Quench process modeling and optimization

We optimize continuous quench process parameters to produce a desired precipitate distribution in aluminum alloy extrudates. To perform this task, an optimization problem is defined and solved using a standard nonlinear programming algorithm. Ingredients of this algorithm include a cost function, constraint functions and their sensitivities with respect to the process parameters. These functions are dependent on the temperature and precipitate size which are obtained by balancing energy to determine the temperature distribution and by using a reaction-rate theory to determine a discrete precipitate particle size distribution. Both the temperature and the precipitate models are solved via the finite element method. Since we use a discrete particle size model, there are as many as 10⁵ degrees-of-freedom per finite element node. After we compute the temperature and precipitate size distributions, we must also compute their sensitivities. This seemingly intractable computational task is resolved by using an element-by-element discontinuous Galerkin finite element formulation and a direct differentiation sensitivity analysis which allows us to perform all of the computations on a PC.