反求工程的NURBS曲面拼接与拟合技术研究
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摘要
随着计算机技术的飞速发展,反求工程通过与计算机技术相结合已广泛应用于航空航天、机械、建筑、模具设计、医疗器械、人体器官再造、复杂产品的精度检测等许多领域。与传统的正向工程设计相比,反求工程可以设计重构出传统设计不能完成的复杂自由曲面模型,并且能够对已有产品进行快速修复和重设计,效率更高。因此这项研究具有重要的理论意义和现实意义。但是反求工程技术还不完善,尤其在曲面重构方面。本文在参考国内外相关研究的基础上,推导了广义细分理论,利用分段NURBS曲面拼接技术获得真正意义上的曲面拼接,提出了分片三角网格模型上快速参数化理论,并且通过试验分析了NURBS曲面拟合精度的影响因素。从而,为反求工程的完善和发展做出了更深层次的探索和研究。本文首先分析了基于均匀B样条理论的曲面细分算法,在分析和研究Catmull-Clark算法基础上,归纳和总结其他各种细分算法的理论基础、影响因子和实际造型的实验结果,提出了广义细分理论,为其他细分算法的分析和研究提供理论基础。通过实际曲面造型观察,分析广义细分算法中各个权值因子对曲面尖锐特征保持的影响,从而得出如何合理地选取各个权值因子已达到提高重构自由曲面的精度。
通过深入分析分片Bezier曲面重构的拼接理论,联系到Bezier曲线是NURBS曲线的特殊格式,提出了借助于分片Bezier曲面的G1连续条件进行NURBS曲面拼接的思想,推导实现了NURBS曲面间的G1光滑拼接,并且利用基于Matlab的COM组件完成了对G1连续条件的线性系统的求解。
在对已有网格划分理论进行深入研究的基础上,发现这些理论对非连续四边区域的划分无能为力。本文创建了一种对三角形网格模型上的四边界区域实现快速数据参数化的算法。经过分析三角网格模型上四边区域参数化的两种方案,选择直接在由折线围成的空间四边区域内进行参数化的方法进行数据参数化。通过坐标变换、点面投影把空间四边区域的数据参数化问题转化为平面四边区域网格划分问题,通过本文提出的平面四边区域双向伸缩网格生成算法快速实现由折线围成的空间四边区域的参数化。
比较了NURBS曲面拟合的两种方法,为了提高NURBS曲面拟合精度、获得更多的NURBS曲面拟合的详细信息,选用张量积的双向数据拟合方法进行曲面拟合。由于分片曲面公共边界和公共交点处对控制顶点进行了G1光滑约束,故通过分片NURBS曲面拟合即可到达整体G1光滑连续的曲面。并且,开发了CAD/CAM数据交换的标准IGES文件接口。
实验分析了分片NURBS拟合曲面的拟合精度影响因素,发现其精度跟三角形网格模型上的分片数量、分片四边界区域内的网格划分数量以及三角形网格模型上特征边界的识辨好坏密切相关,得到了其影响因素对拟合精度的影响趋势。
最后本文通过实例对反求工程的实际应用进行了阐述。从而得出了复杂曲面应用反求工程技术实现重构的一般过程和实际操作方法。
With the rapid development of computer technology, reverse engineering combining with computer has been applied to fields of aerial navigation, aerospace, machine, architecture, mold design, medical instrument, organ reforge, precision measure of the complicated product, etc. Comparing with the traditional engineering, reverse engineering can design and reconstruct complex free surface, and it can repair and redesign the existing product efficiently. So the study of reverse engineering has academic and practical meaning. But it is especially in the field of the free surface construct the technique of the reverse engineering isn’t perfect. On base of the relate research all over the world, the paper deduced the general subdivision theory, obtained the really G1 smooth connecting condition of the surfaces base on the piecewise NURBS surface joint technique and analyzed the precision influence factors of NURBS surface fitting. Finally, the paper made a deeper exploration and research for the perfect and development of the reverse engineering.
Firstly, the subdivision algorithm which is based on the uniform B-spline theory is analyzed, the basic theory, influencing factor and experimental result of the other subdivision algorithm is concluded and summarized, the general subdivision algorithm is proposed. By observing practical surface construction, analyzing the influence factors of the every weight to holding the features on the subdivision algorithm, how to select the weights in order to enhance the precision about the free surface is obtained.
