块代数多重网格算法的研究及其实现技术在在油藏数值模拟软件中的应用
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摘要
本论文的贡献主要可以分为两大部分.
第一部分为块代数多重网格算法的研究.它组成如下：
基于将代数多重网格作为实用迭代法角度出发，我们在此首先假设在最粗层
上使用光滑算子迭代若干步，而不是使用直接法.在此假设下给出代数多重网格
的一般收敛性分析理论.它包括关于先光滑后粗网格校正，先粗网格校正后光滑
以及光滑－粗网格校正－光滑这三种V循环类型收敛因子估计的三个定理.然
后将其推广到块代数多重网格方法中，作为块代数多重网格的收敛性理论.
为构造有效的块代数多重网格算法。我们将块光滑算子和块插值算子分成两
类，针对一些常用矩阵的类型，分别分析了这两类块光滑算子和块插值算子要满
足块多重网格的一般收敛性理论的条件所应具备的若干性质，并且还证明了在多
层循环中的递归应用中，两类插值算子可以确保网格算子本质特性的一致性.
构造块代数多重网格的算法主要包括块代数多重网格的部件设置和循环迭
代求解两部分.在第一部分，针对两类光滑算子和插值算子，我们分别给出了块
块网格粗化算法和块插值算法第二部分与几何多重网格的循环原理基本相似,使
用块光滑算子和V循环进行求解.
数值结果表明这两类的算法对于大型各向同性问题是有效的.
第二部分研究代数多重网格实现技术在给定油藏数值模拟软件中的应用.
组成如下：
我们分析了大型油藏数值模拟软件解法器的矩阵结构特点.定量讨论了井系
数矩阵与网格系数矩阵在构造预条件子时的相互关系.
由发表的文章可以看出，目前将代数多重网格技术应用到解决实际问题的油
藏数值模拟软件产品中是一项创新性的研究工作.
该油藏数值模拟软件是一个大型、复杂并且高效的的油藏数值模拟软件，要
将代数多重网格的实现技术成功的应用其中，并提高其整体性能，其难度很大.
因为首先要寻找正确的技数路线.其次要透彻了解该油藏数值模拟软件与解法器
相关的各个细节，最后要用到各种编程技巧去压缩新增程序的运行开销，同时还
要用Blas技术优化程序的数值计算部分.
针对该大型油藏数值模拟软件，我们将代数多重网格预处理技术成功地应用
到油藏数值模拟的解法器中，提高了解法器和整体软件的效率.
由计算大庆油田实际问题的四组数值结果看出，嵌入代数多重网格预处理技
术的新油藏数值模拟软件提高了解法器的效率，分别减少原解法器的整体墙上时
·2·摘 要
间达13.4％，14.5％，23.1％和66.5％.表明，代数多重网格预处理技术对于大型收
敛问题的求解是非常有效的.
Wenjun Li
Directed by Prof. Jiachang Sun
The contributions of this thesis mainly fall into two parts.
The first part is the study of BAMG's algorithm. It consists of several
parts as following:
Based on the idea that Algebraic Multigrids should be the practically itera-
tive methods. we gite the premise that several iterations with smoothing operator
. instead of direct methods, are performed on the coarsest level. Then we have
the theorv of general convergence of AMG under this premise. It has three the-
orems about the estimates of convergence factors for three kinds of v─cycles
respectively, i.e.. one is smoothing is performed after the coarse grid correction
steps. another is smoothing is done before the coarse grid correction steps, and
the other is smoothing is done both before and after the coarse grid correction.
Then these theorems are extended to the those of Block AMG(BAMG) which
become the theories of general convergence of BAMG.
To construct the effective block Algebraic Multigrid Algorithms, we divide
block srnoothing operators and block interpolation operators into two kinds of
operators respectivelv: then we analyse several properties which two kinds block
smoothing operators and block interpolation operators should satisfv for the the-
orv of general convergence of BAMG, and prove that both of interpolation oper-
ators can guarantee the uniformitv of grid operator's essential properties in the
recursive application of multi-level cycles.
Construction of BAMG Algorithm mainly includes two parts: the setup of
the BAMG's components and the solving process. In the first part of BAMG
algorithnl. we provide block grid coarsening and block interpolation algorithms.
The second part of BAMG algorithm is similar to that of standard Multigrid,
i.e., solving process with block smoothing operators and V--cycle method.
Numerical results show that these two BAMG algorithms are effective in
solving the large scale isotropic problems.
The second part is the application of its realizable techniques in oil reservoir
numerical simulation software .It has the fOllowing several parts'
The structural characters of the matrix in the solver of large oil reservoir nu-
merical simulation software are analysed; the relationship between grid's matrix
and well's matrix are discussed quantitatively fOr construction of preconditioner.
From the published papers, it can be concluded that the application of AMG
technique into oil reservoir numerical simulation software is a new and creative
research work at present.
The given oil reservoir numerical simulation software is large,compIicated
and highly efficient. It is very difficult in applying AMG into the softtvare
successfully in order to improve its total efficiency Firstly the correct technique
line should be found,and all details of the solver and its related parts in the
given oil reservoir numerical simulation software should be understood very well.
In the end, many programming techniques of reducing the run time of the added'
programmes should be used, and Blas technique should be used to optimize the
programmes of numerical computaion parts.
With successful application of AMG pre-dispose technique into the solver of
large scale oil reservoir numerical simulation software 1 the efficiency of the solver
is improved.
The several numerical results coming from the computation of real problems
of DaQing oil field in China show that the new software with AMG pre-dispose
technique improves the efficiency of its solver, and reduce the wall time of the
original solver by 13.4%,l4.5%,23.1% and 66.5%. then the conclusions can be
drawed that .4MG pre-dispose technique is very effective in solving the large
convergent problems.
