## Section: New Results

### Shape and topological optimization methods

#### A two phase optimal design problem for the wave equation

Participant : Grégoire Allaire.

With Alex Kelly, presently a post-doc at CMAP, we are studying a two phase optimal design problem where the state equation is a wave equation. As usual this type of problem is ill-posed, namely it does not admit a solution. Establishing its relaxed formulation is a difficult task, so we simplify the problem by making an assumption on the small amplitude of the contrast. We then perform a second-order asymptotic expansion of the original problem with respect to this small aspect ratio. It is still not a well-posed problem but its relaxation is much more simple, using the notion of H -measures, which is easier to manipulate than homogenization theory. This yields a satisfying existence theory as well as an efficient numerical method for computing the optimal designs. We are currently writing a paper on the topic.

#### 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. We intend to extend this approach to the minimization of other cost functions and are currently working on the multiload case.

#### Numerical simulation of damage evolution

Participant : Grégoire Allaire.

With F. Jouve et N. Van Goethem we worked on the numerical implementation of the Francfort-Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith energy release rate. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation, including kinking and branching. Several numerical examples in 2d and 3d are discussed in [27] and a full article will soon appear.