2. Drag collapse through snapping

The waterbomb base consists of a disc of radius $R$, featuring alternating mountain and valley folds distributed evenly around a central vertex. The present study focuses on the traditional version of the origami waterbomb, comprising a total of eight folds. Additional results for different numbers of folds are presented in the supplementary material available at https://doi.org/10.1017/jfm.2024.985. The unit is rigid-foldable, meaning that facets are considered rigid, with all movement occurring at the joints, thus simplifying the kinematic analysis. In addition, we assume symmetry of motion, based on experimental observation, meaning that all the mountain folds deflect by the same amount, and the same holds for the valley folds. As a result, the mechanism reduces to a one-degree-of-freedom system (Hanna et al. Reference Hanna, Lund, Lang, Magleby and Howell2014, Reference Hanna, Magleby, Lang and Howell2015), where the shape of the entire structure can be determined from a single variable, chosen as the angle $\theta$ between a valley fold and the central axis passing through vertex (see figure 1a).

Figure 1. Drag collapse through flow-induced snap-through of a bistable origami unit. (a) Waterbomb base with radius $R$, composed of eight alternating valley and mountain folds with respective angles $\psi _v$ and $\psi _m$, and with unimodal kinematics described by the angle $\theta$. In the absence of loading, the bistable unit sits in one of two stable equilibria with angle $\theta _0$ and $\theta _0^*$. Evolution of (b) the drag force $F_d$ in a uniform airflow with increasing velocity $U$, (c) the dimensionless frontal area $S/{\rm \pi} R^2$ projected in the plane perpendicular to the flow (see schematics of the experimental set-up in the inset of b) and (d) the drag coefficient defined with respect to the instantaneous area $S$. The drag collapse observed in the snapping unit (with $R=5$ cm, $\kappa =19\times 10^{-3}$ N and $\theta _0 = 41^{\circ }$, blue solid curves) is compared with the quadratic drag of a rigid unit with the same geometry (same $R$ and $\theta _0$, grey dashed curves). Three tests were conducted on the same specimen, with error bars indicating the standard deviation (the grey zone denotes the variation in critical speed). Error bars for the drag coefficient are omitted for readability but are provided in the supplementary material.

Importantly, this unimodal origami mechanism has two stable equilibrium states and can transition between them through snapping. When at rest, the folded unit sits in one of these stable positions with its vertex pointing in one direction (see figure 1a i). Upon applying an external force, the structure flattens out towards a plane that represents an unstable equilibrium. When displaced beyond this plane, it snaps through and moves to a second stable position with the vertex pointing in the opposite direction (see figure 1a ii). These two stable positions are characterised by angles $\theta _0$ and $\theta _0^*$, and although they may look similar in figure 1(a), they are not mirror images of each other. The relationship between the angles of mountain and valley folds, $\varPsi _m$ and $\varPsi _v$ (measured in terms of deviation to straightness, see figure 1a), and $\theta$ also differs on both sides of the unstable equilibrium (Hanna et al. Reference Hanna, Lund, Lang, Magleby and Howell2014).

The origami bistable mechanism presents the advantage of being readily tunable. The stiffness of the folds notably influences the forces required to transition between stable states. One additional tuning parameter is the rest configuration, characterised by the angle $\theta _0$ in the first stable state. A compact unit (with a smaller $\theta _0$) requires more elastic energy to pass through the unstable planar state, as it has to undergo a greater deflection. The second equilibrium state $\theta _0^*$ also changes correspondingly with $\theta _0$, modulating the amplitude of the shape change upon snapping (the relationship between $\theta _0$ and $\theta _0^*$ is illustrated in Supplementary figure 6).

