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A

Matérialiser un roman par l'écriture est une corvée de tâcheron, aussi sombre et fastidieuse au bout de quatre cent pages qu'il était un trouble divin et une ivresse de créateur d'en imaginer, souvent durant des années, les péripéties gratuitement -Blaise Cendrars, L'homme foudroyé is thesis wouldn't have been possible without my supervisor Adrien Bousseau. ank you for giving me the opportunity of doing all of this work, I got to spend four years learning a lot and working on simulation and fabrication and it was all because you chose to bet on me in the rst place. I am also grateful for my co-advisor Mélina Skouras, even though we were not working in the same cities, you taught me so much about simulation, optimization and inverse design, especially during the few visits I got to do in Grenoble.

All the work in this thesis would not have been achieved without Etienne Vouga and Victor Romero who collaborated with us on the di erent research projects. Etienne, you are like a third supervisor for me, I still remember the day you decided to hop on the printing-on-fabric train with us, it was on the beginning of my PhD and you stuck with us ever since. Victor, it was fun doing experiments with you, I hope we will keep on doing cool research about interesting physical phenomena! anks to Stavros, Gilles, Felix, Siddhant, Yorgos, Julien, Johanna, Valentin, Nicolas and all the other (past or present) members of the Graphdeco team I got to meet for making it such a nice environment to work in, I hope I will see you all very soon. One member of the team I want to thank in particular is Emilie Yu, not only you are an integral part of that nice Graphdeco atmosphere I just mentioned, but for me, during this PhD you were so much more. First of all, you helped us in uncountable ways for the two papers we submi ed, taking nice pictures, making cool rendering or animations for illustrative purposes, you taught me how to use Blender, and you even set up the whole photogrammetry system for scanning our fabricated objects! On a more personal level, I have to say that doing a PhD is far from easy, especially during a pandemic, and I want to thank you for your emotional support during times when it was ge ing particularly tough on me, you are a wonderful life partner. I am thankful for my family, and in particular for my parents who helped me nancially pursuing my studies, and encouraged me to study something that I like even though it makes no sense to them. ank you to the members of my thesis commi ee, Maud Marchal, Paolo Cignoni, Bernd Bickel and Olivier Baverel for your insightful comments and very interesting questions, I wish I will get to meet you all in esh soon. Finally, I want to thank the people and organizations who made all this work easier: Alexandra Elbakyan for making it possible to access many of the papers cited therea er which I wouldn't have been able to read otherwise; the creators of open-source libraries that I used during my PhD, libigl [START_REF] Jacobson | libigl: A simple C++ geometry processing library[END_REF] and polyscope [START_REF] Sharp | Polyscope[END_REF]; and the ERC Starting Grant D3 (ERC-2016-STG 714221) which funded most of my PhD.

R´

´ Nous décrivons une technique de fabrication récente perme ant de créer des textiles auto-deployables qui consiste à extruder du plastique sur un tissu étiré au préalable en utilisant des procédés de fabrication additive. Ce e technique d'impression sur tissu a récemment gagné en popularité car c'est une technique de fabrication peu coûteuse perme ant de créer des structures déployables qui prennent automatiquement une forme en 3D lorsque le tissu est relaché.

Le comportement en déformation de ce matériau composite peut s'expliquer par une combination d'e ets géometriques intrinsèques et extrinsèques, le plastique déposé étant à la fois la cause d'un changement de métrique et d'un e et bilame par la combinaison qu'il forme avec la couche sous-jacente de tissu. Nous exploitons ces observations pour créer des métamatériaux qui agissent sur ces deux e ets : la métrique du tissu est modulée par la densité des motifs imprimés, et la exion de la surface est contrôlée par l'épaisseur de plastique imprimé grâce à l'e et bilame. Advances in computer numerically controlled (CNC) machinery, jointly with the development of computational tools that permit the design and processing of freeform geometry allowed to reach an unprecedented level of complexity and customizability for the design and fabrication of objects. However this complexity of form comes with complexity in the manufacturing process: subtractive technologies such as CNC milling generate a lot of waste and are limited in the shapes they can make, while additive manufacturing needs a lot of support material to deal with overhangs. Now that freeform geometry is obtainable, we should strive for e cient ways to manufacture complex geometry. In that context, deployable structures o er an interesting alternative, as they can be manufactured in simple con gurations and then bent or otherwise deformed into shape. Besides the easier fabrication process, at-deployable structures have a compact form-factor which gives them bene ts in the most mundane storage and transport situations as well as the most exotic ones such as space exploration. To manufacture an object at and then deform it to match a speci c shape requires a precise understanding of both the physics of the material in question and the geometry of the deformation, which will be detailed in the following chapters.

Among deployables, self-shaping structures represent an interesting case. ey do not require any kind of actuation force to be deployed but instead morph as a reaction to an energy input which can be heat, light, moisture, a magnetic eld or the energy stored in a pre-stretched substrate. Since their morphing is driven by a physical process (as opposed to manual deformation), they can be actuated more easily and in a more consistent manner. is mechanism can also prove very useful in situations where it is not possible to actuate the shape by pushing or pulling points on the surface, either because the context makes it di cult or because moving individual points on the surface will not achieve the desired result.

ere are many di erent ways of programming curvature into 2D sheets so that they self-actuate into a 3D shape. I investigated a technique to create self-shaping textiles which consists in depositing pa erns of molten plastic into a pre-stretched fabric layer.

e plastic bonds with the fabric layer and cures into a rigid material which will prevent homogeneous contraction of the fabric when it is released and favor buckling into 3D shapes instead (see Fig. 1.1). Compared to other materials that can be used to program self-morphing sheets such as liquid crystal elastomers (LCE) [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF] or swelling gels [START_REF] Kim | Designing responsive buckled surfaces by hal one gel lithography[END_REF], self-shaping textiles have the advantage of being comparatively inexpensive and easy to fabricate. e only equipment needed is a stretchy type of fabric (e.g. spandex) and hobbyist 3D printing hardware to deposit the plastic, which makes this technique accessible to non-experts and opens the door to a wide range of creative practices and shape experimentations. In particular, this allowed architecture and design teachers to organize workshops in fab labs and similar types of facilities where students were able to quickly prototype self-actuated shapes [START_REF] Agkathidis | 4D printing on textiles: Developing a le to fabrication framework for self-forming, composite wearables[END_REF][START_REF] Erioli | informed exible ma er workshop[END_REF][START_REF] Kycia | Self-shaping textiles: Potential of 3D printing for the design of performative textile structures[END_REF].

Contributions

e work presented in this thesis takes both a geometric and design point-of-view to deal with the computational design and fabrication of self-shaping textiles using the aforementioned printing-on-fabric technique.

Geometry. From a di erential geometric standpoint, the deployment of self-shaping textiles can be analyzed through either an intrinsic or extrinsic perspective. In the intrinsic viewpoint, we are interested in how local quantities such as geodesic lengths and angles evolve on the surface. ese lengths and angles can be measured by an object called the metric or the rst fundamental form of the surface 𝐼 𝑝 (𝑢, 𝑣) = 𝑑 𝑓 𝑝 (𝑢), 𝑑 𝑓 𝑝 (𝑣) , which is the scalar product of the pushforward of two vectors 𝑢 and 𝑣 onto the tangent plane of the surface at a point 𝑝. e rst fundamental form is a very powerful object, even though it gives no information about how a surface is embedded in 3D space, it uniquely identi es the Gaussian curvature 𝐾 at every point.

However, in the general case, knowing the rst fundamental form (or the Gaussian curvature) of a surface is not su cient to uniquely determine it because there might be several possible surfaces that are equivalent up to an isometry (see Figure 1.2). is is why the extrinsic perspective is important as well: it can be compelling to only think of deployable materials in terms of how they stretch or expand and how this translates to curvature, but ultimately intrinsic notions give no information as to how a given surface is supposed to be embedded, or in our case, how it is supposed to deploy in 3D space. Extrinsic quantities, -such as the normal curvature 𝜅 𝑁 of a curve, the mean curvature of a surface 𝐻 , or its second fundamental form 𝐼𝐼 𝑝 (𝑢, 𝑣) = -𝑑𝑁 𝑝 (𝑢), 𝑑 𝑓 𝑝 (𝑣) which gives information about how normals around 𝑝 evolve through the di erential 𝑑𝑁 𝑝 -help disambiguate scenarios where a surface with a given metric can have multiple embeddings.

In this thesis, we leverage both intrinsic and extrinsic e ects. By carefully controlling the density of printed plastic pa erns, we can modulate the ability of the fabric to retract back to its original size, and therefore control the underlying metric of the deployed surface.

is allows to program Gaussian, or intrinsic, curvature into the deployable textile. To bias the possible solutions to a speci c embedding, we control the normal curvature of the printed curves by leveraging a bilayer e ect: since the fabric below the printed curves is pre-stretched and the plastic material prevent contraction, the plastic-fabric bilayer will bend with a curvature depending on the thickness of printed plastic. is control on the extrinsic curvature guarantees a theoretical uniqueness of the deployed surface, as there can exist multiple surfaces with the same Gaussian curvature but there is only one surface (up to a global rotation and translation) with the same Gaussian and mean curvature [33, p.239].

Computational design.

is thesis is about developing computational tools for the design of self-shaping textiles. We envision two possible work ows: a direct one and an inverse one. In a direct work ow, the user designs a speci c pa ern and then prints it on stretched fabric. If the deployed shape does not have the expected appearance, the user then has to start all over again. In that work ow, it can be useful to be able to perform form-nding, i.e. to computationally predict what the deployed shape will look like before printing it. is allows for faster and less wasteful iterations loops, because the user does not have to wait for the nal printed object to decide whether they want to discard it or not (Fig. 1.3, le ). e inverse work ow is a bit di erent. Instead of specifying a pa ern to be printed and discovering what kind of shape emerges from that pa ern, the inverse process consists in rst specifying a target shape and then try to nd out which kinds of pa erns can be suitable to obtain such a shape. For example, we can imagine that designers Guberan and Clopath [START_REF] Guberan | Active shoes[END_REF] proceeded in some kind of inverse work ow when they designed their self-shaping shoe: they probably knew beforehand that they wanted to create a shoe, and then they gured out which curve layout to print so that the deployed shape looks like a shoe in the end. In that case, a computational tool which nds an optimal curve layout to be printed for a given shape can be of great help because it seemingly nulli es Fig. 1.2: Two shapes with the same Gaussian curvature can be bent di erently, illustration with a developable surface (𝐾 = 0) and a sphere with positive gaussian curvature (𝐾 > 0) which can be bumped inwards in a similar way to a de ated football.

Pattern speci cation Form-nding Fabrication

Pattern editing

Shape speci cation

Inverse solving Fabrication possible shape editing the need for an iteration loop, even though in a real-life scenario the designer might not be completely satis ed with the result and may want to modify the input shape so as to generate di erent printing pa erns (Fig. 1.3, right).

ese two work ows correspond to di erent design cases because their operate from di erent starting points in the design process. In the direct work ow, a user starts from a pa ern to be printed and then iterates on that pa ern until a satisfying result is found, this may be useful in early stages of the design process when the user tries to explore the possible shapes they can obtain, or for artists who wish to enforce speci c aesthetics to the printed curve pa ern. In the inverse work ow the user starts from the shape to be reproduced, and obtains a pa ern to be printed as a result of an optimization process, the user can then modify the input if the resulting pa ern from the optimization is not satisfying.

In this thesis we address both design work ows, and towards this goal we tackle two main questions. First, is it possible to accurately predict the behavior of these printed-onfabric composite materials? e answer is not obvious because textiles can be extremely diverse in terms of material response, and the process itself is imprecise with unknown variables such as plastic adhesion with the fabric, or over-or under-extrusions changing the result. To simplify this problem, we rst study a particular pa ern composed of tilings of stars which is simple enough so that reduced order simulation models work well to reproduce it, expressive enough so that it is possible to create varied shapes. We then try to predict the nal shape of more general, arbitrary pa erns printed on stretched fabric, by modeling the precise anisotropic behavior of the pre-stretched fabric on one hand, and the bilayer nature of the plastic-fabric composite material on the other. e second question is can we inverse the design process and nd a speci c pa ern that, once printed on stretched fabric, will morph and reproduce a given target shape? Inverse problems are in general ill-posed and this one is no exception: multiple printed pa erns could result to the same shape, while other geometries might not be reproducible at all by self-shaping textiles. Instead of optimizing for any kind of printed pa ern, we focus on parallel ribbon pa erns which allow to control both intrinsic curvature through their widths and spacing, and extrinsic curvature because of the bilayer e ect.

Outline

e rest of the thesis will be as follows:

• Chapter 2 details the work which has inspired and is most related to this thesis.

• Chapter 3 presents a (direct) design tool for fabricating lightweight architectural models based on a tiling of star pa erns, with the dimensions (and hence physical properties) of the individual pa ern elements varying over space. Users of this system design free-form shapes by adjusting the star pa ern; our system then automatically simulates the complex physical coupling between the fabric and stars to translate the design edits into shape variations.

• Chapter 4 describes preliminary work towards extending the form-nding tool to a general-purpose simulation method that can predict the buckling behavior of a variety of printed-on-fabric designs besides star tilings. e full anisotropic, largestrain response of the fabric is reproduced and integrated in a shell simulator which models the bilayer e ect caused by the di erential compression at the interface between the plastic and fabric layers.

• Chapter 5 presents an inverse design tool that is able to reproduce target shapes with printing-on-fabric using a dense pa ern of closely-spaced ribbons. e core of this method is a parameterization algorithm that bounds surface distortions along and across principal curvature directions, along with a pa ern synthesis algorithm that covers a surface with ribbons to match the target distortions and curvature given by the aforementioned parameterization.

Publications

Chapters 3 and 5 are based on the two following peer-reviewed publications: e study of self-shaping textiles is an interdisciplinary subject which requires knowledge in theoretical areas such as di erential geometry, continuum mechanics, and more practical ones like digital fabrication and additive manufacturing. As such, the literature which inspired this thesis covers a wide range of domains such as computer graphics, architectural geometry, human-computer interactions (HCI), and so ma er physics. We organize this chapter in two distinct sections:

• Section 2.1 deals with the computational design of deformable structures and materials, approached from the point of view of computer graphics and architecture, elds which focus on e cient tools and representations. In this body of work we distinguish studies on the forward design of deformable shapes and the related problem of form-nding (2.1.1) from inverse design tools. While form-nding deals with the task of nding the rest shape of a given structure -which we perform in chapters 3 and 4 -inverse design tools aim at programming curvature into the structure of a deformable material, in order to obtain a speci c geometry -which we tackle in chapter 5. In particular we review di erent optimization devices which make this feat possible.

• Section 2.2 focuses on deformable materials which do not need any external actuation to be deformed into shape, but rather deform themselves. e self-actuation can be driven by di erent environmental stimuli such as heat, moisture, or in our case, the residual stress of a stretched substrate, we explain key mechanisms used in many of these projects such as the bilayer e ect. ese self-actuated structures have been studied in the so ma er physics community which is interested in understanding complex behaviors at the intersection between chemistry, physics and geometry. is understanding leads to accurate fabrication methods in controlled environments ; meanwhile computer graphics and HCI communities have devel-oped more practical applications. In particular, we nish by giving an overview of the research in self-shaping textiles.

2.1 Computational design of deformable materials is thesis is a contribution towards the goal of designing deformable structures that can be fabricated at and deformed into a complex, 3D structure. e need to manufacture 3D surfaces out of at sheets of material has motivated signi cant research in various areas such as computer graphics, architectural geometry, and materials science. is section does not a empt to provide an exhaustive analysis of the work from these di erent elds, but rather highlights some of the similarities between their approaches and how they relate to the work that we have done. e computational design of deformable materials in computer graphics is historically linked to the problem of modeling and representing shape. Advances in the geometric understanding of deformations lead to be er representations and algorithms for modeling shapes in the computer as well as methods to fabricate them with a given material. We see a prime example of that link with developable surfaces, which are surfaces with zero Gaussian curvature 𝐾 = 𝜅 1 𝜅 2 -where 𝜅 1 and 𝜅 2 are the maximum and minimum normal curvature. Gauss's theorema egregium [40, p.27-28] shows that such surfaces are locally isometric to the plane. As such, developable surfaces are an appropriate representation for materials that can bend but cannot shear or stretch, like paper. Approaches for modeling developable surfaces can thus represent them as smooth geometric primitives such as splines [START_REF] Tang | Interactive design of developable surfaces[END_REF], or use algorithms based on discrete di erential geometry such as the discrete orthogonal geodesic (DOG) nets of Rabinovich et al. [START_REF] Rabinovich | Discrete geodesic nets for modeling developable surfaces[END_REF].

Forward design and form-nding

𝜅 2 = 0 𝜅 1 > 0
Other types of surface representations allow the modeling of di erent material constraints, for example Chebyshev nets represent wire-like materials which have the ability Fig. 2.2: Pillwein et al. [START_REF] Pillwein | On elastic geodesic grids and their planar to spatial deployment[END_REF] to shear but not to stretch [START_REF] Garg | Wire mesh design[END_REF][START_REF] Andrew | Chebyshev nets from commuting polyvector elds[END_REF] and can be used to model elastic gridshells [START_REF] Baek | Form nding in elastic gridshells[END_REF] which are structures made of bent slender beams a ached together to form a grid. A commonly-found feature of gridshells is that when they are made out of beams whose cross-section is anisotropic, the rod mechanics force them to be geodesics on the surface. is particular geometric property can be exploited to drive e cient algorithms for their design [START_REF] Pillwein | On elastic geodesic grids and their planar to spatial deployment[END_REF]. Once designed, deploying such structures can also be particularly challenging. A common approach is to assemble the grid at and then li it while xing the beam endpoints to a prescribed boundary as was done for the ephemeral cathedral of Créteil [START_REF] Du Peloux | Faith can also move composite gridshells[END_REF] or the Solidays gridshell [START_REF] Baverel | Gridshells in composite materials: Construction of a 300 m 2 forum for the Solidays' festival in Paris[END_REF]. Otherwise, sparse actuation of the structure can be obtained by using a one degreeof-freedom mechanism with scissor-like hinges [START_REF] Pane | X-shells: A new class of deployable beam structures[END_REF][START_REF] Pillwein | On elastic geodesic grids and their planar to spatial deployment[END_REF][START_REF] Soriano | G-shells: Flat collapsible geodesic mechanisms for gridshells[END_REF]. As their name implies, these surfaces have the property of minimizing their area under given boundary constraints.

𝜅 2 = -𝜅 1 𝜅 1 > 0
e numerical computation of such geometries has been a topic of interest for at least three decades [START_REF] Pinkall | Computing discrete minimal surfaces and their conjugates[END_REF] and has, until lately, remained challenging to solve for complex topologies and non-manifold con gurations [START_REF] Ishida | A hyperbolic geometric ow for evolving lms and foams[END_REF][START_REF] Wang | Computing minimal surfaces with di erential forms[END_REF]. But their interest is not only of a mathematical nature, they are useful to model mechanical systems in which tension determines the nal geometry, such as soap lms whose geometry is e ectively that of a minimal surface, or architectural surfaces such as tensile structures and cable nets [START_REF] Wanda | Tension Structures: Form and behaviour[END_REF] whose geometry is dictated by their internal stress. e architect Frei O o famously experimented with soap lms to prototype his tensile structures using a process called form-nding, which consists in experimenting with di erent boundary constraints to see a surface emerge out of the physical constraints [START_REF] Simon | Frei o o: Spanning the future[END_REF]. In general, form-nding designates the process of nding the equilibrium shape of a given structure under a speci c set of forces, it can be performed computationally to explore the shape space of surfaces in equilibrium under internal stress (like tensile surfaces), self-weight (like catenaries and other funicular shapes) [START_REF] Ma | Constructing 3D self-supporting surfaces with isotropic stress using 4D minimal hypersurfaces of revolution[END_REF][START_REF] Tellier | Formnding with isotropic linear weingarten surfaces[END_REF] and bending-active shapes constrained by their endpoints [START_REF] La | Bending-active plates: Form and structure[END_REF].

Computational form-nding tools o en try to solve a problem governed by the laws of physics. A popular approach called Dynamic Relaxation which is implemented in form-nding so ware such as Kangaroo Physics [START_REF] Miki | e geodesic dynamic relaxation method for problems of equilibrium with equality constraint conditions[END_REF][START_REF] Piker | Kangaroo physics[END_REF] consists in integrating the equations of motion with an added damping factor to avoid parasite oscillations, the resulting motion is guaranteed to converge to a static equilibrium even if the motion itself is not physically realistic. In chapters 3 and 4, we employ a di erent strategy for form-nding self-shaping textiles. Instead of integrating the equations of motions we nd the minimum of the potential energy of the system

x = argmin𝑊 (x) ⇐⇒ ∇𝑊 (x) = 0 ∧ ∇ 2 𝑊 (x) 0 (2.1)
is problem is known as the variational approach in nite element analysis, we solve it using a modi ed version of Newton's method where the energy 𝑊 (x) is used as a merit function [START_REF] Nocedal | Numerical Optimization[END_REF]. For a more thorough overview of form-nding techniques we point the reader towards the work of Veenendaal and Block [START_REF] Veenendaal | An overview and comparison of structural form nding methods for general networks[END_REF].

In their particular case of self-shaping textiles (see inset), Pérez et al. [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF] tackle the form-nding problem by identifying that their designs live in a speci c shape space, that of Kirchho -Plateau surfaces -area-minimizing surfaces which are enclosed by exible rods. Notable designs combining tensioned membranes and exible rods, called Bending-Active Tensile Hybrid (BATH) structures [START_REF] Slabbinck | Conceptual framework for analyzing and designing bending-active tensile hybrid structures[END_REF], include the architectural work of Ahlquist and Menges [START_REF] Ahlquist | Frameworks for computational design of textile micro-architectures and material behavior in forming complex force-active structures[END_REF] and Deleuran et al. [START_REF] Holden Deleuran | e tower: Modelling, analysis and construction of bending active tensile membrane hybrid structures[END_REF]. In our case however, (chapters 3 and 5) the tensioned substrate is not completely enclosed by exible rods. erefore, the shapes we are able to create belong to a wider shape space.

Finally, it is relevant to mention that some structures have the ability to encode and control more general types of deformations. ey are usually made out of repetitive, tileable elements that can contract or expand in a speci c manner and thus indue intrinsic curvature by modulating the metric of the surface. Such structures are dubbed metamaterials as their deformation capabilities can be beyond those of the base material they are built with, and they can be de ned by the properties of the individual element that is being tiled.

For example, Konaković et al. [START_REF] Konaković | Beyond developable: Computational design and fabrication with auxetic materials[END_REF] consider triangular auxetic linkages where the opening angle between adjacent triangles dictates the resulting surface curvature, Chen et al. [START_REF] Chen | Bistable auxetic surface structures[END_REF] augment this mechanism by using bistable tiles which can either be completely open or completely close, thus guaranteeing stability of the deployed shape. Malomo et al. [START_REF] Malomo | Flexmaps: Computational design of at exible shells for shaping 3D objects[END_REF] and Laccone et al. [START_REF] Laccone | Flexmaps pavilion: A twisted arc made of mesostructured at exible panels[END_REF] consider repetitive spiral-shaped elastic rods where the amount of twist of the spiral controls surface bending (see inset gure). e tileable pa erns we develop in chapters 3 and 5 can be considered to form a metamaterial as well.

In some cases, it is possible to control how the material will deform by carefully picking the right tileable elements that compose the metamaterial, and thus to create programmable materials which have their deformation behavior encoded within their structure, the process of designing such materials is called inverse design.

