An unconditionally stable semi-implicit FSI finite element method

• We devise an unconditionally stable FSI numerical method based on ALE monolithic formulation.
• On every time step the method is linear.
• Incompressible and slightly compressible neo-Hookean models are numerically compared for a model hemodynamic problem.

The paper addresses the numerical solution of fluid–structure interaction (FSI) problems involving incompressible viscous Newtonian fluid and hyperelastic material. A well known challenge in computing FSI systems is to provide an effective time-marching algorithm, which avoids numerical instabilities due to the loose coupling of fluid and structure motion on the FSI interface. In this work, we introduce a semi-implicit finite element scheme for an Arbitrary Lagrangian–Eulerian formulation of the fluid–structure interaction problem. The approach strongly enforces the coupling conditions on the fluid–structure interface, but requires only a linear problem to be solved on each time step. Further, we prove that the numerical solution to the fully discrete problem satisfies the correct energy balance, and the stability estimate follows without any extra model simplifications or assumptions on the time step. The analysis covers the cases of Saint Venant–Kirchhoff compressible and incompressible neo-Hookean materials. Results of several numerical experiments are included to illustrate the properties of the method and its applicability for the simulation of certain hemodynamic flows. We also experiment with the enforcement of material incompressibility condition in the finite element method via an integral constraint or alternatively letting the Poisson ratio in the compressible model to be close to 12. From these experiments conclusions are drawn concerning the accuracy of flow statistics prediction for incompressible vs. nearly incompressible structure models.