子域积分法及组合网格算法在无压渗流中的应用
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摘要
由于只需要进行一次传导矩阵的集成与分解,流量法在稳定无压渗流分析中获得了广泛的应用。但由于渗透系数修正矩阵在自由面上、下的间断性质,导致初流量法中,与自由面相交的单元中结点初流量的被积函数是不连续的。采用传统的Gauss积分方式时,这些单元内数值积分的过程中可能出现单元部分区域的贡献被忽略、被夸大或自由面在数值积分点上、下的轻微移动导致结点初流量数值积分结果出现突变等问题,降低数值计算的收敛性与稳定性。
初流量法在非稳定流中应用时,也存在着渗透系数修正矩阵和贮水率在自由面上、下的间断性质引起的非连续被积函数。这2项的数值积分计算可能出现较大误差,降低了数值计算的收敛性与稳定性。对这些非连续项的积分计算进行改进,对于提高数值计算的稳定性和收敛性,促进初流量法的进一步推广应用十分有意义。
在无压稳定渗流分析过程中,常常会遇到某一方向尺寸与整体计算区域相比非常小的线形结构。当这些结构中的水流运动不能被忽略的时候,需要对这些结构需要进行精细离散,但这些结构的厚度可达厘米级甚至毫米级,为保持计算精度必然导致其它区域也需要精细的离散,一方面造成网格离散的工作量和计算量大幅增加,另一方面造成有限元求解精度降低、收敛精度较差。针对这些问题,提出一种合理可行的处理方法对于减小网格离散工作量和难度,获取线形结构中渗流场精细分布的同时提高计算的速度和精度有十分重要的意义。
本文针对上述问题进行了研究,提出了子域积分理论、组合网格法在无压渗流中的应用。采用FORTRAN语言编制了相应的计算程序,并对程序及理论的可靠性进行了验证。
本文的主要研究内容和创新性成果包括以下三个方面：
(1)基于子域积分方法,对无压稳定渗流的初流量法进行了改进。将与自由面相交的单元分为自由面上、下两部分,再根据这两个区域的具体形状将其细化为标准形式的区域一子域,然后把每个子域看做一个常规的“单元”,在每个子域内按照标准Gauss积分方式布置数值积分点,从而保证每个数值积分点控制的区域内结点初流量的被积函数都是连续的。避免了结点初流量被夸大、忽略或者自由面在积分点上、下的微小移动导致的数值积分结果出现突变的情况,从而改善了数值计算的稳定性和收敛性。文中给出了与自由面相交的平面4边形和3维6面体单元的基本类型及其具体子域划分方式,同时给出了非基本类型单元和基本类型单元的变换关系。编制了基于子域积分的改进初流量法稳定无压渗流程序IIFM2DS(平面渗流)和IIFM3DS(三维渗流)。
(2)借鉴饱和非饱和渗流理论以及Desai等人的研究,将适用于稳定渗流的初流量法推广到非稳定渗流,根据虚位移原理推导了有限元离散格式,并采用子域积分对其中的非连续项的数值积分进行改进,编制了相应的二维有限元程序IIFM2DT和三维有限元程序IIFM3DT。
(3)将组合网格法应用于无压稳定渗流分析中。采用两套独立的网格进行模拟,对整体区域采用尺寸较大的粗网格进行模拟(不考虑线性结构的影响),对线形结构则采用尺寸比较小的细网格进行模拟。细网格通过流量修正对粗网格渗流场进行调整,计算在粗细两套网格之间迭代进行,直至达到收敛精度。组合网格法适应于非规则网格,并且粗细网格均可独立生成,彼此互不制约。组合网格法减小了网格离散工作量和难度,在获取线形结构中渗流场精细分布的同时提高了计算的速度和精度。根据上述理论编制了三维无压稳定渗流组合网格有限元计算程序IIFM3DS-CGM。
最后,采用数值算例和工程实际应用对本文理论及程序的有效性和可靠性进行了验证。计算结果表明,本文的理论是合理可行,程序是可靠的。
Since the necessity for the assembly and decomposition of the conduction matrix is needed only one time, the Initial Flow Method has been widely used in steady unconfined seepage analysis. However, because of the "jump" of permeability between the upside and underside of the free surface, the integrand of nodal initial flow is discontinuous in elements that intersect with the free surface in this method. If traditional Gaussian integral method is adopted in numerical integration, the contribution of some domain in these elements to the nodal initial flow may be ignored, exaggerated or mutation of nodal initial flow might appear when slight movement of the free surface from the underside of the integration point to the upside, reducing the convergence and stability of numerical simulation.
