面向数万处理器的有限元线性方程组与模态多级算法研究

Linear equations and eigenvalue problem are the core of finite element analysis, which impropriate most of the computational time. However, the existing algorithms cannot stably and efficient enough to actualize the detailed mechanical analysis of complex equipment, such as a plane, with fine grid. Especially, the procedure has a great probability of breaking down, or even can not set-up, when the numerical simulation runs on over tens of thousands of processor. In this project, following the idea of co-design, we will improve stability and expansibility of codes with characteristic of computing cluster. According to the previous research, a combined process-thread bi-parallel mode will be used to re-design the proposed scalable Hierarchical Equilibrium Linear-equations Parallel Solver (HELPS), in which each process is exclusively assigned on a computing node, and a threads-region is created over CPUs and any existing co-processors, such as GPU and Intel Xeon Phi. Also, we will optimize efficiency of parallel solvers of eigenvalue problem according to applications and mechanical theory. First, the spectrum Chebyshev polynomial filter, which succeed in density functional calculations, will be detailed studied and then applied to improve efficiency Krylov- or Davidson-type iterative eigensolvers. Second, a new coarse-fine-nested-grid component mode synthesis will be introduced to parallel modal analysis, which makes components computing totally independent, and keep an acceptable precision on coarse global grid. Finally the algorithms of this project will be integrated in the domestic FEA framework, JAUMIN to investigate the static and dynamic response and the stability qualification of whole and fine-meshed super-large equipment.

面向数万处理器，结合应用特征，发展具备良好并行扩展能力的有限元方程组和模态多级算法，是推动飞机等超大规模工程装备整体精细力学分析实现的关键技术，是国际公认难题。本项目拟围绕这一前沿问题，在协同创新思想指导下，结合国产计算集群、工程应用、理论模型和并行有限元执行特点研究方程组与模态并行算法的理论与实现技术，着重稳定运行、提高并行扩展性和充分发挥硬件能力。工作内容包括设计优化进程-线程二级并行的多级平衡方程组求解方案；探索适合并行有限元特点的几何多重网格预条件方法；研究模态分析的高效滤波方法降低计算复杂度，进而发展高效的模态综合多级求解方法。在此基础上，本项目将瞄准若干重要的实际应用问题，支撑数万处理器的精细有限元分析，具有很强的需求牵引与理论和实际应用价值。

面向数万处理器具备良好并行扩展能力的有限元方程组和模态多级算法，是国际公认难题。在三年项目执行期内，针对结构有限元分析问题，项目组完成了面向数万处理器核数十亿自由度线性方程组和特征值系统高可扩展数值算法研究，并将所发展方法汇总开发了高效数值代数软件包，进而实现了百亿网格三峡大坝静力学、十一亿自由度神光III主体建筑模态和千万自由度电子器件等超大规模仿真，超越商业软件一到二个数量级，达到国内领先国际先进水平。