Optimal control for polymer process modeling.

摘要

In this dissertation, we explore the application of optimization routines to systems of differential equations governing fluid flows which arise in polymer processing applications. We describe properties (called state variables) of a fluid as it flows through a domain using these systems of differential equations. Associated with these systems are initial and boundary conditions, as well as parameters that we control in order to obtain desired properties of the fluid in the flow domain. Some of these desired properties could be added strength (if the fluid is allowed to solidify), thickness, a specific temperature, reduced stagnation region, or to simply reduce waste.; There are many examples of optimization routines. The simplest optimization algorithm, the line search, is ideal for one control parameter. Examples of one-dimensional line searches are the Bisection algorithm and the Golden Section search. If multiple controls are desired, then sensitivity- and adjoint-based optimization algorithms should be considered. Adjoint-based optimization algorithms require only a single solution of a linear system regardless of the amount of control parameters, whereas sensitivity-based algorithms require one solution of a linear system for each control parameter.; The application area motivating this dissertation is polymeric fiber and film processing. Specifically, we consider three problems: the quench and draw-down phases of fiber melt spinning, flow between the die and chill roll for a cast film process, and flow inside a four-to-one contraction domain.; In the first application, we consider applying a line search to equations modeling fiber melt spinning that includes flow induced crystallization. We aim to optimize the orientation in the semi-crystalline phase of the polymer. Also considered is a simple two-parameter grid search, once again with the goal of optimizing orientation. Secondly, we use a sensitivity-based optimization algorithm to control three processing parameters, matching a desired film thickness. Finally, for flow inside a four-to-one contraction domain, we minimize the vortex that occurs in the corner by controlling the heat flux. The energy equation is coupled with the mass, momentum, and constitutive equations through the use of a temperature dependent Newtonian viscosity. Many authors assume a temperature dependent Newtonian viscosity when describing the model equations, but make the simplifying assumption of a constant Newtonian viscosity when carrying out computations; we assume no such simplification for the computations. Our analysis coupled with numerical solution of the problem with temperature-dependent viscosity distinguishes this work from earlier efforts.