Inverse design

In the following, we distinguish between two kinds of inverse design methods. Geometrybased methods, solve the problem by studying its geometry, and use it to drive e cient algorithms. However, some problems may not have a nice mathematical structure which allows for simpli cations. Physics-based methods approach the problem by using optimization machinery which poses static equilibrium as a hard constraint. Deformable materials which are made out of rigid parts, or that are constrained in their deformation are o en well-suited for geometric methods. Flexible materials can also be programmed using a geometric approach (see e.g. [START_REF] Hafner | e design space of plane elastic curves[END_REF]), and some rigid mechanisms can be suitable for physics-based optimization approaches (see e.g. [START_REF] Geilinger | Skaterbots: Optimization-based design and motion synthesis for robotic creatures with legs and wheels[END_REF]). It is also possible to combine a geometric approach with physics-based optimization [START_REF] Pane | Computational inverse design of surface-based in atables[END_REF][START_REF] Ren | 3D weaving with curved ribbons[END_REF].

Physics-based

e physics-based optimization approach to inverse design consists in solving the following problem:

min x,p 𝐸 (x) subject to 𝑓 (x, p) = 0
where x corresponds to geometry variables, p are design variables (e.g. the Young's modulus in elastic parameter optimization [START_REF] Yan | Inexact descent methods for elastic parameter optimization[END_REF]), 𝐸 (x) is the energy to optimize for (usually the distance to a target geometry possibly with added regularizers) and 𝑓 (x, p) is the sum of the forces, which typically depends on both the geometry and the design parameters.

e constraint is usually nonlinear, so this problem is non trivial to solve and requires iterative methods with good initializations to avoid di cult local minima. As an example, Skouras et al. [START_REF] Skouras | Computational design of rubber balloons[END_REF][START_REF] Skouras | Computational design of actuated deformable characters[END_REF] solve this problem with the Augmented Lagrangian Method (ALM) which iteratively updates lagrangian multipliers without having to introduce them in the linear systems, thus avoiding a signi cant increase in the number of variables compared to the traditional method of Lagrange multipliers (see [START_REF] Nocedal | Numerical Optimization[END_REF]Chapter 17] for more details). Skouras et al. [START_REF] Skouras | Designing in atable structures[END_REF] use Sequential adratic Programming (SQP) to nd the optimal layout of at panels such that, when assembled, they form a balloon resembling the input shape.

An alternative to solving this problem using nonlinear constrained optimization tools is sensitivity analysis. Sensitivity analysis reformulates the constraint 𝑓 (x, p) = 0 into an implicit function x (p) which locally approximates the manifold de ned by the constraint. Given variables x 0 , p 0 such that 𝑓 (x 0 , p 0 ) = 0, we can write

𝑓 (x(p), p) -----→ p→p 0 𝑓 (x 0 , p 0 ) + 𝜕𝑓 𝜕x 𝜕x 𝜕p + 𝜕𝑓 𝜕p Δp = 0 ⇐⇒ 𝜕x 𝜕p = - 𝜕𝑓 𝜕x -1
𝜕𝑓 𝜕p e derivative 𝑆 = 𝜕x 𝜕p is called the sensitivity matrix, and can be used to optimize the geometry x as a function of the design variables p. Sensitivity analysis is o en used in conjunction with the adjoint method which consists in avoiding a costly matrix inversion by exploiting the associativity of matrix multiplication when computing the gradient of the energy:

𝜕𝐸 𝜕p = 𝜕𝐸 𝜕x - 𝜕𝑓 𝜕x -1 𝜕𝑓 𝜕p = -𝜕𝐸 𝜕x 𝜕𝑓 𝜕x -1 𝜕𝑓
𝜕p which amounts to simply solving a linear system. Sensitivity analysis has been used for the design of objects made of exible rods [START_REF] Malomo | Flexmaps: Computational design of at exible shells for shaping 3D objects[END_REF][START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF][START_REF] Pérez | Design and fabrication of exible rod meshes[END_REF], silicone inclusions [START_REF] Zehnder | Metasilicone: Design and fabrication of composite silicone with desired mechanical properties[END_REF], polystyrene sculptures made with hot-wire cu ing [START_REF] Duenser | Robocut: Hot-wire cu ing with robot-controlled exible rods[END_REF], objects weaved with curved ribbons [START_REF] Ren | 3D weaving with curved ribbons[END_REF] and in atable structures [START_REF] Pane | Computational inverse design of surface-based in atables[END_REF]. Like other physics-based optimization techniques, its main drawback is that it requires simulating the entire object at every iteration which may be computationally expensive and prevent interactivity in the design process. In many cases, exploiting the geometry of the problem allows to derive more e cient algorithms.

Geometry-based

Because developable materials are only allowed to bend, they are o en well-suited for geometric approaches. Tachi [START_REF] Tachi | Origamizing polyhedral surfaces[END_REF]'s Origamizer is a method to compute a pa ern of folds on a sheet of paper so that when it is folded, the paper will resemble the input polyhedral surface. e idea of the method is that the excess of material can be tucked inside the folds when it is needed to create curvature, a process that has been used empirically before by Ron Resch [START_REF] Resch | Paper and stick lm[END_REF]. e algorithm was proven to work on any polyhedral surface [START_REF] Demaine | Origamizer: A practical algorithm for folding any polyhedron[END_REF]. In contrast to origami where the practitioner is only allowed to fold the material, kirigami [START_REF] Jiang | Freeform quad-based kirigami[END_REF] allows for cuts, which enables less wasteful approaches for creating architectural geometries. A di erent family of approaches start from a smooth surface (or a discretization of it) and optimize for some metric of developability such as the presence of hinges on vertex stars [START_REF] Stein | Developability of triangle meshes[END_REF], the shape operator (i.e. the Hessian in isotropic geometry) being rankde cient [START_REF] Sellán | Developability of height elds via rank minimization[END_REF], or the Gauss image (the map of normals to the unit sphere) being onedimensional [START_REF] Binninger | Developable approximation via Gauss image thinning[END_REF]. Some methods are explicitly made to guarantee fabricability, such as the work of Ion et al. [START_REF] Ion | Shape approximation by developable wrapping[END_REF] based on the discrete orthogonal geodesic (DOG) framework. e method of Sharp and Crane [START_REF] Sharp | Variational surface cu ing[END_REF] optimizes the shape of cuts in arbitrary surfaces such that they can be made from planar patches with the least distortion possible, as showcased by Louis-Rosenberg and Rosenkrantz [START_REF] Louis | Puzzle cell complex[END_REF].

Shapes made out of assemblies of curves are also well-suited for geometric approaches, the case of bending-active structures that I mentioned in the previous section is a good example. Pillwein and Musialski [START_REF] Pillwein | Generalized deployable elastic geodesic grids[END_REF] identify that curves in a gridshell correspond to geodesics, and devise an e cient way to compute them while optimizing for the orientation of the structure. Hafner and Bickel [START_REF] Hafner | e design space of plane elastic curves[END_REF] devise a geometric criterion for all elastic ribbons in equilibrium and use it to optimize the cross-section of such ribbons to match a given curve input. A speci c case relates to gridshells where the elements are weaved together. In this case they are not only geodesics but form a 𝑛-rotationally symmetric (𝑛-RoSy) directional eld. Vekhter et al. [START_REF] Vekhter | Weaving geodesic foliations[END_REF] identify geodesic foliations to be the proper mathematical framework for dealing with weaved straight ribbons, and develop an e cient algorithm to compute such foliations. projected pa ern onto the target surface [START_REF] Konaković-Luković | Rapid deployment of curved surfaces via programmable auxetics[END_REF].

Closest to our inverse design applications (Chapter 5) are methods based on a parameterization approach. Konaković-Luković et al. [START_REF] Konaković-Luković | Rapid deployment of curved surfaces via programmable auxetics[END_REF] leverage the fact that the morphing of their metamaterial from the at plane to a 3D surface preserve angles to use a conformal mapping tool to optimize for the size of elements in their triangular auxetic linkage so that the structure automatically takes the prescribed shape upon actuation by gravity or pneumatic deployment. Chen et al. [START_REF] Chen | Bistable auxetic surface structures[END_REF] also use a conformal map to compute an isotropic scale factor for each individual cell of their metamaterial and match it to a precomputed cell shape which has the desired bistable behavior. e Geodesy+ tool [START_REF] Gu | Inverse design tool for asymmetrical self-rising surfaces with color texture[END_REF] computes an anisotropic parameterization, because the contraction mechanism they exploit favors one direction (along the extrusion path) over the other, this parameterization is then exploited to create self-rising shapes.

Self-shaping materials

Self-shaping materials have the property of not needing any external actuation force to be deformed, instead their shape changes as a result of a variety of stimuli such as temperature, moisture content, internal air pressure or tensile stress. In this section I give an overview of some of the mechanisms that make self-shaping materials possible.

Geometric description

In this section I make the distinction between two kinds of mechanisms to create selfshaping surfaces.

e rst ones act on the lengths of the surface to create Gaussian curvature and are called intrinsic mechanisms because they do not depend on how the surface is embedded in space. e second kind act on extrinsic notions of curvature such as bending by way of bilayer e ects.

A general mechanism for self-actuation that was identi ed by Sharon and Efrati [START_REF] Sharon | e mechanics of non-euclidean plates[END_REF] is that of metric change meaning that some materials can locally expand or contract upon a speci c trigger mechanism. If this contraction or expansion is not uniform across the whole geometry, it will trigger a change in curvature. erefore, by changing the metric (also known as the rst fundamental form) which is the bilinear operator de ning distances on a surface, we can change the curvature of the surface. I previously mentioned Gauss' theorema egregium which states that the Gaussian curvature of a surface does not depend on how the surface is bent in space, which is why we consider that Gaussian curvature is an intrinsic property of surfaces. However, the link between the metric and Gaussian curvature may be more apparent if we introduce a corollary known as the Bertrand-Diguet-Puiseux theorem [15, 131, p.145] which states that Gaussian curvature can be expressed as the limit di erence between circumferences, (or similarly, areas) of a circle in the plane and a (geodesic) circle of the same radius on the surface:

𝐾 = lim 𝑟 →0 3 2𝜋𝑟 -𝐶 (𝑟 ) 𝜋𝑟 3 = 12 𝜋𝑟 2 -𝐴(𝑟 ) 𝜋𝑟 4 , ( 2.2)
this clearly shows how Gaussian curvature is dependent on how the lengths on a surface di er from the lengths on the plane. If the circumference of the geodesic circle is bigger than the circumference of a at circle, the Gaussian curvature will be negative, on the contrary if the circumference of the geodesic circle is smaller, then then Gaussian curvature will be negative. Even though this formula is only true for in nitesimal disks on a surface, we can still see how the same principles can be applied on nite areas to create self-shaping surfaces. For example, Kim et al. [START_REF] Kim | Designing responsive buckled surfaces by hal one gel lithography[END_REF] combine two materials with di erent swelling rates using a hal oning pa ern, a different density of dots meaning a di erent swelling rate (see inset). Since the circumference is mostly covered by red dots which have a low swelling factor while the interior contains a black material with a high swelling factor, the area of the disk will increase faster than its circumference, thus creating positive Gaussian curvature. Some morphing materials [START_REF] Sydney Gladman | Biomimetic 4D printing[END_REF][START_REF] Gu | Geodesy: Self-rising 2.5D tiles by printing along 2D geodesic closed path[END_REF] leverage a different mechanism where the material has been deposited along speci c pathways and shrinks at di erent rates along and across the printing path, which is particularly useful to create this di erence between geodesic radius and circumference.

In particular the Geodesy project [START_REF] Gu | Geodesy: Self-rising 2.5D tiles by printing along 2D geodesic closed path[END_REF][START_REF] Gu | Inverse design tool for asymmetrical self-rising surfaces with color texture[END_REF] focuses on shells printed along spiral pathways whose circumference shrinks faster than their radius, creating positive Gaussian curvature (see inset). A similar mechanism is at play on individual printed stars in Chapter 3, the star branches constrain the circle radius to stay constant while the circumference shrinks. e di erence of shrinking between di erent directions is an example of an anisotropic morphing behavior that can be used e ectively to create self-shaping surfaces [START_REF] Griniasty | Curved geometries from planar director elds: Solving the two-dimensional inverse problem[END_REF]. In particular, this mechanism is at play with nematic elastomers [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF] where the orientation of liquid crystals in uences the swelling behavior and can be programmed by using a parameterization approach similar to ARAP [START_REF] Liu | A local/global approach to mesh parameterization[END_REF]. We take inspiration from nematic elastomers and the general problem of anisotropic morphing to build our own ARAP-like parameterization algorithm for self-shaping textiles in Chapter 5.

An extrinsic mechanism that we exploit in chapters 3 and 5 is called the bilayer e ect. It is the result of binding two layers, with at least one being an active layer which is either expanding or shrinking, the mismatching lengths between the two layers cause the initially planar assembly to bend out (Figure 2.8). is mechanism has been studied at least since Timoshenko [START_REF] Timoshenko | Analysis of bi-metal thermostats[END_REF] who predicted the curvature of a bilayer ribbon made out of two metals with di erent thermal expansion coe cients, we show how their formulation is equivalent to the one we computed for plastic-fabric composites in Chapter 3.

Several papers exploit this mechanism to create self-actuated rods [START_REF] Bodaghi | 4D printing self-morphing structures[END_REF][START_REF] Rüggeberg | Bio-inspired wooden actuators for large scale applications[END_REF][START_REF] Wang | A-line: 4D printing morphing linear composite structures[END_REF], either by binding two layers of di erent materials with di erent properties, or by binding layers of the same material a di erent orientations, this works if the material in question has an anisotropic morphing behavior. Bilayer, self-shaping rods can be combined together to create more complex structures, such as self-shaping meshes [START_REF] Wang | 4DMesh: 4D printing morphing non-developable mesh surfaces[END_REF] (see inset), or in our case, self-shaping rods a ached to a pre-stretched substrate (chapters 3 and 5)

e bilayer e ect can also be used to create self-shaping surfaces that bend along prescribed directions. is can be used to engineer folds along sharp kinks so as to create selffolding versions of the origami and kirigami techniques introduced in section 2.1.2 [START_REF] Santangelo | Extreme mechanics: Self-folding origami[END_REF], or self-shaping periodic metamaterials based on a foldable la ice structure [START_REF] Teunis Van Manen | Kirigamienabled self-folding origami[END_REF][START_REF] Van Manen | eoretical sti ness limits of 4D printed self-folding metamaterials[END_REF].

Passive layer Active layer

Fig. 2.8: Bilayer e ect: the combination of a passive layer resisting deformation and an active layer which is either expanding or shrinking, causes the whole assembly to curve so that one layer becomes smaller than the other one. e curvature of the resulting equilibrium con guration can be computed as a function of the thicknesses of the layers, their Young's moduli, and the amount of expansion or shrinkage the active layer is going through.

More complex bending behavior can also be achieved with the bilayer e ect so as to obtain a wide variety of developable surfaces [START_REF] Byoungkwon An | ermorph: Democratizing 4D printing of self-folding materials and interfaces[END_REF][START_REF] Teunis Van Manen | Programming 2D/3D shape-shi ing with hobbyist 3D printers[END_REF]. is mechanism can be used in conjunction with metric change: if both layers are active, then the metric of the whole assembly is going to change [START_REF] Wim | Mechanics of biomimetic 4D printed structures[END_REF]. In fact, apart from origami techniques, metric change is necessary to create double curvature, as the bilayer e ect itself can only bend the surface and thus cannot change the Gaussian curvature.

Equations for general bilayer shells have been formulated by Pezzulla et al. [START_REF] Ma Eo Pezzulla | Curvaturedriven morphing of non-euclidean shells[END_REF] and van Rees et al. [START_REF] Wim | Growth pa erns for shapeshi ing elastic bilayers[END_REF]. Compared to standard nite element methods where a strain is computed between the current state and a stress-free rest state, it is more complex to simulate self-shaping bilayer shells because there may not exist a shape in which the shell is completely stress-free. Instead it is possible to compute stress-free rst and second fundamental forms separately even if there might not exist a surface with such rst and second fundamental forms, we use the formulas of van Rees et al. [START_REF] Wim | Growth pa erns for shapeshi ing elastic bilayers[END_REF] in Chapter 3 to derive the curvature of a printed-on-fabric bilayer ribbon.

Materials

Self-shaping surfaces can be created using di erent materials and trigger mechanisms, in the following section I review some of them and classify them by their trigger mechanism. Some materials morph when receiving a certain quantity of energy, while some others were already storing energy inside and the trigger merely permits the release of the stored energy, which is the case of self-shaping textiles that I explore in this thesis.

A commonly heard term when talking about self-shaping materials is 4D printing [START_REF] Tibbits | 4D printing: Multi-material shape change[END_REF]. Unfortunately, this expression has no precise or commonly agreed on de nition. In this thesis, the term 4D printing will be used only when referring to self-shaping objects that have been created using additive manufacturing technologies such as fused lament fabrication (FFF), this excludes techniques employing a pre-stretched substrate.

Many 4D printing methods rely on heat to trigger the morphing [START_REF] Byoungkwon An | ermorph: Democratizing 4D printing of self-folding materials and interfaces[END_REF][START_REF] Bodaghi | 4D printing self-morphing structures[END_REF][START_REF] Gu | Geodesy: Self-rising 2.5D tiles by printing along 2D geodesic closed path[END_REF][START_REF] Gu | Inverse design tool for asymmetrical self-rising surfaces with color texture[END_REF][START_REF] Teunis Van Manen | Programming 2D/3D shape-shi ing with hobbyist 3D printers[END_REF][START_REF] Wang | A-line: 4D printing morphing linear composite structures[END_REF][START_REF] Wang | 4DMesh: 4D printing morphing non-developable mesh surfaces[END_REF]. ey exploit a warping e ect that happens when the thermoplastic material that has been used to print the object is heated above its glass transition temperature (usually between 60-80 • C). When a thermoplastic material is deposited using the common FFF technique, the polymer chains are initially aligned along the printing path. Heating the material allows the chains to rearrange along random directions, making the material denser and more compact.

is phenomenon makes the material shorten along the printing path, which is useful to leverage the bilayer e ect to make self-shaping rods [START_REF] Bodaghi | 4D printing self-morphing structures[END_REF][START_REF] Wang | A-line: 4D printing morphing linear composite structures[END_REF][START_REF] Wang | 4DMesh: 4D printing morphing non-developable mesh surfaces[END_REF] and shells [START_REF] Byoungkwon An | ermorph: Democratizing 4D printing of self-folding materials and interfaces[END_REF][START_REF] Teunis Van Manen | Programming 2D/3D shape-shi ing with hobbyist 3D printers[END_REF], it can also be used to trigger more general metric changes [START_REF] Gu | Geodesy: Self-rising 2.5D tiles by printing along 2D geodesic closed path[END_REF][START_REF] Gu | Inverse design tool for asymmetrical self-rising surfaces with color texture[END_REF]. An innovative method combines 4D printing and conductive materials to create tangible user interfaces [START_REF] Cheng | Printed paper actuator: A low-cost reversible actuation and sensing method for shape changing interfaces[END_REF], by controlling current through the material to cause it to heat and morph.

Other types of materials can morph in reaction to a change of temperature, such as the classic case of metals studied by Timoshenko [START_REF] Timoshenko | Analysis of bi-metal thermostats[END_REF]. Boley et al. [START_REF] Boley | Shape-shi ing structured la ices via multimaterial 4D printing[END_REF] combined four di erent elastomeric inks with various expansion coe cents to create la ices which morph out of the plane when heated, and then cure at a high temperature and can stay in place when cooling down to room temperature. is property is particularly desirable as some materials do morph when heated but go back to their initial shape upon cooling down, which means the programmed curvature cannot exist at room temperature. is is the case of Aharoni et al. [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF]'s work, who programmed curvature into Liquid Crystal Elastomers which expand anisotropically when heated but go back to their initial shape at room temperature. A way to circumvent this problem is to prepare the sample at a high temperature and then let it morph by cooling down, even if the mechanism is reversible the sample will hold its shape at room temperature. is is the case for gel lithography [START_REF] Kim | Designing responsive buckled surfaces by hal one gel lithography[END_REF][START_REF] Na | Grayscale gel lithography for programmed buckling of non-euclidean hydrogel plates[END_REF] methods which program a pa ern in a photo-crosslinkable material using light in a controlled environment at a high temperature and then let the material cool down in an aqueous solution at room temperature. Tibbits [START_REF] Tibbits | 4D printing: Multi-material shape change[END_REF] who coined the term 4D printing, explores a di erent mechanism. Using a multi-material 3D printer, they mix a passive material with a hydrophobic one which expands when in contact with water and use this technique to engineer self-folding materials. Wood is also known to expand when wet, since its swelling is anisotropic and follows the material bers, it is possible to assemble wood bilayers whose bers are at di erent orientations to create bending [START_REF] Grönquist | Analysis of hygroscopic self-shaping wood at large scale for curved mass timber structures[END_REF][START_REF] Rüggeberg | Bio-inspired wooden actuators for large scale applications[END_REF]. It is even possible to use additive manufacturing technologies to e ectively 3D print wood and therefore have a higher level of control on the anisotropic swelling behavior [START_REF] Ti Any Cheng | Programming material compliance and actuation: Hybrid additive fabrication of biocomposite structures for large-scale self-shaping[END_REF]. Other materials can have interesting behaviors in reaction to humidity, for example Gladman et al. [START_REF] Sydney Gladman | Biomimetic 4D printing[END_REF] consider hydrogels which have an anisotropic swelling behavior when immersed in water, the results of this experiment were successfully simulated by van Rees et al. [START_REF] Wim | Mechanics of biomimetic 4D printed structures[END_REF].

More unusual trigger mechanisms have been explored, such as cooking, to create selfshaping pasta [START_REF] Tao | Morphing pasta and beyond[END_REF][START_REF] Wang | Transformative appetite: Shape-changing food transforms from 2D to 3D by water interaction through cooking[END_REF] or other our-based foods [START_REF] Tao | Morphlour: Personalized our-based morphing food induced by dehydration or hydration method[END_REF]. Auto-in atables [START_REF] Webb | Auto-in atables: Chemical in ation for pop-up fabrication[END_REF] which are structures that are in ated by a chemical reaction inside the membranes can also lead to interesting developments, as there is a whole body of work on programming curvature into in atable structures [START_REF] Ou | Aeromorph -heat-sealing in atable shape-change materials for interaction design[END_REF][START_REF] Pane | Computational inverse design of surface-based in atables[END_REF][START_REF] Siéfert | Bioinspired pneumatic shape-morphing elastomers[END_REF].

Pre-stretched membranes

e last trigger mechanism for self-shaping surfaces is the most related to this thesis: using the tension of pre-stretched substrates.

e idea is to locally bind some rigid material with a pre-stretched membrane which forms a bilayer where the rigid material has the role of a passive layer resisting deformation, upon release the membrane becomes an active layer which will contract as much as possible.

