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Mesh Refinement

Refinement subdivides every element into smaller sub-elements, increasing the spatial resolution of the mesh while preserving the underlying high-order geometry.

Uniform Refinement

mesh.refine(refinement_factor=n) splits each parametric direction into n sub-intervals, creating n ** ndim sub-elements per original element.

from HOMER import Mesh, MeshNode, MeshElement, H3Basis
import numpy as np

# 1. Build a coarse 3-D mesh
nodes = [MeshNode(loc=[x,y,z], du=np.zeros(3), dv=np.zeros(3), dw=np.zeros(3),
                  dudv=np.zeros(3), dudw=np.zeros(3), dvdw=np.zeros(3), dudvdw=np.zeros(3))
         for x in [0,1] for y in [0,1] for z in [0,1]]
element = MeshElement(node_indexes=list(range(8)),
                      basis_functions=(H3Basis, H3Basis, H3Basis))
mesh = Mesh(nodes=nodes, elements=element)

# 2. Refine: each element is subdivided into 2×2×2 = 8 sub-elements
mesh.refine(refinement_factor=2)
print(f"Elements after refinement: {len(mesh.elements)}")  # → 8

Non-Uniform Refinement

Provide a tuple of xi breakpoint arrays to place element boundaries at specific parametric locations:

# Refine with 2 sub-intervals in u, 3 in v, and 2 in w
mesh.refine(by_xi_refinement=(
    np.array([0, 0.5, 1.0]),   # u: 2 sub-intervals
    np.array([0, 1/3, 2/3, 1.0]),  # v: 3 sub-intervals
    np.array([0, 0.5, 1.0]),   # w: 2 sub-intervals
))

Warning

The two parameters refinement_factor and by_xi_refinement are mutually exclusive. Providing both raises an AssertionError.

Refining a Mesh with Secondary Fields

When the mesh has secondary fields, Mesh.refine() automatically refines all fields simultaneously:

# mesh has a secondary field 'fibre'
mesh.new_field('fibre', field_dimension=3, new_basis=[H3Basis]*3,
               field_locs=data_pts, field_values=fibre_vectors)

# Refine both the geometry and the 'fibre' field
mesh.refine(refinement_factor=2)

# Both mesh geometry and mesh['fibre'] are now at 2× resolution

Visualising Before and After

import pyvista as pv

s = pv.Plotter(shape=(1, 2))
s.subplot(0, 0)
mesh_before.plot(s)
s.subplot(0, 1)
mesh_after.plot(s)
s.show()