【作者】 王佩；

【导师】 朱国荣；

【作者基本信息】 南京大学 ， 水文学及水资源， 2012， 博士

【摘要】 随着地下水数值模拟的深入研究,科研人员对于获取实时、精确、详细和可信任的地下水模拟信息的要求越来越高,对数值模拟软件能够针对具有精细网格剖分、长时间跨度特征问题进行模拟提出了迫切的需求。对研究区域的精细剖分往往导致数据占用内存多、求解效率低的问题。为了解决此类问题,一方面人们从数学模型出发,选择新的数值离散方法如(多尺度有限元、拉普拉斯变换有限层、FAC法等)通过减少剖分单元数来降低方程组维数进而降低内存的占用,在满足一定精度前提下提高求解效率；一方面针对数值离散后形成的大型、稀疏、病态的线性代数方程组,发展了多种高效的求解算法,其中预处理共轭梯度方法(PCG)已成为求解大型稀疏线性代数方程组极为有效的算法,而高效预处理器的构建是预处理共轭梯度方法的关键。近年来,区域分解方法因其独特的优势备受关注。论文首先利用区域分解方法构建预处理器,给出区域分解预处理器(DDP, Domain Decomposition Preconditioner)实现的详细步骤,并与预处理共轭梯度方法结合成为区域分解预处理共轭梯度法(DDP-PCG)。将区域分解预处理共轭梯度法应用于一具有解析解的承压水流问题,验证了方法的可信性。对研究区进行不同规模剖分,分别采用CG、Jacobi-PCG、SSOR-PCG、DDP-PCG方法求解上述均质承压水问题,计算结果表明：在各种网格规模下,DDP-PCG的迭代次数均明显低于其他方法,并且其迭代次数几乎不随网格规模发生变化,说明了DDP-PCG方法具有较强的鲁棒性；然而DDP-PCG方法的CPU耗时却不是最低的。经分析上述现象正是由于子区域问题求解效率低引起的。子区域上的快速算法是区域分解算法的基石。论文基于有限单元法,给出Dirichlet、Neumann边界11种组合模式下矩形区域均质承压水稳定流问题的傅里叶分析方法(FAM, Fourier Analysis method),对于傅里叶分析中的各种变换公式均采用快速傅里叶变换(FFT, Fast Fourier Transform)算法进行计算,并编制了相应的计算机程序,实现了均质承压水稳定流问题的快速求解。接着针对11种组合模式下的模型进行数值试验,将FAM方法的计算结果与解析解或者其他数值解对比,说明了FAM的可信性。采用存在解析解的单井定流量抽水承压水稳定流模型进行数值试验进一步验证了FAM的可信性,并且对该模型研究区进行多种不同模式剖分,分别采用FAM和迭代法(Jacobi、Gauss-Seidel、SOR、PCG)求解,计算结果表明：剖分精度越高,在求解精度相当的情况下FAM的求解效率越显著。此外针对均质承压水稳定流问题,采用FAM不需计算和存储原始系数矩阵,从而节省了大量内存空间。对于DDP-PCG方法,当子区域问题采用FAM求解时,得到基于傅里叶分析的区域分解预处理共轭梯度法(A-DDP-PCG, DDP-PCG based on Fourier Analysis)。文中针对水文地质问题给出子区域划分模式、相应子区域对应的傅里叶分析方法,以及FA-DDP-PCG的求解步骤,并编制了相应的计算机程序。采用具有解析解的均质承压水模型验证了FA-DDP-PCG的可靠性。接着对该模型研究区进行不同规模网格剖分,均采用CG, Jacobi-PCG, SSOR-PCG, FA-DDP-PCG求解,结果表明与CG, Jacobi-PCG, SSOR-PCG相比,在一定精度范围内,FA-DDP-PCG的求解效率更高；并且随着网格规模的增加,A-DDP-PCG的求解效率优势更加显著。对承压含水层介质参数连续变化和突变情况,进行大规模剖分,进一步证明FA-DDP-PCG的求解效率高于其它三种方法。对于介质参数连续变化的非均质模型,通过随机试验证明参数aijk(在水流模型中相当于介质的渗透系数或者导水系数)的取值对于算法求解效率影响并不明显,表明FA-DDP-PCG算法的健壮性。对于介质参数突变情形,研究了介质参数的差异性对算法求解效率的影响。结果表明,FA-DDP-PCG方法可以有效求解强突变介质地下水流问题。因此FA-DDP-PCG方法可以有效求解大规模地下水流问题。更多还原

