定常不可压缩流的若干稳定化方法研究
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摘要
定常不可压缩流可以近似地看做常数的一种流体,它刻划着一些流体的运动规律,如海洋流动、大气运动以及透平机械内部流动等.特别地,它的研究对人们认识和控制湍流至关重要.描述这种流体的控制方程主要是不可压缩Navier–Stokes方程.由于人们对非线性现象本质认识有限,因而数值模拟就成为一种十分重要的研究手段.但直接数值模拟Navier–Stokes方程有一个很大的困难就是巨大的解题规模与有限的计算资源及算法稳定性之间的矛盾.因此,构造和研究具有良好稳定性和收敛性的高效算法就显得尤为重要.本文基于局部高斯积分公式,研究了定常不可压缩流的若干稳定化算法,主要做了下面的工作:
一、基于局部高斯积分公式,提出了求解定常Navier–Stokes方程的一种新亏量校正格式.它主要用一个双线性项来代替原先的人工粘性稳定化项.该双线性项是由二次多项式的精确高斯积分公式和线性多项式的精确高斯积分公式做差得到.不同于通常的亏量校正方法,新格式对参数依赖较小且不依赖于网格.并且通过数值实验发现,用我们提出方法得到的解比通常的亏量校正方法得到的解更为精确一些.
二、提出了基于三种校正算法的两水平二次等阶稳定化有限元方法.采用的P2P2元不满足离散的inf-sup条件,故而用基于局部高斯积分公式的压力投影技巧来稳定.得到了三种算法: Stokes校正、Newton校正和Oseen校正.数值理论分析和计算结果发现Stokes和Newton校正对大粘性问题处理较好,而Oseen校正求解小粘性系数问题则有一定的优越性.接着,建立了两水平亏量校正稳定化方法.在处理大雷诺数问题的时候,采用亏量校正方法来求解,但是往往需要很多的计算时间.而两水平方法是一种高效的算法,它能节约大量的CPU时间.双赢的组合对求解Navier–Stokes方程的大雷诺数问题有了很大的帮助.接着,基于P_1-P_1高斯积分稳定化方法,给出了三种算法: m次亏量步采用Oseen迭代,1次校正步采用Oseen迭代(m-Oseen-1-Oseen); m次亏量步采用Oseen迭代,1次校正步采用Simple迭代(m-Oseen-1-Simple); m次亏量步采用Oseen迭代,1次校正步采用Newton迭代(m-Oseen-1-Newton).
三、采用L2投影方法,对基于P_1-P_1元求解定常Navier–Stokes方程的稳定化非协调有限元方法和稳定化有限体积方法,分别建立了它们的超收敛结果.该方法的主要思想是把数值结果投影到另一个不同的较粗网格空间上.经过后处理,用两种网格尺度的差别来达到超收敛结果.数值例子验证了数值理论的准确性.
四、基于用局部高斯积分来做稳定化的P_1-P_1元,提出了解Stokes特征值问题的一种两水平有限元方法.该方法求得的解和一般的稳定化有限元求得的解具有相同的收敛阶,且我们的方法能够节省大量的计算时间.数值试验验证了理论结果.进而,给出了一些求解Stokes特征值问题的基于最低等阶元的稳定化方法,包括加罚方法、正则化方法、丰富多尺度方法、局部高斯积分方法以及非协调局部高斯积分方法.接着,对这些稳定化混合有限元方法在数值求解Stokes特征值问题时的表现做了比较和分析.最后,指出了相比其他方法,用非协调局部高斯积分方法求解Stokes特征值问题具有较好的稳定性和误差结果.
The stationary incompressible flow can be seen as a flow in steady, which de-scribes the motion law of some fluids. For example, the movement of the atmosphere,the transport in the ocean, the flow in the turbomachinery and so on. Particularly,it is very important to help people to understand and control turbulent. The maingoverning equations are the incompressible Navier–Stokes equations. Because it isnot very easy to realize the essence of the nonlinear phenomena, the numerical meth-ods have played an important role. However, numerical simulation of Navier–Stokesequations has a great difculty, i.e., the contradiction between huge problem sizeand very limited computing ability. Hence, constructing and studying an algorithmwith good stability and convergence are very important. In this thesis, based on thelocal Gauss integration, several stabilized methods for the stationary incompressibleflow are studied as follows:
1、Based on the local Gauss integration, we propose a novel defect-correctionmethod for the stationary Navier–Stokes equations. The method uses a bilinear termto replace an artificial viscosity stabilized term (i.e. adding to the bilinear form thediference between an exact Gaussian quadrature rule for quadratic polynomialsand an exact Gaussian quadrature rule for linear polynomials to ofset the inf-supcondition). Diferent from the common defect-correction method, it is independentof mesh size. Besides, from the numerical experiment, we can see that the results ofour method are much better than those of the common defect-correction method.
