Blending curves for landing problems by numerical differential equations II. Numerical methods

A landing curve of airplane is a blending curve that smoothly joins the two given boundary points, which is described by the parametric functions x(s), y(s), and z(s) governed by a system of ordinary differential equations (ODEs) with certain boundary conditions. In Part I, Mathematical Modelling [1], existence and uniqueness of the ODE system are explored to produce the optimal landing curves in minimum energy. In this paper, numerical techniques are provided by the finite element method (FEM) using piecewise cubic Hermite polynomials, to give the optimal solutions. An important issue is how to deal with infinite solutions occurring in the landing problems reported in [1]. Moreover, error analysis is made, and numerical examples are carried to verify the theoretical results made. This paper displays again the effectiveness and flexibility of the ODE approach to complicated blending curves. Besides, the numerical techniques in this paper can be applied directly to other landing and trajectory problems given in [1], as well as other kinds of blending curves and surfaces of airplane, ships, grand building, and astronautic shuttle-station.