若干自由边界问题的高效算法研究

发布时间：2018-05-24 05:34

本文选题：自由边界问题 + 连续时空有限元方法　；参考：《上海交通大学》2015年博士论文

【摘要】：自由边界问题,是自由边界需要和定解问题的解一起确定的问题,它在科学与工程领域有很广泛的应用.由于它是非线性问题,难以获得解析解,因此数值模拟是研究自由边界问题的重要手段.本文的核心是构造和分析反Stefan问题和超导纳米线单光子探测器热电耦合模型的高效数值算法,并对所提算法进行数值模拟验证其计算效率.首先,我们研究了求解抛物型方程初边值问题的一类有限元方法的稳定性估计.利用文献[1]中的连续型时空有限元方法,对于带Dirichlet边界条件和混合边界条件的抛物型方程初边值问题获得相应的离散化方法.使用内蕴的推导技巧,获得了有限元解的H1(QT)-模估计.这个估计在抛物型方程反问题研究中有重要应用,它也是我们分析后文提出的求解反Stefan问题数值方法解的存在性和收敛性的基础.其次,我们探讨了求解反Stefan问题的基于虚拟Robin边界条件的区域嵌入法.本文借鉴文献[2]中的区域嵌入法思想,将边界变动区域?(t)延拓到固定区域?,在虚拟边界上引入Robin边界条件,用新的函数q替代未知的自由边界函数s(t)作为控制变量,将定义在边界变动区域?(t)上的反Stefan问题描述为一个线性算子方程.考虑到该算子方程的不适定性,我们通过正则化方法将其转化为优化问题,在此基础上使用连续型时空有限元方法实现最优化问题的离散,从而获得求解反Stefan问题的一个高效数值算法.证明了该方法连续情形和有限元情形问题解的存在唯一性,并证明了有限元解收敛到连续问题解.应该指出的是文[2]虽然提出用区域嵌入法求解一维空间变量情形的反Stefan问题,但没有给出任何理论分析结果.对于一维反Stefan问题,还构造了一个时空有限元数值解法,该方法具有执行简便的优点.通过一系列的数值实验验证了这些算法的有效性.最后,我们给出了超导纳米线单光子探测器电热耦合模型数值算法.这是一个关于温度和电流的非线性自由边界问题,由非线性抛物界面方程和二阶常微分方程耦合而成.在这个模型中,超导纳米线单光子探测器近似认为是一维结构,热响应是一个与纳米线电流有关的非线性界面方程.就作者所知,还没有系统的工作研究该模型的数值解法.在本文中,我们对于非线性抛物界面方程我们使用Crank-Nicholson和Implicit-Explicit两种差分格式实现离散,而对于由二阶常微分方程描述的电流方程,先将其表示为一阶常微分方程组,然后利用梯形方法进行离散,最后通过打靶法和相变条件确定自由边界位置.我们提供了一系列数值模拟结果来说明算法的有效性.
[Abstract]:Free boundary problem is a problem that needs to be determined together with the solution of a definite solution problem. It is widely used in the fields of science and engineering. Because it is a nonlinear problem, it is difficult to obtain an analytical solution, so numerical simulation is an important means to study the free boundary problem. The core of this paper is to construct and analyze the inverse Stefan problem and the high-efficient numerical algorithm for thermoelectric coupling model of superconducting nanowire single-photon detector. The numerical simulation of the proposed algorithm is carried out to verify its computational efficiency. First, we study the stability estimation of a class of finite element methods for solving the initial boundary value problem of parabolic equations. By using the continuous space-time finite element method in [1], the corresponding discretization method for the initial boundary value problem of parabolic equation with Dirichlet boundary condition and mixed boundary condition is obtained. In this paper, the H _ 1 Q _ (T) -norm estimation of the finite element solution is obtained by using the implicit derivation technique. This estimate has important applications in the study of inverse problems of parabolic equations. It is also the basis of the analysis of the existence and convergence of the solutions of numerical methods for inverse Stefan problems proposed later. Secondly, we discuss the domain embedding method based on virtual Robin boundary condition for solving anti-Stefan problem. In this paper, using the idea of region embedding method in [2], the boundary variable region is extended to the fixed region, and the Robin boundary condition is introduced on the virtual boundary. The new function Q is used to replace the unknown free boundary function st) as the control variable. The inverse Stefan problem defined on the boundary variable region is described as a linear operator equation. Considering the ill-posed operator equation, we transform it into an optimization problem by regularization method. On this basis, we use the continuous space-time finite element method to realize the discretization of the optimization problem. Thus an efficient numerical algorithm for solving inverse Stefan problem is obtained. The existence and uniqueness of the solution of the problem in the continuous case and the finite element case are proved, and it is proved that the finite element solution converges to the solution of the continuous problem. It should be pointed out that although the region embedding method is proposed to solve the inverse Stefan problem in the case of one-dimensional spatial variables in paper [2], no theoretical analysis results are given. For one dimensional inverse Stefan problem, a spatiotemporal finite element numerical method is also constructed, which has the advantages of simple execution. The validity of these algorithms is verified by a series of numerical experiments. Finally, a numerical algorithm for electrothermal coupling model of superconducting nanowire single photon detector is presented. This is a nonlinear free boundary problem for temperature and current, which consists of a nonlinear parabolic interface equation coupled with a second order ordinary differential equation. In this model, the superconducting nanowire single-photon detector is considered to be a one-dimensional structure, and the thermal response is a nonlinear interface equation related to the nanowire current. As far as the author knows, no systematic work has been done to study the numerical solution of the model. In this paper, we use Crank-Nicholson and Implicit-Explicit difference schemes to discretize the nonlinear parabolic interface equations. For the current equations described by the second order ordinary differential equations, we first express them as the first order ordinary differential equations. Then the trapezoidal method is used to discretize and the free boundary position is determined by the shooting method and the phase transition condition. We provide a series of numerical simulation results to illustrate the effectiveness of the algorithm.
【学位授予单位】：上海交通大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82

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