Traditional Numerical Manifold Method Research Articles

A new procedure for locating free surfaces of complex unconfined seepage problems using fixed meshes

The main challenge in the analysis of unconfined seepage flow is that the position of free surfaces is unknown a priori, which needs to be determined through a series of iterative processes. The numerical manifold method (NMM) is a promising method which uses a dual cover system consisting of both mathematical and physical covers. Compared with those traditional methods, NMM is characterized by meshing convenience, approximation accuracy, and being capable of coping with free boundary value problems. Unlike the traditional NMM where the material interface participates in cutting the mathematical cover while forming the physical cover, the physical patches in this study can contain the material interface. The new weight functions for such physical patches are constructed using the refraction law, followed by the application to the analysis of unconfined seepage flow problems. By comparing with analytical or reference solutions of some classic examples, it is validated that the proposed method can accurately locate the free surface, demonstrating its accuracy and convenience in solving unconfined seepage problems.

Bending of Nonconforming Thin Plates Based on the Mixed-Order Manifold Method with Background Cells for Integration

As it is very difficult to construct conforming plate elements and the solutions achieved with conforming elements yield inferior accuracy to those achieved with nonconforming elements on many occasions, nonconforming elements, especially Adini’s element (ACM element), are often recommended for practical usage. However, the convergence, good numerical accuracy, and high computing efficiency of ACM element with irregular physical boundaries cannot be achieved using either the finite element method (FEM) or the numerical manifold method (NMM). The mixed-order NMM with background cells for integration was developed to analyze the bending of nonconforming thin plates with irregular physical boundaries. Regular meshes were selected to improve the convergence performance; background cells were used to improve the integration accuracy without increasing the degrees of freedom, retaining the efficiency as well; the mixed-order local displacement function was taken to improve the interpolation accuracy. With the penalized formulation fitted to the NMM for Kirchhoff’s thin plate bending, a new scheme was proposed to deal with irregular domain boundaries. Based on the present computational framework, comparisons with other studies were performed by taking several typical examples. The results indicated that the solutions achieved with the proposed NMM rapidly converged to the analytical solutions and their accuracy was vastly superior to that achieved with the FEM and the traditional NMM.

Mathematical cover refinement of the numerical manifold method for the stability analysis of a soil-rock-mixture slope

An improved NMM (Numerical Manifold Method) with mathematical cover refinement (RNMM) is adopted to investigate the stability of a SRM (soil-rock-mixture) slope. In the RNMM, refined mathematical meshes are deployed at specified domains where stress concentration may occur. Compared to the TNMM (Traditional NMM), the computational efficiency of the RNMM is better for slope stability problems. Besides, the recently proposed scheme called MLC (mathematical cover systems with multiple layers) is adopted to further improve the computational efficiency of the RNMM. Based on the proposed RNMM+MLC (RNMM with MLC), stability analyses about two typical examples including a soil slope and a SRM slope are investigated. The numerical results demonstrated that: 1) The FOS (factor of safety) based on the proposed RNMM is better than that based on the TNMM; 2) The FOS based on the RNMM + MLC is almost the same to that based on the RNMM, but the computational efficiency of the RNMM + MLC is better than that of the RNMM; 3) The distribution mode of plastic zones in a SRM slope differs a lot from a soil slope.

An improved numerical manifold method with multiple layers of mathematical cover systems for the stability analysis of soil-rock-mixture slopes

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.

A high-order numerical manifold method with continuous stress/strain field

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.

