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

Supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project UID/MAT/04561/2019

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  • 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.

    Mathematics Subject Classification: Primary: 65K10, 49M15; Secondary: 90C52.


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  • Figure 1.  Periodicity cell with model hole (zoomed)

    Figure 2.  Periodically perforated plane $ {\mathbb{R}}^2 _{\hbox{ perf}} $

    Figure 3.  Optimized microstructures with respect to one direction only

    Figure 4.  Algorithm

    Figure 5.  Initial guess and final microstructure for square periodicity

    Figure 7.  Initial guess and final microstructure for hexagonal periodicity

    Figure 6.  History of convergence, zoom of the first 40 iterations and zoom of the last 6 iterations

    Figure 8.  History of convergence, zoom of the first 40 iterations, zoom of the last 8 iterations

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