基于残余平滑-预处理共轭梯度算法的有限元并行计算

Finite element parallel computing based on minimal residual-preconditioned conjugate gradient method

针对弹塑性问题的有限元分析非常耗时,基于消息传递接口(MPI)集群环境,提出了残余平滑的子结构预处理共轭梯度并行算法。采取区域分解,将子结构通过界面条件处理为独立的有限元模型。整体分析时,每个处理器仅存储与其相关的子结构信息并生成局部刚度矩阵。采用对角存储方式和最小残余平滑法,设计出了结合残余平滑(MR)的并行子结构预处理共轭梯度(PCG)算法。并行算法中对负载平衡进行了探讨,对处理器间的通信进行了优化。利用子步法对弹塑性应力应变进行积分,根据预定的容许值自动调整每个子步的大小来控制积分过程的误差。在工作站集群上实现了数值算例,分析了算法的性能,计算性能与传统的PCG算法进行了比较。算例显示:所提算法具有良好的加速比和效率,优于传统的PCG算法,对弹塑性问题的有限元分析,是一种有效的并行求解算法。

Finite element analysis for elastic-plastic problem is very time-consuming. A parallel substructure Preconditioned Conjugate Gradient（ PCG） algorithm combined with Minimal Residual（ MR） smoothing was proposed under the environment of Message Passing Interface（ MPI） cluster. The proposed method was based on domain decomposition, and substructure was treated as isolated finite element model via the interface conditions. Throughout the analysis, each processor stored only the information relevant to its substructure and generated the local stiffness matrix. A parallel substructure oriented preconditioned conjugate gradient method was developed, which combined with MR smoothing and diagonal storage scheme.Load balance was discussed and interprocessor communication was optimized in the parallel algorithm. A substepping scheme to integrate elastic-plastic stress-strain relations was used. The errors in the integration process were controlled by adjusting the substep size automatically according to a prescribed tolerance. Numerical example was implemented to validate the performance of the proposed PCG algorithm on workstation cluster. The performance of the proposed PCG algorithm was analyzed and the performance was compared with conventional PCG algorithm. The example results indicate that the proposed algorithm has good speedup and efficiency and is superior in performance to the conventional PCG algorithm. The proposed algorithm is efficient for parallel computing of 3D elastic-plastic problems.