Specimens are fabricated using a technique developed in a previous study (Marzin et al. Reference Marzin, de Langre and Ramananarivo2022) and inspired by the literature (Overvelde et al. Reference Overvelde, De Jong, Shevchenko, Becerra, Whitesides, Weaver, Hoberman and Bertoldi2016; Zuliani et al. Reference Zuliani, Liu, Paik and Felton2018), which is based on layer superposition. The rigid facets are laser cut from a $350$-$\mathrm {\mu }{\rm m}$-thick sheet of Mylar, which is then attached to a thinner sheet using double-sided tape, forming the creases. These folds act as elastic hinges of specific stiffness and rest angles. Rest angles are prescribed by utilising the thermoformable property of PET (see Appendix A). As shown in a previous study, the folds then operate as torsional springs, exhibiting elastic behaviour around their assigned rest angle (Marzin et al. Reference Marzin, de Langre and Ramananarivo2022). The stiffness can be adjusted by using different thicknesses of the thin sheet, ranging from $40\unicode{x2013}100\ \mathrm {\mu }{\rm m}$. To measure the stiffness experimentally, a single fold is fabricated using the same process and is tested by tracking the opening angle as an increasing torque is applied to the hinge (see Appendix A and Supplementary figure 1). The value of the stiffness is then obtained from the slope of the torque–angle curve. The fabrication process shows a standard deviation of 8$\%$ for $\kappa$.

We conducted tests on 10 origami units, sequentially varying the radius $R=4\unicode{x2013}7$ cm, the stiffness of the folds $\kappa =8$–$92 \times 10^{-3}$ N, and the rest configuration characterised by the angle $\theta _0=32^{\circ }\unicode{x2013}66^{\circ }$. These units are subjected to a steady air flow generated by an open jet wind tunnel with a square cross-section of width $40$ cm and flow speed varied in $1\unicode{x2013}16\ {\rm m}\ {\rm s}^{-1}$. An elbow arm is used upstream, to hold the unit at its vertex in the centre of the cross-section (see the inset of figure 1b). This arm is connected to a six-component force sensor, measuring the drag component of the fluid force in the direction of the flow. For each measurement, we average the drag over 30 s and subtract the drag on the support system alone. The flow-induced shape changes are characterised through the projected area $S$ of the unit in the plane perpendicular to the flow, captured by a camera placed downstream and extracted through Matlab custom image analysis. The observed kinematics align with the unimodal mechanism described earlier, which is rigidly foldable with motion restricted to the joint and exhibits rotational symmetry of motion. The entire geometry of the unit can be reconstructed from $S$, which will be further used as the observable to measure shape reconfiguration (the relationship between $S$ and $\theta$ is illustrated in Supplementary figure 5).

Figure 1(b) displays the evolution of drag with flow speed for a typical experiment (solid blue curve), alongside the frontal area in figure 1(c), and the drag coefficient calculated using this instantaneous area $C_d=F_d/(1/2 \rho S U^2)$ in figure 1(d), to isolate the effect of shape on drag. Results are compared with the drag behaviour of a rigid cell frozen in its initial configuration (dashed grey curves), which exhibits a classical $U^2$ law, associated with a relatively constant drag coefficient. Initially, the unit is in its first stable position $\theta _0$, with the concave side facing the flow. As the flow speed gradually increases, the cell symmetrically opens up, leading to a faster increase of drag compared with its rigid counterpart. This enhanced drag is attributed solely to the expansion in frontal area, as the drag coefficient actually decreases with the cell adopting a less deep, parachute-like shape, consistent with observations from simpler geometries such as hemispherical caps (Hoerner Reference Hoerner1965; Ganedi et al. Reference Ganedi, Oza, Shelley and Ristroph2018). This drag coefficient reaches approximately $C_d=1.1$ as the cell approaches the flat unstable state (close to the 1.17 value for a flat disc), after which the unit snaps to a more compact and streamlined configuration. Note that slight cell vibrations make it challenging to precisely attain $S/{\rm \pi} R^2=1$. The sudden reconfiguration results in a discontinuity in the drag force on the unit, with a drop by up to $69\,\%$ among the specimens tested in this study, produced by both the significant area reduction in figure 1(c) and shape streamlining in figure 1(d). As $U$ is further increased, the unit closes up. This closing regime has been studied extensively in a previous study (Marzin et al. Reference Marzin, de Langre and Ramananarivo2022); it features an increase of drag that is slower than the quadratic $U^2$ law for rigid objects, owing to the reduction in frontal area exposed to the flow, combined with a decreasing drag coefficient resulting from shape streamlining.

