ALE-based Monolithic Finite Element Strategies for Fluid Structure Interaction Problems

Abstract

Fluid-structure interactions (FSI) represent a class of engineering problems which involve a two-way coupling between the solid motion and the fluid flow. The deformation of the solid caused by the fluid forces at the boundary of the solid generates the boundary condition for the fluid flow. Such interactions are widespread in engineering and also in nature. The fluttering of aircraft wings and compressor blades, flapping of the wings of birds and insects, floating of ships, sound produced by musical instruments like drums and trumpets, inflation of automobile airbags, dynamics of spacecraft parachutes, flowing of blood through the arteries and veins pumped by the human heart are all examples of FSI problems. The non-linearity in the equations of fluid mechanics and large deformation structural mechanics along with the requirement to satisfy the interface conditions between solid and fluid makes it almost impossible for this class of problems to be dealt with analytically. However, with the emergence of computers, there have been significant advances in this class of problems numerically.

In this thesis we present new monolithic finite element strategies for solving various classes of fluid-structure interaction problems. The term `monolithic' means that the governing equations of the solid and fluid along with the interface and boundary conditions are solved simultaneously. The developed strategies are based on the Arbitrary Lagrangian-Eulerian (ALE) formulation for the fluid domain, and the Lagrangian framework for the solid domain.

First, we develop a new monolithic FEM formulation for problems involving a compressible fluid and a hyperelastic structure fully coupled with the thermal field. We develop an `energy-momentum conserving' time-stepping strategy, i.e., in the Lagrangian limit, the time-stepping strategy that we propose conserve the total energy, and the linear and angular momenta. Detailed proofs with numerical validations are provided. We use a displacement-based Lagrangian formulation for the structure, and a velocity-based ALE mixed formulation with appropriately chosen interpolations for the various field variables to ensure stability of the resulting numerical procedure. Apart from physical variables such as displacement, velocity, etc., no new variables are introduced in the formulation.

Next, we present a two-dimensional monolithic FEM-based strategy for FSI problems involving partly immersed (floating) hyperelastic solids in an incompressible fluid. This strategy can be used for studying the dynamics of freely floating bodies as well as the accurate computation of hydrodynamic forces acting on them. Since the portion of the solid immersed in the fluid varies with time, one cannot use the same nodes for the solid and the fluid at the interface. The continuity requirements at the fluid-structure interface have been satisfied in a weak sense using the mortar method. The flexibility of the ALE technique permits us to treat the free surface of the fluid as a Lagrangian entity where the mesh velocity and material velocity are equal. This allows us to model the interface as a contact between the solid and fluid surfaces. The same strategy can be used to analyse sloshing in containers with curved or deformable walls.

Next, motivated by micro-electro-mechanical systems (MEMS), we present a monolithic FEM-based strategy for problems involving the deformation of a hyperelastic solid and an incompressible fluid in the presence of an electrostatic field. The equations of electrostatics are solved on the reference configuration over both the solid and fluid domains, with voltage and electric displacement continuity imposed at the interface. The fluid is assumed to be a non-conductor of electricity so that there is no flow of charge through the fluid. The concept of generalized permittivity is used to model the solid as a general dielectric material.

In the final part of this thesis, we generalize the formulation in the previous chapter by introducing compressibility effects in the fluid flow, and analyse the effect of fluid compressibility on the response of MEMS devices of various geometries.

In all the formulations developed above, a hybrid formulation is used to prevent locking of thin structures. Also, in every formulation, we carry out a consistent linearization resulting in exact tangent stiffness matrices which ensure that the concerned algorithm converges quadratically within each time step. Detailed expressions for the stiffness matrices and load vectors are provided in each chapter so that any interested reader can easily implement any of these strategies. A number of benchmark examples have been presented to illustrate the good performance of the proposed strategies.