  The immersed boundary method is a mathematical framework for modeling
fluid-structure interaction. This formulation describes the momentum,
viscosity, and incompressibility of the fluid-structure system in Eulerian
form, and it uses Lagrangian coordinates to describe the structural
deformations, stresses, and resultant forces. Integral transforms with Dirac
delta function kernels connect the Eulerian and Lagrangian frames. The fluid
and the structure are both typically treated as incompressible materials. Upon
discretization, however, the incompressibility of the structure is only
maintained approximately. To obtain an immersed method for incompressible
hyperelastic structures that is robust under large structural deformations, we
introduce a volumetric energy in the solid region that stabilizes the
formulation and improves the accuracy of the numerical scheme. This formulation
augments the discrete Lagrange multiplier for the incompressibility constraint,
thereby improving the original method's accuracy. This volumetric energy is
incorporated by decomposing the strain energy into isochoric and dilatational
components, as in standard solid mechanics formulations of nearly
incompressible elasticity. We study the performance of the stabilized method
using several quasi-static solid mechanics benchmarks, a dynamic
fluid-structure interaction benchmark, and a detailed three-dimensional model
of esophageal transport. The accuracy achieved by the stabilized immersed
formulation is comparable to that of a stabilized finite element method for
incompressible elasticity using similar numbers of structural degrees of
freedom.
