一种模拟节点达西流速的多尺度有限元-有限元模型

谢一凡; 吴吉春; 王益; 叶逾; 谢春红; 鲁春辉

doi:10.11779/CJGE202201010

一种模拟节点达西流速的多尺度有限元-有限元模型 English Version

谢一凡^{1, 2,},
吴吉春³,
王益^{1, 2},
叶逾^{1, 2},
谢春红⁴,
鲁春辉^{1, 2, 5, ,}

1.
河海大学水文水资源与水利工程科学国家重点实验室, 江苏南京 210024
2.
河海大学水利水电学院, 江苏南京 210024
3.
南京大学地球科学与工程学院, 江苏南京 210023
4.
南京大学数学系, 江苏南京 210093
5.
河海大学长江保护与绿色发展研究院, 江苏南京 210024

基金项目:

中央高校基本业务费项目 B210202018

国家自然科学基金面上项目 51879088

国家自然科学基金青年项目 41702243

详细信息

作者简介:
谢一凡(1987—)，男，副教授，主要从事地下水数值算法方面的研究工作。E-mail：yfxie@hhu.com

通讯作者:
鲁春辉, E-mail：clu@hhu.edu.cn

中图分类号: TU46
计量
- 文章访问数: 306
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出版历程
- 收稿日期: 2021-04-06
- 网络出版日期: 2022-09-22
- 刊出日期: 2021-12-31

Multiscale finite element-finite element model for simulating nodal Darcy velocity

XIE Yi-fan^{1, 2,},
WU Ji-chun³,
WANG Yi^{1, 2},
YE Yu^{1, 2},
XIE Chun-hong⁴,
LU Chun-hui^{1, 2, 5, ,}

1.
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210024, China
2.
College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210024, China
3.
School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, China
4.
Department of Mathematics, Nanjing University, Nanjing 210093, China
5.
Yangtze Institute for Conservation and Development, Hohai University, Nanjing 210024, China

摘要

摘要: 提出了一种能高效模拟节点达西流速并保证其连续性的多尺度有限元-有限元模型(MSFEM-FEM)。该方法先应用多尺度有限元法(MSFEM)框架改进了Yeh的有限元模型的水头模拟部分以提升效率与精度，再将多尺度网格转化为有限元网格，应用Yeh的有限元框架保证流速的连续性。基于多尺度基函数，MSFEM-FEM能够汲取研究区的全局信息并在粗尺度上高效获得精确的水头解。通过将粗尺度网格转换为有限元网格，MSFEM-FEM能够应用Yeh的有限元框架将水头解中的全局信息导入达西流速，提高达西流速的精度并保证其连续性。在获得粗尺度解后，MSFEM-FEM还能应用多尺度基函数对解进行细尺度重构，从而获得研究区内的细尺度水头与流速。数值模拟结果显示MSFEM-FEM能够高效、精确的求解水头，并能够获得连续、精确的达西流速和流量。
- 地下水数值模拟 /
- 多尺度有限元法 /
- Yeh的有限元模型 /
- 连续节点达西流速 /
- 高计算效率
Abstract: A multi-scale finite element-finite element model (MSFEM-FEM) is proposed, and it can effectively simulate the nodal Darcy velocity and ensure the velocity continuity. The MSFEM-FEM employs the multi-scale finite element method (MSFEM) to replace the head simulation part of the Yeh's finite element model, thus to improve the efficiency and accuracy. Then, the MSFEM-FEM transforms the multi-scale grid to the finite element one, thus, it can directly apply the Yeh's finite element model to obtain continuous Darcy velocity. Based on the multi-scale basis function, the MSFEM-FEM can extract the global information of the study area which allows it to obtain the accurate head solution efficiently on the coarse scale. By transforming the coarse-scale grid into the finite element grid, the MSFEM-FEM can directly employ the Yeh's finite element model to import the global information from the head solution into the Darcy velocity, which can also improve the accuracy of Darcy velocity and ensure the velocity continuity. In addition, the MSFEM-FEM can apply multi-scale basis function to reconstruct the solutions, so as to obtain the fine-scale head and velocity solutions in the study area. The simulated results of two-dimensional groundwater problems show that the MSFEM-FEM can efficiently and accurately solve the head, Darcy velocity and flux, which outperforms the MSFEM and the Yeh's finite element model.
- groundwater numerical simulation /
- multiscale finite element method /
- Yeh's finite element model /
- continuous Darcy velocity /
- high computation efficiency

HTML全文

图 1 MSFEM-FEM的研究区Ω的剖分示意图

Figure 1. Study region of MSFEM-FEM

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图 2 MSFEM-FEM的粗网格剖分示意图

Figure 2. Coarse element of MSFEM-FEM

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例2.1中各方法模拟的粗尺度达西流速 ${v_x}$ 的平均相对误差

Relative errors of ${v_x}$ calculated by numerical methods in example 2.1

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图 4 例2.1中解析解以及MSFEM-FEM和MSFEM所计算的通过Ω各垂直截面的流量

Figure 4. Analytical flux and flux calculated by MSFEM-FEM and MSFEM at vertical sections in Ω in example 2.1

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图 5 例2.2中在截面y=5200 m处的参照解以及MSFEM-FEM和Yeh的数值解水头

Figure 5. Reference solution and head calculated by MSFEM-FEM and Yeh at section y=5200 m in example 2.2

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例2.2中在截面y=5200 m处的MSFEM-FEM和Yeh的数值解达西渗透流速 ${v_x}$ 的相对误差

${v_x}$ calculated by MSFEM-FEM and Yeh at section y=5200 m in example 2.2

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图 7 例2.3中在截面y=0.6 [L]处的Yeh-F、MSFEM-FEM和Yeh的水头绝对误差

Figure 7. Absolute errors of head calculated by Yeh-F, MSFEM-FEM and Yeh at section y=0.6 [L] in example 2.3

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例2.3中在截面y=0.6 [L]处的解析解以及Yeh-F、MSFEM-FEM和Yeh的数值解 ${v_x}$

Analytical solution and ${v_x}$ calculated by Yeh-F, MSFEM- FEM and Yeh at section y=0.6 [L] in example 2.3

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表 1 例2.1中各方法的水头相对误差

Table 1 Relative errors of head calculated by numerical methods in example 2.1 (%)

N AS-Yeh MSFEM-FEM Yeh

10 0 0.0067 0.161

20 0 0.0032 0.079

30 0 0.0010 0.024

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表 2 例2.1中MSFEM-FEM与Yeh-F的数值结果对比

Table 2 Numerical results of MSFEM-FEM and Yeh-F in example 2.1

方法水头平均相对误差/% 达西流速平均相对误差/% CPU时间/s

MSFEM-FEM 0.001 0.33 6

Yeh-F 0.001 0.02 8530

下载: 导出CSV

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