【摘要】：網(wǎng)格的質(zhì)量對有限元、有限體積等數(shù)值模擬的精度和收斂性有著很大的影響�，F(xiàn)有的全自動四邊形網(wǎng)格的生成方法,如四(八)叉樹法、前沿推進法、鋪砌法等,在生成網(wǎng)格時都不可避免地會產(chǎn)生質(zhì)量較差的單元,甚至是不合法單元,導(dǎo)致數(shù)值計算精度降低,甚至分析中止。為了提高數(shù)值模擬的精度,有必要對有限元網(wǎng)格進行優(yōu)化,以提高網(wǎng)格質(zhì)量。本文從節(jié)點位置優(yōu)化和網(wǎng)格拓撲優(yōu)化兩個方面開展四邊形網(wǎng)格質(zhì)量優(yōu)化方法的研究,開發(fā)了相應(yīng)的網(wǎng)格優(yōu)化程序,并應(yīng)用到實際的二維洪水分析的網(wǎng)格生成中。在保持網(wǎng)格拓撲不變的前提下,可以通過移動網(wǎng)格的節(jié)點位置來提高網(wǎng)格的質(zhì)量。采用拉普拉斯光滑算法對網(wǎng)格進行光滑處理是最常用的網(wǎng)格質(zhì)量優(yōu)化方法。但是,拉普拉斯光滑算法在處理凹面區(qū)域單元時會出現(xiàn)單元翻轉(zhuǎn)和節(jié)點移到邊界外面的情況,導(dǎo)致單元不合法。針對拉普拉斯算法存在的問題,本文提出了基于最速下降法的節(jié)點位置優(yōu)化方法,建立網(wǎng)格質(zhì)量優(yōu)化目標函數(shù),以節(jié)點位置作為設(shè)計變量,通過優(yōu)化節(jié)點的位置,達到提高網(wǎng)格質(zhì)量的目標。與拉普拉斯光滑算法相比,基于節(jié)點位置優(yōu)化的方法,在處理凹面區(qū)域時不會出現(xiàn)單元翻轉(zhuǎn),保證了凹面區(qū)域的網(wǎng)格質(zhì)量。對于網(wǎng)格單元中節(jié)點角度過大或過小的情況,如邊界上內(nèi)角接近180。的單元、內(nèi)部節(jié)點周圍單元數(shù)目(或度值)大于5或者小于3的單元,通過單純移動節(jié)點位置的方法,很難提高網(wǎng)格的質(zhì)量,需要采用拓撲優(yōu)化的方法,即改變單元節(jié)點的連接,來改善網(wǎng)格的質(zhì)量。本文提出了基于邊界優(yōu)化、形狀優(yōu)化、連接性優(yōu)化、尺寸優(yōu)化的拓撲優(yōu)化方法。在邊界優(yōu)化方面,針對在邊界上出現(xiàn)的接近三角形的單元,提出了將該邊界單元與相鄰單元進行合并,然后對合并的局部網(wǎng)格按照設(shè)定的模板進行重新生成的方法,可以消除邊界上接近三角形的單元。在形狀優(yōu)化方面,針對角度大于160。的質(zhì)量較差的內(nèi)部單元以及不合法單元,提出了將大角度單元與該大角相鄰的單元進行合并和重新生成的方法,可以消除大角度單元和不合法單元。在連接性優(yōu)化方面,總結(jié)了各種節(jié)點連接性模式,根據(jù)節(jié)點及其周圍節(jié)點度值,匹配現(xiàn)有的模式,通過改變單元的節(jié)點連接性,使各節(jié)點的度值接近理想值(內(nèi)部節(jié)點為4),顯著提高網(wǎng)格的質(zhì)量。在尺寸優(yōu)化方面,針對網(wǎng)格單元中最長邊與最短邊之比過大的單元,通過與相鄰單元合并,改變六邊形區(qū)域?qū)蔷�位置或?qū)⒘呅螀^(qū)域分解成三個單元等方法,可有效減小網(wǎng)格尺寸的差異,提高網(wǎng)格的質(zhì)量。在本文提出的節(jié)點位置優(yōu)化和網(wǎng)格拓撲優(yōu)化方法的基礎(chǔ)上,采用C++語言在VS2012平臺上編寫了四邊形網(wǎng)格質(zhì)量優(yōu)化程序,并對實際的四邊形網(wǎng)格進行了優(yōu)化,結(jié)果表明,本文提出的優(yōu)化方法可以有效地消除質(zhì)量較差的單元,顯著地提高了網(wǎng)格質(zhì)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.
【學位授予單位】：山東大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：TB115

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