Solving the Boundary Value Problem of Curves With Prescribed Geodesic Curvature Based on a Cubic B-Spline Element Method

Abstract

Composite manufacturing processes such as ATP and FW put forward specific requirements for the geodesic curvature along the layup paths of the composite tows and tapes. In the present paper, a nonuniform cubic b-spline element method is considered for solving the boundary value problem of curves with prescribed geodesic curvature. The differential equation system of the target curve is discretized through the point collocation method, and a quasi-Newton iteration scheme is adopted to approach the real solution from an initial approximation. The proposed method is proved to have third order accuracy, which shows more superiorities than existing numerical methods. The performance of our approach is investigated on a series of parametric surfaces, and experimental results demonstrate that the method is efficient.The proposed method could cope with the BVP for curves no matter their geodesic curvature vanishes or not. At the same time, the computed curves are natural and smooth, and no interpolation technique is needed to ensure the continuity of target curves. One potential application of this method is trajectory optimization for automated tape placement process.