界面问题的浸入界面有限元方法的一些进展
发布时间：2018-08-27 09:44
【摘要】：界面问题在流体力学、电磁学和材料科学中经常遇到.四十多年来,界面问题的研究越来越受到关注,大量的文献涌现.目前为止,依据离散单元和界面的关系,界面问题的有限元研究主要分为帖体网格的有限元方法和非贴体网格的有限元方法.在本文中,我们考虑使用非匹配网格的浸入界面方法来求解这些界面问题.第一章,我们简单回顾了界面问题.首先,我们给出椭圆界面问题及其应用,给出了使用匹配网格的方法的精度的结果.然后,我们介绍了几种求解界面问题的经典的使用非匹配网格的方法.最后,我们简单介绍了Sobolev空间.第二章,我们研究了提高椭圆界面问题的精度的新方法.不仅是提高解的精度,也提高界面点处通量的精度.对于一维界面问题,我们利用其弱形式得到界面点处通量的二阶精度;对于二维界面问题,思路类似混合有限元方法,通过在界面附近引入一个管和一个未知变量,因此这个方法比标准的有限元计算成本要略高一点.我们给出了一维界面问题的严格的理论分析,证明了解和通量在界面处具有二阶收敛性.二维的数值实验表明解的二阶收敛性和界面处梯度的超收敛性质.第三章,我们提出新的在三角笛卡尔网格上的非协调浸入界面有限元方法求解平面弹性界面问题.该浸入界面有限元方法无论对可压和近不可问题都具有最优逼近性质.新方法的优点在于它的自由度比其他方法的少,并且对于界面的形状和位置是稳定的.理论上,我们证明了浸入界面基函数的唯一可解性和相容性.数值实验表明该数值方法对不同的界面形状和拉梅参数在L2范数,H1半范数上具有最优逼近性.第四章,基于非匹配的网格,我们给出了求解带有不连续系数的四阶微分方程的新有限元方法.对于非界面单元,我们使用标准的Morley元基函数;对于界面单元,我们依据界面位置和跳跃条件,构造了分片的的Morley元的基函数.理论上,我们研究了所构造的Morley浸入界面元方法的性质.数值实验表明所提出方法在L2范数,H1半范数和H2半范数下的最优收敛性.第五章,我们研究了快速的有限差分算法求解带不连续系数的四阶微分方程.基于增广变量方法,通过引入边界上的增广变量,原问题可以分解成两个泊松方程,因此可以使用快速求解器进行求解.数值实验表明该方法在最大范数下具有二阶收敛性.
[Abstract]:Interface problems are often encountered in fluid mechanics, electromagnetism and materials science. Over the past 40 years, more and more attention has been paid to the study of interface problems, and a large number of documents have emerged. Up to now, according to the relationship between the discrete element and the interface, the finite element method of the interface problem is mainly divided into the finite element method of the placer mesh and the finite element method of the non-body-fitted mesh. In this paper, we consider using unmatched meshes immersion interface method to solve these interface problems. In the first chapter, we briefly review the interface problem. First of all, we give the elliptic interface problem and its application, and give the results of the accuracy of the method using matching meshes. Then, we introduce several classical methods for solving interface problems using mismatched meshes. Finally, we briefly introduce the Sobolev space. In chapter 2, we study a new method to improve the precision of elliptic interface problem. It not only improves the accuracy of solution, but also improves the accuracy of flux at interface point. For the one-dimensional interface problem, we use its weak form to obtain the second-order accuracy of flux at the interface point, and for the two-dimensional interface problem, the idea is similar to the hybrid finite element method, by introducing a tube and an unknown variable near the interface. Therefore, the cost of this method is slightly higher than that of the standard finite element method. We give a rigorous theoretical analysis of the one-dimensional interface problem and prove that the solution and flux have second-order convergence at the interface. Two dimensional numerical experiments show the second order convergence of the solution and the superconvergence property of the gradient at the interface. In chapter 3, we propose a new finite element method for solving plane elastic interface problems with non-conforming immersion interface on triangular Cartesian meshes. The immersion interface finite element method has the optimal approximation property for both compressible and nearly incompressible problems. The advantage of the new method is that it has less degree of freedom than other methods and is stable for the shape and position of the interface. Theoretically, we prove the solvability and consistency of the basis function of immersion interface. Numerical experiments show that the numerical method has the optimal approximation property for different interface shapes and Lamy parameters on L _ 2 norm and H _ 1 semi-norm. In chapter 4, we give a new finite element method for solving fourth order differential equations with discontinuous coefficients based on unmatched meshes. For non-interface elements, we use the standard Morley element basis function, and for the interface element, we construct the basis function of the piecewise Morley element according to the interface position and jump condition. Theoretically, we study the properties of the constructed Morley immersion interface element method. Numerical experiments show the optimal convergence of the proposed method under L _ 2-norm / H _ 1 semi-norm and H _ 2 semi-norm. In chapter 5, we study the fast finite difference algorithm for solving fourth order differential equations with discontinuous coefficients. Based on the augmented variable method, by introducing the augmented variable on the boundary, the original problem can be decomposed into two Poisson equations, so the fast solver can be used to solve the problem. Numerical experiments show that the method has the second order convergence under the maximum norm.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.82
