【作者】 张瑗；

【导师】 李寿佛；

【作者基本信息】 湘潭大学 ， 计算数学， 2007， 博士

【摘要】 本文致力于研究惯性约束聚变(ICF)内爆压缩过程数值模拟中需要解决的两个关键问题,一是探索内爆过程中遇到的多介质可压缩大变形流体及流体界面不稳定性的高精度Euler数值模拟方法,二是寻求内爆过程数值模拟所涉及的三温辐射热传导方程的更为有效的数值解法。本文首次系统地探索了高阶加权本质上无振荡格式(weighted essentially nonoscillatoryscheme,简记为WENO)用于ICF内爆压缩过程数值模拟的实际可能性。对于ICF内爆过程所遇到的各种流体不稳定性问题及多介质可压缩大变形流体问题,本文将一律使用经典的高阶WENO方法或者针对具体疑难问题设计基于高阶WENO的新的计算方法进行计算和测试,并与低阶方法进行比较,最终目标是试图在国内外率先研制一个基于高阶WENO方法的高精度ICF内爆程序,为我国国防和现代化建设服务。这既是本文的主旨,也是本文的主要创新之一。本文主要工作如下:(1)提出了求解辐射热传导方程的一类无网格方法(见本文第二章)。这项工作首次将无网格方法成功地应用于求解ICF数值模拟中经常遇到的二维三温方程。理论分析和大量数值试验表明,本文提出的方法在不规则网格上的计算精度以及对于网格不规则程度的适应性均明显优于目前在国内应用最广的九点差分格式。由此可见,本文提出的无网格方法不仅对于求解几何边界复杂的非线性抛物型方程有重要实用价值,而且在ICF数值模拟研究领域具有广阔应用前景。(2)针对多介质可压缩大变形流体,提出了一类基于准守恒γ-模型或准守恒体积分数模型的高阶混合型有限体WENO方法,简记为FV-WENO-MT,而且在求解控制方程的同时,我们用水平集方法(或者Lagrange方法)为多介质问题提供了一张清晰的物质界面(见本文第七章)。这是国内外此前无人涉足的一项创造性成果。这项工作不仅解决了流体混合型方法界面不十分清晰的问题,同时突破了二阶精度方法的局限,为各种高密度比、强激波、大变形、具有多个界面且界面拓扑结构变化十分复杂的高维问题的高精度计算开辟了新的途径。(3)用高阶有限差分WENO格式(简记为FD-WENO)模拟了ICF内爆过程中经常遇到的各种高密度比Rayleigh-Taylor不稳定性(简称RT不稳定性),激光烧蚀RT不稳定性及高马赫数Richtmyer-Meshkov不稳定性(简称RM不稳定性),均获得了令人满意的数值结果,而且通过进行定性及定量比较,表明高阶FD-WENO格式的确明显优于已有的低阶方法,如MUSCL和PPM等(见本文第三章和第四章)。由此证明高阶FD-WENO格式用于内爆压缩过程数值模拟不仅是可行的,而且是十分可取的。(4)在综述多介质可压缩流体界面捕捉方法及界面追踪方法的基础上,对基于高阶FD-WENO格式的流体体积方法(volume of fluid,简记为VOF),基于高阶FD-WENO格式的水平集方法(level set)及界面跟踪方法(front tracking)进行了数值测试、比较和理论分析,表明后二种方法效果很好,但第一种方法欠佳(见本文第六章)。从而得出了一个有指导意义的结论:在使用高阶FD-WENO格式进行ICF数值模拟时,应尽量避免使用VOF方法捕捉界面。(5)为了避免辐射流体力学方程组的表达形式过于复杂和降低求解难度,其中的三温能量方程通常采用内能(而不是总能量)作未知函数,这就导致整个方程组不是守恒型的,以致各种经典的求解双曲守恒律组的数值格式,如高阶FD-WENO格式等在这里不能直接使用。为了解决这个矛盾,我们于本文第五章专门设计了两类用于求解以内能作未知函数的Euler方程组的高阶FD-WENO方法,数值试验表明计算效果较好,这两类新的方法明显优于文献【29】中所介绍的方法。更多还原

