【作者】 张莉；

【导师】 李平；

【作者基本信息】 北京理工大学 ， 力学， 2018， 博士

【摘要】 Level Set方法是处理封闭运动界面随时间演化过程几何拓扑变化的有效工具,已经成为处理广义界面问题最重要的方法之一,广泛应用于各种与曲线演化相关的科学研究和工程计算领域,具有重要的理论研究意义和应用背景。随着科学研究的不断深入,以及工程计算问题提出的更高需求,有效地捕捉和追踪运动界面变得更为重要。多介质界面和图像轮廓界面是广义界面问题中最普遍的问题,本文采用理论分析和数值模拟相结合的方法,探究Level Set方法在重新初始化方面的改进及其在欧拉多介质界面计算问题中的应用;探究Level Set方法在变分理论方面的改进及其在图像轮廓界面捕捉问题中的应用。主要开展工作如下:(1)详细阐述Level Set方法的基本思想和基本理论,分别介绍Level Set方法在多介质界面和图像轮廓界面处理领域中的国内外研究现状及其发展趋势,归纳Level Set方法求解问题的一般步骤,分析Level Set方法的优势与不足。(2)针对重新初始化造成的界面位置偏离问题,提出一种固定界面Level Set方法,主要改进在于将传统Level Set方法中固定的光滑参数修正为变化的光滑参数,通过固定界面有效地改善了Level Set方法的质量守恒性,具有较高的分辨率。固定界面Level Set方法的基本思想是在重新初始化相邻两次迭代过程中,以固定界面为基本原则,构建保持界面不动满足的等式条件,从而推导光滑参数满足的计算公式。讨论和分析了重新初始化、常数流场、旋转流场和剪切流场数值算例,结果表明,与传统Level Set方法相比,固定界面Level Set方法改善了流体体积的守恒性,能够捕捉到尖锐、狭窄、细长的界面,且界面的几何变化没有影响到周围流场的速度变化。(3)讨论固定界面Level Set方法在计算欧拉多介质流体计算问题中的应用,有效地抑制了流体界面附近产生的数值振荡和耗散,提高了间断处的分辨率,改善了气泡的质量守恒性,具有较强地捕捉流体界面能力。简述相关基本理论、控制方程及其数值方法,归纳固定界面Level Set方法求解欧拉多介质流体问题的详细步骤。讨论和分析了气-气激波管、气-液激波管和激波作用气泡问题的数值模拟,结果表明,与传统Level Set方法相比,固定界面Level Set方法有效地消除了激波处的数值耗散,密度、等压和速度的数值解更加吻合精确解,能够更准确地捕捉激波、膨胀波和接触间断的位置,能够更精细地描述气泡在激波作用下的变形过程,并降低气泡质量的损失。(4)讨论固定界面Level Set方法在爆轰波阵面传播计算问题中的应用,有效地避免了由于误差累积导致的爆轰波阵面偏离现象,具有较高的计算精度和分辨率,能够较为准确地描述爆轰波阵面的传播情况。简述相关基本理论、控制方程及其数值方法,归纳固定界面Level Set方法求解爆轰波阵面传播问题的详细步骤。讨论和分析了中心一点起爆、一点起爆、两点起爆和圆弧炸药平面起爆问题的数值模拟,结果表明,与传统Level Set方法相比,固定界面Level Set方法能够更为准确地捕捉到合理地爆轰波阵面,数值计算结果精度更高。(5)针对传统变分Level Set方法处理图像轮廓界面出现的边界粘连问题,提出一种自适应变分Level Set方法,主要改进在于将外部能量泛函中的常数修正为依赖于图像信息的函数,自适应变分Level Set方法的演化速度快,数值计算稳定,有效性和适用性较强。新参数函数借助于图像的二阶导数值,自动确定零等值线演化的方向,自动调整零等值线的演化速度,能够确保演化曲线的长度和面积同时取得全局最小值。讨论和分析了人造图像、医学图像、多目标图像轮廓界面等数值算例,结果表明,与传统变分Level Set方法相比,自适应变分Level Set方法对演化曲线的初始位置不敏感,可以实现自适应运动,能够同时捕捉图像的内外轮廓界面、狭窄深度凹陷区域边界以及多目标物体的边界,特别适用于小距离间隔多目标物体边界轮廓及其弱边界的图像轮廓界面捕捉。更多还原

