反求工程中自由形状模型的美化技术
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摘要
作为现代设计方法学之一，反求工程已经成为产品创新设计和快速开发的重要技术手段。本文在基于特征的反求工程建模策略下，深入研究了以自由曲线和曲面模型的形状优化为核心的美化理论和方法。
针对包含异常数据的点云的切片结果，提出通过基于直接几何修改的自动光顺方法来恢复原始曲线的特征结构与形状信息，并利用新的节点矢量配置方法以达到自由曲线形状的优化拟合。
利用可微流形的无穷小变形技术，实现紧公差约束下对点云逼近的多张B-样条曲面的全局美化。首先，根据微分流形上的Beltrami-Laplace算子定义反映曲面整体形状变化的变形能量泛函，并得到泛函的唯一极小解．然后，给出了两张单节点B-样条曲面间G~1连续的充要条件以及公共边界曲线控制顶点的本征方程，并利用局部格式构造整体收敛的G~1光滑拼接的边界表示模型获得变形映射族的特解。特解的构造不可避免地导致模型在缝合区域的形状产生局部瑕疵，影响了反求模型的保形性。最后，通过特解来构造能量泛函的极小解，使得这类似应力集中的效应被逐步松弛到曲面的内部，从而改善了模型的整体形状。
针对反求工程中由多张B-样条曲面构成的边界表示模型不可避免地在局部区域会出现的形状不规则性，提出利用局部美化技术来消除这种外形的不规则性的方法。首先根据边界表示模型的拓扑给出局部美化的模式，并阐述曲面光顺与模型局部美化之间的区别和联系。然后，通过分析可接受的形状优化准则及其局部微分几何特性，给出局部美化的数学模型。最后，在利用形状分析方法构造局部形状不规则性区域的方法后，借鉴离散的非线性优化方法来保证形状修改的局部性和保持模型的拓扑。
提出综合利用CAGD中的曲线曲面理论和方法，通过构造合理的局部区域来裁剪N边汇交曲面，并利用汇交曲面的原始信息解决局部NURBS曲面的协调设计。在新生成曲面与裁剪汇交曲面之间保证处处G~1连续，同时光滑地逼近于局部区域的特征走向。局部协调设计能够较好地解决模型的整体连续性和保形性。
与自由形状模型的美化相关的主要算法已经嵌入到基于特征的反求建模系统—RE-SOFT中。给出的应用实例揭示了模型美化在反求工程中的意义。在总结全文工作的基础上，对未来的研究工作进行展望。
As a modern design methodology, reverse engineering has become an essential tool for industrial product development and innovation. Feature based reverse modeling strategy can be used to capture the original design intention accurately and to guide the reverse modeling process conveniently. The beautification theories and methods, focusing on the freeform models, are intensively explored to improve the shape of the final reverse B-rep model in this dissertation.
The sectional profile converted by slicing technique from unorganized point cloud may contain unpleasant points wandering away too far from the real-life object and may deteriorate the quality of the final curve/curves. Therefore, an automatic fairing method based on directly geometric modification is proposed to recover the feature structure and shape information of the original curve. Relying on the faired results, a novel knot placement algorithm is presented to improve the shape in the freeform curve fitting process.
The multiple surfaces approximated to point cloud subject to tight error are beautified globally by using the infinitesimal deformation technique of differentiable manifold. The deformation energy functional reflecting the overall shape of a surface is defined by using the Beltrami-Laplace operator on the manifold. Also the unique solution of the minimum of the energy functional is formulated according to the property of harmonic function. Then, the necessary and sufficient conditions of G~1 continuity between two B-spline surfaces with single knots are given and simplified, as well as the intrinsic equations of control points of the common boundary curve. Based on the local scheme of convergent G~1 smooth surfaces, a special solution of the family of deformation maps is constructed. The special solution is represented by the smoothly stitched B-rep model. The inevitable local imperfection at the stitching regions caused by constructing the special solution greatly influences on the shape preservation of reverse engineered model. Finally, the final solution is constructed such that the deformation energy clustering round the stitching regions is released gradually to the surface interior. Consequently, the shape of the model is improved.
Boundary representation model composed of multiple B-spline surfaces reconstructed from point cloud suffers from various local shape imperfections inevitably caused by noise in the point data and the reverse modeling strategies. A beautification technique is proposed to remove the local shape irregularities of such a model. Based on the topologies of the B-rep model, the basic patterns of local beautification are classified firstly. The shape beautification is compared with the traditional surface fairing according to the pattern classification. Then, the mathematical model of the beautification is given after analyzing acceptable shape optimization criteria and their intrinsic characteristics of differential geometry. Finally, the local irregularities on the model regions are identified by using shape interrogation methods. As the beautifying variables, each control point associated with the regions is modified solely subject to the tolerance constraints and/or G~0 continuity constraints. Simultaneously, an optimization method is used to preserve an approximated G~1 continuity in the local regions. Consequently, the shape is improved without destroying the model topologies.
A new local consistent fitting method is presented for dealing with the confluence region, where multiple surfaces meet, to reach global smoothness and keep local feature of normal vector trends. Based on the basic theories of curve and surface in CAGD such as geometric continuity, intersection, bridge, discretization and fitting etc., a reasonable local region definition and feature sensitive surface fitting technique subject to complex boundary constraints are discussed in detail. The final consistent surface, which keeps the fitting precision and reflects the feature trend of the original surfaces, satisfies G~1 continuity with the whole surface model.
