Optimal design for polymer extrusion. Part II: Sensitivity analysis for weakly-coupled nonlinear steady-state systems

The die design methodology presented in Part I is extended to include performance measures that are functions of material residence time. We solve a hyperbolic differential equation using the velocity field computed from the Hele-Shaw pressure analysis to evaluate the material residence time in polymer melts. The residence time governing equation lacks natural diffusion, therefore, we employ the streamline upwind Petrov-Galerkin (SUPG) finite element method to compute a spatially stable residence time field. In the design problem, we derive design sensitivities for steady-state nonlinear weakly-coupled systems and include design variables that parameterize essential boundary conditions. Design sensitivities are derived via the direct differentiation and adjoint methods. Special consideration is given to the SUPG weighting function since it is a function of the design. To demonstrate the methodology, sheet extrusion dies are designed to simultaneously minimize the exit velocity variation and the exit residence time variation.