The efficiency of solving equations plays an important role in implicit-scheme discontinuous deformation analysis (DDA). A systematic investigation of six iterative methods, namely, symmetric successive over relaxation (SSOR), Jacobi (J), conjugate gradient (CG), and three preconditioned CG methods (ie, J-PCG, block J-PCG [BJ-PCG], and SSOR-PCG), for solving equations in threedimensional sphere DDA (SDDA) is conducted in this paper. Firstly, simultaneous equations of the SDDA and iterative formats of the six solvers are presented. Secondly, serial and OpenMP-based parallel computing numerical tests are done on a 16-core PC, the result of which shows that (a) for serial computing, the efficiency of the solvers is in this order: SSOR-PCG > BJ-PCG > J-PCG > SSOR>J > CG, while for parallel computing, BJ-PCG is the best solver; and (b) CG is not only the most sensitive to the ill-condition of the equations but also the most time consuming under both serial and parallel computing. Thirdly, to estimate the effects of equation solvers acting on SDDA computations, an application example with 10 000 spheres and 200 000 calculation steps is simulated on this 16-core PC using serial and parallel computing. The result shows that SSOR-PCG is about six times faster than CG for serial computing, while BJ-PCG is about four times faster than CG for parallel computing. On the other hand, the whole computation time using BJ-PCG for parallel computing is 3.37 hours (ie, 0.061 s per step), which is about 36 times faster than CG for serial computing. Finally, some suggestions are given based on this investigation result.
K E Y W O R D Shigh efficiency, iterative method, linear equation solver, three-dimensional discontinuous deformation analysis, sphere DDA
| INTRODUCTIONBoth the discontinuous deformation analysis (DDA) method 1 and discrete element method (DEM) 2 are proposed to simulate the mechanical behaviors of discrete block systems. The DDA and DEM have extensive prospects in engineering application since they can automatically trace and simulate block rotation, fracture opening, and complete detachments, which are often included in real jointed rock masses failure process. 3,4 Compared with the DEM, the DDA has more rigorous mathematical theory and higher computational accuracy. 5,6 Although the DDA is relatively young, it has been widely applied in various aspects of geotechnical engineering, such as tunnel engineering, 7 rock hydraulic fracturing, 8,9 rock burst analysis, 10 and landslide. 11 However, the computational efficiency of the DDA is much lower than that in the DEM, although there has been great progress within DDA researches over the past three decades. 12 The main reason for this is that the DEM uses an explicit time integration scheme and the second law of Newton is used to calculate the displacement of each block independently, while the DDA often uses an implicit solution method in which simultaneous equilibrium equations are assembled by applying the minimum potential energy principle and solved in each calcul...
