MODIFIED CHEBYSHEV-PICARD ITERATION METHODS FOR SOLUTION
OF INITIAL VALUE AND BOUNDARY VALUE PROBLEMS
A Dissertation by XIAOLI BAI
Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY
August 2010
Major Subject: Aerospace Engineering MODIFIED CHEBYSHEV-PICARD ITERATION METHODS FOR SOLUTION OF INITIAL VALUE AND BOUNDARY VALUE PROBLEMS
A Dissertation by
XIAOLI BAI
Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
Approved by:
Chair of Committee, John L. Junkins Committee Members, David C. Hyland Srinivas Rao Vadali Johnny Hurtado Aniruddha Datta Head of Department, Dimitris Lagoudas
August 2010
Major Subject: Aerospace Engineering iii
ABSTRACT
Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value and Boundary Value Problems. (August 2010) Xiaoli Bai, B.S., Beijing University of Aeronautics and Astronautics;
M.S., Beijing University of Aeronautics and Astronautics Chair of Advisory Committee: Dr. John L. Junkins
The solution of initial value problems (IVPs) provides the evolution of dynamic system state history for given initial conditions. Solving boundary value problems (BVPs) requires finding the system behavior where elements of the states are defined at different times. This dissertation presents a unified framework that applies modi- fied Chebyshev-Picard iteration (MCPI) methods for solving both IVPs and BVPs. Existing methods for solving IVPs and BVPs have not been very successful in exploiting parallel computation architectures. One important reason is that most of the integration methods implemented on parallel machines are only modified ver- sions of forward integration approaches, which are typically poorly suited for parallel computation. The proposed MCPI methods are inherently parallel algorithms. Using Cheby- shev polynomials, it is straightforward to distribute the computation of force functions and polynomial coefficients to different processors. Combining Chebyshev polynomi- als with Picard iteration, MCPI methods iteratively refine estimates of the solutions until the iteration converges. The developed vector-matrix form makes MCPI meth- ods computationally efficient. The power of MCPI methods for solving IVPs is illustrated through a small per- turbation from the sinusoid motion problem and satellite motion propagation prob- lems. Compared with a Runge-Kutta 4-5 forward integration method implemented in iv
MATLAB, MCPI methods generate solutions with better accuracy as well as orders of magnitude speedups, prior to parallel implementation. Modifying the algorithm to do double integration for second order systems, and using orthogonal polynomi- als to approximate position states lead to additional speedups. Finally, introducing perturbation motions relative to a reference motion results in further speedups. The advantages of using MCPI methods to solve BVPs are demonstrated by addressing the classical Lambert’s problem and an optimal trajectory design problem.
MCPI methods generate solutions that satisfy both dynamic equation constraints and boundary conditions with high accuracy. Although the convergence of MCPI methods in solving BVPs is not guaranteed, using the proposed nonlinear transformations, linearization approach, or correction control methods enlarge the convergence domain.
Parallel realization of MCPI methods is implemented using a graphics card that provides a parallel computation architecture. The benefit from the parallel implemen- tation is demonstrated using several example problems. Larger speedups are achieved when either force functions become more complicated or higher order polynomials are used to approximate the solutions. v
Dedicated to my parents for their unconditional love and always b