MHD Example

examples/mhd.jl

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+# The model in this file is copied from the stream function formulation introduced here:
+# https://algebraicjulia.github.io/Decapodes.jl/dev/navier_stokes/ns/
+# , but updated with terms representing β being advected by the fluid. The initial conditions
+# setup code is copied outright.
+
+# The main reference used for developing this model is:
+# "Magnetohydrodynamics Simulation via Discrete Exterior Calculus", Gillespie, M.
+# https://markjgillespie.com/Research/MHD/MHD_Simulation_with_DEC.pdf
+# Note that the Gillespie paper does not use a stream function-vorticity formulation.
+
+@info "Loading Dependencies"
+
+#=
+# Dependencies can be installed with the following command:
+using Pkg
+Pkg.add(["ACSets", "CairoMakie", "CombinatorialSpaces", "ComponentArrays",
+  "CoordRefSystems", "DiagrammaticEquations", "GeometryBasics", "JLD2",
+  "LinearAlgebra", "Logging", "LoggingExtras", "OrdinaryDiffEq", "SparseArrays",
+  "StaticArrays", "TerminalLoggers"])
+=#
+
+# Saving
+using JLD2
+
+# other dependencies
+using MLStyle
+using Statistics: mean
+
+begin # Dependencies
+# AlgebraicJulia
+using ACSets
+using CombinatorialSpaces
+using Decapodes
+using DiagrammaticEquations
+
+# Meshing
+using CoordRefSystems
+using GeometryBasics: Point3
+Point3D = Point3{Float64};
+
+# Visualization
+using CairoMakie
+
+# Simulation
+using ComponentArrays
+using LinearAlgebra
+using LinearAlgebra: factorize
+using Logging: global_logger
+using LoggingExtras
+using OrdinaryDiffEq
+using SparseArrays
+using StaticArrays
+using TerminalLoggers: TerminalLogger
+global_logger(TerminalLogger())
+
+# Saving
+using JLD2
+
+# other dependencies
+using MLStyle
+using Statistics: mean
+end # Dependencies
+
+@info "Defining models"
+# Beta is out-of-plane:
+mhd_out_of_plane = @decapode begin
+    ψ::Form0
+    η::DualForm1
+    (dη,β)::DualForm2
+    # δ = ⋆d⋆
+    ∂ₜ(dη) == -1*(∘(⋆₁, dual_d₁)((⋆(dη) ∧₀₁ ♭♯(η)) + (⋆(β) ∧₀₁ ♭♯(∘(⋆, d, ⋆)(β)))))
+    ∂ₜ(β) == -1*(∘(⋆₁, dual_d₁)(⋆(β) ∧₀₁ ♭♯(η)))
+    # solve for stream function
+    ψ == ∘(⋆, Δ⁻¹)(dη)
+    η == ⋆(d(ψ))
+end;
+
+# Beta lies in-plane:
+mhd = @decapode begin
+    ψ::Form0
+    (η,β)::DualForm1
+    dη::DualForm2
+
+    ∂ₜ(dη) == -1*(∘(⋆₁, dual_d₁)((⋆(dη) ∧₀₁ ♭♯(η)) + (♭♯(⋆(β)) ∧ᵈᵈ₁₀ ∘(⋆, d, ⋆)(β))))
+    ∂ₜ(β) == -1*(∘(⋆₂, dual_d₀)(⋆(β) ∧ᵖᵈ₁₁ η))
+
+    ψ == ∘(⋆, Δ⁻¹)(dη)
+    η == ⋆(d(ψ))
+end;
+
+@info "Allocating Mesh and Operators"
+const RADIUS = 1.0;
+sphere = :ICO7;
+s = @match sphere begin
+    :ICO5 => loadmesh(Icosphere(4, RADIUS));
+    :ICO6 => loadmesh(Icosphere(6, RADIUS));
+    :ICO7 => loadmesh(Icosphere(7, RADIUS));