One way to create self-shaping mechanisms is by altering the pre-stretched substrate itself, either by depositing resin onto the membrane to locally sti en it, or by integrating rigid elements such as rods within the fabric. Oxman and Louis-Rosenberg [START_REF] Oxman | Material-based design computation: An inquiry into digital simulation of physical material properties as design generators[END_REF] deposit resin into a pre-stretched latex membrane, forming di erent pa erns and experimenting with simple form-nding algorithms using the Processing environment. Ahlquist et al. [START_REF] Ahlquist | Post-forming composite morphologies: Materialization and design methods for inducing form through textile material behavior[END_REF] and Sharmin and Ahlquist [START_REF] Sharmin | Knit architecture: Exploration of hybrid textile composites through the activation of integrated material behavior[END_REF] use pre-stretched machine kni ed textiles locally reinforced with epoxy resin and experiment with simple base forms and heuristic simulation models. Aldinger et al. [START_REF] Lo E Aldinger | Tailoring self-formation: Fabrication and simulation of membrane-actuated sti ness gradient composites[END_REF] sew sti ber reinforced polymer rods into a pre-stretched tulle mesh fabric, they are able to control the curvature of the result depending on the pa ern the rods form and whether they are on the top side or the bo om side of the fabric. Another popular technique is to laminate a layer of sti material onto the pre-stretched substrate. e layer itself has usually a sparse geometry to let the membrane contract and can be either 3D printed [START_REF] Agkathidis | Active membranes: 3D printing of elastic bre pa erns on pre-stretched textiles[END_REF][START_REF] Berdos | Architectural hybrid material composites: Computationally enabled techniques to control form generation[END_REF][START_REF] Sebastien | Hyperbolic origamiinspired folding of triply periodic minimal surface structures[END_REF][START_REF] Guseinov | Programming temporal morphing of self-actuated shells[END_REF][START_REF] Guseinov | Curveups: Shaping objects from at plates with tension-actuated curvature[END_REF] or laser cut from materials such as wood [START_REF] Ahlquist | Post-forming composite morphologies: Materialization and design methods for inducing form through textile material behavior[END_REF][START_REF] Berdos | Architectural hybrid material composites: Computationally enabled techniques to control form generation[END_REF]. Some authors experimented with pa erns forming a grid, for example Ahlquist et al. [START_REF] Ahlquist | Post-forming composite morphologies: Materialization and design methods for inducing form through textile material behavior[END_REF] glues laser cut Voronoi patterns onto a pre-stretched textile and examine their folding behavior, Agkathidis et al. [START_REF] Agkathidis | Active membranes: 3D printing of elastic bre pa erns on pre-stretched textiles[END_REF] and Berdos et al. [START_REF] Berdos | Architectural hybrid material composites: Computationally enabled techniques to control form generation[END_REF] print pa erns following principal stress directions which are important for structural stability at a large scale. A problem with this type of approach is that a dense grid will constrain the metric of the nal geometry so that it will only be possible to obtain developable surfaces since the composite material can barely stretch or expand. Moreover, since the curves have an anisotropic cross-section, all the laminated lines are constrained to be geodesics on the surface. Another approach is to control the geometry of the material to obtain speci c bending modes. Callens et al. [START_REF] Sebastien | Hyperbolic origamiinspired folding of triply periodic minimal surface structures[END_REF] glue 3D printed structures with speci c hinges on a latex membrane so that upon release the hinges make a 90°angle and the overall structure forms a triply periodic minimal surface. Guseinov et al. [START_REF] Guseinov | Curveups: Shaping objects from at plates with tension-actuated curvature[END_REF] propose a more general method to program curvature into pre-stretched membranes, which consists in 'sandwiching' 3D printed frustum-like elements between two rubber sheets, the speci c geometry of the rigid elements allowing to control bending angles on the surface. [START_REF] Guseinov | Programming temporal morphing of self-actuated shells[END_REF] show how to also control the temporal morphing behavior of these structures by using heat as a second trigger mechanism.

Instead of laminating a given structure onto the pre-stretched substrate, it is possible to use a 3D printer to directly print onto the fabric material and the molten plastic will bond with the textile bers upon cooling down. is technique has been also experimented for printing onto non-stretched textiles for composite applications, in particular there has been a lot of studies into the adhesion of 3D printed pa erns onto textiles. See [START_REF] Dereje Berihun Sitotaw | Additive manufacturing and textiles: State-of-the-art[END_REF] for an overview on 3D printing on non-stretched fabric.

Directly 3D printing on stretched fabric to create self-shaping textiles has been used in many di erent applications such as to design shoes [START_REF] Guberan | Active shoes[END_REF], architectural shell prototypes [START_REF] Al-Badry | Lobster shell[END_REF], circular shading panels [START_REF] Kycia | Hybrid textile structures as means of material-informed design strategy[END_REF], sound-absorbing panels (using tilings of stars similar to the one showcased in Chapter 3) [START_REF] Clasen | Sonogrid". Purmundus Challenge[END_REF] and even wearable tangible user interfaces [START_REF] Goudswaard | Fabriclick: Interweaving pushbu ons into fabrics using 3D printing and digital embroidery[END_REF], and for good reason: fabric is an ubiquitous and very versatile material, and has countless uses in fashion, sound absorption and architecture (see Fig. 2.10).

However, the morphing behavior of these assemblies is di cult to anticipate just by looking at the 2D printed pa ern. e deployed shape is the result of a stress-minimizing process which can be simulated, therefore researchers experimented with form-nding tools to predict the e ect of speci c pa erns. If some architects experimented with heuristic form-nding [START_REF] Christie | Spatial textile hybrids: Computing a self-forming behavior[END_REF][START_REF] Kycia | Self-shaping textiles -a material platform for digitally designed, material-informed surface elements[END_REF], more systematic and physics-based methods have been developed as well: experiments on a simple rectangle shape have been done using the FEM package Abaqus [START_REF] Sco | Finite element modeling to predict the steady-state structural behavior of 4D textiles[END_REF], but the most notable and closely related to this thesis is the Fig. 2.10: Self-shaping textiles. From le to right: shading panels [START_REF] Kycia | Hybrid textile structures as means of material-informed design strategy[END_REF], architectural prototype [START_REF] Al-Badry | Lobster shell[END_REF], wearable user interface [START_REF] Goudswaard | Fabriclick: Interweaving pushbu ons into fabrics using 3D printing and digital embroidery[END_REF].

interactive form-nding program of Pérez et al. [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF].

Perez et al. 's method focuses of Kirchho -Plateau surfaces which are piecewise-minimallike shapes that appear when printing sparse networks of closed curves. If the network is more dense like a gridshell, the structure will become more constrained and only bending will be possible. is gives rise to developable or almost developable surfaces, as shown in the recent results by Agkathidis and Varinlioglu [START_REF] Agkathidis | 4D printing on textiles: Developing a le to fabrication framework for self-forming, composite wearables[END_REF]. But closed networks are not a requirement, in fact one of the strengths of this method is that by printing directly of the substrate we can easily print any pa ern included disconnected ones. Indeed, experiments in workshops such as the one of Erioli and Naldoni [START_REF] Erioli | informed exible ma er workshop[END_REF] show great examples of that.

To control the metric in a more methodic way, it might be interesting to take inspiration from the whole body of work cited previously: even if they do not use the same materials, much of the same geometric principles can be applied. For example, a pa ern of dots where the spacing between the dots varies can be a good way to control the metric in an isotropic, or conformal way [START_REF] Kim | Designing responsive buckled surfaces by hal one gel lithography[END_REF], and indeed such a pa ern was shown to work for self-shaping textiles [START_REF] Fields | Self forming structures: An exploration into 3D printing on pre-stretched fabric[END_REF].

Since the imprinted plastic layers form a bilayer with the fabric, it is convenient to exploit the bilayer e ect to control the bending behavior of the surface. Starting with simple pa erns such as parallel ribbons [START_REF] Kycia | Self-shaping textiles -a material platform for digitally designed, material-informed surface elements[END_REF] or lamellae [START_REF] Neuß | Interaction between 3D deformation of textile fabrics and imprinted lamellae[END_REF], it is possible to identify which parameters have an impact on the resulting curvature. Parameters such as the amount of stretch of the fabric or the thickness of the plastic elements are known to control the bending behavior -for example, Kycia [START_REF] Kycia | Material form-nding of modular textile structures[END_REF] exploited some of the capabilities of a 3D printer by varying the thickness of the printing pa ern, this allowed them to locally modulate the amount of curvature along a printed element. More surprising ndings are that the number of parallel elements and their spacing also control the curvature: the more spaced the elements, the bigger the area of fabric which acts on the elements; and the bigger the number of elements, the larger the total force applied on the fabric [START_REF] Kycia | Self-shaping textiles -a material platform for digitally designed, material-informed surface elements[END_REF][START_REF] Neuß | Interaction between 3D deformation of textile fabrics and imprinted lamellae[END_REF]. In Chapter 5, we exploit both the printed thickness and the ribbon spacing to modulate the curvature of our elements. Another interesting nding by Kycia [START_REF] Kycia | Self-shaping textiles -a material platform for digitally designed, material-informed surface elements[END_REF] is how slender elements can either bend and roll into a cylinder, or make wrinkles depending on the thickness and the spacing of the elements. ese two modes (wrinkling and bending) are related to the sti ness limits of bilayers and the theoretical limit between the two modes can be computed as a function of geometric and material parameters [START_REF] Van Manen | eoretical sti ness limits of 4D printed self-folding metamaterials[END_REF].

A limitation of using a 3D printer to deposit plastic on stretched fabric is that access to both sides of the substrate is di cult, which becomes limiting when one wants to exploit the bilayer e ect: if only one side is covered by plastic pa ern they will all be biased towards bending upwards. Christie [START_REF] Christie | Spatial textile hybrids: Computing a self-forming behavior[END_REF] used a 6 degrees of freedom (DoF) robot arm to print both onto and under a pre-stretched textile, the printed shapes are not forming bilayers but are tridimensional mechanisms allowing some metric change as well as some bending. Physical models are an important tool architects use to explore ideas and communicate them to clients and collaborators [START_REF] Dunn | Architectural Modelmaking[END_REF]. Architects have a long tradition of exploiting a ordable and easy-to-assemble materials for fabricating small-scale prototypes, including cardboard, foam and wood cutouts. With the rise of maker spaces, technologies for milling, laser cu ing, and 3D printing are increasingly accessible and powerful tools for architectural prototyping. Doubly-curved freeform surfaces, however, remain a signi cant challenge: even with modern additive manufacturing techniques, prototypes of thin curved structures cannot be easily fabricated without formwork, support structures, or other artifacts.

is chapter explores the use of 3D printing on stretched fabric to create self-shaping architectural models. is fabrication technique can be a powerful medium for architects who wish to explore new shape ideas, as was demonstrated by recent experimentations [START_REF] Agkathidis | 4D printing on textiles: Developing a le to fabrication framework for self-forming, composite wearables[END_REF][START_REF] Erioli | informed exible ma er workshop[END_REF][START_REF] Kycia | Material form-nding of modular textile structures[END_REF]. Our work follows in the line of several recent explorations of combining fabric under tension with networks of elastic rods to control shape [START_REF] Agkathidis | Active membranes: 3D printing of elastic bre pa erns on pre-stretched textiles[END_REF][START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF]. e resulting fabric and curve structure behaves as a so-called Kirchho -Plateau Surface, where the fabric forms minimal surface patches bounded by the rods. Yet, a wider variety of shapes are possible if one does not restrict to printing closed curves and instead uses dense repetitive pa erns of plastic rods [START_REF] Erioli | informed exible ma er workshop[END_REF][START_REF] Kycia | Material form-nding of modular textile structures[END_REF]. Our main contribution is the use of dense 3-pointed star pa erns for fabricating freeform structures with large-scale curvature variation. Our choice of 3-pointed stars is based on two important physical observations: rst, the arms of the stars bend to form a small bump localized around the star, which contracts the surface locally. Varying the thickness of the stars o ers control on the amount of contraction in their neighborhood. Second, since the pa ern is formed of disconnected elements, the fabric contracts in-between these elements, bringing them closer together, with the amount of contraction controlled by the local spacing of the stars. In both cases, if the amount of contraction varies spatially, it induces in the initially-at fabric a new rest state with non-Euclidean metric [START_REF] Kim | Designing responsive buckled surfaces by hal one gel lithography[END_REF][START_REF] Konaković-Luković | Computational design of deployable auxetic shells[END_REF][START_REF] Sharon | e mechanics of non-euclidean plates[END_REF], and the fabric bends in space to resolve the metric frustration. Due to the combination of these e ects, our plastic pa ern act as a metamaterial, whose local geometry dictates the overall shape. Varying the two degrees of freedom o ered by the stars thickness and spacing allows us to achieve a variety of shapes, including cylindrical and doubly-curved surfaces with negative Gaussian curvature (Fig. 3.9). An additional bene t of our star-based approach is that the pa ern enriches the surface with a texture, allowing architects to explore a design space that combines shape and appearance. e precise amount of contraction within and between the stars in a star pa ern is determined by the nonlinear interaction of the sti plastic with the fabric substrate. To avoid the need for laborious trial-and-error with physical 3D-printed models, we present a form-nding tool that simulates the deployment of our printed-on-fabric metamaterials, allowing users to quickly iterate on virtual prototypes before investing time in fabrication. To perform our simulation, we introduce a physical model of plastic pa erns printed on stretched fabric. Our model goes beyond existing ones [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF] by accounting for the bilayer structure of the plastic-on-fabric assemblies and for the speci c mechanical behavior of the structure at the boundary between the pa erns and the fabric, especially near the star arms tips, which we found to be critical to accurately reproduce the shapes we target. We also detail our fabrication protocol used to achieve accurate, reproducible physical prototypes using a commodity 3D printer.

In summary, we make the following contributions:

• A description of the main physical phenomena that contribute to the emergence of curved surfaces when printing dense plastic pa erns over pre-stretched fabric.

• A physical simulator of dense plastic pa erns printed on pre-stretched fabric, allowing accurate reproduction of these phenomena.

• A form-nding tool based on this simulator, which we used to design and fabricate a variety of architectural forms based on a simple 3-pointed star pa ern.

is chapter is mainly based on the following publication:

• David Jourdan, Mélina Skouras, Etienne Vouga, Adrien Bousseau, 2021. Printingon-Fabric Meta-Material for Self-Shaping Architectural Models. In Advances in Architectural Geometry 2020, pages 264-285.

Physical Model

To fabricate our self-shaping textile models, we take a rectangle of thin fabric, pre-stretch it along each dimension by a factor 𝑠 by pinning the fabric boundary to a wooden frame, place the fabric on the print bed of a conventional FFF 3D-printer, and print small threepointed stars over the fabric (see Figure 3.6). e melted plastic deposited by the printer adheres to the surface of the fabric, resulting in strong bonding of the two materials to each other. A er the plastic has cured, we remove the fabric from the frame, trim excess fabric, and pose the structure by pinning several points of the boundary to the ground. See Section 3.3 for full details of the fabrication process.

We parameterize the fabric by a rectangle Ω in the plane. Each star's center is placed at the nodes of a regular hexagonal tiling of Ω, with distance 𝑑 mm between neighboring centers, with the star's three arms aligned to the symmetry axes of the tiling. ree scalar Geometric intuition behind our approach, at several length scales. At the scale of a single star, the contraction of the pre-stretched fabric back to its rest dimensions is halted by the plastic star bonded to the fabric. e star arms buckle to form a small bump (a); neighborhoods of thicker stars contract less due to the star arms bending less. At the scale of a neighborhood of several stars, the fabric contracts unimpeded, since stars are not connected (b). A sparser star pa ern with smaller stars (and hence more spacing between them) allows more contraction. At the scale of a large patch of metamaterial area, several factors control the surface shape: di erential contraction due to variations in star thickness and spacing induces buckling of the surface to relieve metric frustration; boundary conditions can impose additional contraction and buckling (c); and if stars are laid out in a regular pa ern, there is global coupling in how each star breaks symmetry while buckling, introducing large-scale curvature (d).

sizing elds specify the design of the star pa ern: ℓ : Ω → [0, 1] speci es the length of the star's arms at di erent locations on the fabric, with ℓ = 0 indicating no star at all and ℓ = 1 a star with arm lengths 𝑑 (so that the star touches its neighbors); and ℎ, 𝑤 : Ω → R specify the thickness (in the direction perpendicular to the fabric and printing plane) of the stars and (in-plane) width of the star arms, both in millimeters.

To summarize, a star pa ern design consists of a choice of:

1. fabric tension 𝑠 and star spacing 𝑑, both global to the entire pa ern;

2. three functions ℓ, ℎ, 𝑤 over Ω; which encode variations in the star sizing;

3. boundary conditions for how the border of the pa ern should be pinned to the ground a er printing.

e geometry of the metamaterial

A er the star pa ern has been printed and the fabric is allowed to relax to static equilibrium, the surface buckles into a 3D structure with residual internal stress. Figure 3.1 illustrates how the choice of design parameters provides several means of li ing the resulting surface into controllable shapes. In the neighborhood of each individual star, the star arms bend to form a bump under the action of the fabric's compressive forces (Figure 3.1a). e size of this bump depends on the fabric tension 𝑠 and the length and thickness of the star arms, which control the star's resistance to bending and thus nal curvature. In between stars, the fabric contracts unimpeded (Figure 3.1b), by an amount proportional to the length of the star arms.

At a coarse scale much larger than that of an individual star, we can treat the metamaterial as a homogenized smooth surface without the bumps around each star. In this homogenized view, the e ect of each bump is to change the surface area of a neighborhood of the star at equilibrium, where the spatially-varying amount of surface contraction depends on the thickness, width, and length of the stars. erefore ℓ, ℎ, 𝑤 equip the homogenized surface with a rest state described by a non-Euclidean metric [START_REF] Sharon | e mechanics of non-euclidean plates[END_REF]. In addition to changing the local surface area of the homogenized surface, the stars modify the surface's rest extrinsic curvature, since the stars are printed on top of the fabric (rather than embedded within it); in other words the metric of the metamaterial varies in the thickness as well as the curvilinear directions. e di erential contraction described by this metric causes the surface to buckle out of plane, in order to exchange large amounts of stretching strain for slight bending strain (Figure 3.1c). is relationship between change of metric and buckling has also been exploited by related self-shaping fabrication technologies based on swelling [START_REF] Kim | Designing responsive buckled surfaces by hal one gel lithography[END_REF] or auxetic linkages [START_REF] Konaković-Luković | Computational design of deployable auxetic shells[END_REF]. In our case, the precise relationship between the surface metric and the values of ℓ, ℎ, and 𝑤 depends on a complex physical coupling between the fabric and the printed plastic ribbons.

We also observed a coupling phenomenon between neighboring stars, where the bending of each individual star propagates to adjacent stars through deformation of the fabric in between (Figure 3.1d). It is unclear whether this behavior is a consequence of, or an additional e ect independent of the induced non-Euclidean metric. is coupling leads to globally consistent symmetry-breaking in the surface, a phenomenon that is especially visible when we do not x the boundary of the domain, since in this case the accumulation of local bending makes the entire surface fold on itself to form a tube, as shown in Figure 3.2.

Given the complexity of the physical phenomena involved, from local contraction of the surface to global propagation of bending, we propose a dedicated numerical simulation model to predict the shape that user-provided star pa erns would take. In the following sections we present the di erent material models used, they have all been implemented in C++ and their implementation is available on GitHub [START_REF] Jourdan | fabsim: Tools for fabrication & simulation[END_REF].

Fabric's material model

We model the fabric as a plate discretized using constant strain triangles and edge-based bending hinges following the popular Discrete Shells model of Grinspun et al. [START_REF] Grinspun | Discrete shells[END_REF]. e energy describing the behavior of the fabric is computed as a sum of a exural term associated to bending, and a membrane term associated to stretch, it is optimized to nd the static equilibrium con guration (see Section 3.2.4).

For the exural term, we followed the implementation suggested by Tamstorf and Grinspun [START_REF] Tamstorf | Discrete bending forces and their jacobians[END_REF] and compute it as a sum over edges of the mesh:

𝑊 𝐵 (𝑥) = 𝑘 𝐵 ∑ 𝑖 3 𝑒 𝑖 2 Ā𝑖 2 tan 𝜃 𝑖 2 2 (3.1)
where 𝜃 𝑖 is the dihedral angle, 𝑒 𝑖 is the initial length of edge 𝑒 𝑖 , Ā𝑖 is the sum of the initial areas of the adjacent triangles, 𝑘 𝐵 = 𝐸ℎ 3 24(1-𝜈 2 ) is the bending sti ness, ℎ is the thickness of the membrane, 𝐸 its Young's modulus and 𝜈 its Poisson's ratio. While bending forces are typically much smaller than membrane forces in Kirchho -Plateau surfaces and can be mostly neglected [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF], we found that accounting for the nite exural resistance of the material at the boundary between the stars and fabric was necessary to limit the tangent discontinuities at the boundary and to reproduce the global curvature of the surface that we observed on the real artifacts (see Figure 3.5).

Our structures exhibit high localized stresses and large deformations near the tips of the stars' arms, due to the arm tips bending and "digging into" the fabric substrate. Using a linear material model for the fabric causes elements to completely compress and cause so-called altitude collapses [START_REF] Tamstorf | Discrete bending forces and their jacobians[END_REF] which leads to division by zero in the Discrete Shells model. To prevent this problem, we use a neo-Hookean material model to approximate the nonlinear deformation of the fabric near the rod tips. e membrane energy contribution writes as

𝑊 𝑀 (𝑥) = ℎ 2 ∑ 𝑖 Ā𝑖 𝜇 (tr(𝐹 𝑇 𝐹 ) -2 -2 ln 𝐽 ) + 𝜆(ln 𝐽 ) 2 (3.2)
where 𝜆 = 𝐸𝜈 1-𝜈 2 and 𝜇 = 𝐸 2(1+𝜈) are the Lamé coe cients, 𝐹 is the deformation gradient, and 𝐽 = √ det 𝐹 T 𝐹 . Note that this formula is slightly di erent than the equivalent one for volumetric elements.

Despite the orthotropic behavior exhibited by the fabric that we used, we noticed that in the case of this star tiling pa ern it was su cient to model it as an isotropic material.

Rods material model

We use the Discrete Elastic Rods (DER) model [START_REF] Bergou | Discrete viscous threads[END_REF][START_REF] Bergou | Discrete elastic rods[END_REF][START_REF] Lestringant | A discrete, geometrically exact method for simulating nonlinear, elastic or non-elastic beams[END_REF] for modeling the arms of the stars.

is model is based on a reduced centerline representation, where a rod is represented as a strip of vertices with additional degrees of freedom to represent twist. For a rod with a rectangular cross-section with thickness 𝑤 𝑛 and width 𝑤 𝑏 , the energy contribution of a rod writes as:

𝑊 𝑅 = 1 2 ∑ 𝑖 𝐸𝐴 8 𝑒 𝑖-1 2 -l2 𝑖-1 2 l3 𝑖-1 + 𝑒 𝑖 2 -l2 𝑖 2 l3 𝑖 + 2 l𝑖-1 + l𝑖 𝐸𝐼 1 𝜅 1 𝑖 -κ1 𝑖 2 + 𝐸𝐼 2 𝜅 2 𝑖 -κ2 𝑖 2 + 𝜇 𝐼 1 + 𝐼 2 2 𝜏 2 𝑖 (3.3)
where

𝐼 1 = 𝑤 3 𝑛 𝑤 𝑏 12 , 𝐼 2 = 𝑤 3 𝑏 𝑤 𝑛
12 are the geometric moments of inertia, 𝐴 = 𝑤 𝑛 𝑤 𝑏 is the cross-sectional area,𝜏 𝑖 is the discrete twist associated with vertex 𝑖, 𝜅 1 𝑖 and 𝜅 2 𝑖 are the two components of the curvature binormal projected onto the local material basis. For more details, see [START_REF] Lestringant | A discrete, geometrically exact method for simulating nonlinear, elastic or non-elastic beams[END_REF].

Coupling. We enforce the coupling between the plate and the rods via colocation: all centerline degrees of freedom of the rod are also vertices of the plate. is strategy implies that the fabric mesh must have edges that align with the star arms: we rst discretize the rods constituting each star (by choosing the resolution of the rod centerline) and then we generate the fabric mesh using constrained Delaunay triangulation [START_REF] Richard | Triangle: Engineering a 2D quality mesh generator and delaunay triangulator[END_REF]. Within a star, the rods should also be coupled together because they are rigidly linked at the center of the star.

