Diffusion Equation on an Annulus

In this tutorial, we consider a diffusion equation on an annulus:

\[\begin{equation} \begin{aligned} \pdv{u(\vb x, t)}{t} &= \grad^2 u(\vb x, t) & \vb x \in \Omega, \\ \grad u(\vb x, t) \vdot \vu n(\vb x) &= 0 & \vb x \in \mathcal D(0, 1), \\ u(\vb x, t) &= c(t) & \vb x \in \mathcal D(0,0.2), \\ u(\vb x, t) &= u_0(\vb x), \end{aligned} \end{equation}\]

demonstrating how we can solve PDEs over multiply-connected domains. Here, $\mathcal D(0, r)$ is a circle of radius $r$ centered at the origin, $\Omega$ is the annulus between $\mathcal D(0,0.2)$ and $\mathcal D(0, 1)$, $c(t) = 50[1-\mathrm{e}^{-t/2}]$, and

\[u_0(x) = 10\mathrm{e}^{-25\left[\left(x+\frac12\right)^2+\left(y+\frac12\right)^2\right]} - 10\mathrm{e}^{-45\left[\left(x-\frac12\right)^2+\left(y-\frac12\right)^2\right]} - 5\mathrm{e}^{-50\left[\left(x+\frac{3}{10}\right)^2+\left(y+\frac12\right)^2\right]}.\]

The complicated task for this problem is the definition of the mesh of the annulus. We need to follow the boundary specification from DelaunayTriangulation.jl, discussed here. In particular, the outer boundary must be counter-clockwise, the inner boundary be clockwise, and we need to provide the nodes as a Vector{Vector{Vector{Int}}}. We define this mesh below.

using DelaunayTriangulation, FiniteVolumeMethod, CairoMakie
R₁ = 0.2
R₂ = 1.0
θ = collect(LinRange(0, 2π, 100))
θ[end] = 0.0 # get the endpoints to match
x = [
    [R₂ .* cos.(θ)], # outer first
    [reverse(R₁ .* cos.(θ))] # then inner - reverse to get clockwise orientation
]
y = [
    [R₂ .* sin.(θ)], #
    [reverse(R₁ .* sin.(θ))]
]
boundary_nodes, points = convert_boundary_points_to_indices(x, y)
tri = triangulate(points; boundary_nodes)
A = get_total_area(tri)
refine!(tri; max_area=1e-4A)
triplot(tri)

mesh = FVMGeometry(tri)
FVMGeometry with 8226 control volumes, 16007 triangles, and 24233 edges

Now let us define the boundary conditions. Remember, the order of the boundary conditions follows the order of the boundaries in the mesh. The outer boundary came first, and then came the inner boundary. We can verify that this is the order of the boundary indices as follows:

fig = Figure()
ax = Axis(fig[1, 1])
outer = [get_point(tri, i) for i in get_neighbours(tri, -1)]
inner = [get_point(tri, i) for i in get_neighbours(tri, -2)]
triplot!(ax, tri)
scatter!(ax, outer, color=:red)
scatter!(ax, inner, color=:blue)
fig

So, the boundary conditions are:

outer_bc = (x, y, t, u, p) -> zero(u)
inner_bc = (x, y, t, u, p) -> oftype(u, 50(1 - exp(-t / 2)))
types = (Neumann, Dirichlet)
BCs = BoundaryConditions(mesh, (outer_bc, inner_bc), types)
BoundaryConditions with 2 boundary conditions with types (Neumann, Dirichlet)

Finally, let's define the problem and solve it.

