A Newton multigrid framework for optimal control of fluid–structure interactions

Abstract

We present a monolithic parallel Newton-multigrid solver for nonlinear nonstationary three dimensional fluid-structure interactions in arbitrary Lagrangian Eulerian (ALE) formulation. We start with a finite element discretization of the coupled problem, based on a remapping of the Navier–Stokes equations onto a fixed reference framework. The strongly coupled fluidstructure interaction problem is discretized with finite elements in space and finite differences in time. The resulting nonlinear and linear systems of equations are large and show a very high condition number. We present a novel Newton approach that is based on two essential ideas: First, a condensation of the solid deformation by exploiting the discretized velocitydeformation relation dtu = v, second, the Jacobian of the fluid-structure interaction system is simplified by neglecting all derivatives with respect to the ALE deformation, an approximation that has shown to have little impact. The resulting system of equations decouples into a joint momentum equation and into two separate equations for the deformation fields in solid and fluid. Besides a reduction of the problem sizes, the approximation has a positive effect on the conditioning of the systems such that multigrid solvers with simple smoothers like a parallel Vanka-iteration can be applied.We demonstrate the efficiency of the resulting solver infrastructure on a well-studied 2d test-case and we also introduce a challenging 3d problem.