New trends and challenges in the mathematics of optimal design
Monday 10th June 2019 to Friday 14th June 2019
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 3d printed materials
We address the grand question of identifying the set of possible elasticity tensors (including anisotropic ones) of 3dprinted 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 3delasticity tensors live in an 18dimensional 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 CamarEddine, 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 NeumannLaplacian 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 nontrivial one is maximized by the ball, the result being due to Szego
and Weinberger in the fifties. Concerning the second nontrivial 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 DuisburgEssen) Effective Maxwell's equations in a geometry with flat splitrings 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 splitrings 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 cellproblem. 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 splitring 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 levelset 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 
AncaMaria Toader (Universidade de Lisboa); (Universidade de Lisboa) Optimization of bodies with locally periodic microstructure by varying the shape, the topology and the periodicity pattern
Mimicking nature, an optimization method that makes the
link between microstructure and macrostructure is considered. Homogenization
theory is used to describe the macroscopic (effective) elastic properties of
the body.
The already known alternate optimization of shape and
topology of the model cell is a procedure that gives a limited flexibility to
the microstructure for adapting to the macroscopic loads.
Beyond that, one may vary the periodicity cell itself
during the optimization process, thus allowing the microstructure to adapt more
freely to the given loads.
What we propose is a method that combines the three
optimization techniques :
the shape, the topology and the periodicity pattern.
By combining variations of these three ingredients, the
obtained optimal design approaches the homogenized structure of the body,
giving one the possibility to obtain a manufacturable design with smooth
transition of material properties as in functionally graded materials.
Numerical examples will be presented.
The problem is numerically heavy, since the optimization
of the macroscopic problem is performed by optimizing in simultaneous hundreds
or even thousands of periodic structures, each one using its own finite element
mesh on the periodicity cell. Parallel computation is used in order to
alleviate the computational burden.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Benedikt Wirth (Westfalische WilhelmsUniversitat Munster) Variational models for transportation networks: old and new formulations
A small number of models for transportation networks
(modelling street, river, or vessel networks, for instance) has been studied
intensely during the past decade, in particular the socalled branched
transport and the socalled urban planning. They assign to each network the
total cost for transporting material from a given initial to a prescribed final
distribution and seek the costoptimal network. Typically, the considered
transportation cost per mass is smaller the more mass is transported together,
which leads to highly patterned and ramified optimal networks. I will present
novel formulations of these models which allow a better interpretation as an
optimal design problem.

INI 1  
12:30 to 13:30  Lunch at Murray Edwards College  
13:30 to 14:30 
Jeroen Peter Groen (Technical University of Denmark) Simple singlescale interpretations of optimal designs in the context of extremal stiffness
It is wellknown that rankN laminates can reach the
theoretical bounds on strain energy in the context of linear elasticity. The
theory of homogenizationbased topology optimization using this class of
composite materials is welldeveloped, and can therefore be used to find an
overall optimal material distribution at low computational cost. A downside of
these optimal multiscale designs is that features exist at several
lengthscales limiting the manufacturability. The main contribution of the
presented work is to develop and extend on new methods, to interpret these
designs on a single scale, while still being close to what is theoretically
possible.
Using these methods highresolution near optimal designs
can be achieved on a standard PC at low computational cost. Several
modifications are given, such as a method to locally adapt microstructure
spacing and a method to interpret the singlescale designs as a frame structure.
Furthermore, simple microstructures are presented that
are optimized for multiple anisotropic loading conditions. This is done by
approximating optimal microstructures on a singlescale, resulting in a
performance that is close (e.g. 1015%) to the theoretical bounds. When used as
starting guess for topology optimization these proposed microstructures can be
further improved, outperforming topology optimized designs using classical
starting guesses both in performance and simplicity.
Finally, a class of simple periodic truss lattice
structures is presented that exhibits nearoptimal performance in the high
porosity limit. The performance difference between closed and openwalled
microstructures is presented for anisotropic loading situations, where it is
demonstrated that the maximum difference occurs when isotropic microstructures
are considered.

