采用Biot固结方程作为支配方程的渗流分析有限单元法
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摘要
在水利水电工程中,渗流计算是经常碰到的一个常规计算项目,应用范围非常广泛。在多孔介质中,稳定渗流的有限单元法计算日趋成熟,不论是在支配方程的选取,还是在有限单元法的应用上。但是非稳定渗流由于其支配方程的多样性,在数学上的建模相应也存在着多样性,而且在有限单元法的应用上也存在着一些困难。本文在深入分析多孔介质中的非稳定渗流计算物理过程的基础上,采用最能反映渗流本质的比奥(Biot)固结方程作为支配方程结合边界条件建立数学模型。在有限单元法的应用上,由于变网格法自身缺陷,采用固定网格法,但单元传导矩阵调整法与非稳定渗流支配方程的融合过程中又存在着一定的矛盾。
首先,简单介绍了渗流的基本原理和渗流分析的几种研究方法,并比较了它们的优点和缺点。在本文所关注的有限单元法中,从最基本的可压缩流体在复杂应力状态作用下的可变形各向异性渗流介质中的流动问题着手,推导了种种适应于某特定条件的简化支配方程,并较系统地说明了渗流中各支配方程所基于的物理模型、基本假设、各自的适应条件以及它们之间的相互联系。
接着,对变网格法和几种有影响的固定网格法,即剩余流量法、单元传导矩阵调整法和初流量法进行了分析。在这个基础上,提出了在非稳定渗流计算中的某些数学模型与它们之间的矛盾。由于几种固定网格法提出的算法是以稳定渗流为基础的,这样直接应用于非稳定渗流时,不可避免的要产生矛盾。
比奥(Biot)固结理论是比较成熟的应用于多孔介质中的流固耦合理论,比奥于1941 年建立比奥方程以来,由于计算问题,一直未在工程中广泛应用。
In hydraulic and hydropower engineering, the seepage calculation is a routine subject often confronted and applied in wide range. In porous medium, the F.E.M. for steady seepage is mature little by little, no matter in the field of choice of control equation, or of application of F.E.M..However, the unsteady seepage calculation is various in establishing mathematic model for its control equation is also various. Moreover, there are difficulties in application of F.E.M..Based on deep analysis of unsteady seepage’s physical process in the porous medium, mathematic model has been established with Biot consolidation equation as the control equation, which could most reflect the seepage. In the application of F.E.M., fixed mesh method is adopted because of the self-goodness, but there are some discrepancies in the blending process of Bathe Method and the unsteady seepage’s control equations.
First, the basic principle of seepage and methods used in seepage analysis has been described briefly, and then their merits and their demerits are compared. In F.E.M. for seepage calculation, beginning from the condensable flow in the deformable and aeolotropism medium, which is in the complex stress, various
simplified control equations have been deduced, and then their physical model, basic assumption, application condition and mutual relation are explained systemically. Second, several methods common in use are introduced, including A.M.M., Residual Schemes Method, Bathe Method and Initial Flow Method. On the base of this, it is put forward that the discrepancies between some mathematic models using in unsteady seepage analysis and them. Owing to several F.M.M. based on the steady seepage, it is inevitable that there are discrepancies when they are applied to the unsteady-state directly. Biot consolidation equation is more mature coupling fluid and solid theory applied in porous medium. Since it was established in 1941, it has not been applied to engineering because of calculation’s difficulties. With the development of the F.E.M., Biot consolidation is just began to apply to. In the end, the program ME1SEEP for coupling fluid and solid analysis has been coded in C++ language, which uses Biot consolidation equation as control equation. And the planning and developing processes of software ME1SEEP are described. After analyzing seepage in the dam base, the results demonstrate that it is feasible.
