On the similarity between Dirichlet–Neumann with interface artificial compressibility and Robin–Neumann schemes for the solution of fluid-structure interaction problems

Abstract

In this note, the similarity between two implicit, partitioned solution techniques for fluidstructure interaction (FSI) problems is analyzed using finite volume discretization of the flow equations. Fluid-structure interaction refers to the mutual interaction between a fluid flow and a flexible structure. Partitioned solution techniques solve the flow equations and the structural equations separately. These techniques are classified as implicit (or strongly coupled) if they satisfy the interaction conditions on the fluid-structure interface in each time step and as explicit (or loosely coupled) if they do not. Both techniques analyzed in this note use block Gauss-Seidel (GS) iterations, meaning that the flow equations and the structural equations are solved consecutively within a time step until some convergence tolerance is reached. As the flow and structural equations are solved separately, the interaction conditions on the fluid-structure interface have to be converted into boundary conditions on the common boundary of the fluid and structure subdomains. Several types of boundary conditions can be applied, resulting in different decompositions. In the case of Dirichlet-Neumann (DN) decomposition, the flow equations are solved with a Dirichlet boundary condition (given velocity) on the fluid-structure interface, while the structural equations are solved with a Neumann boundary condition (given stress) on the interface. Conversely, RobinNeumann (RN) decomposition, introduced in [1], denotes a Robin boundary condition on the fluid side of the interface and a Neumann boundary condition on the structure side. The first technique in this comparison is block Gauss-Seidel iterations applied to the monolithic system previously multiplied by a suitable permutation matrix, leading to a Robin-Neumann decomposition (GS-RN). This first technique includes a simplified version of the structural model in the flow equations by means of a Robin boundary condition to accelerate the convergence of the GS iterations [1, 2]. The second technique is block Gauss-Seidel iterations with DirichletNeumann decomposition and Interface Artificial Compressibility (GS-DN-IAC). This second