A dual-mesh multiscale finite element method for simulating nodal Darcy velocities in aquifers
-
摘要: 提出了一种用于模拟节点达西渗透流速的双重网格多尺度有限单元法（D-MSFEM）。该方法是多尺度有限单元法（MSFEM）与VedatBatu双重网格技术（D-FEM）的有机结合,不仅可以应用双重网格技术获得连续、精确的水头一阶导数,还可以应用多尺度基函数直接在粗网格上求解水头和达西渗透流速,从而突破了传统有限元基础框架的限制,具有极高的计算效率。同时,D-MSFEM还可以应用粗尺度节点的达西渗透流速和多尺度基函数直接获得细尺度节点的达西渗透流速,而无需在精细尺度上求解,能够节约大量的计算消耗。应用D-MSFEM和多种传统达西渗透流速计算方法对地下水稳定流和非稳定流进行了模拟,结果显示D-MSFEM具有极高的模拟效率和精度。该方法可为高效计算地下水达西渗透流速问题提供新途径。Abstract: A dual-mesh multiscale finite element method (D-MSFEM) is developed to simulate nodal Darcy velocities in aquifers. It is a combination of the multiscale finite element method (MSFEM) and the dual-mesh finite element method (D-FEM). D-MSFEM can obtain continuous first-order head derivatives and solve the head and nodal Darcy velocities directly on the coarse grid without the necessity for solving Darcy equation specifically. Therefore, it breaks through the limitations of the traditional finite element basic framework and improves the computational efficiency extremely in comparison to the traditional methods for nodal Darcy velocities. D-MSFEM can also directly obtain the fine-scale nodal Darcy velocities by using the coarse-scale nodal Darcy velocities and the multiscale base functions, which can save a lot of computational cost. D-MSFEM is compared with some traditional methods for nodal Darcy velocities in the simulation of groundwater steady flow and transient flow. The results show that D-MSFEM achieves higher simulation efficiency and accuracy. This study may provide a new approach to simulate nodal Darcy velocities in aquifers efficiently.
-
-
![]()
图 3 各数值方法所计算的水头平均相对误差与粗网格单元尺度关系
Figure 3. Relationship between coarse element scale and relative error of H
![]()
图 4 各数值方法所需计算时间与粗网格单元尺度关系
Figure 4. Relationship between coarse element scale and computation time
![]()
图 5 各数值方法所计算的
平均相对误差与粗网格单元尺度关系 Figure 5. Relationship between coarse element scale and relative error of
![]()
图 7 在截面y=0.6处各数值方法所计算达西渗透流速的绝对误差
Figure 7. Absolute errors of velocities calculated by numerical methods at section y=0.6
表 1 D-MSFEM（1600,50）计算的细尺度节点在x方向的达西渗透流速
Table 1 Fine-scale nodal Darcy velocities in x direction calculated by D-MSFEM（1600,50）
节点坐标 解析解 D-MSFEM 相对误差/% （0.805,0.605） 0.149241 0.145775 2.38 （0.810,0.605） 0.149300 0.148165 0.77 （0.815,0.605） 0.149277 0.150554 0.85 （0.820,0.605） 0.149343 0.152944 2.35 （0.805,0.610） 0.148530 0.145119 2.35 （0.810,0.610） 0.148604 0.147498 0.75 （0.815,0.610） 0.148565 0.149877 0.88 （0.820,0.610） 0.148643 0.152256 2.37 （0.805,0.615） 0.147825 0.144433 2.35 （0.810,0.615） 0.147902 0.146801 0.75 （0.815,0.615） 0.147863 0.149168 0.87 （0.820,0.615） 0.147937 0.151536 2.37 （0.805,0.620） 0.147125 0.143716 2.37 （0.810,0.620） 0.147191 0.146072 0.77 （0.815,0.620） 0.147167 0.148428 0.85 （0.820,0.620） 0.147226 0.150784 2.36 
下载: 导出CSV
表 2 各数值方法所计算的达西渗透流速的平均相对误差
Table 2 Relative errors of velocities calculated by numerical methods
(%) 数值方法 截面平均相对误差 全局平均相对误差 D-FEM 0.325 0.356 D-MSFEM 0.080 0.086 Yeh-F 0.034 0.054 AS-D-MSFEM 0.005 0.006
下载: 导出CSV
-
