Blending curves for landing problems by numerical differential equations, III. Separation techniques

K e y w o r d s B l e n d i n g curves, Ordinary differential equation (ODE), Separation techniques, Fundamental solutions, Finite element method, Computer geometric aided design. We thank the reviewer for one's valuable comments and suggestions on this manuscript. *Author to whom all correspondence should be addressed. 0895-7177/05/$ see front matter @ 2005 Elsevier Ltd. All rights reserved. Typeset by .A.A/IS-~X doff 10.1016/j.mcm.2005.01.024 1352 H.-T. HUANO AND Z.-C. LI 1. I N T R O D U C T I O N This paper is continued study in [1-3] of the blending curves for landing problems by numerical differential equations. A landing curve of airplane is a smooth curve described by three functions x(s), y(s), and z(s) which are governed by a system of linear ordinary differential equations (ODEs) with certain boundary conditions. The separation techniques are proposed in this paper first for the landing curves of the airplane to the airport. The separation techniques are particularly significant for nonlinear boundary conditions. The separation techniques may greatly simplify the numerical algorithms and error analysis, because the blending solutions may be found by the parametric functions, x(s), y(s), and z(s), obtained separately, and then combined together to satisfy the boundary conditions, linear or nonlinear. When a space ship is approaching a shuttle station, only a few nonlinear algebraic equations are needed to deal with the landing curves. This displays significance of the separation techniques, compared to the numerical approaches given in [2], where a large number of linear and nonlinear algebraic equations are involved in, and then solved together. The motivation of the study of blending curves and the related references are provided in [1,3]. We propose the following separation techniques, to greatly simplify the computational algorithms and error analysis. The separation techniques consist of two steps. STEP I. The solutions are first obtained by the linear boundary problems of (ODEs), and the fundamental solutions are explored under different linear boundary conditions. STEP II. The landing curve of the space ship is expressed as a linear combination of the solutions obtained from Step I, and the optimal expansion coefficients in the linear combination are sought by minimizing the global energy of the landing curve under the nonlinear constraints. Only a few nonlinear algebraic equations are obtained together; they can be solved independently. The error analysis for the nontrivial landing problem is made to derive three significant results: (1) The exact blending curves by the separation techniques with the fundamental solutions. (2) The errors O(h 1/2) and O(h 5/2) in H 2 norms of the blending curves by the separation techniques with the cubic Hermite solutions and their a posterior{ interpolant, respectively. (3) The errors O(h 2"~) in H 2 norms by those with the fifth-order Hermite method for general cases. Numerical experiments for nonlinear boundary conditions are provided in this paper to display the efficiency of the separation techniques, and to verify the error analysis made. This paper is organized as follows. In the next section, we describe the landing problems by a system of ordinary differential equations (ODEs), and derive the natural boundary conditions. In Section 3, we propose the separation techniques and extend them to ODEs with general linear conditions. In Section 4, we apply the separation techniques for the blending curves of the space ship to the shuttle station, which involve nonlinear boundary conditions. The Lagrange multipliers method and the Hermite finite element methods (FEMs) are used for the nonlinear boundary equations and the linear ODE systems, respectively. In Section 5, the fundamental solutions of ODEs are explored. In Section 6, error analysis is made for the blending curves by the separation techniques with the fundamental solutions, and with the Hermite FEMs. In Section 7, numerical experiments are reported, to display the significance of the algorithms proposed in this paper, and to verify the error analysis made. In the last section, concluding remarks are made. 2. D I F F E R E N T I A L E Q U A T I O N S A N D V A R I A T I O N A L F O R M S Consider an airplane to be landing at an airstrip constructed on the given l ine/ '= E E ~, where the ending point E is unknown, but E E ~, see Figure 1. To represent a landing curve, we choose the parametric functions W ( s ) = (x, y, z) T -~ (x(s), y(s), z(8)) T, 0 % S ~ 1, for the 3D curve. Denote the straight lines B B ~ and E E ~ as in Figure 2 by B B ' = (x~os + Xo, y~o s + Yo, Z~o s + zo) T "EE' = (X~N s + a, JNS + ~, Zrg s + ~/)T Blending Curves for Landing Problems 1353 where a, fl, and 3' are constant , and the boundary values of the solutions are given by x(0) = x0, y(0) = yo, z(0) = zo, x(1) = x g , y(1) = YN, z(1) = Zg , ' Z'(0) ' X'(1) = X~V, y ' (1) = y~r, Z'(1) = Z;V. x ' ( 0 ) = Xo, y ' ( 0 ) = = Zo,