Numerical stability analysis of a coupled fluid-elastic structure system using the Method of Regularized Stokeslets.

The method of regularized Stokeslets (MRS) is a widely used numerical method for simulating fluid-structure interaction in the Stokes flow regime (Reynolds number Re=0). The method works by desingularizing forces distributed over a curve or surface discretization of the structure, resulting in a smooth velocity field that can be directly evaluated at any point in the domain. While several works have been devoted to improving the error that arises from the spatial discretization or regularization, relatively fewer works have worked on issues of numerical stability. This is particularly important in the modeling of deformable bodies (e.g. flagella or cilia), where a common approach is to model elasticity by connecting points on the body with virtual Hookean springs. This approach results in a stiff system that requires extremely small time steps to maintain stability. Here, we present work from our investigation into the nature of these stability issues. We start with a linear stability analysis of an elastic membrane immersed in a Stokes flow subject to small perturbations from its configuration at equilibrium. For a particular choice of regularized delta function and time stepping scheme, this allows us to estimate a critical time step at which the system becomes unstable. The analysis is accompanied by numerical simulations where we initialize the elastic membrane with small deformations for validation, and with larger deformations to understand how our insights can be used in practice.