# Programming developable surfaces using multilayer inflatables

Ofir Mirkin, Nathan Vani, Etienne Reyssat, José Bico, Mélina Skouras

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Ofir Mirkin, Nathan Vani, Etienne Reyssat, José Bico, Mélina Skouras. Programming developable surfaces using multilayer inflatables. SCF'25 - 10th ACM Symposium on Computational Fabrication, Nov 2025, Cambridge (MA), United States. pp.1 - 13, 10.1145/3745778.3766669. hal-05477653

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# Programming developable surfaces using multilayer inflatables

Ofir Mirkin*

Univ. Grenoble Alpes, Inria, CNRS,
Grenoble INP, LJK
Grenoble, France
ofir.mirkin@inria.fr

Nathan Vani*

PMMH, ESPCI Paris, PSL Univ., CNRS,
Sorbonne Univ., Univ. Paris Cité
Paris, France
nathan.vani@espci.fr

Etienne Reyssat

PMMH, ESPCI Paris, PSL Univ., CNRS,
Sorbonne Univ., Univ. Paris Cité
Paris, France
etienne.reyssat@espci.fr

José Bico

PMMH, ESPCI Paris, PSL Univ., CNRS,
Sorbonne Univ., Univ. Paris Cité
Paris, France
jose.bico@espci.fr

Mélina Skouras

Univ. Grenoble Alpes, Inria, CNRS,
Grenoble INP, LJK
Grenoble, France
melina.skouras@inria.fr

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Given a developable ruled surface (a), our method outputs two welding layouts (b) that can be used to seal three stacked membranes forming an inflatable bilayer consisting of staggered tubes with varying radii that best approximate the input surface when inflated (c). These layouts can then be used to fabricate an actual prototype made of TPU-coated nylon (d) whose deployed shape closely matches simulations.</figcaption></figure>

## Abstract

We present a novel type of inflatable structure, made of three stacked membranes forming *inflatable bilayers*, consisting of staggered cylindrical or conical tubes. Our structures are fabricated flat using a quasi-inextensible material and bend out-of-plane when the two networks of tubular chambers are inflated to the same pressure. We program the local curvature of the structure by locally adjusting the width of the welding lines used to seal each pair of membranes.

We leverage this new architected inflatable structure in an inverse design algorithm that allows us to generate inflatable ribbons of a desired target shape. Our method takes as input a developable surface, represented as a 2D polyline or a 3D quad strip, and outputs the layouts of the three membranes to be welded together. Various prototypes made of thermosealable Thermoplastic PolyUrethane (TPU) or TPU-coated nylon are fabricated to demonstrate the capabilities of our simulation approach. Inflatable bilayers present as a

promising platform for shape-morphing and soft robotics applications.

## CCS Concepts

* **Computing methodologies → Modeling and simulation; Shape modeling; Physical simulation.**

## Keywords

Inflatables, developable sheets, bilayers, computational design, computational fabrication

## ACM Reference Format:

Ofir Mirkin, Nathan Vani, Etienne Reyssat, José Bico, and Mélina Skouras. 2025. Programming developable surfaces using multilayer inflatables. In *ACM Symposium on Computational Fabrication (SCF '25), November 20-21, 2025, Cambridge, MA, USA*. ACM, New York, NY, USA, 13 pages. https://doi.org/10.1145/3745778.3766669

## 1 Introduction

Inflatable structures are used in various applications from soft robotics [Wang et al. 2021] to architecture [Mélançon et al. 2021], through haptic devices [Zook et al. 2024] and aerospace [Veldman and Vermeeren 2001]. We are here interested in structures fabricated flat by stacking thin membranes made of a quasi-inextensible material welded according to an architected network of channels. Such

<figure><img src="image_3.png" /></figure>

<footnote>*Joint first authors</footnote>
structures typically form arrangements of tubular chambers which maximize the stiffness of the structure in one direction, i.e., along the tubes, at the cost of a much weaker rigidity in the transverse direction [Siéfert et al. 2020]. Therefore, getting an inflated structure with some bending rigidity in all directions implies modifying its design, either by changing the shape of the welding patterns such as zigzags [Gao et al. 2020], dashed lines, stars or holes [Ren et al. 2024], 6-fold symmetric patterns [He et al. 2025], by using vertical walls as typically done for inflatable mattresses [Chandler et al. 2025], or by using yarns as exploited in the drop-stitch technology [Lacorre et al. 2022; Niiyama et al. 2020]. While very effective, these latter techniques significantly complicate the fabrication process and are restricted to the design of structures that remain planar in their inflated state.

The anisotropy of assemblies of inflated tubes also reflects in the metric change of the midplane of the structure, which only contracts perpendicularly to the tubes. 3D inflatable shells can thus be obtained by locally changing the direction of the tubes so as to spatially vary the metric of the structure and generate metric frustration released through out-of-plane deformation, according to Gauss's *Theorema Egregium* [Gauss 1828]. This effect has been successfully exploited by Siéfert et al. [2020] and Panetta et al. [2021] but is limited to the design of surfaces with non-zero Gaussian curvature, leaving the reproduction of curved developable surfaces out of reach.

Interestingly, metric frustration can also act across the thickness of initially planar structures, as investigated and exploited in material science. In this case, the designed structures typically consist of two fully bonded sheets differing in material, forming so-called *bilayers*. The deformation of the structure from its flat state is then triggered by an external stimulus, e.g., a change of humidity [Poppinga et al. 2018; Reyssat and Mahadevan 2009] or temperature [Jourdan et al. 2023; Timoshenko 1925], that induces differential extension of the layers resulting in the extrinsic curvature of the device. Metric frustration can also be achieved by using the same material for the two layers and pre-stretching one of the two layers (possibly anisotropically) before bonding it to the other [Armon et al. 2011]. As their actuation often relies on thermal or solutal diffusion [Pezzulla et al. 2016], they are generally slow and sometimes difficult to revert (e.g., when pre-stretched materials are used). Faster actuation can be achieved with pneumatic bilayers relying on the inflation of off-centered chambers in compliant beams. Such systems are widely used in elastomer-based actuators for soft robotics applications [Shepherd et al. 2011]. Nevertheless bilayers structures are usually very slender, which makes them very flexible in bending and limits their size to a few centimeters.