There are also discontinuous intgrands related to the "jump" of permeability and specific storage coefficient around the free surface in the Initial Flow Method when it is adopted in the unconfined transient seepage analysis. Once again, the existence of these two discontinuous terms greatly reduced the convergence rate and stability of numerical simulation as large integration error may be caused. Improvement for the integration of these discontinuous terms is of great value to improvement of the algorithom's stability and convergence and of great importance for the further application of Initial Flow Method.
In unconfined steady seepage analysis,"line-style structure" that has a very small size in some direction compared with the whole domain is often encountered. When the flow in these structures can not be ignored, they have to be finely discreted. However, the thickness of these structures may be up to the centimeter or millimeter level, this inevitably leads to the fine discretization also in other regions so as to maintain accuracy, greatly increasing the difficulty and the workload of discretization as well as computation and degrading the precision and convergence of Finite Element simulation. To solve these problems, it is of great significance that a reasonable and feasible approach for reducing the discretization, deserving the seepage characteristics in these structures and improving the caculation speed and accuracy is presented.
Theoretical study is carried out on the problems mentioned above in this paper. The application of the subdomain integration theory, the composite grid method in the unconfined seepage analysis and the extension of Initial Flow Method to unconfiend transient seepage analysis is presented., the initial flow method extended to the non-steady seepage calculation. Corresponding computer programs writen with FORTRAN language is coed accordingly to verify their validation.
The main content and innovative achievements include the following three aspects:
(1) The Initial Flow Method is improved based on the sub-domain integration method. The elements intersecting with the free surface are divided into two parts, one part is below the free surface while the other not. Then these two parts are refined to standard finer regions named sub-domains according to their specific shape. Thereafter, each sub-domain is treated as if it is a "conventional element" and Gaussian integration points are arranged in each sub-domain as in the standard Gaussian integration points sampling process. By this means, the integrand of nodal initial flow is continuous within the control volme/area of each integration points. Exaggeration/ignorance of some domain of the element's contribution to nodal initial flow and mutation of nodal initial flow that might appear when slight movement of the free surface from the underside of the integration point to its upside is avoided. Therefore, stability and convergence of the Initial Flow Method is improved. The basic types of quadrilateral and hexahedron that intersect with the free surface are given and their detailed modes of sub-domain subdivision is presented in the case of two-dimension and three-dimensional seepage analysis. The transformation between these basic types of elements and others is also listed in this paper. The programs IIFM2DS (Improved Initial Flow Method2D with sub-domain improvement, S stands for steady) and IIFM3DS are coded for two-dimensional and three-dimensional unconfined steady seepage analysis seperately.
(2)Drawing lessons from saturated-unsaturated seepage theory and research of Desai et al., the Initial Flow Method is extended to unconfined transient seepage analysis from steady case. The Finite Element discretization scheme is derived base on virtual displacement principle and the discontinuous terms are still improved by sub-domain integration method as in the steady case. The2D program IIFM2DT and3D program IIFM3DT are developed independently.
(3) Composite Grid Method is applied in unconfined steady seepage analysis. Two sets of grid are used in numerical simulation, with the coarse grid of larger size simulating the entire region without consideration of "line-style structure" while the fine grid of relatively small size simulating the "line-style structure". Solution in coarse grid is adjusted by the one from fine grid through discharge correction and iterations are carried out between the two sets of grids and until desired convergence precision is achieved. The Composite Grid Method can be adapted to non-regular grid. In other words, the coaser grid and fine grid can be generated independently without any restriction from each other. A three-dimensional unconfined steady seepage analysis program named IIFM3DS-CGM is developed accordingly.