[1] 薛禹群, 谢春红. 地下水数值模拟[M]. 北京: 科学出版社, 2007: 175-178. XUE Yu-qun, XIE Chun-hong. Numerical simulation for Groundwater[M]. Beijing: Science Press, 2007: 175-178. (in Chinese)
[2] ZHOU Q, BENSABAT J, BEAR J. Accurate calculation of specific discharge in heterogeneous porous media[J]. Water Resources Research, 2001, 37(12): 3057-3069. doi: 10.1029/1998WR900105
[3] 王铁行, 罗扬, 张辉. 黄土节理二维稳态流流量方程[J]. 岩土工程学报, 2013, 35(6): 1115-1120. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201306019.htm WANG Tie-hang, LUO Yang, ZHANG Hui. Two-dimensional steady flow rate equation for loess joints[J]. Chinese Journal of Geotechnical Engineering, 2013, 35(6): 1115-1120. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201306019.htm
[4] WOUTER Z. Finite-Element methods based on a transport velocity representation for groundwater motion[J]. Water Resources Research, 1984, 20(1): 137-145. doi: 10.1029/WR020i001p00137
[5] 谢一凡, 吴吉春, 薛禹群, 等. 一种模拟非均质介质中地下水流运动的快速提升尺度法[J]. 水利学报, 2015, 46(8): 918-924. doi: 10.13243/j.cnki.slxb.20141475 XIE Yi-fan, WU Ji-chun, XUE Yu-qun, et al. A fast upscaling method for simulating groundwater flow in heterogeneous media[J]. Journal of Hydraulic Engineering, 2015, 46(8): 918-924. (in Chinese) doi: 10.13243/j.cnki.slxb.20141475
[6] EFENDIEV Y R, WU X H. Multiscale finite element for problems with highly oscillatory coefficients[J]. Numerische Mathematik, 2002, 90(3): 459-486. doi: 10.1007/s002110100274
[7] 薛禹群, 叶淑君, 谢春红, 等. 多尺度有限元法在地下水模拟中的应用[J]. 水利学报, 2004, 35(7): 7-13. doi: 10.3321/j.issn:0559-9350.2004.07.002 XUE Yu-qun, YE Shu-jun, XIE Chun-hong, et al. Application of multi-scale finite element method to simulation of groundwater flow[J]. Journal of Hydraulic Engineering, 2004, 35(7): 7-13. (in Chinese) doi: 10.3321/j.issn:0559-9350.2004.07.002
[8] HOU T Y, WU X H. A multiscale finite element method for elliptic problems in composite materials and porous media[J]. Journal of Computational Physics, 1997, 134(1): 169-189. doi: 10.1006/jcph.1997.5682
[9] YE S, XUE Y, XIE C. Application of the multiscale finite element method to flow in heterogeneous porous media[J]. Developments in Water Science, 2004, 55(9): 337-348.
[10] 贺新光, 任理. 求解非均质多孔介质中非饱和水流问题的一种自适应多尺度有限元方法——Ⅰ.数值格式[J]. 水利学报, 2009, 40(1): 38-45. doi: 10.13243/j.cnki.slxb.2009.01.017 HE Xin-guang, REN Li. Adaptive multiscale finite element method for unsaturated flow in heterogeneous porous media: I numerical scheme[J]. Journal of Hydraulic Engineering, 2009, 40(1): 38-45. (in Chinese) doi: 10.13243/j.cnki.slxb.2009.01.017
[11] 贺新光, 任理. 求解非均质多孔介质中非饱和水流问题的一种自适应多尺度有限元方法——Ⅱ.数值结果[J]. 水利学报, 2009, 40(2): 138-144. doi: 10.3321/j.issn:0559-9350.2009.02.002 HE Xin-guang, REN Li. Adaptive multiscale finite element method for unsaturated flow in heterogeneous porous media: II numerical result[J]. Journal of Hydraulic Engineering, 2009, 40(2): 138-144. (in Chinese) doi: 10.3321/j.issn:0559-9350.2009.02.002
