基于分层发散思想的四边形网格生成算法研究与实现
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摘要
近十几年来，随着计算机技术的飞速发展，数值模拟技术在理论和应用上都取得了巨大的成功。数值模拟技术作为工程领域复杂问题求解不可或缺强有力手段，已得到工程人员足够高重视，成为推动工程技术发展重要工具。网格划分是数值模拟技术关键技术之一，在CAE分析之前，网格划分需要花费大量时间。同时，数值模拟技术准确性和分析速度都直接受到网格质量影响。所以研究快速生成高质量网格，对数值模拟技术而言，具有不可替代的意义。
铺路法作为四边形网格经典算法之一，已得到广泛应用。但是由于铺路法存在网格生成过程操作复杂而且生成的网格均匀性较低的缺陷，本文提出了分层发散网格划分法。为有效保证所生成的四边形网格的质量，在三角不等式的基础上，推导出一种新的网格质量评价因子。在对网格平滑过程中，结合被搜索节点周围网格形成区域的凹凸性，对等参Laplacian网格平滑法进行了修正。
本文围绕以上几个目标开展了如下的工作：
文章介绍了网格生成的国内外研究状况，在四边形网格生成的背景以及研究意义的基础上，提出了论文研究的主要内容。
在铺路法的思想基础上，提出的一种基于分层发散思想的有限元网格生成方法。详细地介绍了这种分层发散网格生成过程中的初始条件确定、布点、点的存贮等网格划分预处理的关键内容，最后给出了布点生成实例。
在分层发散网格划分法产生的节点基础上进行网格生成，研制并实现了基于分层发散网格生成法的四边形网格生成器，给出了网格生成流程图和大量网格生成实例，最后对分层发散算法进行了分析。
为保证所生成网格合理性，在三角不等式的基础上，推导了一种四边形网格质量判断因子。然后在经典Laplacian平滑法和等参Laplacian平滑法的基础上，结合被搜索节点邻近的网格的凹凸性，对等参Laplacian平滑法进行了修正。
对分层发散的四边形网格划分方法在空间曲面上进行推广，讨论了空间曲面的连续区域和非连续区域布点，空间曲面网格生成等内容。
In the past few years, with the development of computer technology, the method of the numerical simulation has received great success both in theory and in practice. It has powerful means to complicated problems in engineering community. The mesh generation, which is a key technology for numerical simulation, is very complicated and easily be mistaken. It takes up a major part of the total time of the numerical simulation. At the same time, the quality of a mesh directly affects both the reliability of finite element and the time that the analysis process cost. As a result, how to generate high quality mesh rapidly is extremely valuable for the numerical simulation.
Paving method has received wide acceptance, which generates all-quadrilateral meshes in an arbitrary 2D region. However, it has several defects, such as the complexity of making program and irregular mesh. In order to avoid those defects of the paving, a new algorithm, named delamination & divergence algorithm, has been brought forward. Base on the triangle inequality, a new judgment parameter for mesh quality has been advanced to guarantee the mesh quality. Based on the irregular mesh around a node, a modification is made to the IsoLaplacian smooth method in the mesh smoothing.
In this dissertation, research work related to the content that discusses above has been introduced.
Firstly, the dissertation expatiates the status about mesh generation research. After analyse the significances of researching quadrilateral meshes, the structure of the dissertation details is presented.
Secondly, Based on the thought of the delamination & divergence, the dissertation expatiates the pre-dispose of the FEM method. It introduces how to prepare for the pre-condition of mesh generation and how to distribute and store the points by which the mesh is generated.
Thirdly, the dissertation introduces the details of how to generate the meshbased on the thought of the delamination & divergence. FreeDraw software, which generates quadrilateral meshesbased on the delamination &' divergence, mesh generation algorithm, has been developed.
Fourthly, Base on the triangle inequality, a new judgment parameter of mesh
quality has been advanced to protect the mesh quality. Based on the irregular mesh around a node, a modification to the IsoLaplacian smooth method is presented.
Fifthly, the delamination & divergence mesh generation algorithm extends to the space facet. It expatiates how to generate node in space facet and how to mesh space facet.