本文编号：2206894
[Abstract]:Interface problems are often encountered in fluid mechanics, electromagnetism and materials science. Over the past 40 years, more and more attention has been paid to the study of interface problems, and a large number of documents have emerged. Up to now, according to the relationship between the discrete element and the interface, the finite element method of the interface problem is mainly divided into the finite element method of the placer mesh and the finite element method of the non-body-fitted mesh. In this paper, we consider using unmatched meshes immersion interface method to solve these interface problems. In the first chapter, we briefly review the interface problem. First of all, we give the elliptic interface problem and its application, and give the results of the accuracy of the method using matching meshes. Then, we introduce several classical methods for solving interface problems using mismatched meshes. Finally, we briefly introduce the Sobolev space. In chapter 2, we study a new method to improve the precision of elliptic interface problem. It not only improves the accuracy of solution, but also improves the accuracy of flux at interface point. For the one-dimensional interface problem, we use its weak form to obtain the second-order accuracy of flux at the interface point, and for the two-dimensional interface problem, the idea is similar to the hybrid finite element method, by introducing a tube and an unknown variable near the interface. Therefore, the cost of this method is slightly higher than that of the standard finite element method. We give a rigorous theoretical analysis of the one-dimensional interface problem and prove that the solution and flux have second-order convergence at the interface. Two dimensional numerical experiments show the second order convergence of the solution and the superconvergence property of the gradient at the interface. In chapter 3, we propose a new finite element method for solving plane elastic interface problems with non-conforming immersion interface on triangular Cartesian meshes. The immersion interface finite element method has the optimal approximation property for both compressible and nearly incompressible problems. The advantage of the new method is that it has less degree of freedom than other methods and is stable for the shape and position of the interface. Theoretically, we prove the solvability and consistency of the basis function of immersion interface. Numerical experiments show that the numerical method has the optimal approximation property for different interface shapes and Lamy parameters on L _ 2 norm and H _ 1 semi-norm. In chapter 4, we give a new finite element method for solving fourth order differential equations with discontinuous coefficients based on unmatched meshes. For non-interface elements, we use the standard Morley element basis function, and for the interface element, we construct the basis function of the piecewise Morley element according to the interface position and jump condition. Theoretically, we study the properties of the constructed Morley immersion interface element method. Numerical experiments show the optimal convergence of the proposed method under L _ 2-norm / H _ 1 semi-norm and H _ 2 semi-norm. In chapter 5, we study the fast finite difference algorithm for solving fourth order differential equations with discontinuous coefficients. Based on the augmented variable method, by introducing the augmented variable on the boundary, the original problem can be decomposed into two Poisson equations, so the fast solver can be used to solve the problem. Numerical experiments show that the method has the second order convergence under the maximum norm.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.82
【参考文献】
相关期刊论文 前2条
1 ;Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions[J];Numerical Mathematics:Theory,Methods and Applications;2010年01期
2 石钟慈;关于Morley元的误差估计[J];计算数学;1990年02期
，本文编号：2206894
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