Path-following methods for shape optimal design of periodic microstructural materials

We are concerned with optimal shape design of composite materials with periodic microstructures. The homogenization approach is applied to obtain a computationally feasible macromodel. The microstructural geometrical details of the microcells (such as lengths and widths of the different layers forming the cell walls) are considered as design parameters. They have a tremendous impact on the macroscopic behaviour of the final produced composites. Our purpose is to find the best material-and-shape combination in order to achieve the optimal performance of the materials. The objective functional depends on the state variables describing the operational mode and the design parameters determining the shape. Our partial differential equation (PDE) constrained optimization routine is based on the elasticity problem as a state equation and additional equality and inequality constraints this are technically or physically motivated. The discretization of the PDE-constrained optimization problem typically gives rise to a large-scale nonlinear programming problem. Primal–dual Newton-type interior-point methods are used for the numerical solution. The inequality constraints are treated by parameterized logarithmic barrier functions. The algorithm relies on an application of the adaptive path-following predictor–corrector method, inexact Newton solvers, and the all-at-once optimization approach. Numerical experiments are presented and discussed.