求解二阶椭圆界面问题的二次浸入型有限元方法

发布时间：2019-06-16 19:05

【摘要】：本文提出了二次浸入型有限元方法,该方法用来求解二维空间中二阶椭圆界面问题.标准有限元方法可以解决均匀材质问题,如连续介质力学中的微分方程,电磁场中的Maxwell方程等.对于非均匀材质问题,如电磁体问题,航天工程中的空间带电问题等,一般有限元方法的网格剖分需要沿着界面曲线进行,但此方法需要花费大量的计算量来进行网格剖分,因此本文提出了浸入型有限元方法.浸入型有限元方法是非协调元方法并且它的网格剖分不受界面曲线的控制,因此可以选择固定网格,如笛卡尔网格剖分,从而节省大量的时间和存储量.一般的界面问题有限元形状函数的形成需要满足自由度与界面跳跃条件,但要唯一确定二次浸入型有限元基函数,还需要添加额外的限制条件.本文给出两类扩展界面条件,并证明此基函数的存在唯一性.文章先给出线性界面的两种类型扩展条件,进而给出一般界面曲线的扩展条件.接下来给出此方法构造的分片二次有限元空间的最优插值误差,即在L2空间中的收敛阶为θ(h3),在间断的H1空间中的收敛阶为θ(h2),并给出数值算例.但此方法构造的分片二次有限元空间的有限元插值误差在L2空间和在间断的H1空间中均不能达到最优误差,因此最后提出加局部惩罚项的浸入型有限元方法,具体方法就是内部界面边上加惩罚项构成了局部惩罚方法从而达到与插值误差相同的收敛阶.对于线性或双线性浸入型有限元空间均不需要加惩罚项即能达到最优误差估计.
[Abstract]:In this paper, a quadratic immersion finite element method is proposed, which is used to solve the second order elliptical interface problem in two dimensional space. The standard finite element method can solve the problem of uniform material, such as differential equation in continuum mechanics, Maxwell equation in electromagnetic field and so on. For non-uniform material problems, such as electromagnet problem, space electrification problem in space engineering and so on, the mesh generation of the general finite element method needs to be carried out along the interface curve, but this method needs to spend a lot of computation to carry on the grid generation, so this paper proposes the immersive finite element method. Immersive finite element method is nonconforming element method and its mesh generation is not controlled by interface curve, so fixed grid, such as Cartesian mesh generation, can be selected, thus saving a lot of time and storage. The formation of finite element shape function of general interface problem needs to meet the conditions of degree of freedom and interface jump, but in order to determine the quadratic immersion finite element basis function, it is necessary to add additional constraints. In this paper, two kinds of extended interface conditions are given, and the existence and uniqueness of the basis function are proved. In this paper, two types of extension conditions of linear interface are given, and then the extension conditions of general interface curve are given. Next, the optimal interpolation error of piecewise quadratic finite element space constructed by this method is given, that is, the convergence order is theta (h3) in L2 space and theta (H2) in interrupted H 1 space, and a numerical example is given. However, the finite element interpolation error of piecewise quadratic finite element space constructed by this method can not achieve the optimal error in L2 space and intermittent H 1 space, so the immersive finite element method with local penalty term is proposed, that is, the local penalty method is formed by adding penalty term on the edge of the internal interface, so as to achieve the same convergence order as the interpolation error. For linear or bilinear immersion finite element spaces, the optimal error estimation can be achieved without the need of penalty term.
【学位授予单位】：郑州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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