Extrinsic Operators

The extrinsic module implements the tangent-projector discretisation from Zhu, Saintillan, and Chern (2025). Instead of discretising the intrinsic connection on tensor bundles (which is difficult for dim > 2), it works in the ambient and projects to the tangent plane per simplex.

This gives dimension-agnostic operators that generalise to k-manifolds embedded in without ever computing connection coefficients.

Tangent Projector

For each face with unit normal , the tangent projector is:

This is idempotent () and kills the normal component ().

Killing Operator

The Killing operator maps vertex velocities to per-face symmetric strain rates:

where is the FEM gradient of the hat function at vertex , component .

Derived Operators

All other operators are obtained from by adjoint or trace:

Viscosity Laplacian

where is the diagonal face-area matrix. Symmetric negative semi-definite, with a 6D kernel (3 translations + 3 rotations for surfaces in ).

Usage

use cartan_dec::extrinsic::ExtrinsicOperators;
 
let ops = ExtrinsicOperators::from_mesh(&mesh);
 
// Apply divergence to a velocity field (3*nv -> nv).
let div_u = ops.apply_div(&velocity);
 
// Apply gradient to a pressure field (nv -> 3*nv).
let grad_p = ops.apply_grad(&pressure);
 
// Apply viscosity Laplacian (3*nv -> 3*nv).
let lu = ops.apply_viscosity_lap(&velocity);

Advantages Over Intrinsic DEC for Tensor Fields

• No parallel transport between dual cells needed
• The embedding + projection gives coordinate-free tensor calculus without connection coefficients
• Works on arbitrary genus without special treatment
• Generalises to k-manifolds by replacing the normal projector

The trade-off: requires an embedding. Purely intrinsic computations (e.g., the Bochner Laplacian on line bundles) should use the line_bundle module instead.

References

• Zhu, Saintillan, Chern. "Stokes flow of an evolving fluid film with arbitrary shape and topology." JFM 1003, R1, 2025. Section 3.