  Dirac-delta distributions are often crucial components of the solid-fluid
coupling operators in immersed solution methods for fluid-structure interaction
(FSI) problems. This is certainly so for methods like the Immersed Boundary
Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta
distributions are approximated via smooth functions. By contrast, a truly
variational formulation of immersed methods does not require the use of
Dirac-delta distributions, either formally or practically. This has been shown
in the Finite Element Immersed Boundary Method (FEIBM), where the variational
structure of the problem is exploited to avoid Dirac-delta distributions at
both the continuous and the discrete level. In this paper, we generalize the
FEIBM to the case where an incompressible Newtonian fluid interacts with a
general hyperelastic solid. Specifically, we allow (i) the mass density to be
different in the solid and the fluid, (ii) the solid to be either viscoelastic
of differential type or purely elastic, and (iii) the solid to be and either
compressible or incompressible. At the continuous level, our variational
formulation combines the natural stability estimates of the fluid and
elasticity problems. In immersed methods, such stability estimates do not
transfer to the discrete level automatically due to the non- matching nature of
the finite dimensional spaces involved in the discretization. After presenting
our general mathematical framework for the solution of FSI problems, we focus
in detail on the construction of natural interpolation operators between the
fluid and the solid discrete spaces, which guarantee semi-discrete stability
estimates and strong consistency of our spatial discretization.