To estimate the measurement uncertainty, we conducted three repetition of the experiments on the same specimen, re-annealed between each iteration to ensure a consistent initial configuration. Error bars in figure 1 represent the standard deviation, indicating relative variation in drag force and frontal area within 2 %. Small vibrations of the unit, especially as it approaches the unstable flat state, likely contribute to variations in the critical snapping velocity, which however remain within 2 %. Variability in the experiments were found to mainly results from the origami fabrication process. Additional tests conducted on five distinct units, all fabricated with identical parameters, exhibited a higher but reasonable dispersion, with about 10 % for the drag data, 9.5 % for the frontal area and 5 % for the critical snapping velocity (see Supplementary figure 3 and the supplementary material for more details on sources of uncertainties and measurement precision).

We thus observe a non-continuous evolution of drag with flow speed with a marked collapse. This behaviour can be described by selected observable parameters, namely the critical flow velocity of the snapping $U_c$, the maximum drag force reached just before snapping $F_{d,max}$ and the subsequent drag drop $\Delta F_d$. These parameters are prominent features characterising the discontinuity but are also relevant for practical applications, such as designing protection devices or valves. They determine a maximum load, the amount of load reduction, or the critical speed at which the structure would operate. In the following section, we investigate how those quantities are related to the origami geometrical and mechanical properties, specifically its size $R$, rest configuration $\theta _0$ and fold stiffness $\kappa$.

3. Aeroelastic mechanism

We first vary the radius $R$ of the unit, while keeping the stiffness at $\kappa =19\times 10^{-3}$ N and the rest angle at $\theta _0=41^{\circ }$. Note that fabrication produces a small variability in the rest angle among specimens, which is visible in figure 2(b) in the $6\,\%$ dispersion in the dimensionless frontal area in the absence of flow ($U=0$). As shown in figure 2(a,b), all units exhibit the characteristic behaviour described earlier, that is a non-continuous evolution of drag with flow speed with a sudden drop associated with the snap-through of the origami unit. However, bigger units (darker orange curves) initially experience a higher fluid loading, because of the larger area exposed to the flow. As a result, they reach the flat state $S/{\rm \pi} R^2=1$ faster and flip at a lower critical speed (see figure 2b). Interestingly, in figure 2(a), the maximum drag $F_{d,max}$ experienced by the units before snapping is the same, and they feature similar jumps amplitude $\Delta F_d$. Changing the radius of the unit thus primarily shifts the critical velocity $U_c$ at which the drag collapse occurs, while preserving $F_{d,max}$ and $\Delta F_d$.

Figure 2. Influence of the origami structural parameters on drag. Evolution of (a,c,e) the drag force $F_d$ and (b,d,f) the dimensionless frontal area $S/ {\rm \pi}R^2$ as a function of flow velocity $U$, for origami units with different (a,b) radii $R$, (c,d) fold stiffness $\kappa$ and (e,f) rest angle $\theta _0$. The unit $R=5$ cm, $\kappa =19 \times 10^{-3}\ {\rm N}$ and $\theta _0=41^{\circ }$ is common to all graphs.

Figure 2(c,d) shows the results for origami cells with different fold stiffness, but the same size $R=5$ cm and rest angle $\theta _0=41^{\circ }$ (with a similar degree of variability as earlier). Stiffer units (darker blue curves) deploy slower with increasing $U$ owing to larger resistance to deformation, and reach the flat state at larger critical velocities (see figure 2d). The most rigid one does not reach this unstable point within the range of flow covered here. Snapping at larger speeds $U_c$ results both in higher drag peak values $F_{d,max}$ and larger jumps $\Delta F_d$ (see figure 2c). Stiffness thus affects all three drag features, offering an additional lever for control.

