ICF中流体不稳定性的数值模拟研究与Hamilton-Jacobi方程的运动网格方法

【作者】 葛全文；

【导师】 江松；

【作者基本信息】 中国工程物理研究院 ， 应用数学， 2003， 博士

【摘要】 Hamilton-Jacobi(简称H-J)方程出现于最优控制、计算流体力学、计算机图形图像、微分几何、晶体生长、网格生成等许多领域。近年来，越来越多的数学家关注和研究H-J方程数学理论与数值模拟，并取得了一系列重要成果。一般来说，H-J的解析解是难以求出的，并且其弱解不唯一，解的导数会出现间断，这些特点给方程的数值模拟带来不少困难。本文第一部分研究H-J方程的基于水平集方法的自适应运动网格数值方法，研究此方法对尖点(解出现奇性的地方)的分辨能力等问题。 在惯性约束聚变(ICF)研究中，流体不稳定性是一个十分重要的问题，它的存在会直接影响ICF内爆压缩和点火燃烧。为了提高内爆效率，节省点火所要求的激光器驱动能量，ICF通常使用大收缩比和高形状因子的聚变靶丸。初始外表面或激光非均匀性引起的扰动，经瑞利—泰勒(RT)不稳定性的指数放大耦合到内表面，在减速阶段内表面扰动进一步发展，从而对聚变点火和燃烧带来不利因素。当烧蚀阶段RT增长严重时，热的气泡穿透烧蚀壳，进入并加热DT燃料，进而影响DT燃料的高压缩，影响聚变点火。在ICF流体不稳定性的研究中，数值模拟是重要的研究手段。本文第二部分研究ICF流体不稳定性的数值模拟。 本文共分三章。第一章为绪言，简要介绍H-J方程数学理论与数值方法的研究状况与ICF中流体不稳定性数值模拟的研究进展，以及本文研究的问题和结论。第二章讨论结构网格上用基于水平集方法的自适应运动网格方法解二维H-J方程。该方法的特点是：网格自动局部调整疏密跟踪间断的移动，只需增加少量计算量，即可提高计算精度和对间断的分辨能力。数值例子表明此方法计算H-J方程解的“极值”和“尖点”的有效性和精确性。第三章用水平集方法和三阶本质无振荡(ENO)方法对ICF中二维多介质激光烧蚀驱动内界面不稳定性进行数值模拟研究，改进了一些相关的数值模拟结果。更多还原

【Abstract】 Hamilton-Jacobi equations have been widely applied to various fields, such as, optimal control theory, computational fluid dynamics, computing vision, differential geometry, crystal growth, mesh generation. In the past years, the research on the Hamilton-Jacobi equations has attracted much attention. It is well-known that the solutions of the Hamilton-Jacobi equations are continuous but with discontinuous derivatives, even the initial values and the Hamiltonian are smooth. And such solutions are in general not unique. The first part of this thesis is concerned with the numerical solution of the Hamilton-Jacobi equations. A numerical scheme is presented for solving the Hamilton- Jacobi equations by applying adaptive moving mesh methods based on level-set-based deformation. Numerical examples are given, which demonstrate the accuracy and efficiency of computing "extremes" and "spikes" of solution to the Hamilton-Jacobi equations. Hydrodynamic instability is one of the most important issues arising from Inertial Confinement Fusion (ICF) ignition, and influences implosion and inertial confinement fusion ignition. To enhance implosion efficiency and decrease energy of laser facility, ICF in general utilizes large convergent ratio, high aspect fusion targets. The growth of perturbation in fusion targets is driven by acceleration/deceleration (Rayleigh-Taylor), influencing implosion and the ICF ignition. With the development of perturbation, thermal spikes feedthrough thin shell, penetrating and then heating up DT material, damaging high compressibility of the DT material, therefore affecting the ignition. Numerical simulations of the laser ablative interface instability of the inertial confinement fusion are the second part of this thesis.The thesis consists of three chapters. In Chapter 1, we briefly review the developments on theoretical analysis and numerical methods for the Hamilton-Jacobi equations and on numerical simulation of Inertial Confinement Fusion laser ablative interface instability, then the main results of this thesis are presented. In Chapter 2, a numerical scheme is presented for solving Hamilton-Jacobi equations by applying adaptive moving mesh methods based on level-set-based deformation. The advantages of the adaptive moving mesh methods based on level-set-based deformation are that the local adjusted meshes trace discontinuities automatically and can improve the accuracy and resolution. Numerical examples illustrate these advantages. In Chapter 3, we study numerical simulations of the two-dimensional hydrodynamic instability and the laser ablative interface instability of the ICF by applying the level-set method and ENO (Essentially Non- Oscillatory) on a adaptive moving mesh, comparing with the Lared-s code , the numerical results show the improvement in interface accuracy and resolution of capturing interfaces.更多还原