Mixed Basis Functions

HOMER supports mixing different 1-D basis functions across parametric directions within the same element. This is useful when you need smooth interpolation in some directions but only linear continuity (or fewer degrees of freedom) in others.

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L2 × L2 Surface Mesh

A quadratic-Lagrange surface mesh requires 3 × 3 = 9 nodes per element. No derivative fields are needed on the nodes.

import numpy as np
from HOMER import Mesh, MeshNode, MeshElement, L2Basis

# 9-node quadratic patch (xi=0,0.5,1 in each direction)
nodes = [
    MeshNode(loc=[0, 0, 1]),   # (0,0)
    MeshNode(loc=[0, 0, 0.5]), # (0,0.5)
    MeshNode(loc=[0, 0, 0]),   # (0,1)
    MeshNode(loc=[0, 0.5, 1]), # (0.5,0)
    MeshNode(loc=[0.5, 0.5, 0.5]), # middle
    MeshNode(loc=[0, 0.5, 0]), # (0.5,1)
    MeshNode(loc=[0, 1, 1]),   # (1,0)
    MeshNode(loc=[0, 1, 0.5]), # (1,0.5)
    MeshNode(loc=[0, 1, 0]),   # (1,1)
]

element = MeshElement(
    node_indexes=list(range(9)),
    basis_functions=(L2Basis, L2Basis),
)
mesh = Mesh(nodes=nodes, elements=element)
mesh.plot()

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H3 × L2 Mixed Surface Mesh

Use H3Basis in the xi_0 direction for smooth derivatives and L2Basis in the xi_1 direction for simpler parametric variation:

from HOMER import H3Basis, L2Basis

# 2 × 3 = 6 nodes per element
# Nodes at xi_u ∈ {0, 1} and xi_v ∈ {0, 0.5, 1}
element = MeshElement(
    node_indexes=[0, 1, 2, 3, 4, 5],
    basis_functions=(H3Basis, L2Basis),
)

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Choosing the Right Basis

Requirement	Recommended basis
C¹ smooth geometry, shape optimisation	H3Basis
Simple coarse mesh before rebasing	L1Basis
Mid-order accuracy, fewer DoF than H3	L2Basis or L3Basis
High-accuracy Lagrange interpolation	L4Basis

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Rebasing Between Bases

Any mesh can be converted to a different basis with rebase():

# Start with a coarse linear mesh
linear_mesh = Mesh(nodes=nodes, elements=MeshElement(
    node_indexes=list(range(4)),
    basis_functions=(L1Basis, L1Basis),
))

# Convert to cubic Hermite
from HOMER import H3Basis
smooth_mesh = linear_mesh.rebase([H3Basis, H3Basis])

See the Basis conversion guide for full details.