Mesh element height along a given direction

General
1

Hi,

I am implementing SUPG for a magnetostatic problem as presented in this work. There is a stabilization parameter \tau which depends upon the height of each mesh element parallel to the direction of a given velocity field. Following is the expression I have to evaluate:

h_E = \frac{2}{\sum_n|\frac{\mathbf{v}}{v} \cdot \nabla \tilde{V}_n|},

where \tilde{V}_n is the test function of the unknown scalar potential evaluated and summed on each node n of the mesh element.

I would appreciate if someone could show how it is done. Below is the MWE in which this step can be added.

import dolfinx, ufl, gmsh, mpi4py, petsc4py
import numpy as np
from dolfinx.io import gmshio

gmsh.initialize()
gmsh.clear()
occ = gmsh.model.occ
occ.addCylinder(0, 0, 0, 0, 0, 1, 0.5)

occ.synchronize()

gdim = 3
all_doms = occ.getEntities(gdim)
frag, _ = occ.fragment([all_doms[0]], all_doms[1:])
occ.synchronize()

# add_sub physical tags
all_doms = occ.getEntities(gdim)
for j, dom in enumerate(all_doms):
    gmsh.model.addPhysicalGroup(dom[0], [dom[1]], j + 1)  # create the main group/node

# number all boundaries
all_edges = gmsh.model.getEntities(gdim - 1)
for j, edge in enumerate(all_edges):
    gmsh.model.addPhysicalGroup(edge[0], [edge[1]], edge[1])  # create the main group/node

gmsh.model.mesh.generate()

mpi_rank = 0
mesh, ct, ft = gmshio.model_to_mesh(gmsh.model, mpi4py.MPI.COMM_WORLD, mpi_rank, gdim)

Omega_V = dolfinx.fem.functionspace(mesh, ("CG", 1))
Vt = ufl.TestFunction(Omega_V)

x_ufl, y_ufl, z_ufl = ufl.SpatialCoordinate(mesh)
velocity_vec = ufl.as_vector((-y_ufl, x_ufl, 0)) # known velocity field

# calculate mesh height along velocity field in each element
2

Hi, you can try with ufl.Jacobian which gives the Jacobian from the reference element to the real element mapping. This should give you the “size” of the element along different directions

3

Thanks very much for your quick feedback. Unfortunately I am unable to see how to process it . Could you share an example of how it can be used?