Finally, we vary the degree of opening of the cell rest state $\theta _0$, while keeping the size and stiffness constant as $R=5$ cm and $\kappa =19\times 10^{-3}$ N, respectively. Starting from a more compact rest state (lighter green curves) delays the flipping towards higher flow velocities, as depicted in figure 2(f). Intuitively, reaching the flat state that is further away requires greater work from fluid loading. As was the case for the $\kappa$-series of units, snapping at greater $U_c$ results in larger maximal drag $F_{d,max}$ in figure 2(e). As mentioned earlier and illustrated in the schematics of figure 2(f), changing $\theta _0$ also affects the degree of closure of the second stable state. Small $\theta _0$ (light green) results in a larger collapse in the frontal area post-snapping compared with units initially more opened (dark green) in figure 2(f). It correspondingly leads to larger jumps in drag $\Delta F_d$ in figure 2(e). The rest angle thus completes the set of three parameters that will allow us to adjust the three drag characteristics $U_c$, $F_{d,max}$ and $\Delta F_d$.

The trends observed through variations in parameters reflect the interplay between external fluid loading and elastic restoring forces of the origami unit, which governs the unit's reconfiguration process. This fluid–elastic competition can be analysed in terms of Cauchy number $C_y$, as introduced in a previous study (Marzin et al. Reference Marzin, de Langre and Ramananarivo2022) and frequently used in the literature (Alben et al. Reference Alben, Shelley and Zhang2002; Gosselin et al. Reference Gosselin, de Langre and Machado-Almeida2010; de Langre et al. Reference de Langre, Gutierrez and Cossé2012). This dimensionless number examines the relative magnitudes of the work done by fluid forces, $\rho U^2 R^3$, and the elastic energy of folding, $\kappa R$:

(3.1)\begin{equation} C_y = \frac{\rho U^2 R^2}{\kappa}. \end{equation}

In figure 3(a), drag measurements are re-plotted in the dimensionless form $\overline {F_d}=F_d/\kappa$ as a function of $C_y$, using the fold stiffness $\kappa$ as a characteristic force scale, in line with Marzin et al. (Reference Marzin, de Langre and Ramananarivo2022). Here, we only report data for units with different stiffness $\kappa$ and radius $R$, which share the same initial rest configuration $\theta _0$. Data collapse onto a master curve, with a drag peak and subsequent jump that are solely set by the fold stiffness, and which occur at a critical Cauchy number of $C_{y,c} \approx 14$ for all units. Alternatively, the reconfiguration number $\mathcal {R}=F_d/F_{d,r}$ is often used to discuss the effect of deformability on drag (Schouveiler & Boudaoud Reference Schouveiler and Boudaoud2006; Gosselin et al. Reference Gosselin, de Langre and Machado-Almeida2010). It compares the drag force $F_d$ on the deforming cell to that on a rigid unit with the same geometry $F_{d,r}$, which is evaluated by fitting a $U^2$ law to drag measurements at low speeds where deformation is minimal. The curve of $\mathcal {R}$ as a function of $C_y$ in figure 3(a) highlights distinct drag scaling regimes. At low $C_y$, where fluid loading is small compared with structural rigidity, the drag behaves like that of a rigid body, and $\mathcal {R} \sim 1$. As $C_y$ increases, $\mathcal {R}$ rises due to the larger frontal area of the cell, then drops sharply below 1 at the critical Cauchy number $C_{y,c}$, and continues to decrease continuously with further deformation. In the large-deformation regime, $\mathcal {R}$ declines with a logarithmic slope approaching $-1$, consistent with the plateau-like behaviour for drag observed in a previous study on the waterbomb unit (Marzin et al. Reference Marzin, de Langre and Ramananarivo2022). We previously showed that the dimensionless quantities $C_y$ and $\overline {F_d}$ capture the effect of origami size and stiffness in fluid–elastic mechanisms. However, it does not account for the more intricate influence of the rest angle $\theta _0$. As shown in figure 3(b i), the latter affects the peak and jump value of $\overline {F_d}$, as well as the critical Cauchy number for snapping.

