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HOMER – High Order Mesh Embeddings and Refinement

HOMER is a Python library for constructing, fitting, evaluating, and visualising high-order finite-element meshes using JAX for fast, differentiable computations. It supports arbitrary tensor-product basis functions (cubic Hermite, linear/quadratic/cubic/quartic Lagrange) in 2-D and 3-D, and provides tools for:

  • Building manifold surface meshes and volume meshes
  • Fitting meshes to point clouds via nonlinear least squares
  • Embedding arbitrary points into the parametric coordinates of a mesh
  • Refining mesh resolution while preserving the underlying geometry
  • Converting between basis functions (rebasing)
  • Storing and evaluating secondary vector/scalar fields over the mesh topology
  • Computing Green-Lagrange strain tensors
  • Saving and loading meshes to/from JSON

Quick Example

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

# 1. Create four corner nodes for a flat 2-D patch
node0 = MeshNode(loc=np.array([0., 0., 0.]), du=np.zeros(3), dv=np.zeros(3), dudv=np.zeros(3))
node1 = MeshNode(loc=np.array([1., 0., 0.]), du=np.zeros(3), dv=np.zeros(3), dudv=np.zeros(3))
node2 = MeshNode(loc=np.array([0., 1., 0.]), du=np.zeros(3), dv=np.zeros(3), dudv=np.zeros(3))
node3 = MeshNode(loc=np.array([1., 1., 0.]), du=np.zeros(3), dv=np.zeros(3), dudv=np.zeros(3))

# 2. Link the nodes through a bicubic-Hermite element
element = MeshElement(node_indexes=[0, 1, 2, 3], basis_functions=(H3Basis, H3Basis))

# 3. Create the mesh
mesh = Mesh(nodes=[node0, node1, node2, node3], elements=element)

# 4. Evaluate the surface at a 10×10 grid
xis = mesh.xi_grid(10)                          # (100, 2)
pts = mesh.evaluate_embeddings_in_every_element(xis)  # (100, 3)

# 5. Visualise
mesh.plot()

Getting Started

Installation

pip install HOMER

For the optional documentation extras:

pip install "HOMER[docs]"

Key Concepts

Term Description
MeshNode Physical location + Hermite derivative vectors
MeshElement Connects nodes via tensor-product basis functions
MeshField Collection of nodes + elements; evaluates/fits a field
Mesh Primary coordinate field that can carry secondary fields
Basis 1-D interpolation building block: H3, L1–L4
xi Parametric coordinates in [0, 1]ⁿ

Supported Basis Functions

Class Type Nodes/dir C⁰ Node fields
H3Basis Cubic Hermite 2 C¹ du, dv, …
L1Basis Linear Lagrange 2 C⁰ –
L2Basis Quadratic Lagrange 3 C⁰ –
L3Basis Cubic Lagrange 4 C⁰ –
L4Basis Quartic Lagrange 5 C⁰ –

Workflow

The core workflow demonstrated in the test suite is:

  1. Create nodes – instantiate MeshNode objects with physical coordinates and (for Hermite bases) derivative vectors.
  2. Create elements – combine nodes with a tuple of basis classes.
  3. Build the mesh – pass nodes and elements to Mesh(...).
  4. Evaluate – call evaluate_embeddings(), evaluate_jacobians(), etc.
  5. Fit – use linear_fit() or point_cloud_fit() to update node parameters.
  6. Refine – call mesh.refine(2) to subdivide elements.
  7. Save / load – mesh.save('path.json') / load_mesh('path.json').

See the How-To Guides for detailed walk-throughs.

Next Steps