基于四叉树的复杂边界四边形网格自适应生成方法研究

发布时间：2018-03-11 15:54

本文选题：四叉树　切入点：四边形网格　出处：《东北大学》2012年硕士论文　论文类型：学位论文

【摘要】：网格生成是有限元分析中不可缺少的前处理阶段,被处理对象只有在被网格化之后才能运用有限元方法进行求解。通常典型的适体网格生成方法不适于对复杂边界进行自适应网格生成,作为一种典型的非适体网格生成方法,笛卡尔网格越来越受人们的青睐。目前,笛卡尔切割网格法中的边界切割的形式仅仅局限于单边切割,对于边界切割单元的处理也多局限于合并处理。某些情况下不方便甚至不可能将其划分至单边切割的形式。此外,还存在如对于切割单元的合并处理会导致有限元分析中不要的悬点的产生等情况。本文以上述问题为切入点,对笛卡尔切割网格进行了深入研究。本文首先将平衡四叉树网格生成的终止条件之一修改为双边切割。为了便于分析和判断,对两线段的位置关系和网格类型做了详细的分类,并给出严格的说明和定义；在此基础上,给出了两线段位置关系的判别算法和网格节点内外类型判别方法。在判定网格是否被域边切割时,创新性地提出了网格对角线标准相交判别法。其次,本文深入研究了悬点的特点和规律,提出了悬点环和悬点链的概念,并用来处理初始生成的网格中的悬点。在这个过程中,本文给出了悬点环和悬点链的悬点处理模板。为进一步消除边界顶点式悬点和小面积网格,本文对悬点处理后的网格进行了移点处理。第三,在边界切割处理环节,本文研究了多种边界切割情况,并从中总结出了边界切割规律。本文提出了单边切割和域点切割两种基本网格切割类型,其他网格切割类型都可看成这两种基本切割类型的组合。通过这两种基本切割类型,本文有效地求解了三边以下的网格切割,得到了8种切割单元类型。针对每种切割单元类型,本文都给出了相应的四边形化模板,应用这些模板将切割单元进行四边形化不会产生新的悬点,并通过多个实例验证了其中较为常用的模板的正确性。最后,本文采用了拉普拉斯方法对网格进行优化,并将优化后的网格应用于卡门涡街和后台阶流动两个典型的实例,验证了本文网格划分方法可行性和有效性。
[Abstract]:Mesh generation is an indispensable pre-processing stage in finite element analysis. Usually the typical adaptive mesh generation method is not suitable for adaptive mesh generation of complex boundaries, which can be used as a typical non-adaptive mesh generation method. Descartes grid is more and more popular. At present, the form of boundary cutting in Cartesian mesh method is limited to unilateral cutting. In some cases, it is not convenient or even possible to divide the boundary cutting unit into the form of unilateral cutting. In addition, There are also cases such as the combination of cutting elements will lead to the generation of unwanted suspensions in finite element analysis and so on. In this paper, the Cartesian mesh cutting is deeply studied from the above point of view. In this paper, one of the termination conditions for balanced quadtree mesh generation is first modified as bilateral cutting. The location relation and mesh type of the two segments are classified in detail, and the strict explanation and definition are given. In this paper, an algorithm for judging the position relationship between two segments and a method for judging the type of grid nodes inside and outside are presented, and a standard grid diagonal intersection method is proposed to determine whether the grid is cut by the edge of the grid. Secondly, in this paper, the characteristics and laws of suspension point are deeply studied, and the concepts of suspension ring and catenary are put forward, which are used to deal with the suspending point in the initial generated mesh. In this paper, the template of catenary processing for suspensions and catenary is given. In order to eliminate the boundary vertex suspensions and small area meshes, this paper deals with the meshes after the suspensions are processed. Thirdly, in the process of boundary cutting, this paper studies many kinds of boundary cutting, and summarizes the law of boundary cutting. In this paper, two kinds of basic mesh cutting types, single edge cutting and domain point cutting, are proposed. Other mesh cutting types can be regarded as a combination of these two basic cutting types. Eight cutting unit types are obtained. For each cutting unit type, the corresponding quadrilateral template is given in this paper. Several examples are given to verify the correctness of the more commonly used templates. Finally, the Laplace method is used to optimize the grid, and the optimized mesh is applied to two typical examples of Carmen vortex street and back step flow, which proves the feasibility and effectiveness of the mesh generation method in this paper.
【学位授予单位】：东北大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：TH123

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