This paper introduced a parallel implementation of numerical manifold method based on multiprocessor platforms. First, the computing performance for a class of rock engineering problems was analyzed. ...For solving simultaneous equations is the most time consuming, we choose parallelized Jacobi's iterative method to speed up the computing process. The implementation of parallel Jacobi's iterative method with OpenMP was introduced. A series of experiments shows that our parallel algorithm is an effective way to improve computing performance for such class of engineering problems.
The inherent complexities in the mechanical behavior of rock masses come from the discontinuous nature of rock masses, which exist in many forms such as fissures, cleavages, beddings, joints, and ...even faults. The scale effects of jointed rock masses, rock fracture propagation, and the effects of groundwater are postulated to be paradigmatic components of these complexities, and a proper incorporation of these components into an analysis is crucial to construction and design involving jointed rock masses. This study presents the development of a unified numerical framework based on the numerical manifold method (NMM) for modeling these three complexities in jointed rock masses. In this study, the modeling of scale effects was tackled by a proposed two-scale approach through the use of the primary joint set and the secondary joint set. The primary joint set, which has wider joint spacing, was modeled as physical discontinuities to preserve the rock mass kinematics, while the secondary joint set, which has narrower joint spacing, was modeled as an equivalent continuum. The modeling of fracture propagation was based on the theory of linear elastic fracture mechanics. The stress intensity factors were computed by both displacement-based and energy-based methods. The maximum stress criterion was employed for fracture propagation. The modeling of fracture propagation was accomplished within the NMM by merely changing the physical mesh to describe the evolving cracks. Furthermore, the modeling of the effects of groundwater on joints was introduced by considering water as pressure acting on the joint surface. The methodology proposed in this study has been examined through the calculation of a number of numerical examples and comparison with available analytical and experimental results. Application examples were also provided to demonstrate the applicability of the developed numerical framework. The results obtained show that: (1) the proposed two-scale concept has the potential for describing very complex behavior of jointed rock masses and may be employed in assessing large-scale rock mass strength and the problems of slope stability with many joints; (2) the study explores the simplicity of modeling fracture propagation by merely changing the physical mesh to describe evolving cracks, which alleviates the difficulty of the requirement of the spatial discretization that accommodates the changing topology of a problem domain; and (3) the ability to capture an entire process of a slope failure evolution is demonstrated from an initial phase of a tensile crack on a slope surface, to the intermediate phase of forming a failure surface, and to the final phase of the post-failure of a failed soil block sliding down the slope.
The numerical manifold method (NMM) provides a unified approach to address continuum-discontinuum problems in geotechnical engineering. Owing to the dynamic nature of its governing equations, the NMM ...should be suitable for modelling dynamic problems such as those associated to earthquakes. However, due to the current limitations in far-field boundary conditions implemented in the original NMM formulation, NMM has not been used to carry out seismic response analysis. In the present study, three new boundary conditions have been developed to extend the capability of the NMM to conduct seismic response analysis: (1) the classical viscous boundary condition, which allows for the absorption of the seismic wave energy at the boundaries, based on the viscous boundaries, and the seismic motion input method is also proposed; (2) the free field boundary condition, which captures the free field motion and absorption of the reflected waves at the sides of the model. The algorithms to generate free field mesh and its coupling calculations with the main mesh are also presented; (3) the static-dynamic unified boundary, which models the transition from fixed boundary condition in static state to free field boundary condition in seismic state, thus ensuring the accuracy and consistency of the numerical simulation. Finally, five numerical examples are shown to validate the proposed methods. The numerical results indicate that the improved NMM can be successfully adopted for seismic response analysis.
AbstractThe numerical manifold method (NMM) is suitable for the solution of both continuous and discontinuous problems in geotechnical engineering. In the conventional NMM, the contact between blocks ...is treated with the open-close iteration, which needs to fix or remove spurious springs between two blocks in contact and to assume properly the normal stiffness and the tangential stiffness (the penalty parameters). Unreasonable values of stiffness would result in numerical problems. To avoid the penalty parameters, contacts are treated in a direct way in which contact forces are primal variables. Numerical examples have confirmed the correctness and feasibility of the proposed procedure.
