diego domenzain

April 2021 @ Colorado School of Mines

These scripts are an example of solving the Helmholtz-Hodge Decomposition on a two-dimensional vector field.

Let u be a 2D vector field,

u = grad φ + rot ψ + h

u = ∇φ + ∇ × ψ + h

where,

∇φ = (∂x φ, ∂y φ),

∇ × ψ = (∂y ψ, -∂x ψ),

∇ × h = ∇⋅h = 0.

We solve for ψ and φ given by,

∇⋅∇ φ = ∇ ⋅ u

∇⋅∇ ψ = -∇⋅J u

h = u - ∇φ - ∇ × ψ

where J is rotation by π/2,

J = [0 -1]
    [1  0]

There are a couple of different approaches for finding φ and ψ. Here, I follow Harsh Bhatia et al.

The idea is to incorporate open-flow boundary conditions in a natural way, whatever that means.

This accomplished by "solving" for φ and ψ by putting ∇⋅∇ on the other side of the equality as ∫ g ⋅ dΩ,

φ(xo) = ∫ g ∇ ⋅ u dΩ

ψ(xo) = - ∫ g ∇⋅J u dΩ

where g is the Green function that solves the Laplace equation on Ω with a source at xo.

Note: I am not entirely sure the boundaries are well recovered. I should check that.

All these implement this idea:

hhd_simple.m (60 seconds)
hhd_cool.m   (60 seconds)
hhd_cooler.m the integral operator is coded in Fortran-Mex (no speed-up 😢)
hhd_faster.m the integral operator is stored in memory (lots of memory, my computer did not like it)

Discretize the operator ∇⋅∇ and solve for:

∇⋅∇ φ = ∇ ⋅ u

∇⋅∇ ψ = -∇⋅J u

This one implements this idea,

hhd_fast.m (0.5 seconds, light on memory)

The Laplacian operator is built using the alles/projects/graph-alg/mesher/ project.

1. The Natural Helmholtz-Hodge Decomposition for Open-Boundary Flow Analysis. Harsh Bhatia, Valerio Pascucci, Peer-Timo Bremer. IEEE Transactions on Visualization and Computer Graphics, 2014.
2. The Helmholtz-Hodge Decomposition — A Survey. Harsh Bhatia, Gregory Norgard, Valerio Pascucci, Peer-Timo Bremer. IEEE Transactions on Visualization and Computer Graphics, 2013.

Below are φ and ψ.