De ning energies that allow for proper transfer of bending and twisting forces from one rod to the other is, in the general case, a challenging problem. Even though several models exist for computing connections between rods using within the DER framework [START_REF] Lestringant | Modeling of Flexible Beam Networks and Morphing Structures by Geometrically Exact Discrete Beams[END_REF][START_REF] Pérez | Design and fabrication of exible rod meshes[END_REF], they can be complex to implement and expensive to run. We instead exploit the 3-folded symmetry of stars and the fact that the rods mostly bend about their width axis: we compute twist and bending at the connection as if each star arm was split into two "copies" of the same rod with the same geometry and each was bonded to one rod copy on each of the two other arms (see Figure 3.3). e elastic energy contributed by the connection is then the sum of the three pairwise bending and twisting energies, weighted by 1 2 since the cross-sectional area of each "rod copy" is half that of the full rod. is approximation gives the proper weighting only in the case where rods only bend about their width axis at the center connection, that is 𝜅 2 𝑖 = 0. In that case, both connection contributions are weighted by 𝐼 1 = 𝑤 3 𝑛 𝑤 𝑏 24 and thus sum to the expected total.

Bilayer ribbon model. Simply colocating the vertices representing star arms to the thin shell model representing the fabric does not correctly capture the geometry of the ribbon-on-fabric assembly: in the printed assembly, the stars are on top of the fabric, while in the colocation model, rods are embedded within the fabric. Yet the bilayer structure of the assembly is precisely what causes the rods to bend in a privileged direction, because of the di erences in stresses between the traction-free top surface of the ribbon and the compressive forces applied by the fabric to the bo om surface. We can easily observe this so-called bilayer e ect on a at ribbon printed on top of a strip of fabric of the same width: the structure will not stay at when released and will consistently bend in the same direction (see Figure 3.4, le ). Furthermore, the assembly also exhibits some plastic deformation: when we separate the ribbon from the fabric layer, the initially straight ribbon does not completely recover its original shape (see Figure 3.4, right) and has a non-zero rest curvature κ.

Van Rees et al. [START_REF] Wim | Growth pa erns for shapeshi ing elastic bilayers[END_REF] show that a bilayer shell made of two monolayers of respective Young's moduli 𝐸 1 , 𝐸 2 , thicknesses ℎ 1 , ℎ 2 , and where the strain-free rest state of each layer is given by a di erent metric (i.e. rst fundamental form) a 𝑟 1 and a 𝑟 2 , is energetically equivalent to a monolayer with non-zero rest curvature. ey give formulas for the rest rst and second fundamental forms a 𝑟 and b 𝑟 of this equivalent monolayer (see [147, Supplemental material section S1.3.3]). To account for ribbon plasticity, we extend their formula to the case where each of the two layers in the bilayer have non-at rest shape with the same second fundamental form b. 1 Here we derive new formulas for a 𝑟 and b 𝑟 that account for this bilayer rest curvature.

To this end, we make the following ansatz:

a 𝑟 1 = a 𝑟 + 𝛼 1 (b 𝑟 -b) a 𝑟 2 = a 𝑟 -𝛼 2 (b 𝑟 -b), (3.4)
where 𝛼 1 and 𝛼 2 are factors to be determined so that the elastic energy of the e ective monolayer and that of the bilayer agree up to a constant. Note that for any choice of these factors, Equation (3.4) is satis ed by se ing

a 𝑟 = a 𝑟 1 + a 𝑟 2 2 - 𝛼 1 -𝛼 2 2(𝛼 1 + 𝛼 2 ) (a 𝑟 1 -a 𝑟 2 ) (3.5) b 𝑟 = b + a 𝑟 1 -a 𝑟 2 𝛼 1 + 𝛼 2 . ( 3.6)
Like van Rees et al. [START_REF] Wim | Growth pa erns for shapeshi ing elastic bilayers[END_REF], we de ne the elastic inner product associated to a material with Poisson's ratio 𝜈 as

< A, B >= 𝜈 1 -𝜈 2 tr(A) tr(B) + 1 1 + 𝜈 tr(AB) (3.7)
and the elastic energy norm as

||A|| 2 =< A, A >= 𝜈 1 -𝜈 2 tr 2 (A) + 1 1 + 𝜈 tr(A 2 ). (3.8)
Le ing a 𝑐 and b 𝑐 denote, respectively, the rst and second fundamental forms of the midsurface of the shell in the current con guration, we can de ne the energy of the bilayer by

𝑊 𝐵𝐿 = 1 2 ∫ 𝑈 𝐸 1 ℎ 1 8 ||a -1 𝑟 1 a 𝑐 -I|| 2 + ℎ 3 1 24 ||a -1 𝑟 1 (b 𝑐 -b)|| 2 + ℎ 2 1 8 < a -1 𝑟 1 a 𝑐 -I, a -1 𝑟 1 (b 𝑐 -b) > √ det a 𝑟 1 d𝑥 d𝑦 + 1 2 ∫ 𝑈 𝐸 2 ℎ 2 8 ||a -1 𝑟 2 a 𝑐 -I|| 2 + ℎ 3 2 24 ||a -1 𝑟 2 (b 𝑐 -b)|| 2 - ℎ 2 2 8 < a -1 𝑟 2 a 𝑐 -I, a -1 𝑟 2 (b 𝑐 -b) > √ det a 𝑟 2 d𝑥 d𝑦 (3.9)
and that of the equivalent monolayer by

𝑊 𝑀𝐿 = 1 2 ∫ 𝑈 𝐸 1 ℎ 1 + 𝐸 2 ℎ 2 8 ||a -1 𝑟 a 𝑐 -I|| 2 + 𝐸 1 ℎ 3 1 + 𝐸 2 ℎ 3 2 24 ||a -1 𝑟 (b 𝑐 -b 𝑟 )|| 2 + 𝐸 1 ℎ 2 1 -𝐸 2 ℎ 2 2 8 < a -1 𝑟 a 𝑐 -I, a -1 𝑟 (b 𝑐 -b 𝑟 ) > √ det a 𝑟 d𝑥 d𝑦, (3.10)
where 𝑈 is the parameterization domain of the shell midsurface parameterized by curvilinear coordinates (𝑥, 𝑦).

A er plugging (3.4) in Equation (3.9), expanding the terms inside the norms, discarding all terms which do not depend on a 𝑐 nor b 𝑐 (and therefore do not change the equilibrium state), and equating all the remaining terms, we nd that 𝑊 𝐵𝐿 matches 𝑊 𝑀𝐿 (up to a constant) when the following system of equations is satis ed:

𝐸 2 ℎ 2 𝛼 2 -𝐸 1 ℎ 1 𝛼 1 4 = 1 8 (𝐸 2 ℎ 2 2 -𝐸 1 ℎ 2 1 ) 𝐸 1 ℎ 2 1 𝛼 1 + 𝐸 2 ℎ 2 2 𝛼 2 8 = 1 12 (𝐸 2 ℎ 3 2 + 𝐸 1 ℎ 3 1 ). (3.11)
Solving System (3.11) for 𝛼 1 and 𝛼 2 gives us

𝛼 1 = 4ℎ 3 1 𝐸 1 + 3ℎ 2 1 ℎ 2 𝐸 1 + ℎ 3 2 𝐸 2 6ℎ 2 1 𝐸 1 + 6ℎ 1 ℎ 2 𝐸 1 𝛼 2 = ℎ 3 1 𝐸 1 + 3ℎ 1 ℎ 2 2 𝐸 2 + 4ℎ 3 2 𝐸 2 6ℎ 1 ℎ 2 𝐸 2 + 6ℎ 2 2 𝐸 2 , (3.12)
which allows us to write

b 𝑟 -b = a 𝑟 1 -a 𝑟 2 𝛼 1 + 𝛼 2 = 6ℎ 1 ℎ 2 𝐸 1 𝐸 2 (ℎ 1 + ℎ 2 ) ℎ 4 1 𝐸 2 1 + 2ℎ 1 ℎ 2 𝐸 1 𝐸 2 (2ℎ 2 1 + 3ℎ 1 ℎ 2 + 2ℎ 2 2 ) + ℎ 4 2 𝐸 2 2 (a 𝑟 1 -a 𝑟 2 ). (3.13)
In our case, the quantities 𝐸 1 , ℎ 1 and a 𝑟 1 correspond to the plastic layer, and 𝐸 2 , ℎ 2 and a 𝑟 2 correspond to the fabric layer. e plastic material that we use is much sti er than the fabric, i.e. 𝐸 1 >> 𝐸 2 , so we can further simplify this expression:

b 𝑟 -b ≈ 6ℎ 2 (ℎ 1 + ℎ 2 )𝐸 2 ℎ 3 1 𝐸 1 (a 𝑟 1 -a 𝑟 2 ). (3.14)
We assume the plastic printed on top of the stretched fabric has no residual strain, so that the metric of the plastic is a 𝑟 1 = 1 0 0 1 . Since the fabric is stretched from rest by a factor 𝑠,

a 𝑟 2 = 1 𝑠 2 0 0 1 𝑠 2 . We then have b 𝑟 -b ≈ 6ℎ 2 (ℎ 1 + ℎ 2 )𝐸 2 ℎ 3 1 𝐸 1 1 -1 𝑠 2 0 0 1 -1 𝑠 2 , (3.15)
i.e.,

b 𝑟 ≈ 6ℎ 2 (ℎ 1 + ℎ 2 )𝐸 2 ℎ 3 1 𝐸 1 1 -1 𝑠 2 0 0 1 -1 𝑠 2 + b. (3.16)
e bending energy of the ribbon-on-fabric assembly will penalize deviation of the assembly's second fundamental form b 𝑐 away from b 𝑟 . Since the arms of our stars are much longer than they are wide, we neglect bending deformations in the width direction, and assume that the arms are at rest when the curvature 𝜅 in the longitudinal direction matches the normal curvature in that direction prescribed by b 𝑟 :

𝜅 = 6ℎ 2 (ℎ 1 + ℎ 2 )𝐸 2 ℎ 3 1 𝐸 1 1 - 1 𝑠 2 + κ, (3.17)
where κ is the rest curvature of the plastically deformed rod in the longitudinal direction.

We can show that, in the case of κ = 0, this result is consistent with previous models for bilayers such as the classic Timoshenko model for bilayer metal thermostats [START_REF] Timoshenko | Analysis of bi-metal thermostats[END_REF] which states that the curvature is given by We use this expression to set the rest curvature of the rods about their width axis, κ1 𝑖 in Equation 3.3.

𝜅 = 6(ℎ 1 + ℎ 2 ) 3(ℎ 1 + ℎ 2 ) 2 + ℎ 2 1 + ℎ 2 2 + ℎ 3 2 𝐸 2 𝐸 1 + ℎ 3 1 ℎ 2 𝐸 1 𝐸 2 1 - 1 𝑠 2 ------→ 𝐸 1 >>𝐸 2 6ℎ 2 (ℎ 1 + ℎ 2 )𝐸 2 ℎ 3 1 𝐸 1 1 - 1 𝑠 2 (
is model allows us to accurately reproduce the global curling up of a uniform star pa ern into a rolled tube; see Figure 3.2. We provide in Figure 3.5 a comparison between our model and a model that ignores this bilayer e ect, showing that our model be er captures the behavior of a real-world fabricated surface.

Optimization

De ning the total energy 𝑊 (x) = 𝑊 𝐵 (x) + 𝑊 𝑀 (x) + 𝑊 𝑅 (x) + 𝑊 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑜𝑛𝑠 (x) where x is a vector stacking all degrees of freedom, we compute the equilibrium shape of a star pa ern by minimizing the total energy and nding x = argmin𝑊 (x) using the Newton-Raphson procedure with line search [START_REF] Nocedal | Numerical Optimization[END_REF].

Starting from a at initial guess x 0 , the Newton-Raphson algorithm iteratively looks for x such that ∇𝑊 (x) = 0 and ∇ 2 𝑊 (x) 0. To use this algorithm, we need to compute the rst and second derivatives of 𝑊 , which are implemented in our open-source library [START_REF] Jourdan | fabsim: Tools for fabrication & simulation[END_REF].

is algorithm iteratively solves a linear system involving the Hessian matrix ∇ 2 𝑊 (x), this matrix may be non-positive, when that happens we regularize it by adding a multiple of the identity matrix. We report the runtime performances of this minimization procedure for all the examples shown in this chapter in Table 3.1.

Fig. 3.6: Our 3D printing setup. We stretch the fabric and clamp it between a wooden frame and a plexiglas plate, which we a ach to the build plate with magnets (le ). e printer deposits plastic on the surface of the stretched fabric (right).

Fabrication

We now describe the hardware setup we used to produce our physical models, shown in Figure 3.6.

Our substrate is an elastic polyurethane fabric. We used a stretch factor of 𝑠 = 1.35 (in both directions) in all of our experiments. To hold the fabric in place under tension, we built a custom, 26 by 26cm wooden frame and clamp the fabric between the frame and a plexiglass plate of similar dimensions (see Figure 3.6), which we x to the print bed with magnets. To con rm that the fabric has been installed in its frame at the proper tension, we use a fabric marking pencil to draw a square with side length 10 cm near the center of the fabric before stretching it, and check that the square has side length 10𝑠 cm once the fabric is clamped in its frame.

In contrast to prior work [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF] that used standard PLA printing lament, we performed our experiments using a more exible TPU95A plastic to allow ner-grained control and higher range of star bending sti ness. We employ a UltiMaker2+ printer, which performs well with exible lament. e printer must be calibrated to account for the thickness of the fabric and plate, e ectively raising the height of the print bed. Indeed, we observed that without proper calibration to ensure that the nozzle begins directly above the fabric when extruding the rst star layer, the melted plastic does not stick well to the fabric.

e material properties of printing lament can vary widely even between di erent spools of the same material purchased from the same manufacturer. To avoid relying on inaccurate material parameters from a material datasheet, we used Equation 3.17 to directly calibrate the relative Young's modulus of the plastic with respect to the fabric. We printed 4 × 10 strips of plastic of known thicknesses ℎ 1 ranging from 0.3 mm to 0.45 mm over the fabric, and trimmed the fabric around each strip to produce a beam whose cross-section is a fabric-plastic bilayer. is beam exes out of plane into a circular arc of approximately constant curvature; we photographed the side view (see Figure 3.4, le ) and estimated the beam's natural curvature 𝜅 from the dimensions of the bounding rectangle in the image plane. We then carefully removed the bo om fabric layer from the plastic rod and measured its rest curvature κ the same way (see Figure 3.4, right). Since we know the fabric's thickness ℎ 2 = 0.8 mm from measurements, the stretch factor 𝑠 = 1.35, and the thickness ℎ 1 of each printed plastic strip, we use linear regression and Equation 3.17 to deduce the ratio of Young's moduli 𝐸 1 /𝐸 2 = 3.2 × 10 3 .

Evaluation

We rst present several simulation experiments that illustrate how our model behaves with typical parameter se ings. We then demonstrate the potential of our approach for architectural modeling by designing and fabricating a variety of freeform surfaces.

E ect of parameters. As described in section 3.2, the amount of contraction of the surface is in uenced both by the arm length ℓ and thickness ℎ of the stars. Figure 3.7 illustrates the e ect of these two parameters when we vary them according to a linear radial gradient over a hexagonal domain. Se ing ℓ smaller at the boundary of the domain and larger in the center gives more room for contraction at the boundary than in the middle, inducing the surface to buckle into a dome-like shape. A similar e ect is achieved by using thinner stars on the boundary. In contrast, using long or thick stars at the boundary prevents contraction, so that the surface shape near the center is a er. Using thick stars also results in smoother surfaces overall, while thin stars decorate the surface with multiple small bumps. While both thickness and arm length can be varied at the same time, we observed that ℓ usually has a greater e ect on curvature, and so we used a xed thickness of ℎ = 0.3 mm for all other results. We also experimented with variations in arm width 𝑤, but this parameter had less impact that ℎ on the surface shape -consistent with the fact that the bending sti ness of the star arms scales cubically in the arm thickness, but only linearly in its width. Equipped with the ability to vary surface contraction locally, we can achieve di erent amounts of curvature by adjusting the rate of change of surface contraction. Figure 3.8 illustrates this control on a hexagonal domain xed at its boundary, where we vary the size of the stars according to radial gradients of di erent pro les to produce surfaces ranging from a bell shape to a cone. For illustration purposes, we use a stronger stretching factor 𝑠 = 1.75 in this experiment.

Example designs. Figure 3.9 showcases several architectural models that we created with our approach. For each result, we provide the input star pa ern, the simulated surface, and a picture of the fabricated model. In all cases, we observe a close agreement between our simulation and reality. e rst two rows on top show singly-curved, cylinder-like shapes with rst a curved cylindrical tunnel composed of a uniform eld of stars which we xed to the ground along two of its edges to prevent it to fold on itself, and second a cylindrical section with varying star sizes supported by six needles to model a small roof. e third row shows the result of a layout where large stars form a U-shaped boundary, while smaller stars contract the surface in its center. e resulting surface takes the shape of an amphitheater. Only points along the U-shaped boundary were xed. e last row shows a doubly-curved surface that we obtained using a hexagonal domain where we made the stars smaller in the middle and towards its three xed corners. e three other corners are covered by bigger stars, which prevent them to contract as much as their surrounding, yielding buckling. We also applied a textile strengthener (Powertex) so that this surface supports its own weight.

Printing our models took 20 minutes on average, see Table 3.1 for simulation runtimes.

Discussion.

e examples shown demonstrate that the speci c pa ern of disjointed stars we employed was well-suited to create positively curved surfaces and thus move away from the appearance of minimal surfaces that is prevalent in general tensioned structures. e tiling pa ern of stars was exible enough so that, by varying the size of the stars, we were able to create a variety of shapes, both singly-and doubly-curved. To aid in the design and shape exploration of this self-shaping textile medium, I developed a form-nding tool which allowed for easy visualization of the resulting shape. e formnding method was inspired from existing work [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF] but also modeled more complex behavior such as the bilayer e ect of the plastic-fabric composite assembly. is formnding method shows good agreement with the printed results, but the method may not be accurate for simulating some pa erns di erent than tilings of stars. In the next chapter I will present a general-purpose simulation method for form-nding deployable, printed-on-fabric structures which is intended to work well on a variety of printed pa erns.

Star pa ern

Simulation Fabrication Fig. 3.9: Various architectural models created using our system, from top to bo om: a U-shape, a tunnel and a doubly-curved roof. For each result we provide the star layout to be printed (le ), the simulated surface (middle), and the printed result (right). Making simplifying assumptions about the behavior of materials is common practice when implementing simulation tools. Even the simple act of choosing a scale at which to observe a given physical phenomenon naturally leads to ignoring certain e ects because they are not observable at the considered scale. ese simplifying assumptions are o en necessary as they allow to focus the computational e ort on the phenomena which actually contribute to the results, but simpli cations that are valid in one context might not be in another where they would introduce noticeable discrepancies between the actual object and its simulated counterpart. In the previous chapter, two of such simplifying assumptions were made, the fabric was thought to have an almost isotropic behavior and the rods' width was assumed to be negligible compared to their length. In this chapter, we are interested in developing a simulator that is not only capable of predicting the deployment of the shapes presented before, but also more general pa erns of rod networks, general parallel curves, or even pa erns which do not look like curves at all (see examples below). With that goal in mind, we can revisit those assumptions and see if they are still valid approximations.

Guberan and Clopath [51]

Erioli and Naldoni [START_REF] Erioli | informed exible ma er workshop[END_REF] Fields [START_REF] Fields | Self forming structures: An exploration into 3D printing on pre-stretched fabric[END_REF] Impact of width In the previous chapter -and similarly to Pérez et al. [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF] -we used the Discrete Elastic Rods (DER) method to simulate the plastic curves. e DER method is based on a one-dimensional, reduced centerline representation which models the cross-section of the rods implicitly in the constitutive equations but does not explicitly model its geometry nor the impact it can have on the underlying substrate. is geometric representation proved to be e ective in the case of sparse networks of connected curves [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF] as well as in our case of 3-pointed stars. However, ignoring the width of curves can be detrimental when one wishes to modulate the retraction of the fabric in order to create some metric distortion. is can be easily seen by considering a simple pa ern of parallel straight ribbons printed on a fabric that was stretched uniaxially along the 𝑥-axis. If the ribbons are modeled as thin curves, the fabric will be free to retract back to its original width 𝑙 0 :

However, if the ribbons have their surface properly modeled, the fabric will only be able to properly retract in the black regions where it is not impeded by the white plastic. e nal width will therefore be larger than its initial width 𝑙 0 .

It is therefore necessary to have a proper surface model for the printed pa erns so that the retraction of the fabric can be tracked accurately. A surface representation for the plastic has also the advantage of being able to generalize to pa erns which are not curves such as hexagon tilings [START_REF] Fields | Self forming structures: An exploration into 3D printing on pre-stretched fabric[END_REF].

Orthotropic behavior For the case of 3-pointed stars in the previous chapter, the fabric was modeled as an isotropic membrane which means that the simulated textile had the same properties in every direction. erefore, the simulation was essentially rotationindependent: no ma er how the stars were oriented, the results would be the same. And indeed we can observe experimentally that the approximation was valid in that case: when printing individual stars or groups of stars in di erent orientations, the results are qualitatively the same (see inset). But this is not the case for all pa erns, a pa ern of parallel straight ribbons for example might give very di erent results depending on its orientation. I did an experiment where I printed two sets of parallel ribbon pa erns with di erent thicknesses. e rst set (le ) was printed in one orientation, while the second (right) was printed at a 90°angle. e resulting shapes bent along two di erent axes:

A possible explanation for this experimental result would be that the samples have two di erent bending modes, one along the ribbons and one across, and which one dominates the other depends on the tensile forces applied by the surrounding fabric to the ribbons.

In that case, the fabric having an orthotropic response -which means it has possibly di erent responses when rotated by a 90°angle -could explain this surprising result.

Materials which have a one-axis re ection symmetry are known to exhibit orthotropic properties [START_REF] Schumacher | Mechanical characterization of structured sheet materials[END_REF], and indeed looking at the structure of a typical elastic textile (which are usually kni ed following speci c stitch pa erns for be er compliance) we can see the kni ing pa ern exhibits such a re ection symmetry (see Fig. 4.1).

For a simulation method to work on as many printing-on-fabric scenarios as possible, it is therefore necessary to take into account both the exact width of the printed curves and the direction-dependent behavior of the stretched fabric. However, to the extent of my knowledge, there is no simulator for self-shaping textiles which takes into account both of these e ects. As I mentioned before, both our method from Chapter 3 and the previous work of Pérez et al. [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF] used a one-dimensional representation for the plastic curves coupled to an isotropic membrane model. e work of Stapleton et al. [START_REF] Sco | Finite element modeling to predict the steady-state structural behavior of 4D textiles[END_REF] who used the FEM so ware Abaqus to simulate a simple rectangular shape goes further: by modeling explicitly the surface of the plastic as well as the fabric, their model seems be er suited to accurately track local metric variations induced by the width of the printed lines. However, they still modeled the fabric as an isotropic material which can be problematic, as highlighted above.

