流体及流体耦合问题的有限元方法研究

发布时间：2018-04-04 09:49

本文选题：流体及流体耦合问题　切入点：特征线　出处：《南京师范大学》2016年博士论文

【摘要】：在本文中,我们研究几类流体及流体耦合问题的有限元方法。流体及流体耦合问题在海洋学、地球物理学以及流体力学中经常遇到。例如,低速运动的气流,水流,地下水污染问题以及大气-海洋耦合问题等。当应用通常的有限元方法来数值求解这些问题,由于对流占优特性、高雷诺数问题及线性和非线性耦合条件,通常的有限元方法会使数值方法的有效性变差。本文的目的是综合运用特征方法、变分多尺度方法及稳定化有限元方法,设计有效求解此类问题的数值方法,给出相应的稳定性分析和误差估计。第一章,我们提出了求解对流占优对流扩散反应问题的特征变分多尺度方法。该格式的构造综合了特征线方法和变分多尺度方法,给出了相应格式的稳定性分析和误差估计。该格式不仅降低了时间截断误差、可以应用较大的时间步长而且还保持了良好的稳定性和高精度。二维和三维数值试验表明了该格式的有效性。第二章,基于最低阶的等阶协调有限元子空间,我们给出了一类新的特征稳定化有限元方法数值求解不可压的Navier-Stokes方程。我们运用动量方程的残量和散度自由方程定义稳定化项。特征线方法和稳定化有限元方法的自然组合保持了两类方法的最优特点。严格推出了相应格式的稳定性和误差估计。最后,数值试验验证了该方法数值求解非稳态Navier-Stokes方程的有效性。第三章,我们研究了逼近流体耦合问题的数值方法,考虑一个简化模型,两个对流占优对流扩散反应方程通过界面条件耦合,提出了隐显时间步进流线扩散法求解该类问题,得到了相应的稳定性分析和误差估计,数值试验证明了该方法的有效性。第四章,我们分析了局部投影稳定化特征解耦格式求解流体耦合问题。我们应用特征线方法克服非线性项导致的困难,应用局部投影稳定化方法来控制伪振荡,应用几何平均的思想解耦耦合问题。给出相应格式的稳定性分析,数值试验证明了该方法的有效性。
[Abstract]:In this paper, we study the finite element method for several kinds of fluid and fluid coupling problems.Fluid and fluid coupling problems are often encountered in oceanography, geophysics, and fluid dynamics.For example, low-speed moving air flow, water flow, groundwater pollution and atmospheric-ocean coupling problem and so on.When the conventional finite element method is used to solve these problems, due to the convection dominance, the high Reynolds number problem and the linear and nonlinear coupling conditions, the general finite element method will make the effectiveness of the numerical method worse.The purpose of this paper is to design an effective numerical method for solving this kind of problems by means of characteristic method, variational multi-scale method and stabilized finite element method, and give the corresponding stability analysis and error estimation.In chapter 1, we propose a characteristic variational multiscale method for convection-dominated convection-diffusion reaction problems.The construction of the scheme combines the eigenline method and the variational multi-scale method, and gives the stability analysis and error estimation of the corresponding scheme.This scheme not only reduces the time truncation error, but also keeps good stability and high precision.Two-dimensional and three-dimensional numerical experiments show the effectiveness of the scheme.In chapter 2, based on the lowest order equal-order conforming finite element subspace, we give a new type of eigen-stabilized finite element method to solve the incompressible Navier-Stokes equation numerically.We define the stabilization term by using the residual and divergence free equations of momentum equation.The natural combination of the eigenline method and the stabilized finite element method preserves the optimal characteristics of the two methods.The stability and error estimation of the corresponding scheme are strictly derived.Finally, numerical experiments show that the method is effective in solving unsteady Navier-Stokes equations.In chapter 3, we study the numerical method of approximate fluid coupling problem, and consider a simplified model. Two convection-dominated convection-diffusion reaction equations are coupled by interfacial conditions, and a implicit time step streamline diffusion method is proposed to solve this kind of problems.The corresponding stability analysis and error estimation are obtained, and the effectiveness of the method is proved by numerical experiments.In chapter 4, we analyze the characteristic decoupling scheme of local projection stabilization to solve the fluid coupling problem.We apply the eigenline method to overcome the difficulties caused by nonlinear terms, apply the method of local projection stabilization to control the pseudo oscillation, and apply the idea of geometric mean to decouple the coupling problem.The stability analysis of the corresponding scheme is given, and the effectiveness of the method is proved by numerical experiments.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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