反求工程中基于几何约束的模型重建理论及应用研究
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摘要
作为先进制造技术的一个重要组成部分，反求工程已从最初的原型复制技术逐步发展成为支持产品创新设计和新产品开发的重要技术手段。与这一趋势相对应，反求工程重建的CAD模型已不再是传统的实体模型，而是更具创新设计能力、支持产品全生命周期的特征模型。在面向特征造型的反求工程技术背景下，本文研究基于几何约束的曲线曲面模型重建理论和方法。
解析曲线曲面(直线、圆弧、平面、圆柱面、球面等)和自由曲线曲面(样条曲线、四边域样条曲面)常见于工业产品的几何外形中，本文研究这些曲线曲面在局部几何约束条件下的重建问题。文中讨论曲线曲面的表达形式，给出曲线曲面用于拟合的目标函数和曲线曲面之间各种几何约束的代数表达式。分析各曲线曲面的目标函数在数值优化中的稳定性问题，阐述应用相似变换改善稳定性的可行性和具体方法。
对称是工业产品几何形状的一项重要性质，重建对称模型的关键是对称约束的处理。对称约束是整体几何约束，难以用少数几个代数方程表达，其处理方式不同于局部几何约束。对于解析曲线曲面对称模型的重建，采用二维和三维空间各种常见对称形式的显式表达，推导数据点到对称模型的距离函数，进而导出用于最小二乘拟合的目标函数，并讨论对称变换参数初值的获取方法。对于B样条表达的自由曲线曲面对称模型的重建，基于从点云提取的对称轴或对称平面，给出B样条曲线曲面对称节点矢量的构造方法，并推导控制顶点的对称约束方程，用拉格朗日乘子法求解对称约束下的B样条曲线曲面拟合问题。
任意边域曲面填充有时会出现于反求工程复杂CAD模型重建中，基于边界条件的离散化处理和裁剪B样条曲面模型，本文给出一种实用的任意边域曲面填充方法。对复杂边界条件离散化处理中需要遵循的准则作较为详细的讨论，并给出任意边域曲面填充的数学模型及其求解方法。
介绍反求工程CAD建模软件RE-SOFT中几何约束的处理模块及其实现方法，并给出部分数据结构的伪代码。最后以工业汽轮机叶片的反求建模为例，展示应用RE-SOFT并结合商业CAD建模软件进行反求建模的基本方法。
As one of the most important components of advanced manufacturing technologies, reverse engineering is no longer simply for prototype duplication, but has gradually developed into a key supporting tool for product innovative design and new product development. In such a situation, the CAD models reverse engineering creates are no longer the traditional solid models, but the feature models, which are with more capability of innovative design and support the product life time cycle. Under the background of feature modeling oriented reverse engineering technologies, research work regarding the theory and application of geometric constraint based model reconstruction is presented in this thesis.
Both analytic shapes (line, circular arc, plane, cylinder, sphere etc.) and freeform shapes (spline curve, 4-sided spline surface) appear frequently in industrial products, the problem of reconstructing these curves and surfaces under local geometric constraints is tackled in this thesis. The representations of curves and surfaces are discussed; the objective functions for curve and surface fitting are deduced, as well as the algebraic representations of the geometric constraints between these curves and surfaces. The stability issue of the objective functions of curves and surfaces in numerical optimization is analyzed, and the feasibility of improvement by using similarity transformation is presented, as well as detailed algorithmic steps.
Symmetry is an important property of the shape of industrial products, and to deal with the symmetry constraint properly is of key importance to the reconstruction of symmetric models. Symmetry constraint is a global constraint, which cannot be represented by a few algebraic equations. As a result, the resolution of symmetry constraint is different from local geometric constraints. For the reconstruction of symmetric models composed of analytic curves and surfaces, explicit representations are adopted, distance functions between data points and symmetric models are deduced, as well as the objective functions for least squares model fitting, and methods to obtain initial values of parameters for symmetry transformations are also discussed. For the reconstruction of symmetric freeform curves and surfaces (represented in B-Spline model), a method based on the symmetry axis or symmetry plane extracted from point cloud is presented to construct symmetric knot vector, and symmetry constraint equations regarding the control points are deduced, problem of B-Spline curve or surface fitting under symmetry constraint equations is sloved by the Lagrange multiplier method.
N-Sided hole filling appears occasionally in the reconstruction of complex CAD models in reverse engineering, this thesis presents a practical N-Sided hole filling method, which is based on discretization of boundary conditions and trimmed B-Spline surface model. Criteria for boundary discretization are discussed in detail, and mathematical model for N-Sided hole filling is given, as well as its solving method.
