界面問題的浸入界面有限元方法的一些進展
發(fā)布時間：2018-08-27 09:44
【摘要】：界面問題在流體力學、電磁學和材料科學中經(jīng)常遇到.四十多年來,界面問題的研究越來越受到關(guān)注,大量的文獻涌現(xiàn).目前為止,依據(jù)離散單元和界面的關(guān)系,界面問題的有限元研究主要分為帖體網(wǎng)格的有限元方法和非貼體網(wǎng)格的有限元方法.在本文中,我們考慮使用非匹配網(wǎng)格的浸入界面方法來求解這些界面問題.第一章,我們簡單回顧了界面問題.首先,我們給出橢圓界面問題及其應用,給出了使用匹配網(wǎng)格的方法的精度的結(jié)果.然后,我們介紹了幾種求解界面問題的經(jīng)典的使用非匹配網(wǎng)格的方法.最后,我們簡單介紹了Sobolev空間.第二章,我們研究了提高橢圓界面問題的精度的新方法.不僅是提高解的精度,也提高界面點處通量的精度.對于一維界面問題,我們利用其弱形式得到界面點處通量的二階精度;對于二維界面問題,思路類似混合有限元方法,通過在界面附近引入一個管和一個未知變量,因此這個方法比標準的有限元計算成本要略高一點.我們給出了一維界面問題的嚴格的理論分析,證明了解和通量在界面處具有二階收斂性.二維的數(shù)值實驗表明解的二階收斂性和界面處梯度的超收斂性質(zhì).第三章,我們提出新的在三角笛卡爾網(wǎng)格上的非協(xié)調(diào)浸入界面有限元方法求解平面彈性界面問題.該浸入界面有限元方法無論對可壓和近不可問題都具有最優(yōu)逼近性質(zhì).新方法的優(yōu)點在于它的自由度比其他方法的少,并且對于界面的形狀和位置是穩(wěn)定的.理論上,我們證明了浸入界面基函數(shù)的唯一可解性和相容性.數(shù)值實驗表明該數(shù)值方法對不同的界面形狀和拉梅參數(shù)在L2范數(shù),H1半范數(shù)上具有最優(yōu)逼近性.第四章,基于非匹配的網(wǎng)格,我們給出了求解帶有不連續(xù)系數(shù)的四階微分方程的新有限元方法.對于非界面單元,我們使用標準的Morley元基函數(shù);對于界面單元,我們依據(jù)界面位置和跳躍條件,構(gòu)造了分片的的Morley元的基函數(shù).理論上,我們研究了所構(gòu)造的Morley浸入界面元方法的性質(zhì).數(shù)值實驗表明所提出方法在L2范數(shù),H1半范數(shù)和H2半范數(shù)下的最優(yōu)收斂性.第五章,我們研究了快速的有限差分算法求解帶不連續(xù)系數(shù)的四階微分方程.基于增廣變量方法,通過引入邊界上的增廣變量,原問題可以分解成兩個泊松方程,因此可以使用快速求解器進行求解.數(shù)值實驗表明該方法在最大范數(shù)下具有二階收斂性.
[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.
【學位授予單位】：南京師范大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O241.82
本文編號：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.
【學位授予單位】：南京師范大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O241.82
【參考文獻】
相關(guān)期刊論文 前2條
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 石鐘慈;關(guān)于Morley元的誤差估計[J];計算數(shù)學;1990年02期
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