Finite Element Interpolation (FEM)#

This subsection specifies the characteristics of the finite element method used in the simulations. The polynomial degrees for velocity, pressure and other physics are specified. An additional option is also present to enable the use of high-order mapping throughout the entire mesh. At the moment, all variables are interpolated using Lagrange elements. On quad/hex meshes, FE_Q elements are used whereas on simplex meshes (triangles, tetrahedron), FE_P elements are used.

subsection FEM
  # interpolation degree pressure
  set pressure degree     = 1

  # interpolation degree velocity
  set velocity degree     = 1

  # interpolation degree temperature
  set temperature degree  = 1

  # interpolation degree tracer
  set tracer degree       = 1
  set tracer uses dg      = false

  # interpolation degree CLS
  set CLS degree          = 1

  # interpolation degree cahn hilliard
  set phase cahn hilliard degree     = 1
  set potential cahn hilliard degree = 1

  # bubble enrichment function
  set enable bubble function velocity = false
  set enable bubble function pressure = false
end

  • velocity degree specifies the polynomial degree for velocity.

  • pressure degree specifies the polynomial degree for pressure. For the lethe-fluid family of solvers, the polynomial degree for pressure can be equal or lower than that of velocity. For the lethe-fluid-block family of solvers, the polynomial degree for pressure has to be one degree lower than that of velocity.

  • temperature degree specifies the polynomial degree for temperature for the heat transfer physics.

  • tracer degree specifies the polynomial degree for the tracer physics.

  • tracer uses dg specifies if the Discontinuous Galerkin (DG) formulation is used instead of the Continuous Galerkin (CG) that is used by default, for the tracer physics. This formulation allows discontinuities between elements and at boundaries, and only penalizes them; this prevents oscillations and enforces local conservation of mass, at the cost of more numerous degrees of freedom.

Important

The DG formulation is sensitive to the CFL value. Use a small time step to keep the tracer value bounded. From our experience, a CFL of 1 or lower is recommended.

  • CLS degree specifies the polynomial degree for the CLS phase indicator. It is not recommended to use higher polynomial degrees for the CLS method as this may conflict with the bounding and the sharpening mechanism used therein.

  • phase cahn hilliard degree and potential cahn hilliard degree specify the polynomial degree for the phase order parameter and the chemical potential in the Cahn-Hilliard equations. The degrees chosen should be equal. They are left as two separate parameters for debugging purposes.

  • enable bubble function velocity and enable bubble function pressure specifies if the bubble enrichment function is used in the velocity and pressure fields, respectively. This is a polynomial enrichment function centered at the mid-point of the cell and that vanishes at the element boundary. It can be used to improve accuracy and stability in Galerkin FEM, similarly to SUPG stabilization; we refer the reader to the work of Franca and Farhat 1995 and Brezzi et al 1992 for more detail.

Warning

The bubble enrichment function FE_Q_Bubbles is not compatible with simplex meshes.