09:30 to 09:50 Registration 09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director) 10:00 to 11:00 Graeme Milton (University of Utah)Optimizing the elastic response of 3-d printed materials We address the grand question of identifying the set of possible elasticity tensors (including anisotropic ones) of 3d-printed materials constructed from a given elastic material with known elastic constants. We identify many almost optimal geometries with elasticity tensors arbitrarily near the boundary of what one can achieve. We characterize many parts of the surface of the set of possible elasticity tensors. This is no easy task as completely anisotropic 3d-elasticity tensors live in an 18-dimensional space of invariants, much more than the two invariants (bulk and shear moduli) that characterize isotropic elasticity tensors. We completely characterize the set of possible (average stress, average strain) pairs that can exist in these porous materials. Unfortunately, the geometries we find are rather extreme but this should motivate the search for more realistic ones that come close to having the desired elasticity tensors. Also, not all parts of the surface are characterized for elastically isotropic composites. Further progress will require new ideas. This is joint work with Marc Briane, Mohamed Camar-Eddine, and Davit Harutyunyan. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Dorin Bucur (Université de Savoie)Spectral shape optimization problems with Neumann conditions on the free boundary In this talk I will discuss the question of the maximization of the $k$-th eigenvalue of the Neumann-Laplacian under a volume constraint. After an introduction to the topic I will discuss the existence of optimal geometries. For now, there is no a general existence result, but one can prove existence of an optimal {\it (over) relaxed domain}, view as a density function. In the second part of the talk,  I will focus on the low eigenvalues. The first non-trivial one is maximized by the ball, the result being due to Szego and Weinberger in the fifties. Concerning the second non-trivial eigenvalue, Girouard, Nadirashvili and  Polterovich proved that the supremum in the family of planar simply connected domains of $R^2$ is attained by the union of two disjoint, equal discs. I will show that a similar statement holds in any dimension and without topological restrictions. INI 1 12:30 to 13:30 Lunch at Murray Edwards College 14:30 to 15:30 Agnes Lamacz (Universität Duisburg-Essen)Effective Maxwell's equations in a geometry with flat split-rings and wires Propagation of light in heterogeneous media is a complex subject of research. Key research areas are photonic crystals, negative index metamaterials, perfect imaging, and cloaking.   The mathematical analysis of negative index materials, which we want to focus on in this talk, is connected to a study of singular limits in Maxwell's equations. We present a result on homogenization of the time harmonic Maxwell's equations in a complex geometry. The homogenization process is performed in the case that  many (order $\eta^{-3}$) small (order $\eta^1$), flat (order $\eta^2$) and highly conductive (order $\eta^{-3}$) metallic split-rings are distributed in a domain $\Omega\subset \mathbb{R}^3$. We determine the effective behavior of this metamaterial in the limit $\eta\searrow 0$. For $\eta>0$, each single conductor occupies a simply connected domain, but the conductor closes to a ring in the limit $\eta\searrow 0$. This change of topology allows for an extra dimension in the solution space of the corresponding cell-problem. Even though both original materials (metal and void) have the same positive magnetic permeability $\mu_0>0$, we show that the effective Maxwell system exhibits, depending on the frequency, a negative magnetic response. Furthermore, we demonstrate that combining the split-ring array with thin, highly conducting wires can effectively provide a negative index metamaterial. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 Beniamin Bogosel (École Polytechnique)Optimization of support structures in additive manufacturing Support structures are often necessary in additive manufacturing in order to ensure the quality of the final built part. These additional structures are removed at the end of the fabrication process, therefore their size should be reduced to a minimum in order to reduce the material consumption and impression time, while still preserving their requested properties.   The optimization of support structures is formulated as a shape and topology optimization problem. Support structures need to hold all overhanging parts in order to assure their manufacturability, they should be as rigid as possible in order to prevent the deformations of the structure part/support and they should not contain overhanging parts themselves. In processes where melting metal powder is involved, high temperature gradients are present and support structures need to prevent eventual deformations which are a consequence of these thermal stresses.   