铺路法作为四边形网格经典算法之一，已得到广泛应用。但是由于铺路法存在网格生成过程操作复杂而且生成的网格均匀性较低的缺陷，本文提出了分层发散网格划分法。为有效保证所生成的四边形网格的质量，在三角不等式的基础上，推导出一种新的网格质量评价因子。在对网格平滑过程中，结合被搜索节点周围网格形成区域的凹凸性，对等参Laplacian网格平滑法进行了修正。
本文围绕以上几个目标开展了如下的工作：
文章介绍了网格生成的国内外研究状况，在四边形网格生成的背景以及研究意义的基础上，提出了论文研究的主要内容。
在铺路法的思想基础上，提出的一种基于分层发散思想的有限元网格生成方法。详细地介绍了这种分层发散网格生成过程中的初始条件确定、布点、点的存贮等网格划分预处理的关键内容，最后给出了布点生成实例。
在分层发散网格划分法产生的节点基础上进行网格生成，研制并实现了基于分层发散网格生成法的四边形网格生成器，给出了网格生成流程图和大量网格生成实例，最后对分层发散算法进行了分析。
为保证所生成网格合理性，在三角不等式的基础上，推导了一种四边形网格质量判断因子。然后在经典Laplacian平滑法和等参Laplacian平滑法的基础上，结合被搜索节点邻近的网格的凹凸性，对等参Laplacian平滑法进行了修正。
对分层发散的四边形网格划分方法在空间曲面上进行推广，讨论了空间曲面的连续区域和非连续区域布点，空间曲面网格生成等内容。
In the past few years, with the development of computer technology, the method of the numerical simulation has received great success both in theory and in practice. It has powerful means to complicated problems in engineering community. The mesh generation, which is a key technology for numerical simulation, is very complicated and easily be mistaken. It takes up a major part of the total time of the numerical simulation. At the same time, the quality of a mesh directly affects both the reliability of finite element and the time that the analysis process cost. As a result, how to generate high quality mesh rapidly is extremely valuable for the numerical simulation.
Paving method has received wide acceptance, which generates all-quadrilateral meshes in an arbitrary 2D region. However, it has several defects, such as the complexity of making program and irregular mesh. In order to avoid those defects of the paving, a new algorithm, named delamination & divergence algorithm, has been brought forward. Base on the triangle inequality, a new judgment parameter for mesh quality has been advanced to guarantee the mesh quality. Based on the irregular mesh around a node, a modification is made to the IsoLaplacian smooth method in the mesh smoothing.
In this dissertation, research work related to the content that discusses above has been introduced.
Firstly, the dissertation expatiates the status about mesh generation research. After analyse the significances of researching quadrilateral meshes, the structure of the dissertation details is presented.
Secondly, Based on the thought of the delamination & divergence, the dissertation expatiates the pre-dispose of the FEM method. It introduces how to prepare for the pre-condition of mesh generation and how to distribute and store the points by which the mesh is generated.
Thirdly, the dissertation introduces the details of how to generate the meshbased on the thought of the delamination & divergence. FreeDraw software, which generates quadrilateral meshesbased on the delamination &' divergence, mesh generation algorithm, has been developed.
Fourthly, Base on the triangle inequality, a new judgment parameter of mesh
quality has been advanced to protect the mesh quality. Based on the irregular mesh around a node, a modification to the IsoLaplacian smooth method is presented.
Fifthly, the delamination & divergence mesh generation algorithm extends to the space facet. It expatiates how to generate node in space facet and how to mesh space facet.
