混凝土细观力学分析程序中的快速算法与并行算法设计

针对一套混凝土细观力学分析程序,在分析其计算方法与计算效率的不足之后,提出了采用稀疏矩阵与稀疏向量技术来高效实现有限元刚度矩阵装配过程的算法,并采用双门槛不完全Cholesky分解预条件技术与CG法相结合来高效地求解稀疏线性方程组。之后,从整体上提出了一个将有限单元分布与未知量分布有机结合的并行算法设计方案,并分别针对刚度矩阵装配、双门槛不完全Cholesky分解、稀疏矩阵与稠密向量相乘、稀疏向量相加等核心算法,进行了相应的并行算法设计。最后,在由每节点2 CPU的8个Intel Xeon节点采用千兆以太网连成的机群上,针对两个混凝土数值试样进行了数值实验,第一个试样含44117个网格点与53200个有限单元,第二个试样含71013个网格点与78800个有限单元;对第一个试样,原串行程序进行全程567次加载计算需要984.83小时约41天,采用文中串行算法后,模拟时间减少到22531秒约6.26小时,采用并行算法在16个CPU上的模拟时间进一步降为3860秒约1.07小时。对第二个试样,原串行程序进行全程94次加载计算需要467.19小时约19.5天,采用文中串行算法后,模拟时间减少到11453秒约3.18小时,采用并行算法在16个CPU上的模拟时间进一步降为1704秒约28.4分钟。串行算法的改进与并行算法的设计大大缩短了计算时间,对加快混凝土力学性能的分析研究具有重要意义。

Fast and parallel algorithm design in the program for meso-mechanics analysis of concrete

For meso-mechanics analysis of concrete.This paper provides a new assembling process based on sparse matrix techniques and sparse vector techniques to generate the global stiff-matrix.The incomplete Cholesky factorization with double thresholds is incorporated into the conjugate gradient method to solve the sparse linear system.Then,the parallelization is considered in several aspects.First,an overall framework is provided to accommodate the problem.The tasks are scheduled by finite elements in the whole,but in solving sparse linear systems,the scheduling by unknowns is used.Efficient parallel algorithms are introduced for the kernels respectively,including the assembly process of the stiff matrix,incomplete Cholesky factorization with double threshold,multiplication of a sparse matrix with a dense vector,and the addition of two sparse vectors.Finally,experiments are done on a cluster of Intel Xeon processors connected with Ethernet network for two concrete samples. The first one contains 44117 discrete points and 53200 finite elements,and the second one contains 71013 discrete points and 78800 finite elements.For the first sample,the overall simulation requires 567 load steps,and the original program takes about 41 days for the whole simulation.With the improved algorithms provided in the paper,it is decreased to 6.26 hours,and with the parallel program,it is decreased further to 1.07 hours when 16 processors are used.For the second one,the overall simulation requires 94 steps,and the original program takes about 19.5 days.With the improved algorithms,it is decreased to 3.18 hours,and with the parallel program,it is decreased further to 28.4 minutes when 16 processors are used.The improvement of serial algorithms and the design of the parallel algorithms reduce the simulation time greatly,which is significant in the studies of mechanics characteristics of concrete materials.