Splitting schemes for the stress formulation of fluid–structure interaction problems

Abstract

In this article we demonstrate that the novel stress formulation of the Navier–Stokes equations proposed in Minev and Vabishchevich (2018) can be extended to the case of fluid–structure interaction problems. This formulation allows for an easy treatment of the fluid–structure interface boundary conditions. Furthermore, we propose a first order (in time) splitting scheme for this formulation and study its stability in the linear case. It utilizes a level set approach for the interface tracking and regularization of the interface problem. We also demonstrate how this scheme can be extended to the nonlinear case of a neo-Hookean elastic material. The computational complexity of the resulting problem seem to be comparable or better than most available schemes that treat the problem in primitive variables. A downside of such an approach is that it requires a higher than the traditional formulations in terms of primitive unknowns degree of smoothness of the solution for the stress. However, in addition to the solution for the velocity and the stress, it also yields information about the stress tensor, computed with an optimal accuracy.The scheme is demonstrated on two benchmark problems borrowed by other authors, and the results, although computed with a purely linear model look very similarly to the results of other authors that are based on a nonlinear neo-Hookean model.