Boundary pressure projection for partitioned solution of fluid-structure interaction with incompressible Dirichlet fluid domains

Partitioned solutions to fluid-structure interaction problems often employ a Dirichlet-Neumann decomposition, where the fluid equations are solved subject to Dirichlet boundary conditions on velocity from the structure, and the structure equations are solved subject to forces from the fluid. In some scenarios, such as an elastic balloon filling with air, an incompressible fluid domain may have pure Dirichlet boundary conditions, leading to two related issues which have been described as the incompressibility dilemma. First, the Dirichlet boundary conditions must satisfy the incompressibility constraint for a solution to exist. However, the structure solver is unaware of this constraint and may supply the fluid solver with incompatible velocities. Second, the constant fluid pressure mode lies in the null space of the fluid pressure solver, but must be determined to apply to the structure. Previously proposed solutions to the incompressibility dilemma have included modifying the fluid solver, the structure solver, or the Dirichlet-Neumann coupling interface between them. In this paper, we present a boundary pressure projection method which alleviates the incompatibility while maintaining the Dirichlet-Neumann structure of the decomposition and without modification of the fluid or solid solvers. Our method takes incompatible velocities from the structure solver and projects them to be compatible while in the process computing the constant pressure modes for the Dirichlet regions. The compatible velocities are then used as Dirichlet boundary conditions for the fluid while the constant pressure modes are added to the fluid-solver-computed pressures to be applied to the structure. The intermediate computation performed in the boundary pressure projection method is small, with the number of unknowns equal to the number of Dirichlet regions. We show that the boundary pressure projection method can be used to solve a variety of scenarios including inflation of an elastic balloon and the action of a hydraulic press. We also demonstrate the method on multiple coupled Dirichlet regions. The method offers a simple approach to overcome the incompressibility dilemma using a small intermediate computation that requires no additional knowledge of the black box fluid and solid solvers.