Shape and topology optimization for periodic problems

Abstract

In the present paper we deduce formulae for the shape and topological derivatives for elliptic problems in unbounded domains subject to periodicity conditions. Note that the known formulae of shape and topological derivatives for elliptic problems in bounded domains do not apply to the periodic framework. We consider a general notion of periodicity, allowing for an arbitrary parallelepiped as periodicity cell. Our cal- culations are useful for optimizing periodic composite materials by gradient type methods using the topo- logical derivative jointly with the shape derivative for periodic problems. Important particular cases of func- tionals to minimize/maximize are presented. A numer- ical algorithm for optimizing periodic composites using the topological and shape derivatives is the subject of a second paper (Barbarosie and Toader, Struct Multidis- cipl Optim, 2009).