INI 1  
14:30 to 15:30 
Perle Geoffroy (École Polytechnique) Topology optimization of modulated and oriented periodic microstructures by the homogenization method in 2d and in 3d
The work presented here is motivated by the optimization of socalled lattice materials which are becoming increasingly popular in the context of additive manufacturing. We propose a method for topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented in the working domain. This method is made of three steps. The first step amounts to compute the homogenized properties of an adequately chosen parametrized microstructure (here, a cubic lattice with varying bar thicknesses). The second step optimizes the homogenized formulation of the problem, which is a classical problem of parametric optimization. The third, and most delicate, step projects the optimal oriented microstructure at a desired length scale. In 2d case, rotations are parametrized by a single angle, to which a conformality constraint can be applied. A conformal diffeomorphism is then computed from the orientation field, thanks which each periodic cell is well oriented in the final structure. The 3d case is more involved and requires new ingredients. In particular, the full rotation matrix is regularized (instead of just one angle in 2d) and the projection map which deforms the periodic lattice is computed component by component. 
INI 1 
10:00 to 11:00 
Jesus MartinezFrutos (Universidad Politécnica de Cartagena) Levelset topology optimization for robust design of structures under internal porosity constraints
Porosity is a wellknown 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 threedimensional 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 posttreatment 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 microstructure. 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 GinzburgLandau 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 GL 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

10:00 to 11:00 
Martin Rumpf (University of Bonn); (Universität Bonn) MultiScale and Risc Averse Stochastic Shape Optimization
This talk discusses the optimization for elastic materials and elastic microstructures under different and in particular stochastic loading scenarios. To this end, on the one hand we transfers concepts from finitedimensional stochastic programming to elastic shape optimization. Thereby, the paradigm of stochastic dominance allows for flexible risk aversion via comparison with benchmark random variables, Rather than handling risk aversion in the objective, this enables risk aversion by including dominance constraints that single out subsets of nonanticipative shapes which compare favorably to a chosen stochastic benchmark. On the other hand, we investigate multiscale shape optimization using mechanically simple, parametrized microscopic supporting structure those parameters have to be optimized. An posteriori analysis of the discretization error and the modeling error is investigated for a compliance cost functional in the context of the optimization of composite elastic materials and a twoscale linearized elasticity model. This error analysis includes a control of the modeling error caused when replacing an optimal nested laminate microstructure by this considerably simpler microstructure. Furthermore, an elastic shape optimization problem with simultaneous and competitive optimization of domain and complement is discussed. Such a problem arises in biomechanics where a bioresorbable polymer scaffold is implanted in place of lost bone tissue and in a regeneration phase new bone tissue grows in the scaffold complement via osteogenesis. In fact, the polymer scaffold should be mechanically stable to bear loading in the early stage regeneration phase and at the same time the new bone tissue grown in the complement of this scaffold should as well bear the loading. The talk is based on joint work with Sergio Conti, Patrick Dondl, Benedikt Geihe, Harald Held, Rüdiger Schultz, Stefan Simon, and Sascha Tölkes. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Samuel Amstutz (University of Avignon) Gradientfree perimeter approximation for topology optimization and domain partitioning
I will present a Gammaconvergence approximation of the perimeter of a set built upon the solution of an elliptic PDE. I will discuss the advantages and drawbacks of this approach compared with other functionals, at first to address topology optimization problems with perimeter control. I will emphasize the specific mathematical properties and algorithmic issues, showing in particular how the variational formulation of the PDE can be exploited to design alternating minimizations schemes. Then I will explain how those results and methods, through combinatorial and duality techniques, can be adapted to multiphase optimal partitioning problems with an energy term consisting of a weighted sum of measures of interfaces. Problems of hydrostatics with surface tensions will be shown as examples.

