四边形网格质量优化的研究

发布时间：2018-12-07 18:34

【摘要】：网格的质量对有限元、有限体积等数值模拟的精度和收敛性有着很大的影响。现有的全自动四边形网格的生成方法,如四(八)叉树法、前沿推进法、铺砌法等,在生成网格时都不可避免地会产生质量较差的单元,甚至是不合法单元,导致数值计算精度降低,甚至分析中止。为了提高数值模拟的精度,有必要对有限元网格进行优化,以提高网格质量。本文从节点位置优化和网格拓扑优化两个方面开展四边形网格质量优化方法的研究,开发了相应的网格优化程序,并应用到实际的二维洪水分析的网格生成中。在保持网格拓扑不变的前提下,可以通过移动网格的节点位置来提高网格的质量。采用拉普拉斯光滑算法对网格进行光滑处理是最常用的网格质量优化方法。但是,拉普拉斯光滑算法在处理凹面区域单元时会出现单元翻转和节点移到边界外面的情况,导致单元不合法。针对拉普拉斯算法存在的问题,本文提出了基于最速下降法的节点位置优化方法,建立网格质量优化目标函数,以节点位置作为设计变量,通过优化节点的位置,达到提高网格质量的目标。与拉普拉斯光滑算法相比,基于节点位置优化的方法,在处理凹面区域时不会出现单元翻转,保证了凹面区域的网格质量。对于网格单元中节点角度过大或过小的情况,如边界上内角接近180。的单元、内部节点周围单元数目(或度值)大于5或者小于3的单元,通过单纯移动节点位置的方法,很难提高网格的质量,需要采用拓扑优化的方法,即改变单元节点的连接,来改善网格的质量。本文提出了基于边界优化、形状优化、连接性优化、尺寸优化的拓扑优化方法。在边界优化方面,针对在边界上出现的接近三角形的单元,提出了将该边界单元与相邻单元进行合并,然后对合并的局部网格按照设定的模板进行重新生成的方法,可以消除边界上接近三角形的单元。在形状优化方面,针对角度大于160。的质量较差的内部单元以及不合法单元,提出了将大角度单元与该大角相邻的单元进行合并和重新生成的方法,可以消除大角度单元和不合法单元。在连接性优化方面,总结了各种节点连接性模式,根据节点及其周围节点度值,匹配现有的模式,通过改变单元的节点连接性,使各节点的度值接近理想值(内部节点为4),显著提高网格的质量。在尺寸优化方面,针对网格单元中最长边与最短边之比过大的单元,通过与相邻单元合并,改变六边形区域对角线位置或将六边形区域分解成三个单元等方法,可有效减小网格尺寸的差异,提高网格的质量。在本文提出的节点位置优化和网格拓扑优化方法的基础上,采用C++语言在VS2012平台上编写了四边形网格质量优化程序,并对实际的四边形网格进行了优化,结果表明,本文提出的优化方法可以有效地消除质量较差的单元,显著地提高了网格质量。
[Abstract]:The quality of mesh has great influence on the accuracy and convergence of finite element and finite volume numerical simulation. Existing automatic quadrilateral mesh generation methods, such as quadrilateral tree method, forward propulsion method, paving method and so on, will inevitably produce poor quality elements, even illegal elements. The accuracy of numerical calculation is reduced, and even the analysis is stopped. In order to improve the accuracy of numerical simulation, it is necessary to optimize the finite element mesh to improve the mesh quality. In this paper, the quadrilateral mesh quality optimization method is studied from two aspects of node location optimization and grid topology optimization, and the corresponding mesh optimization program is developed, which is applied to the mesh generation of the actual two-dimensional flood analysis. On the premise of keeping the grid topology unchanged, the quality of the grid can be improved by moving the node position of the grid. Laplace smoothing algorithm is the most commonly used mesh quality optimization method. However, the Laplace smoothing algorithm can flip the cells and move the nodes out of the boundary during the processing of the concave area cells, which leads to the illegal of the units. Aiming at the problems of Laplace algorithm, this paper proposes a node location optimization method based on the steepest descent method. The objective function of mesh quality optimization is established. The node position is taken as the design variable, and the node position is optimized by optimizing the node position. Achieve the goal of improving grid quality. Compared with Laplace smoothing algorithm, the mesh quality of concave area can be guaranteed by the method based on node position optimization. In the case of too large or too small a node angle in a grid cell, for example, the inner angle on the boundary is close to 180. The number (or degree) of the elements around the internal node is greater than 5 or less than 3. It is difficult to improve the quality of the mesh by simply moving the position of the node. That is to change the connection of cell nodes to improve the quality of the grid. In this paper, a topology optimization method based on boundary optimization, shape optimization, connectivity optimization and dimension optimization is proposed. In the aspect of boundary optimization, a method of combining the boundary element with the adjacent element is proposed for the element near the triangle on the boundary, and then the merged local mesh is regenerated according to the set template. The element near the triangle on the boundary can be eliminated. For shape optimization, the angle is greater than 160. The method of merging and regenerating the large angle unit and the adjacent unit with large angle is proposed, which can eliminate the large angle unit and the illegal unit. In the aspect of connectivity optimization, various node connectivity modes are summarized. According to the degree values of nodes and their surrounding nodes, the existing patterns are matched. By changing the node connectivity of the unit, the degree of each node is close to the ideal value (internal node is 4). The quality of grid is improved significantly. In the aspect of dimension optimization, by merging with adjacent elements, the diagonal position of hexagonal region can be changed or the hexagonal region can be decomposed into three elements in view of the element whose ratio of the longest edge to the shortest edge is too large. It can effectively reduce the difference of mesh size and improve the quality of mesh. Based on the methods of node location optimization and mesh topology optimization proposed in this paper, a quadrilateral mesh quality optimization program based on VS2012 is developed with C language, and the actual quadrilateral mesh is optimized. The results show that, The optimization method proposed in this paper can effectively eliminate the units with poor quality and improve the mesh quality significantly.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：TB115

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