A variational finite element method for stationary nonlinear fluid—solid interaction

Abstract

We consider the problem of the interaction of a stationary viscous fluid with an elastic solid that undergoes large displacement. The fluid is modeled by the stationary incompressible Navier-Stokes equations in an Eulerian frame of reference, while a Lagrangian reference frame and large displacement-small strain theory is used for the solid. A variational formulation of the problem is developed that ensures satisfaction of continuity of interface tractions and velocities. The variational formulation is approximated by a Galerkin finite element method, yielding a system of nonlinear algebraic equations in unknown fluid velocities and pressures and solid displacements. A Newton-like method is introduced for solution of the discrete system. The method employs a modified Jacobian that enables decomposition into separate fluid and solid subdomains. This domain decomposition avoids possible ill-conditioning of the Jacobian, as well as the need to compute and store geometric coupling terms between fluid and interface shape. The capability of the methodology is illustrated by solution of a problem of the flow-induced large displacements of an elastic infinite cylinder.