Figure 3. Aeroelastic mechanism. (a) Evolution of the dimensionless drag $\overline {F_d}=F_d/\kappa$ as a function of the Cauchy number $C_y=\rho U^2R^2/\kappa$ for origami units with different radii $R$ and fold stiffness $\kappa$ (with the same rest angle $\theta _0=41^{\circ }$). The inset shows the reconfiguration number $\mathcal {R}=F_d/F_{d,r}$, which compares the drag force $F_d$ with that of a rigid unit having the same geometry $F_{d,r}$. (b) Plot of $\overline {F_d}$ as a function of $C_y$ for units with different initial degree of opening $\theta _0$ (with the same $R=5$ cm and $\kappa =19 \times 10^{-3}$ N). Experimental force data are compared with theoretical predictions (solid curves).

Next, we develop a simplified theoretical model to further explore and corroborate the link between the origami unit's characteristics and drag. This model will also be used for the inverse problem. It employs the same energy approach as previous work on the closing dynamics of the waterbomb unit in a flow (Marzin et al. Reference Marzin, de Langre and Ramananarivo2022) and has been extended to account for the snapping. We provide a brief overview of the main elements of the model here, and additional details can be found in Appendix A and in Marzin et al. (Reference Marzin, de Langre and Ramananarivo2022). The static equilibrium of the unit in a flow is determined through the minimisation of its energy $E_{el}-W$, comprising both the elastic potential energy associated with folding $E_{el}$ and the work done by fluid forces $W$ as the unit deforms:

(3.2)\begin{align} E_{el}-W =4\frac{\kappa R}{2}[(\psi_m - \psi_m^0)^2 + (\psi_v - \psi_v^0)^2] - 8 \int_{\theta_0}^{\theta} \int_{S_f} \rho(\boldsymbol{U}\boldsymbol{\cdot}\boldsymbol{n}_{\theta'})^2 \,{\rm d}S_f \boldsymbol{n}_{\theta'}\boldsymbol{\cdot}{\rm d} \boldsymbol{X}_{\theta'}, \end{align}

where $\rho$ is the air density and $S_f={\rm \pi} R^2/8$ is the facet area (with other quantities introduced subsequently). The first term of (3.2) corresponds to the elastic potential energy of the four mountain and four valley folds, modelled as elastic torsional springs with stiffness $\kappa$. It scales with the square of the deviation of their respective angles $\psi _m$ and $\psi _v$ to their rest values $\psi _{m}^{0}$ and $\psi _{v}^{0}$ (Hanna et al. Reference Hanna, Lund, Lang, Magleby and Howell2014). The second term corresponds to the work done by fluid forces acting on the unit facets, which is the same for all eight facets owing to the rotational symmetry of motion. The Reynolds number for our experiments falls within the range $Re=UR/\nu \approx 10^3\unicode{x2013}10^5$, where form drag predominates over friction drag, and the latter is neglected. As discussed in Marzin et al. (Reference Marzin, de Langre and Ramananarivo2022), friction forces likely become significant in regimes with substantial deformation, where facets nearly align with the flow. However, this particular regime is not reached in our present study. Consistent with earlier studies on the reconfiguration of flexible structures in a flow (Schouveiler & Boudaoud Reference Schouveiler and Boudaoud2006; Gosselin et al. Reference Gosselin, de Langre and Machado-Almeida2010; Schouveiler & Eloy Reference Schouveiler and Eloy2013; Marzin et al. Reference Marzin, de Langre and Ramananarivo2022), pressure is computed using conservation of momentum principles. It is proportional to the square of the component of flow momentum perpendicular to the facet, denoted as $\rho (\boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {n}_{\theta })^2$, where $\boldsymbol {n}_{\theta }$ represents the normal unit vector. While simplified, this model has demonstrated effectiveness in capturing fluid–elastic behaviours (Schouveiler & Boudaoud Reference Schouveiler and Boudaoud2006; Gosselin et al. Reference Gosselin, de Langre and Machado-Almeida2010; Schouveiler & Eloy Reference Schouveiler and Eloy2013; Marzin et al. Reference Marzin, de Langre and Ramananarivo2022). It offers a reasonable approximation of more complex potential flow expressions (Alben, Shelley & Zhang Reference Alben, Shelley and Zhang2004; Marzin et al. Reference Marzin, de Langre and Ramananarivo2022), which would be challenging to implement for three-dimensional geometries. The pressure force is integrated over the facet area $S_f$ and its work is computed along the facet trajectory, with $d \boldsymbol {X}_{\theta }$ the local displacement. By expressing the energy in a dimensionless form as $\overline {E_{el}-W}=(E_{el}-W)/\kappa R$, the previously introduced Cauchy number $C_y$ emerges as the pertinent fluid–elastic parameter:

(3.3)\begin{align} \overline{E_{el}-W} =2\big[(\psi_m - \psi_m^0)^2 + (\psi_v - \psi_v^0)^2\big] - 8 C_y \int_{\theta_0}^{\theta} \int_{\bar{S}_f} \rho(\boldsymbol{e}_z\boldsymbol{\cdot}\boldsymbol{n}_{\theta'})^2 \,{\rm d}\bar{S}_f \boldsymbol{n}_{\theta'}\boldsymbol{\cdot}{\rm d} \bar{\boldsymbol{X}}_{\theta'}, \end{align}

where $\bar {\cdot }$ denotes dimensionless quantities, using the radius as characteristic length, and $\boldsymbol {e}_z$ is the unit vector in the direction of the flow. In (3.3), all kinematic variables of the one-degree-of-freedom mechanism depend on a single parameter $\theta$. However, these relationships differ on each side of the flat state. The equilibrium angle $\theta$, which satisfies ${\rm d}\overline {(E_{el}-W)}/{\rm d}\theta =0$, is determined numerically while gradually increasing the Cauchy number, first addressing the opening phase of the kinematics. As the system approaches the flat state at $\theta =90^{\circ }$ beyond a specific criterion, the kinematic relations are modified to describe the closing phase. From the static equilibrium angle $\theta$, we can determine the drag force acting on the unit as the component of fluid pressure forces in the direction of the flow (see Appendix A).

Theoretical drag results are presented in figure 3(a) in dimensionless form, considering a given rest angle $\theta _0=41^{\circ }$ based on experiments (black solid line). The simplified model successfully captures experimental features, displaying a peak in drag that remains unaffected by the unit's size, followed by a sudden drop at a critical Cauchy number. Although the model tends to overestimate $F_{d,max}$ and $\Delta F_d$, the quantitative agreement is reasonable, considering that no adjustable parameters were used and the simplifications made. As depicted in figure 3(b ii), the model also effectively captures the effect of varying the initial rest angle $\theta _0$ on drag. A more compact unit exhibits a higher drag peak and jump, along with a higher critical Cauchy number. This model will be valuable for the inverse design of origami units to achieve the desired drag, as we discuss in the next section.