The G1 smooth merging theory of the piecewise Bezier surface is analyzed, the Bezier curve is a special NURBS curve is considered, the idea of resort to Bezier surface G1 continuous condition for solving the G1 continuous condition of NURBS surfaces is presented. G1 continuous condition of the NURBS is deduced and realized. Linear system of G1 continuous condition is solved by the COM components about Matlab.
The existing grid generation theory is studied, incapability of grid generation that these theory to the four sided region constructed by four discontinuous broken line is found. A rapid parametrization algorithm which is used to spatial four-sided region on triangular mesh is presented in the paper. By comparing two kinds of parametrization projects of the spatial four-sided region, directly grid generation algorithm is chose. By coordinates transform and point project to the basic plane, the problem of grid generation about spatial four-sided region changed to planar four-sided region grid generation problem. A planar two-directional flexing grid generation algorithm is presented which can quickly parametrizing the planar four-sided region, in fact, a spatial four-sided region is parametrizated.
Comparing two NURBS surface fitting methods, in order to enhance the fitting precision and obtain more detail information of the NURBS surface fitting, two-directional fitting method is adopted to fitting the NURBS surface. For the points on the common boundaries and corners is restricted base on the G1 smooth connection condition. So by fitting every patches and a whole G1 smooth surface is obtained. A read-write interface is developed in the paper.
By analyzing the precision influencing factors about piecewise NURBS fitting, relationship of the number of the triangular mesh, grid number of every patches and whether well distinguished sharp features of the triangular mesh to precision is considered and influencing trend is obtained.
Finally, practical application of reverse engineering is illuminated by using reverse engineering to an aerial part and common process and practical operating methods is concluded by using reverse engineering to complex surface construction.
通过深入分析分片Bezier曲面重构的拼接理论,联系到Bezier曲线是NURBS曲线的特殊格式,提出了借助于分片Bezier曲面的G1连续条件进行NURBS曲面拼接的思想,推导实现了NURBS曲面间的G1光滑拼接,并且利用基于Matlab的COM组件完成了对G1连续条件的线性系统的求解。
在对已有网格划分理论进行深入研究的基础上,发现这些理论对非连续四边区域的划分无能为力。本文创建了一种对三角形网格模型上的四边界区域实现快速数据参数化的算法。经过分析三角网格模型上四边区域参数化的两种方案,选择直接在由折线围成的空间四边区域内进行参数化的方法进行数据参数化。通过坐标变换、点面投影把空间四边区域的数据参数化问题转化为平面四边区域网格划分问题,通过本文提出的平面四边区域双向伸缩网格生成算法快速实现由折线围成的空间四边区域的参数化。
比较了NURBS曲面拟合的两种方法,为了提高NURBS曲面拟合精度、获得更多的NURBS曲面拟合的详细信息,选用张量积的双向数据拟合方法进行曲面拟合。由于分片曲面公共边界和公共交点处对控制顶点进行了G1光滑约束,故通过分片NURBS曲面拟合即可到达整体G1光滑连续的曲面。并且,开发了CAD/CAM数据交换的标准IGES文件接口。
实验分析了分片NURBS拟合曲面的拟合精度影响因素,发现其精度跟三角形网格模型上的分片数量、分片四边界区域内的网格划分数量以及三角形网格模型上特征边界的识辨好坏密切相关,得到了其影响因素对拟合精度的影响趋势。
最后本文通过实例对反求工程的实际应用进行了阐述。从而得出了复杂曲面应用反求工程技术实现重构的一般过程和实际操作方法。
With the rapid development of computer technology, reverse engineering combining with computer has been applied to fields of aerial navigation, aerospace, machine, architecture, mold design, medical instrument, organ reforge, precision measure of the complicated product, etc. Comparing with the traditional engineering, reverse engineering can design and reconstruct complex free surface, and it can repair and redesign the existing product efficiently. So the study of reverse engineering has academic and practical meaning. But it is especially in the field of the free surface construct the technique of the reverse engineering isn’t perfect. On base of the relate research all over the world, the paper deduced the general subdivision theory, obtained the really G1 smooth connecting condition of the surfaces base on the piecewise NURBS surface joint technique and analyzed the precision influence factors of NURBS surface fitting. Finally, the paper made a deeper exploration and research for the perfect and development of the reverse engineering.
Firstly, the subdivision algorithm which is based on the uniform B-spline theory is analyzed, the basic theory, influencing factor and experimental result of the other subdivision algorithm is concluded and summarized, the general subdivision algorithm is proposed. By observing practical surface construction, analyzing the influence factors of the every weight to holding the features on the subdivision algorithm, how to select the weights in order to enhance the precision about the free surface is obtained.
