Un método de optimización de tipo bundle no diferenciable aplicado al flujo de materiales viscoplásticos.

Abstract

In the present work, a nonsmooth optimization method applied to the Herschel- Bulkley flow model is developed. The motion of these fluids is characterized as a minimization problem of a convex and nonsmooth functional. To solve it, we follow the “discretize, then optimize” approach. We use the finite element method to obtain an approximation of the problem of interest, then, by means of the Moreau-Yosida regularization, we obtain a smooth optimization problem that is equivalent to minimizing the nonsmooth functional. Applying this type of regularization is a challenging task, since, it is necessary to solve an auxiliary optimization problem at each iteration. Following the ideas of Bundle methods, we develop a strategy that allows to solve this auxiliary problem, approximately. We build a method that mainly follows the BFGS update scheme, and furthermore, solves the auxiliary problem at each iteration as a subroutine. It is shown that the proposed algorithm converges globally to the minimum, in addition, it possesses a superlinear convergence rate for the case of shear-thickening fluids. Finally, several numerical experiments are presented to verify the efficiency of the algorithm and its convergence properties.