A monolithic fluid-structure interaction method, application to a piston problem

A monolithic FSI method is presented. A standard piston problem is considered as test case. The piston problem's fluid domain is represented by a closed tube filled with air. One end of the fluid tube is formed by a piston connected to a spring. We use the Euler equations of gas dynamics as well as a linear simplification of these, the acoustic equations, to model the gas dynamics in the tube. A Discontinuous Galerkin method is applied to discretize the fluidflow equations, together with an immersed-boundary method to account for the moving piston. A monolithic formulation of the coupled system is derived and analyzed. It is proven that the semidiscrete formulation is stable, if two correction terms are used at the coupling interface. We use Lyapunov functions to prove stability of the semi-discrete monolithic formulation. Further, different time-integration methods are considered, analyzed and tested. The numerical results are very accurate; they correspond very well to analytical approximations. The theoretical prediction of the eigenfrequency can be reproduced very accurately. Moreover, the amplitude of the spring oscillation is conserved very well.