Overview. In this chapter, I will present preliminary work towards developing a general method to simulate the nal deployed shape of printed-on-fabric pa erns, by modeling both the nite width of the printed curves and the non-linear, orthotropic response of the fabric. e method is based on a physically accurate shell simulator which naturally takes into account the geometry of printed ribbons because of its surface representation, and is tailored to reproducing bilayer e ects such as the one encountered when a aching a rigid material to a pre-stretched substrate [START_REF] Wim | Growth pa erns for shapeshi ing elastic bilayers[END_REF]. Instead of computing explicitly the curvature of the plastic-fabric bilayer and se ing it as the rest curvature in a rod-based framework, our shell simulator directly integrates the bilayer e ect within its constitutive equations. erefore, this shell simulation method takes into account both intrinsic behaviors such as metric frustration due to the printed rods having nite width, and extrinsic behaviors such as bilayer e ects due to the layers of plastic and fabric having di erent reference geometries.

To give accurate results, the simulator needs to be properly calibrated: the geometry of the nal deployed shape can be very sensitive to changes in material properties such as Young's modulus of the di erent materials and in geometric properties such as the thickness of the di erent layers. We performed extensive measurements on two materials: an elastic textile material which can be used for printing-on-fabric on one hand, and a exible lament used for 3D printing on the other hand. Using these measurements, I developed a data-driven material law which reproduces well the high-strain behavior of the stretched fabric when it is being released.

In summary, the two main contributions of this chapter are (1) a general-purpose simulation method able to accurately reproduce a variety of printed-on-fabric designs, and (2) a set of experimental results used to calibrate this simulation tool to particular materials. is chapter is part of an ongoing collaboration between Victor Romero from Inria Grenoble, Etienne Vouga from UT Austin, and my supervisors Adrien Bousseau and Mélina Skouras. Victor Romero helped me perform the physical experiments described in the rest of this chapter.

Background

Bilayer shell simulation. Van Rees et al. [START_REF] Wim | Growth pa erns for shapeshi ing elastic bilayers[END_REF] proposed a shell simulation method capable of modeling combinations of layers with di erent metrics by de ning an elastic energy inner product. ey showed how to compute the rst and second rest fundamental forms of the bilayer as a function of the rest fundamental forms of the individual layers. Chen et al. [START_REF] Chen | Physical simulation of environmentally induced thin shell deformation[END_REF] used this method to simulate environmental e ects such as moisture or temperature gradients. We adopt this framework to model the bilayer formed by the combination of plastic and fabric whose respective rst fundamental forms and material properties can be combined into a reduced shell representation (see Section 4.3). Even though this model was based on a small-strain analysis, we show how to adapt it in our case where the fabric is stretched well past its linear regime.

Fabric simulation. Computer graphics has had a long history of trying to simulate textiles, notably because of the need to realistically animate the clothes of virtual characters. In that context, the deformations are mostly restricted to the small-strain regime and approximate models such as the popular approach of Bara and Witkin [START_REF] Bara | Large steps in cloth simulation[END_REF] were shown to work well in many cases.

However, the textiles are subject to extreme deformation in our case, for the simulation to match real-life experiments we need to accurately reproduce the full stress-strain curve and not just its derivative at 0. Unfortunately, it is impossible to derive a simple stressstrain relationship for say, kni ed fabrics, because fabric are heterogeneous materials made from a multitude of bers arranged into threads. erefore it is di cult to describe the material response of textiles in terms of continuum mechanics, as we can do for e.g. rubbers with the neo-Hookean or Mooney-Rivlin models. Instead, there are two popular options: either simulate the fabric at a smaller, micro-or meso-scale, or use a data-driven material model optimized for a given fabric material. Simulating at smaller scales is the approach of yarn-based methods [START_REF] Cirio | Yarnlevel simulation of woven cloth[END_REF], the idea is to simulate the textile at the level of individual threads where each thread is modeled as an elastic rod [START_REF] Bergou | Discrete elastic rods[END_REF] and contact and friction between threads is tracked as well. Unfortunately, this type of method can become prohibitively expensive when considering a large area of fabric because of the number of contact points involved. Homogenization-based methods can mitigate the issue by combining the level of detail of yarn-based methods with the speed of shell simulations [START_REF] Sperl | Homogenized yarn-level cloth[END_REF] but they still need proper calibration and measurements to be physically accurate.

Another approach is to directly measure the elastic response of the material and derive a data-driven material model from those measurements. A way to do it is by measuring so-called stress-strain curves, expressing the 2nd Piola-Kirchho stress 𝝈 as a nonlinear function of the Green strain 𝜺 = 1 2 (𝐹 T 𝐹 -𝐼 ), where 𝐹 is the deformation gradient. Volino et al. [START_REF] Volino | A simple approach to nonlinear tensile sti ness for accurate cloth simulation[END_REF] showed that we can express the entries of the stress matrix 𝜎 𝑖 𝑗 as a nonlinear function of the strains 𝜀 𝑖 𝑗 to derive an orthotropic data-driven material model. We take inspiration from their method, even though they do not take into account transverse e ects, i.e. couplings between strains in both directions due to the Poisson e ect.

Material model

Our printed-on-fabric metamaterial is modeled as an inhomogeneous shell which is comprised of two di erent materials: a fabric material and a bilayer material, i.e. each point on the surface is either in a fabric-only area, or a bilayer area (see Fig. 4.2). Both materials are modeled as thin shells but with di erent material models: the bilayer shell model is based on the work of van Rees et al. [START_REF] Wim | Growth pa erns for shapeshi ing elastic bilayers[END_REF] which models the curvature of bilayers by looking at metric di erences between the two layers; and the fabric is modeled using a custom data-driven material model which is ed against the experimental data described in section 4.4.

Bilayer material Similar to the previous chapter (Eq. 3.9), the elastic energy of a bilayer composed of two materials with rest rst fundamental forms a 𝑟 1 and a 𝑟 2 , Young's

Fabric material

Bilayer material

Fig. 4.2: Example of modeling a 3-pointed star shape similar to the ones in Chapter 3: the black area corresponds to the fabric and is modeled using a custom data-driven material model, the white area corresponds to a plastic-fabric bilayer modeled using the method of van Rees et al. [START_REF] Wim | Growth pa erns for shapeshi ing elastic bilayers[END_REF].

moduli 𝐸 1 and 𝐸 2 , and thicknesses ℎ 1 , ℎ 2 is de ned as:

𝑊 𝐵𝐿 = 1 2 ∫ 𝑈 𝐸 1 ℎ 1 8 ||a -1 𝑟 1 a 𝑐 -I|| 2 + ℎ 3 1 24 ||a -1 𝑟 1 b 𝑐 || 2 + ℎ 2 1 8 < a -1 𝑟 1 a 𝑐 -I, a -1 𝑟 1 b 𝑐 > √ det a 𝑟 1 d𝑥 d𝑦 + 1 2 ∫ 𝑈 𝐸 2 ℎ 2 8 ||a -1 𝑟 2 a 𝑐 -I|| 2 + ℎ 3 2 24 ||a -1 𝑟 2 b 𝑐 || 2 - ℎ 2 2 8 < a -1 𝑟 2 a 𝑐 -I, a -1 𝑟 2 b 𝑐 > √ det a 𝑟 2 d𝑥 d𝑦 (4.1)
where < A, B >= 𝜈 1-𝜈 2 tr(A) tr(B) + 1 1+𝜈 tr(AB) is the elastic inner product associated to a material with Poisson's ratio 𝜈, and

||A|| 2 =< A, A >= 𝜈 1-𝜈 2 tr 2 (A) + 1 1+𝜈 tr(A 2
) is the elastic energy norm.

a 𝑟 1 and a 𝑟 2 are respectively the rst fundamental forms of the fabric and plastic layers. Depending on how the fabric was stretched, a 𝑟 1 will have di erent entries. Two use cases are common when fabricating self-shaping textiles, the rst case is uniaxial strain, meaning the fabric was stretched only in one direction, the second case is isotropic stretch which means the fabric was stretched equally in both directions. If we call 𝑠 the ratio of lengths between the undeformed and the deformed state, we will have a 𝑟 1 = 1 𝑠 2 0 0 1 for uniaxial strain and a 𝑟 1 = 1 𝑠 2 I for isotropic strain. Since the plastic is initially strain-free, its rst fundamental form corresponds to the identity matrix: a 𝑟 2 = I a 𝑐 and b 𝑐 are respectively the rst and second fundamental forms of the current con guration. We discretize Equation 4.1 so that they are constant on each triangle. Following Chen et al. [START_REF] Chen | Physical simulation of environmentally induced thin shell deformation[END_REF], we compute a 𝑐 on the triangle formed by vertices 𝑣 𝑖 , 𝑣 𝑗 , 𝑣 𝑘 as:

a 𝑐 = 𝑣 𝑗 -𝑣 𝑖 2 (𝑣 𝑗 -𝑣 𝑖 ) • (𝑣 𝑘 -𝑣 𝑖 ) (𝑣 𝑗 -𝑣 𝑖 ) • (𝑣 𝑘 -𝑣 𝑖 ) 𝑣 𝑘 -𝑣 𝑖 2
b 𝑐 is computed with the "triangle with aps" stencil of Grinspun et al. [START_REF] Grinspun | Computing discrete shape operators on general meshes[END_REF]:

b 𝑐 = (𝑛 𝑗 -𝑛 𝑖 ) • (𝑣 𝑗 -𝑣 𝑖 ) (𝑛 𝑗 -𝑛 𝑖 ) • (𝑣 𝑘 -𝑣 𝑖 ) (𝑛 𝑘 -𝑛 𝑖 ) • (𝑣 𝑗 -𝑣 𝑖 ) (𝑛 𝑘 -𝑛 𝑖 ) • (𝑣 𝑘 -𝑣 𝑖 )
𝑛 𝑖 , 𝑛 𝑗 , 𝑛 𝑘 are normals de ned on the edge opposite 𝑣 𝑖 , 𝑣 𝑗 , 𝑣 𝑘 respectively, and computed by averaging the normals of their adjacent faces.

Equation 4.1 gives the elastic energy of a bilayer composed of homogeneous materials with the same Poisson's ratio 𝜈. In our case, we use the average value of the Poisson's ratios of the two layers: the fabric's Poisson's ratio is measured from videos of a uniaxial tensile test as the ratio of transverse contraction to axial strain and is estimated to be about 0.3; the plastic material used (thermoplastic polyurethane) is known to be almost incompressible [START_REF] Hang | Stress-strain behavior of thermoplastic polyurethanes[END_REF] which means a value of 0.5.

Moreover, the fabric layer is not isotropic, we will see in section 4.4 that we can compute its Young's modulus 𝐸 1 by averaging the tangent of the stress-strain curve upon unloading over a sampling of di erent orientations.

Fabric material model As is standard for modelling cloth [START_REF] Miguel | Data-driven estimation of cloth simulation models[END_REF][START_REF] Wang | Data-driven elastic models for cloth: Modeling and measurement[END_REF], we separate the bending and membrane contributions of the energy:

𝑊 𝐹 = 𝑊 bending + 𝑊 membrane . For the bending contribution, as in the previous chapter (Eq. 3.1) we use the Discrete Shell model of Grinspun et al. [START_REF] Grinspun | Discrete shells[END_REF]:

𝑊 bending = 𝑘 𝐵 ∑ 𝑖 3 𝑒 𝑖 2 Ā𝑖 2 tan 𝜃 𝑖 2 2
For the membrane energy, since textiles are inhomogeneous materials made from a multitude of bers, it is di cult to describe their material response with a simple law derived from continuum mechanics as is possible for e.g. rubbers with the neo-Hookean or Mooney-Rivlin models. Instead, we de ne a custom parametric material model designed to reproduce well the nonlinear response of the fabric especially when released a er a given amount of pre-stretch. For each triangle we can write the elastic energy as a function of its Green strain tensor wri en in Voigt notation as 𝜺 = 𝜀 11 𝜀 22 2𝜀 12 T :

Ψ(𝜺) = 1 1 -𝜈 2 𝜺 T 𝛼 1 √ 𝛼 1 𝛼 2 𝜈 0 √ 𝛼 1 𝛼 2 𝜈 𝛼 2 0 0 0 𝛼 3 (1 -𝜈 2 ) 𝜺 -𝜑 1 (𝜀 11 ) -𝜑 2 (𝜀 22 ) (4.2)
where 𝜑 1 and 𝜑 2 are symmetric log-barrier functions:

𝜑 𝑖 (𝜀 𝑖𝑖 ) = 𝛽 𝑖 (𝜀 𝑖𝑖 -𝛾 𝑖 ) log 𝛾 𝑖 -𝜀 𝑖𝑖 𝛾 𝑖 -𝜀 𝑖𝑖 if 𝜀 𝑖𝑖 > 0, = -𝛽 𝑖 (𝜀 𝑖𝑖 + 𝛾 𝑖 ) log 𝛾 𝑖 + 𝜀 𝑖𝑖 𝛾 𝑖 -𝜀 𝑖𝑖 if 𝜀 𝑖𝑖 < 0. (4.3)
and

𝑊 membrane = ℎ ∑ 𝑖 𝐴 𝑖 Ψ(𝜺 𝑖 ).
In the small strain regime, 𝜑 𝑖 ≈ 0 and the model has the form of an orthotropic StVK energy [START_REF] Li | Stable orthotropic materials[END_REF], it then increases exponentionally in the high strain regime to match the measured stress-strain curves in section 4.4. Since the function is symmetric with respect ot the yaxis, it also penalizes exponentionally negative strains which is useful to prevent triangle inversions. e model is parameterized by seven coe cients: 𝛼 1 , 𝛼 2 , 𝛼 3 , 𝛽 1 , 𝛽 2 , 𝛾 1 , 𝛾 2 which are optimized to match the experimentally measured response, and a Poisson's ratio 𝜈 measured experimentally.

Measurements and ing

In this section, I will describe the di erent experiments we conducted and how we used these measurements to calibrate the simulation model presented above. We considered two materials: an elastic textile which is representative of the kinds of materials used for 3D printing on stretched fabric, and a exible 3D printing plastic lament. e textile is a nely-kni ed lycra fabric composed of 80 % polyamide and 20 % elastane, while the printing lament material is a thermoplastic polyurethane known as Ultimaker TPU 95A. We conducted three types of measurements: uniaxial stretch tests (performed on both the fabric and plastic material), shear tests and cantilever tests (performed on the textile only).

Uniaxial stretch

For this test, samples were prepared in a standard dogbone shape, the plastic was 3D printed into a 15cm long by 1cm wide shape, while several fabric samples were laser cut out of a 11cm long by 2cm wide template. e fabric samples were cut out at di erent orientations to measure the orthotropy of the material, we measured the tensile response of samples oriented at 0, 15, 30, 45, 60, 75, and 90°respectively (measured from the horizontal axis in Fig. 4.1). e tensile measurements were made using an Instron 5865 machine with a 50 N force sensor which tracks both the displacement 𝑑 of the clamped endpoints and the force applied 𝑓 (called the response of the material). ese force-displacement curves then have to be converted to stress-strain curves in order to be able to extract useful calibration information such as the Young's modulus of the materials. We convert these values into entries of the second Piola-Kirchho stress tensor 𝝈 = 𝜕Ψ 𝜕𝜺 and the Green strain tensor 𝜺 which are accurate quantities for large strains. For uniaxial stretch, the diagonal entries of these 2 × 2 matrices can be computed from the force-displacement curves of either the 0°or 90°samples:

𝜀 𝑖𝑖 = 𝑑 𝐿 + 𝑑 2 2𝐿 2 𝜎 𝑖𝑖 = 1 𝑤 𝑓 𝐿 𝑑 + 𝐿 (4.4)
the 𝑖 = 1 values can be computed from the 0°sample and the 𝑖 = 2 values can be computed from the 90°one. e testing machine stretched and released the samples by performing load-unload cycles, which created loops characteristic of a hysteresis behavior when plo ing the stress-strain curves. is hysteresis, or pathdependent behavior, is likely caused by the internal friction between bers of the fabric which rearrange as the textile gets stretched. Since we want to model the behavior of the textile once it has been stretched and gets released, we are really interested in the unloading part of the curve (in green in Fig. 4.3).

We rst use the stress-strain curves of the plastic and the di erent fabric samples to calibrate the bilayer material model (Equation 4.1), and in particular, to measure the tangent at 0 measured data Fig. 4.4: le : stress-strain curve for the plastic material, the slope of its tangent at 0 is the material's Young's modulus; right: stress-strain curve for the fabric material, here we are interested in the slope of the tangent upon unloading.

Young's moduli 𝐸 1 and 𝐸 2 of the two layers. Classically, the Young's modulus is the slope of the tangent at 0 of the stress-strain curve, it is a linear approximation of the stress-strain behavior for small strains, we can therefore compute the plastic's Young's modulus 𝐸 1 by nding the tangent at zero of its stress-strain curve (Fig. 4.4, le ). For the fabric substrate however the situation is a bit di erent, since it was pre-stretched when the plastic-fabric bilayer was formed, it does not make much sense to consider an approximation of its response for strains close to zero since the plastic layer prevents it from ever going back to its original length. Instead, the linear approximation should start from the maximum strain value, 𝐸 2 is therefore computed as a pseudo "Young's modulus" by nding the slope of the tangent at the point of release (Fig. 4.4, right). Since this value is measured at the point of unloading, it may depend on how far the sample was stretched, to account for a dependency on the initial strain of the fabric, we performed two tests in which the fabric samples were stretched up to 45 % and 70 % of their initial lengths, and linearly interpolate between the two measured values to simulate self-shaping textiles pre-stretched with a di erent strain value.

We also use the uniaxial stretch data to nd the coe cients of the fabric material model. We can compute the stress tensor by di erentiating Equation 4.2 (assuming 𝜀 𝑖𝑖 > 0): of 𝜀 22 :

𝝈 = 𝜕Ψ 𝜕𝜺 = 1 1 -𝜈 2 𝛼 1 √ 𝛼 1 𝛼 2 𝜈 0 √ 𝛼 1 𝛼 2 𝜈 𝛼 2 0 0 0 𝛼 3 (1 -𝜈 2 ) 𝜀 11 𝜀 22 2𝜀 12 - 𝛽 1 log 𝜀 11 -𝛾 1 𝛾 1 𝛽 2 log
𝜎 11 = 1 1 -𝜈 2 𝛼 1 - √ 𝛼 1 𝛼 2 𝜈 2 𝜀 11 -𝛽 1 log 𝜀 11 -𝛾 1 𝛾 1 (4.6) 𝜎 22 = 1 1 -𝜈 2 𝛼 2 - √ 𝛼 1 𝛼 2 𝜈 2 𝜀 22 -𝛽 2 log 𝜀 22 -𝛾 2 𝛾 2 . (4.7)
We then t a function of the form 𝜎 𝑖𝑖 (𝜀) = 𝑎𝜀 𝑖𝑖 -𝑏 log 𝜀 𝑖𝑖 -𝑐 𝑐 which allows to directly retrieve 𝛽 1 , 𝛽 2 , 𝛾 1 and 𝛾 2 . For 𝛼 1 and 𝛼 2 , solving a simple nonlinear system allows to compute them as a function of 𝜈. ese parameters also depend on how much was the fabric stretched, we can also linearly interpolate the ed values for di erent amounts of stretch as was done for 𝐸 2 .

Shear tests

e last coe cient needed for the membrane parametric model is 𝛼 3 which corresponds to the shear modulus. To measure this coe cient we clamp two parallel edges of a square sample and move one of its edges laterally while probing the response (Fig. 4.5, le ). We performed the measurement twice on the same sample in two di erent orientations, 90°from each other. e results at 0°and 90°showed an approximately linear response which corresponds well with the choice of expressing 𝜎 12 as a linear function of 𝜀 12 in the parametric model. e two shearing experiments correspond to the same strain value 𝜀 12 and therefore should give the same stress 𝜎 12 as a result. However, a surprising experimental result was that the response was not the same at a 0°and 90°orientation (Fig. 4.5, right), this could either be due to errors in the experimental process or to more Matching measurements of cantilevered shapes against the "master curve" of Romero et al. [START_REF] Romero | Physical validation of simulators in computer graphics: A new framework dedicated to slender elastic structures and frictional contact[END_REF].

complex e ects occurring within the fabric which our model cannot reproduce. Another possibility is that the curves are diverging a er the small strain regime because of the orthotropy of the material: at large strains, the deformation is not pure shearing anymore but a mix of shearing and stretching, and since the stretching response is direction dependent this could explain the deviation between the curves. Since our model cannot explain this divergence between the two curves, we compute 𝛼 3 as the average linear regression of each curve.

Bending tests

e last coe cient needed for the fabric's material model is the exural coe cient 𝑘 𝐵 of the bending energy. is coe cient is usually expressed as a function the Young's modulus 𝐸, the thickness ℎ and the Poisson's ratio 𝜈 for a homogenous material:

𝑘 𝐵 = 𝐸ℎ 2 12(1 -𝜈 2 )
but we can directly measure it by performing a cantilever test in which a fabric ribbon sample is clamped horizontally and a length 𝐿 of itself is submi ed to gravity. Romero et al. [START_REF] Romero | Physical validation of simulators in computer graphics: A new framework dedicated to slender elastic structures and frictional contact[END_REF] show the relationship between a unitless gravito-bending parameter Γ = 𝜌𝑔 𝑘 𝐵 𝐿 3where 𝜌 is the mass density and 𝑔 is the acceleration of gravity -and the aspect ratio 𝑦/𝑥 𝐸 of the bounding rectangle of the cantilevered sample (see Fig. 4.6, inset). is relationship is universal in the sense that the pair (Γ, 𝑦/𝑥) will always be on a speci c curve, called the master curve, no ma er what the material properties of the sample.

To nd the value of 𝑘 𝐵 , we therefore measure 𝑦/𝑥 for di erent values of 𝐿 and nd the coe cient 𝑘 𝐵 such that the di erent points ( 𝜌𝑔 𝑘 𝐵 𝐿 3 , 𝑦/𝑥) are as close as possible to the curve in the least squares sense. We performed the test on two di erent orientations of the fabric (0°and 90°) and tested them both front side up and back side up, for a total of 4 di erent experiments. e results (Fig. 4.6) show a di erence in 𝑘 𝐵 between the 0°and 90°o rientation, which is not surprising given the structure of the kni ed textile (Fig. 4.1). e di erence between bending front side up and back side up was found to be negligible in the 0°case -meaning the resistance to bending is essentially symmetric in that direction -but for the 90°case the exural coe cient is almost 3 times bigger on one side compared to the other, which can be explained by the fact that kni ed textiles in general do not exhibit mirror symmetry between their front and back sides and therefore can have fairly di erent responses between bending upwards and downwards. e Discrete Shells model that we use to model bending does not account for these direction-dependent e ects and is only weighted by one exural coe cient (𝑘 𝐵 ), therefore we compute 𝑘 𝐵 as mean value between the 4 measured ones. All the di erent parameters used in the bilayer model (𝐸 1 , 𝐸 2 ), the membrane energy (𝛼 1 , 𝛼 2 , 𝛼 3 , 𝛽 1 , 𝛽 2 , 𝛾 1 , 𝛾 2 ) and the bending energy (𝑘 𝐵 ) are summed up in table 4.1.

Preliminary results and validation

To validate our simulation method, I performed a series of basic tests, comparing the simulation results against experimental data and simulating simple shapes. To check that the fabric's material model reproduces well the measured behavior, I ran several simulations in which a rectangle was uniaxially stretched. Depending on how the rectangle was oriented in global coordinates, the integrated forces along each edge were di erent and matched well with the experimental data (Fig.

Discussion and limitations

e goal of this chapter was twofold: be er understanding, by means of physical experiments, the mechanics of pre-stretched textiles when released and their interaction with thermoplastics printed on top, and deriving a simulation model from those measurements and experiments which should be both accurate and e cient.

