Shape optimization of periodic structures

Abstract

This paper describes a numerical approach to the optimization of effective properties of periodic perforations in an infinite body, in the frameworks of heat conduction and of linear elasticity. We implement a special finite element mesh in order to deal with the periodic nature of the problem. We compute the gradient of the functional to be minimized. We describe the process of mesh deformation and mesh regeneration. We give several numerical examples, some of them having practical relevance.

Optimization of bodies with locally periodic microstructure by varying the periodicity pattern

“…Cantilever with three independent loads, optimized structure very rare to see more than one hole in the periodicity cell. 3 The only exception is shown in Figure 14, zoom b, where a very small hole can be seen. If the optimization process were continued, it is likely that this hole would have been destroyed by subsequent shape optimization steps.…”

“…The periodicity conditions in (15) are implemented by identifying the opposite sides of Y and by keeping track of the linear part A of w A . For more details, see [7], [6] and [3].…”

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

“…Consider θ : R 2 → R 2 a vector field defining the deformation; note that θ itself should be periodic in order to preserve the periodic character of the microstructure under study. Then the variation induced by this deformation in the quantity C H A, B is (see [7,Section 6] and [6])…”

“…[33,4]. It has been described in [6] without proof of convergence; its local convergence has been proven in [9].…”

“…where γ is a scalar function computed as a linear combination of the integrands in (6). If a steepest descent method were used, one should choose a deformation of the interface ∂T equal to θ = −γ n (multiplied by some positive constant η which controls the speed of the process) in order to decrease the value of J.…”