+    :ICO8 => loadmesh(Icosphere(8, RADIUS));
+    :flat => triangulated_grid(10, 10, 0.2, 0.2, Point3D)
+    :UV => begin
+        s, _, _ = makeSphere(0, 180, 2.5, 0, 360, 2.5, RADIUS);
+        s;
+    end
+end;
+sd = EmbeddedDeltaDualComplex2D{Bool,Float64,Point3D}(s);
+subdivide_duals!(sd, Circumcenter());
+
+Δ0 = Δ(0,sd);
+fΔ0 = factorize(Δ0);
+d0 = dec_differential(0,sd);
+d1 = dec_differential(1,sd);
+dd0 = dec_dual_derivative(0,sd);
+dd1 = dec_dual_derivative(1,sd);
+δ1 = δ(1,sd);
+s0 = dec_hodge_star(0,sd,GeometricHodge());
+s1 = dec_hodge_star(1,sd,GeometricHodge());
+s2 = dec_hodge_star(2, sd);
+s0inv = dec_inv_hodge_star(0,sd,GeometricHodge());
+♭♯_m = ♭♯_mat(sd);
+@info "    Differential operators allocated"
+
+function generate(s, my_symbol; hodge=GeometricHodge())
+  op = @match my_symbol begin
+    :Δ⁻¹ => x -> begin
+      y = fΔ0 \ x
+      y .- minimum(y)
+    end
+    :♭♯ => x -> ♭♯_m * x
+    _ => default_dec_matrix_generate(s, my_symbol, hodge)
+  end
+  return (args...) -> op(args...)
+end;
+
+sim = gensim(mhd);
+open("mhd_sim.jl", "w") do f
+    write(f, string(gensim(mhd)))
+end
+sim = include("mhd_sim.jl")
+f = sim(sd, generate);
+
+constants_and_parameters = (
+  μ = 0.001,)
+
+begin # ICs
+@info "Setting Initial Conditions"
+
+"""    function great_circle_dist(pnt,G,a,cntr)
+Compute the length of the shortest path along a sphere, given Cartesian coordinates.
+"""
+function great_circle_dist(pnt1::Point3D, pnt2::Point3D)
+  RADIUS * acos(dot(pnt1,pnt2))
+end
+
+abstract type AbstractVortexParams end
+
+struct TaylorVortexParams <: AbstractVortexParams
+  G::Real
+  a::Real
+end
+
+struct PointVortexParams <: AbstractVortexParams
+  τ::Real
+  a::Real
+end
+
+"""    function taylor_vortex(pnt::Point3D, cntr::Point3D, p::TaylorVortexParams)
+Compute the value of a Taylor vortex at the given point.
+"""
+function taylor_vortex(pnt::Point3D, cntr::Point3D, p::TaylorVortexParams)
+  gcd = great_circle_dist(pnt,cntr)
+  (p.G/p.a) * (2 - (gcd/p.a)^2) * exp(0.5 * (1 - (gcd/p.a)^2))
+end
+
+"""    function point_vortex(pnt::Point3D, cntr::Point3D, p::PointVortexParams)
+Compute the value of a smoothed point vortex at the given point.
+"""
+function point_vortex(pnt::Point3D, cntr::Point3D, p::PointVortexParams)
+  gcd = great_circle_dist(pnt,cntr)
+  p.τ / (cosh(3gcd/p.a)^2)
+end
+
+taylor_vortex(sd::HasDeltaSet, cntr::Point3D, p::TaylorVortexParams) =
+  map(x -> taylor_vortex(x, cntr, p), point(sd))
+point_vortex(sd::HasDeltaSet, cntr::Point3D, p::PointVortexParams) =
+  map(x -> point_vortex(x, cntr, p), point(sd))
+
+"""    function ring_centers(lat, n)
+Find n equispaced points at the given latitude.
