Thomas A. Manteuffel

Stephen F. McCormick

Abstract

Numerous mathematical models for the mechanical coupling between
a moving fluid and an elastic solid have been developed. The models
are inherently nonlinear because the shape of the Eulerian fluid domain
is not known *a priori* -- it is at least partially determined by the
displacement of the elastic solid. Different iteration techniques have been
developed to solve the nonlinear system of equations. For example,
one approach is to iteratively solve the fluid equations on a fixed domain,
apply the fluid stresses from the fluid solution to an elastic solid, and
remap the fluid domain based on the displacement of the elastic solid.
At the other extreme, the full system of equations (fluid, elastic, and mapping/meshing)
can be solved simultaneously using a Newton iteration. There are advantages
and disadvantages to each approach, and the choices effect the performance
of the solver (AMG) and the accuracy of the solution differently.

The performance of different iteration techniques will be presented for
a model of a linear elastic solid coupled to a Newtonian fluid using a FOSLS
formulation. In this approach, the system of non-linear partial differential
equations is recast as a linearized first-order system of equations, and
the solution is found using least squares minimization principles. A finite
element discretization of the FOSLS formulation results in a SPD matrix
problem that is solved using AMG. With AMG, only a single Newton iteration
is required for each refinement beyond the course grid to achieve an accurate
discrete solution on the fine grid. However, the convergence rate of
the AMG cycles depends on the iteration technique.