Finite calculus formulations for finite element analysis of incompressible flows. Eulerian, ALE and Lagrangian approaches

We present a general formulation for incompressible fluid flow analysis using the finite element method (FEM). The standard Eulerian formulation is described first. The necessary stabilization for dealing with convective effects and the incompressibility condition are introduced via the finite calculus (FIC) method. A simple extension of the fluid flow equations to an arbitrary Lagrangian–Eulerian (ALE) frame adequate for treating fluid–structure interaction problems is briefly presented. A fully Lagrangian formulation called the particle finite element method (PFEM) is also described. The PFEM is particularly attractive for fluid–structure interaction problems involving large motions of the free surface and breaking waves. Examples of application of the Eulerian, the ALE and the fully lagrangian PFEM formulations are presented.