In contrast, our goal is to design *stiff* structures of possibly large dimensions, that are fabricated flat and remain flat when not actuated, while being easily and quickly deformed for easy deployment and returning to their initial planar state for easy folding, storage and transportation. To do so, we rely on arrangements of aligned inflatable tubes using an inextensible material such as thermosellable fabrics, as done in prior works [Ou et al. 2016; Panetta et al. 2021; Ren et al. 2024; Siéfert et al. 2020]. However, instead of limiting ourselves to two membranes, we stack *three membranes* to create *inflatable bilayers*. We program the metric change of each layer by tuning the width of the welding lines between each pair of

membranes. Wider seam lines reduce the active fraction of the layer and its effective contraction upon inflation. The contrast in dimensional change between layers results in the curvature of the overall structure when inflated. Taken individually, each layer formed by two membranes has a very low bending rigidity, as the welding lines act as weak hinges between tubes. This relative rotational degree of freedom is eliminated by arranging the two layers such that the tubes form staggered rows. This provides an increase in the effective moment of inertia and bending stiffness of the structure.

Our technique allows us to reliably program the normal curvature of the structure perpendicularly to the tubes by controlling the metric change of each layer. We leverage this mechanism to propose a simple and effective algorithm for the inverse design of developable sheets of custom target shapes that solely relies on geometric principles and does not require any costly optimization.

## 2 Related work

Thin-sheet inflatables offer a promising avenue for shape-morphing materials, providing rapid deployment, simplified fabrication, and enhanced mechanical stiffness compared to their elastomer-based counterparts commonly used in soft robotics. The seminal works of [Niiyama et al. 2015] on *pouch motors* and [Ou et al. 2016] on *AeroMorph* have sparked significant interest for thin-sheet inflatables across the Human Computer Interactions, Computer Graphics and Mechanical Engineering communities. Often made from TPU-coated nylon sheets, the inflatables are conventionally fabricated by 2D-welding patterns with a soldering CNC-rig [Choi and Ishii 2021]. Alternative fabrication processes have been developed, either through a *roll-to-roll* non-planar welding [Sareen et al. 2017], planar welding without contact using a laser cutter [Lu et al. 2019], bonding with glue [Boutemy et al. 2022] to increase the strength of the seam lines [Broshkevitch et al. 2025], or using custom shaped heat-sealing tools for rapid prototyping [Gohlke et al. 2023; Niiyama et al. 2015]. However, these techniques primarily consider the fabrication of inflatables made from two sheets only. We extend the concept of thin-sheet inflatables to *bilayers* made from three sheets using a conventional planar fabrication method.

Programming the shape of thin-sheet inflatables is a complex problem due to the large design space of soldering lines. One approach involves designing chains of squares to create kinks that act as hinges in the structure [Ou et al. 2016; Yang et al. 2024]. Nonetheless, creating flat inflatables with this method is difficult and the bending direction relies on a delicate asymmetry of the welded seams. Another approach exploits tubular structures in which parallel soldering lines define tubes. The anisotropic shrinking of tubes during their inflation leads to geometrical frustration [Siéfert et al. 2020], which can be harvested to program complex shapes [He et al. 2025; Panetta et al. 2021; Ren et al. 2024]. However, these approaches do not allow for the programming of developable shapes, including flat surfaces. Another related strategy is to design bent tubes, whose curvature is increased upon inflation. This can be used to program any curvilinear shape in the plane [Siéfert et al. 2019] or in a star-shaped singular tube to act as a gripper [Andrade-Silva and Marthelot 2023]. The resulting shapes are nonetheless closer to vine-like robots rather than surfaces. Using *bilayer* inflatables allows us to program any developable shape into our surfaces. The
<figure><img src="image_4.png" /><figcaption>Figure 2: Internal structure of an inflatable bilayer. (a) Welding layouts corresponding to a cylindrical (left) and a conical (right) bilayer. The welding curves between the bottom and the middle membranes are shown in purple, while those between the middle and the top membranes are shown in orange. (b) Cross-section along the dotted lines of the bilayers in their deflated state. The membranes are shown in black, pink and yellow, respectively, while the welded regions are depicted in purple and orange. (c) Cross-section along the dotted lines of the bilayers in their inflated state.</figcaption></figure>

resulting inflatables also have an increased bending stiffness compared to conventional networks of tubes, as the thickness of the inflated shape is more homogeneous, with no weak points along the seam lines.

### 3 Anatomy of an inflatable bilayer

We describe inflatable bilayers made of interconnected tubular chambers arranged in two staggered rows (see Figure 2). These inflatables are fabricated flat by stacking three membranes that are welded together according to two welding layouts: the first one (shown in purple in the picture) is used to seal the bottom membrane to the middle one, while the second (in orange) is used to seal the middle membrane to the upper one. Each pair of welded membranes thus defines a *layer* of our inflatable structure.

The considered seam lines have the shapes of rectangles or circular sectors and control the radial dimensions of the inflated tubes, i.e., wider welds lead to thinner tubes and, in turn, to a smaller in-plane contraction for the layer. When the two layers are inflated to the same pressure, the non-welded parts of the middle membranes define planar walls between the air chambers, which shape into truncated cylindrical or conical tubes. Tuning the widths of the welds of the two welding layouts simultaneously allows us to modulate the differential contraction of the layers and program the normal curvature of the structure in the direction perpendicular to the seams. In geometrical terms, the seams define the rulings of the developable inflated structures. Note that inflating the structure also affects the average metric of the bilayer which exhibits in-plane contraction.

<figure><img src="image_5.png" /><figcaption>Figure 3: Fabrication method. (a) CNC-mounted soldering iron as used for selective bonding of the material. (b) Stack of materials taped to the tempered glass as used in the first (left) and second (right) soldering steps.</figcaption></figure>

To summarize, the orientations of the welds define the ruling directions of the inflated bilayer while their transverse dimensions define the in-plane contraction of the structure as well as its normal curvature perpendicularly to the rulings, which we exploit to fully program the shape of the inflatable.

In the following, we start by characterizing the macroscopic geometry of bilayers corresponding to periodic structures made of parallel tubes, before explaining how we leverage the obtained relations to convert an arbitrary developable ruled surface into welding pattern geometries.

### 4 Fabrication of bilayer inflatables

We fabricate airtight networks of inflatable channels by heat-sealing planar sheets of TPU or TPU-coated nylon using a CNC-mounted soldering iron. While the fabrication method is now well-established, the novelty of our work lies in the choice of materials and in the precise control of soldering parameters, enabling selective bonding of only the two topmost sheets within a multilayer material stack.