At the moment, this method is still very much work in progress and needs to be validated against a variety of printed pa erns to assess its accuracy. To check that the bending behavior caused by the bilayer e ect is modeled correctly, we will compare the curvature of simulations of parallel ribbon pa erns such as the one shown in Fig. 4.8 with the measured curvature of their fabricated counterparts. To check that the intrinsic contraction behaves similarly to printed pa erns, we can print a pa ern from next chapter which reproduces half a torus by having a lower density of curves on the inside of the a ened shape and a higher density of curves on the outside (Fig. 5.2). A er performing these validations we will try to simulate our own printed pa erns and a variety of other curve layouts inspired from previous work (such as the self-shaping textiles shown in introduction).

However, we mentioned in chapter 3 that the printed plastic material exhibited a certain amount of plasticity (Fig. 3.4) which has not yet been taken into account in a systematic way in the model, and therefore might cause the simulated results to be sti er than the printed ones, this could prevent the method to accurately reproduce extrinsic curvature in particular. Another issue with the method is the amount of experimental data it requires to nd the parameters for given plastic and fabric materials, it would be useful to have a way to easily calibrate the simulation method to other unseen materials (provided they are not too di erent). One way to do this would be to propose a set of tests that would each be used to measure a speci c parameter. For example, measuring the curvature of an individual bilayer ribbon can be used to derive the Young's moduli ratio 𝐸 1 /𝐸 2 or measuring a series of parallel ribbons at di erent orientations gives information about the orthotropic fabric response. e previous chapters were about the form-nding of self-shaping textiles created by 3D printing on stretched fabric: given a speci c pa ern to be printed, the goal of the form-nding is to nd a deployed shape which satis es an equilibrium of forces between the pre-stretched fabric and the more rigid plastic printed on top. In this chapter we tackle the inverse problem instead: given a target surface that we want to reproduce, the goal is to nd an optimal layout to be printed so that the deployed shape is as close as possible to the input. is is a di cult problem as it amounts to nding a mapping between a 3D surface and a a ened one which preserves the dimensions of all the printed shapes, since the plastic is essentially incompressible during deployment.

To solve this problem, we focus on printing pa erns of almost-parallel ribbons on both sides of a pre-stretched elastic textile.

is pa ern allows to control both the metric by modulating the density of printed ribbons, and extrinsic curvature via the bilayer e ect (Figure 5.1). We examine the interplay of geometry and physics that governs the behavior of the ribbon-fabric ensemble, and use those insights to develop an algorithm for inverse design and a fabrication pipeline for approximating a given freeform surface by a self-actuating deployable structure.

Relation to previous plastic-on-fabric techniques Several recent methods have experimented with the idea of printing plastic on fabric to rapidly prototype lightweight curved surfaces [START_REF] Guberan | Active shoes[END_REF][START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF]. e main di erence of our approach is that we propose to print dense, broken ribbons of plastic, rather than the sparse networks of closed curves found in prior work.

Forces over empty fabric

Forces along a ribbon

While both strategies involve printing plastic on fabric, there are deep di erences in the mechanics of how the plastic-fabric ensemble buckles, and in the ability of the ribbons to control the shell geometry. Speci cally, in a sparse assembly of long ribbons (as found in prior work), most of the shape change during self-actuation is the result of compression of the pre-stretched fabric in the regions in between the curves. e curves buckle and bend to release this stress [START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF] (inset, top), and serve as boundary conditions for the fabric which shrinks to a Plateau-like surface. On the one hand, this mechanism allows for signi cant coarse-scale shape change with only a small amount of added plastic material.

On the other hand, the sparse ribbons provide only limited control over the geometry of the fabric patches (which always have negative intrinsic and approximately zero extrinsic curvature).

In contrast, in our dense assemblies, the plastic ribbons cover a signi cant fraction of the fabric surface area: not only does the shrinking fabric exert in-plane stress on the plastic ribbons, but bilayer e ects become important in the regions covered by plastic. As the fabric relaxes and shrinks to relieve stretch, it exerts stress on the plastic along the plastic-fabric interface. Since the plastic is incompressible yet elastic, the ribbon buckles to form an arc at the equilibrium state (inset, bo om). e curvature of this arc depends on the pre-stress of the fabric as well as on the thicknesses of the elastic and plastic layers. At the coarse scale, our dense assemblies thus form a type of metamaterial, whose geometry is directly controlled by the spacing, orientation, and thickness of the plastic ribbons. Given a sheet of fabric with constant pre-stress, our key idea is to program 𝜎 2 Fig. 5.2: Overview of our method. Given a target surface (a), we rst compute its principal curvatures, along which plastic ribbons will be placed. We compute a a ening of the 3D surface into the plane, with bounded stretching (b; colors indicate the magnitude of stretching 𝜎 2 in the ribbon direction). We then generate a staggered stripe pa ern aligned with these directions; the deformation desired for deploying from the a ened state to 3D determines the spacing and density of the ribbon pa ern (c). We generate plastic ribbons from this pa ern, and adjust their thicknesses according to the target extrinsic curvature magnitude (d). Printing these plastic ribbons on pre-stretched fabric results in a self-actuated assembly that deploys to a shape approximating the target surface when released (e).

the curvature of a target surface by optimizing these parameters to leverage the above mechanisms of shape change.

Note that in contrast to methods that mainly rely on metric frustration to control intrinsic curvature [START_REF] Chen | Bistable auxetic surface structures[END_REF][START_REF] Konaković-Luković | Rapid deployment of curved surfaces via programmable auxetics[END_REF], our approach provides control of extrinsic curvature as well, as we can force the surface to bend along prescribed directions by aligning ribbons with those directions. e ability to manipulate extrinsic curvature is particularly important when fabricating nearly-developable surfaces, when prescribing Gaussian curvature is not enough to reliably achieve the desired shape.

Overview Our inverse design tool takes as input a triangle mesh embedded in R 3 representing the desired target (deployed) surface shape, and computes (1) a a ening of this surface into the plane, and (2) a set of ribbons over this planar domain, so that 3D-printing the ribbons onto fabric with constant pre-stress, and cu ing the fabric along the boundary of the planar domain, yields an assembly whose static shape deploys to match the target surface (see Figure 5.2). e task of programming the desired surface curvature is divided between these two steps: during a ening, we optimize for a ribbon layout that will achieve the desired intrinsic curvature, and aligns with directions of principal extrinsic curvature to prevent torsion of the ribbons. en, we nd the optimal ribbon thickness to obtain the extrinsic curvature magnitude.

In contrast to the direct design tools presented in the previous chapters, the inverse design method presented in this chapter is based on geometric principles rather than physical simulation: we compute the a ening by optimizing for a 2D parameterization of the target surface, subject to certain fabricability constraints:

1. the metric distortion of the parameterization remains below the maximum stretch of the fabric;

2. the metric distortion of the parameterization is close to 1 along the direction of maximum curvature.

e rst condition encodes that the fabric can, at most, remain in its pre-stretched state, and cannot stretch further. e second condition ensures that the assembly will undergo li le compression along directions of high curvature, which allows us to print closelyspaced, incompressible plastic ribbons along those directions to maximally exploit the bilayer e ect to induce the necessary surface curvature. We solve for a a ening that satis es these properties using a local/global iterative algorithm akin to As-Rigid-As-Possible parameterization [START_REF] Liu | A local/global approach to mesh parameterization[END_REF]. While recent methods relied on a similar procedure to bound distortion for other fabrication techniques [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF][START_REF] Pane | Computational inverse design of surface-based in atables[END_REF], we introduce a hard constraint on the direction of distortion to align our plastic ribbons with the principal directions of curvature of the target surface.

We then position the plastic ribbons over the a ened surface to form a dense pa ern with spatially-varying spacing and thickness. As explained above, the resulting pa ern acts as a homogenized metamaterial, where spacing between the ribbons controls the average stretch of the assembly as dictated by the parameterization, while thickness controls the magnitude of directional curvature. We generate G-code for printing this ribbon pa ern by tracing orthogonal families of curves along smooth curvature-aligned direction elds [START_REF] Knöppel | Globally optimal direction elds[END_REF][START_REF] Knöppel | Stripe pa erns on surfaces[END_REF].

Since we use a 3D printer to deposit plastic on top of the stretched fabric, the bilayer e ect strongly biases the surface to curve away from the printer bed, so that the curvature of the assembly a er it has relaxed to its static shape is almost always positive in the direction of the ribbons. We circumvent this limitation by introducing a custom, reversible frame structure that allows us to print plastic on both sides of the stretched fabric. is simple hardware further expands the range of surfaces we can reproduce (Figure 5.11).

Contributions In summary, we introduce:

• a new mechanism for self-actuated structures, based on closely-spaced plastic ribbons printed on pre-stretched fabric, which provides controllable directional distortion and curvature;

• an inverse design tool to fabricate low-cost, lightweight prototypes of freeform surfaces using this mechanism;

• a parameterization algorithm that bounds surface distortions along and across prescribed directions, along with a pa ern synthesis algorithm that covers a surface with ribbons to match the target distortions and curvature given by the aforementioned parameterization;

• a simple hardware setup to reliably print plastic ribbons on both sides of a prestretched sheet of elastic fabric, with minimum manual intervention.

We applied our approach to fabricate a number of freeform surfaces, and evaluate our method by measuring the agreement between the fabricated and target 3D shapes.

is chapter is mainly based on the following publication:

• David Jourdan, Mélina Skouras, Etienne Vouga, Adrien Bousseau, 2022. Computational Design of Self-Actuated Surfaces by Printing Plastic Ribbons on Stretched Fabric.

Computer Graphics Forum (Proc. EUROGRAPHICS, conditionally accepted).

Related work

Out of the various methods for designing deployable structures already introduced in chapter 2, the in atable structures of Pane a et al. [START_REF] Pane | Computational inverse design of surface-based in atables[END_REF], present an interesting case due to their similar approach to solving the inverse design problem. Two thin, at sheets of elastic material are fused along a network of curves, creating air pockets between the layers in the form of tube-like channels. Pressurizing the interstitial space causes the channels to in ate and contract tranversally. At the highest level, the design problem for in atables solved by Pane a et al. shares similar features to the problem we solve for ribbon networks on fabric: in both cases, actuation results from programming anisotropic residual strain in a at sheet, parameterized by a network of curves, and subject to fabricability constraints on the maximum strain. But crucial di erences prevent using the method of Pane a et al. for designing ribbon layouts: in ated channels have zero normal curvature, whereas ribbons exhibit strong extrinsic curvature (due to the bilayer e ect) which can be controlled and must be accounted for; moreover, whereas in ated channels have circular cross-section, plastic ribbon cross-sections are rectangular and a ribbon layout must account for potential axial twisting of the ribbons due to geodesic torsion (see Section 5.3.1 and Figure 5.4).

Surface parameterization is a classic problem in geometry processing [START_REF] Hormann | Mesh parameterization: eory and practice[END_REF], with many available algorithms that seek to minimize angle [START_REF] Sawhney | Boundary rst a ening[END_REF][START_REF] She | Abf++: Fast and robust angle based a ening[END_REF] or length distortion [START_REF] Zhou | Iso-charts: Stretchdriven mesh parameterization using spectral analysis[END_REF], while possibly also considering ancillary complications such as seam placement, local and global injectivity, atlas decomposition, etc. We do not give a complete survey here; note that while conformal mapping in particular is popular for texture mapping and similar applications, and for fabrication techniques that exploit isotropic scaling [START_REF] Guseinov | Curveups: Shaping objects from at plates with tension-actuated curvature[END_REF][START_REF] Konaković | Beyond developable: Computational design and fabrication with auxetic materials[END_REF], we speci cally require an anisotropic parameterization with bounded distortion. Few prior methods approached the problem of computing a a ening whose distortion is bounded in one arbitrary direction [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF]. Closest to our approach is Pane a et al. [START_REF] Pane | Computational inverse design of surface-based in atables[END_REF]'s a ening step, which includes a so penalty term to favor alignment of the channels perpendicularly to the principal direction of curvature. In contrast, we treat curvature direction alignment as a hard constraint, which allows us to build our a ening algorithm on top of As-Rigid-As-Possible (ARAP) parameterization [START_REF] Liu | A local/global approach to mesh parameterization[END_REF], a classic and easily-extendable localglobal algorithm for distortion-minimizing parameterization. See Section 5.3.2 for more discussion of the pros and cons of the two approaches.

e last step of our method consists in generating a pa ern of plastic ribbons over the a ened shape, where the local orientation and spacing of the ribbons is dictated by the curvature and distortion elds of the target surface. Producing such a graded pa ern of discrete, elongated elements arises in other fabrication scenarios, including the design of metal frame and wire structures [START_REF] Ma | Designing robotically-constructed metal frame structures[END_REF][START_REF] Miguel | Computational design of stable planar-rod structures[END_REF], kni ing paths [START_REF] Narayanan | Automatic machine kni ing of 3D meshes[END_REF], microstructures [START_REF] Ibault Tricard | Freely orientable microstructures for designing deformable 3D prints[END_REF], weaves [START_REF] Ren | 3D weaving with curved ribbons[END_REF][START_REF] Vekhter | Weaving geodesic foliations[END_REF] and nets [START_REF] Andrew | Chebyshev nets from commuting polyvector elds[END_REF], ber-reinforced composites [START_REF] Boddeti | Optimal design and manufacture of variable sti ness laminated continuous ber reinforced composites[END_REF], and the aforementioned in atable structures [START_REF] Pane | Computational inverse design of surface-based in atables[END_REF]. Many of the algorithms in geometry processing on vector-eld integration could be used to generate ribbon curves from our curvature-aligned vector eld; see the survey by Vaxman et al. [START_REF] Vaxman | Directional Field Synthesis, Design, and Processing[END_REF] for a review of some of these alternatives.

We chose the method of Knöppel et al. [START_REF] Knöppel | Stripe pa erns on surfaces[END_REF], originally designed for computing stripe textures on surfaces, due to the ease of specifying the frequency of the reconstructed pa ern in the transverse direction, and availability of source code.

Programming intrinsic and extrinsic curvature

In this section, we rst describe the mechanics of plastic ribbons embedded in stretched fabric, then we derive the a ening formulation in the continuous se ing, before discretizing it. We then show how to implement the parameterization algorithm and how to nd the optimal thickness to print so as to match a given normal curvature.

e mechanics of plastic ribbons embedded in stretched fabric

e key idea behind our approach is to balance between two e ects that occur when plastic ribbons are bound to a stretched fabric substrate. First, the ribbons frustrate the contraction of the fabric back to its rest state, forcing residual strain in the fabric and creating intrinsic curvature. Second, as mentioned in the introduction, due to di erential compression of the plastic and fabric layers in the direction of the ribbons, there is a bilayer e ect that induces each ribbon to buckle into an arc when the assembly is released. We seek to exploit both phenomena to program the desired surface curvature. To that end, we de ne a pa ern of plastic ribbons that behaves like a homogeneous metamaterial with controllable stretch and curvature, illustrated as inset. is pa ern will be printed on fabric that has been uniformly stretched by A empting to orient the ribbons away from the direction of maximum curvature still yields a cylinder as ribbons resist torsion (a, bo om). On a curved surface, curvature lines are the only curves with zero geodesic torsion, as visualized by sliding a Darboux frame along the curve and observing its rotation around the curve tangent (b, a er [START_REF] Iarussi | Bend elds: Regularized curvature elds from rough concept sketches[END_REF]). a factor 𝑠. In this pa ern, the plastic ribbons are separated by empty fabric to form dashed stripes. Parallel stripes are also separated by empty fabric. Note that we shi every two stripes by half the period of the dashes to form a staggered pa ern, which prevents the emergence of long bands of empty fabric transverse to the plastic ribbons. Varying the quantity of empty fabric in-between consecutive and parallel plastic ribbons controls the average contraction 1/𝜎 1 and 1/𝜎 2 of the metamaterial when deployed, while varying the thickness 𝜏 of plastic deposited on the ribbons impacts the strength of their bilayer e ect.

Given a target surface S, our key idea is to design an appropriate ribbon pa ern in three stages:

1.

e bilayer e ect induces signi cant extrinsic curvature in the direction of the ribbons, and the plastic ribbons resist torsion along that direction (Fig. 5.4a). Based on these two observations, we align the ribbon pa ern (horizontal axis in Fig. 5.3) in the direction of maximum magnitude principal curvature k 1 , which maximally exploits the bilayer e ect and minimizes torsion along the ribbons since curvature lines are characterized by vanishing geodesic torsion [START_REF] Biard | Construction of rational surface patches bounded by lines of curvature[END_REF][START_REF] Manfredo | Di erential Geometry of Curves and Surfaces[END_REF][START_REF] Iarussi | Bend elds: Regularized curvature elds from rough concept sketches[END_REF] (Fig. 5.4b). It follows that the transverse direction of the pa ern aligns to the other principal curvature direction k 2 .

2. When the plastic-fabric ensemble is released and allowed to relax to equilibrium, the fabric will contract. At most, it will return to its original shape; and at least, it will not contract at all (if the fabric is covered completely in plastic). e fabric pre-stretch factor 𝑠, together with fabricability constraints on the minimum and maximum values of ribbon length and width 𝑙 𝑟 , 𝑤 𝑟 and spacing 𝜇 1 , 𝜇 2 , determine the range of possible contractions in the ribbon and transverse direction. We compute a a ening of the target surface S to the 2D plane, which satis es these constraints (Section 5.3.2).

3. To control extrinsic bending of the surface, we adjust the ribbon thickness 𝜏, using a data-driven law for the relationship between 𝜏 and curvature derived from physical experiments (Section 5.3.5).

Fla ening with bounded directional scaling

Let the mapping 𝜑 : Ω ⊂ R 2 → R 3 describe our (given) target deployed surface S, and let φ : Ω ⊂ R 2 → R 2 describe its (currently unknown) a ened, pre-stretched counterpart S, with Ω denoting an arbitrary surface parametric domain with coordinates (𝑢, 𝑣). We write k 1 (𝑢, 𝑣), k 2 (𝑢, 𝑣) for the vector elds Ω → R 3 in the direction of maximum-magnitude and minimum-magnitude principal curvature of S.

We seek a a ening φ ] is determined by 𝑠, the geometry of the ribbon pa ern, and limits on the accuracy and resolution of the 3D printer; we discuss how to determine these ranges in Section 5.3.4, and assume they are given for now (roughly speaking, 𝜎 1 must be close to inextensible, while there is more exibility in the choice of 𝜎 2 , since the ribbons are longer than they are wide).

More precisely, let 𝐽 φ = dφ ∈ R 2×2 denote the Jacobian of the mapping φ, and likewise for 𝐽 𝜑 . We are looking for a mapping φ such that the pushforward d( φ • 𝜑 -1 ) has the form

d( φ • 𝜑 -1 ) = 𝑅 S Σ𝑅 T S , ∀(𝑢, 𝑣) ∈ Ω, (5.1)
where 𝑅 S (𝑢, 𝑣) is an arbitrary rotation matrix; 𝑅 S = k1 k2 3×2 rotates the Euclidean plane to the tangent plane of S, with the Euclidean axes mapping to the principal curvature directions; and Σ(𝑢, 𝑣) = 𝜎 1 (𝑢,𝑣) 0 0 𝜎 2 (𝑢,𝑣) encodes stretching, subject to the bound constraints

𝜎 𝑖 ∈ [𝜎 min 𝑖 , 𝜎 max 𝑖 ].
Since satisfying (5.1) exactly for all (𝑢, 𝑣) ∈ Ω might be impossible, we seek the closest solution in the least square sense. Denoting • 𝐹 the Frobenius norm, we cast our minimization problem as

min φ,𝜎 1 ,𝜎 2 ,𝑅 S ∫ Ω 𝐽 φ -𝑅 S Σ𝑅 T S 𝐽 𝜑 2 𝐹 d𝑢d𝑣, (5.2)
whose unknowns are the elds φ :

Ω → R 2 , 𝜎 1 : Ω → [𝜎 min 1 , 𝜎 max 1 ], 𝜎 2 : Ω → [𝜎 min 2 , 𝜎 max 2
] and 𝑅 S : Ω → 𝑆𝑂 (2).

While this formulation is reminiscent of other a ening methods with bounded scaling [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF][START_REF] Pane | Computational inverse design of surface-based in atables[END_REF], unique to our approach is the use of the xed matrix 𝑅 S to strictly constrain the directions of maximal strain. In contrast, Pane a et al. [START_REF] Pane | Computational inverse design of surface-based in atables[END_REF] employ a so regularizer to penalize alignment of their air channels with directions of high normal curvature.

While their formulation gives additional freedom to trade curvature alignment for be er scaling distribution, the strong resistance of plastic ribbons to torsion (Fig. 5.4) demands close curvature alignment and limits the usefulness of this tradeo . Moreover, hard constraints allow the use of an ARAP-style local-global solver (described next), which is more e cient (parallelizes trivially) and easier to implement than the black-box nonlinear optimization using Newton's method that would be required for so constraints.

Discrete formulation

We discretize (5.2) using triangular elements. To this end, we represent the surfaces S and S using triangle meshes having same number of vertices 𝑛 V and faces 𝑛 T , and same topology, and we stack the coordinates of their vertices into the vectors x and x respectively. Approximating φ by a piecewise linear function makes 𝐽 φ constant on each triangle. We denote by 𝐽 𝑒 the value of 𝐽 φ on element 𝑒, and likewise for other quantities such as We then compute its mapping φ to the a ened con guration S using a local/global algorithm (top). is mapping φ is built so as to maximally stretch the surface S along its direction of maximum curvature k 1 (bo om).

Σ 𝑒 = 𝜎 𝑒 1 0 0 𝜎 𝑒 2 , etc. S ⊂ R 3 Ω 𝑒 ⊂ R 2 S ⊂ R 2 𝑥 𝑦 𝑧 𝑢 𝑣 𝑢 𝑣 k 1 a 𝑢 a 𝑣 x 1 x 2 x 3 𝜃 𝜑 X 1 X 2 X 3 d𝜑 -1 (k 1 )
Note that in practice we do not need to build a full (and consistent) parametrization of the surface S on the entire domain Ω at once to compute the Jacobians 𝐽 𝑒 ; we can instead locally parametrize the surface on a per triangle basis. More speci cally, we de ne the preimage Ω 𝑒 of a given element 𝑒 of S with 3D vertex coordinates x 𝑒 1≤𝑘 ≤3 as a 2D triangle of same shape and size with vertex coordinates X 𝑒 𝑘 , whose edge vector X 𝑒 2 -X 𝑒 1 is aligned with the 𝑢-axis of our 2D Cartesian coordinate system (see Figure 5.5). We can then express the Jacobian 𝐽 𝑒 as

𝐽 𝑒 = [x 𝑒 2 -x𝑒 1 x3 -x𝑒 1 ] [X 𝑒 2 -X 𝑒 1 X 𝑒 3 -X 𝑒 1 ] -1
. Moreover, on each triangle, the parameterization of S is now an isometry, with 𝑅 T S 𝐽 𝜑 = 𝑅 𝜃 a 2 × 2 rotation matrix by angle 𝜃 , where 𝜃 is the angle from d𝜙 -1 k 1 to the 𝑢-axis.

We can now rewrite our optimization problem (5.2) as

min x,𝝈 1 ,𝝈 2 ,𝑹 𝑛 T ∑ 𝑖=1 𝐽 𝑖 (x) -𝑅 𝑖 S Σ 𝑖 𝑅 𝑖 𝜃 2 𝐹 𝐴 𝑖 𝐸 (x,𝝈 1 ,𝝈 2 ,𝑹) , (5.3)
where

𝝈 1 = {𝜎 𝑖 1 ∈ [𝜎 min 1 , 𝜎 max 1 ]}, 𝝈 2 = {𝜎 𝑖 2 ∈ [𝜎 min 2 , 𝜎 max 2
]}, 𝑹 = {𝑅 𝑖 S ∈ 𝑆𝑂 (2)}, and 𝐴 𝑒 is the area of element 𝑒 in the mesh corresponding to S.

