模拟介质交界面处达西流速的跳跃多尺度有限元法  

Multiscale finite element method with JUMP vector for simulating Darcy velocity at the interface of media

作　　者：谢一凡 苏振宁 南统超 吴吉春[4] 谢春红[5] 鲁春辉 XIE Yifan;SU Zhenning;NAN Tongchao;WU Jichun;XIE Chunhong;LU Chunhui(The National Key Laboratory of Water Disaster Prevention,Hohai University,Nanjing 210098,China;College of Water Conservancy and Hydropower Engineering,Hohai University,Nanjing 210098,China;Yangtze Institute for Conservation and Development,Hohai University,Nanjing 210098,China;School of Earth Sciences and Engineering,Nanjing University,Nanjing 210023,China;Department of Mathematics,Nanjing University,Nanjing 210093,China;College of Hydrology and Water Resource,Hohai University,Nanjing 210098,China)

机构地区：[1]河海大学水灾害防御全国重点实验室,江苏南京210098 [2]河海大学水利水电学院,江苏南京210098 [3]河海大学长江保护与绿色发展研究院,江苏南京210098 [4]南京大学地球科学与工程学院,江苏南京210023 [5]南京大学数学系,江苏南京210093 [6]河海大学水文水资源学院,江苏南京210098

出　　处：《水利学报》2024年第12期1439-1447,1460,共10页Journal of Hydraulic Engineering

基　　金：国家重点研发计划项目(2021YFC3200500);国家自然科学基金面上项目(42277190)。

摘　　要：提出一种用于模拟介质交界面处达西流速的跳跃多尺度有限元法(MSFEM-J),可解决有限元等惯用方法模拟含交界面地下水问题时的两个缺点:一是现有方法需要精细剖分刻画介质非均质性来保证解的精度,会消耗巨大的运算资源;二是现有数值方法无法保证介质交界面处的达西流速满足折射定律,即无法保证法向流速连续、切向流速按渗透系数呈比例。MSFEM-J可通过多尺度有限元(MSFEM)的基函数显著提升地下水水头和达西流速两个过程的计算效率;然后基于折射定律构造JUMP向量,迭代修正交界面弱侧节点的右端项并直至收敛,从而令交界面处达西流速满足折射定律。数值试验表明,本文提出的MSFEM-J所模拟的达西流速在介质交界面处符合折射定律,在研究区其他位置具有连续性。同时,该方法的计算精度高于精细剖分的惯用方法,且计算消耗仅为同效方法的2%左右。A JUMP vector multiscale finite element method(MSFEM-J)for simulating Darcy flow velocity at the interface of media is proposed,which can solve two shortcomings of current methods such as finite element method when simulating groundwater problems with interfaces.First,the current method needs to accurately dissect and characterize the heterogeneity of media to ensure the accuracy of solution,and consumes huge computational resources;Secondly,the current numerical method cannot ensure that the Darcy velocity at the interface of the medium obeys the refraction law,that is,the normal velocity is continuous and the tangential velocity is proportional to the permeability coefficient.MSFEM-J first significantly improves the calculation efficiency of the groundwater head and Darcy velocity processes through the multi-scale finite element(MSFEM)basis function;Then,the JUMP vector is constructed based on the refraction law,and the right end term of the weak side node of the interface is iteratively corrected until convergence,so that the Darcy velocity at the interface meets the refraction law.The numerical experiments show that the Darcy velocity simulated by MSFEM-J in this paper conforms to the refraction law at the interface of the medium,and has continuity at other locations in the study area.The accuracy of this method is higher than that of the current method of fine dissection,and its calculation consumption is only about 2%of that of the same method.

关 键 词：地下水数值模拟 多尺度有限元法 达西流速 介质交界面 JUMP向量 

分 类 号：P641.2[天文地球—地质矿产勘探]