A Monolithic FEM Multigrid Penalty-Projection Solver for Incompressible Fluid-Structure Interaction

In this paper we present the results of computations of fluid-structure interaction of an incompressible elastic object and a laminar incompressible viscous flow with the use of the penalty-projection method. The equations of incompressible solid mechanics are based on a mixed formulation where pressure and displacement variables are coupled in a saddle-point manner, similar to the incompressible Navier-Stokes system. The discretization and numerical solution of these problems is a challenging task both from the algorithmic and from the computational point of view. We propose to reformulate the solid problem from a displacement-pressure formulation into a velocity-pressure one and to reduce the solution of this saddle-point problem into a system of decoupled elliptic equations for velocity and pressure with a projection method. In order to constrain the divergence of the solid velocity, a penalty method is implemented. This approach leads in a natural manner to a monolithic treatment of the coupling between solid and fluid. Therefore, the continuity of the velocity at the interface is ensured as well as the energy balance. This formulation is particularly attractive in the field of fluid-structure interaction, due to a large number of applications that can be found in biomechanics, such as the mechanical interaction of incompressible biological elastic tissues with blood or organic fluids. The FSI problem is solved in the Arbitrary Lagrangian-Eulerian framework with a conforming finite element method using a piecewise-quadratic function space for the velocity and piecewise linear for the pressure. The decoupled system is advanced in two steps: in a first stage the momentum balance is solved to obtain an intermediate velocity, which is corrected in the second stage by projecting it onto a divergence-free space. A multigrid algorithm is implemented together with an interface with PETSc parallel computing libraries in order to reduce the CPU time. We show the results of a series of simulations concerning some typical FSI applications, such as blood flow in a compliant vessel.