边界面法后处理研究与程序实现
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摘要
新近提出的边界面法中,边界积分和场变量插值都是在以边界表征的实体边界曲面的参数空间里进行。该法不仅比边界元的计算精度更高,而且避免了计算几何误差。后处理可视化在边界面法的应用中扮演非常重要的作用。虽然有限元的后处理已经发展成熟,但是在边界面法后处理中不能照搬有限元的后处理方法。因此,有必要对边界面法的后处理进行研究。
本文对边界面法分析数据的预处理、等值线图和云图的生成算法及程序实现进行了详细研究。具体研究内容如下：
1.以边界面法求解势问题为例,分析边界面法的基本原理特点,提出自适应树结构求解任意点场量值。
2.在分析等值线生成理论的基础上,对传统的四边形规则网格法进行改进。提出在等值线跟踪过程中,加入基于网格单元的自适应算法,以改进等值线出现折线的缺点。同时,对传统的三角单元不规则网格法进行分析,将等值线的追踪限制在三角单元内,避免了区分边界等值线与内部等值线的问题。
3.在分析云图生成理论的基础上,对基于拓扑关系的等值线填充算法和基于三角单元的等值线填充算法进行讨论。基于拓扑关系的等值线填充关键在于等值线间拓扑关系的形成。而对基于三角单元的等值线填充算法进行改进,直接在单元内生成Patch,以Patch为单位完成等值线填充。对比两种方法,后者具有程序实现简单,适应性强的优点。
4.在熟悉UG二次开发理论及相关技术的研究基础上,研究了边界面法后处理程序在UG上实现的方法。数值算例表明边界面法后处理可视化取得了很好的视觉效果。
In the newly developed boundary face method (BFM), both boundary integration and variable approximation are performed in the 2-D parametric space of the boundary surface of solid represented by the boundary. The proposed approach can not only provide much more accurate results than the boundary element method (BEM), but also avoid geometric error in the BFM. Visualization in post-processing plays an important role in the application of the BFM. Although there are already abundant of matured post-processing tools existing for the finite element method (FEM), these codes can not be directly applied to the BFM. Therefore, it is necessary to investigate new algorithms that are especially suitable for the BFM post-processing.
In this paper, the algorithms of the data preparation and isoline generation in the BFM post-processing were studied in detail. Our work involves four aspects as follows:
(1) For instance of solving a potential problem by the BFM, according to the speciality of the BFM, an adaptive tree structure for calculating the value of variable at any point is proposed.
(2) Based on the theory of isoline generation, the traditional rectangular grid method has been improved. For the shortage that broken line would be created when the linear interpolation was taken, based on the rectangular grid method, some self-adaptive algorithm using grid subdividing was presented. Furthermore, the traditional irregular triangular grid method was analyzed; and the isoline tracking was limited in the traditional element to avoid the problem of distinction between the boundary isoline and internal isoline.
(3)Based on the theory of the contour generation, two contour filling algorithms were discussed respectively, one of which is based on the topological relation and the other one of which is based on triangular cell. The formation of the topological relation for isolines is crucial to the contour filling algorithm based on the topology relationship. The contour filling algorithm which is based on the triangular cell has been improved, and patches were directly generates in the cell to fulfill contour filling in the patch. Comparison of two methods showed that the latter algorithm is better than the former in the program implemented and adaptability.
(4) After the study on the theory of secondary development of UG and its related technology, the project of post-processing of BFM which is UG-NX oriented has been developed. Numerical examples showed that excellent visualization can be obtained by proposed methods.
本文对边界面法分析数据的预处理、等值线图和云图的生成算法及程序实现进行了详细研究。具体研究内容如下：
1.以边界面法求解势问题为例,分析边界面法的基本原理特点,提出自适应树结构求解任意点场量值。
2.在分析等值线生成理论的基础上,对传统的四边形规则网格法进行改进。提出在等值线跟踪过程中,加入基于网格单元的自适应算法,以改进等值线出现折线的缺点。同时,对传统的三角单元不规则网格法进行分析,将等值线的追踪限制在三角单元内,避免了区分边界等值线与内部等值线的问题。
3.在分析云图生成理论的基础上,对基于拓扑关系的等值线填充算法和基于三角单元的等值线填充算法进行讨论。基于拓扑关系的等值线填充关键在于等值线间拓扑关系的形成。而对基于三角单元的等值线填充算法进行改进,直接在单元内生成Patch,以Patch为单位完成等值线填充。对比两种方法,后者具有程序实现简单,适应性强的优点。
4.在熟悉UG二次开发理论及相关技术的研究基础上,研究了边界面法后处理程序在UG上实现的方法。数值算例表明边界面法后处理可视化取得了很好的视觉效果。
In the newly developed boundary face method (BFM), both boundary integration and variable approximation are performed in the 2-D parametric space of the boundary surface of solid represented by the boundary. The proposed approach can not only provide much more accurate results than the boundary element method (BEM), but also avoid geometric error in the BFM. Visualization in post-processing plays an important role in the application of the BFM. Although there are already abundant of matured post-processing tools existing for the finite element method (FEM), these codes can not be directly applied to the BFM. Therefore, it is necessary to investigate new algorithms that are especially suitable for the BFM post-processing.