We show how to enforce the support of overhanging parts and to maximize the rigidity of the supports using linearized elasticity systems. In a second step we show how a functional depending on the gradient of the signed distance function allows us to efficiently prevent overhang regions in the support structures. The optimization is done by computing the corresponding shape derivatives with the Hadamard method. In order to simulate the build process we also consider models in which multiple layers of the part and of the support are taken into account.   The models presented are illustrated with numerical simulations in dimension two and three. The goal is to obtain algorithms which are computationally cheap, while still being physically relevant. The numerical framework used is the level-set method and the numerical results are obtained with the freeware software FreeFem++ and other freely available software like Advect and Mshdist from the ISCD Toolbox.This work was done in the project SOFIA in collaboration with Grégoire Allaire. INI 1 17:00 to 18:00 Welcome Wine Reception and posters at the INI
 10:00 to 11:00 Jesus Martinez-Frutos (Universidad Politécnica de Cartagena)Level-set topology optimization for robust design of structures under internal porosity constraints Porosity is a well-known phenomenon occurring during various manufacturing processes (casting, welding, additive manufacturing) of solid structures, which undermines their reliability and mechanical performance. The main purpose of this talk is to introduce a new constraint functional of the domain which controls the negative impact of porosity on elastic structures in the framework of shape and topology optimization. The main ingredient of our modeling is the notion of topological derivative, which is used in a slightly unusual way: instead of being an indicator of where to nucleate holes in the course of the optimization process, it is a component of a new constraint functional which assesses the influence of pores on the mechanical performance of structures. The shape derivative of this constraint is calculated and incorporated into a level set based shape optimization algorithm. This approach will be illustrated by several two- and three-dimensional numerical experiments of topology optimization problems constrained by a control on the porosity effect. These works have been conducted together with Grégoire Allaire, Charles Dapogny and Francisco Periago. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Olivier Pantz (Université de Nice Sophia Antipolis)Singular lattices, regularization and dehomogenization method The deshomogenization method consists in reconstructing a minimization sequence of genuine shapes converging toward the optimal composite. We introduced 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 as a post-treatment of the classical homogenization method. The output of the (periodic) homogenization method is : - An orientation field of the periodic cells - Geometric parameters describing the local micro-structure. From this output, the deshomogenization method allows to construct a sequence of genuine shapes, converging toward the optimal, (almost) suitable for 3D printers. The sequence of shapes is defined via a so called "grid map", which aim is to ensure the correct alignment of the cells with respect to the orientation. field. It also enforce the connectivity of the structure between neighboring cells. If the orientation field is regular and the optimization domain $D$ is simply connect, the grid map can be defined as local diffeomorphism from $D$ into $R^n$ (with n=2 or 3). If those requirements are not met, the definition of the grid map is much more intricate. Moreover, a minimal kind of regularity is needed to be able to ensure the convergence of the sequence of shapes toward the optimal composite : it is necessary to regularize the orientation field but still allow for the presence of singularities. This is done by a penalization of the cost function based on the Ginzburg-Landau theory. In this talk, we will present 1/ A general definition of the grid map based on the introdcution of an abstract manifold. 2/ A regularization of the orientation field based on G-L theory. 3/ Numerical applications in 2D and 3D. This talk is based on a joint work by G. Allaire, P. Geoffroy and K. Trabelsi. INI 1 12:30 to 13:30 Lunch at Murray Edwards College 19:30 to 22:00 Formal Dinner at Westminster College   LOCATION Westminster CollegeMadingley Rd, Cambridge CB3 0AADRESS CODESmart casualMENU   Starters Slow-cooked guinea fowl leg, Iberian ham and a red currant salad   Quinoa, rocket, pomegranate and heritage tomato salad with a lemon and Balsamic dressing (Vegan)  Main course Salmon with samphire and mussels, crushed new potatoes and a warm butter sauce   Roasted vegetable stack, sautéed Jerusalem artichokes, butternut squash purée, split lentils, salsa verde and a Brussel sprout and sun-dried tomato salad (Vegan)  PuddingEton mess cheesecake, with strawberry coulis and a fruit of the forest compote