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基于改进FPM法高维复杂多相分离过程的GPU并行计算研究

甄玉洁 胥康 蒋涛 任金莲

甄玉洁,胥康,蒋涛,任金莲. 基于改进FPM法高维复杂多相分离过程的GPU并行计算研究 [J]. 应用数学和力学,2023,44(1):93-104 doi: 10.21656/1000-0887.430147
引用本文: 甄玉洁,胥康,蒋涛,任金莲. 基于改进FPM法高维复杂多相分离过程的GPU并行计算研究 [J]. 应用数学和力学,2023,44(1):93-104 doi: 10.21656/1000-0887.430147
ZHEN Yujie, XU Kang, JIANG Tao, REN Jinlian. GPU Parallelization Computation of High-Dimensional Multi-Phase Separation in Complex Domains Based on the Corrected FPM[J]. Applied Mathematics and Mechanics, 2023, 44(1): 93-104. doi: 10.21656/1000-0887.430147
Citation: ZHEN Yujie, XU Kang, JIANG Tao, REN Jinlian. GPU Parallelization Computation of High-Dimensional Multi-Phase Separation in Complex Domains Based on the Corrected FPM[J]. Applied Mathematics and Mechanics, 2023, 44(1): 93-104. doi: 10.21656/1000-0887.430147

基于改进FPM法高维复杂多相分离过程的GPU并行计算研究

doi: 10.21656/1000-0887.430147
基金项目: 国家自然科学基金(11501495);中国博士后科学基金(2015M581869;2015T80589);江苏省自然科学基金(BK20150436);国家科技支撑计划 (2015BAD24B02-02)
详细信息
    作者简介:

    甄玉洁(1996—),女,硕士生(E-mail:1056700190@qq.com

    蒋涛(1978—),男,副教授,博士(通讯作者. E-mail:jtrjl_2007@126.com

  • 中图分类号: O246

GPU Parallelization Computation of High-Dimensional Multi-Phase Separation in Complex Domains Based on the Corrected FPM

  • 摘要:

    为了高效、准确地模拟高维多元Cahn-Hilliard (C-H)方程描述的复杂相分离过程,该文发展了一种基于纯无网格改进有限点集法(corrected finite pointset method, CFPM) 和CPU-GPU异构的快速并行算法 (简称为CFPM-GPU)。CFPM-GPU的构造过程为:① 基于Taylor展开和加权最小二乘思想,采用Wendland权函数推导出空间一/二阶导数的有限点集法(finite pointset method, FPM)格式;② 将多元C-H方程中四阶导数分为两个二阶导数,依次运用FPM对其离散得到C-H的改进FPM法(CFPM);③ 基于CUDA的单个GPU架构,首次给出了CFPM的一种并行算法以提高计算效率。 数值研究中,首先对二维径向或三维球对称C-H方程描述的相分离基准算例进行了求解,并与可靠结果作对比验证了提出的并行算法的准确性和高效性,单个CPU-GPU异构并行计算效率约是串行情况的160倍;其次,运用CFPM-GPU对复杂区域上二维/三维的两相或三相分离现象进行数值预测,并与其他方法结果做比较。数值结果表明,给出的CFPM-GPU能准确、高效地模拟二维/三维复杂区域上的多相分离过程。

  • 图  1  二维圆环收缩现象CFPM-GPU数值模拟结果:(a) t=0;(b) t=1.31;(c) t=2.38

    注 为了解释图中的颜色,读者可以参考本文的电子网页版本,后同。

    Figure  1.  The 2D shrinking annuli obtained with the CFPM-GPU: (a) t=0; (b) t=1.31; (c) t=2.38

    图  2  三维球壳收缩现象CFPM-GPU数值模拟结果:(a) t=0;(b) t=1.192

    Figure  2.  The 3D spherical shells obtained with the CFPM-GPU: (a) t=0; (b) t=1.192

    图  3  二维圆盘域上二相分离现象CFPM-GPU模拟结果

    Figure  3.  The 2-phase separation obtained with the CFPM-GPU in the 2D disk domain

    图  4  二维脑剖面原图及其复杂脑剖面轮廓[10]:(a)脑剖面原图;(b)脑剖面计算区域图

    Figure  4.  The 2D brain section picture and the complex brain shape domian[10]: (a) the brain section picture; (b) the brain section calculation domain

    图  5  二维脑剖面域上二相分离现象CFPM-GPU模拟结果

    Figure  5.  The 2-phase separation obtained with the CFPM-GPU in the 2D brain shape domain

    图  6  二维星形域上三相分离现象CFPM-GPU模拟结果

    Figure  6.  The 3-phase separation obtained with the CFPM-GPU in the 2D star domain

    图  7  三维环体域上相分离现象CFPM-GPU模拟结果

    Figure  7.  The phase separation obtained with the CFPM-GPU in the 3D torus domain

    图  8  三维Schwarz-D区域上相分离现象CFPM-GPU模拟结果

    Figure  8.  The phase separation obtained with the CFPM-GPU in the 3D Schwarz-D domain

    表  1  不同节点数下,相邻节点标定消耗计算时间对比

    Table  1.   The consumed computing time for the calibration of neighbor nodes under different node numbers

    node numbercomputing time
    CPU TCPU/sGPU TGPU/sδSur
    $65 \times 65 \times 65$833.554.73176.2
    $129 \times 129 \times 129$51513.65287.69179.1
    $257 \times 257 \times 257$3226951.3217907.61180.2
    下载: 导出CSV

    表  2  不同节点数下,每个时间层里物理量更新循环所需平均计算时间对比

    Table  2.   The average computing time costs at each time step under different node numbers

    node numbercomputing time
    CPU TCPU/sGPU TGPU/sδSur
    $65 \times 65 \times 65$10.3320.065158.95
    $129 \times 129 \times 129$86.3110.539160.13
    $257 \times 257 \times 257$725.2814.525160.28
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-04-26
  • 修回日期:  2022-06-13
  • 网络出版日期:  2023-01-06
  • 刊出日期:  2023-01-15

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