+"""
+function ring_centers(lat, n)
+  ϕs = range(0.0, 2π; length=n+1)[1:n]
+  map(ϕs) do ϕ
+    v_sph = Spherical(RADIUS, lat, ϕ)
+    v_crt = convert(Cartesian, v_sph)
+    Point3D(v_crt.x.val, v_crt.y.val, v_crt.z.val)
+  end
+end
+
+"""    function vort_ring(lat, n_vorts, p::T, formula) where {T<:AbstractVortexParams}
+Compute vorticity as primal 0-forms for a ring of vortices.
+Specify the latitude, number of vortices, and a formula for computing vortex strength centered at a point.
+"""
+function vort_ring(lat, n_vorts, p::T, formula) where {T<:AbstractVortexParams}
+  sum(map(x -> formula(sd, x, p), ring_centers(lat, n_vorts)))
+end
+
+"""    function vort_ring(lat, n_vorts, p::PointVortexParams, formula)
+Compute vorticity as primal 0-forms for a ring of vortices.
+Specify the latitude, number of vortices, and a formula for computing vortex strength centered at a point.
+Additionally, place a counter-balance vortex at the South Pole such that the integral of vorticity is 0.
+"""
+function vort_ring(lat, n_vorts, p::PointVortexParams, formula)
+  Xs = sum(map(x -> formula(sd, x, p), ring_centers(lat, n_vorts)))
+  Xsp = point_vortex(sd, Point3D(0.0, 0.0, -1.0), PointVortexParams(-1*n_vorts*p.τ, p.a))
+  Xs + Xsp
+end
+
+X =  # Six equidistant points at latitude θ=0.4.
+  # "... an additional vortex, with strength τ=-18 and a radius a=0.15, is
+  # placed at the south pole (θ=π)."
+  vort_ring(0.4, 6, PointVortexParams(3.0, 0.15), point_vortex)
+
+"""    function solve_poisson(vort::VForm)
+Compute the stream function by solving the Poisson equation.
+"""
+function solve_poisson(vort::VForm)
+  ψ = fΔ0 \ vort.data
+  ψ = ψ .- minimum(ψ)
+end
+solve_poisson(vort::DualForm{2}) =
+  solve_poisson(VForm(s0inv * vort.data))
+
+ψ = solve_poisson(VForm(X))
+
+# Compute velocity as curl (⋆d) of the stream function.
+curl_stream(ψ) = s1 * d0 * ψ
+divergence(u) = s2 * d1 * (s1 \ u)
+RMS(x) = √(mean(x' * x))
+
+integral_of_curl(curl::DualForm{2}) = sum(curl.data)
+# Recall that s0 effectively multiplies each entry by a solid angle.
+# i.e. (sum ∘ ⋆₀) computes a Riemann sum.
+integral_of_curl(curl::VForm) = integral_of_curl(DualForm{2}(s0*curl.data))
+
+u₀ = ComponentArray(dη = s0*X, β = zeros(ne(sd)))
+
+constants_and_parameters = (
+  μ = 0.0,)
+
+@info "RMS of divergence of initial velocity: $(∘(RMS, divergence, curl_stream)(ψ))"
+@info "Integral of initial curl: $(integral_of_curl(VForm(X)))"
+end # ICs
+
+@info("Solving")
+tₑ = 1.0; 
+
+prob = ODEProblem(f, u₀, (0, tₑ), constants_and_parameters)
+soln = solve(prob,
+  Tsit5(),
+  dtmax = 1e-3,
+  dense=false,
+  progress=true, progress_steps=1);
+
+function visualize_dynamics(file_name, soln)
+    time = Observable(0.0)
+    fig = Figure()
+    Label(fig[1, 1, Top()], @lift("...at $($time)"), padding = (0, 0, 5, 0))
+    ax = CairoMakie.Axis(fig[1,1])
+    msh = CairoMakie.mesh!(ax, s,
+      color=@lift(s0inv*soln($time).dη),
+      colormap=Reverse(:redsblues))
+    Colorbar(fig[1,2], msh)
+    record(fig, file_name, soln.t[1:10:end]; framerate = 10) do t
+      time[] = t
+    end
+end
+visualize_dynamics("mhd.mp4", soln)
+