Software module and its implementation regarding the resolution of constraints in RE-SOFT (a reverse engineering software) are introduced; and pseudo code of some data structures is presented. Finally, an example for the modeling of an industrial gas turbine blade is given to show the basic methods of reverse modeling in RE-SOFT with the assistance of commercial CAD software.
解析曲线曲面(直线、圆弧、平面、圆柱面、球面等)和自由曲线曲面(样条曲线、四边域样条曲面)常见于工业产品的几何外形中，本文研究这些曲线曲面在局部几何约束条件下的重建问题。文中讨论曲线曲面的表达形式，给出曲线曲面用于拟合的目标函数和曲线曲面之间各种几何约束的代数表达式。分析各曲线曲面的目标函数在数值优化中的稳定性问题，阐述应用相似变换改善稳定性的可行性和具体方法。
对称是工业产品几何形状的一项重要性质，重建对称模型的关键是对称约束的处理。对称约束是整体几何约束，难以用少数几个代数方程表达，其处理方式不同于局部几何约束。对于解析曲线曲面对称模型的重建，采用二维和三维空间各种常见对称形式的显式表达，推导数据点到对称模型的距离函数，进而导出用于最小二乘拟合的目标函数，并讨论对称变换参数初值的获取方法。对于B样条表达的自由曲线曲面对称模型的重建，基于从点云提取的对称轴或对称平面，给出B样条曲线曲面对称节点矢量的构造方法，并推导控制顶点的对称约束方程，用拉格朗日乘子法求解对称约束下的B样条曲线曲面拟合问题。
任意边域曲面填充有时会出现于反求工程复杂CAD模型重建中，基于边界条件的离散化处理和裁剪B样条曲面模型，本文给出一种实用的任意边域曲面填充方法。对复杂边界条件离散化处理中需要遵循的准则作较为详细的讨论，并给出任意边域曲面填充的数学模型及其求解方法。
介绍反求工程CAD建模软件RE-SOFT中几何约束的处理模块及其实现方法，并给出部分数据结构的伪代码。最后以工业汽轮机叶片的反求建模为例，展示应用RE-SOFT并结合商业CAD建模软件进行反求建模的基本方法。
As one of the most important components of advanced manufacturing technologies, reverse engineering is no longer simply for prototype duplication, but has gradually developed into a key supporting tool for product innovative design and new product development. In such a situation, the CAD models reverse engineering creates are no longer the traditional solid models, but the feature models, which are with more capability of innovative design and support the product life time cycle. Under the background of feature modeling oriented reverse engineering technologies, research work regarding the theory and application of geometric constraint based model reconstruction is presented in this thesis.
Both analytic shapes (line, circular arc, plane, cylinder, sphere etc.) and freeform shapes (spline curve, 4-sided spline surface) appear frequently in industrial products, the problem of reconstructing these curves and surfaces under local geometric constraints is tackled in this thesis. The representations of curves and surfaces are discussed; the objective functions for curve and surface fitting are deduced, as well as the algebraic representations of the geometric constraints between these curves and surfaces. The stability issue of the objective functions of curves and surfaces in numerical optimization is analyzed, and the feasibility of improvement by using similarity transformation is presented, as well as detailed algorithmic steps.
Symmetry is an important property of the shape of industrial products, and to deal with the symmetry constraint properly is of key importance to the reconstruction of symmetric models. Symmetry constraint is a global constraint, which cannot be represented by a few algebraic equations. As a result, the resolution of symmetry constraint is different from local geometric constraints. For the reconstruction of symmetric models composed of analytic curves and surfaces, explicit representations are adopted, distance functions between data points and symmetric models are deduced, as well as the objective functions for least squares model fitting, and methods to obtain initial values of parameters for symmetry transformations are also discussed. For the reconstruction of symmetric freeform curves and surfaces (represented in B-Spline model), a method based on the symmetry axis or symmetry plane extracted from point cloud is presented to construct symmetric knot vector, and symmetry constraint equations regarding the control points are deduced, problem of B-Spline curve or surface fitting under symmetry constraint equations is sloved by the Lagrange multiplier method.
N-Sided hole filling appears occasionally in the reconstruction of complex CAD models in reverse engineering, this thesis presents a practical N-Sided hole filling method, which is based on discretization of boundary conditions and trimmed B-Spline surface model. Criteria for boundary discretization are discussed in detail, and mathematical model for N-Sided hole filling is given, as well as its solving method.
Software module and its implementation regarding the resolution of constraints in RE-SOFT (a reverse engineering software) are introduced; and pseudo code of some data structures is presented. Finally, an example for the modeling of an industrial gas turbine blade is given to show the basic methods of reverse modeling in RE-SOFT with the assistance of commercial CAD software.
引文
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