Curved Geometries from Planar Director Fields: Solving the Two-Dimensional Inverse Problem

Thin nematic elastomers, composite hydrogels, and plant tissues are among many systems that display uniform anisotropic deformation upon external actuation. In these materials, the spatial orientation variation of a local director field induces intricate global shape changes. Despite extensive recent efforts, to date there is no general solution to the inverse design problem: How to design a director field that deforms exactly into a desired surface geometry upon actuation, or whether such a field exists. In this work, we phrase this inverse problem as a hyperbolic system of differential equations. We prove that the inverse problem is locally integrable, provide an algorithm for its integration, and derive bounds on global solutions. We classify the set of director fields that deform into a given surface, thus paving the way to finding optimized fields.

Figure 1

Integration of the director field. Left: Initialization. Given a surface with curvature $K$ the initial condition consists of two perpendicular curves that will become integral curves of the actuated director and its perpendicular. The corresponding unactuated integral curves inherit the geodesic curvatures $b={\lambda }^{-\nu }{\kappa }_{g1}$ and $s=\lambda {\kappa }_{g2}$, respectively, and we are free to set $\alpha =1$ along the former and $\beta =1$ along the latter. We next integrate to obtain $\beta$ and $s$ along the director integral curve completing the initialization of the SOE. Right: Iterative integration step. (i) $\alpha ,\widehat{\mathbf{n}},\mathbf{r}$, and $b$ are integrated a $dv$ step along ${\widehat{\mathbf{n}}}_{\perp }$ on the flat sheet. ${\widehat{\mathbf{n}}}_A$ and ${\mathbf{r}}_A$ are integrated a step $dv$ along ${\widehat{\mathbf{n}}}_{A\perp }$ on the desired curved surface. (ii) $\beta$ and $s$ are integrated along $\widehat{\mathbf{n}}$ according to the Gaussian curvature $K_{\mathcal{M}}$ pulled back from the desired surface. With the information at hand this step can now be reiterated.

Figure 2

Solutions of the SOE: Director curves on a flat sheet and the shapes they take when actuated, found by integrating the SOE. Left: Sphere with Gaussian curvature $K_{\mathcal{M}}=1$. Center: Constant negative curvature, $K_{\mathcal{M}}=-1$, body of rotation. Right: Anisotropic Gaussian surface, with varying Gaussian curvature.

Figure 3

Distinct director fields (bottom) deforming into the same section of the unit sphere (top). The solutions are characterized by the geodesic curvatures of the initial curves, the $u$ baseline (thick blue) with ${\kappa }_{g1}$ and the $v$ baseline (dashed red) with ${\kappa }_{g2}$, from which they are integrated.

Figure 4

Surface mimicking a human face. The director field is found by integrating the SOE according to the iterative algorithm depicted in Fig. 1.