Local Refinement

Local Refinement#

Local refinement can be perormed both in FEniCSx or in GMSH. Be aware that in case of complex geometries, a refinement based on a surface definition is more stable than a refinement from a square. This is for instance the case here with a cylinder inside of a cube (please refer to: Lavigne et al., 2023 appendices where the method is described). In case of a choice of a local refinement inside of FEniCSx, be careful with the facet tag definition which might contain mismatches or not cover the whole surface (for instance with the boundary of the cylinder).

Using FEniCSx#

The idea for local refinement within FEniCSx is to identify the boundaries to be refined using a (marker,locator) tuple.

refine_boudaries = [(11, lambda x: test_on_boundary(x))]

Then, a loop corresponding to the number of operations we require is created to refine the mesh as many times as required:

for _ in numpy.arange(2):
    # Refinement 
    refine_indices, refine_markers = [], []
    fdim = mesh.topology.dim-2
    for (marker, locator) in refine_boudaries:
        facets = dolfinx.mesh.locate_entities(mesh, fdim, locator)
        refine_indices.append(facets)
        refine_markers.append(numpy.full_like(facets, marker))
    refine_indices = numpy.hstack(refine_indices).astype(numpy.int32)
    refine_markers = numpy.hstack(refine_markers).astype(numpy.int32)
    # indices in meshtag must be sorted
    sorted_facets_refine = numpy.argsort(refine_indices)
    refine_tag = dolfinx.mesh.meshtags(mesh, fdim, refine_indices[sorted_facets_refine], refine_markers[sorted_facets_refine])
    mesh.topology.create_entities(fdim)
    mesh = dolfinx.mesh.refine(mesh, refine_indices[sorted_facets_refine])

Using GMSH#

The mesh refinement in GMSH relies on a distance field to an existing surface of the mesh. A precise tutorial is available here.

# Create a distance field from which a local resolution is defined for the refinement
indenter_interface = surfaces[0][1]
distance = gmsh.model.mesh.field.add("Distance")
gmsh.model.mesh.field.setNumbers(distance, "FacesList", [indenter_interface])
# A threshold function is defined:
resolution = r/10
threshold = gmsh.model.mesh.field.add("Threshold")
gmsh.model.mesh.field.setNumber(threshold, "IField", distance)
gmsh.model.mesh.field.setNumber(threshold, "LcMin", resolution)
gmsh.model.mesh.field.setNumber(threshold, "LcMax", 5*resolution)
gmsh.model.mesh.field.setNumber(threshold, "DistMin", 0.6*r)
gmsh.model.mesh.field.setNumber(threshold, "DistMax", r)
# If several fields are defined:
minimum = gmsh.model.mesh.field.add("Min")
gmsh.model.mesh.field.setNumbers(minimum, "FieldsList", [threshold]) # add other fields in the list if needed
gmsh.model.mesh.field.setAsBackgroundMesh(minimum)
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