2、The two-level quadratic equal-order stabilized finite element method for thestationary Navier–Stokes equations based on the local Gauss integration is consid-ered. The method includes three corrections: Stokes correction, Newton correctionand Oseen correction. The theoretical analysis and numerical results confirm thatthe Stokes and Newton corrections are useful for large viscosity and the Oseen cor-rection is the best method for small viscosity. Moreover, two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokesequations based on the local Gauss integration are considered. The methods combinethe defect-correction method and the two-level strategy with the locally stabilizedmethod. It includes three algorithms: m defect steps by Oseen iteration and oncecorrection by Oseen iteration (m-Oseen-1-Oseen); m defect steps by Oseen itera-tion and once correction by Simple iteration (m-Oseen-1-Simple); m defect steps byOseen iteration and once correction by Newton iteration (m-Oseen-1-Newton).
3、Applying an L2-projection, we present the superconvergence results for thestationary Navier–Stokes equations by the stabilized nonconforming finite elementmethod and the stabilized finite volume method. The basic idea is to project theapproximate solution to another finite dimensional space on a diferent, but coarsermesh. The diference in the two mesh sizes can be used to achieve a superconvergenceafter the post-processing procedure. Numerical results are shown to support thedeveloped theory analysis.
4、A two-level stabilized finite element method for the Stokes eigenvalue prob-lem based on the local Gauss integration is considered. It provides an approximatesolution with the convergence rate of same order as the usual stabilized finite ele-ment solution. Hence, our method can save a large amount of computational time.Numerical tests confirm the theoretical results of the presented method. Moreover,several stabilized finite element methods for the Stokes eigenvalue problem basedon the lowest equal-order finite element pair are numerically investigated. They arepenalty, regular, multiscale enrichment, local Gauss integration, and nonconforminglocal Gauss integration method. Comparisons between them are carried out, whichshow that the nonconforming local Gauss integration method has good stability,efciency and accuracy properties and it is a favorite method among these methodsfor the Stokes eigenvalue problem.
一、基于局部高斯积分公式,提出了求解定常Navier–Stokes方程的一种新亏量校正格式.它主要用一个双线性项来代替原先的人工粘性稳定化项.该双线性项是由二次多项式的精确高斯积分公式和线性多项式的精确高斯积分公式做差得到.不同于通常的亏量校正方法,新格式对参数依赖较小且不依赖于网格.并且通过数值实验发现,用我们提出方法得到的解比通常的亏量校正方法得到的解更为精确一些.
二、提出了基于三种校正算法的两水平二次等阶稳定化有限元方法.采用的P2P2元不满足离散的inf-sup条件,故而用基于局部高斯积分公式的压力投影技巧来稳定.得到了三种算法: Stokes校正、Newton校正和Oseen校正.数值理论分析和计算结果发现Stokes和Newton校正对大粘性问题处理较好,而Oseen校正求解小粘性系数问题则有一定的优越性.接着,建立了两水平亏量校正稳定化方法.在处理大雷诺数问题的时候,采用亏量校正方法来求解,但是往往需要很多的计算时间.而两水平方法是一种高效的算法,它能节约大量的CPU时间.双赢的组合对求解Navier–Stokes方程的大雷诺数问题有了很大的帮助.接着,基于P_1-P_1高斯积分稳定化方法,给出了三种算法: m次亏量步采用Oseen迭代,1次校正步采用Oseen迭代(m-Oseen-1-Oseen); m次亏量步采用Oseen迭代,1次校正步采用Simple迭代(m-Oseen-1-Simple); m次亏量步采用Oseen迭代,1次校正步采用Newton迭代(m-Oseen-1-Newton).