引文
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[2]J.R. Shewchuk, Delaunay Refinement mesh generation [PH.D. dissertation]. Computer Science department, Canegie Mellon Univ. 1997
[3]Steven J. Owen, M, L. Staten, S.A. Canann. Q-Morph: An Approch to Advancing Front Quad Meshing. International Journal for Numerical Methods in Engineering, 1999,44(9): 1317-1340
[4]Steven E. Benzley, Ernest Perry, Karl Merkley, Brett Clark and Greg Sjaardema. A Comparision of All-Hexahedral and All-Tetrahedral Finite Elemnet Meshes for Elastic and Elasto-Plastic Analysis, Proceedings 4th International Meshing Roundtable, 1995,179-191
[5]A.O Cifuentes, and A. Kalbag. A Performance Study of Tetrahedral and Hexahedral Elements in 3-D Finite Element Structural Anasis, Finite Elements in analysis and Design, 1992,12,313-318
[6]7th International Meshing Roundtable, Dearborn, Michigan, U.S.A, October 1998
[7]郑志镇，李尚健，李志刚．复杂曲面上的四边形网格生成方法．计算机辅助设计与图形学学报，1999，11(6)：521-524
[8]施云生，沈国强．基于边界适应的有限元网格自动生成及局部调整技术．锻压技术，1997，(4)：28-30
[9]肖双九，张树生，年泽阳等．离散点集3D三角划分算法在裁剪曲面中的实现．计算机应用，2001，21(9)：18-19
[10]孙玉文，王晓明，刘健．密集散乱数据的三角形网格曲面逼近方法．计算机辅助设计与图形学学报，2000，12(4)：281-285
[11]王建华，王卫中．平面任意区域自适应网格生成技术．计算机辅助设计与图形学学报，1997，9(4)：295-301
[12]杨春太，杨晓东，申长雨等．任意平面区域的变尺寸有限元网格划分．计算力学学报，2000，17(1)：105-108
[13]黄志超，周天瑞等，有限元网格划分技术研究．南昌大学学报：工科版．2001，23(4)．-25-31，44
[14]杜平安，有限元网格划分的基本原则．机械设计与制造．2000，(1)．-34-36
[15] 李水乡，袁明武，平面有限元网格生成的自适应技术．计算机工程与设计．1999，20(4)．-51-55
[16] 郑志镇，曲面网格划分算法的分类与比较．计算机辅助工程．1998，7(1)．-53-58
[17] C. Zilenkiewicz, D.V. Philips.An Automatic Mesh Generation Scheme for Plane and Curve Surface by Isoparamatic Coordinates. International Journal for Numerical Methods in Engineering, 1971,3,519-528
[18] W.J. Gorden, M.S. Hall. Construction of Curcilinear Coordinate Systems and Appliications to Mesh Generation. International Journal for Numerical Methods in Engineering, 1973,5,461-477
[19] Matthew L. Staten, Scott A. Canann, and Steve J.Owen. BMSWEEP: Locating Interior Nodes During Sweeping. Proceedings, 7th International Meshing Roundtable, Dearborn, Michigan, U.S.A, October, 1998,7-18
[20] Mingwu, Lai, Steven E. Benzley, Greg Sjaardema and Tautges. A Multiple Source and Target Sweeping Method for Generating All-Hexahedral Meshes. Proceedings, 5th International Meshing Roundtable, 1996,217-228
[21] Ted D. Blacker, The Cooper Tool. Proceedings, 5th International Meshing Roundtable, 1996,13-29
[22] M.A. Yerry. M.S. Shephard. A modified Quatree Approach to Finite Element Mesh Generation. IEEE Computer Graphics and Applications, 1983,3, 39-46
[23] Yerry M.A.. Shephard M.S. Automatic three-dimensional mesh generation by the modified octree technique. International Journal for Numerical Methods in Engineering, 1984,20:1965-1990
[24] A. Kela. Hierachical octree approximations for boundary representationbased geometric models, CAD, 1989,21(6): 255-262
[25] Marshall Bern, David Eppetein, Jphn R. Gilbert. Provably Good Mesh Generation. 31st Annual Symposium on Foundation of Computer Science. IEEE Computer Socity Press, 1990,231-241
[26] 杨名生，张立京。基于四叉树的有限元网格自动剖分。大连理工大学学报。1997，37(5)：615-616
[27] 杨名生，王冬。有限元网格全自动生成的四分法。计算结构力学及其应用。1989，6(4)：61-68
[28] 杨名生，吴京宁。基于四叉树的有限元网格自动方法及凝聚方法。计算结构力学及其应用。1995，12(4)：409-416
[29] Rainald Lohner Parikh and Clyde Gumbert, Interactive Generation of Unstructured Grid for Three Dimensional Problems. Numerical Grid Generation in Computational Mechanics' 88, Pineridge Press, 1988,687-697
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