The fabrication set-up is shown in Figure 3 (a). A commercially available soldering iron (Weller WSP80) is controlled in motion at speed $v$ by a three-axis arm (OpenBuilds Workbee 1510, typically at 100 mm/min) and in temperature $T$ by a soldering station (Weller WE 1010, ranging from 50 to 450°C). The force applied to the stack of material is controlled by mounting the iron on a freely sliding rail along the vertical direction – the applied force thus depends only on the weight of the iron rather than on the force applied by the z-axis motor. Additional weights are clamped to the iron to increase the iron weight and impose a total force $F = mg$, where $m$ is the total weight and $g$ the gravitational constant. The iron is fitted with a rounded conical tip (Weller ETA 1.6 mm) to minimize shear stress during soldering.

Materials are layered on tempered glass which serves as a soldering mat, as shown in Figure 3 (b). A carbon-coated Teflon sheet (TVTP200 from Plastiques Elastomeres) is used as a non-adhesive,
protective layer. Teflon provides improved soldering quality compared to conventionally used baking paper, while being reusable for up to 30 soldering operations. In a first soldering step, two sheets are straightforwardly soldered. In a second step, a third sheet is stacked on top of the first two. By selecting optimal soldering parameters (*T*, *v*, *F*), the two uppermost sheets are soldered without bonding the two bottom ones. Parameters are optimized empirically and strongly depend on the considered materials.

The inflatables are made from either Thermo-Polyurethane (TPU) sheets (Advanced Creations Supplies) or TPU-coated sheets (ExtremTextil), which appear respectively as translucent black and yellow in the pictures of the samples we present. The process requires the sheets to be fusible on both sides, the nylon sheets are thus double-coated. We successfully found parameters allowing for selective bonding for sheets that were sufficiently thick (more than 150 µm thick for PU sheets, more than 200 µm thick for nylon sheets). We provide optimal parameters for the considered materials in Table 1.

<table><caption>**Table 1: Soldering parameters for TPU and Nylon.**</caption><thead><tr><td>Material</td><td>Force (N)</td><td>Speed (mm/min)</td><td>Temperature (°C)</td></tr></thead><tbody><tr><td>TPU, 220 g/m²</td><td>2</td><td>100</td><td>450</td></tr><tr><td>Nylon, 400 g/m²</td><td>3</td><td>100</td><td>300</td></tr></tbody></table>

The actuating pressure in the inflatables is set with a pressure controller (SMC IR2000) and measured by an Arduino sensor (Adafruit MPRLS). The numerical method used does not account for bending stiffness in the sheets – the pressure must thus be large enough for this assumption to apply. Given *Y* and *t* the Young’s modulus and thickness of a given sheet, this requires the inflatable pressure to be larger than $Yt^3/(\min(L-w))^3$ where $\min(L-w)$ is the typical size of the smallest tube in the system. Another standard assumption for thin-sheet inflatables is to neglect the extension of the membranes – as used in the following analytical model. This assumption requires that the inflatable pressure be smaller than $Yt/\max(L-w)$ where $\max(L-w)$ is the typical size of the largest tube in the system. We verify in experiments that we remain in this doubly asymptotic regime, with typical *L* in the range of 2 to 5 cm and inflating pressure in the range of 5 to 50 kPa.

Concerning the former assumption, a recent work by some of the authors of this work has shown that bending stiffness affects the shape of the junction of sheets near soldering lines regardless of the inflating pressure – possibly strongly influencing the macroscopic shape [Vani et al. 2025]. In the pressure regime considered here, the size of the boundary layers are expected to be small enough not to affect the macroscopic shape of the inflatable. The sheets must however be symmetric in stiffness so as not to rotate the junctions. For fabrication by soldering, one should thus be careful of using identical sheets oriented in the same direction since the materials are anisotropic. An alternative fabrication method consists of gluing a pair of layers assembled independently with no risk of bonding the bottom channels while welding the upper layer. However, the resulting asymmetric inflatable would have a central sheet with a

<figure><img src="image_6.png" /><figcaption>**Figure 4: Unit cell of an inflatable periodic multilayer in its deflated configuration (top) and inflated configuration (bottom).**</figcaption></figure>

doubled stiffness and the junction of sheets at the edges of soldering lines could have a direct impact on the geometry after inflation.

## 5 Mapping welds widths to curvature

We first consider periodic inflatable structures made of parallel cylindrical tubes, fabricated by welding the membranes according to rectangular patterns. In the case of such periodic structures, the width of these rectangular welds is the same for all the welds between the bottom membrane and the middle membrane and is denoted by $w_1$. Similarly, the welds between the middle and the upper membranes have all the same width, denoted by $w_2$. Therefore, the final inflated structure consists of two series of tubes with alternating opening sizes and can be analyzed by considering a representative unit corresponding to two tubes. Furthermore, we assume the length of the tubes is much longer than their widths so that we can consider the problem in 2D and restrict our analysis to that of the deformation of the cross-section of the structure (see Figure 4). In this work, we also assume that the material used is quasi-inextensible and that the pressure inside the inflatable is large enough for the shape of the inflated structure to only depend on the geometry of the welds independently of the value of the pressure.

We denote by *L* and *l* the widths of this representative unit element in the initial (i.e. deflated) and inflated configurations, respectively. The effective in-plane contraction of the structure corresponds to the ratio $l/L$, which is a function of the relative widths $w_1/L$ and $w_2/L$. Note that $w_1/L$ and $w_2/L$ also impact the contraction of each individual layer and therefore the effective curvature of the bilayer.

Our goal in this step is to determine the relation between the weld widths $w_1$ and $w_2$, controlling the deformations of the tubes, and therefore their metric, and the in-plane contraction and curvature of the mid-surface of the assembly of tubes, which we define as the surface passing halfway between the middle of the welds.

