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.

中文翻译：

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求解三维球面不连续变形分析线性方程组的高效迭代方法

求解方程的效率在隐式不连续变形分析（DDA）中起着重要作用。系统研究了六种迭代方法，即对称连续松弛（SSOR），雅可比（J），共轭梯度（CG）和三种预处理CG方法（即J‐PCG，块J‐PCG [BJ‐PCG] ，以及SSOR-PCG），用于解决三维球体DDA（SDDA）中的方程。首先，给出了SDDA的联立方程和六个求解器的迭代格式。其次，在16核PC上进行了基于串行和基于OpenMP的并行计算数值测试，结果表明（a）对于串行计算，求解器的效率按以下顺序排列：SSOR-PCG> BJ-PCG > J‐PCG> SSOR> J> CG，而对于并行计算，BJ‐PCG是最好的求解器；（b）CG不仅对方程的病态最敏感，而且在串行和并行计算下都是最耗时的。第三，为了估计方程求解器对SDDA计算的影响，在这台16核PC上使用串行和并行计算模拟了一个具有10,000个球体和20万个计算步骤的应用示例。结果表明，对于串行计算，SSOR-PCG约比CG快六倍，而对于并行计算，BJ-PCG约比CG快四倍。另一方面，使用BJ-PCG进行并行计算的整个计算时间为3.37小时（即每步0.061 s），这比串行计算的CG快36倍。最后，根据调查结果提出了一些建议。