Unified Lagrangian formulation for solid and fluid mechanics and FSI problems

We present a Lagrangian monolithic strategy for solving fluid-structure interaction (FSI) problems. The formulation is called Unified because fluids and solids are solved using the same solution scheme and unknown variables. The method is based on a mixed velocity-pressure formulation. Each time step increment is solved via an iterative partitioned two-step procedure. The Particle Finite Element Method (PFEM) is used for solving the fluid parts of the domain, while for the solid ones the Finite Element Method (FEM) is employed. Both velocity and pressure fields are interpolated using linear shape functions. For quasi-incompressible materials, the solution scheme is stabilized via the Finite Calculus (FIC) method. The stabilized elements for quasi-incompressible hypoelastic solids and Newtonian fluids are called VPS/S-element and VPS/F-element, respectively. Other two non-stabilized elements are derived for hypoelastic solids. One is based on a Velocity formulation (V-element) and the other on a mixed Velocity-Pressure scheme (VP-element). The algorithms for coupling the solid elements with the VPS/F fluid element are explained in detail. The Unified formulation is validated by solving benchmark FSI problems and by comparing the numerical solution to the ones published in the literature.