三、采用L2投影方法,对基于P_1-P_1元求解定常Navier–Stokes方程的稳定化非协调有限元方法和稳定化有限体积方法,分别建立了它们的超收敛结果.该方法的主要思想是把数值结果投影到另一个不同的较粗网格空间上.经过后处理,用两种网格尺度的差别来达到超收敛结果.数值例子验证了数值理论的准确性.
四、基于用局部高斯积分来做稳定化的P_1-P_1元,提出了解Stokes特征值问题的一种两水平有限元方法.该方法求得的解和一般的稳定化有限元求得的解具有相同的收敛阶,且我们的方法能够节省大量的计算时间.数值试验验证了理论结果.进而,给出了一些求解Stokes特征值问题的基于最低等阶元的稳定化方法,包括加罚方法、正则化方法、丰富多尺度方法、局部高斯积分方法以及非协调局部高斯积分方法.接着,对这些稳定化混合有限元方法在数值求解Stokes特征值问题时的表现做了比较和分析.最后,指出了相比其他方法,用非协调局部高斯积分方法求解Stokes特征值问题具有较好的稳定性和误差结果.
The stationary incompressible flow can be seen as a flow in steady, which de-scribes the motion law of some fluids. For example, the movement of the atmosphere,the transport in the ocean, the flow in the turbomachinery and so on. Particularly,it is very important to help people to understand and control turbulent. The maingoverning equations are the incompressible Navier–Stokes equations. Because it isnot very easy to realize the essence of the nonlinear phenomena, the numerical meth-ods have played an important role. However, numerical simulation of Navier–Stokesequations has a great difculty, i.e., the contradiction between huge problem sizeand very limited computing ability. Hence, constructing and studying an algorithmwith good stability and convergence are very important. In this thesis, based on thelocal Gauss integration, several stabilized methods for the stationary incompressibleflow are studied as follows:
1、Based on the local Gauss integration, we propose a novel defect-correctionmethod for the stationary Navier–Stokes equations. The method uses a bilinear termto replace an artificial viscosity stabilized term (i.e. adding to the bilinear form thediference between an exact Gaussian quadrature rule for quadratic polynomialsand an exact Gaussian quadrature rule for linear polynomials to ofset the inf-supcondition). Diferent from the common defect-correction method, it is independentof mesh size. Besides, from the numerical experiment, we can see that the results ofour method are much better than those of the common defect-correction method.
2、The two-level quadratic equal-order stabilized finite element method for thestationary Navier–Stokes equations based on the local Gauss integration is consid-ered. The method includes three corrections: Stokes correction, Newton correctionand Oseen correction. The theoretical analysis and numerical results confirm thatthe Stokes and Newton corrections are useful for large viscosity and the Oseen cor-rection is the best method for small viscosity. Moreover, two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokesequations based on the local Gauss integration are considered. The methods combinethe defect-correction method and the two-level strategy with the locally stabilizedmethod. It includes three algorithms: m defect steps by Oseen iteration and oncecorrection by Oseen iteration (m-Oseen-1-Oseen); m defect steps by Oseen itera-tion and once correction by Simple iteration (m-Oseen-1-Simple); m defect steps byOseen iteration and once correction by Newton iteration (m-Oseen-1-Newton).
3、Applying an L2-projection, we present the superconvergence results for thestationary Navier–Stokes equations by the stabilized nonconforming finite elementmethod and the stabilized finite volume method. The basic idea is to project theapproximate solution to another finite dimensional space on a diferent, but coarsermesh. The diference in the two mesh sizes can be used to achieve a superconvergenceafter the post-processing procedure. Numerical results are shown to support thedeveloped theory analysis.
4、A two-level stabilized finite element method for the Stokes eigenvalue prob-lem based on the local Gauss integration is considered. It provides an approximatesolution with the convergence rate of same order as the usual stabilized finite ele-ment solution. Hence, our method can save a large amount of computational time.Numerical tests confirm the theoretical results of the presented method. Moreover,several stabilized finite element methods for the Stokes eigenvalue problem basedon the lowest equal-order finite element pair are numerically investigated. They arepenalty, regular, multiscale enrichment, local Gauss integration, and nonconforminglocal Gauss integration method. Comparisons between them are carried out, whichshow that the nonconforming local Gauss integration method has good stability,efciency and accuracy properties and it is a favorite method among these methodsfor the Stokes eigenvalue problem.
引文
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