To solve problem (5.3), we adapt the local/global optimization algorithm proposed by Liu et al. [START_REF] Liu | A local/global approach to mesh parameterization[END_REF] and alternate between local steps in which we optimize the per-triangle rotations and stretches 𝑅 𝑒 S and Σ 𝑒 , and global steps where we minimize the cost function 𝐸 over the positions x while keeping all the matrices 𝑅 𝑒 S and Σ 𝑒 xed. We detail below how we solve these local and global problems.

Computation of stretching matrices Σ 𝑒 . For each element 𝑒, the minimizer of 𝐸 with respect to 𝜎 𝑒 1 and 𝜎 𝑒 2 can be obtained by solving the local problem

( σ𝑒 1 , σ𝑒 2 ) = argmin (𝜎 1 ,𝜎 2 ) 𝜎 min 1 ≤𝜎 1 ≤𝜎 max 1 , 𝜎 min 2 ≤𝜎 2 ≤𝜎 max 2 𝐽 𝑒 (x) -𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 2 F 𝐸 (𝜎 1 ,𝜎 2 )
.

(5.4)

Noting that the Frobenius norm of any matrix 𝐴 is invariant under transposition and rotation of the matrix, i.e. 𝐴 T 2 F = 𝐴 2 F and 𝑅𝐴 2 F = 𝐴𝑅 2 F = 𝐴 2 F for any rotation matrix 𝑅, we rewrite the cost function in Equation (5.4) as

𝐸 (𝜎 1 , 𝜎 2 ) = (𝑅 𝑒 S ) T (𝐽 𝑒 -𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 )(𝑅 𝑒 𝜃 ) T 2 F = (𝑅 𝑒 S ) T 𝐽 𝑒 (𝑅 𝑒 𝜃 ) T -Σ 𝑒 2 F = 𝑅 𝑒 𝜃 (𝐽 𝑒 ) T 𝑅 𝑒 S -Σ 𝑒 2 F .
(5.5) Computation of 2D node positions x. Following Liu et al. [START_REF] Liu | A local/global approach to mesh parameterization[END_REF] and denoting 𝑀 (𝑒) = 𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 , we rewrite our energy 𝐸 using cotangent weights as

𝐸 (𝜎
𝐸 = ∑ (𝑖,𝑗)∈H𝑒 cot 𝛼 𝑖 𝑗 2 x 𝑖 -x 𝑗 -𝑀 (T (𝑖,𝑗) )(X 𝑖 -X 𝑗 ) 2 , (5.13)
where H𝑒 denotes the set of all half-edges of the meshes, T (𝑖,𝑗) the triangle incident to the half-edge (𝑖, 𝑗), and 𝛼 𝑖 𝑗 the angle opposite to the half-edge (𝑖, 𝑗) in the triangle T (𝑖,𝑗) . e minimum of (5.13) with respect to x can then be obtained by solving the linear system

∑ 𝑗 ∈N (𝑖) cot 𝛼 𝑖 𝑗 + cot 𝛼 𝑗𝑖 (x 𝑖 -x 𝑗 ) = ∑ 𝑗 ∈N (𝑖)
cot 𝛼 𝑖 𝑗 𝑀 (T (𝑖,𝑗) ) + cot 𝛼 𝑗𝑖 𝑀 (T ( 𝑗,𝑖) ) (X 𝑖 -X 𝑗 )

(5. [START_REF] Bergou | Discrete elastic rods[END_REF] for all vertices 𝑖, where N (𝑖) represents the set of vertices adjacent to 𝑖.

Implementation details

Computation of curvature directions k 1 . We compute the directions of maximum curvature k 1 on the target mesh S using the method of Knöppel et al. [START_REF] Knöppel | Globally optimal direction elds[END_REF] that we slightly modi ed to output a line eld aligned with the curvature directions having largest magnitude in terms of absolute value. For many examples, we kept the default value 𝜆 = 0 for the parameter controlling the tradeo between faithfulness to the curvature directions and smoothness of the output eld (smoothness is necessary to regularize the eld when the principal curvature directions have similar magnitudes). However, in some cases, we found necessary to slightly decrease this value to avoid oversmoothing (see Table 5.1).

Setting the principal stretch bounds 𝝈 min , 𝝈 max . e upper bound in the transverse direction is determined by how much spacing we allow between neighboring ribbons. We

x this distance to 𝜇 max 2 = 𝑙 𝑟 to avoid producing large areas free of plastic. From this limit spacing value, we deduce 𝜎 max 2 ≈ 1.51 using Equation 5.15 described in Section 5.4.1. For the lower bound, we set the value 𝜎 min 2 = 1.3 as a safeguard against fusion of neighboring ribbons. Along k 1 , we set 𝜎 min 1 = 1 and 𝜎 max 1 = 1.3 to give room to the fabric to contract a bit along the principal direction of curvature if necessary to a en the surface.

Initialization of x, 𝝈 1 , 𝝈 2 and 𝑹. We initialize the vertex positions x of the a ened mesh by computing a Tu e's embedding a er regularly distributing the boundary vertices on a unit disk. We then run the implementation of Scalable locally Injective Mappings [START_REF] Rabinovich | Scalable locally injective mappings[END_REF] (as implemented within libigl [START_REF] Jacobson | libigl: A simple C++ geometry processing library[END_REF]). e initial stretch values 𝜎 𝑒 1 , 𝜎 𝑒 2 are set to 1. e rotations matrices 𝑅 𝑒 S are initialized using Formula (5.12). Update scheme and termination. We alternate between local computations of 𝑹, 𝝈 1 and 𝝈 2 and global computation of x (in this order). e loop updating 𝑹, 𝝈 1 and 𝝈 2 is run twice before switching to x. e algorithm is stopped when the average change of both 𝜎 1 and 𝜎 2 is below 10 -6 . Note that the principal stretches and stretch directions that will be used subsequently to compute ribbon layout are those of the Jacobians 𝐽 𝑒 at termination, which might not exactly satisfy the scaling bounds if the optimal residual of Equation 5.3 is nonzero. See Table 5.1 for detailed statistics about the behavior of the optimization in practice.

Controlling normal curvature

As discussed in the previous section, we align the ribbon pa ern along the direction of maximum principal curvature k 1 to take advantage of the bilayer e ect: the tendency of plastic printed on top of pre-stressed fabric to bend about the axis transverse to the ribbon, to relieve the di erential stress in the thickness direction at the bilayer interface.

In chapter 3, we derived an expression for the curvature expected for an isolated plastic ribbon printed on fabric, as a function of the ribbon thickness. However, when we tested the theory using physical prototypes, we observed that curvature is also a ected by the quantity of empty fabric surrounding the ribbons, as this fabric exerts additional forces on the ribbons and modi es their equilibrium shape. We conducted an experiment where we printed several regular pa erns that roll in a cylindrical shape (inset), with ribbons of constant thickness, width and length, but varying 𝜇 1 and 𝜇 2 . We then measured the curvature 𝜅 1 of each cylinder, as plo ed in Figure 5.6. is experiment reveals that curvature is a ected primarily by the amount of empty fabric in the transverse direction, 𝜇 2 , and not by the spacing in the ribbon direction, 𝜇 1 . We conclude from this experiment that the forces applied by the fabric along the ribbons depend on the area of fabric across the ribbon (controlled by 𝜇 2 ).

Given the complex interplay between the fabric and the ribbons forming our assemblies, we chose to adopt a data-driven approach to relate the target curvature to the thickness and spacing of the ribbons. Concretely, we printed a series of regular pa erns with varying plastic thickness 𝜏 and spacing 𝜇 2 , while holding 𝜇 1 and the ribbon width and length xed. We then measured the curvature 𝜅 1 along the k 1 direction of the resulting cylinders, as reported in Figure 5.7. Since the function is monotonic, tabulating and interpolating this data allows us to obtain, for a target curvature and spacing, the thickness 𝜏 that should be used for each ribbon. While we only measured this data on cylindrical shapes produced by parallel pa erns, our results demonstrate its generalization to more complex shapes ( When a empting to simulate a similar pa ern as in Fig. 5.2, the surface bends properly along the ribbons due to the bilayer e ect (a), but it contracts uniformly in the direction transverse to the ribbons, despite the varying density of plastic along the interior and exterior boundary (b). As a result, the simulation predicts a deployed shape that does not match the target half-torus.

We also considered physical simulation to optimize the ribbon parameters such that they reproduce the target shape. However, our experiments with the rod-based model presented in chapter 3 revealed that it cannot reliably predict the coarse-scale behavior of our ribbon pa erns. Because the plastic ribbons bond to the fabric along their entire width, the presence of ribbons causes the surface to resist transverse contraction, by a factor that depends on the ribbon width 𝑤 𝑟 (See Eq. 5.15 below). Rod-shell coupling models that assume that the ribbons are in nitesimally wide fail to capture this phenomenon. Fig. 5.8 shows the result of our rod-based model when simulating a similar pa ern as in Fig. 5.2. e simulation predicts that the shell contracts uniformly during deployment, yielding an equilibrium shape di erent from the target half-torus. (In reality, the shell contracts more along the interior boundary than along the exterior one.) Other e ects that are not captured by existing simulations, that we hypothesize are important to accurately predicting the deployed shape, include nonlinearity of the fabric (which is kni ed, and stretched well outside its linear regime); plasticity of the ribbons; residual stress in the ribbons induced by the printing process; printer fabrication errors; and seeping of the plastic partially into the fabric during printing. e more complex model presented in chapter 4 might be able capture these important e ects, however we have not yet fully tested it against a variety of printed samples and therefore cannot guarantee that it is capable of computing the correct curvature values for the whole range of parameters.

Front-and-back asymmetry We found that the curvature varied signi cantly for similar values of 𝜏 and 𝜇 2 depending on the side of the fabric we printed on: the same Fig. 5.9: Illustration of the front-and-back asymmetry of the textile: even though the nozzle is at the same height, the amount of extruded plastic can be bigger if the surface is rougher because the extruded plastic generally lls up all the available volume, including the small crevices in the surface of the fabric. ribbons, when printed on the stitch front, had a 50 to 60 % higher curvature than the ones printed on the stitch back. is could be due to several factors, one explanation might be that the bending resistance of the fabric is stronger in one direction than in the other as was measured in chapter 4 (section 4.4.3), but in general the sti ness of the fabric is negligible compared to that of the plastic. Another explanation could be a geometric di erence between the two sides of the fabric, where one side has a rougher surface than the other, causing the 3D printer to print a bigger thickness of fabric on the rougher side (see Fig. 5.9).

To account for this di erence in curvature, we computed two di erent tables of 𝜇 2 vs. 𝜅 1 vs. 𝜏, one for each side of the fabric, each of these tables has 8 × 6 data points in total.

Ribbon layout

We are now equipped with a a ened surface S, along with the directions k𝑖 = d φ • 𝜑 -1 k 𝑖 and magnitudes 𝜎 𝑖 of principal stretch of the a ening map. Our goal is to map these quantities to the parameters 𝑙 𝑟 , 𝑤 𝑟 , 𝜇 1 , and 𝜇 2 of the discrete ribbon pa ern that will be printed on the fabric (see inset gure in Section 5.3.1), and compute curves on S that trace the centerline of each ribbon. Together with the thickness 𝜏 computed in Section 5.3.5, these parameters and curves fully determine the structure to be fabricated.

Determining the Pa ern Parameters

In theory, we can vary the quantity of empty fabric locally by adjusting the length 𝑙 𝑟 and width 𝑤 𝑟 of the ribbons, or the spacing 𝜇 1 and 𝜇 2 between the centers of neighboring ribbons, or both. In other words, a dense pa ern of thin ribbons yields as much empty fabric as a sparse pa ern of large ribbons. Since varying the spacing between ribbons is less susceptible to limitations on printer accuracy and resolution than a empting to vary the ribbon dimensions, we eliminate this redundancy by xing 𝑙 𝑟 and 𝑤 𝑟 , leaving spacing between the ribbons along 𝜇 1 and 𝜇 2 as the only parameters that control stretch. We experimentally set 𝑙 𝑟 = 15mm and 𝑤 𝑟 = 1.5mm as a trade-o between the resolution of the pa ern and the adherence of the ribbons, as smaller ribbons would increase resolution but adhere less to fabric.

Given the fabric pre-stretch 𝑠, we compute the values of spacing 𝜇 1 and 𝜇 2 to achieve the target stretching values 𝜎 1 and 𝜎 2 by considering individual stripes of ribbon dashes. More speci cally, we assume that, a er we release the fabric, a periodic motif of initial length 𝜇 1 contracts to an average length μ1 = 𝑙 𝑟 + and 𝜎 2 = 𝜇 2 μ2 , from which we deduce

𝜇 1 = 𝑙 𝑟 𝑠 -1 𝑠 -𝜎 1 𝜎 1 , 𝜇 2 = 𝑤 𝑟 𝑠 -1 𝑠 -𝜎 2 𝜎 2 .
(5.15)

Note that these equations assume that the empty fabric surrounding the ribbons e ectively contracts by a factor of 𝑠 on deployment. In practice, the presence of staggered ribbons on each side of the gaps along k1 might prevent Equation (5.15) from holding for small values of 𝜇 2 . Nevertheless, we found this e ect to be negligible once the bound 𝜎 min 2 on 𝜎 2 was enforced during optimization. As discussed above, we reduced the design space of our ribbon pa ern to three parameters that correspond to the spacing 𝜇 1 and 𝜇 2 of the ribbons, and their thickness 𝜏. We now need to place ribbons over the a ened surface S so that their local spacing agrees with the target stretching values 𝜎 1 and 𝜎 2 according to Equation (5.15). We achieve this goal by leveraging the fact that the ribbons in our staggered pa ern lie on two grid layouts with cell size 𝜇 1 × 2𝜇 2 , one grid being shi ed by half a cell with respect to the other in a brick-like pa ern (inset). Each such grid can further be decomposed into two families of nearly-parallel curves with spacing 𝜇 1 and 2𝜇 2 respectively. We generate these families of curves over S using the stripe pa ern algorithm of Knöppel et al. [START_REF] Knöppel | Stripe pa erns on surfaces[END_REF], which provides local control on stripe spacing and orientation. Since this algorithm requires spacing values per vertex, we compute these values by area-weighted averaging of the per-triangle values given by our a ening algorithm. In practice, we rst run the stripe pa ern algorithm twice with the spacing elds 𝜇 1 and 2𝜇 2 and the direction elds k1 and k2 , respectively, to form one grid of our pa ern. e output of the algorithm of Knöppel et al. [START_REF] Knöppel | Stripe pa erns on surfaces[END_REF] is two 𝑆 1 -valued unit complex elds over S, 𝜃 and 𝜓 : the zero isolines of arg(𝜃 ) give the centerlines of the ribbons on one copy of the staggered grid (dark blue lines in inset gure), and the zero isolines of arg(𝜓 ) (light blue lines) intersect those arg(𝜃 )-isolines at the ribbon midpoints. We generate the shi ed second grid (orange lines) simply by extracting the 𝜋 isolines of both elds. e curvature 𝜅 1 along these lines (used to de ne the thicknesses of the associated ribbons) is obtained by linearly interpolating the values of 𝜅 1 at the crossing points between the lines and the edges of the mesh. e ribbon centerline curves can be directly computed from the staggered grid described above, by cu ing the 𝜃 -isolines into pieces of length 𝑙 𝑟 enclosed by the grid intersection points. We clip the ribbons to the boundary of S, and we delete ribbons that are shorter than 2mm. Also, we noticed that near singularities, the stripes can deviate signi cantly from the prescribed direction eld to merge or split. We detect these cases by measuring the angle between the ribbon centerline and the direction eld, and trim the ribbon when this angle exceeds 25 • . Finally, we walk along each ribbon and assign its segments to either the front or back of the fabric depending on the sign of curvature 𝜅 1 . In cases where the sign of the curvature changes sign, we split the ribbon in two pieces.

Generating the ribbon layout

Frames Fabric Base Removable corner Fig. 5.10: Our frame structure is composed of two rigid frames, in between which the fabric is clamped in tension. ese frames sit on a base that a aches to the printer bed. e four corners of the frames can be removed to be xed to the sheet of fabric before stretching.

Fabrication

Since fabricating our structures requires printing onto fabric that is under uniform, prescribed amount of tension, and requires careful control over the width and thickness of the ribbons being printed, we designed a custom frame structure that a aches to the printer bed, holds the pre-stretched fabric in place, and allows us to reliably fabricate our results. e frame structure was designed with several goals in mind:

• it should be easy to place the fabric under the prescribed tension, and the frame structure should maintain that tension throughout the printing process;

• it should support the fabric from below to prevent the printer extrusion nozzle from deforming the fabric during printing;

• it should be xed to the printer bed to prevent sliding during printing;

• it should be easily reversible to print on both sides of fabric, without allowing the pa ern on one side to become miscalibrated with respect to that on the other due to sliding or rotation of the fabric.

Figure 5.10 provides an exploded-view drawing of our design. e structure is composed of two frames that clamp down on the fabric to maintain it under uniform tension. To ease setup of the fabric, the four corners of the frames are removable. We cut a rectangular piece of (unstretched) fabric of the size of the framed scaled by a factor 1/𝑠, where 𝑠 is the desired stretching factor. We then detach the four corner pieces from the frame and independently clamp them onto the four corner of the fabric. e corners then snap back into place on the frame, stretching the fabric by the factor 𝑠. Once the corners are pulled back to their location on the frames, we x additional clamps along all sides of the frame to distribute the tension uniformly.

e frames sit on a base that is glued to the printer bed. is base has the same size as the inner boundary of the frames, and of the same thickness as one frame (so that the frame snaps into place ush with the base, which ensures that the fabric is supported by the base once the frames are in place). Since the base perfectly ts within the bo om frame, the whole assembly cannot slide. And since the two frames have equal thickness, we can print on both sides of the fabric simply by ipping the frame structure around and placing it back on the base.

We produced all our results with a desktop Ultimaker 2 FDM printer equipped with a large printing nozzle (0.8mm) and con gured at low speed (15mm/s), which we found to improve adhesion of plastic on fabric. We used TPU 95A as plastic lament, which is more exible than standard PLA. We used a nely kni ed spandex material (80 % polyamide, 20 % elastane) as fabric, which binds well with the melted plastic and o ers enough elasticity to be stretched by a factor 𝑠 = 1.6 in our experiments. .14 illustrate results produced with our method, including several architectural models and fashion items. Note that all these shapes deployed without manual placement of their boundaries, except the ones shown in Figure 5.13 and Figure 5.14 for which we used additional support or a textile strengthener, as discussed below.

Results and discussion

Our results exhibit various degrees of curvature, including positive and negative curvature achieved by printing on both sides of the fabric (Figure 5.1) and nearly developable surfaces (Skirt in Figure 5.11). is la er example highlights the bene t of being able to control extrinsic curvature (via the bilayer e ect), since developables cannot be fabricated by controlling the intrinsic curvature alone. Fig. 5.12 plots the curvature values sampled over some of our target shapes, which shows that while the bilayer e ect allows us to reproduce nearly-developable surfaces (k 2 close to 0), metric frustration also provides a way to reproduce shapes with moderate positive and negative Gaussian curvature.

We also visualize for each result in Figure 5.11 its deviation from the target surface: we reconstruct the geometry of the deployed structure from a photogrammetry scan of the fabricated object [START_REF]Realitycapture[END_REF], rigidly register it to the target surface S, and for each point on the reconstructed surface, compute the distance to its closest point on S. Errors are expressed as percentages of the bounding box diagonal of S.

Table 5.1 reports the computation time for each model shown in the paper, along with the average and maximum deviation of the photogrammetry scan from the target surface. Note that we did not scan some models that require additional intervention to support .12: Visualization of the two principal curvature values sampled over some of our target surfaces. While we expect our method to be most e ective at controlling extrinsic curvature where the target shape is nearly-developable (k 2 ≈ 0, such as throughout the Skirt), most of the surfaces we reproduced also exhibit positive and negative Gaussian curvature (Mask, Vault and Torus), or even solely negative Gaussian curvature (Octopus Saddle).

Table 5.1: For each result shown in this chapter we report the computation time, curvature eld smoothing 𝜆, average and maximum deviation between the scanned fabricated shape and the target (99 𝑡ℎ percentile to remove scanning outliers), average deviation of the ribbon direction from the target direction k 1 , maximum deviation from the bounds on 𝜎 1 and 𝜎 2 , percentage of triangles that violate the bounds, and average residual energy a er convergence. While some shapes exhibit a large portion of triangles outside the bounds, this deviation remains small. e two shapes that have the highest deviation in direction and percentage of triangles out of the bounds are Octopus saddle (Fig. 5.1) and Vault (Fig. 5.14). We note that these two models contain an umbilical region in their center, where the direction eld exhibits a singularity.

their own weight, as discussed below. While the plastic ribbons produce small-scale relief over the fabricated surface, our method captures the overall shape well with an average deviation of around 1.4%, which corresponds to 1.6 mm on such small-scale objects (11 cm average bounding box diagonal). is level of accuracy is on par with the one of prior methods that exploit similar mechanisms for the lightweight fabrication of deployable structures [START_REF] Pane | Computational inverse design of surface-based in atables[END_REF][START_REF] Pérez | Computational design and automated fabrication of Kirchho -Plateau surfaces[END_REF]. Table 5.1 also provides the average deviation from the prescribed direction k 1 , as well as the percentage of triangles that exceed the stretching bounds, along with the maximum deviation from the bounds. is deviation remains small for all models, even though some models reach the bounds on a relatively large portion of the surface.

Limitations. e maximum length distortion achievable using our technique, as the surface deploys from the at to curved state, is bounded by the fabric pre-stretch factor (which is 𝑠 = 1.6 for the kni ed fabric we used). Consequently, our method cannot reproduce surfaces with patches of large total Gaussian curvature. (One workaround would be to introduce cuts in the parameterization, and stitch the fabric along those cuts to form the surface a er printing [START_REF] Guseinov | Curveups: Shaping objects from at plates with tension-actuated curvature[END_REF][START_REF] Schüller | Shape representation by zippables[END_REF].) Although our a ening algorithm is guaranteed to converge, running it on surfaces that cannot be a ened while satisfying the principal stretch bounds will yield a a ening with signi cant cost function residual (Equation 5.3): in this case, either the right singular vectors of the resulting mapping 𝐽 do not align well with the prescribed directions, i.e. the a ening does not stretch the surface along the curvature directions; or the singular values of 𝐽 do not lie within the prescribed bounds and the desired a ening violates the ribbon minimal or maximal spacing constraints.

Our choice of aligning to the direction of principal curvature is particularly e ective for nearly-developable surfaces, where the di erence between principal curvatures is large.

