本文選題：非匹配網格 + 混合有限體積元法��； 參考：《吉林大學》2016年博士論文

【摘要】：有限體積元法是求解偏微分方程的主要方法之一,其優(yōu)點是不僅能夠更靈活地處理復雜的幾何區(qū)域和邊界條件,而且能夠保持某些物理量的守恒性,因此近些年得到了較大的發(fā)展,但是現有的研究結果幾乎都集中在常規(guī)的單介質問題,也就是非界面問題.然而,在許多實際問題中,所遇到的模型是具有多介質、多物理性質的界面問題,這類問題的數值求解具有更大的挑戰(zhàn),同時也更具有應用價值.在本文中,我們針對三類界面問題,研究了其相關的有限體積元法.首先考慮的界面問題是以壓力函數和速度函數為未知函數的Darcy流模型,該模型的特點是計算區(qū)域被分解為有限個非重疊的子區(qū)域,在任意兩個相鄰子區(qū)域間的交界面上擴散系數矩陣可能是間斷的.這種區(qū)域結構常用于帶有斷層的多孔介質流的模擬,另外為了在局部上獲得高精度的梯度近似,也會用到這種區(qū)域結構.對于這類問題的數值求解,在每個子區(qū)域上需要獨立地定義網格剖分,那么在相鄰子區(qū)域間的交界面的兩側,網格節(jié)點是非匹配的.在這種非匹配的網格上,構造數值格式并進行相應的收斂性分析是有很大難度的.本文在混合有限體積元法的框架下,研究了該問題的數值求解.接下來考慮的界面問題是輻射三溫能量方程組.輻射三溫能量方程組屬于非平衡輻射擴散方程組,在慣性約束聚變、磁約束聚變、天體物理、高超聲速等領域中有廣泛的應用.三溫指的是電子、離子、光子的溫度函數,該方程組用來描述輻射能在多介質物理區(qū)域內的傳輸過程以及電子、離子、光子這三者之間的能量交換過程.由于這類方程組的系數是強間斷、強非線性及高度耦合的,因此相關的數值求解非常困難.首先要求得到的數值解具有良好的單調性(保正性)和守恒性.此外由于計算量較大,因此如何提高計算效率也是大家需要考慮的問題之一.本文將用于求解二階橢圓方程的有限體積元法,推廣到輻射三溫能量方程組,研究了兩種有限體積元格式并給出了一種網格自適應算法.最后考慮的是浸入界面問題.這類問題的求解區(qū)域被一條充分光滑的曲線劃分為兩個非重疊的子區(qū)域,這條曲線被稱為界面,方程的擴散系數在每個子區(qū)域上是連續(xù)的,但在界面上是間斷的.在做網格剖分時,不考慮界面的存在,在求解區(qū)域內獨立地進行網格剖分,這時會出現界面穿過剖分單元的情形,此時如果采用傳統的有限元法來進行數值求解,所得的計算結果無法達到最優(yōu)階的數值精度.為了改善這一缺陷,一些學者構造了一種新的有限元空間,即浸入界面有限元空間.若采用浸入界面有限元空間作為試探函數空間,所得的浸入界面有限元法的收斂階可以接近最優(yōu).雖然這類方法在實際應用中發(fā)展迅速,但是相應的收斂性分析非常困難,至今仍有許多問題亟待解決.本文將浸入界面有限元空間應用到有限體積元法框架下,研究了一種帶有懲罰項的浸入界面有限體積元法.本文前一部分共分五章,最后一部分是結論,前五章的內容包括：第一章是緒論,先簡單介紹有限體積元法,然后介紹本文研究內容的背景和發(fā)展狀況,即非匹配網格上Darcy流的混合法,非平衡輻射擴散方程以及浸入界面問題的數值解法.第二章考慮的是非匹配網格上的Darcy流模型,即前面提到的第一類界面問題.在每個子區(qū)域上,采用的是三角形網格剖分,其相應的對偶剖分為重心對偶剖分.在每個子區(qū)域網格上,選取最低階的Raviart-Thomas空間來近似速度函數和壓力函數,并按照標準的混合有限體積元法來離散原方程.由于網格的不匹配,近似速度空間在界面上不再滿足法向流連續(xù),并且變分方程中會涉及壓力函數在界面上的跡.在本章中,我們在界面上引入線性Mortar元空間來近似壓力函數的跡,并添加了一個用于提高近似速度流連續(xù)性的界面條件.這樣所得的數值格式稱為Mortar元混合有限體積元法,我們從理論和數值實驗兩個方面證明了該格式按L2范數具有最優(yōu)的收斂階.第三章考慮的仍然是非匹配網格上的Darcy流模型.在每個子區(qū)域上,采用的網格剖分、對偶剖分以及近似速度函數空間和近似壓力函數空間與前一章中的相同,并仍然按照標準的混合有限體積元法來離散原方程.但是在界面上,本章采用雙重拉格朗乘子空間來近似壓力函數的跡.相對于每個子區(qū)域,其上的網格剖分在界面上會誘導出一個一維的網格,在這個網格上引入一個分片常數函數空間來近似壓力函數的跡,這樣便得到一個雙重取值的拉格朗日乘子空間.此外,我們添加一個Robin型界面條件來增強近似函數在界面處的連續(xù)性.這樣所得的數值格式稱為非Mortar元混合有限體積元法.同樣,我們從理論和數值實驗兩個方面證明了該格式按L2范數具有最優(yōu)的收斂階.第四章考慮的是輻射三溫能量方程組,即前面提到的第二類界面問題.對于這類問題的數值求解,需要克服的主要問題有兩個,分別是單調性和守恒性.本章從輻射三溫能量方程組的守恒形式出發(fā),采用合理的數值積分公式和近似方法來處理非線性項和間斷系數.借助不同的積分公式,構造了兩種守恒的有限體積元格式.由單調性分析和數值實驗來看,第一種格式在許多網格上是單調的,我們推導出了相應的網格限制條件.然而,第二種格式不可能保持單調性并且在數值模擬的一開始便會產生大量的負數溫度,這與實際問題不相符.因此,第二種格式通常被認為是不可用的.但是我們設計了兩種后處理技術來克服這一問題,包括全局修補技術和截斷法.數值結果表明,這兩種后處理技術都具有較好的計算效果.最后,我們設計了一個基于殘量型后驗誤差估計的自適應算法,使網格能夠靈活地局部加細和粗化,這在很大程度上提高了計算效率.第五章針對浸入界面問題,即前面提到的第三類界面問題,構造了一種新的浸入界面有限體積元法.稱被界面穿過的網格單元為界面單元,稱不被界面穿過的網格單元為非界面單元.浸入界面有限元空間的構造方式為：在非界面單元上選取以節(jié)點為自由度的多項式,在界面單元上利用界面條件重新構造一個分片多項式.當選取浸入界面有限元空間作為試探函數空間時,相對應的有限體積元法被稱為浸入界面有限體積元法.本章在已有的浸入界面有限體積元法的基礎上,將原有的格式作了修正,在界面分劃和與界面相交的邊上,添加了兩個懲罰項,分別用來限制函數值和法向流在界面上的跳躍.數值結果表明,這樣所得的數值格式具有較好的穩(wěn)定性,即使當擴散系數在界面上的跳躍較大時,數值解仍然能保持最佳的收斂性.通過嚴格的理論分析,我們論證了修正后的浸入界面有限體積元法的穩(wěn)定性,進而得到該方法解的存在性與唯一性.
[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.
【學位授予單位】：吉林大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O241.82

