| Fluid-structure interaction problems, in which the dynamics of a deformable structure are coupled to the dynamics of a fluid, are prevalent in biological applications. The immersed boundary method is a popular method for simulating such problems. In this thesis we present an immersed boundary method in which forces are computed from an energy functional using a finite-element-like discretization. Unlike previous versions of the immersed boundary method that use a spring network to discretize the elastic structure, it is simple to incorporate material models from continuum mechanics into the framework of this new method. Also, unlike finite element methods for elasticity, we do not explicitly use stress tensors in the formulation of the elastic force density, and we do not solve a linear system involving a mass matrix. The method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. Incompressibility is well maintained, as indicated by area conservation in this 2D problem. We study convergence of the velocity field, the deformation map, and the Eulerian force density for this problem. The numerical results indicate that the velocity field and deformation map converge strongly at second order and the Eulerian force density converges weakly at second order. We provide an analytical proof of the weak convergence for the Eulerian force density that holds in two and three dimensions. This proof assumes that the viscoelastic material deforms according to a given deformation map that is twice continuously differentiable. Finally, the method is applied to a three-dimensional fluid-structure interaction problem with two different materials: an isotropic neo-Hookean model and an anisotropic fiber-reinforced model. This example illustrates the simplicity in swapping one material model for another within the framework of the method. |
