基于优化的NURBS曲面G¹连续拼接实现

Research on G¹ Continuity Optimization Theory Based on NURBS Representation Surface

摘要: 航天飞机、汽车数字模型等零件对拼接处要求C¹/G¹连续，因此本文提出面向NURBS曲面进行G¹连续性优化算法理论。首先根据NURBS曲面求取连续性方程并作为约束，通过设置控制点与权值作为优化目标，利用拉格朗日乘子法进行约束求解得到G¹连续后的控制点和权值，将得到的控制点和权值带回到原曲面实现在拼接处C¹/G¹连续。为了保证曲面在优化后变化尽可能的小，将变化前后的控制点与权值乘积作为优化问题，控制点前后变化的差值为优化条件，通过设计算法将数值经过数次迭代得到NURBS曲面C¹/G¹拼接时的最优解。结果表明，该方法能够实现控制点及权值的数值优化，并对于NURBS曲面高阶连续优化、多片拼接以及NURBS三维模型的连续性优化提供借鉴意义。

Abstract: Space shuttle, automobile digital model and other parts require C¹/G¹ continuity at the splice, so this paper proposes G¹ continuity optimization algorithm theory for NURBS surfaces. First, the con-tinuity equation is obtained according to the NURBS surface and used as the constraint. By setting control points and weights as the optimization objective, the constraint solution is performed using the Lagrange multiplier method to obtain the control points and weights after G¹ continuity. The control points and weights are brought back to the original surface to achieve C¹/G¹ continuity at the splice. In order to ensure that the surface changes as little as possible after optimization, the product of the control points before and after the change and the weight value is taken as the opti-mization problem, and the difference between the changes of the control points before and after the change is taken as the optimization condition. The optimal solution for NURBS surface C¹/G¹ splicing is obtained through several iterations of the design algorithm. The results show that the method can realize the numerical optimization of control points and weights, and provide reference for the higher-order continuous optimization of NURBS surfaces, multi piece splicing, and continuity opti-mization of NURBS three-dimensional models.