A Volumetrically Stabilized Mixed Formulation of the Finite Element Immersed Boundary Method for Fluid Structure Interaction with Fully Incompressible Hyperelastic Solids
The finite element immersed boundary method (FE IBM) with a distributed Lagrange multiplier is an attractive framework for fluid structure interaction (FSI) because it couples an Eulerian description of an incompressible fluid to a Lagrangian description of an immersed solid on two independent, nonconforming meshes, avoiding the costly remeshing required by body fitted arbitrary Lagrangian Eulerian methods. When the immersed solid is modelled as a fully incompressible hyperelastic material, however, a direct finite element discretization of the deviatoric solid stress fails to enforce the Lagrangian incompressibility constraint, and the computed structure exhibits spurious volumetric instabilities and locking. In this work we present a mixed formulation of the distributed Lagrange multiplier FEIBM that restores volumetric stability. Following the theory of nearly incompressible hyperelasticity, we augment the solid stress with a volumetric contribution derived from a dilatational strain energy and introduce an additional solid pressure field, enforced weakly, that plays the role of the Lagrange multiplier for the Lagrangian incompressibility constraint J = 1. The resulting formulation is discretized in space by finite elements and in time by an unconditionally stable semi implicit scheme, and is implemented as a reusable Immersed Boundary physics kernel within the GRINS multiphysics framework, built on the libMesh finite element library. The method is verified on three FSI benchmarks, an elliptically displaced thick ring returning to equilibrium, a radially stretched incompressible ring, and a disk falling under gravity in a viscous fluid, for which the mixed formulation removes the volumetric failure observed with the unstabilized formulation and reproduces the analytical terminal velocity of the falling disk to within 1%.