# A nearest-neighbour discretisation of the regularized stokeslet boundary integral equation

## TL;DR

This paper introduces a meshless nearest-neighbour discretisation method for the regularized stokeslet boundary integral equation, significantly reducing computational costs and increasing accuracy in biological fluid dynamics simulations.

## Contribution

A novel nearest-neighbour interpolation approach that decreases degrees of freedom needed for discretisation, enhancing efficiency and accuracy over classical methods.

## Key findings

- Over 10x reduction in computational cost compared to standard discretisation
- More accurate results with less force discretisation
- Effective application to complex biophysical problems

## Abstract

The method of regularized stokeslets is extensively used in biological fluid dynamics due to its conceptual simplicity and meshlessness. This simplicity carries a degree of cost in computational expense and accuracy because the number of degrees of freedom used to discretise the unknown surface traction is generally significantly higher than that required by boundary element methods. We describe a meshless method based on nearest-neighbour interpolation that significantly reduces the number of degrees of freedom required to discretise the unknown traction, increasing the range of problems that can practically solved, without excessively complicating the task of the modeller. The nearest-neighbour technique is tested against the classical problem of rigid body motion of a sphere immersed in very viscous fluid, then applied to the more complex biophysical problem of calculating the diffusion timescales of a macromolecular structure modelled by three closely-spaced non-slender rods. A heuristic for finding the required density of force and quadrature points by numerical refinement is suggested. Matlab/GNU Octave code for the key steps of the algorithm are given, which predominantly use basic linear algebra operations, with a full implementation being provided on github. Compared with the standard Nystr\"om discretisation more accurate and substantially more efficient results can be obtained by de-refining the force discretisation relative to the quadrature discretisation: a cost reduction of over 10 times with improved accuracy is observed. This improvement comes at minimal additional technical complexity. Future avenues to develop the algorithm are then discussed.