# A stable finite element method for low inertia undulatory locomotion in three dimensions

## TL;DR

This paper introduces a novel stable finite element method for simulating the low-inertia, three-dimensional undulatory locomotion of microswimmers, accurately capturing elastic and viscous deformations with preserved geometric properties.

## Contribution

The authors develop a new finite element scheme that ensures stability and geometric fidelity for modeling low-inertia microswimmer dynamics in three dimensions.

## Key findings

- The method is stable and preserves frame orthonormality up to machine precision.
- Numerical experiments validate the method's accuracy and applicability.
- The scheme effectively simulates undulatory locomotion in low-inertia regimes.

## Abstract

We present and analyse a numerical method for understanding the low-inertia dynamics of an open, inextensible viscoelastic rod - a long and thin three dimensional object - representing the body of a long, thin microswimmer. Our model allows for both elastic and viscous, bending and twisting deformations and describes the evolution of the midline curve of the rod as well as an orthonormal frame which fully determines the rod's three dimensional geometry. The numerical method is based on using a combination of piecewise linear and piecewise constant functions based on a novel rewriting of the model equations. We derive a stability estimate for the semi-discrete scheme and show that at the fully discrete level that we have good control over the length element and preserve the frame orthonormality conditions up to machine precision. Numerical experiments demonstrate both the good properties of the method as well as the applicability of the method for simulating undulatory locomotion in the low-inertia regime.