界面问题的有限体积元法研究

发布时间：2018-05-23 16:22

本文选题：非匹配网格 + 混合有限体积元法　；参考：《吉林大学》2016年博士论文

【摘要】：有限体积元法是求解偏微分方程的主要方法之一,其优点是不仅能够更灵活地处理复杂的几何区域和边界条件,而且能够保持某些物理量的守恒性,因此近些年得到了较大的发展,但是现有的研究结果几乎都集中在常规的单介质问题,也就是非界面问题.然而,在许多实际问题中,所遇到的模型是具有多介质、多物理性质的界面问题,这类问题的数值求解具有更大的挑战,同时也更具有应用价值.在本文中,我们针对三类界面问题,研究了其相关的有限体积元法.首先考虑的界面问题是以压力函数和速度函数为未知函数的Darcy流模型,该模型的特点是计算区域被分解为有限个非重叠的子区域,在任意两个相邻子区域间的交界面上扩散系数矩阵可能是间断的.这种区域结构常用于带有断层的多孔介质流的模拟,另外为了在局部上获得高精度的梯度近似,也会用到这种区域结构.对于这类问题的数值求解,在每个子区域上需要独立地定义网格剖分,那么在相邻子区域间的交界面的两侧,网格节点是非匹配的.在这种非匹配的网格上,构造数值格式并进行相应的收敛性分析是有很大难度的.本文在混合有限体积元法的框架下,研究了该问题的数值求解.接下来考虑的界面问题是辐射三温能量方程组.辐射三温能量方程组属于非平衡辐射扩散方程组,在惯性约束聚变、磁约束聚变、天体物理、高超声速等领域中有广泛的应用.三温指的是电子、离子、光子的温度函数,该方程组用来描述辐射能在多介质物理区域内的传输过程以及电子、离子、光子这三者之间的能量交换过程.由于这类方程组的系数是强间断、强非线性及高度耦合的,因此相关的数值求解非常困难.首先要求得到的数值解具有良好的单调性(保正性)和守恒性.此外由于计算量较大,因此如何提高计算效率也是大家需要考虑的问题之一.本文将用于求解二阶椭圆方程的有限体积元法,推广到辐射三温能量方程组,研究了两种有限体积元格式并给出了一种网格自适应算法.最后考虑的是浸入界面问题.这类问题的求解区域被一条充分光滑的曲线划分为两个非重叠的子区域,这条曲线被称为界面,方程的扩散系数在每个子区域上是连续的,但在界面上是间断的.在做网格剖分时,不考虑界面的存在,在求解区域内独立地进行网格剖分,这时会出现界面穿过剖分单元的情形,此时如果采用传统的有限元法来进行数值求解,所得的计算结果无法达到最优阶的数值精度.为了改善这一缺陷,一些学者构造了一种新的有限元空间,即浸入界面有限元空间.若采用浸入界面有限元空间作为试探函数空间,所得的浸入界面有限元法的收敛阶可以接近最优.虽然这类方法在实际应用中发展迅速,但是相应的收敛性分析非常困难,至今仍有许多问题亟待解决.本文将浸入界面有限元空间应用到有限体积元法框架下,研究了一种带有惩罚项的浸入界面有限体积元法.本文前一部分共分五章,最后一部分是结论,前五章的内容包括：第一章是绪论,先简单介绍有限体积元法,然后介绍本文研究内容的背景和发展状况,即非匹配网格上Darcy流的混合法,非平衡辐射扩散方程以及浸入界面问题的数值解法.第二章考虑的是非匹配网格上的Darcy流模型,即前面提到的第一类界面问题.在每个子区域上,采用的是三角形网格剖分,其相应的对偶剖分为重心对偶剖分.在每个子区域网格上,选取最低阶的Raviart-Thomas空间来近似速度函数和压力函数,并按照标准的混合有限体积元法来离散原方程.由于网格的不匹配,近似速度空间在界面上不再满足法向流连续,并且变分方程中会涉及压力函数在界面上的迹.在本章中,我们在界面上引入线性Mortar元空间来近似压力函数的迹,并添加了一个用于提高近似速度流连续性的界面条件.这样所得的数值格式称为Mortar元混合有限体积元法,我们从理论和数值实验两个方面证明了该格式按L2范数具有最优的收敛阶.第三章考虑的仍然是非匹配网格上的Darcy流模型.在每个子区域上,采用的网格剖分、对偶剖分以及近似速度函数空间和近似压力函数空间与前一章中的相同,并仍然按照标准的混合有限体积元法来离散原方程.但是在界面上,本章采用双重拉格朗乘子空间来近似压力函数的迹.相对于每个子区域,其上的网格剖分在界面上会诱导出一个一维的网格,在这个网格上引入一个分片常数函数空间来近似压力函数的迹,这样便得到一个双重取值的拉格朗日乘子空间.此外,我们添加一个Robin型界面条件来增强近似函数在界面处的连续性.这样所得的数值格式称为非Mortar元混合有限体积元法.同样,我们从理论和数值实验两个方面证明了该格式按L2范数具有最优的收敛阶.第四章考虑的是辐射三温能量方程组,即前面提到的第二类界面问题.对于这类问题的数值求解,需要克服的主要问题有两个,分别是单调性和守恒性.本章从辐射三温能量方程组的守恒形式出发,采用合理的数值积分公式和近似方法来处理非线性项和间断系数.借助不同的积分公式,构造了两种守恒的有限体积元格式.由单调性分析和数值实验来看,第一种格式在许多网格上是单调的,我们推导出了相应的网格限制条件.然而,第二种格式不可能保持单调性并且在数值模拟的一开始便会产生大量的负数温度,这与实际问题不相符.因此,第二种格式通常被认为是不可用的.但是我们设计了两种后处理技术来克服这一问题,包括全局修补技术和截断法.数值结果表明,这两种后处理技术都具有较好的计算效果.最后,我们设计了一个基于残量型后验误差估计的自适应算法,使网格能够灵活地局部加细和粗化,这在很大程度上提高了计算效率.第五章针对浸入界面问题,即前面提到的第三类界面问题,构造了一种新的浸入界面有限体积元法.称被界面穿过的网格单元为界面单元,称不被界面穿过的网格单元为非界面单元.浸入界面有限元空间的构造方式为：在非界面单元上选取以节点为自由度的多项式,在界面单元上利用界面条件重新构造一个分片多项式.当选取浸入界面有限元空间作为试探函数空间时,相对应的有限体积元法被称为浸入界面有限体积元法.本章在已有的浸入界面有限体积元法的基础上,将原有的格式作了修正,在界面分划和与界面相交的边上,添加了两个惩罚项,分别用来限制函数值和法向流在界面上的跳跃.数值结果表明,这样所得的数值格式具有较好的稳定性,即使当扩散系数在界面上的跳跃较大时,数值解仍然能保持最佳的收敛性.通过严格的理论分析,我们论证了修正后的浸入界面有限体积元法的稳定性,进而得到该方法解的存在性与唯一性.
[Abstract]:The finite volume element method is one of the main methods to solve the partial differential equation. Its advantage is not only to deal with complex geometric regions and boundary conditions more flexibly, but also to maintain the conservation of some physical quantities, so it has been greatly developed in recent years, but the existing research results are mostly concentrated on the conventional single medium problem. However, in many practical problems, the model is a multi medium and multi physical interface problem. The numerical solution of this kind of problem has a greater challenge and more practical value. In this paper, we have studied the finite volume element method for the three types of interface problems. First of all, we have studied the finite volume element method. The interface problem is a Darcy flow model with an unknown function of pressure function and velocity function. The characteristic of the model is that the calculation area is decomposed into a limited non overlapping subregion, and the diffusion coefficient matrix may be discontinuous at the intersection of any two adjacent subregions. This regional structure is often used in porous media with a fault. The simulation of flow, in addition to obtaining high precision gradient approximation for locally, also uses this regional structure. For the numerical solution of this kind of problem, the mesh generation needs to be defined independently on each subregion, then the grid nodes are non matched on