The algorithms about beautification is embedded in the feature-based reverse modeler—RE-SOFT. Typical industrial examples are used to illustrate the validation of the proposed beautification schemes and to reveal the practical value in reverse engineering. Finally, some topics for future work are discussed briefly.
针对包含异常数据的点云的切片结果，提出通过基于直接几何修改的自动光顺方法来恢复原始曲线的特征结构与形状信息，并利用新的节点矢量配置方法以达到自由曲线形状的优化拟合。
利用可微流形的无穷小变形技术，实现紧公差约束下对点云逼近的多张B-样条曲面的全局美化。首先，根据微分流形上的Beltrami-Laplace算子定义反映曲面整体形状变化的变形能量泛函，并得到泛函的唯一极小解．然后，给出了两张单节点B-样条曲面间G~1连续的充要条件以及公共边界曲线控制顶点的本征方程，并利用局部格式构造整体收敛的G~1光滑拼接的边界表示模型获得变形映射族的特解。特解的构造不可避免地导致模型在缝合区域的形状产生局部瑕疵，影响了反求模型的保形性。最后，通过特解来构造能量泛函的极小解，使得这类似应力集中的效应被逐步松弛到曲面的内部，从而改善了模型的整体形状。
针对反求工程中由多张B-样条曲面构成的边界表示模型不可避免地在局部区域会出现的形状不规则性，提出利用局部美化技术来消除这种外形的不规则性的方法。首先根据边界表示模型的拓扑给出局部美化的模式，并阐述曲面光顺与模型局部美化之间的区别和联系。然后，通过分析可接受的形状优化准则及其局部微分几何特性，给出局部美化的数学模型。最后，在利用形状分析方法构造局部形状不规则性区域的方法后，借鉴离散的非线性优化方法来保证形状修改的局部性和保持模型的拓扑。
提出综合利用CAGD中的曲线曲面理论和方法，通过构造合理的局部区域来裁剪N边汇交曲面，并利用汇交曲面的原始信息解决局部NURBS曲面的协调设计。在新生成曲面与裁剪汇交曲面之间保证处处G~1连续，同时光滑地逼近于局部区域的特征走向。局部协调设计能够较好地解决模型的整体连续性和保形性。
与自由形状模型的美化相关的主要算法已经嵌入到基于特征的反求建模系统—RE-SOFT中。给出的应用实例揭示了模型美化在反求工程中的意义。在总结全文工作的基础上，对未来的研究工作进行展望。
As a modern design methodology, reverse engineering has become an essential tool for industrial product development and innovation. Feature based reverse modeling strategy can be used to capture the original design intention accurately and to guide the reverse modeling process conveniently. The beautification theories and methods, focusing on the freeform models, are intensively explored to improve the shape of the final reverse B-rep model in this dissertation.
The sectional profile converted by slicing technique from unorganized point cloud may contain unpleasant points wandering away too far from the real-life object and may deteriorate the quality of the final curve/curves. Therefore, an automatic fairing method based on directly geometric modification is proposed to recover the feature structure and shape information of the original curve. Relying on the faired results, a novel knot placement algorithm is presented to improve the shape in the freeform curve fitting process.
The multiple surfaces approximated to point cloud subject to tight error are beautified globally by using the infinitesimal deformation technique of differentiable manifold. The deformation energy functional reflecting the overall shape of a surface is defined by using the Beltrami-Laplace operator on the manifold. Also the unique solution of the minimum of the energy functional is formulated according to the property of harmonic function. Then, the necessary and sufficient conditions of G~1 continuity between two B-spline surfaces with single knots are given and simplified, as well as the intrinsic equations of control points of the common boundary curve. Based on the local scheme of convergent G~1 smooth surfaces, a special solution of the family of deformation maps is constructed. The special solution is represented by the smoothly stitched B-rep model. The inevitable local imperfection at the stitching regions caused by constructing the special solution greatly influences on the shape preservation of reverse engineered model. Finally, the final solution is constructed such that the deformation energy clustering round the stitching regions is released gradually to the surface interior. Consequently, the shape of the model is improved.
Boundary representation model composed of multiple B-spline surfaces reconstructed from point cloud suffers from various local shape imperfections inevitably caused by noise in the point data and the reverse modeling strategies. A beautification technique is proposed to remove the local shape irregularities of such a model. Based on the topologies of the B-rep model, the basic patterns of local beautification are classified firstly. The shape beautification is compared with the traditional surface fairing according to the pattern classification. Then, the mathematical model of the beautification is given after analyzing acceptable shape optimization criteria and their intrinsic characteristics of differential geometry. Finally, the local irregularities on the model regions are identified by using shape interrogation methods. As the beautifying variables, each control point associated with the regions is modified solely subject to the tolerance constraints and/or G~0 continuity constraints. Simultaneously, an optimization method is used to preserve an approximated G~1 continuity in the local regions. Consequently, the shape is improved without destroying the model topologies.
A new local consistent fitting method is presented for dealing with the confluence region, where multiple surfaces meet, to reach global smoothness and keep local feature of normal vector trends. Based on the basic theories of curve and surface in CAGD such as geometric continuity, intersection, bridge, discretization and fitting etc., a reasonable local region definition and feature sensitive surface fitting technique subject to complex boundary constraints are discussed in detail. The final consistent surface, which keeps the fitting precision and reflects the feature trend of the original surfaces, satisfies G~1 continuity with the whole surface model.
The algorithms about beautification is embedded in the feature-based reverse modeler—RE-SOFT. Typical industrial examples are used to illustrate the validation of the proposed beautification schemes and to reveal the practical value in reverse engineering. Finally, some topics for future work are discussed briefly.
引文
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