initial_condition_f = (x, y) -> begin
    10 * exp(-25 * ((x + 0.5) * (x + 0.5) + (y + 0.5) * (y + 0.5))) - 5 * exp(-50 * ((x + 0.3) * (x + 0.3) + (y + 0.5) * (y + 0.5))) - 10 * exp(-45 * ((x - 0.5) * (x - 0.5) + (y - 0.5) * (y - 0.5)))
end
diffusion_function = (x, y, t, u, p) -> one(u)
initial_condition = [initial_condition_f(x, y) for (x, y) in each_point(tri)]
final_time = 2.0
prob = FVMProblem(mesh, BCs;
    diffusion_function,
    final_time,
    initial_condition)
FVMProblem with 8226 nodes and time span (0.0, 2.0)
using OrdinaryDiffEq, LinearSolve
sol = solve(prob, TRBDF2(linsolve=KLUFactorization()), saveat=0.2)
retcode: Success
Interpolation: 1st order linear
t: 11-element Vector{Float64}:
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 1.2
 1.4
 1.6
 1.8
 2.0
u: 11-element Vector{Vector{Float64}}:
 [-1.6918979226151304e-9, -2.6824824758316947e-8, -3.508918015123032e-7, -3.7478900338780008e-6, -3.2376946343892575e-5, -0.0002242588591250139, -0.0012357868125636464, -0.005380630921148495, -0.01840119235904892, -0.04918487388571497  …  2.0918713433458237e-14, 3.458890574190645, -0.005558506394611075, 2.8396329156286174e-11, -5.049929362733235e-9, -1.913445529149473e-10, 5.667754383376993e-12, 5.945251247871394e-15, -3.6470661441248834e-9, -4.3485569041014065e-5]
 [0.044776239431783424, 0.016726907958835927, -0.0098190205049255, -0.03492537473095343, -0.0589525739059889, -0.0811913566499606, -0.10109478437577493, -0.11900339123181591, -0.13396061458347966, -0.1461851369688485  …  0.4604244890762291, 1.3158333385225314, 0.03518599997718615, 0.5534220548094115, 0.1796198426114264, 0.4147202809074338, 0.5840259694640018, 0.40954910091969754, 0.04562251319893571, 0.9448599818948571]
 [1.8560465047556662, 1.8442231107917193, 1.8341189228750276, 1.8251569018804588, 1.816408153091307, 1.8085637692014882, 1.8020486458198512, 1.7959648309553493, 1.791372066121246, 1.7875907024842426  …  2.196699182934059, 2.8835895560671783, 1.9837736621895719, 2.1237680104471393, 2.0188237615834073, 2.336190392410584, 2.1341487006708393, 2.1752212900606858, 1.8575547389410483, 3.3565016059477526]
 [4.3844956626170575, 4.378545995804241, 4.374092325972138, 4.370387141064735, 4.3662874071770155, 4.362603895840999, 4.359819933554336, 4.356828842769086, 4.35488647000348, 4.353144723776922  …  4.683981092075278, 5.335520809924663, 4.574811590140766, 4.5093274584343686, 4.561387785181804, 4.928043263091946, 4.513379397177754, 4.682838874718729, 4.386373146810551, 6.220764297797526]
 [7.289327439041294, 7.28610944721683, 7.284274530281505, 7.282993724561839, 7.281021171345756, 7.2792279122974035, 7.27812763415356, 7.276514233941954, 7.275741413179119, 7.2748812088794015  …  7.56978289683622, 8.210574460320776, 7.509439242066505, 7.347218975767628, 7.472416305980833, 7.8631216439680465, 7.34892232564419, 7.57840988719536, 7.291384124443954, 9.281115142352236]
 [10.349864716591316, 10.347912278241836, 10.347271966947286, 10.347090447534972, 10.346094140489061, 10.345168069281993, 10.344831718149985, 10.343852416686643, 10.343607493414781, 10.343147040875175  …  10.617624243370118, 11.243632500674538, 10.579916645649323, 10.377257248218204, 10.532839929231029, 10.927997080853823, 10.377979056274048, 10.630463513128609, 10.351971017975025, 12.371462230126909]
 [13.419167926910074, 13.41781113227301, 13.41770333448452, 13.418001717997312, 13.417452175610725, 13.416923875934378, 13.416925900666152, 13.41624460883248, 13.416233433836018, 13.415956482628237  …  13.674755314320933, 14.278332981604084, 13.647771159994207, 13.432726776104372, 13.59751623843466, 13.984541055121042, 13.433018976571283, 13.689089544317357, 13.421243539051773, 15.399209554116393]
 [16.40294230710082, 16.40188299839002, 16.40200429392593, 16.402495938860138, 16.402153898874715, 16.401811637664757, 16.40195682477512, 16.401424387472126, 16.401512470372563, 16.40132234395567  …  16.644810996179245, 17.218389394051336, 16.62327237704967, 16.41019551417071, 16.573358562372473, 16.943861406638472, 16.410297683581025, 16.659295197514986, 16.404935650365353, 18.29702290179812]
 [19.24550938695191, 19.244614269912603, 19.244827126650037, 19.24538151788817, 19.245139224838063, 19.244887712131817, 19.245087524240414, 19.244637314733378, 19.244763727309568, 19.2446170682584  …  19.47263752298845, 20.012541835470465, 19.454182010436305, 19.249848757295258, 19.406263119436968, 19.756137462874126, 19.24986755719106, 19.486647895613643, 19.247394042446498, 21.035917198515495]
 [21.91732633217701, 21.916535089163556, 21.916777735472504, 21.917336394554113, 21.917145265619478, 21.916941502462457, 21.91715622359347, 21.916758068269093, 21.916895476678704, 21.91677273681172  …  22.129486281255307, 22.635814101696177, 22.113024295751757, 21.920279408360912, 22.067778541636255, 22.395955057938913, 21.92026264374806, 22.1427658888403, 21.9190913367679, 23.608729555731376]
 [24.399479988903376, 24.398770157110683, 24.39901295282689, 24.399545747360502, 24.399384865274424, 24.399210544172945, 24.399420750245497, 24.39906340742901, 24.399198503471773, 24.39909155511398  …  24.59530361538348, 25.065981088430707, 24.58043249620733, 24.401722824192777, 24.538428062491725, 24.842757086613155, 24.401692408661553, 24.607662262491655, 24.40110929509522, 25.99198334271778]
fig = Figure(fontsize=38)
for (i, j) in zip(1:3, (1, 6, 11))
    ax = Axis(fig[1, i], width=600, height=600,
        xlabel="x", ylabel="y",
        title="t = $(sol.t[j])",
        titlealign=:left)
    tricontourf!(ax, tri, sol.u[j], levels=-10:2:40, colormap=:matter)
    tightlimits!(ax)
end
resize_to_layout!(fig)
fig