【Abstract】 With the in-depth study of the numerical simulation of groundwater, researchers make higher requirements for real-time, accurate, detailed and credible information of groundwater simulation, and have pressing needs for the numerical simulation software that can efficiently solve problems with the characteristics of fine meshes and long time span.The fine subdivision of the study area often leads to a higher memory and lower efficiency problem. In order to solve the mentioned problem, on the one hand, researchers select new discretization methods for the mathematical model, such as (multi-scale finite element method, Laplace transform finite layer method, and the FAC method, etc.) to reduce the equations dimension and memory storage with less meshes, and to improve the efficiency under certain precision. On the other hand, reaearchers pay much attention on efficient algorithms for solving the large, sparse, morbid linear algebraic equations. And so far the preconditioned conjugate gradient method (PCG) has already been an extremely efficient algorithm for solving large sparse linear algebraic equations. Furthermore, the construction of efficient preprocessor is the key for PCG.In recent years, domain decomposition method is popular because of its unique advantages. In the paper, the preprocessor is built with domain decomposition methods and detailed implementation processes of the domain decomposition preprocessor (DDP) are given. Domain decomposition preconditioned conjugate gradient method (DDP-PCG) is made with the combination of DDP and PCG method. A confined groundwater problem with the analytical solution is used to verify the credibility of DDP-PCG. Under different discretizations of the study area,CG Jacobi-PCG, SSOR-PCG, DDP-PCG methods are used respectivly to solve the above-mentioned problem. Results show that the number of iterations of DDP-PCG are significantly lower than other methods in a variety of grid scale, and almost independent of the grid scale indicating the robustness of DDP-PCG. However, the CPU time-consuming of DDP-PCG is not the lowest which is caused by the low efficiency for solving the sub-region problems.A fast algorithm for the sub-region problems is the cornerstone of the domain decomposition algorithm. Based on the finite element theory, fourier analysis method(FAM) is studied for the homogeneous confined steady flow problem in a rectangular area with11kinds of combination of the Dirichlet and Neumann boundary conditions. A variety of transformation formulas in the FAM are calculated using the FFT algorithm and the corresponding computer program are prepared to realize the fast solution for the homogeneous confined steady flow problems. Then, the results of FAM are compared to the analytical solution or other numerical solution for11numerical experiments to indicate the credibility of FAM. And the numerical experiment for the confined steady flow model with the single well of fiexed pumping flow rate which has the analytic solution further validates the credibility of FAM. Under different subdivision of the study area, FAM and iterative methods (Jacobi, Gauss-Seidel, SOR, PCG) are used to solve the above problem. And calculation results show that:under a certain precision, the finer the discretization of the study aera is, the more significant is the efficiency of FAM. In addition, for the homogeneous confined steady flow problem, FAM does not need the calculation and storage of the original coefficient matrix, which makes much savement of the memory space.When the subregion problems of DDP-PCG are calculated with FAM, we get the domain decomposition preconditioned conjugate gradient method based on fourier analysis(FA-DDP-PCG). As for hydrogeological problems, the sub-region division modes, corresponding FAM for the subregions, as well as the solving steps of FA-DDP-PCG are given and corresponding computer program are prepared. FA-DDP-PCG is applied for homogeneous confined flow model with the analytical solution to verify the reliability of FA-DDP-PCG. Then CG, Jacobi-PCG, SSOR-PCG, FA-DDP-PCG are used to solve the model under different subdivisions, which show that FA-DDP-PCG is of higher efficiency compared to other methods and the finer the subdivision is, the more notable is FA-DDP-PCG’s efficiency under a certain precesion. Finally, confined aquifer with continuous coefficients and abrupt coefficients are caculated under finely subdivision, which further proves that FA-DDP-PCG’s efficiency are the highest of all. For the case with continuous coefficients, randomized trials explaine that akij (which represents the permeability coefficient or transmissibility) has little effect on the algorithm’s efficiency, which indicates FA-DDP-PCG is of strong robustness. For the case with abrupt coefficients, experiments with different parameters are caculated, which show that FA-DDP-PCG can effectively solve groundwater model with strongly abrupt coefficients. In all, FA-DDP-PCG is an effective method for solving large-scale groundwater flow problems.更多还原