G. Allaire
France
ON THE USE OF LAMINATE COMPOSITE MATERIALS
IN SHAPE OPTIMIZATION
Grégoire ALLAIRE
Laboratoire d'Analyse Numérique
Université Paris 6
4, place Jussieu
75252 Paris Cedex 05, France
The homogenization method for topology optimization in structural design is
by now well established (cf. the works of Bendsoe, Cherkaev, Kikuchi,
Kohn, Lurie, Murat, Tartar, as well as many others including the author).
The theory is fairly complete for compliance or eigenfrequency optimization
(in the single or multiple loadings case). Indeed, enlarging the class of
admissible designs by allowing for generalized porous designs (which are
precisely reiterated laminate composite materials) yields a full relaxation
of the original ill-posed shape optimization problem. It is also the basis
for efficient numerical algorithms that are frequently called topology
optimization methods. However, in the case of general objective functions,
this approach is not satisfying since the optimal microstructures are
unknown in full generality. Of course, in numerical practice,
many generalizations have appeared: they often rely on the use of
fictitious materials (so-called power-law materials) or sub-optimal materials
(for example, obtained by homogenization of a perforated periodic cell).
Working with a subclass of microstructures is called a partial relaxation
of the problem. This subclass needs to be rich enough in order to
approximate as much as possible the true optimal microstructures, which
yields good numercial properties (fast convergence, global minima). On the
other hand it must be as explicit as possible for a good efficiency.
We describe such a procedure for the class
of sequential laminate composites which are delivered
by an explicit formula and are optimal in a number of important
cases. We describe the numerical implementation of this method of
partial relaxation and discuss its application on several examples.
