A novel form of interpolating and reproducing kernel method and numerical studies on the nonlinear dynamics of pipe whip

摘要

A novel discretization scheme based on the reproducing kernel interpolating (RKI) approximations of functions is proposed. It is a novel form of RK interpolation that non-trivially improves upon the earlier version of the strategy by Chen et al.14 where, the primitive function had to attain its maximum value of unity over a typically small support size thereby leading to an ill-conditioning of the algorithm. The presently proposed interpolating strategy effectively resolves such numerical difficulties by taking a linear combination of two different families of RK basis functions and determines the coefficients of the linear combination through interpolating conditions. In the process, the twin objectives of polynomial reproduction and interpolation are together met by the two families of basis functions and thus, unlike the approach by Chen et al. (2003). none of these two families of functions has to separately satisfy the Kronecker delta property and there are no imposed restrictions on the support sizes, The proposed RKI method is then applied to obtain strong solutions of highly nonlinear partial differential equations (PDE-s) representing elastic-plastic nonlinear vibration of a cantilever pipe under a follower load at its tip. While Reid et a/.15 have employed a central difference (CD) scheme for space and time discretizations of the governing PDE-s, this approach showed numerical instability (especially over the plastic regions where curvature profiles exhibit highly localized peaks) and a lack of convergence as the number of nodes for spatial discretization increases. The new RKI strategy developed in this study is shown to be quite suited to negotiate such problems. Indeed, a major observation of this paper has been that spurious and high frequency bursts in numerical computations of curvature profiles and their derivatives through central differencing or the RKI method by .Chen et al.13, may be significantly arrested, and even nearly eliminated, by this method.