A monolithic, finite element-based strategy for solving fluid structure interaction problems coupled with electrostatics

In this work, we present a monolithic finite-element-based strategy for problems involving the deformation of a hyperelastic solid, an incompressible fluid and electrostatics. We use a two-field hybrid Lagrangian formulation for the structure, and a velocity-based ALE mixed formulation for the fluid with appropriately chosen interpolations for the various field variables to ensure stability of the resulting numerical procedure. 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. Keeping in view that the thickness of typical MEMS structures is small, a stress-based hybrid formulation is used to prevent locking. The use of a monolithic strategy provides a robust and stable algorithm, and allows the user to take large time steps. The consistent linearization ensures a quadratic rate of convergence of the non-linear iterations at each time step. Detailed expressions for the tangent stiffness and associated matrices for the three-way coupled problem are provided. The robustness and accuracy of the proposed method are demonstrated by solving several benchmark problems from the literature.