摘要
网格生成是诸如有限单元法(Finite Element Method,FEM)等各类数值方法的前处理过程,是计算机辅助工程(Computer Aided Engineering,CAE)的共性支撑技术之一,所用网格的质量和生成速度跟数值方法的精度、效率甚至成败都密切相关。作为几何网格的一个子分支,表面网格(包括曲面网格和平面网格)有其特殊性和重要性,不但它的生成跟几何造型直接相关,其自身也是体网格生成程序的输入和各类边界条件的直接加载对象,因而它对最终得到的体网格质量和数值计算结果都有很大影响。在对工程中所用的很多薄壳结构进行壳分析时也要求有高质量的表面网格来支持。此外表面网格在其他很多领域中也有广泛应用。
为满足工程与科学计算中对高质量曲面网格的需求,本文系统研究了曲面网格生成、重生成和优化等相关热点和难点问题,提出或改进了一系列相应算法;随着大规模工程与科学计算需求的日益迫切,并行网格生成已成为一个新的研究热点,我们还实现了一个通用的并行平面网格生成框架。具体地说,本文从以下几个方面展开论述:
第1章是绪论,简述了本文的研究背景和意义、研究内容及基本框架。
第2章是研究综述,回顾了有限元网格生成方法的国内外研究现状,其中跟本文密切相关的几个部分是论述重点,包括网格单元的质量控制与网格过渡、并行网格生成、曲面网格生成和重生成以及网格优化。
第3章给出了一些平面全四边形有限元网格的拓扑优化策略。由于单纯的光滑化效果受到网格拓扑结构的限制,在进行网格优化时常常需要将拓扑优化和光滑化结合起来使用以得到更高质量的网格。基于有限元网格的局部拓扑结构,这部分给出的拓扑优化策略被组织成“型—操作”的形式,其中“型”是指一类满足一定约束条件的局部区域网格,而“操作”则是指与特定型相对应的拓扑变换,它能优化局部网格中某些节点的度值,从而优化该局部网格质量,最终实现整体网格的优化。
第4章给出了一个通用的并行平面网格生成框架,它包括如下模块:序列化或并行几何分解器、子域图管理模块、并行平面网格生成器以及可选的网格重划分模块。这个并行框架具有可扩展、稳定和高效等基本特性,同时还具有其他一些良好性质:它可对任意序列化及并行几何分解算法、序列化平面平格生成算法实现完全的代码复用;通过引入子域图的概念,并结合静态或动态图划分策略,可使得并行生成的分布式网格具有很好的划分质量,从而减少或消除传统方法中所需要的网格重划分代价,进而提高整个并行模拟过程的效率。存该框架基础上,实现了一个高效、可扩展的并行平面Delaunay网格生成器PDMG-2D,它能利用中等规模并行资源在几分钟内生成包含上亿三角形单元的平面网格。
第5章给出了一个复杂组合参数曲面网格生成框架。从系统层面来看,它有自己的面向网格生成的几何模型及其拓扑结构定义,并有相应的几何建模功能,且能和商业CAD文件进行数据交换,并具备常用的几何验证和修复功能。从算法层面来看,它包含两种应用不同参数平面网格生成技巧的间接表面网格生成方法,分别是已有的基于变换矩阵的前沿推进法(Advancing Front Technique,AFT)和新提出的基于黎曼度量的Delaunay方法,它们所生成的表面网格质量都较高。本章详细介绍了框架和算法的各个方面,包括几何模型及其拓扑结构定义;物理空间和参数空间上的黎曼度量和变换矩阵及其相互关系,它们可用来控制参数平面上的网格生成;边界裁剪曲线离散;边界裁剪曲线离散节点在参数平面上的投影计算;参数平面上边界节点方向调整;参数平面网格生成以及对最终得到的整体曲面网格进行方向调整。这些不同环节之间只要满足一定的数据交换标准便可进行模块化封装。
第6章给出了一个改进的曲面网格拉普拉斯光滑化算法。拉普拉斯光滑化算法是最基本也最流行的一种方法,它实现简单,效率很高,但也存在一些致命缺陷,如可能产生畸形单元或单元自交现象,会使曲面收缩及曲面几何特征消失。本章针对这些问题一一提出了相应的改进方法,它检测曲面的几何特征并在后面的光滑化过程中进行特别处理以保证它们不被丢失;它考虑了一些的新的几何因素,并根据微分几何中的极小曲面原理,通过求解一个带约束的优化问题来防止无效单元的产生;通过投影算法保证光滑化后的网格节点仍落在原始网格上,因此曲面也不会被收缩。
第7章给出了一个保特征的曲面网格重生成方法,它可以根据用户指定的密度通过拓扑变换对网格的不同区域分别进行粗化细化,这些拓扑变换直接在离散曲面上操作而没有一个连续曲面支持,因而实现较为简单,最后通过边交换操作和本文第6章中给出的改进的拉普拉斯光滑化算法对网格质量进行优化,得到最终高质量的计算网格。
本文最后的第8章总结全文,并展望了可以进一步开展的研究工作。
Mesh generation is the pre-process of various numerical methods such as Finite Element Method (FEM), and one of the universal supporting techniques of Computer Aided Engineering (CAE). The ability to generate high-qualified meshes in an efficient way is vital to the success of these numerical methods. In this field, surface meshes play a special and important role. They are the input of volume mesh generators and the boundary conditions will be attached on them, thus they will significantly influence the quality of volume meshes and further the numerical results. High-quality surface meshes are also needed for the analysis of shell structures, and their generation is directly related to geometric modelling. Surface meshes also have many other applications in various fields.