Finally, numerical examples and practical application in engineering are taken advantage to verify the validity and reliability of these theory and programs. The results show that the theory is reasonable and the programs are reliable.
初流量法在非稳定流中应用时,也存在着渗透系数修正矩阵和贮水率在自由面上、下的间断性质引起的非连续被积函数。这2项的数值积分计算可能出现较大误差,降低了数值计算的收敛性与稳定性。对这些非连续项的积分计算进行改进,对于提高数值计算的稳定性和收敛性,促进初流量法的进一步推广应用十分有意义。
在无压稳定渗流分析过程中,常常会遇到某一方向尺寸与整体计算区域相比非常小的线形结构。当这些结构中的水流运动不能被忽略的时候,需要对这些结构需要进行精细离散,但这些结构的厚度可达厘米级甚至毫米级,为保持计算精度必然导致其它区域也需要精细的离散,一方面造成网格离散的工作量和计算量大幅增加,另一方面造成有限元求解精度降低、收敛精度较差。针对这些问题,提出一种合理可行的处理方法对于减小网格离散工作量和难度,获取线形结构中渗流场精细分布的同时提高计算的速度和精度有十分重要的意义。
本文针对上述问题进行了研究,提出了子域积分理论、组合网格法在无压渗流中的应用。采用FORTRAN语言编制了相应的计算程序,并对程序及理论的可靠性进行了验证。
本文的主要研究内容和创新性成果包括以下三个方面：
(1)基于子域积分方法,对无压稳定渗流的初流量法进行了改进。将与自由面相交的单元分为自由面上、下两部分,再根据这两个区域的具体形状将其细化为标准形式的区域一子域,然后把每个子域看做一个常规的“单元”,在每个子域内按照标准Gauss积分方式布置数值积分点,从而保证每个数值积分点控制的区域内结点初流量的被积函数都是连续的。避免了结点初流量被夸大、忽略或者自由面在积分点上、下的微小移动导致的数值积分结果出现突变的情况,从而改善了数值计算的稳定性和收敛性。文中给出了与自由面相交的平面4边形和3维6面体单元的基本类型及其具体子域划分方式,同时给出了非基本类型单元和基本类型单元的变换关系。编制了基于子域积分的改进初流量法稳定无压渗流程序IIFM2DS(平面渗流)和IIFM3DS(三维渗流)。
(2)借鉴饱和非饱和渗流理论以及Desai等人的研究,将适用于稳定渗流的初流量法推广到非稳定渗流,根据虚位移原理推导了有限元离散格式,并采用子域积分对其中的非连续项的数值积分进行改进,编制了相应的二维有限元程序IIFM2DT和三维有限元程序IIFM3DT。
(3)将组合网格法应用于无压稳定渗流分析中。采用两套独立的网格进行模拟,对整体区域采用尺寸较大的粗网格进行模拟(不考虑线性结构的影响),对线形结构则采用尺寸比较小的细网格进行模拟。细网格通过流量修正对粗网格渗流场进行调整,计算在粗细两套网格之间迭代进行,直至达到收敛精度。组合网格法适应于非规则网格,并且粗细网格均可独立生成,彼此互不制约。组合网格法减小了网格离散工作量和难度,在获取线形结构中渗流场精细分布的同时提高了计算的速度和精度。根据上述理论编制了三维无压稳定渗流组合网格有限元计算程序IIFM3DS-CGM。
最后,采用数值算例和工程实际应用对本文理论及程序的有效性和可靠性进行了验证。计算结果表明,本文的理论是合理可行,程序是可靠的。
Since the necessity for the assembly and decomposition of the conduction matrix is needed only one time, the Initial Flow Method has been widely used in steady unconfined seepage analysis. However, because of the "jump" of permeability between the upside and underside of the free surface, the integrand of nodal initial flow is discontinuous in elements that intersect with the free surface in this method. If traditional Gaussian integral method is adopted in numerical integration, the contribution of some domain in these elements to the nodal initial flow may be ignored, exaggerated or mutation of nodal initial flow might appear when slight movement of the free surface from the underside of the integration point to the upside, reducing the convergence and stability of numerical simulation.
