基于GPU的快速有限元法求解密度场

李果阳; 严华; 张征宇; 陈沁梅; 祝福顺

doi:10.13700/j.bh.1001-5965.2020.0346

基于GPU的快速有限元法求解密度场

doi: 10.13700/j.bh.1001-5965.2020.0346

1.
四川大学电子信息学院, 成都 610065
2.
中国空气动力研究与发展中心高速空气动力研究所, 绵阳 621000

基金项目:

国家自然科学基金 11872069

详细信息

通讯作者:
严华, E-mail: yanhua@scu.edu.cn

中图分类号: TP301.6
计量
- 文章访问数: 425
- HTML全文浏览量: 100
- PDF下载量: 127
- 被引次数: 0
出版历程
- 收稿日期: 2020-07-21
- 录用日期: 2020-10-16
- 网络出版日期: 2021-10-20

A fast finite element method based on GPU to solve density field

1.
College of Electronics Information Engineering, Sichuan University, Chengdu 610065, China
2.
High Speed Aerdynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

Funds:

National Natural Science Foundation of China 11872069

More Information

Corresponding author: YAN Hua, E-mail: yanhua@scu.edu.cn

摘要

摘要:
为了快速计算分析利用视频测量方法测得的高速风洞试验密度场在扰动流场作用下的实验数据，针对密度场的数值求解问题，经过光线偏折理论分析密度场得到的二阶偏微分方程，对其研究实现了CPU串行有限元法求解。在此基础上提出了基于GPU的快速有限元求解密度场的方法，该方法经过对串行有限元法求解过程效率分析后，将耗时的神经网络拟合、总刚度矩阵和总载荷向量的求解进行了基于GPU的并行加速。实验结果表明：在精度满足实际工程要求的前提下，相对于CPU串行求解方法，所提方法可大大提高求解效率，且随着网格剖分成倍加密，其加速比成倍增加。
- 密度投影场 /
- 有限元 /
- 并行加速 /
- 二阶偏微分方程 /
- 刚度矩阵
Abstract:
In order to quickly calculate and analyze the experimental data under the action of turbulence field, which are measured by video measurement method in high-speed wind tunnel test density projection field, and aimed at the numerical solution of density projection field, the second-order partial differential equation of density projection field is analyzed by ray deflection theory. And the CPU serial finite element method is realized to solve the problem. On this basis, a GPU-based fast finite element method for solving density field is proposed. After analyzing the efficiency of the serial finite element method solving process, the time-consuming neural network fitting, total stiffness matrix and total load vector are solved and then GPU-based parallel acceleration are carried out. The experimental results show that, under the premise that the accuracy meets the actual engineering requirements, the proposed method can greatly improve the solving efficiency compared with the CPU serial solution method, and the acceleration ratio is multiplied with the grid subdivision and multiplied encryption.
- density projection field /
- finite element /
- parallel acceleration /
- second-order partial differential equation /
- stiffness matrix

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图 1 光线的偏折角分析示意图

Figure 1. Schematic diagram of ray deflection angle analysis

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图 2 网格剖分平面示意图

Figure 2. Grid subdivision plan sketch

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图 3 神经网络拟合模型

Figure 3. Neural network fitting model

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图 4 CUDA线程组织结构模型

Figure 4. CUDA thread organization structure model

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图 5 神经网络并行模型

Figure 5. Neural network parallel model

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图 6 剖分的任一网格位置结点编号

Figure 6. Node number of any grid position of subdivision

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图 7 单元刚度矩阵与总刚度矩阵位置对应关系

Figure 7. Location of element stiffness matrix corresponding to total stiffness matrix

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图 8 结点存储位置

Figure 8. Node storage location

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图 9 不同网格精度下主要模块占总运行时间的比例

Figure 9. Proportion of main modules in total running time with different grid precision

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图 10 不同网格精度下求解密度场的加速比

Figure 10. Acceleration ratio of solved density field with different grid precision

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图 11 不同网格精度下求解密度场主要模块的加速比

Figure 11. Acceleration ratio of main modules of solved density field with different grid precision

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图 12 有限元串并行精度比较

Figure 12. Serial and parallel precision comparison of finite element method

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表 1 不同模块下CPU串行求解时间

Table 1. CPU serial solution time under different modules

网格精度	求解时间/s
网格精度	网络训练	网络预测	总刚度矩阵和总载荷向量	密度场
249×226	632.067	3.50	155.14	792.45
496×450	648.152	3.98	1 045.29	1 700.77
745×674	643.10	5.208	3 295.803	3 937.379
993×898	661.75	6.247	7 718.611	8 396.45

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表 2 不同模块下GPU并行求解时间

Table 2. GPU parallel solving time under different modules

网格精度	求解时间/s
网格精度	网络训练	网络预测	总刚度矩阵和总载荷向量	密度场
249×226	27.05	1.01	0.91	30.94
496×450	27.38	1.16	3.91	35.24
745×674	27.51	1.18	10.04	45.86
993×898	27.58	1.19	23.03	64.18

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