第一部分为块代数多重网格算法的研究.它组成如下：
基于将代数多重网格作为实用迭代法角度出发，我们在此首先假设在最粗层
上使用光滑算子迭代若干步，而不是使用直接法.在此假设下给出代数多重网格
的一般收敛性分析理论.它包括关于先光滑后粗网格校正，先粗网格校正后光滑
以及光滑－粗网格校正－光滑这三种V循环类型收敛因子估计的三个定理.然
后将其推广到块代数多重网格方法中，作为块代数多重网格的收敛性理论.
为构造有效的块代数多重网格算法。我们将块光滑算子和块插值算子分成两
类，针对一些常用矩阵的类型，分别分析了这两类块光滑算子和块插值算子要满
足块多重网格的一般收敛性理论的条件所应具备的若干性质，并且还证明了在多
层循环中的递归应用中，两类插值算子可以确保网格算子本质特性的一致性.
构造块代数多重网格的算法主要包括块代数多重网格的部件设置和循环迭
代求解两部分.在第一部分，针对两类光滑算子和插值算子，我们分别给出了块
块网格粗化算法和块插值算法第二部分与几何多重网格的循环原理基本相似,使
用块光滑算子和V循环进行求解.
数值结果表明这两类的算法对于大型各向同性问题是有效的.
第二部分研究代数多重网格实现技术在给定油藏数值模拟软件中的应用.
组成如下：
我们分析了大型油藏数值模拟软件解法器的矩阵结构特点.定量讨论了井系
数矩阵与网格系数矩阵在构造预条件子时的相互关系.
由发表的文章可以看出，目前将代数多重网格技术应用到解决实际问题的油
藏数值模拟软件产品中是一项创新性的研究工作.
该油藏数值模拟软件是一个大型、复杂并且高效的的油藏数值模拟软件，要
将代数多重网格的实现技术成功的应用其中，并提高其整体性能，其难度很大.
因为首先要寻找正确的技数路线.其次要透彻了解该油藏数值模拟软件与解法器
相关的各个细节，最后要用到各种编程技巧去压缩新增程序的运行开销，同时还
要用Blas技术优化程序的数值计算部分.
针对该大型油藏数值模拟软件，我们将代数多重网格预处理技术成功地应用
到油藏数值模拟的解法器中，提高了解法器和整体软件的效率.
由计算大庆油田实际问题的四组数值结果看出，嵌入代数多重网格预处理技
术的新油藏数值模拟软件提高了解法器的效率，分别减少原解法器的整体墙上时
·2·摘 要
间达13.4％，14.5％，23.1％和66.5％.表明，代数多重网格预处理技术对于大型收
敛问题的求解是非常有效的.
Wenjun Li
Directed by Prof. Jiachang Sun
The contributions of this thesis mainly fall into two parts.
The first part is the study of BAMG's algorithm. It consists of several
parts as following:
Based on the idea that Algebraic Multigrids should be the practically itera-
tive methods. we gite the premise that several iterations with smoothing operator
. instead of direct methods, are performed on the coarsest level. Then we have
the theorv of general convergence of AMG under this premise. It has three the-
orems about the estimates of convergence factors for three kinds of v─cycles
respectively, i.e.. one is smoothing is performed after the coarse grid correction
steps. another is smoothing is done before the coarse grid correction steps, and
the other is smoothing is done both before and after the coarse grid correction.
Then these theorems are extended to the those of Block AMG(BAMG) which
become the theories of general convergence of BAMG.
To construct the effective block Algebraic Multigrid Algorithms, we divide
block srnoothing operators and block interpolation operators into two kinds of
operators respectivelv: then we analyse several properties which two kinds block
smoothing operators and block interpolation operators should satisfv for the the-
orv of general convergence of BAMG, and prove that both of interpolation oper-
ators can guarantee the uniformitv of grid operator's essential properties in the
recursive application of multi-level cycles.
Construction of BAMG Algorithm mainly includes two parts: the setup of
the BAMG's components and the solving process. In the first part of BAMG
algorithnl. we provide block grid coarsening and block interpolation algorithms.
The second part of BAMG algorithm is similar to that of standard Multigrid,
i.e., solving process with block smoothing operators and V--cycle method.
Numerical results show that these two BAMG algorithms are effective in
solving the large scale isotropic problems.
The second part is the application of its realizable techniques in oil reservoir
numerical simulation software .It has the fOllowing several parts'
The structural characters of the matrix in the solver of large oil reservoir nu-
merical simulation software are analysed; the relationship between grid's matrix
and well's matrix are discussed quantitatively fOr construction of preconditioner.
From the published papers, it can be concluded that the application of AMG
technique into oil reservoir numerical simulation software is a new and creative
research work at present.
The given oil reservoir numerical simulation software is large,compIicated
and highly efficient. It is very difficult in applying AMG into the softtvare
successfully in order to improve its total efficiency Firstly the correct technique
line should be found,and all details of the solver and its related parts in the
given oil reservoir numerical simulation software should be understood very well.
In the end, many programming techniques of reducing the run time of the added'
programmes should be used, and Blas technique should be used to optimize the
programmes of numerical computaion parts.
With successful application of AMG pre-dispose technique into the solver of
large scale oil reservoir numerical simulation software 1 the efficiency of the solver
is improved.
The several numerical results coming from the computation of real problems
of DaQing oil field in China show that the new software with AMG pre-dispose
technique improves the efficiency of its solver, and reduce the wall time of the
original solver by 13.4%,l4.5%,23.1% and 66.5%. then the conclusions can be
drawed that .4MG pre-dispose technique is very effective in solving the large
convergent problems.
引文
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