both sides of the adjacent subregions. The numerical solution of the numerical scheme and the corresponding convergence analysis is very difficult. In this paper, the numerical solution of the problem is studied under the framework of the mixed finite volume element method. The next consideration is the radiation three temperature energy equation group. The radiation three temperature energy equation group belongs to the nonequilibrium radiation diffusion equation group, in the inertial confinement fusion, There are extensive applications in magnetic confinement fusion, astrophysics, hypersonic speed and other fields. Three temperature refers to the temperature function of electrons, ions, and photons. The equations are used to describe the transfer process of radiant energy in a multi medium physical region and the energy exchange process between the three groups of electrons, ions and photons. The coefficients of these equations are strong. Discontinuous, strong nonlinear and highly coupled, so the relative numerical solution is very difficult. First, the numerical solution is required to have good monotonicity and conservation. In addition, because of the large amount of calculation, how to improve the calculation efficiency is one of the questions that everyone needs to consider. This paper will be used to solve the two order elliptic equation. The finite volume element method is extended to the radiation three temperature energy equation group. Two finite volume element schemes are studied and a mesh adaptive algorithm is given. Finally, the immersion interface problem is considered. The solution area of this kind of problem is divided into two non overlapping subregions by a fully smooth curve. This curve is called the interface and the equation. The diffusion coefficient is continuous on each subregion, but it is discontinuous on the interface. In the mesh generation, it does not consider the existence of the interface and dissecting the mesh in the solution area independently. At this time, the interface through the division unit will appear. At this time, if the traditional finite element method is used to solve the numerical results, the calculated results will be obtained. In order to improve the numerical accuracy of the optimal order, in order to improve this defect, some scholars have constructed a new finite element space, that is, the finite element space of the immersion interface. If the finite element space of the immersion interface is used as the exploratory function space, the convergence order of the finite element method of the immersion interface can be close to the optimal. It is very rapid in use, but the corresponding convergence analysis is very difficult, so far, there are still many problems to be solved. In this paper, a finite volume element method with the finite volume element method is applied to the finite volume element method. In this paper, a finite volume element method with a penalty term is studied. The first part of this paper is divided into five chapters, the last part is the conclusion, the first five The contents of chapter include: the first chapter is the introduction, first briefly introducing the finite volume element method, and then introducing the background and development of the research content, that is, the mixed method of Darcy flow on the non matched grid, the non-equilibrium radiation diffusion equation and the numerical solution of the immersion interface problem. The second chapter considers the Darcy flow model on the non matched grid, that is, The first type of interface problem mentioned above is a triangular mesh generation on each subregion, and its corresponding dual section is divided into the barycentric duality. On each subregion grid, the lowest order Raviart-Thomas space is selected to approximate the velocity function and pressure function, and the standard mixed finite volume element method is used to discrete the original square. In this chapter, we introduce linear Mortar element space on the interface to approximate the trace of pressure function, and add an interface to improve the continuity of approximate velocity flow in this chapter. The