First-Order System Least Squares (FOSLS) and AMG for Fluid-Elastic Problems

Jeffrey J. Heys

Thomas A. Manteuffel

Stephen F. McCormick


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.