检索矩形结构化网格中裁剪单元的数值算法
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摘要
裁剪等几何分析的首要任务就是检索出裁剪曲面中的裁剪单元,为此提出一种快速检索矩形结构化网格中裁剪单元的数值算法.首先将网格中每个单元的边界划分为12段区间;然后在剪裁曲线上选取适当的参数点,并将得到的离散曲线代替原剪裁曲线,对矩形结构化网格进行剪裁;根据离散剪裁曲线与单元边界交点位置的不同,将裁剪单元划分为156种不同的类型.该算法还可以根据不同情况来获取剪裁曲线上的点,当采用闭合逆时针矩形剪裁曲线对裁剪NURBS曲面参数网格进行剪裁时,该算法能够快速、有效地检索到裁剪单元,并得到剪裁曲线曲率变化大的点以及裁剪单元在物理空间中的像;悬臂梁的最优拓扑结构算例证明了该算法能够快速、有效地检索出任意矩形结构化网格中的裁剪单元.
The primary task of trimmed isogeometric analysis is searching trimmed elements of trimmed surfaces, so a numerical algorithm searching trimmed elements of a rectangular structured grid is proposed in this paper. Firstly, the boundary of every element in the grid was segmented into 12 parts. Some parametric points on the trimming curve were selected, and then the grid was trimmed by the obtained discrete trimming curve rather than the original trimming curve. Therefore, all the trimmed elements fell into 156 categories according to the intersections of the discrete trimming curves and boundaries of elements. Besides, specific points on the trimming curve can also be obtained depending on situations using the algorithm. When the parametric grid of a trimmed NURBS surface was trimmed by a closed anticlockwise rectangular curve, all the trimmed elements can be detected using the algorithm. The point at which the curvature changes greatly and the images in physical space of trimmed elements can also be obtained. The optimal topology structure of a short cantilever beam demonstrates that the algorithm can searching trimmed elements of any rectangular structured grid effectively and efficiently.
The primary task of trimmed isogeometric analysis is searching trimmed elements of trimmed surfaces, so a numerical algorithm searching trimmed elements of a rectangular structured grid is proposed in this paper. Firstly, the boundary of every element in the grid was segmented into 12 parts. Some parametric points on the trimming curve were selected, and then the grid was trimmed by the obtained discrete trimming curve rather than the original trimming curve. Therefore, all the trimmed elements fell into 156 categories according to the intersections of the discrete trimming curves and boundaries of elements. Besides, specific points on the trimming curve can also be obtained depending on situations using the algorithm. When the parametric grid of a trimmed NURBS surface was trimmed by a closed anticlockwise rectangular curve, all the trimmed elements can be detected using the algorithm. The point at which the curvature changes greatly and the images in physical space of trimmed elements can also be obtained. The optimal topology structure of a short cantilever beam demonstrates that the algorithm can searching trimmed elements of any rectangular structured grid effectively and efficiently.
引文
[1]Hughes T J R,Cottrell J A,Bazilevs Y.Isogeometric analysis:CAD,finite elements,NURBS,exact geometry and mesh refinement[J].Computer Methods in Applied Mechanics and Engineering,2005,194(39-41):4135-4195
[2]Austin Cottrell J,Hughes T J R,Bazilevs Y.Isogeometric analysis:toward integration of CAD and FEA[M].Chichester:John Wiley&Sons,2009
[3]Xu Gang,Li Xin,Huang Zhangjin,et al.Geometric computing for isogeometric analysis[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(4):570-581(in Chinese)(徐岗,李新,黄章进,等.面向等几何分析的几何计算[J].计算机辅助设计与图形学学报,2015,27(4):570-581)
[4]Kim H J,Seo Y D,Youn S K.Isogeometric analysis for trimmed CAD surfaces[J].Computer Methods in Applied Mechanics and Engineering,2009,198(37-40):2982-2995
[5]Kim H J,Seo Y D,Youn S K.Isogeometric analysis with trimming technique for problems of arbitrary complex topology[J].Computer Methods in Applied Mechanics and Engineering,2010,199(45-48):2796-2812
[6]Seo Y D,Kim H J,Youn S K.Shape optimization and its extension to topological design based on isogeometric analysis[J].International Journal of Solids and Structures,2010,47(11/12):1618-1640
[7]Seo Y D,Kim H J,Youn S K.Isogeometric topology optimization using trimmed spline surfaces[J].Computer Methods in Applied Mechanics and Engineering,2010,199(49-52):3270-3296
[8]Kang P,Youn S K.Isogeometric topology optimization of shell structures using trimmed NURBS surfaces[J].Finite Elements in Analysis and Design,2016,120:18-40
[9]Schmidt R,Wüchner R,Bletzinger K U.Isogeometric analysis of trimmed NURBS geometries[J].Computer Methods in Applied Mechanics and Engineering,2012,241-244:93-111
[10]Nagy A P,Benson D J.On the numerical integration of trimmed isogeometric elements[J].Computer Methods in Applied Mechanics and Engineering,2015,284:165-185
[11]Zhu X F,Hu P,Ma Z D.B++splines with applications to isogeometric analysis[J].Computer Methods in Applied Mechanics and Engineering,2016,311:503-536
[12]Zhang W S,Yang W Y,Zhou J H,et al.Structural topology optimization through explicit boundary evolution[J].Journal of Applied Mechanics,2017,84(1):Article No.011011
[2]Austin Cottrell J,Hughes T J R,Bazilevs Y.Isogeometric analysis:toward integration of CAD and FEA[M].Chichester:John Wiley&Sons,2009
[3]Xu Gang,Li Xin,Huang Zhangjin,et al.Geometric computing for isogeometric analysis[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(4):570-581(in Chinese)(徐岗,李新,黄章进,等.面向等几何分析的几何计算[J].计算机辅助设计与图形学学报,2015,27(4):570-581)
[4]Kim H J,Seo Y D,Youn S K.Isogeometric analysis for trimmed CAD surfaces[J].Computer Methods in Applied Mechanics and Engineering,2009,198(37-40):2982-2995
[5]Kim H J,Seo Y D,Youn S K.Isogeometric analysis with trimming technique for problems of arbitrary complex topology[J].Computer Methods in Applied Mechanics and Engineering,2010,199(45-48):2796-2812
[6]Seo Y D,Kim H J,Youn S K.Shape optimization and its extension to topological design based on isogeometric analysis[J].International Journal of Solids and Structures,2010,47(11/12):1618-1640
[7]Seo Y D,Kim H J,Youn S K.Isogeometric topology optimization using trimmed spline surfaces[J].Computer Methods in Applied Mechanics and Engineering,2010,199(49-52):3270-3296
[8]Kang P,Youn S K.Isogeometric topology optimization of shell structures using trimmed NURBS surfaces[J].Finite Elements in Analysis and Design,2016,120:18-40
[9]Schmidt R,Wüchner R,Bletzinger K U.Isogeometric analysis of trimmed NURBS geometries[J].Computer Methods in Applied Mechanics and Engineering,2012,241-244:93-111
[10]Nagy A P,Benson D J.On the numerical integration of trimmed isogeometric elements[J].Computer Methods in Applied Mechanics and Engineering,2015,284:165-185
[11]Zhu X F,Hu P,Ma Z D.B++splines with applications to isogeometric analysis[J].Computer Methods in Applied Mechanics and Engineering,2016,311:503-536
[12]Zhang W S,Yang W Y,Zhou J H,et al.Structural topology optimization through explicit boundary evolution[J].Journal of Applied Mechanics,2017,84(1):Article No.011011