numerical scheme is called the Mortar element mixed finite volume element method. We prove that the scheme has the best convergence order according to the L2 norm from two aspects of theory and numerical experiment. The third chapter is still the Darcy flow model on the non matched grid. The velocity function space and the approximate pressure function space are the same as in the previous chapter, and the original equation is still discrete according to the standard mixed finite volume element method. But on the interface, this chapter uses double Lagrangian multiplier subspace to approximate the trace of the pressure function. In this grid, a piecewise constant function space is introduced to approximate the trace of the pressure function, and then a dual value Lagrange multiplier space is obtained. In addition, we add a Robin type interface condition to enhance the continuity of the approximate function at the interface. The numerical scheme is called non Mortar element. In the same way, we prove that the scheme has the best convergence order according to the L2 norm in two aspects of theory and numerical experiment. The fourth chapter considers the radiation three temperature energy equation group, that is, the second types of interface problems mentioned earlier. There are two main problems to be overcome for the numerical solution of this kind of problems. In this chapter, based on the conservation of the radiation three temperature energy equations, this chapter uses a reasonable numerical integration formula and an approximate method to deal with the nonlinear term and the discontinuity coefficient. With the help of the different integral formulas, two conservation finite volume element schemes are constructed. The first form is shown by the monotonicity analysis and numerical experiments. Many grids are monotonous, and we deduce the corresponding grid constraints. However, the second formats can not maintain monotonicity and produce a large number of negative temperatures at the beginning of the numerical simulation, which is not consistent with the actual problem. Therefore, the second formats are generally considered unavailable. But we have designed two kinds of post. Processing technology to overcome this problem, including global repair and truncation. Numerical results show that the two post-processing techniques have good computational results. Finally, we design an adaptive algorithm based on the residual error estimation of the residual type, so that the mesh can be flexibly fined and coarsened, which is greatly improved. In the fifth chapter, a new finite volume element method is constructed for the third types of interface problems mentioned above, which is called the interface element, which is called the interface unit, which is called the non interface element which is not passed by the interface. The structure of the finite element space of the impregnated interface is: On the non interface element, the polynomial of the node is chosen as the degree of freedom, and a piecewise polynomial is rebuilt by the interface condition on the interface unit. When the finite element space of the immersion interface is selected as the exploratory function space, the corresponding finite volume element method is called the finite volume element method of the immersion interface. This chapter is limited in the existing immersion interface. On the basis of the volume element method, the original format is modified, and two penalty terms are added to the interface division and the interface with the interface, which are used to restrict the function value and the jump of the normal flow on the interface. The numerical results show that the numerical scheme is better stable, even if the diffusion coefficient is jumping on the interface. When the numerical solution is larger, the optimal convergence is maintained. Through the rigorous theoretical analysis, we demonstrate the stability of the modified finite volume element method for the immersion interface, and then obtain the existence and uniqueness of the solution.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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