In this paper, the algorithms of the data preparation and isoline generation in the BFM post-processing were studied in detail. Our work involves four aspects as follows:
(1) For instance of solving a potential problem by the BFM, according to the speciality of the BFM, an adaptive tree structure for calculating the value of variable at any point is proposed.
(2) Based on the theory of isoline generation, the traditional rectangular grid method has been improved. For the shortage that broken line would be created when the linear interpolation was taken, based on the rectangular grid method, some self-adaptive algorithm using grid subdividing was presented. Furthermore, the traditional irregular triangular grid method was analyzed; and the isoline tracking was limited in the traditional element to avoid the problem of distinction between the boundary isoline and internal isoline.
(3)Based on the theory of the contour generation, two contour filling algorithms were discussed respectively, one of which is based on the topological relation and the other one of which is based on triangular cell. The formation of the topological relation for isolines is crucial to the contour filling algorithm based on the topology relationship. The contour filling algorithm which is based on the triangular cell has been improved, and patches were directly generates in the cell to fulfill contour filling in the patch. Comparison of two methods showed that the latter algorithm is better than the former in the program implemented and adaptability.
(4) After the study on the theory of secondary development of UG and its related technology, the project of post-processing of BFM which is UG-NX oriented has been developed. Numerical examples showed that excellent visualization can be obtained by proposed methods.
引文
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[7]Zhang JM, Yao Z H. A hybrid boundary node method. Int. J. Numer. Meth. Engng, 2002,53:751-763.
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[33]M.Weiler, R.Westermann, C.Hansen,et al, Level-of-detail valume rendering via 3d textures.2000,35-50.
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[35]章勤,李品,张继顺.一种基于二次等参单元的等值线图生成算法.华中科技大学学报,2002,33：992-994.
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[37]林毅,金烨,马登哲等.正规网格等值线的虚路径扫描算法.计算机工程与应用,2001,92：91-94.
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[39]韦美雁,杜丹蕾.基于规则网格的等值线的生成研究.湖南科技学院学报,2007,28：39-41.
[40]张梅华,梁文康.一个三角形网格上等值线图的绘制方法.计算机辅助设计与图形学报,1997,9：213-218.
[41]成建梅,除崇希,孙红林.三角网络等值线自动生成方法及程序实现.水利学报,1998,10：23-27.
[42]郑盛贵,颜七笙,黄临平.基于点的三角形构网算法及等值线自动生成方法.计算机与现代化,2004,5：7-9.
[43]吴自银,高金耀.一种基于网格的快速等值线填充算法.测绘学报,1999,28：350-351.
[44]罗伟锦,刘汉龙,高玉峰.一种基于确定区域填充点的等值线填色算法.河海大学学报(自然科学版),2003,31(5)：584-588.
[45]戴常英,李昕,李凌博.等值线图区域填充的边界扫描算法.微机发展,2004,14(1)：23-25.
[46]庞世明,蔡玉华,勒文芳.等值线图的彩色填充方法.计算机应用,2004,24(1)：60-62.
[47]龙述尧编著.边界单元法概论.中国科学文化出版社,2002.23-126.
[48]Zhang JM, Numerical Simulation of 3-D Potential Problems by Regular Hybrid Boundary Node Method. International Journal for Computational Methods in Engineering Science and Mechanics,2008,9:111-120.
[49]Zhang JM, Yao ZH, MTanaka. The meshless regular hybrid boundary node method for 2-D linear elasticity. Engineering Analysis with Boundary Elements, 2003,27:259-268.
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[2]C.R.Liu, S.S.Quek著,龙述尧,何淑娟等译.有限元法实用教程.长沙：湖南大学出版社,2004,79-213.
[3]C.A.Brebbia等著,龙述尧等译.边界单元理论和工程应用.长沙：国防业出版社,1988,81-121.
[4]Yoshida T, Nishimura N, Kobayashi S. Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D. International Journal for Numerical methods in Engineering,2001,50:525-547.
[5]Zhang J M, Tanaka M. Adaptive spatial decomposition on fast multiple method. J. Comput. Phys,2007,226:17-28.
[6]Zhang J M, Yao Z H. Meshless regula hybrid boundary node method comput model.Eng. Sci,2001,2:307-318.
[7]Zhang JM, Yao Z H. A hybrid boundary node method. Int. J. Numer. Meth. Engng, 2002,53:751-763.
[8]Zhang J M, Qin X Y, Han X, Li G Y. A boundary face method for potential problems in three dimensions. International Journal for Numerical Methods in Engineering,2009,80:320-337.
[9]杨晓松.三维有限元数据场可视化技术研究与实现.大连理工大学博士论文,2000,2-19.
[10]I.Fredholm. Sur une classed' Equations Fonctionelles, Sweden:Acta Mathematica,1903,365-390.
[11]田中正隆等,郎德宏译.边界元法的基础与应用.北京：煤炭工业出版社,1987,98-167.