To satisfy the requirements for high-quality surface meshes oriented to engineering and scientific computation, this thesis systematically studies the related problems such as surface meshing, remeshing and optimization with corresponding algorithms implemented. Besides, with the fast development of supercomputers and the ever larger problems arising in such areas as Computational Fluid Dynamics (CFD) and Computational Electro-Magnetics (CEM), close attention has been paid to parallel mesh generation to overcome the bottlenecks of serial mesh generation in terms of time and memory consuming. Thus we also have developed a general framework for parallel planar mesh generation. Concretely, the contents of the thesis are arranged as follows.
The background, significance and framework of the thesis are briefly introduced in the first chapter. Then comes the review of finite element mesh generation methods, with the emphasis on closely related topics, such as mesh quality control and mesh gradation, parallel mesh generation, surface meshing and remeshing, and mesh optimization.
Chapter 3 presents several procedures for improving the topology of unstructured quadrilateral meshes. Based on the local topological structures of the meshes, these procedures are organized in the form of "case-operation". A case is a kind of local meshed region, which satisfies some pre-defined constrained conditions, and its topology or the degrees of its vertices can be optimized by means of topological transformation.
The general framework for parallel planar mesh generation is given in Chapter 4, which includes several modules, i.e. serial or parallel geometry decomposer, SubDomain Graph (SDG) Manager, parallel planar mesh generator and an optional mesh repartitioner. Besides the basic features required by parallel algorithms such as scalability, stability and high parallel efficiency, this framework also possesses some other good traits: It can reuse the existed codes of serial geometry decomposition and mesh generation algorithms; With the help of SDG and its static or dynamic partitioning, the parallel generated distributed mesh is of high partitioning quality, thereby the cost of mesh repartitioning, which is required by traditional methods, can be eliminated or reduced, consequently, the efficiency of the whole parallel simulation will be improved; A parallel planar Delaunay mesh generator PDMG-2D, which is integrated in this framework, can generate hundreds of millions of elements in minutes with medium sized parallel resources.
Chapter 5 presents a system for complex composite parametric surface mesh generation. From the perspective of system, it has its own geometric definition oriented to mesh generation and the ability to generate such geometric objects, which reveals the effort to bridge the gap between the CAD and mesh generation systems. Meanwhile it also has the functions of CAD/CAE conversion and CAD repair, which brings it the capability of generating meshes on a mass of CAD models. From the perspective of algorithm, it includes two indirect surface mesh generation algorithms. One is an existed Advancing Front Technique (AFT) based on the transformation matrix, and the other a new Delaunay method based on Riemannian metric. The surface meshes generated by them are of high quality. The chapter introduces these algorithms and the framework in detail.
Chapter 6 gives an improved Laplacian smoothing approach for surface meshes. The geometric features are first detected by a simple procedure and then treated carefully to prevent them from disappearing. All the nodes of smoothed meshes are guaranteed to be on the original discrete surface by a projection algorithm, thus the shrinkage problem of Laplacian smoothing is avoided. What's more important, the fatal flaw of Laplacian smoothing for generating extremely abnormal or even inverted elements is settled by solving a constrained optimization problem, which is based on the principle of minimum surface in differential geometry.
Chapter 7 presents a surface remeshing method with feature preservation. According to the specified density, different regions on a mesh are coarsened or refined by topological transformations, which operate directly on the discrete surface without a continuous supporting surface. Thus it is easy to implement and of high efficiency. The mesh is finally optimized by methods of edge swapping and the improved Laplacian smoothing approach given in Chapter 6.
Finally, Chapter 8 concludes the thesis and suggests the directions for future research work.
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