triangulax

Overview

This Python package provides data-structures for triangular meshes and a geometry processing toolkit based on JAX, fully compatible with automatic differentiation and just-in-time compilation. Additionally, triangulax provides tools for simulating membranes, thin shells, and tissue sheets.

Design

triangulax is designed for modularity and flexibility. The library includes a suite of geometry processing tools based on discrete differential geometry (Voronoi duals, curvatures, Laplace operator, …) and represents surfaces as half-edge meshes.

Automatic differentiation and just-in-time compilation with JAX

The main feature of triangulax is (forward- and reverse-mode) automatic differentiation to compute of derivatives of any mesh-based function. triangulaxuses JAX, a “Python library for accelerator-oriented array computation and program transformation, designed for high-performance numerical computing and large-scale machine learning.” Most triangulax functions are compatible with JAX’s just-in-time (JIT) compilation. JIT provides high performance in Python (rather than C++) and allows running on GPUs.

To simulate dynamics or minimize energies, triangulax integrates with JAX ecosystem libraries like optimistix (optimization) and diffrax (ODE integration), creating end-to-end differentiable simulations.

Prerequisites: triangulax assumes familiarity with triangular meshes (tutorial 0), and basic JAX usage (tutorial 1)

Use cases

Triangular meshes are ubiquitous in computer graphics and in scientific computing. Examples in soft-matter and biophysics include:

1. Cell-resolved models of two-dimensional tissue sheets like the self-propelled Voronoi model (tutorial 3).
2. Reaction-diffusion systems on 3D curved surfaces (tutorial 4)
3. Mechanics of membranes and thin elastic shells in 3D (tutorial 5)

These tasks revolve around a mesh-based “energy”. JAX automatically computes their gradients, making it easy to optimize energies or to simulate forces. For “multiphysics” simulations (e.g. reaction-diffusion systems on deforming surfaces), it suffices to specify the combined energy of the system - all forces and cross-terms are calculated automatically.

Inverse problems

Since triangulax is fully JAX-compatible, one can differentiate a simulation with respect to its parameters. This means one can apply gradient-based optimization to inverse problems (tutorial 2). Effectively, a simulation becomes a “neural network” which maps initial conditions to simulation results. The parameters of the simulation can be fitted to data, or optimized to find a mechanism that generates a desired shape.

Documentation

Documentation can be found hosted on GitHub pages. Jupyter notebooks tutorials can be found in the nbs/tutorials/ folder.

Installation instructions

The triangulax package is hosted on PyPI. Install it as follows:

1. (Recommended) Initialize a virtual environment, for instance with conda:

$ conda env create -n triangulax 
$ conda activate triangulax

See the JAX documentation for how to install JAX with GPU support.

2. Install with pip:

$ pip install triangulax 

(Optional) Install optional dependencies for running the tutorials:

$ pip install triangulax[tutorials]

3. Verify installation:

conda run -n triangulax python -c "from triangulax import mesh, geometry"

4. (Optional) Download jupyter notebooks to run tutorials interactively from GitHub

Usage

triangulax comprises the following modules (see full documentation for details):

• trigonometry: trigonometry and linear algebra utilities
• triangular: input/output for triangular meshes
• mesh: half-edge data structure for triangular meshes, compatible with JAX
• topology: topological modifications (edge flip, collapse, and split)
• adjacency: vertex-vertex, vertex-face, and face-face adjacency operators
• geometry: angles, edge lengths, triangle areas, Voronoi areas, curvatures
• periodic: geometry computations under periodic boundary conditions
• linops: discrete differential operators (Laplacian, mass matrix, gradient)
• interp: linear interpolation and closest-point queries
• elastic: discrete elastic energies for shells and membranes
• algorithms: Delaunay flipping, mesh quality improvement
• simulation: Utilities for time-dependent simulations and checkpointing

Minimal example

import jax
import jax.numpy as jnp
from triangulax import triangular, mesh, geometry

# load example mesh and convert to half-edge mesh

vertices, faces = triangular.read_obj("test_meshes/disk.obj", dim=2)
hemesh = mesh.HeMesh.from_triangles(vertices.shape[0], faces)

# with the half-edge mesh, you can carry out various operations, for example
# compute the coordination number by summing incoming half-edges per vertex

coord_number = jnp.zeros(hemesh.n_vertices)
coord_number = coord_number.at[hemesh.dest].add(jnp.ones(hemesh.n_hes))
print("Mean coordination number:", coord_number.mean())

# Let's define a simple geometric function and compute its gradient with JAX

def mean_voronoi_area(vertices: jax.Array, hemesh: mesh.HeMesh) -> jax.Array:
    """Compute the mean Voronoi area per vertex."""
    voronoi_areas = geometry.get_voronoi_areas(vertices, hemesh)
    return jnp.mean(voronoi_areas)

value, gradient = jax.value_and_grad(mean_voronoi_area)(vertices, hemesh)
print("Mean gradient norm:", jnp.linalg.norm(gradient, axis=1).mean())

Warning: readOBJ() ignored non-comment line 3:
  o flat_tri_ecmc

Mean coordination number: 5.40458
Mean gradient norm: 0.0003638338

See also

• libigl Geometry processing library with Python bindings. You can use libigl functions on triangulax meshes via the .faces attribute

• VertAX JAX-based simulations of 2D tissues.

Developer guide

This package is developed based on Jupyter notebooks, which are converted into python modules using nbdev. Details are in .github/copilot-instructions.md.

Install triangulax in Development mode

1. Clone the GitHub repository

$ git clone https://github.com/nikolas-claussen/triangulax.git

2. Create a conda environment with all Python dependencies

$ conda env create -n triangulax -f environment.yml
$ conda activate triangulax

3. Install the triangulax package

# make sure triangulax package is installed in development mode
$ pip install -e .

4. Install nbdev package

pip install nbdev

5. Optional: edit the package notebooks and export the code into Python modules

# make changes under nbs/ directory
# ...

# export to have changes apply to triangulax
$ nbdev_export