We conduct our analysis using numerical simulation relying on a linear material model for the membranes. The assumption of quasi-inextensibility for the membranes imposes a low pressure-to-membrane stiffness ratio while the negligible impact of the pressure on the shape of the tubes imposes a large pressure-to-bending-stiffness ratio. In practice, we set the ratio between pressure and
<figure><img src="image_7.png" /><figcaption>Figure 5: Impact of the widths of the lower and upper welds $w_1/L$ and $w_2/L$ on the integrated curvature of the inflatable $\kappa L$ (a) and on its in-plane contraction $l/L$ (b) for periodic structures (the plots correspond to numerical simulations). (c) Inflatables with the same curvature $\kappa L$, i.e., corresponding to pairs $(w_1, w_2)$ in the same level set (here, the white dotted line in (a)). From top to bottom: inflatables corresponding to $(w_1/L, w_2/L)$ equal to (0, 0.3), (0.1, 0.35), (0.15, 0.4) and (0.2, 0.5).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Impact of the width of the upper welds $w_2/L$ on the curvature of the inflatable $\kappa L$, for a width of the lower welds $w_1/L = 0$ and a periodic bilayer. (a) Comparisons of the obtained mappings using different methods: analytical model based on volume maximization (black dashed line), numerical simulations (blue line), and experiments (markers). All relations agree very well in the contact-free regime. For $\kappa L \ge 0.5$, the curvature saturates experimentally due to self-contact between air chambers, a mechanism captured neither by the theoretical model nor the simulation. (b) Simulation of periodic inflatables and corresponding prototypes ($L = 2$ cm) (c).</figcaption></figure>

membrane stiffness to 0.01 to satisfy our first requirement and entirely neglect the bending resistance of the material to comply with the second. We discretize the cross-section of the structure using edge elements with constant strain and adopt a variational approach consisting in minimizing the total energy of the system. The corresponding induced curvature and in-plane contraction are shown in Figure 5.

We validate our simulator by comparing the obtained relation from welds widths ($w_1/L, w_2/L$) to dimensionless curvature $\kappa L$ to a theoretical law analytically derived for the case when the welds between the bottom and middle layers are infinitesimally thin (i.e.

$w_1/L = 0$), as detailed in the appendix. For these derivations, we assume inextensibility of the membranes and look for the geometric configuration of the structure that maximizes the volume of its chambers. As shown in Figure 6, the numerical and theoretical models show excellent agreement.

We also compare these results to experimental data, obtained by measuring the curvature of inflatables fabricated using TPU sheets, as described in Section 4, still for the case $w_1/L = 0$. Here, the relation between the relative welding width $w_2/L$ and the relative curvature $\kappa L$ matches very well the prediction until it reaches a plateau at around $w_2/L = 0.5$, which we attribute to contacts
between the chambers that are not accounted for in the numerical and theoretical models. For the inverse design procedure, which we describe next, we stick to the contact-free region of the map, i.e., we use $w_2/L$ (resp. $w_1/L$) values that range from 0 to 0.4 for $w_1/L = 0$ (resp. $w_2/L = 0$).

# 6 Inverse design algorithm

Our algorithm takes as input a developable ruled surface specified as a 2D polyline (planar case) or a 3D quad strip with planar quads, and outputs the layouts (external contour and shapes of the welded areas) of three superimposed membranes forming an inflatable bilayer whose midsurface, once inflated, approximates the input target surface. Our method exploits the pre-computed mappings for width of the welds to in-plane contraction and normal curvature of the periodic structures described in Section 5, and proceeds in three steps:

* We flatten the input developable surface using an anisotropic transformation that preserves the distances along the rulings while stretching the surface in the transverse direction to counteract the contraction across the tubes when they are inflated.

* We compute the shapes of the welds between, respectively, the bottom and the middle membranes, and the middle and the upper membranes, based on the normal curvature of the input developable surface, perpendicularly to the rulings, using the mapping computed in the first step, and we add the weld contours to the layouts. These welds are rectangle-shaped if the input strip corresponds to an extruded 2D curve, while they take the shape of circular sectors if the ruling directions vary.

* We generate the outer contour of the layouts from the flattened quad-mesh using rounded curves so as to facilitate air flow from one chamber to the other.

We first describe our algorithm for designing inflatables whose silhouette matches a given 2D profile and then its generalization to the case of arbitrary 3D ruled surfaces.

## 6.1 Inverse design of extruded 2D curves

Given an input 2D curve, we first sample it at the desired resolution, keeping in mind that one edge of the resulting polyline will be converted into two staggered tubes in the final inflatable structure. Note that regular curvilinear sampling is often desired to obtain structures with an even thickness. However, this is not a strong requirement as our algorithm can accommodate varying input edge sizes.

We then compute the integrated curvature binormal $\kappa l$ at each vertex of the resulting polyline, using the discrete formulation proposed by Bergou et al. [2008] which approximates it as

$$ \kappa l \approx \frac{2e_i \times e_{i+1}}{\|e_i\| \|e_{i+1}\| + e_i \cdot e_{i+1}}, \quad (1) $$

where $e_i = x_i - x_{i-1}$ and $e_{i+1} = x_{i+1} - x_i$ are the two edge vectors on either side of the considered vertex $x_i$ (see Figure 7). The norm of the vector, $\kappa l = \|\kappa l\| = 2 \tan \frac{\theta_i}{2}$, gives the magnitude of the integrated curvature associated with vertex $x_i$, while the curvature

<figure><img src="image_9.png" /><figcaption>Figure 7: (a)-(b) Notations for the vertices and edges used in the computation of the discrete integrated curvature associated with vertex $x_i$ (Equation 1). (c) In the case of 3D developable surfaces with non parallel rulings, the “widths” $w_1(s)$ and $w_2(s)$ of the welds, as well as the size $L(s)$ of the unit cell, vary along the rulings.</figcaption></figure>

sign can be trivially obtained by computing the sign of the triple product $e_i \cdot (e_{i+1} \times \kappa l)$.

The next step consists of mapping the signed value $\kappa l$ to a weld widths pair ($w_1/L, w_2/L$) using the map computed in Section 5 and illustrated in Figure 5, left. As illustrated in Figure 5, right, this problem is under-constrained, as multiple weld widths pairs correspond to the same curvature value $\kappa l$. For our prototypes, we decided to use the two (symmetric) curves $(0, w_2/L) \mapsto \kappa l$ (for positive curvatures) and $(w_1/L, 0) \mapsto \kappa l$ (for negative curvatures) as this allows us to get monotonic relations which we can easily invert, and that span a wide range of curvature values, while maximizing the thickness, and therefore the stiffness, of the resulting inflatable. Note that other choices of curves can also be exploited as long as they are monotonic.

