The Cahn-Hilliard Equation

For this example, Decapodes will model the Cahn-Hilliard equation. This equation describes the evolution of a binary fluid as its two phases separate out into distinct domains. Below is a high resolution preview of this model. Notice how the fluid has separated into distinct regions (blue and red) as well as the presence of a transition region.

Formulating the Equation

We first load in our dependencies

# AlgebraicJulia Dependencies
using Catlab
using CombinatorialSpaces
using Decapodes
using DiagrammaticEquations

# External Dependencies
using CairoMakie
using ComponentArrays
using GeometryBasics
using LinearAlgebra
using MLStyle
using OrdinaryDiffEq
using Random
Point3D = Point3{Float64};

and then proceed to describe our physics using Decapodes.

CahnHilliard = @decapode begin
    C::Form0
    (D, γ)::Constant
    ∂ₜ(C) == D * Δ(C.^3 - C - γ * Δ(C))
end

to_graphviz(CahnHilliard)

In this equation C will represent the concentration of the binary fluid, ranging from -1 to 1 to differentiate between different phases. We also have a diffusion constant D and a constant γ whose square root is the length of the transition regions. This formulation of the Cahn-Hilliard equation was drawn from the Wikipedia page on the topic found here.

Loading the Data

We now generate the mesh information. We'll run the equation on a triangulated grid. We hide the mesh visualization code for clarity.

s = triangulated_grid(100, 100, 0.5, 0.5, Point3D);
sd = EmbeddedDeltaDualComplex2D{Bool, Float64, Point3D}(s);
subdivide_duals!(sd, Circumcenter());

The Cahn-Hilliard equation starts with a random concentration holding values between -1 and 1. For both D and γ constants we choose 0.5.

Random.seed!(0)

C = rand(Float64, nv(sd)) * 2 .- 1
u₀ = ComponentArray(C=C)
constants = (D = 0.5, γ = 0.5);

Colorbar(fig[1,2], msh)

We'll now create the simulation code representing the Cahn-Hilliard equation. We pass nothing in the second argument to sim since we have no custom functions to pass in.

sim = eval(gensim(CahnHilliard))
fₘ = sim(sd, nothing, DiagonalHodge());

(::Main.var"#f#5"{PreallocationTools.FixedSizeDiffCache{Vector{Float64}, Vector{ForwardDiff.Dual{nothing, Float64, 12}}}, PreallocationTools.FixedSizeDiffCache{Vector{Float64}, Vector{ForwardDiff.Dual{nothing, Float64, 12}}}, PreallocationTools.FixedSizeDiffCache{Vector{Float64}, Vector{ForwardDiff.Dual{nothing, Float64, 12}}}, PreallocationTools.FixedSizeDiffCache{Vector{Float64}, Vector{ForwardDiff.Dual{nothing, Float64, 12}}}, PreallocationTools.FixedSizeDiffCache{Vector{Float64}, Vector{ForwardDiff.Dual{nothing, Float64, 12}}}, PreallocationTools.FixedSizeDiffCache{Vector{Float64}, Vector{ForwardDiff.Dual{nothing, Float64, 12}}}, SparseArrays.SparseMatrixCSC{Float64, Int32}}) (generic function with 1 method)

Getting the Solution

Now that everything is set up and ready, we can solve the equation. We run the simulation for 200 time units to see the long-term evolution of the fluid. Note we only save the solution at intervals of 0.1 time units in order to reduce the memory-footprint of the solve.

tₑ = 200
prob = ODEProblem(fₘ, u₀, (0, tₑ), constants)
soln = solve(prob, Tsit5(), saveat=0.1);
soln.retcode

ReturnCode.Success = 1

And we can see the result as a gif.

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