Investigation of stress wave induced cracking behavior of underground rock mass by the numerical manifold method

The present study investigated the coupling effects of cracking and wave propagation on the underground rock mass by the numerical manifold method (NMM). The traditional NMM was developed for the simulation of dynamic cracking behavior of rock mass. One-dimensional wave propagation was utilized to validate the present code. Subsequently, the transient stress field in the rock mass with an inclined crack was investigated. Finally, two cases of underground caverns with symmetrical or unsymmetrical discontinuities under stress wave were simulated to validate the application potentials. The results show that the transient stress field is not only influenced by the stress concentration around the pre-existing crack, but also influenced by the characteristics of stress wave, e.g. reflection, transmission and diffraction, which are different from the static situation. The results also show that the cracking significantly affects the stress wave duration and amplitude. On the other hand, the stress wave induced cracking depends on the wave propagation distance significantly. Moreover, the NMM can be used to simulate the cracking behavior of underground rock mass under stress wave efficiently.

A mass lumping scheme for the second-order numerical manifold method

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.

An edge-based smoothed numerical manifold method and its application to static, free and forced vibration analyses

As a partition of unity (PU) method, the numerical manifold method (NMM) is able to solve continuous and discontinuous problems in a unified manner. However, the accuracy obtained through the traditional NMM (t-NMM) is far from satisfactory, because the strain distribution is constant in each manifold element. To improve the performance of the t-NMM without adding extra nodes or DOFs, the edge-based smoothing technique, which has been applied in the edge-based finite element method (ES-FEM), is incorporated into the t-NMM, resulting in an edge-based smoothed numerical manifold method (ES-NMM). Formulation of the ES-NMM for static, free and forced vibration analyses are presented in great detail. The numerical examples of static, free and forced vibration problems show that the ES-NMM possesses several advantages over the t-NMM, including the close-to-exact stiffness, superconvergence, and ultra-accuracy.

Coupled Time Domain Integration Algorithm for Numerical Manifold Method

AbstractA temporal coupled explicit–implicit time integration algorithm is proposed to improve the computational efficiency of the numerical manifold method (NMM) for seismic stability analysis of rock slope. It includes a coupled time integration scheme, a phase transfer criterion, and an associated contact algorithm. To calibrate the proposed algorithm, a block sliding along a slope under seismic excitation and a block rocking under half-sine pulse shaking are simulated. The traditional limit equilibrium method (LEM) for slope stability analysis determines the factor of safety (FoS) without considering the time-dependent effect. The developed algorithm is able to simulate jointed rock slope seismic stability taking advantage of the NMM for continuous and discontinuous deformation analysis. An open-pit mine slope excited by the El Centro earthquake wave is studied. The simulated results are in good agreement with the results based on the traditional NMM, whereas the computational efficiency is improved s...

An explicit time integration scheme of numerical manifold method

The traditional numerical manifold method (NMM) has the advantage of simulating a continuum and a discontinuum in a unified framework based on a dual cover system. However, since an implicit time integration algorithm is used, the computational efficiency of the original NMM is very low, especially when more contacts are involved. The present study proposes an explicit version of the NMM. Since a lumped mass matrix is used for the manifold element, the accelerations by the corresponding physical covers can be solved explicitly without forming a global stiffness matrix. The open–close iteration is still applied to ensure computational accuracy. The developed method is first validated by two examples, and a highly fractured rock slope is subsequently simulated. Results show that the computational efficiency of the proposed explicit NMM has been significantly improved without losing the accuracy. The explicit NMM is more suitable for large-scale rock mass stability analysis and it deserves to be further developed for engineering computations in rock engineering.

Modelling P-wave transmission across rock fractures by particle manifold method (PMM)

In the study of wave attenuation across fractured rock masses different methods are applied to deal with various properties of rock fractures. The particle manifold method (PMM) is a new numerical method for coupled continuum-discontinuum analysis. It simplifies the geometrical operation in the traditional numerical manifold method (NMM) by adopting a unique particle representation of the physical domain. The contact description in the PMM is explicit. In this paper, the PMM is used to model P-wave transmission across rock fractures with different properties. Both continuous and discontinuous models are implemented into the PMM. By comparisons with the theoretical solutions, the PMM is validated for modelling wave transmission across rock fractures treated continuously and discontinuously.