Finally, we need to convert the ($w_1^i/L, w_2^i/L$) pair assigned to the vertex $x_i$ to the geometry of the corresponding “elementary unit” of the inflatable, i.e., define the exact positions of the welds in the structure. For simplicity, we decided to assign the computed vertex properties to the unit located immediately to the left of the vertex (see Figure 7, bottom), although a more symmetric treatment could be implemented as well. We first compute the local spacing between the welds of the bottom layer in the undeformed configuration, i.e. the distance $L_i$ between the vertices $X_{i-1}$ and $X_i$ associated to $x_{i-1}$ and $x_i$, using the mapping from welds widths to contraction ratios $l/L$ shown in Figure 5, right. We center the weld between the bottom and the middle membranes at $X_i$ and set its width to $w_1^i$. We position the weld between the middle and the upper membranes at mid-distance between $X_{i-1}$ and $X_i$ and set its width to $w_2^i$.

## 6.2 Inverse design of 3D developable surfaces

We generalize the algorithm described above to the case where the target shape is a 3D developable ruled surface. Here, we assume the input is represented by a quad strip with planar faces. The steps of our algorithm are illustrated in Figure 8.

In the general case where the rulings are not parallel (which in the discrete setting translates to a quad strip with non-parallel inner edges), the welds we compute take the shape of circular sectors. The “widths” $w_1$ and $w_2$ of the welds, as well as the “width” $L$ of the unit elements in the deflated state, vary along the rulings, i.e.
<figure><img src="image_10.png" /><figcaption>Figure 8: Steps for generating the shapes of the welds of an inflatable whose inflated shape matches a given quad strip.</figcaption></figure>

they are function of the curvilinear abscissa $s$ (see Figure 7(c)), but the ratios $w_1(s)/L(s)$ and $w_2(s)/L(s)$, and the relative size of the inflated tubes $l(s)/L(s)$, are constant with respect to $s$, which we exploit in our algorithm, as detailed below.

We first compute the discrete integrated curvature in the direction orthogonal to the internal edges of the input strip using a generalized version of Equation 1 that approximates the curvature binormal for planar quad strips [Rodriguez et al. 2024], i.e., we define

$$ \kappa l \approx \frac{2\mathbf{n}_i \times \mathbf{n}_{i+1}}{\|\mathbf{n}_i\|\|\mathbf{n}_{i+1}\| + \mathbf{n}_i \cdot \mathbf{n}_{i+1}}, \quad (2) $$

where $\mathbf{n}_i$ and $\mathbf{n}_{i+1}$ are the normals of the two faces incident to the internal edge $(\mathbf{x}_i^1 \mathbf{x}_i^2)$ (see Figure 8).

We then convert the signed curvature value associated with the edge (i.e. the magnitude of the binormal vector) to a $(w_1^i/L_i, w_2^i/L_i)$ pair as described for the 2D case and we evaluate the contraction factor $l_i/L_i$ associated with these welds sizes.

We isometrically flatten the input quad strip by successively rotating each quad along the edge connecting it to the previous quad so that the normals of these two quads coincide (see Figure 8(b)).

We note $\hat{\mathbf{X}}_i^j$ the position of the vertex $\mathbf{x}_i^j$ after this transformation.

We then stretch azimuthally each quad $(\hat{\mathbf{X}}_{i-1}^1, \hat{\mathbf{X}}_{i-1}^2, \hat{\mathbf{X}}_i^2, \hat{\mathbf{X}}_i^1)$ so that the length of any arc defined between the two boundary edges $(\hat{\mathbf{X}}_{i-1}^1, \hat{\mathbf{X}}_{i-1}^2)$ and $(\hat{\mathbf{X}}_i^1, \hat{\mathbf{X}}_i^2)$ is stretched by the factor $L_i/l_i$ (see Figure 8(c)). We note $\mathbf{X}_i^j$ the position of the vertex corresponding to $\hat{\mathbf{X}}_i^j$ after this transformation.

We position the weld between the bottom and middle membranes along the edge $(\mathbf{X}_i^1, \mathbf{X}_i^2)$ and set its angle to $\alpha_{w1} = \alpha_L \cdot w_1^i/L_i$ so that its width relative to the width of the left quad is $w_1^i/L_i$, i.e. $\alpha_{w1}/\alpha_L = w_1^i/L_i$, using the notations in Figure 8(c). Similarly, we position the weld between the middle and top membranes at the middle of the quad and set its angle to $\alpha_{w2} = \alpha_L \cdot w_2^i/L_i$.

Finally, we crop all the generated circular sectors (shown in purple and orange in Figure 8(c)) to only keep the parts that strictly fit inside the stretched quads $(\mathbf{X}_{i-1}^1, \mathbf{X}_{i-1}^2, \mathbf{X}_i^2, \mathbf{X}_i^1)$, as shown in Figure 8(d). We then add the circular curves joining every other welds as done in the 2D case.

<table><caption>Table 2: 2D model statistics. From left to right: model name, number of tubes (per layer), width of a unit cell relative to the diagonal D of the bounding box, curvature error relative to the width of a unit cell, maximum $L_2$ distance relative to the diagonal of the bounding box. The curvature and distance errors are computed on the simulated structures, after rigid alignment.</caption><thead><tr><th>Model</th><th># Tubes</th><th>L/D</th><th>max(|κ - κ<sub>t</sub>|) · L</th><th>max(L<sub>2dist</sub>)/D</th></tr></thead><tbody><tr><td>wave</td><td>11</td><td>0.147</td><td>0.269</td><td>0.028</td></tr><tr><td>sine curve</td><td>28</td><td>0.065</td><td>0.112</td><td>0.030</td></tr><tr><td>snail</td><td>29</td><td>0.103</td><td>0.014</td><td>0.032</td></tr><tr><td>deck chair</td><td>19</td><td>0.085</td><td>0.067</td><td>0.018</td></tr></tbody></table>

# 7 Results

We experimentally validated our framework and fabrication technique by designing and fabricating a number of bilayers of diverse shapes made of either TPU or TPU-coated nylon.

