A variational framework for fluid–solid interaction problems based on immersed domains: Theoretical bases

Abstract

In this work, the theoretical bases to handle the fluid–solid interaction problem based on the conception of immersed domains are developed starting from a well-known governing variational principle from the continuum mechanics. The purpose of the present work is twofold. Firstly, a variational principle to model the interaction between a fluid and a thin structure, characterized as a shell for which its volume is geometrically neglected, is devised by incorporating suitable kinematical assumptions into the original problem. This leads to a rather general formulation for the fluid–shell interaction problem that embraces methodologies such as the immersed boundary method. Secondly, no longer is the volume of the solid disregarded and the variational principle is manipulated in order to attain a formulation that comprises the problem of an augmented fluid, which is extended as a fictitious fluid to the whole domain of analysis, interacting with an immersed solid. In some parts, the formulation resembles the immersed finite element method, as well as the recently presented immersed continuous method, due to the appearance of a volume force defined over the fictitious fluid domain. One of the main differences lies in the fact that, in the present approach, the volume force is the Lagrange multiplier that imposes the continuity of the velocity field along the whole overlapped domain. The other noteworthy difference dwells in the handling of the mass conservation for the artificial fluid. Three aspects of this setting must be pointed out as original contributions of the work: (i) a consistent generalization of the theoretical continuous setting of immersed boundary methods is carried out, (ii) the variational basis for the finite element approximations is clearly posed/written and easy to understand, and (iii) when dealing with an immersed solid, a more general formulation valid for all classes of combinations fluid/solid is developed, eliminating the restriction to compressible fluid–compressible solid of the immersed continuous method and the incompressible fluid–incompressible solid of the immersed finite element method.