[12]杨德全,赵忠生.边界元理论及应用.北京：北京理工大学出版社,2002,2-116.
[13]Jaswon M A. Integral equation methods in potential theory. Proceedings of the royal society mathematical physics and engineering sciences,1963:24-32.
[14]Cruse T A, Rizzo F J. A direct formulation and numerical solution of the general transient elastodynamic problem.Journal of mathematical analysis and applications,1968,244-256.
[15]Cruse T A. Application of the boundary-integral equation method to three dimensional stress analysis. Computers and structures,1973,509-527.
[16]C.A.Brebbia. The Boundary Element Method for Engineers. Pentech Press. London:1978.
[17]Belytchko T, Lu YY, Gu L. Element free Galerkin methods. International Journal for Numerical Methods in Engineering,1994,37:229-256.
[18]Zhu T, Zhang J, Atluri SN. A local boundary integral equation method in computation methanics and a meshless discretization approach. Computational Mechanics,1998,21:223-235.
[19]Atluri SN, Zhu T. A new meshless local Petrov-Galerkin approach in computational mechanics. Computational Mechanics,1998,22:117-127.
[20]Mukherjee YX, Mukherjee S. The boundary node method for potential problems. International Journal for Numerical Methods in Engineering,1997,40:797-815.
[21]Chati MK, Mukherjee S. The boundary node method for three-dimensional problems in potential theory. International Journal for Numerical Methods in Engineering,2000,47:1523-1547.
[22]B.H.McCormick, T.A.DeFanti and M.D.Brown. Visualization in Scientific Computing. Computer Graphics,1987,21(6):Special Issue.
[23]石教英,蔡文立.科学计算可视化算法与系统.北京：科学出版社,1996,6-86.
[24]刘晓强.科学可视化的研究现状与发展趋势.工程图学学报.1997,3：132-126.
[25]史元.从" CAD/CG93 "看计算机图形学的新发展.北方工业大学学报,1995.7(1)：87-92.
[26]贾春华,肖卫国,贾霖.可视化技术及研究方向.西安工业学院学报.1997,2：46-53.
[27]唐泽圣等.三维数据场可视化.北京：清华大学出版社,1999,76-222.
[28]齐同军.科学计算可视化在结构有限元分析中的应用.大连理工大学硕士学位论文,2001,3-20.
[29]M.Levoy. Display of Surfaces from Volume Data. IEEE Computer Graphics and Applications.1988,8(5):29-37.
[30]P.Lacroute, M.Levoy. Fast Volume Rendering Using a Shear-Warp Factorization of the Viewing Transform. Computer & Graphics.1994,28(3):451-459.
[31]M.A.Westenberg, J.B.T.M.Roerdink. Frequency Domain Volume Rendering by the Wavelet X-Ray Transform. IEEE Transactions on Image Processing.2000, 9(7):1249-1261.
[32]E.C.LaMar, B.Hamann, K.I.Joy. Multiresolution techniques for interactive texture-based volume visualization. D.Ebert, M.Gross, B.Hamann. IEEE Visualization'99. San Francisco,1999:355-362.
[33]M.Weiler, R.Westermann, C.Hansen,et al, Level-of-detail valume rendering via 3d textures.2000,35-50.
[34]刘永军,范维澄,姚斌.绘制等值线的一种离散方法.计算机应用研究,2004,8：126-127.
[35]章勤,李品,张继顺.一种基于二次等参单元的等值线图生成算法.华中科技大学学报,2002,33：992-994.
[36]袁政强,祝家麟,白绍良.等值线图形的扫描线法.重庆大学学报(自然科学版),2001,24：89-93.
[37]林毅,金烨,马登哲等.正规网格等值线的虚路径扫描算法.计算机工程与应用,2001,92：91-94.
[38]宋丽娟,龚晓峰,钟猛.基于网格法的等值线绘制方法.现代电子技术,2005,14：65-68.
[39]韦美雁,杜丹蕾.基于规则网格的等值线的生成研究.湖南科技学院学报,2007,28：39-41.
[40]张梅华,梁文康.一个三角形网格上等值线图的绘制方法.计算机辅助设计与图形学报,1997,9：213-218.
[41]成建梅,除崇希,孙红林.三角网络等值线自动生成方法及程序实现.水利学报,1998,10：23-27.
[42]郑盛贵,颜七笙,黄临平.基于点的三角形构网算法及等值线自动生成方法.计算机与现代化,2004,5：7-9.
[43]吴自银,高金耀.一种基于网格的快速等值线填充算法.测绘学报,1999,28：350-351.
[44]罗伟锦,刘汉龙,高玉峰.一种基于确定区域填充点的等值线填色算法.河海大学学报(自然科学版),2003,31(5)：584-588.
[45]戴常英,李昕,李凌博.等值线图区域填充的边界扫描算法.微机发展,2004,14(1)：23-25.
[46]庞世明,蔡玉华,勒文芳.等值线图的彩色填充方法.计算机应用,2004,24(1)：60-62.
[47]龙述尧编著.边界单元法概论.中国科学文化出版社,2002.23-126.
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