Multivalued Inverse Design: Multiple Surface Geometries from One Flat Sheet

Designing flat sheets that can be made to deform into three-dimensional shapes is an area of intense research with applications in micromachines, soft robotics, and medical implants. Thus far, such sheets were designed to adopt a single target shape. Here, we show that through anisotropic deformation applied inhomogeneously throughout a sheet, it is possible to design a single sheet that can deform into multiple surface geometries upon different actuations. The key to our approach is development of an analytical method for solving this multivalued inverse problem. Such sheets open the door to fabricating machines that can perform complex tasks through cyclic transitions between multiple shapes. As a proof of concept, we design a simple swimmer capable of moving through a fluid at low Reynolds numbers.

Figure 1

Inverse design of pluripotent sheets. Left, target surfaces with defined Gaussian curvatures $K$. Center, inverse designed sheet, where $\widehat{\mathbf{n}}$ specifies deformation’s orientation and $\epsilon$ is an experimentally accessible system feature controlling the deformation’s magnitude, which is designed with $\widehat{\mathbf{n}}$ such that the sheet deforms into the target surfaces. Right, actuated sheet deforming into multiple target surfaces in response to different values of the stimulus $\mathrm{\Lambda }$.

Figure 2

Inverse design scheme of a flat uniaxial sheet with a designable elastic response [${\lambda }_1(\epsilon )\ne \text{const}]$ for two target surfaces with Gaussian curvatures $K_1$ and $K_2$ in response to stimuli ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$. $(u,v)$ coordinates given by integral curves of deformation’s anisotropy axis $\widehat{\mathbf{n}}$ or ${\widehat{\mathbf{n}}}_{\perp }$ are shared by all surfaces and allow derivation of a system of equations describing the inverse problem. Numerical integration of this inverse problem is given by iteration of the following integration steps: Bottom left, given complete data on a director integral curve, $u\mathrm{data}=({\mathbf{r}}_{\mathrm{\Lambda }},{\widehat{\mathbf{n}}}_{\mathrm{\Lambda }},\alpha ,b,\epsilon ,q)$ is propagated a step $dv$ along the director perpendicular, forming a new director integral curve. Top, curvature of target surfaces along new director integral curve is obtained. Bottom right, initial data for $\beta$ and $s$, given on initial director-perpendicular curve, are integrated along new director integral curve to complete the data on it.

Figure 3

Inverse design of multiple shapes. A flat uniaxial sheet with two designable system features [${\lambda }_i=\exp ({\epsilon }_i{\mathrm{\Lambda }}^i),i\in \{1,2\}$] is designed to deform into three target shapes in response to two stimuli ${\mathrm{\Lambda }}^1$ and ${\mathrm{\Lambda }}^2$. The sheet deforms into a sphere and then a facelike mask in response to ${\mathrm{\Lambda }}^1$ and into a wavy sheet in response to ${\mathrm{\Lambda }}^2$. Maximal strains are below 300% and are within experimental reach [45].

Figure 4

(a) Bilayered uniaxial sheet composed of a top layer with an azimuthal director pattern and a bottom layer with the orthogonal radial director pattern. Top and bottom layers deform into cones and anticones upon actuation, respectively [47]. (b) Nonreciprocal cycle of shape transformations between multiple flat and conical shapes in the bilayered design through the use of two independently controllable stimuli ${\mathrm{\Lambda }}^1$ and ${\mathrm{\Lambda }}^2$ and the corresponding cycle in parameter space. (c) Locomotion of a swimmer at low Reynolds number with net displacement $\mathrm{\Delta }x$ achieved during each cycle. (d) Bounds on work [$E$ from Eq. (8)] that system can perform on its surroundings at each stage of cycle.