The dynamics relaxation algorithm is used to solve quasi-static problem in Numerical Manifold Method (NMM). However, the mechanism of energy dissipation of NMM is unclear. In this study, the viscous type damping is adopted to absorb the kinetic energy caused by the oscillation of system. The equilibrium equations containing damping term are derived by minimizing the total potential energy. In order to improve the calculation efficiencies of NMM, convergence criterion including displacement and acceleration criterion are defined to determine whether the system reach new equilibrium. The example of strip foundation settlement is calculated to demonstrate the capability of the NMM with new extensions. The numerical results show that the improved NMM has high accuracy and good convergence performance.
The Numerical Manifold Method (NMM) [1–3], proposed by Dr. Shi Genhua in 1990s, has been known as an analysis method that unifies both the method of the finite element method (FEM) and discontinuous deformation analysis (DDA). Using a different combination of cover, the approximate function in the problem domain is created by linking the covers together by the weight function. Compared with those methods such as the analytical method, finite element and discontinuous deformation analysis, NMM can solve complex problems in comprehensive aspects. Since NMM was proposed, a number of scholars improve it from different aspects [4–10], and this method gradually developed into one of the main numerical method in the area of Rock and Soil Mechanics.
Nevertheless, the theoretical system and analytical methods needed to be further improved. At present, NMM control the calculation process by entering the specified calculation steps instead of giving a convergence criterion of calculation. If the input number of time steps is less, the system is likely to be unbalanced and resulting in subsequent calculations error. And if the number of time steps is too large, the computation will be inefficient. In addition, since dynamic relaxation method is employed in NMM for solving quasi-static problems, the inertial force term is inevitably introduced in the calculation. If kinetic energy could not be effectively absorbed, the system would oscillate and would not quickly converge to a stable state, thereby reducing computational efficiency and the reliability of computational results.
In this paper, two basic problems about energy dissipation and convergence criteria, which are encountered in applications of the NMM in practical engineering, are discussed and the corresponding solution and calculation format are given as well.