有自由面渗流分析的加密高斯点单元传导矩阵调整法
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摘要
在水利水电工程中,常常会遇到有自由面的无压渗流问题,由于该类问题的自由面边界和逸出面边界事先未知,需要在迭代的过程中逐步形成,因此属于边界非线性问题。如何准确、快捷地确定自由面和逸出面的位置历来都是无压渗流分析的一个关键,也是一个难点。目前,有自由面的渗流问题主要是通过有限单元法来求解,其中的固定网格法凭借自身的优势,在无压渗流分析中逐渐取代了变网格法的传统地位,应用日益广泛。本文的加密高斯点单元传导矩阵调整法即是在已有固定网格法——单元传导矩阵调整法的基础上,对高斯点法的一种改进。
首先,简单地介绍了渗流的基本原理和渗流计算有限单元方法。接着对变网格法和现有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。单元传导矩阵调整法以自由面为分界线,自由面以下的饱和区域的渗透系数采用实际数值,自由面以上的非饱和区域的渗透系数则折减一个系数。高斯点法针对有限单元法采用高斯积分公式近似计算单元传导矩阵这一特点,直接根据高斯点处水头与位置高程的关系来调整渗透系数,不用每一次都计算自由面的位置。本文通过加密复合单元的高斯积分点个数,对高斯点法做了改进,得到了加密高斯点单元传导矩阵调整法,使复合单元的单元传导矩阵计算值更趋精确。文中还对逸出面边界进行了处理,实现了逸出面边界节点的可逆转换。
最后,采用C++语言编制了稳定渗流分析的有限元程序MESeep,从软件工程的角度对MESeep软件的计划和开发过程进行了介绍。运用MESeep软件对有
压、无压,二维、三维,均质、非均质等情况进行了稳定渗流分析,结果表明,加密高斯点单元传导矩阵调整法和根据该法开发的稳定渗流分析有限元软件MESeep能进行各种情况的稳定渗流分析,是可行的。
In hydraulic and hydropower engineering, the unconfined seepage problem with free surface is often confronted. Because the position of free surface and overflow boundaries are both unknown in advance, an iterative process is consequently required, it is a boundary nonlinear problem. At all times, how to locate the free surface and overflow boundaries reliably and efficiently is where the shoe pinches for the unconfined seepage field analysis. Currently, Finite Element Method (FEM) is the primary means for numerical analysis of seepage. Fixed Mesh Method (FMM), one of the FEM, owing to advantages of itself, has a very broad application, and is taking the place gradually of?the Altered Mesh Method (AMM), the traditional finite element method in this realm which need to modify the mesh when the free surface changes during the iteration process. Augmented Gauss Points Method proposed in this paper is an improved Gauss Points Method, and it is a FMM.
First, the basic principle of seepage and FEM for seepage calculation has been described briefly. And then several methods common in use were introduced, including AMM, Residual Schemes Method, Bathe Method and Initial Flow Method. The Bathe Method suggests divide the whole domain by free surface into two parts, saturated region and unsaturated region under and beyond the free surface, the element coefficient of permeability within the unsaturated region is
discounted to a very small value while keeps unchanged within the saturated region. Because that FEM adopt the Gaussian integrating formula to calculate the element conductivity matrix approximately, Gauss Points Method suggests adjust the permeability coefficient at Gaussian integrating point according to the relation between its hydraulic head and elevation, as a result, the calculation for the position of free surface is avoided. In order to raise the calculation precision of element conductivity matrix of the compound elements, Augmented Gauss Points Method has been adopted in this paper. According to this method, the number of Gaussian integrating points increases within the compound elements while keeps invariable within the others. At the same time, the treatment for the overflow boundary conditions was introduced in this paper.
In the end, the program MESeep for steady-state seepage analysis has been coded in C++ language. And the planning and developing processes of software MESeep were described. Employing the software MESeep, steady-state seepage analysis for cases of confined and unconfined, 2-D and 3-D, homogeneous and unhomogeneous, isotropic and anisotropic have been carried out. And the results demonstrate that Augmented Gauss Points Method and software MESeep are feasible.
