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

A FEM-DEM technique for studying the motion of particles in non-Newtonian fluids. Application to the transport of drill cuttings in wellbores

“…, where j ranges from one to the number of element sides and l j is the vector defining the jth element side [12].…”

“…(37b) and (39) h ij and h are characteristic length distances that are expressed in terms of the finite element dimensions and δ is a time parameter [12,16,19,37].…”

“…In our work we use a stabilized form of the momentum mass balance equations obtained via the Finite Calculus (FIC) technique [12,16,19,37] written as…”

“…Another source of instability, however, remains in the numerical solution of Lagrangian flows such as PFEM, that due to the treatment of the incompressibility constraint which still requires using a stabilized numerical method. In this work we use a PFEM formulation based on a residual-based stabilized expression of the mass balance equation [10][11][12][13][14][15][16]. The excellent mass preservation feature of this formulation has been demonstrated previously [7,16].…”

“…The coupling effects between the particles and the fluid are introduced via an immersed technique [3][4][5]. The fluid motion is modelled either with an Eulerian stabilized FEM formulation using a fixed mesh, or using a Lagrangian formulation using the Particle Finite Element Method (PFEM) [4,[6][7][8][9][10][11][12][13][14][15][16] for which the mesh evolves in time. For both the Eulerian and the Lagrangian formulations we use a mixed finite element formulation with an equal order linear interpolation for the velocities and the pressure variables.…”