**2D examples.** We first experimented with inflatables made of parallel tubes. In addition to the periodic structures shown in Figure 5, we generated the welding layouts for four models with varying curvature: a surface with monotonically increasing curvature, the *Snail* model, and three surfaces with both positive and negative curvature, the *Wave*, *Sine Curve* and *Deck Chair* models, shown in Figure 11. We provide statistics on these models in Table 2. The shapes of the target surfaces are overall very well approximated by our inflatable bilayers. Since curvature is the parameter programmed by our method, we first evaluated the errors between the target shapes and the programmed inflatables in terms of curvature. These errors, easily quantifiable in simulations, are generally highest in regions of largest curvature (see Figure 11), especially for models with coarser discretization. As shown in Figure 9, the error in curvature decreases when the resolution of the bilayer increases, as expected. We also report in Table 2 the maximum $L_2$ distance between the vertices of the target polylines and those of the midcurves of the simulated structures, after rigidly aligning the two curves. Note that the location of highest error in terms of vertex distances is not necessarily the same as for curvatures, as small local curvature deviations can have a larger impact further away in the structure (see Figure 11, last row).
<figure><img src="image_11.png" /><figcaption>**Figure 9:** Impact of the discretization on the *wave model*. By refining the discretization of the input curve, both the curvature errors and the $L_2$ distance errors decrease. The black curve corresponds to the target curve while the colored curve corresponds to the midcurve of the inflated structure with colors matching the distribution of the absolute curvature error $|\kappa - \kappa_{\text{target}}|$.</figcaption></figure>

The comparisons between the simulated structures and the fabricated prototypes also show good agreement, although we noted that the actual bilayers tended to bend slightly less than predicted, especially for structures made out of plain TPU and in regions with higher curvature. We believe this might be caused by self-contact between the chambers, slight extension of the membranes and bending rigidity of the material, which are not accounted for in our simulator. These small errors are reduced when using TPU-coated nylon sheets, which are stiffer and can be inflated to a higher pressure.

While our algorithm for 2D inverse design assumes the provided profile is orthogonal to the ruling directions of the structure, the exact position of the lateral boundaries can be positioned arbitrarily in the local tangent plane of the surface, i.e., the tubes can have arbitrary lengths. We played with this possibility to design and fabricate the curling ribbons shown in Figure 10. The radius of the created helices corresponds to the 2D radius of curvature $\kappa^{-1}$, and its pitch corresponds to $2\pi\kappa^{-1} \tan \alpha$ where $\alpha$ is the angle between the welding lines of the tubes and the normal to the lateral boundary of the strip. Note that such 3D surfaces could also be obtained using the 3D version of our algorithm.

**3D examples.** We also tested our inverse design tool to program the shape of 3D non-cylindrical developable surfaces: two cones with different opening angles and a spiraling one (see Figure 12). Such structures were designed with radial seams, which locally ensure uniform values of the relative width of the welds $w_i/L$. As a consequence, we expect the local curvature $\kappa$ to be proportional to the width of the tube $L$ and as such to vary linearly with the radial position.

**Applications.** The prototyped inflatables function as lightweight yet structurally rigid elements. Figure 13 shows a comparison between a conventional monolayer of tubes made from two sheets and a bilayer made from three sheets. The former is barely able to

<figure><img src="image_12.png" /><figcaption>**Figure 10:** Helical strips. From top to bottom (resp. left to right): welding layouts, prototypes.</figcaption></figure>

sustain its own weight as it bends easily along the welding lines, whereas the former is considerably stiffer. The increased stiffness in the direction transverse to the tubes opens numerous potential applications in engineering as it removes the strong mechanical anisotropy of thin-sheet inflatables.

We illustrate a few use cases in Figure 14. The large stiffness of the bilayers could make them suitable for load-bearing furniture such as a deck chair as shown at a small scale version in (a). The rather homogeneous thickness provides effective cushioning for packaging or protective equipment as shown in (b) – here the helical shape of the balloon leads to a tighter fit of the object inside. 3D surfaces can be used for a variety of applications, such as sunshades or lightweight but voluminous garments as shown in (c) and (d).

## 8 Limitations and future work

We presented a simple and effective method to generate surfaces with desired curvature profiles. At the core of our approach is the use of a novel type of inflatable structure, *bilayers* made of three stacked membranes welded in pairs whose fabrication is amenable to standard manufacturing techniques. Our proposed inverse design algorithm relies on a precomputed map from welding pattern geometries to surface curvatures, and is thus purely geometric and scales very well as it does not require solving any complex optimization problem.

While very powerful, our method nevertheless comes with some limitations that leave opportunities for future work. In particular, our inflatables owe their stiffness to their tube-shaped chambers and their staggered arrangement. This internal structure limits the space of the shapes that we can program to developable surfaces (with ruling along the seams). Investigating welding patterns that would allow us to create inflatable bilayers with double curvature is a promising research direction.

Also, while our tool could in principle be used to compute the shape of curved surfaces of arbitrary sizes, addressing the design of very small or very large structures would certainly require some adaptations, e.g., accounting for the bending stiffness of the sheets
<figure><img src="image_13.png" /><figcaption>**Figure 11:** 2D results. From top to bottom: target profiles, welding layouts between, respectively, the top and the middle membranes (black) and the middle and bottom membranes (orange), simulated inflated shapes, errors between the target (black) and the simulated (colored) shapes (the colors indicate the distribution of the relative curvature error $|\kappa - \kappa_{\text{target}}| \cdot L$), actual prototypes.</figcaption></figure>

in the first case, and considering gravity in the second. We are also limited in the maximum curvature $\kappa_l$ that we can reproduce: structures with high curvature require small tubes and are therefore less stiff than less curved structures.

Moreover, neither our numerical simulator nor our theoretical model considers contact between the tubes that arise for very curved configurations. While sticking to the collision-free regime is sufficient for applications focusing on shape approximation, exploiting the collision regime could be interesting for other applications such as soft robotics. More generally, we did not study quantitatively the mechanical properties of our inflatables in this work which would also be worth future investigation.