INI 1  
12:30 to 13:30  Lunch at Murray Edwards College  
13:30 to 14:30 
Julian Panetta (EPFL  Ecole Polytechnique Fédérale de Lausanne) Computational Design of Robust Elastic Metamaterials and Deployable Structures
My talk will present some computational design tools targeting various classes of structures and fabrication technologies. In the first half, I will present a method for designing elastic metamaterials that can be fabricated with consumerlevel single material 3D printers to achieve custom deformation behaviors. These metamaterials cover a wide range of elastic properties and are optimized for robustness in generic use, experiencing minimal stresses under the worstcase load. Our coarsescale design optimization can then automatically assign these metamaterials to an input geometry so that the printed object undergoes a userspecified deformation under applied loads. In the second half, I will introduce a new class of deployable elastic gridshell structures. These structures consist of flat, conveniently assembled layouts of elastic beams coupled by rotational joints that can be deployed to programmed 3D curved shapes by a simple expansive actuation. During deployment, the coupling imposed by the joints forces the beams to twist and buckle out of plane, allowing interesting 3D forms to emerge. However the simulation and optimization of these structures is challenging, especially due to the frequent unstable equilibria encountered in the deployment path; I will discuss the efficient algorithms we have developed to assist the design of these structures. This talk is based on joint work with Denis Zorin, Mark Pauly, and Florin Isvoranu.

INI 1  
14:30 to 15:30 
Charles Dapogny (Université de Grenoble) About new constraints induced by additive manufacturing technologies on the shape optimization process
However they allow, in principle, to assemble arbitrarily complex structures  thereby arousing much enthusiasm within the engineering community  modern additive manufacturing technologies (also referred to as 3d printing) raise new difficulties which have to be taken into account from the early stages of the construction, and notably at the level of the design optimization. In this presentation, we shall deal with the modeling and the understanding of two such major challenges related to additive construction methodologies. The first one of these is to avoid the emergence of overhanging regions during the shape optimization process, that is, of large, nearly horizontal regions hanging over void, without sufficient support from the lower structure. The second difficulty addressed in this presentation is related to the fact that the use of an additive technique to realize a structure entails a significant alteration of the mechanical performance of the constituent material of the assembled shape: this material turns out to be inhomogeneous, and it presents anisotropic properties, possibly depending on the global shape itself. These works have be conducted together with Grégoire Allaire, Rafael Estevez, Alexis Faure and Georgios Michailidis. 
INI 1 
10:00 to 11:00 
Antonin Chambolle (CNRS (Centre national de la recherche scientifique)); (École Polytechnique) Remarks on the discretizations of the perimeter
I will discuss some results on finite differences and finite element approximations of the total variation for possibly discontinuous functions. In particular the talk will focus on the differences between various types of approximations, both qualitatively and quantitatively. This is based on joint works with Thomas Pock (TU Graz) and Corentin Caillaud (CMAP, Ecole Polytechnique & CNRS, Palaiseau) 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
H Alicia Kim (University of California, San Diego) Optimization for Multiscale Material Design
Topology
optimization is able to provide unintuitive and innovative design solutions and
a performance improvement (e.g. weight savings) in excess of 50% is not
uncommonly demonstrated in a wide range of engineering design problems. With
the rise of advance materials and additive manufacturing, topology optimization
is attracting much attention in the recent years. This presentation will
introduce topology optimization in structural design, fiber composites and
architected material. It will also include more recent advances topology
optimization, multiscale design optimization breaking down the barrier between
material and structural designs. Another direction of interests in largescale
topology optimization using the latest sparse data structures tailored to novel
level set method. We have demonstrated an order of magnitude improvements on
both the memory footage and the computation time. These efforts represent a
pathway to applying topology optimization for complex multiphysics
multifunctional structures, which may be too complex to rely on designers’
intuition.

INI 1  
12:30 to 13:30  Lunch at Murray Edwards College 