  The immersed boundary method is a mathematical formulation and numerical
method for solving fluid-structure interaction problems. For many biological
problems, such as models that include the cell membrane, the immersed structure
is a two-dimensional infinitely thin elastic shell immersed in an
incompressible viscous fluid. When the shell is modeled as a hyperelastic
material, forces can be computed by taking the variational derivative of an
energy density functional. A new method for computing a continuous force
function on the entire surface of the shell is presented here. The new method
is compared to a previous formulation where the surface and energy functional
are discretized before forces are computed. For the case of Stokes flow, a
method for computing quadrature weights is provided to ensure the integral of
the elastic spread force density remains zero throughout a dynamic simulation.
Tests on the method are conducted and show that it yields more accurate force
computations than previous formulations as well as more accurate geometric
information such as mean curvature. The method is then applied to a model of a
red blood cell in capillary flow and a 3D model of cellular blebbing.