The devised bilayers open new avenues for shape-morphing materials. First, the present work is restricted to multilayers made of two networks of tubes with three individual membranes. The fabrication process can however be extended by adding additional soldering steps with corresponding sheets allowing the making of networks of any number $n > 2$ of stacked tubes. While the design space of networks with similar welding patterns with $n > 2$ would remain limited to developable surfaces, we expect the bending stiffness of the tubular structures to scale with the number of chambers – a direction to be perhaps considered for inflatables at larger scales.
<figure><img src="image_14.png" /><figcaption>**Figure 12:** 3D results. From top to bottom: target surfaces, welding layouts between, respectively, the top and the middle membranes (black) and the middle and bottom membranes (orange), actual prototypes.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 13:** Mechanical properties. (a) The stiffness of a monolayer made of two sheets is dominated by the very high flexibility of the welding lines. When supported on its extremities, a 15 cm sample of 6 tubes barely sustains its own weight. (b) By contrast, a bilayer bridge of similar width not only holds its own weight but also sustains an additional load of three times its own weight.</figcaption></figure>

Another extension of multilayer inflatables is the differential pressurization of the tubular networks. Rather than setting the pressures as equal as done throughout this study, controlling each independent pressure is possible as each tubular layer has its own air supply. We show in Fig. 15 that a difference between the two inflating pressures $P_1$ and $P_2$ leads to the curling of a structure designed to remain flat if both pressures are similar ($w_1 = w_2 = 0$). In the shown examples, bending stiffness cannot be neglected in the modeling of the cross-section as the shape of the central membrane is non-trivial. The asymptotic case of $P_2 = 0$ corresponds to a monolayer of tubes made of two sheets differing in stiffness, whose curvature is ruled by asymmetric bending boundary layers [Vani et al. 2025]. Differential pressurization appears as an exciting avenue for soft robotics as the shape of the inflatables is precisely controlled

<figure><img src="image_16.png" /><figcaption>**Figure 14:** Examples of applications. Pneumatic developable surfaces can be used for inflatable furniture (a), inflatable stretcher (b), light deployable items such as umbrellas and sunshades (c), and sag-free garments (d).</figcaption></figure>

in real time via $P_1 – P_2$. It would nonetheless require adapting the existing model to account for bending stiffness.

## Acknowledgments

This work received financial support from the French Agence Nationale de la Recherche (ANR) under grant agreement ANR-22-CE51-0024 (MatAIRialS project). N.V. received funding from a doctoral fellowship awarded by École Normale Supérieure Paris-Saclay. We thank Antoine Garine-Wichatitsky for preliminary experiments, Samantha Kucher for technical assistance, as well as Benoît Roman and Arthur Lebée for valuable discussions.

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## 9 Appendix: analytical derivations

We thereafter present the analytical derivation of the geometry of individual tubes found in bilayer inflatables through volume maximization. Conceptualizing these tubes as unit cells, we predict the homogeneous curvature of multilayer inflatables with constant soldering widths.

**Geometry of individual tubes.** The following derivations assume the doubly asymptotic regime in which there is no extension of the sheets ($P \ll Yt/L$, with $P$ the inflating pressure, $Y$ and $t$ respectively the Young's modulus and thickness of the sheets, and $L$ the typical width of a tube), as well as no bending resistance from the sheets ($P \gg Yt^3/L^3$). In this regime, the minimization of elastic energy amounts to the maximization of the volume of each pressurized chamber with fixed metrics of the sheets.

To determine the shape of a cell with given length scales, it is thus simply necessary to express its volume – or its cross-section area in the present case of translation invariant tubes –, and to find the set of geometrical parameters that maximize the volume. As shown in Fig. 16, the bilayer is composed of a first set of tubes with a curved sheet of arc length $L - w_2$ connected to a wedge formed by two segments of length $(L - w_2)/2$, and a second set of tubes with two curved sheets of respective arc lengths $L$ and $w_2$, connected to a pair of segments of length $(L - w_2)/2$. We successively describe both configurations.

<figure><img src="image_18.png" /><figcaption>**Figure 16: Sketches of a pressurized tubes with a single (a) or two (b) curved interfaces.** The circular arcs correspond to the part of the chamber in contact with the atmospheric pressure, while the straight walls separate the tubes from neighboring ones.</figcaption></figure>

**Tubes with a single curved sheet.** The cross-section of the first considered tube corresponds to a soldering width $w_1 = 0$ at the bottom. The section is made of one circular arc of length $L - w_2$ and two straight lines of lengths $(L - w_2)/2$ as sketched in Fig. 16 (a). Laplace's law indeed dictates that the curved sheet – in contact with exterior pressure – has a homogeneous curvature $R$. Conversely, the two other edges separate the considered cell from neighboring ones. Equal pressures are applied on each side of the sheets, which thus remain straight. We introduce $\phi_1$ as the half-angle between the two straight sheets, and $\alpha$ as the tangent angle to the circular arc – which corresponds to half of its angular span. The radius of the circular arc is $R = (L - w_2)/2\alpha$. We then determine the area of the cross-section.

The area of the two triangles is $\frac{1}{4}(L - w_2)^2 \sin \phi_1 \cos \phi_1$.

The area of the circular arc is $R^2(\alpha - \sin \alpha \cos \alpha)$ where $2R\alpha = L - w_2$. In addition, we can express the half length of the chord as $\frac{L - w_2}{2} \sin \phi_1$ and also as $R \sin \alpha$. Expressing $\phi_1$ as a function of $\alpha$ gives:

$$ \sin \phi_1 = \frac{\sin \alpha}{\alpha}. \qquad (3) $$

The area $\mathcal{A}$ of the cross-section can thus be expressed as a function of $\alpha$ only:

$$ \mathcal{A} = \frac{(L - w_2)^2}{4\alpha} \left( 1 - \frac{\sin \alpha \cos \alpha}{\alpha} + \sin \alpha \sqrt{1 - \frac{\sin^2 \alpha}{\alpha^2}} \right). \quad (4) $$

We find the value of $\alpha$ which maximizes the cross-section area using the optimize module from the SciPy Python library. We find an optimal value $\alpha \approx 84.7^\circ$, with a corresponding angle $\phi_1 \approx 42.3^\circ$. Note that since the lengths of all walls are proportional to $L - w_2$, these values do not depend on $w_2$ or $L$: $\phi_1$ remains independent of any geometrical parameter in what follows.

The asymptotic case of a flat multilayer inflatable with both soldering widths equal to zero is described by the repetition of this unit cell with a periodic bottom-top flip. The shrinking ratio $l/L$ in this case corresponds straightforwardly to $\sin \phi_1 \approx 0.67$.

**Tubes with two curved sheets.** The other configuration corresponds to a soldering width $w_2 > 0$ on the top side. In this case, the cell is still delimited by two straight segments of length $(L - w_2)/2$ but now two circular arcs, one of length $w_2$ and the other of length $L$, as sketched in Fig. 16 (b). We introduce the respective tangent angles as $\gamma$ and $\beta$.