【Abstract】 This dissertation is devoted to the study of two subjects which are important for the numerical simulation of implosion compression process of inertial confinement fusion(ICF).The first subject is to search for high accuracy order Eulerian methods for the numerical simulation of multicomponent compressible large distortion flow problems and fluid interface instability problems in implosion compression process,the other one is to find more efficient numerical methods for solving three-temperature radiation heat conduction equations related to the implosion compression process.We are among the first to investigate systematically the feasibility of high order weighted essentially non-oscillatory(WENO) schemes when applied to the numerical simulation of ICF implosion compression process.To every kind of fluid instability problems and multicomponent compressible large distortion flow problems,which will be met in the implosion compression process of ICF,we will all use classical high order WENO schemes or design new numerical methods based on high order WENO schemes to compute them,and then compare our high order methods with the commonly used lower order methods.Our final goal is trying to develop a high accuracy applied software for the numerical simulation of ICF implosion process for serving national defense and modernization construction,which is also one of the main innovations of the dissertation.The main work in the dissertation are as follows:(1) A class of meshiess methods for heat conduction equations are presented (see Chapter 2).The methods are first successfully applied to solve two-dimension three-temperature equations which are often met in the numerical simulation of ICF.Theoretical analysis and lots of numerical simulations show that our methods are superior to the nine-point difference schemes——the most popular in domestic at present,both in the calculation accuracy and the adaptability to irregular degree of meshes.Thus it can be seen that the meshless methods not only have important practical value for solving nonlinear parabolic equations with complex geometric boundary but have broad application prospects in the researching fields of ICF numerical simulation.(2) To multicomponent compressible large distortion flows,a class of high order mixture type finite volume WENO(FV-WENO-MT) methods are developed on the quasi-conservativeγ-based model or the quasi-conservative volume-fraction model,also when solving the control equations,a clear material interface is provided for the multicomponent problems by using the level set methods or Lagrange methods(see Cheaper 7).The original work solves the not quite clear interface’s problem and breaks through the second-order accuracy methods’ limitation, that open a new approach for the high accuracy calculation of the high dimensional problems which with high density-ratio,strong shock,large distortion, multi-interface and very complex changing of interface topological structures.(3) With high order finite difference WENO(FD-WENO) schemes,the satisfactory results are obtained in the numerical simulation of high density-ratio Rayleigh-Taylor(RT) instability problems,laser ablative RT instability problems and high march number Richtmyer-Meshkov(RM) instability problems,which are often met in the implosion compression process of ICF.Furthermore,according the qualitative and quantitative comparison,it is observed that high order FD-WENO schemes are obviously better than the lower order schemes,such as MUSCL and PPM(see Chapter 3 and Chapter 4).It proved that applying the high order FD-WENO schemes in the numerical simulation of the implosion compression process is feasible and quite advisable.(4) Based on the summary of the interface capturing methods and the interface tracking methods of multicomponent compressible flows,three methods including volume of fluid(VOF) methods,level set methods based on FD-WENO schemes and front tracking methods are tested.Numerical results and theoretical analysis show that the first method is not very suitable and the other two are much more appropriate(see Chapter 6).Then there is a significant conclusion that we should try to avoid using VOF methods to capture interface while applying the high order FD-WENO schemes in the numerical simulation of ICF.(5) In order to avoid forming too complicated formulas in radiation hydrodynamic equations and reduce difficulties in calculation,the internal energy(but not the total energy) is usually used as an unknown function in the three-temperature energy equations.This causes the whole system of equaions not conservative, which make the classical numerical schemes for solving hyperbolic conservation laws can’t be used directly here,e.g.high order FD-WENO schemes.To deal with the contradictions,in Chapter 5,two classes of high order FD-WENO methods are designed for solving the nonconservative Euler equations.The numerical experiments show that the calculation effects are good and these two classes of new methods are obviously superior to the method introduced in[29].更多还原