基于有限元求解均质承压水稳定流问题的傅里叶分析方法  

Fourier analysis method for steady groundwater flow in homogeneous confined aquifer based on FEM

作　　者：王佩[1] 朱国荣[1] 江思珉[2] 孙斌堂 

机构地区：[1]南京大学水科学系,江苏南京210093 [2]同济大学水利工程系,上海200092 [3]中国冶金地质工程勘察总局山东局,山东济南250014

出　　处：《水利学报》2012年第3期372-378,共7页Journal of Hydraulic Engineering

摘　　要：规则区域均质承压水稳定流问题在只考虑Dirichlet、Neumann边界时具备11种边界类型组合模式。不同组合模式在有限元剖分下具有不同的稀疏线性方程组。本文对不同组合模式下的稀疏线性方程组进行傅里叶分析,在分析过程中对各种变换公式均采用快速傅里叶变换算法进行计算,实现了均质承压水稳定流问题的快速求解。文中通过具有解析解的水流基准问题验证了傅里叶分析方法的可信性,并以此为基础,在不同剖分模式下分别应用傅里叶分析法与迭代法(Jacobi、Gauss-Seidel、SOR、PCG)进行算法的性能比较。计算结果表明:剖分密度越高(即剖分结点数多),在求解精度相当的情况下傅里叶分析方法的求解效率优势越显著。该方法的另一大优势在于不需计算和存储原始系数矩阵,从而节省了大量内存空间。Steady groundwater flow problem in homogeneous confined aquifer over regular domain with Dirichlet,Neumann boundary conditions can form 11 kinds of sub-problems.Fourier analysis method（FAM） is used to discrete the above sub-problems and to solve the corresponding sparse linear equations.During Fourier analysis process,FFT algorithm were adopted to calculate all kinds of transform formula and therefore an efficient direct method has been developed for the steady groundwater flow problem.Then a flow model with analytical solution is used to validate the FAM.Further more,inconsideration of various kinds of grid sizes,the paper solve the problems with the FAM and iterative methods（Jacobi,Gauss-Seidel,SOR,and PCG） separately.The calculation results show that with similar precision,the higher the grid sizes are,the more notable is the efficiency of the FAM.Another advantage of this method is that much memory can be saved as the FAM which does not need to compute and storage coefficient matrix for steady groundwater flow problem in homogeneous confined aquifer.

关 键 词：有限元 傅里叶分析方法 快速傅里叶变换 稳定流 

分 类 号：P641[天文地球—地质矿产勘探]