【Abstract】 Level set method,an effective tool for dealing with the geometric topology changes of the closed moving interface over time,has become one of the most important methods for conducting the problem of generalized interface.It has been widely used in the scientific research and engineering calculation fields related to various curve evolution,having important theoretical research significance and application background.With the constant deepening of scientific research and the higher demands of engineering calculation problem,it has become more important to effectively capture and track the moving interface.The most common problem in generalized interface refers to multi-medium interface and image contour interface,so this paper combines theoretical analysis and numerical simulation to explore the improvement of level set method in reinitialization and the application of that in the problem of Euler multi-medium interface calculation;and the improvement of level set method in variational theory and the application of it in the problem of image contour interface capturing.The main work is described as follows:(1)The basic idea and basic theory of level set method is explained in detail,and it also introduces the current research status at home and abroad and the development trend of level set method in the field of multi-medium interface and image contour interface processing.In addition,it also concludes the general steps of solving the problem through level set method,and analyzes the advantages and disadvantages of level set method.(2)A fixed-interface level set method is proposed on the basis of the problem of the deviation of the interface position caused by reinitialization,in which the main improvement is about correcting the fixed smooth parameters in the traditional level set method to the changed smooth parameters.It has effectively improved the mass conservation of level set method through the fixed interface,which has a higher resolution.The basic idea of the fixed-interface level set method is to construct a equation condition that keeps the interface immobile based on the the basic principle of the fixed interface in the process of the reinitialization of the adjacent two iterations,in which the calculation formula satisfied by the smooth parameters has been consequently deduced.And the numerical examples of reinitialization,constant flow field,rotating flow field and shear flow field are also discussed and analyzed.It could be shown from the result that the fixed-interface level set method improves the conservation of fluid volume and could capture the sharp,narrow,slender interface compared with the traditional level set method,in which the geometrical changes of the interface do not affect the speed of the surrounding flow field.(3)It discusses the application of the fixed-interface level set method in the calculation of the Euler multi-medium fluid calculation,effectively suppressing the numerical oscillation and dissipation generated near the fluid interface,improving the resolution at the discontinuity and improving the mass conservation of the bubble and having stronger ability to capture the fluid interface.It not only briefly describes the relevant basic theories,governing equations and numerical methods,but also summarizes the detailed steps of the Euler multi-medium fluid solved by the fixed-interface level set method.The numerical simulations of the bubble problems such as gas-gas shock tube,gas-liquid shock tube and shock-acting are also discussed and analyzed.It could be shown from the result that the fixed-interface level set method effectively eliminates the numerical dissipation of the shock wave compared with the traditional level set method,in which the numerical solution of density,isobaric pressure and velocity is more consistent with the exact solution.In addition,it could more accurately capture the position of shock wave,expansion wave and contact discontinuity,and could more finely describe the deformation process of bubble under shock wave and reduce the loss of bubble quality.(4)It has effectively avoided the deviation of detonation wave front caused by error accumulation through the discussion about the application of the fixed-interface level set method in the calculation of detonation wave front spreading.The method could conduct a comparatively accurate description about the spreading of the detonation wave front due to its higher calculation accuracy and resolution.It not only briefly describes the relevant basic theories,governing equations and numerical methods,but also summarizes the detailed steps of the detonation wave front spreading solved by fixed-interface level set method.In addition,the numerical simulation of the center one-point detonation,one-point detonation,two-point detonation and the detonation with arc explosive plane is discussed and analyzed.It could be shown from the result that compared with the traditional level set method,the fixed-interface level set method would more accurately capture the reasonable detonation wave front with higher numerical calculation results.(5)An adaptive variational level set method is proposed on the basis of the problem of boundary sticking appearing in the image contour interface that is handled by the traditional variational level set method,which has mainly modified the constant in the external energy functional to the function that depends on the image information.The adaptive variational level set method is characterized by fast evolution rate,stable numerical calculation,and strong validity and applicability.The new parameter function could automatically determine the direction of the zero contour evolution by means of the second derivative of the image,and automatically regulate the evolution speed of the zero contour to ensure the length and area of the evolution curve with a global minimum simultaneously.Furthermore,numerical examples of the contour interfaces of artificial images,medical images,and multi-target images are discussed and analyzed.The results show that,compared with the traditional variational level set method,the adaptive variational level set method is not sensitive to the initial position of the evolution curve and could complete adaptive moving.It could also capture the inner and outer contour interface of the image,the boundary of the concave region with narrow depth and the boundary of the multi-target object at the same time,which is especially suitable for the contour interface capture of the boundary contour of the multi-target object with small-distance interval as well as the image contour of its weak boundary.更多还原