-processing stage. In addition, the “linear dependence” (LD) issue does not exist in the HONMM, and traditional equation solvers can be employed to solve the simultaneous algebraic equations. A number of numerical examples including four linear elastic continuous problems and five cracked problems are solved with the proposed method. The results show that the proposed HONMM performs much better than the TNMM.
A three-node triangular element fitted to the numerical manifold method with continuous nodal stress called Trig3-CNS (NMM) element for accurately modeling two-dimensional linear elastic fracture ...problems is presented. By adopting two cover systems, namely, the mathematical cover and physical cover, the numerical manifold method (NMM) could easily solve continuous and discontinuous problems in a unified way. In contrast to the three-node triangular element (Trig3), the Trig3-CNS element has higher order of 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 based methods with high order approximations. The purpose of the present work is to synergize the advantages of both the recently developed Trig3-CNS element and the NMM to precisely model two-dimensional linear elastic fracture problems. A number of numerical examples indicate the accuracy and robustness of the present Trig3-CNS (NMM) element.
Diametral compression of disc-like specimens with pre-existing cracks is widely used in the study of rock fracturing failures. In the present study, a unified numerical method, namely, numerical ...manifold method (NMM), is further enhanced for better simulations of rock cracking by effectively considering the friction on the new crack surfaces. Simulations of disc and semi-disc specimens with a single pre-existing crack of different inclinations prove that the improved NMM can reproduce the crack development path which agrees better with the theoretical and experimental results. Thereafter, disc specimens with vertically or horizontally distributed parallel pre-existing cracks of different numbers and distances are simulated, respectively. The derived fracturing failure patterns of the discs match the corresponding results by other numerical methods or physical experiments well. The NMM simulations also indicate that the increase of the pre-existing crack number will reduce the strength of the disc, but the number of the vertically distributed pre-existing cracks shows a much more obvious influence. Meanwhile, the increase of the pre-existing crack distance will increase the strength of the disc with horizontally distributed pre-existing cracks, while the pre-existing crack distance shows a nonmonotonic influence on the disc strength for discs with vertically distributed cracks.
AbstractThis paper presents an investigation of wave propagation through microfractured rock mass. The effects of microfracture on wave propagation were observed by a series of scanning electron ...microscope (SEM) tests and wave-velocity measurements. A spectrum analysis was introduced to analyze the attenuation coefficient and the wave number of seismic waves propagating through microfractured rock mass. The effects of fracture length, fracture quantity, and frequency of incident wave on the attenuation rate, effective velocity, attenuation coefficient, and wave number were numerically simulated and discussed. The results demonstrate that the attenuation rate, effective velocity, attenuation coefficient, and wave number are significantly influenced by the geometrical parameters of microfracture (e.g., length and quantity). In addition, the numerical manifold method (NMM) was validated as a method for investigating the dynamic behavior of heavy microfractured rock mass efficiently.
Grouting is a typical HM (hydro-mechanical) coupling process between slurry and rock and widely used in underground engineering to reinforce the strength of fractured rock mass. In present work, ...based on numerical manifold method (NMM), an NMM-HM grouting model is proposed to investigate slurry flowing and analyze grouting efficiency. In this NMM-HM grouting model, the slurry flows in fractures and grouting pressure can cause fracture aperture variation and fracture propagation. Then, two numerical examples for central embedded crack and hydraulic fracturing are adopted to validate the presented model. Furthermore, grouting simulations are conducted on a multi-crack and a roadway with fracture networks. Results show that Grouting pressure in fractures can increase fracture aperture, which is beneficial for grouting efficiency. However, fluid pressure could also cause closure of neighboring fractures or even fracture propagation and failure of rock mass, which will weaken grouting effectiveness. Therefore, the coupled HM impacts on grouting efficiency and grouting pressure should be seriously considered and optimized to achieve the expected grouting performance in practice.
An improved numerical manifold method (INMM) with multiple layers of mathematical cover (MC) systems is employed for a sequential excavation analysis of SRM (soil-rock-mixture) slopes. For the ...purpose of obtaining the FOS (factor of safety) of a SRM slope, an enhanced SSRT (shear strength reduction technique) is used in the INMM. Furthermore, two algorithms, i.e., an algorithm to identify the excavated MEs (manifold elements) and an algorithm to remove the excavated MEs, are incorporated into the INMM to study the effects of excavation of the SRM slopes. With the INMM, two examples, including a soil slope and two SRM slopes subjected to sequential excavation construction, are investigated. The simulation results show that 1) the excavation processes, as well as the FOSs of the slopes, can be accurately predicted with the INMM; 2) the higher the content of rock blocks, the larger the FOS of the SRM slope will be; and 3) the FOSs assessed from the INMM are the same as those from the traditional NMM (TNMM), but the memory consumption and computational cost of the INMM are smaller.