[12] 叶淑君, 吴吉春, 薛禹群. 多尺度有限单元法求解非均质多孔介质中的三维地下水流问题[J]. 地球科学进展, 2004, 19(3): 437-442. https://www.cnki.com.cn/Article/CJFDTOTAL-DXJZ200403014.htm YE Shu-jun, WU Ji-chun, XUE Yu-qun. Application of multiscale finite element method to three dimensional groundwater flow problems in heterogeneous porous media[J]. Advance in Earth Sciences, 2004, 19(3): 437-442. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-DXJZ200403014.htm
[13] 于军, 吴吉春, 叶淑君, 等. 苏锡常地区非线性地面沉降耦合模型研究[J]. 水文地质工程地质, 2007(5): 11-16. doi: 10.16030/j.cnki.issn.1000-3665.2007.05.017 YU Jun, WU Ji-chun, YE Shu-jun, et al. Research on nonlinear coupled modeling of land subsidence in Suzhou, Wuxi and Changzhou areas, China[J]. Hydrogeology & Engineering Geology, 2007(5): 11-16. (in Chinese) doi: 10.16030/j.cnki.issn.1000-3665.2007.05.017
[14] 谢一凡, 吴吉春, 薛禹群, 等. 一种模拟节点达西渗透流速的三次样条多尺度有限单元法[J]. 岩土工程学报, 2015, 37(9): 1727-1732. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201509030.htm XIE Yi-fan, WU Ji-chun, XUE Yu-qun, et al. A cubic-spline multiscale finite element method for simulating nodal Darcy velocities[J]. Chinese Journal of Geotechnical Engineering, 2015, 37(9): 1727-1732. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201509030.htm
[15] 谢一凡, 吴吉春, 薛禹群, 等. 结合有限单元法运用三次样条技术求解达西渗透流速[J]. 水文地质工程地质, 2015, 42(5): 1-5. XIE Yi-fan, WU Ji-chun, XUE Yu-qun, et al. Combination of finite element method and cubic-spline technique to solve Darcy velocities[J]. Hydrogeology & Engineering Geology, 2015, 42(5): 1-5. (in Chinese)
[16] XIE Y, WU J, XIE C. Cubic-spline multiscale finite element method for solving nodal Darcian velocities in porous media[J]. Journal of Hydrologic Engineering, 2015, 20(11): 04015030. doi: 10.1061/(ASCE)HE.1943-5584.0001222
[17] 谢一凡. 改进多尺度有限单元法求解二维地下水流问题[D]. 南京: 南京大学, 2015. XIE Yi-fan, Improved Multiscale Finite Element Method for Solving Two-Dimensional Groundwater Flow Problems[D]. Nanjing: Nanjing University, 2015. (in Chinese)
[18] XIE Y F, WU J C, XUE Y Q, et al Combination of multiscale finite-element method and Yeh's finite-element model for solving nodal Darcian velocities and fluxesin porous media[J]. Journal of Hydrologic Engineering, 2016, 21(12): 04016048. doi: 10.1061/(ASCE)HE.1943-5584.0001456
[19] VEDAT B. A finite element dual mesh method to calculate Nodal Darcy velocities in nonhomogeneous and anisotropic aquifers[J]. Water Resources Research, 1984, 20(11): 1705-1717. doi: 10.1029/WR020i011p01705
[20] YEH G T. On the computation of Darcian velocity and mass balance in the finite element modeling of groundwater flow[J]. Water Resources Research, 1981, 17(5): 1529-1534. doi: 10.1029/WR017i005p01529
-
期刊类型引用(6)
1. 张志红，王树青，杨凡. 基于Furnas方程和分形理论的级配优化方法及试验验证. 土木工程学报. 2023(01): 109-118 . 
百度学术
2. 张文周，何忠明，刘正夫，刘洋. 炭质泥岩粗粒土动态回弹模量及预估模型研究. 水利水电技术(中英文). 2023(02): 161-169 . 百度学术
3. 陈晓斌，郭云鹏，蔡德钩，尧俊凯，肖源杰. 铁路工程粗颗粒土路基填料研究现状与发展综述. 路基工程. 2021(03): 1-11 . 百度学术
4. 王萌，肖源杰，王小明，蔡德钩，陈晓斌. 道砟压实质量与颗粒运动关联特征及内在机制研究. 铁道科学与工程学报. 2021(08): 2055-2065 . 百度学术
5. 王萌，肖源杰，卢小永，畅振兴，陈晓斌，古牧，叶新宇. 重载铁路道砟旋转压实特性及参数优化试验研究. 铁道科学与工程学报. 2020(10): 2503-2515 . 百度学术
6. 肖建章，蔡红，孙平，魏然，严俊. 不良级配颗分曲线的方程表述. 水利水电技术. 2020(S2): 416-421 . 百度学术
其他类型引用(10)
计量
- 文章访问数: 275
- HTML全文浏览量: 19
- PDF下载量: 93
- 被引次数: 16
