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Highly efficient iterative methods for solving linear equations of three‐dimensional sphere discontinuous deformation analysis
Author(s) -
Huang GangHai,
Xu YuanZhen,
Yi XiongWei,
Xia Ming,
Jiao YuYong,
Zhang Shu
Publication year - 2020
Publication title -
international journal for numerical and analytical methods in geomechanics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.419
H-Index - 91
eISSN - 1096-9853
pISSN - 0363-9061
DOI - 10.1002/nag.3062
Subject(s) - solver , conjugate gradient method , iterative method , computer science , computation , parallel computing , mathematics , relaxation (psychology) , discontinuous deformation analysis , computational science , block (permutation group theory) , linear equation , algorithm , mathematical optimization , mathematical analysis , finite element method , geometry , psychology , social psychology , physics , thermodynamics
Summary 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 three‐dimensional 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.