# Curved geometries from planar director fields - Solving the two-dimensional inverse problem

## TL;DR

This paper formulates and solves the inverse problem of designing director fields in thin anisotropic materials to achieve specific 2D surface geometries upon actuation, providing a mathematical framework and algorithms.

## Contribution

It introduces a hyperbolic differential equation approach to the inverse design problem and offers a classification of solutions for targeted surface deformations.

## Key findings

- The inverse problem is locally integrable.
- An algorithm for integrating the inverse problem is provided.
- Bounds on global solutions are derived.

## Abstract

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.