Force balance of the cross-section along its vertical axis of symmetry shows that the tensions in the two curved sheets are equal. Since they are subjected to the same internal pressure with the same tension, it follows from Laplace's law that they have the same radius of curvature $R$. From the expression of the chords of the two half circles, we thus have:

$$ 2R\gamma = w_2 \text{ and } 2R\beta = L \qquad (5) $$

so that

$$ \gamma = \frac{w_2}{L} \beta. \qquad (6) $$

The area of the lower circular arc is:

$$ \mathcal{A}_{\text{lower}} = R^2(\beta - \sin \beta \cos \beta) $$

$$ \mathcal{A}_{\text{lower}} = \frac{L^2}{4\beta} \left( 1 - \frac{\cos \beta \sin \beta}{\beta} \right). \qquad (7) $$

Similarly, the area of the upper circular arc is:

$$ R^2(\gamma - \sin \gamma \cos \gamma) $$

$$ \mathcal{A}_{\text{upper}} = \frac{w_2^2}{4\gamma} \left( 1 - \frac{\cos \gamma \sin \gamma}{\gamma} \right). \qquad (8) $$

The half width of the cell verifies:

$$ R \sin \beta = R \sin \gamma + \frac{L - w_2}{2} \sin \phi_2. $$

Inputting Eq. 5, we obtain:

$$ \sin \phi_2 = \frac{\frac{\sin \beta}{\beta} - \frac{w_2}{L} \frac{\sin \gamma}{\gamma}}{1 - \frac{w_2}{L}}. \qquad (9) $$
The area of the trapezoid is:

$$ \mathcal{A}_{\text{trap}} = \frac{(L - w_2)^2}{4} \sin \phi_2 \cos \phi_2 + 2 \frac{L - w_2}{2} \cos \phi_2 R \sin \gamma $$

$$ \mathcal{A}_{\text{trap}} = \frac{L^2(1 - \frac{w_2}{L})}{4} \cos \phi_2 \left( \left(1 - \frac{w_2}{L}\right) \sin \phi_2 + 2 \frac{w_2}{L} \frac{\sin \gamma}{\gamma} \right) $$

which finally gives:

$$ \mathcal{A}_{\text{trap}} = \frac{L^2}{4} \sqrt{\left(1 - \frac{w_2}{L}\right)^2 - \left(\frac{\sin \beta}{\beta} - \frac{w_2}{L} \frac{\sin \gamma}{\gamma}\right)^2} \left(\frac{\sin \beta}{\beta} + \frac{w_2}{L} \frac{\sin \gamma}{\gamma}\right) $$

The total area of the cross-section is thus:

$$ \mathcal{A} = \mathcal{A}_{\text{upper}} + \mathcal{A}_{\text{lower}} + \mathcal{A}_{\text{trap}}, \quad (10) $$

which can be expressed as a function of a single variable, $\gamma$ or $\beta$. Similarly to the previous configuration, we determine the value of the angles which maximizes the volume of the cell. Note that here this optimal angle depends on a single geometrical parameter, the relative size of the soldering width $w_2/L$.

We show the shape of individual tubes as predicted through volume maximization for increasing values of $w_2/L$ in Fig. 17 (a). We plot the evolution of the three angles $\gamma$, $\beta$ and $\phi_2$ as a function of the relative soldering width $w_2/L$ in Fig. 17 (b). Qualitatively, the shapes appear to agree fairly well with the ones predicted through numerical simulations. The asymptotic case $w_2/L \to 0$ corresponds to the previously studied cell with a single curved sheet, and thus in this limit $\phi_2$ tends toward $\phi_1$.

<figure><img src="image_19.png" /><figcaption>Figure 17: (a) Shapes of the pressurized cells for increasing relative soldering width $w_2/L$ as predicted through volume maximization. (b) Evolution of the angles $\beta$, $\gamma$, $\phi_2$ as a function of $w_2/L$.</figcaption></figure>

Resulting macroscopic curvature. Having determined the shape of each pressurized tube, we consider the geometry of an inflatable with one soldering width equal to zero ($w_1 = 0$) and the other soldering width with a constant non-zero value ($w_2 > 0$). Fig. 18 shows a sketch of three assembled tubes as used for the following derivation. The assembly serves as a unit cell for a larger inflatable.

<figure><img src="image_20.png" /><figcaption>Figure 18: Sketch of three neighboring tubes, two of which have a corresponding soldering width equal to zero.</figcaption></figure>

The rotation between the symmetry axes of the two consecutive smaller tubes is $2(\phi_1 - \phi_2)$, where $\phi_2$ is a function of the ratio $w_2/L$ in contrast to the constant $\phi_1$. Both angles were determined in the previous Section.

We aim to determine the radius of the circular arc shown in blue in the figure, delimited by the two axes of symmetry and such that its chord passes through the midpoints of the straight sheets. This arbitrary choice appears to be a fair candidate to define the typical radius of curvature of the overall structure. The length $c$ of the chord of the circular arc is:

$$ c = \frac{\delta_p + \delta_m}{2} + 2 \frac{L - w_2}{4} \sin \phi_2 + \frac{L - w_2}{2} \cos \phi_2 \tan (\phi_1 - \phi_2), \quad (11) $$

with $\delta_p$ and $\delta_m$ the bases of the central trapezoid. The arc length of the corresponding circular arc (in blue) is:

$$ \mathcal{L} = c \frac{\phi_1 - \phi_2}{\sin (\phi_1 - \phi_2)}. \quad (12) $$

The expected dimensionless curvature of the structure $\kappa L$ is thus:

$$ \kappa L = 2 (\phi_1 - \phi_2) \frac{L}{\mathcal{L}}, \quad (13) $$

where $\phi_2$ and $\mathcal{L}$ are now both known functions of $w_2/L$. We plot the evolution of this dimensionless curvature as a function of the relative soldering width $w_2/L$ in Fig. 6 (a) of the main paper. The analytical model based on volume maximization successfully captures the quantitative behavior of both numerical simulations and experiments.

The choice of the considered circular arc was arbitrary, and other definitions of the typical curvature could have been considered (e.g., using the circle directly passing through the midpoints of the straight edges, or the one passing through the midpoints of vertices joining the middle of the two soldering widths as done for simulations). We have checked that the theoretically predicted radius of curvature is not significantly impacted by this choice, down to a few percent in the range of interest $w_2/L < 0.6$.