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Timetable (DNMW04)

New trends and challenges in the mathematics of optimal design

Monday 10th June 2019 to Friday 14th June 2019

Monday 10th 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
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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Dorin Bucur
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.
12:30 to 13:30 Lunch at Murray Edwards College
14:30 to 15:30 Agnes Lamacz
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.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Beniamin Bogosel
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.
17:00 to 18:00 Welcome Wine Reception and posters at the INI
Tuesday 11th June 2019
10:00 to 11:00 Anca-Maria Toader
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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Benedikt Wirth
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 so-called branched transport and the so-called 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 cost-optimal 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.
12:30 to 13:30 Lunch at Murray Edwards College
13:30 to 14:30 Jeroen Peter Groen
Simple single-scale interpretations of optimal designs in the context of extremal stiffness
It is well-known that rank-N laminates can reach the theoretical bounds on strain energy in the context of linear elasticity. The theory of homogenization-based topology optimization using this class of composite materials is well-developed, and can therefore be used to find an overall optimal material distribution at low computational cost. A downside of these optimal multi-scale designs is that features exist at several length-scales 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 high-resolution 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 single-scale 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 single-scale, resulting in a performance that is close (e.g. 10-15%) 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 near-optimal performance in the high porosity limit. The performance difference between closed and open-walled microstructures is presented for anisotropic loading situations, where it is demonstrated that the maximum difference occurs when isotropic microstructures are considered.
14:30 to 15:30 Perle Geoffroy
Topology optimization of modulated and oriented periodic microstructures by the homogenization method in 2-d and in 3-d
The work presented here is motivated by the optimization of so-called 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 2-d 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 3-d case is more involved and requires new ingredients. In particular, the full rotation matrix is regularized (instead of just one angle in 2-d) and the projection map which deforms the periodic lattice is computed component by component.
Wednesday 12th June 2019
10:00 to 11:00 Jesus Martinez-Frutos
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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Olivier Pantz
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.

12:30 to 13:30 Lunch at Murray Edwards College
19:30 to 22:00 Formal Dinner at Westminster College


Westminster College
Madingley Rd, Cambridge CB3 0AA

Smart casual

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)  
Eton mess cheesecake, with strawberry coulis and a fruit of the forest compote

Thursday 13th June 2019
10:00 to 11:00 Martin Rumpf
Multi-Scale 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 finite-dimensional 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 two-scale 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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Samuel Amstutz
Gradient-free perimeter approximation for topology optimization and domain partitioning
I will present a Gamma-convergence 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.
12:30 to 13:30 Lunch at Murray Edwards College
13:30 to 14:30 Julian Panetta
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 consumer-level 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 worst-case load. Our coarse-scale design optimization can then automatically assign these metamaterials to an input geometry so that the printed object undergoes a user-specified 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.
14:30 to 15:30 Charles Dapogny
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.
Friday 14th June 2019
10:00 to 11:00 Antonin Chambolle
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)
11:00 to 11:30 Morning Coffee
11:30 to 12:30 H Alicia Kim
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 large-scale 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. 
12:30 to 13:30 Lunch at Murray Edwards College
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons