A gradient-type algorithm for constrained optimization with application to microstructure optimization

• Autor(a) principal: Barbarosie, Cristian
• Data de Publicação: 2020
• Outros Autores: Toader, Anca-Maria, Lopes, S.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.21/11459
• topic: Nonlinear programming; Constrained minimization; Worst case design; Optimization of microstructures; Porous materials; Microstructure; Auxetic materials

We propose a method to optimize periodic microstructures for obtaining homogenized materials with negative Poisson ratio, using shape and/or topology variations in the model hole. The proposed approach employs worst case design in order to minimize the Poisson ratio of the (possibly anisotropic) homogenized elastic tensor in several prescribed directions. We use a minimization algorithm for inequality constraints based on an active set strategy and on a new algorithm for solving minimization problems with equality constraints, belonging to the class of null-space gradient methods. It uses first order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (minimizing the objective functional) and a correction step related to the Newton method (aiming to solve the equality constraints). The linear combination between these two steps involves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satisfied in the limit (after convergence). A local convergence result is proven for a general nonlinear setting, where both the objective functional and the constraints are not necessarily convex functions.