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 automated tape placement (ATP) and filament winding process (FW) put forward specific requirements for the geodesic curvature along the layup paths of the composite tows and tapes. In this paper, a non-uniform cubic b-spline element method is proposed 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 comparing with existing numerical methods. Simulations and experiments on a series of parametric surfaces are performed to investigate the performance of the proposed approach and the results verify the high efficiency. 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 such that interpolation technique is unnecessary to ensure the continuity of target curves. One potential application of this method is trajectory optimization for automated tape placement process.