The G1 smooth merging theory of the piecewise Bezier surface is analyzed, the Bezier curve is a special NURBS curve is considered, the idea of resort to Bezier surface G1 continuous condition for solving the G1 continuous condition of NURBS surfaces is presented. G1 continuous condition of the NURBS is deduced and realized. Linear system of G1 continuous condition is solved by the COM components about Matlab.
The existing grid generation theory is studied, incapability of grid generation that these theory to the four sided region constructed by four discontinuous broken line is found. A rapid parametrization algorithm which is used to spatial four-sided region on triangular mesh is presented in the paper. By comparing two kinds of parametrization projects of the spatial four-sided region, directly grid generation algorithm is chose. By coordinates transform and point project to the basic plane, the problem of grid generation about spatial four-sided region changed to planar four-sided region grid generation problem. A planar two-directional flexing grid generation algorithm is presented which can quickly parametrizing the planar four-sided region, in fact, a spatial four-sided region is parametrizated.
Comparing two NURBS surface fitting methods, in order to enhance the fitting precision and obtain more detail information of the NURBS surface fitting, two-directional fitting method is adopted to fitting the NURBS surface. For the points on the common boundaries and corners is restricted base on the G1 smooth connection condition. So by fitting every patches and a whole G1 smooth surface is obtained. A read-write interface is developed in the paper.
By analyzing the precision influencing factors about piecewise NURBS fitting, relationship of the number of the triangular mesh, grid number of every patches and whether well distinguished sharp features of the triangular mesh to precision is considered and influencing trend is obtained.
Finally, practical application of reverse engineering is illuminated by using reverse engineering to an aerial part and common process and practical operating methods is concluded by using reverse engineering to complex surface construction.
引文
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108施法中.计算机辅助几何设计与非均匀有理B样条(CAGD&NURBS).高等教育出版社. 2001
109李亚利,秦新强,童小红,何怡. B样条曲面的节点插入算法.西安工业学院学报. 2005, 25(1):79~83
110李建坤.多边参数域曲面设计方法的实现及应用研究.吉林大学硕士论文. 2003
111潘日晶,潘日红,姚志强. B样条曲线同时插入多个节点的快速算法. 2003,24(12):2295~2298
112 Xiquan Shi, Tianjun Wang, Peiru Wu, Fengshan Liu. Reconstruction of Convergent G1 Smooth B-Spline Surfaces. CAGD. 2004,(21): 893~913
113 Bennis C., Vézien J. M., and Iglésias G. Piecewise Surface Flattening for Non-Distorted Texture Mapping. Computer Graphics (Proceedings of SIGGRAPH’91). 1991:237~246
114李基拓,陆国栋.基于边折叠和质点-弹簧模型的网格简化优化算法.计算机辅助设计与图形学学报. 2006,18(3):426~432
115甘家付,杨勋年,赵艳.一种网格参数化的优化算法.浙江大学学报(理学版).2004,31(5):538-543
116 Robert T.Bailey, C.K. Hsieh and H. Li: Grid Generation in Two Dimensions Using the Complex Variable Boundary Element Method. Applied Mathematical Modelling. 1995, 19(6): 322~332
117 Ma W., Kruth J. P. Parameterization of Randomly Measured Points for Least Squares Fitting of B-Spline Curves and Surfaces. Computer-Aided Design. 1995,(27): 663~675
118 Piegl L. A., Tiller W. Parameterization for Surface Fitting in Reverse Engineering. Computer-Aided Design. 2001,(33):593~603
119 LIN Hongwei, WANG Guojin, Liu Ligang, BAO Hujun. Parameterization for Fitting Triangular Mesh. Progress in Natural Science. 2006, 16(11):1214~1221
120 L. Chacon, G. Lapenta. A Fully Implicit, Nonlinear Adaptive Grid Strategy. Journal of Computational Physics. 2006, (212):703~717
121 B.A. Souza, E.M. Matos, L.T. Furlan, J.R. Nunhez. A Simple Two-Dimensional Method for Orthogonal and Nonorthogonal Grid Generation. Computers and Chemical Engineering. 2007, (31):800~807
122陆邦春,张立武,张冠华.逃逸发动机数控加工与仿真技术.航天制造技术. 2003, (5):3~6