首先,简单介绍了渗流的基本原理和渗流分析的几种研究方法,并比较了它们的优点和缺点。在本文所关注的有限单元法中,从最基本的可压缩流体在复杂应力状态作用下的可变形各向异性渗流介质中的流动问题着手,推导了种种适应于某特定条件的简化支配方程,并较系统地说明了渗流中各支配方程所基于的物理模型、基本假设、各自的适应条件以及它们之间的相互联系。
接着,对变网格法和几种有影响的固定网格法,即剩余流量法、单元传导矩阵调整法和初流量法进行了分析。在这个基础上,提出了在非稳定渗流计算中的某些数学模型与它们之间的矛盾。由于几种固定网格法提出的算法是以稳定渗流为基础的,这样直接应用于非稳定渗流时,不可避免的要产生矛盾。
比奥(Biot)固结理论是比较成熟的应用于多孔介质中的流固耦合理论,比奥于1941 年建立比奥方程以来,由于计算问题,一直未在工程中广泛应用。
In hydraulic and hydropower engineering, the seepage calculation is a routine subject often confronted and applied in wide range. In porous medium, the F.E.M. for steady seepage is mature little by little, no matter in the field of choice of control equation, or of application of F.E.M..However, the unsteady seepage calculation is various in establishing mathematic model for its control equation is also various. Moreover, there are difficulties in application of F.E.M..Based on deep analysis of unsteady seepage’s physical process in the porous medium, mathematic model has been established with Biot consolidation equation as the control equation, which could most reflect the seepage. In the application of F.E.M., fixed mesh method is adopted because of the self-goodness, but there are some discrepancies in the blending process of Bathe Method and the unsteady seepage’s control equations.
First, the basic principle of seepage and methods used in seepage analysis has been described briefly, and then their merits and their demerits are compared. In F.E.M. for seepage calculation, beginning from the condensable flow in the deformable and aeolotropism medium, which is in the complex stress, various
simplified control equations have been deduced, and then their physical model, basic assumption, application condition and mutual relation are explained systemically. Second, several methods common in use are introduced, including A.M.M., Residual Schemes Method, Bathe Method and Initial Flow Method. On the base of this, it is put forward that the discrepancies between some mathematic models using in unsteady seepage analysis and them. Owing to several F.M.M. based on the steady seepage, it is inevitable that there are discrepancies when they are applied to the unsteady-state directly. Biot consolidation equation is more mature coupling fluid and solid theory applied in porous medium. Since it was established in 1941, it has not been applied to engineering because of calculation’s difficulties. With the development of the F.E.M., Biot consolidation is just began to apply to. In the end, the program ME1SEEP for coupling fluid and solid analysis has been coded in C++ language, which uses Biot consolidation equation as control equation. And the planning and developing processes of software ME1SEEP are described. After analyzing seepage in the dam base, the results demonstrate that it is feasible.
引文
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[19] 王均星,吴雅峰,百呈富.有自由面渗流分析的流形单元法[J].水电能源科学2003,21(4):23-25.
[20] 张有天,陈平,王镭.有自由面渗流分析的初流量法[J].水利学报,1988,(8):18-26.
[21] 王媛.求解有自由面渗流问题的初流量法的改进[J].水利学报,1998,(3):68-73.
[22] 梁业国,熊文林,周创兵.有自由面渗流分析的子单元法[J].水利学报,1997,(8):34-38.
[23] 黄蔚,刘迎曦,周承芳.三维无压渗流场的有限元算法研究[J].水利学报,2001,(6):33-36.
[24] 速宝玉,沈振中,赵坚.用变分不等式理论求解渗流问题的截止负压法[J].水利学报,1996,(3):22-29.
[25] 李广信,葛镜宏,介玉新.有自由面渗流的无单元法[J].清华大学学报(自然科学版),2002,42(11):1552-1553.
[26] 葛镜宏,李广信,介玉新.无单元法在有自由面渗流计算中的应用[J].计算力学学报,2003,20(2):241-254.
[27] 王贤能,黄润秋.有自由面渗流分析的高斯点法[J].水文地质工程地质,1997,(6):1-4.
[28] 朱军,刘光廷.改进的单元渗透矩阵调整法求解无压渗流场[J].水利学报,2001,(8):49-52.
[29] 吴梦喜,张学勤.有自由面渗流数值分析的虚单元法[J].水利学报,1994,(8):67-71.
[30] 吴梦喜,高莲士.饱和-非饱和土体非稳定渗流数值分析[J].水利学报,1999,(12):38-42.
[31] 丁留谦,许国安.三维非达西渗流的有限元分析[J].水力学报,1990,(10):49-54.
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