【相似文獻】

相關期刊論文 前10條

1 耿加強;畢春加;;二階雙曲方程的間斷有限體積元方法[J];煙臺大學學報(自然科學與工程版);2009年02期

2 賈保敏;楊青;;非線性擬雙曲方程的有限體積元方法[J];科學技術與工程;2009年16期

3 陳國榮;王雪玲;熊之光;;一類參數識別問題的有限體積元計算[J];衡陽師范學院學報;2011年03期

4 張本良;3-動量體積元的局域洛侖茲形變及減除噴注中粒子測定的背景[J];四川師范大學學報(自然科學版);1990年04期

5 豐連海;求解二階橢圓型偏微分方程的一種有限體積元格式[J];工程數學學報;2002年04期

6 高夫征;賈尚輝;;一類完全非線性拋物方程組的高次有限體積元方法及分析[J];高等學校計算數學學報;2005年S1期

7 朱愛玲;;拋物方程的擴展混合體積元方法[J];山東師范大學學報(自然科學版);2006年04期

8 陳長春;;四階波動方程的有限體積元法[J];中國海洋大學學報(自然科學版);2007年01期

9 楊素香;;二維不可壓縮無粘流動問題的特征混合體積元的數值模擬[J];山東科學;2007年05期

10 郭偉利;王同科;;兩點邊值問題基于應力佳點的一類二次有限體積元方法[J];應用數學;2008年04期

相關會議論文 前2條

1 張陽;;一類非線性拋物型方程高次有限體積元的預測-校正格式及其最優(yōu)L~2模誤差估計[A];第四屆全國青年計算物理學術會議論文摘要集[C];2006年

2 曾志;李君利;許振華;邱睿;;質子劑量的Monte Carlo計算方法[A];中國生物醫(yī)學工程學會第六次會員代表大會暨學術會議論文摘要匯編[C];2004年

相關博士學位論文 前10條

1 朱玲;兩類界面問題的有限體積元方法[D];南京師范大學;2015年

2 閆金亮;波方程中一些新的能量守恒有限體積元方法[D];南京師范大學;2016年

3 高艷妮;界面問題的有限體積元法研究[D];吉林大學;2016年

4 王翔;三角形網格上高次有限體積元法的L~2估計和超收斂[D];吉林大學;2016年

5 田萬福;混合有限體積元法[D];吉林大學;2010年

6 王全祥;流體力學中幾類波方程的有限體積元方法[D];南京師范大學;2013年

7 方志朝;發(fā)展型方程的混合有限體積元方法及數值模擬[D];內蒙古大學;2013年

8 丁玉瓊;解二階橢圓型方程的高次有限體積元法的若干研究[D];吉林大學;2010年

9 楊e，

本文編號：1925484