## Section: New Results

### Shape and topological optimization methods

#### Topological optimal design problems for unsteady state equations

Participant : GrĂ©goire Allaire.

Homogenization is one of possible methods of topological optimization and its principle is as follows. Consider a structural optimization problem in which it is to find how to place two different materials (eg, a solid but heavy one and a light but fragile one). The homogenization method consists in not only optimize the interface between the two phases, but also to consider all "fine" blends of the two phases, that is to say all composite materials that can be built with. Therefore, the new optimization variable will be the local percentage of the two phases (also with the effective tensor that represents the composite underlying microstructure). This view is radically different from the classical point of view which is more geometric: we switch from a method of shape tracking to a method of shape capturing where the topology may change without any constraints. The success of this approach are immense for about twenty years now, but still there is a need for many generalizations or mathematical justifications.

For instance, in the article [8] we considered this type of shape optimization problem where the state equation is unsteady of parabolic type. We show in particular that the relaxation process by homogenization and the stationary limit for large time commute. In other words, the optimal composites in the parabolic case (which have a more complicated structure than in the stationary case) become simpler as time goes to infinity.

In the article [5] we consider another unsteady state equation of the wave type. An additional difficulty is the consideration of objective functions depending not only on the solution of the equation but also on its gradient. In this case the method of homogenization cannot be implemented due to lack of knowledge of all possible pairs of effective tensors and correctors for the gradients for composites obtained by homogenization. It is a difficult open problem under investigation of many researchers. We propose to solve it under an important simplification: we assume that both initial phases properties are very close (we then say that their contrast is small). It is then possible, based on the (small) contrast setting, to perform a Taylor expansion of the problem up to the second order. We can then relax the problem and in this case the homogenization method is replaced by the theory of H -measures (easier ultimately) introduced by P. Gerard and L. Tartar. In addition to obtaining rigorous relaxation results for this structural optimization problem "with small contrast", we propose a numerical method of topological type for evaluating the optimal shapes.

#### Post-treatment of the homogenization method

Participant : Olivier Pantz.

In most shape optimization problems, the optimal solution does not belong to the set of genuine shapes but is a composite structure. The homogenization method consists in relaxing the original problem thereby extending the set of admissible structures to composite shapes. From the numerical viewpoint, an important asset of the homogenization method with respect to traditional geometrical optimization is that the computed optimal shape is quite independent from the initial guess (even if only a partial relaxation is performed). Nevertheless, the optimal shape being a composite, a post-treatment is needed in order to produce an almost optimal non-composite (i.e. workable) shape. The classical approach consists in penalizing the intermediate densities of material, but the obtained result deeply depends on the underlying mesh used and the details level is not controllable. We proposed in a joint work with K. Trabelsi a new post-treatment method for the compliance minimization problem of an elastic structure. The main idea is to approximate the optimal composite shape with a locally periodic composite and to build a sequence of genuine shapes converging toward this composite structure. This method allows us to balance the level of details of the final shape and its optimality. Nevertheless, it was restricted to particular optimal shapes, depending on the topological structure of the lattice describing the arrangement of the holes of the composite. We lifted this restriction in order to extend our method to any optimal composite structure for the compliance minimization problem in previous works (see bibliography of 2009).Since that time, the method has been improved and a new article presenting the last results is in preparation. Moreover, we intend to extend this approach to other kinds of cost functions. A first attempt, based on a gradient method, has been made. Unfortunately, it was leading to local minima. Thus a new strategy has to be worked out. It will be mainly based on the same ideas than the one developed for the compliance minimization problem, but some difficulties are still to be overcome.

#### A new Liouville type Rigidity Theorem

Participant : Olivier Pantz.

We have recently developed a new Liouville type Rigidity Theorem. Considering a cylindrical shaped solid, we prove that if the local area of the cross sections is preserved together with the length of the fibers, then the deformation is a combination of a planar deformation and a rigid motion. The results currently obtained are limited to regular deformations and we are currently working with B. Merlet to extend them. Nevertheless, we mainly focus on the case where the conditions imposed to the local area of the cross sections and the length of the fibers are only "almost" fulfilled. This will enable us to derive rigorously new nonlinear shell models combining both membrane and flexural effects that we have obtained using a formal approach.