ON THE OPTIMIZATION OF DIFFERENTIAL-ALGEBRAIC SYSTEMS OF EQUATIONS IN CHEMICAL ENGINEERING (ORTHOGONAL COLLOCATION, FINITE ELEMENTS, DISCRETIZATION).

摘要

This thesis develops a method for optimizing problems that have differential and algebraic equation models. The approach used constructs polynomial approximations of the continuous profiles and uses orthogonal collocation on finite elements to discretize the differential equations. The resulting set of algebraic equations and unknown polynomial coefficients are then included in a Nonlinear Program which solves the approximation and optimization problems simultaneously. Lagrange polynomial basis functions are used so that the coefficients represent physically meaningful quantities such as temperature or concentration. This allows bounds on the continuous profiles to be enforced by writing bounds on the polynomial coefficients.; The topics of approximation error and optimization error are also addressed. In order to ensure that the error of the approximation is minimized, an additional set of equations, involving the approximations and the locations of the finite elements knots, are developed. These are then included into the Nonlinear Program as equality constraints, and when solved, position the knots so as to minimize the approximation error. In order to address the accuracy of the optimization, conditions are established under which solutions of the Nonlinear Programming approach converge to the solution of the original problem. In addition, an extra level of elements, super-elements, is introduced so that problems which contain profile discontinuities can be solved.; The main contributions of this research include the following. A Nonlinear Programming method which accurately solves differential-algebraic optimization problems that contain steep profiles and profile discontinuities has been successfully implemented on a number of example problems. Approximation accuracy has been guaranteed by developing a set of knot placement equations which can be included into the Nonlinear Program. Conditions have been established under which solutions of the Nonlinear Program converge to the solution of the original problem. Also, an equivalence between the method of orthogonal collocation on finite elements and a fully implicit Runge-Kutta numerical integration scheme which uses Gaussian roots has been illustrated. Lastly, an extra level of approximation, which uses super-elements, has been introduced so that problems with discontinuous profiles can be solved.