An ALE method for simulations of axisymmetric elastic surfaces in flow

TL;DR

This paper introduces a novel Arbitrary Lagrangian-Eulerian (ALE) method for simulating axisymmetric elastic surfaces in fluid flow, enabling accurate and efficient modeling of biophysical membrane dynamics.

Contribution

The paper develops a new ALE approach combining surface finite differences and finite elements for axisymmetric elastic surfaces in Navier-Stokes flow, with an implicit coupling strategy for stability.

Findings

Accurate simulation of shape oscillations of microswimming shells.

Modeling of biological cell cortex compression during AFM experiments.

Efficient computations achieving results in minutes on a single CPU.

Abstract

The dynamics of membranes, shells and capsules in fluid flow has become an active research area in computational physics and computational biology. The small thickness of these elastic materials enables their efficient approximation as a hypersurface which exhibits an elastic response to in-plane stretching and out-of-plane bending, possibly accompanied by a surface tension force. In this work, we present a novel ALE method to simulate such elastic surfaces immersed in Navier-Stokes fluids. The method combines high accuracy with computational efficiency, since the grid is matched to the elastic surface and can therefore be resolved with relatively few grid points. The focus of this work is on axisymmetric shapes and flow conditions which are present in a wide range of biophysical problems. We formulate axisymmetric elastic surface forces and propose a discretization with surface finite-differences coupled to evolving finite elements. We further develop an implicit coupling strategy to reduce time step restrictions. We show in several numerical test cases that accurate results can be achieved at computational times on the order of minutes on a single core CPU. We demonstrate two state-of-the-art applications which to our knowledge cannot be simulated with any other numerical method so far: We present first simulations of the observed shape oscillations of novel microswimming shells and the uniaxial compression of the cortex of a biological cell during an AFM experiment.