•An improved NMM (INMM) is proposed for the sequential excavation analysis of the SRM slopes.•The excavation algorithms are implemented to simulate the sequential excavation processes.•The improved shear strength reduction technique is adopted and implemented into the INMM.•Compared to the traditional NMM, computational cost of the INMM is smaller.•The advantages of the INMM in discretization and accuracy are demonstrated.
•The NMM is used to conduct stability analysis of soil-rock-mixture (SRM) slopes.•An improved shear strength reduction technique is implemented into the NMM.•Influence of rock blocks on the stability ...of SRM slopes can be captured by the NMM.•Failure pattern of SRM slopes is different from that of a homogeneous slope.•The advantages of the NMM in discretization and accuracy are demonstrated.
Recently, fruitful results of geotechnical problems solved by the numerical manifold method (NMM) have been gained. In the present work, the shear strength reduction technique (SSRT) together with the ϕ–υ inequality is implemented into the NMM to carry out stability analysis of soil-rock-mixture (SRM) slopes. Several examples are used to validate the correctness and effectiveness of the developed numerical model. The results from these examples indicate that the proposed model can predict the factor of safety (FOS) of slopes with high accuracy, and capture the failure patterns of slopes. In addition, the influence of rock blocks on the stability of SRM slopes can be captured.
This paper presents a multi-temporal series high-accuracy numerical manifold method (NMM) for fracture analysis of solid materials under highly time-dependent thermal loadings. In the present method, ...the framework of NMM is used to perform space discretization of the thermo-mechanical coupling system, due to its advantages in accurately calculating field variables and describing discontinuity. An explicit direct integration scheme based on the precise time step integration method (or PTSIM scheme for short) is constructed for discretization in the time domain. In the present PTSIM scheme, the time-dependent thermal loadings are expressed by means of the basic function in a polynomial form. Highly accurate calculation of the response matrix for the load item is achieved by introducing the exponential transfer matrix. The results of stability, error and convergence analyses indicate that the proposed method is unconditionally stable and convergent. The superior advantages of present method are firstly verified by one example for transient thermal analysis, then four numerical examples with different time-dependent thermal boundary conditions and crack configurations are used for fracture analysis. The results show that the present method is unconditionally stable and time-independent, and high-accuracy can be achieved even for highly time-dependent thermal loadings and large time-step sizes.
The numerical manifold method (NMM), a Galerkin-type numerical method, has been successful in the solution of problems with finite definition domains, yet it has never been applied to problems with ...unbounded domains, or exterior problems. This study aims to fill the big gap by constructing infinite patches, together with the finite patches, to cover the unbounded domain. The local approximations of infinite patches can take the asymptotic estimations of the solutions at infinity, which are available for all those well-established boundary value problems. Compared with the infinite element methods in the finite element method (FEM), the construction of the trial functions by NMM is more elegant in theory and more systematical in methodology, resulting in more accurate solutions. Some typical examples in potential and half-space elasticity problems are investigated to illustrate the applicability and accuracy of the proposed method.
•The infinite patches and their weight functions in a cover are constructed.•Discrepancy between finite element method and numerical manifold method in constructing trial functions is ascertained.•Excellence of numerical manifold method in the solution of exterior problems is exhibited.
The numerical manifold method (NMM) has been widely utilized to solve problems involving complicated boundaries, cracks, and interfaces. Recently, strain or gradient smoothing techniques have been ...incorporated into NMM to improve its performance. The resulting smoothed NMMs (SNMMs) normally possess enhanced numerical properties, for example, higher accuracy, convergence, and efficiency. A challenging issue rooted in NMM and other enhanced finite element methods using unfitted meshes is the ill‐conditioning induced by extremely small cut element. In this study, we investigate the behaviors of SNMMs in the case of small cut element. It is demonstrated that the cut‐induced ill‐conditionings also exist for two types of SNMMs, namely, edge‐based SNMM and physical‐patch‐based SNMM. Furthermore, a preconditioner based on the normalization of basis functions is proposed to resolve the ill‐conditioning. Several benchmark problems demonstrate the performances of SNMMs regarding accuracy, convergence, and stability.
