基于PHG平台的非结构四面体网格欧拉方程间断有限元并行求解器

doi:10.12288/szjs.s2020-0714

数值计算与计算机应用 ›› 2021, Vol. 42 ›› Issue (2): 155-168.DOI: 10.12288/szjs.s2020-0714

基于PHG平台的非结构四面体网格欧拉方程间断有限元并行求解器

杜玉龙¹, 徐凯文^2,3, 赵昆磊^2,3, 袁礼^2,3

1. 北京航空航天大学数学科学学院, 北京 100091;
2. 中国科学院数学与系统科学研究院计算数学所, 科学与工程计算国家重点实验室, 北京 100190;
3. 中国科学院大学数学科学学院, 北京 100090

收稿日期:2020-11-08 出版日期:2021-06-15 发布日期:2021-06-03
基金资助:
国家自然科学基金（91641107，91852116，12071470）和工信部专项（MJ-F-2012-04）资助.

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杜玉龙, 徐凯文, 赵昆磊, 袁礼. 基于PHG平台的非结构四面体网格欧拉方程间断有限元并行求解器[J]. 数值计算与计算机应用, 2021, 42(2): 155-168.

Du Yulong, Xu Kaiwen, Zhao Kunlei, Yuan Li. A PARALLEL DGM SOLVER FOR THE EULER EQUATIONS ON UNSTRUCTURED TETRAHEDRAL GRIDS BASED ON THE TOOLBOX PHG[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(2): 155-168.

A PARALLEL DGM SOLVER FOR THE EULER EQUATIONS ON UNSTRUCTURED TETRAHEDRAL GRIDS BASED ON THE TOOLBOX PHG

Du Yulong¹, Xu Kaiwen^2,3, Zhao Kunlei^2,3, Yuan Li^2,3

1. School of Mathematical Sciences, Beihang University, Beijing 10091, China;
2. LSEC and Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100090, China;
3. School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing 100090, China

Received:2020-11-08 Online:2021-06-15 Published:2021-06-03

针对计算流体力学对高性能计算的需求,基于三维并行自适应有限元程序开发平台PHG (Parallel Hierarchical Grid)开发了在非结构四面体网格上求解可压缩流欧拉方程的间断有限元法并行求解器（Libdgphg库）. 该求解器以C++函数库的形式实现数值方法中各项功能. 实施了模态基一次间断有限元, 采用低耗散的MLP(Multi-dimensional Limiting Process)限制器来抑制间断附近的数值振荡. 由于MLP限制器需要所有与当前单元共享顶点的邻近单元的信息, 模板较宽, 这给程序设计带来一定的困难. 我们通过引入辅助向量收集共享顶点的所有单元中的最大、最小单元积分平均值, 并归属到单元数据结构上, 从而利用PHG内在的通信机制实现MPI分区间的信息交换.通过几个数值算例测试了Libdgphg库的数值结果以及并行性能. 算例表明: 该求解器能得到理论精度阶和较高分辨率, 同时有良好的并行性能, 在千核测试中可达到60%以上的并行效率，可用于流体问题的大规模计算.

关键词： 双曲守恒律, 并行计算, PHG平台, MLP限制器, 间断有限元法

We develop a C++ library libdgphg for solving the Euler equations on unstructured tetrahedral grids with a discontinuous Galerkin method (DGM), based on the 3D finite element code development toolbox PHG (Parallel Hierarchical Grid). The solver uses C++ classes for different modules of the DGM. We have implemented the modal base P¹ DG method. The multi-dimensional limiting process (MLP) technique is applied to suppress numerical oscillations near discontinuities. To overcome the drawback of the non-compact stencil of the MLP limiter, we develop a strategy in which an auxiliary vector is introduced to collect the max-min values from cell averages of all vertex-sharing neighbour cells. Then, we distribute the max-min values in the auxiliary vector to the element object of PHG. In this way, we can realize the MPI communication automatically through the neighbour interface of PHG. Finally, we show that the MLP limiter can obtain expected accuracy and superior resolution, and the solver can attain 60 % parallel efficiency on 1024 cores.

Key words: Hyperbolic conservation law, Parallel computing, Multi-dimensional limiting process, Parallel hierarchial grid, Discontinuous Galerkin method

MR(2010)主题分类:

65D17

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摘要

基于PHG平台的非结构四面体网格欧拉方程间断有限元并行求解器

A PARALLEL DGM SOLVER FOR THE EULER EQUATIONS ON UNSTRUCTURED TETRAHEDRAL GRIDS BASED ON THE TOOLBOX PHG

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