4. Inverse design approach

We have characterised how the waterbomb unit features influence its drag behaviour. We now pose the question: ‘Can we identify the specific geometrical and mechanical origami parameters that would lead to a targeted drag collapse in a uniform fluid flow?’. More specifically, we formulate the inverse problem where the objective is to find the optimal set of parameters $(R,\kappa,\theta _0)$ that will result in a collapse of drag, with a predefined target peak drag force $F_{d,max}^t$ and jump $\Delta F_d^t$ value at a critical flow speed $U_c^t$. To address this, we use the model presented earlier and validated experimentally, which establishes a relationship between the parameters $(R, \kappa, \theta _0)$ and the resulting drag characteristics $(F_{d,max}, \Delta F_d, U_c)$. The optimisation is implemented numerically in Matlab using the trust-region-dogleg algorithm of fsolve solver. It solves the system of equations $F_d-F_{d,max}^t=0$, $\Delta F_d-\Delta F_d^t=0$ and $U_c-U_c^t=0$ with respect to the origami unit's features $(R,\kappa,\theta _0)$. The algorithm finds a single solution, satisfying the constraints by a maximum residual of $O(10^{-19})$ within a hundred iterations.

To validate the inverse design approach, we test our prediction experimentally by fabricating the optimised design and measuring the evolution of its drag with flow speed. While the specific radius $R$ and rest angle $\theta _0$ can be easily implemented through laser-cutting and an adjustable mount, achieving an arbitrary fold stiffness $\kappa$ is more challenging. This limitation arises from the finite number of sheet thicknesses available for manufacturing the folds. To overcome this constraint, we introduce slits in the folds. These slits modulate the effective length of the folds and, consequently, their stiffness, enabling a controlled variation of $\kappa$ by adjusting their number (see Appendix A and Supplementary figure 2).

To demonstrate our ability to control the drag behaviour, we create sets of specimens with one parameter gradually changing while keeping the other two fixed. We first vary the targeted critical flow speed for snapping, as $U_c^t=[4, 7, 10]\ {\rm m}\ {\rm s}^{-1}$, while prescribing constant peak drag value $F_{d,max}^t=0.8$ N and jump $\Delta F_d^t=0.6$ N. The corresponding optimised origami parameters $(R,\kappa,\theta _0)$ are provided in Appendix A. Consistent with the differences observed earlier in figure 3 between experiments and theory, the experimental realisations exhibit lower values in figure 4(b), namely $U_c=[3.7,6.0,8.8]\ {\rm m}\ {\rm s}^{-1}$, $F_{d,max}=0.29$ N and $\Delta F_d=0.10$ N. Nevertheless, we successfully achieved a gradual variation in $U_c$ while keeping the other two parameters unchanged. Note that the optimised designs have the same rest angle and fold stiffness, but varying radius, which is consistent with the previous results shown in figure 2(a). We indeed demonstrated that $F_{d,max}$ and $\Delta F_d$ are size-independent and, therefore, determined by the specific combination of $(\kappa,\theta _0)$. In figure 4(c,d), we systematically vary the jump $\Delta F_d$ and peak $F_{d,max}$, respectively (target values are reported in Appendix A). Similarly, although the physical samples have drag quantities below the target values, they still showcased the intended selective variations. Our results confirm the validity of our inverse design approach, but there are limitations to how closely we can match desired drag behaviours. These limitations arise primarily from the theoretical model employed in the optimisation procedure. While it accurately captures the effect of origami parameters on drag, it only provides semi-quantitative agreement with the experimental data. Note that deviations between the targeted drag features and those of the resulting optimised design in figure 4 align with differences reported in prior experimental–theoretical comparisons of figure 3 (see Supplementary figures 11 and 12). Refining the model, notably using a more realistic representation for the fluid dynamics, is thus expected to produce better quantitative results.

Figure 4. Targeting drag collapse features. (a) The inverse design approach consists of finding the set of origami structural parameters ($R$, $\kappa$, $\theta _0$) that will lead to a collapse of drag with a targeted maximum before snapping $F_{d,max}$, jump $\Delta F_d$ and occurring at a defined critical flow speed $U_c$. Evolution of drag with flow speed for physical prototypes of optimised origami designs, which are calculated to produce a selective variation of (b) $U_c$, (c) $\Delta F_d$ and (d) $F_{d,max}$, while keeping the two remaining features identical. Respective target values of ($U_c$, $\Delta F_d$, $F_{d,max}$) are indicated in Appendix A.