There are also discontinuous intgrands related to the "jump" of permeability and specific storage coefficient around the free surface in the Initial Flow Method when it is adopted in the unconfined transient seepage analysis. Once again, the existence of these two discontinuous terms greatly reduced the convergence rate and stability of numerical simulation as large integration error may be caused. Improvement for the integration of these discontinuous terms is of great value to improvement of the algorithom's stability and convergence and of great importance for the further application of Initial Flow Method.
In unconfined steady seepage analysis,"line-style structure" that has a very small size in some direction compared with the whole domain is often encountered. When the flow in these structures can not be ignored, they have to be finely discreted. However, the thickness of these structures may be up to the centimeter or millimeter level, this inevitably leads to the fine discretization also in other regions so as to maintain accuracy, greatly increasing the difficulty and the workload of discretization as well as computation and degrading the precision and convergence of Finite Element simulation. To solve these problems, it is of great significance that a reasonable and feasible approach for reducing the discretization, deserving the seepage characteristics in these structures and improving the caculation speed and accuracy is presented.
Theoretical study is carried out on the problems mentioned above in this paper. The application of the subdomain integration theory, the composite grid method in the unconfined seepage analysis and the extension of Initial Flow Method to unconfiend transient seepage analysis is presented., the initial flow method extended to the non-steady seepage calculation. Corresponding computer programs writen with FORTRAN language is coed accordingly to verify their validation.
The main content and innovative achievements include the following three aspects:
(1) The Initial Flow Method is improved based on the sub-domain integration method. The elements intersecting with the free surface are divided into two parts, one part is below the free surface while the other not. Then these two parts are refined to standard finer regions named sub-domains according to their specific shape. Thereafter, each sub-domain is treated as if it is a "conventional element" and Gaussian integration points are arranged in each sub-domain as in the standard Gaussian integration points sampling process. By this means, the integrand of nodal initial flow is continuous within the control volme/area of each integration points. Exaggeration/ignorance of some domain of the element's contribution to nodal initial flow and mutation of nodal initial flow that might appear when slight movement of the free surface from the underside of the integration point to its upside is avoided. Therefore, stability and convergence of the Initial Flow Method is improved. The basic types of quadrilateral and hexahedron that intersect with the free surface are given and their detailed modes of sub-domain subdivision is presented in the case of two-dimension and three-dimensional seepage analysis. The transformation between these basic types of elements and others is also listed in this paper. The programs IIFM2DS (Improved Initial Flow Method2D with sub-domain improvement, S stands for steady) and IIFM3DS are coded for two-dimensional and three-dimensional unconfined steady seepage analysis seperately.
(2)Drawing lessons from saturated-unsaturated seepage theory and research of Desai et al., the Initial Flow Method is extended to unconfined transient seepage analysis from steady case. The Finite Element discretization scheme is derived base on virtual displacement principle and the discontinuous terms are still improved by sub-domain integration method as in the steady case. The2D program IIFM2DT and3D program IIFM3DT are developed independently.
(3) Composite Grid Method is applied in unconfined steady seepage analysis. Two sets of grid are used in numerical simulation, with the coarse grid of larger size simulating the entire region without consideration of "line-style structure" while the fine grid of relatively small size simulating the "line-style structure". Solution in coarse grid is adjusted by the one from fine grid through discharge correction and iterations are carried out between the two sets of grids and until desired convergence precision is achieved. The Composite Grid Method can be adapted to non-regular grid. In other words, the coaser grid and fine grid can be generated independently without any restriction from each other. A three-dimensional unconfined steady seepage analysis program named IIFM3DS-CGM is developed accordingly.
Finally, numerical examples and practical application in engineering are taken advantage to verify the validity and reliability of these theory and programs. The results show that the theory is reasonable and the programs are reliable.
引文
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