Energy conservation under incompatibility for fluid-structure interaction problems

Concurrent numerical methods for fluid-structure interaction problems are typically based on partitioned solution procedures. However, such partitioned methods are inherently non-conservative. In the present work, we investigate the conservation properties of monolithic discretisations for fluid-structure interaction problems. We consider a prototypical fluid-structure interaction problem, viz., the piston problem. A variational formulation allows us to establish precisely the conservation properties of the continuum problem and its discretisation by the finite-element method. We show that the conservation of energy by monolithic discretisations is only trivially maintained under restrictive compatibility conditions on the approximation spaces in the fluid and the structure. Moreover, we introduce a new discretisation based on coincidence conditions which ensures energy conservation under incompatibility. Numerical results which illustrate the effectiveness of the new discretisation are presented.