We intr oduced this method a few years ago. Since\, it has gain some interest -- see the works of JP. Groen and O. Sigmund -- thanks to the rise of additive manufacturing. Bascillay\, it can be considered a s a post-treatment of the classical homogenization method.

The output of the (periodic) homogen ization method is :

- An orientation field of the periodic cells

- Geometric parameters des cribing the local micro-structure.

From this o utput\, the deshomogenization method allows to con struct a sequence of genuine shapes\, converging t oward the optimal\, (almost) suitable for 3D prin ters.

The sequence of shapes is defined v ia a so called "grid map"\, which aim is to ensure the correct alignment of the cells with respect t o the orientation. field.

It also enforce the connectivity of the structure between neighboring cells. If the orientation field is regular and th e optimization domain $D$ is simply connect\, the grid map can be defined as local diffeomorphism fr om $D$ into $R^n$ (with n=2 or 3). If those requir ements are not met\, the definition of the grid ma p is much more intricate.

Moreover\, a mi nimal kind of regularity is needed to be able to e nsure the convergence of the sequence of shapes to ward the optimal composite : it is necessary to re gularize the orientation field but still allow for the presence of singularities. This is done by a penalization of the cost function based on the Gin zburg-Landau theory.

In this talk\, we wi ll present

1/ A general definition of the grid map based on the introdcution of an abstract mani fold.

2/ A regularization of the orientation f ield based on G-L theory.

3/ Numerical applica tions in 2D and 3D.

This talk is based on a joint work by G. Allaire\, P. Geoffroy and K. T rabelsi.

LOCATION:Seminar Room 1\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR