Discrete Exterior Calculus

cartan-dec bridges continuous Riemannian geometry (cartan-core) to discrete operators for PDE solvers on simplicial meshes. It implements the Desbrun et al. DEC framework: topology is encoded metric-free in the exterior derivative; all metric information flows through the Hodge star.

As of v0.3, the crate also provides complex line bundle sections for k-atic fields on 2-manifolds, extrinsic operators (Killing, divergence, gradient) for surfaces in , and an augmented Lagrangian Stokes solver for active nematohydrodynamics on arbitrary manifolds.

The DEC Pipeline

Mesh (vertices, edges, triangles)
  |
  |-- ExteriorDerivative: d_0: V->E,  d_1: E->T   (topology only)
  |
  |-- HodgeStar:          *_0, *_1, *_2            (metric: primal/dual volumes)
  |
  |-- Operators:          Laplace-Beltrami, Bochner, Lichnerowicz
  |                       Advection, Divergence
  |
  |-- line_bundle:        Section<K>, BochnerLaplacian<K>, defect charges
  |
  |-- extrinsic:          Killing operator, DIV, GRAD, viscosity Laplacian
  |
  |-- stokes:             Augmented Lagrangian Stokes solver
  |
  +-- mesh_quality:       Delaunay predicates, edge flips, Lloyd smoothing

Modules

Module	Struct / function	Purpose
`mesh`	`Mesh`	K-generic simplicial complex; vertices, edges, triangles, adjacency
`mesh_quality`	`is_delaunay`, `make_well_centered`	Mesh quality predicates and improvement operators
`mesh_gen`	`icosphere`, `torus`	Mesh generators with optional well-centred output
`exterior`	`ExteriorDerivative`	Sparse incidence matrices ,
`hodge`	`HodgeStar`	Diagonal , , (barycentric or circumcentric dual)
`laplace`	`Operators`	Laplace-Beltrami, Bochner, Lichnerowicz Laplacians
`line_bundle`	`Section<K>`, `BochnerLaplacian<K>`	Complex sections of line bundles for k-atic fields
`extrinsic`	`ExtrinsicOperators`	Tangent-projector discretisation: Killing, DIV, GRAD, viscosity Laplacian
`stokes`	`StokesSolverAL`	Augmented Lagrangian Stokes solver on triangle meshes
`divergence`	`apply_divergence`	Covariant divergence of vector/tensor fields
`advection`	`apply_scalar_advection`	Upwind covariant advection

Quick Start

use cartan_dec::{FlatMesh, Operators};
use nalgebra::DVector;
 
// 4x4 uniform grid on [0,1]^2
let mesh = FlatMesh::unit_square_grid(4);
let ops  = Operators::from_mesh(&mesh, &Euclidean::<2>);
 
// Scalar Laplacian applied to a vertex field
let f  = DVector::from_element(mesh.n_vertices(), 1.0_f64);
let lf = ops.apply_laplace_beltrami(&f);

Two Discretisation Paths

Intrinsic (line bundles): For 2-manifolds, k-atic fields (nematics, tetratics, hexatics) are represented as complex sections of the line bundle with connection . The discrete Bochner Laplacian uses parallel-transport phase factors and cotangent weights. Topological defect charges are computed exactly (discrete Poincare-Hopf holds).

Extrinsic (Killing operator): For surfaces in (and eventually k-manifolds in ), the tangent-plane projection bypasses connection discretisation entirely. The Killing operator, its adjoints (DIV, GRAD), and the viscosity Laplacian are assembled from FEM gradients and projectors.

Both paths are available on 2-manifolds for cross-validation. The extrinsic path generalises to higher-dimensional manifolds where the complex line bundle structure does not exist.

References

Desbrun, Hirani, Leok, Marsden. "Discrete Exterior Calculus." arXiv:math/0508341, 2005.
Hirani. "Discrete Exterior Calculus." PhD thesis, Caltech, 2003.
Zhu, Saintillan, Chern. "Active nematic fluids on Riemannian 2-manifolds." arXiv:2405.06044, 2024.
Zhu, Saintillan, Chern. "Stokes flow of an evolving fluid film with arbitrary shape and topology." JFM 1003, R1, 2025.