# Semi-implicit methods for the dynamics of elastic sheets

## TL;DR

This paper introduces semi-implicit algorithms for simulating elastic sheet dynamics, significantly increasing stability and efficiency over explicit methods, with applications to active gels and self-assembly.

## Contribution

It develops semi-implicit time stepping algorithms that improve stability and efficiency for elastic sheet simulations, avoiding the complexity of fully-implicit methods.

## Key findings

- Semi-implicit methods allow larger stable time steps, up to 2-3 orders of magnitude greater than explicit schemes.
- The algorithms are stable for all time steps in overdamped dynamics.
- Transitions from quasi-periodic to chaotic behavior are observed as parameters vary.

## Abstract

Recent applications (e.g. active gels and self-assembly of elastic sheets) motivate the need to efficiently simulate the dynamics of thin elastic sheets. We present semi-implicit time stepping algorithms to improve the time step constraints that arise in explicit methods while avoiding much of the complexity of fully-implicit approaches. For a triangular lattice discretization with stretching and bending springs, our semi-implicit approach involves discrete Laplacian and biharmonic operators, and is stable for all time steps in the case of overdamped dynamics. For a more general finite-difference formulation that can allow for general elastic constants, we use the analogous approach on a square grid, and find that the largest stable time step is two to three orders of magnitude greater than for an explicit scheme. For a model problem with a radial traveling wave form of the reference metric, we find transitions from quasi-periodic to chaotic dynamics as the sheet thickness is reduced, wave amplitude is increased, and damping constant is reduced.