In umbilic regions, the principal curvature direction is not well-de ned, and currently our approach chooses one direction arbitrarily in such regions. As a consequence, the ribbon layout is sometimes suboptimal near umbilic points; notice for instance that the Octopus Saddle (Fig. 5.1) and Vault (Fig. 5.14), which feature prominent umbilic regions, also have the largest shape deviation among our examples (Table 5.1). A potential future improvement would be to relax the curvature-alignment constraint (similar to the so penalty approach of Pane a et al. [START_REF] Pane | Computational inverse design of surface-based in atables[END_REF]), perhaps weighting alignment by a factor depending on |𝜅 1 -𝜅 2 |, though the optimization would also need to include terms to prevent shape distortion due to ribbon torsion. Another promising idea for future work is to add anisotropy of the ribbon pa ern as an optimization variable, so that the ribbon shape can adjust in response to the ratio 𝜅 1 𝜅 2 (with ribbons degenerating to disks or squares in umbilic regions, with only their spacing and not orientation encoding intrinsic curvature in those regions, similar to the experiments by Fields [START_REF] Fields | Self forming structures: An exploration into 3D printing on pre-stretched fabric[END_REF].) Finally, our method solves the inverse design task purely geometrically, without simulat- ing the complex interactions between the elastic fabric and the plastic ribbons, nor the e ect of external forces like gravity or additional load. Figure 5.13 and Figure 5.14 show results on shapes that tend to sag under the e ect of gravity, which we corrected for by adding external support or by applying a textile strengthener (Powertex). Nevertheless, our geometric approach also has its strengths, as it achieves a good agreement with the target surface in a ma er of seconds when physics-based optimization o en takes hours.

is allows the method to be usable as part of a design loop where users can quickly visualize the resulting pa ern and make edits if necessary. We also note that any simulation-based algorithm would require a good ribbon pa ern initialization to converge to a good solution, which our method provides. C 6

C

In this thesis, I presented and used a method consisting in extruding plastic curves onto pre-stretched fabric so as to create self-shaping textiles which automatically deploy when the underlying fabric is released. Two distinct metamaterials made out of repetitive and controllable pa erns were presented, rst a metamaterial arising from a pa ern of stars which we exploited to create self-shaping architectural models, and then a metamaterial composed of parallel ribbons in which we leveraged the stretch in-between the ribbons and the bilayer e ect at the ribbon-fabric interface to program the intrinsic and extrinsic curvature of the deployable surfaces.

In terms of the computational tools developed, they can be classi ed into two distinct design work ows. First, chapters 3 and 4 proposed a direct form-nding work ow in which the user speci es a pa ern to be printed on stretched fabric and can then pre-visualizes the deployed shape which rises from the speci ed pa ern before actually printing it. A direct work ow can be interesting in early stages of the design process when the user is exploring possible shape ideas, we apply it to the architectural modelmaking context. en, in Chapter 5, an inverse design tool was presented. e work ow was di erent: the user was rst asked to specify a target shape to be reproduced, and the system found an optimal layout to be printed so as to obtain a deployed shape similar to the input. e inverse design work ow might seem more productive because the designer does not need to go through a trial-and-error process to obtain a speci c shape, but in reality some of their requirements might not be satis ed by the inverse solver. e proposed inverse design tool was fast enough so that users would have no issue iterating on the input shape if necessary.

Both types of computational design work ows have been presented and studied, because I believe both have their merits and use cases. In a way, they only fundamentally di er Fig. 6.1: Integrating the direct and inverse work ows: the user can start from either the shape or the pa ern, and make edits on either of these two.

from their starting point in the design process: in the direct work ow, a user starts from a pa ern to be printed and then iterates on that pa ern until a satisfying result is found; in the inverse work ow they start from the shape to be reproduced and then modify it if the resulting pa ern from the optimization is not satisfying. An interesting development would be to be able to join both the direct and the inverse design work ows in a system where the user can start from either the pa ern or the target shape and then have the possibility to edit either one or the other and see these edits propagate to the other end (see Fig. 6.1). Such a work ow would suit both artists who wish to obtain a certain aesthetic result with printed curves, and designers who focus more on the speci c shape they want to create, and would allow both types of users to make edits on either end.

Exciting potential directions for improving the simulation of printed-on-fabric deployable structures include the ongoing work presented in Chapter 4 which could be expanded into a tool that would allow users to simulate the deployment of arbitrary pa erns printed on stretched fabric instead of just the tiling of stars on which we focused in Chapter 3.

e goal of Chapter 4 was to circumvent some of the limitations of the form-nding method of Chapter 3 by explicitly modeling the impact of the width of printed curve, as well as modeling the orthropic and nonlinear response of the pre-stretched textile.

Modeling the imprecisions in the printing process would be also greatly bene cial as a variety of defects can occur, either because the plastic sometimes does not adhere well to fabric (sometimes due to under-extrusions), or because it leaks from one printed element to the next as the printing head moves over the surface (over-extrusions, as in inset). ese imprecisions could be modeled in an explicit way by using a thermomechanical simulation of the plastic extrusion process to more precisely understand the interaction between the extruded plastic and the textile surface, or in an implicit way by using statistical tools to assess the impact of unknown factors such as, e.g. the variation in the dimensions of the printed curves, and by propagating such uncertainty within the simulation of the fabric-plastic assembly.

Many of the imprecisions we had to deal with were also ampli ed because we were working at quite a small scale.

e commercial 3D printer we used restricted us to 20 cm-wide designs at best, which means we had to print very ne details in order for the metamaterials to assume detailed curvature variations. Being able to create larger scale self-shaping textiles would therefore be an interesting avenue for future work. It would be possible, for example, to mount an extruder on a larger 2D CNC machine so as to increase the available footprint, or to apply the same principles developed in this thesis but on slightly di erent fabrication techniques such as gluing plywood panels to a large pre-stretched latex sheet [START_REF] Berdos | Architectural hybrid material composites: Computationally enabled techniques to control form generation[END_REF] or embedding exible rods inside a kni ed textile [START_REF] Lo E Aldinger | Tailoring self-formation: Fabrication and simulation of membrane-actuated sti ness gradient composites[END_REF].

An interesting challenge which would arise when fabricating larger shapes is how to make a so material such as fabric withstand its own weight. Traditionally, textiles in architecture are o en kept pre-stretched to prevent them from sagging, but in our case it is the release of the membrane tension which gives the textile its shape. A possible solution could be to sti en the fabric by coating it with a material such a concrete, a technique that has been successfully employed by Popescu et al. [START_REF] Popescu | Structural design, digital fabrication and construction of the cable-net and kni ed formwork of the knitcandela concrete shell[END_REF]. Another solution could be to automatically generate support structures, a problem that was the topic of my master's thesis, and for which I developed an algorithm based on a sparsity-inducing norm to nd a small set of poles that would support a tensile structure [START_REF] Jourdan | Optimizing support structures for tensile architecture[END_REF].

Fig. 1 . 1 :

 11 Fig.1.1: Overview of the printing-on-fabric technique: (a) a square piece of fabric is stretched on the buildplate of a conventional (FDM) 3D printer,(b) the printer prints a shape as it would normally do, (c) the shape buckles out of the plane a er being cut out from the pre-stretched fabric.

Fig. 1 . 3 :

 13 Fig. 1.3: Illustration of the two design work ows explored in this thesis.

Fig. 2 .

 2 Fig. 2.1: A developable surface, 𝜅 1 is the inverse of the radius of the osculating circle.

Fig. 2 .

 2 Fig. 2.3: A minimal surface.

Fig. 2 . 4 :

 24 Fig. 2.4: Fabricating a doubly curved surface (right) by assembling planar patches (le ) [80].

Fig. 2 .

 2 Fig. 2.5: le : a ened auxetic pa ern, right:

Fig. 2 . 6 :

 26 Fig. 2.6: Illustration of the Bertrand-Diguet-Puiseux theorem: the three circles have the same (geodesic) radius, but di erent circumferences due to the Gaussian curvature of the surfaces they are drawn on.

Fig. 2 . 7 :

 27 Fig. 2.7: Geodesy[START_REF] Gu | Geodesy: Self-rising 2.5D tiles by printing along 2D geodesic closed path[END_REF]

Fig. 2 . 9 :

 29 Fig.2.9: CurveUps[START_REF] Guseinov | Curveups: Shaping objects from at plates with tension-actuated curvature[END_REF]

  Fig.3.1: Geometric intuition behind our approach, at several length scales. At the scale of a single star, the contraction of the pre-stretched fabric back to its rest dimensions is halted by the plastic star bonded to the fabric. e star arms buckle to form a small bump (a); neighborhoods of thicker stars contract less due to the star arms bending less. At the scale of a neighborhood of several stars, the fabric contracts unimpeded, since stars are not connected (b). A sparser star pa ern with smaller stars (and hence more spacing between them) allows more contraction. At the scale of a large patch of metamaterial area, several factors control the surface shape: di erential contraction due to variations in star thickness and spacing induces buckling of the surface to relieve metric frustration; boundary conditions can impose additional contraction and buckling (c); and if stars are laid out in a regular pa ern, there is global coupling in how each star breaks symmetry while buckling, introducing large-scale curvature (d).

Fig. 3 . 2 :

 32 Fig.3.2: Our simulation tool (le ) correctly predicts that a uniform star pa ern, with free boundary, rolls up into a cylinder at static equilibrium (right). is complex behavior is due to the extrinsic curvature induced by the bilayer structure of the ribbon-and-fabric assembly, and the coupling of the bending of neighboring stars causing global symmetry-breaking in the pa ern.

Fig. 3 . 3 :

 33 Fig. 3.3:To de ne a bending and twisting energy at the connection between three rods, we split each rod into two copies and divide the volume of the connection into three equal parts, here shown in di erent colors.

Fig. 3 .

 3 Fig. 3.4: A plastic ribbon printed on top of a strip of fabric. Due to the bilayer e ect, the ribbon will naturally bend upwards (le ). e plasticity of the material causes the upper layer not to recover its initially at shape when detached from the fabric layer (right).

  a) No bending (b) No bilayer (c) Our model (d) Fabrication Fig. 3.5: Impact of critical components of our model when simulating a uniform eld of stars. (a) Neglecting bending forces in the fabric makes the stars contract independently of each other, preventing the emergence of global curvature. (b) Using a at rest shape for the rods makes the stars less curved than in reality. (c, d) Our complete model be er reproduces the fabricated surface.

Fig. 3 . 7 :

 37 Fig.3.7: We show the e ect of two design parameters on the equilibrium shape of the star metamaterial: the length of the star arms ℓ and their thickness ℎ. For this experiment we print 37 stars on a hexagon of fabric with side length 67 mm. In the top row we xed ℓ = 10 mm and set, from le to right, ℎ with a decreasing radial gradient from 0.6 mm to 0.2 mm, a constant value of 0.4 mm and an increasing gradient from 0.2 to 0.6 mm. In the bo om row we x ℎ = 0.3 mm and prescribe three di erent radial gradients for ℓ: a decreasing gradient from 10 mm to 4 mm, and constant value of 7 mm and an increasing gradient from 4 mm to 10 mm.

Fig. 3 . 8 :

 38 Fig. 3.8: E ect of di erent arm-length gradient pro les on the equilibrium shape of a hexagonal star pa ern. Each row corresponds to a di erent function ℓ (𝑥, 𝑦): on the le , a plot of ℓ as a function of distance to the hexagon center; in the middle, a diagram of the corresponding star pa ern design; and on the right, the simulated equilibrium shape.

Fig. 4 . 1 :

 41 Fig. 4.1: Microscope view of a nely-kni ed spandex textile (80 % polyamide, 20 % elastane).

Fig. 4 . 3 :

 43 Fig. 4.3: le : uniaxial stretch testing setup, right: the stress-strain curve of a fabric sample forms a cycle because the path followed is not the same between loading and unloading.

Fig. 4 .

 4 Fig. 4.3 shows a picture of the rig setup.e testing machine stretched and released the samples by performing load-unload cycles, which created loops characteristic of a hysteresis behavior when plo ing the stress-strain curves. is hysteresis, or pathdependent behavior, is likely caused by the internal friction between bers of the fabric which rearrange as the textile gets stretched. Since we want to model the behavior of the textile once it has been stretched and gets released, we are really interested in the

𝜀 22 -𝛾 2 𝛾 2 0. ( 4 . 5 )Fig. 4 . 5 :

 24545 Fig. 4.5: le : shear testing setup right: stress-strain curves for a square sample oriented at 0°and 90°, and 𝛼 3 as a linear approximation of the average of both curves.

  Fig. 4.6: Matching measurements of cantilevered shapes against the "master curve" of Romero et al. [108].

Fig. 4 . 7 :

 47 Fig. 4.7: le : simulated sample undergoing uniaxial strain, right comparison between measured force values and simulated ones for di erent orientations.

Fig. 4 . 8 :

 48 Fig. 4.8: Simulation of a pa ern of parallel ribbons rolling into a cylinder.

Fig. 5 . 1 :

 51 Fig.5.1: Given a target 3D surface (a), our method computes a at layout of plastic ribbons (b) which, when printed on pre-stretched fabric, causes the surface to deploy into 3D when the fabric is released (c). We designed a custom frame to easily print ribbons on both sides of the fabric (b, red and blue), allowing us to reproduce surfaces with positive and negative extrinsic curvature.

2 𝜇 1 Fig. 5 . 3 :

 2153 Fig. 5.3: Ribbon pa ern.

  (a) Fabricated ribbon patterns (b) Darboux frames Fig. 5.4: Due to the bilayer e ect, parallel plastic ribbons roll to form a cylinder (a, top).

1 𝜎 2 Fig. 5 . 5 :

 1255 Fig.5.5: Overview of the a ening algorithm. Each triangle of the target mesh S is rst mapped to the 2D plane by aligning its (x 1 , x 2 ) edge with the 𝑢-axis of our 2D Cartesian coordinate system. We then compute its mapping φ to the a ened con guration S using a local/global algorithm (top).is mapping φ is built so as to maximally stretch the surface S along its direction of maximum curvature k 1 (bo om).

𝜇 2 = 4 . 5 𝜇 2 = 5 . 1 𝜇 2 = 6 𝜇 2 = 7 . 2

 24525126272

Fig. 5 . 7 :

 57 Fig. 5.7: Curvature 𝜅 1 of our metamaterial as a function of plastic thickness 𝜏 and spacing 𝜇 2 (in mm).

Figure 5 . 1 ,

 51 Figure 5.11). For many surfaces, the smooth curvature elds extracted by the methods of Knöppel et al. [64] are locally parallel.(a) Front view (b) Side view

Fig. 5 .

 5 Fig. 5.8: e rod-based model from chapter 3 does not account for the width of the plastic ribbons.When a empting to simulate a similar pa ern as in Fig.5.2, the surface bends properly along the ribbons due to the bilayer e ect (a), but it contracts uniformly in the direction transverse to the ribbons, despite the varying density of plastic along the interior and exterior boundary (b). As a result, the simulation predicts a deployed shape that does not match the target half-torus.

𝜇 1 -

 1 𝑙 𝑟 𝑠 . Likewise, we estimate the average width of parallel motifs of initial width 𝜇 2 to μ2 = 𝑤 𝑟 + 𝜇 2 -𝑤 𝑟 𝑠 . Taking the ratios of initial to contracted lengths (respectively widths) gives us the average stretch values 𝜎 1 = 𝜇 1 μ1

2𝜇 2 𝜇 1

 21

2 Fig. 5 . 11 :

 2511 Fig.5.11: Representative results produced with our method, including freeform doubly-curved surfaces (Neumunster and Shell) and a nearly-developable surface achieved by printing on both sides of the fabric (Skirt). We visualize the deviation of the scanned fabricated shape from the target surface, expressed as percentages of the bounding box diagonal of the target.

Figure 5 . 1 ,

 51 Figure 5.1, Figure 5.11 and Figure5.14 illustrate results produced with our method, including several architectural models and fashion items. Note that all these shapes deployed without manual placement of their boundaries, except the ones shown in Figure5.13 and Figure5.14 for which we used additional support or a textile strengthener, as discussed below.

k 2 ≈ 0 k 1 .k 2 > 0 k 1 1 Fig. 5

 12115 Fig.5.12: Visualization of the two principal curvature values sampled over some of our target surfaces. While we expect our method to be most e ective at controlling extrinsic curvature where the target shape is nearly-developable (k 2 ≈ 0, such as throughout the Skirt), most of the surfaces we reproduced also exhibit positive and negative Gaussian curvature (Mask, Vault and

Fig. 5 .

 5 Fig. 5.13: Limitation. Without additional support, this architectural model sags under its own weight (b). Adding a cardboard support underneath the structure brings it closer to the target surface (c).

Fig. 5 . 14 :

 514 Fig. 5.14: Architectural model (Vault) and fashion items (Skirt and Mask) prototyped with our method.

  • David Jourdan, Mélina Skouras, Etienne Vouga, Adrien Bousseau, 2021. Printingon-Fabric Meta-Material for Self-Shaping Architectural Models. In Advances in

	C	2
	Architectural Geometry 2020, pages 264-285. • David Jourdan, Mélina Skouras, Etienne Vouga, Adrien Bousseau, 2022. Computa-B R W
	tional Design of Self-Actuated Surfaces by Printing Plastic Ribbons on Stretched
	Fabric. Computer Graphics Forum (Proc. EUROGRAPHICS, conditionally accepted).
	Chapter 4 presents preliminary results of an ongoing work that has not been published
	yet.

Table 3 .

 3 1: Runtime performance of the simulation on a PC with an Intel Xeon Gold 5118 processor running at 2.60GHz (single-threaded).

	Name tunnel roof U shape doubly-curved 40m0.4s Runtime Nb of vertices 23m57.8s 8020 17m27.7s 7237 28m22.7s 10551 12140

Table 4 .

 4 1 (MPa) 𝐸 2 (MPa) 1: Parameters measured on a TPU95A thermoplastic lament and a spandex textile stretched up to 70 % of its initial length.

	72.3	1.05	𝛼 1 24161 90890 29676 20512 0.9626 0.9637 63419 0.0192 𝛼 2 𝛼 3 𝛽 1 𝛽 2 𝛾 1 𝛾 2 𝑘 𝐵 (N)

  • 𝜑 -1 : S → S with the following properties: (1) the principal stretch directions are aligned with k 1 , k 2 ; (2) the two principal stretches 𝜎 𝑖 are in the range 1 ≤ 𝜎 min

	𝑖	≤ 𝜎 𝑖 ≤ 𝜎 max 𝑖	. As discussed above, the achievable range of principal
	stretches [𝜎 min 𝑖	, 𝜎 max 𝑖

  1 , 𝜎 2 ) is thus equal to the sum of the squared entries of 𝑅 𝑒 𝜃 (𝐽 𝑒 ) T 𝑅 𝑒 S -Σ 𝑒 :𝐸 (𝜎 1 , 𝜎 2 ) = (𝜎 1 -[𝑅 𝑒 𝜃 (𝐽 𝑒 ) T 𝑅 𝑒 S ] 11 ) 2 + (𝜎 2 -[𝑅 𝑒 𝜃 (𝐽 𝑒 ) T 𝑅 𝑒 S ] 22 ) 2 + 𝐶 (5.6)where 𝐶 is constant with respect to 𝜎 1 and 𝜎 2 . e minima subject to𝜎 min 𝜕𝐸 𝜕𝜎 𝑖 (𝜎 1 , 𝜎 2 ) = 0, i.e. σ𝑒 𝑖 = [𝑅 𝑒 𝜃 (𝐽 𝑒 ) T 𝑅 𝑒 S ] 𝑖𝑖 . 𝐽 𝑒 -𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 ) 𝑡 (𝐽 𝑒 -𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 ) = tr (𝐽 𝑒 ) 𝑡 𝐽 𝑒 -2 tr (𝐽 𝑒 ) 𝑡 𝑅 𝑒 S Σ 𝑒 𝑅 𝑒We use the signed version of the singular value decomposition of Σ 𝑒 𝑅 𝑒 𝜃 (𝐽 𝑒 ) 𝑡 𝑡 = 𝑈 𝑆𝑉 𝑡 such that |𝑆 1 | < |𝑆 2 | with 𝑆 1 being possibly negative and 𝑈𝑉 𝑡 ∈ 𝑆𝑂 (2) [79] tr 𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 (𝐽 𝑒 ) 𝑡 = tr 𝑅 𝑒 S𝑉 𝑆𝑈 𝑡 = tr 𝑈 𝑡 𝑅 𝑒 S𝑉 𝑆 =𝑆 1 𝐻 11 + 𝑆 2 𝐻 22 , 𝐻 = 𝑈 𝑡 𝑅 𝑒 𝐻 ∈ 𝑆𝑂 (2) therefore 𝐻 11 = 𝐻 22 = cos 𝜃 for some 𝜃 in R e maximum of tr 𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 (𝐽 𝑒 ) 𝑡 is thus reached when 𝐻 = 𝐼 , i.e. R𝑒 S = 𝑈𝑉 𝑡 , 𝑈 𝑆𝑉 𝑡 = 𝐽 𝑒 (𝑅 𝑒 𝜃 ) 𝑡 Σ 𝑒 (5.12)

	𝑖 S . Minimizing 𝐸 with respect to 𝑅 𝑒 ≤ 𝜎 𝑒 𝑖 ≤ 𝜎 max 𝑖 𝑖 < 𝜎 min if σ𝑒 𝑖 , if 𝜎 min 𝑖 ≤ σ𝑒 𝑖 ≤ 𝜎 max 𝑖 , 𝑖 = {1, 2}, if 𝜎 max 𝑖 < σ𝑒 𝑖 , (5.7) S amounts to S = argmin 𝜎 min 𝑖 σ𝑒 𝑖 𝜎 max 𝑖 R𝑒 𝑅 𝑒 S ∈𝑆𝑂 (2) 𝐽 𝑒 -𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 2 F . (5.8) which is known as the Orthogonal Procustes problem [113], the following proof is given are then given by 𝜎 𝑒 𝑖 =              where σ𝑒 solving here for completeness but can be found in a similar form in [44, Chapter 4]: Expanding the right hand side of equation (5.8) gives us 𝐽 𝑒 -𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 2 𝐹 = tr (𝜃 + tr (𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 ) 𝑡 𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 . (5.9) Discarding terms that do not depend on 𝑅 𝑒 S we can write R𝑒 S = argmax 𝑅 𝑒 S ∈𝑆𝑂 (2) tr (𝐽 𝑒 ) 𝑡 𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 = argmax 𝑅 𝑒 S ∈𝑆𝑂 (2) tr 𝑅 𝑒 S Σ 𝑒 𝑅 𝑒 𝜃 (𝐽 𝑒 ) 𝑡 (5.10) S𝑉 𝑖 is solution of Computation of rotation matrices 𝑅 𝑒 (5.11)

  Fig.5.6: e curvature 𝜅 1 of our metamaterial varies not only as a function of plastic thickness 𝜏, but also as a function of spacing 𝜇 2 in the transverse direction (le , with 𝜇 1 = 22.5mm and 𝜏 = 0.5mm). In contrast, varying spacing 𝜇 1 in the ribbon direction has li le impact (right, with 𝜇 2 = 6mm).

	𝜅 1 (mm -1 )	𝜅 1 (mm -1 )
	0.12	0.12
	0.10	0.10
	0.08	0.08
	0.06	0.06
	0.04	0.04
	0.02	0.02
	3.5 4 4.5 5 5.5 6 6.5 7 7.5 𝜇 2 (mm)	15	17	19	21 𝜇 1 (mm) 23	25	27

Nous utilisons ces métamatériaux pour proposer deux types d'outils de design aidant à la conception de structures auto-déployables légères : d'un côté des outils de design direct, ou de recherche de forme, qui perme ent de prédire la forme déployée de la structure à partir du motif à imprimer, de l'autre un outil de design inverse qui prend en entrée une forme donnée et trouve les paramètres optimaux du métamatériau (notamment l'épaisseur et la densité) pour reproduire au mieux la forme cible.

Note that in our case, only the plastic ribbon exhibits signi cant plasticity. However, assuming that the fabric has the same curved rest state as the ribbon helps simplifying the formulas. Since our fabric is highly exible, we believe that the error that we introduce by doing so is negligible.

Model

Nb.