首先,简单地介绍了渗流的基本原理和渗流计算有限单元方法。接着对变网格法和现有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。单元传导矩阵调整法以自由面为分界线,自由面以下的饱和区域的渗透系数采用实际数值,自由面以上的非饱和区域的渗透系数则折减一个系数。高斯点法针对有限单元法采用高斯积分公式近似计算单元传导矩阵这一特点,直接根据高斯点处水头与位置高程的关系来调整渗透系数,不用每一次都计算自由面的位置。本文通过加密复合单元的高斯积分点个数,对高斯点法做了改进,得到了加密高斯点单元传导矩阵调整法,使复合单元的单元传导矩阵计算值更趋精确。文中还对逸出面边界进行了处理,实现了逸出面边界节点的可逆转换。
最后,采用C++语言编制了稳定渗流分析的有限元程序MESeep,从软件工程的角度对MESeep软件的计划和开发过程进行了介绍。运用MESeep软件对有
压、无压,二维、三维,均质、非均质等情况进行了稳定渗流分析,结果表明,加密高斯点单元传导矩阵调整法和根据该法开发的稳定渗流分析有限元软件MESeep能进行各种情况的稳定渗流分析,是可行的。
In hydraulic and hydropower engineering, the unconfined seepage problem with free surface is often confronted. Because the position of free surface and overflow boundaries are both unknown in advance, an iterative process is consequently required, it is a boundary nonlinear problem. At all times, how to locate the free surface and overflow boundaries reliably and efficiently is where the shoe pinches for the unconfined seepage field analysis. Currently, Finite Element Method (FEM) is the primary means for numerical analysis of seepage. Fixed Mesh Method (FMM), one of the FEM, owing to advantages of itself, has a very broad application, and is taking the place gradually of?the Altered Mesh Method (AMM), the traditional finite element method in this realm which need to modify the mesh when the free surface changes during the iteration process. Augmented Gauss Points Method proposed in this paper is an improved Gauss Points Method, and it is a FMM.
First, the basic principle of seepage and FEM for seepage calculation has been described briefly. And then several methods common in use were introduced, including AMM, Residual Schemes Method, Bathe Method and Initial Flow Method. The Bathe Method suggests divide the whole domain by free surface into two parts, saturated region and unsaturated region under and beyond the free surface, the element coefficient of permeability within the unsaturated region is
discounted to a very small value while keeps unchanged within the saturated region. Because that FEM adopt the Gaussian integrating formula to calculate the element conductivity matrix approximately, Gauss Points Method suggests adjust the permeability coefficient at Gaussian integrating point according to the relation between its hydraulic head and elevation, as a result, the calculation for the position of free surface is avoided. In order to raise the calculation precision of element conductivity matrix of the compound elements, Augmented Gauss Points Method has been adopted in this paper. According to this method, the number of Gaussian integrating points increases within the compound elements while keeps invariable within the others. At the same time, the treatment for the overflow boundary conditions was introduced in this paper.
In the end, the program MESeep for steady-state seepage analysis has been coded in C++ language. And the planning and developing processes of software MESeep were described. Employing the software MESeep, steady-state seepage analysis for cases of confined and unconfined, 2-D and 3-D, homogeneous and unhomogeneous, isotropic and anisotropic have been carried out. And the results demonstrate that Augmented Gauss Points Method and software MESeep are feasible.
引文
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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.
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.
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.
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,
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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.
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.
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.
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.
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.
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.
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.
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.
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.
彭华,罗谷华.有自由面渗流分析的子单元及其应用[J].水电站设计,1996,12(4):24-27.
李春华.稳定渗流的有限元计算和决定自由面表面位置的一种方法.水利水电科学院论文集第8集(岩土工程),北京:水利电力出版社,1982:148-156.
陈士俊等.工程渗流有限元计算方法研究[J].人民黄河,2000,22(9):34-36.
徐书平,喻国安,扬剑飞.渗流分析边界条件处理的算法实现[J].土工基础,2002,16(2):51-54.
肖明.三维非均质岩体各项异性渗流场分析[J].武汉水利电力大学学报,1995,28(4):419-424.
杜延龄,许国安.复杂岩基三维渗流有限元分析研究[J].水利学报,1991,(7):
19-26.
李春华.稳定渗流有限元计算时采用固定网格法的初步研究.第三届全国渗流力学学术讨论会论文汇编(3),长江科学院,1986.
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彭华,曹定胜等.有自由面渗流分析的弃单元法及其应用[J].人民长江,1997,28(8):38-40.
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