To finish this example, let us consider how natural neighbour interpolation can be applied here. The application is more complicated for this problem since the mesh has holes. Before we do that, though, let us show how we could use pl_interpolate, which could be useful if we did not need a higher quality interpolant. Let us interpolate the solution at $t = 1$, which is sol.t[6]. For this, we need to put the ghost triangles back into tri so that we can safely apply jump_and_march. This is done with add_ghost_triangles!.

add_ghost_triangles!(tri)

Now let's interpolate.

x = LinRange(-R₂, R₂, 400)
y = LinRange(-R₂, R₂, 400)
interp_vals = zeros(length(x), length(y))
u = sol.u[6]
last_triangle = Ref((1, 1, 1))
for (j, _y) in enumerate(y)
    for (i, _x) in enumerate(x)
        T = jump_and_march(tri, (_x, _y), try_points=last_triangle[])
        last_triangle[] = indices(T) # used to accelerate jump_and_march, since the points we're looking for are close to each other
        if DelaunayTriangulation.is_ghost_triangle(T) # don't extrapolate
            interp_vals[i, j] = NaN
        else
            interp_vals[i, j] = pl_interpolate(prob, T, sol.u[6], _x, _y)
        end
    end
end
fig, ax, sc = contourf(x, y, interp_vals, levels=-10:2:40, colormap=:matter)
fig

Let's now consider applying NaturalNeighbours.jl. We apply it naively first to highlight some complications.

using NaturalNeighbours
_x = vec([x for x in x, y in y]) # NaturalNeighbours.jl needs vector data
_y = vec([y for x in x, y in y])
itp = interpolate(tri, u, derivatives=true)
Natural Neighbour Interpolant
    z: [10.349864716591316, 10.347912278241836, 10.347271966947286, 10.347090447534972, 10.346094140489061, 10.345168069281993, 10.344831718149985, 10.343852416686643, 10.343607493414781, 10.343147040875175  …  10.617624243370118, 11.243632500674538, 10.579916645649323, 10.377257248218204, 10.532839929231029, 10.927997080853823, 10.377979056274048, 10.630463513128609, 10.351971017975025, 12.371462230126909]
    ∇: [(0.0167759898313112, -0.01878578243151147), (-0.032908955775623944, -0.00262729515068905), (-0.0499857484041637, -0.028310041468257337), (-0.0036232656312899718, -0.012179801096744552), (0.001988309479592199, -0.002205287158034077), (-0.0033426705601698322, -0.018184635905077507), (0.023677745363512127, -0.004738085134243245), (-0.016326170141901342, -0.013441410165932562), (0.013140605190489733, -0.0073889162622241935), (-0.003534308098465609, 0.0032229658783444644)  …  (1.7559371616569213, -2.4109153545203887), (5.176488791237448, 2.751464523042365), (-2.7584085170587995, -0.7684092321135257), (-0.1448942568797414, 0.2181202955857769), (0.2627845167491605, -2.47053927005044), (1.3111000316616632, -4.4848132772296125), (0.2106258934891403, -0.15947438204654413), (-2.5845247658419055, 1.6660687236508438), (-0.2671645494903941, -0.01917277216052589), (0.3936531616440942, -9.97875863146146)]
    H: [(16.288141655107403, -0.6394535341044831, 0.6854575492474814), (14.335584652134871, 0.6955755751292197, 0.8702733271567972), (10.831669262884455, 0.35107866273802235, 1.2388893791228002), (13.562816107328018, -0.4753557730912129, 2.6914162879265797), (13.299434352775776, 0.7098705301318029, 3.978005563335226), (12.961763952972248, 1.3081852166380359, 3.967516058279129), (13.378086826768516, 1.9628398235215962, 5.599097606392703), (11.023722943553466, 2.781116547355478, 5.312298304241565), (12.168900954138454, 2.947587603587016, 6.256466484912792), (9.922031664521095, 4.55885688686574, 7.003956074571443)  …  (4.528619693662909, 11.03739306944645, -10.724310463496717), (16.313060788112068, -1.6223355249800782, 13.425380303917457), (17.220186845167305, -1.8028616548100926, 5.806155464599244), (2.913297168551773, 10.80774838792608, -6.136927800991009), (-2.7193537362874736, 17.909425855572504, -2.402873271088078), (-4.038603125437703, 19.22114948500803, -7.597607167701472), (10.222014842704965, 3.353644951280155, -6.480285894682976), (11.740698189055745, 2.785718339489214, -10.337729524010104), (15.465098540749937, -0.49936621092941696, 0.12810171693728375), (-18.826376615132325, 32.1891065646984, -2.03364083283624)]
itp_vals = itp(_x, _y; method=Farin())
160000-element Vector{Float64}:
 10.397841454574205
 10.397881486052796
 10.391546417236441
 10.391529995958052
 10.391513574679607
  ⋮
 10.334195891351527
 10.334057991849193
 10.333920092346858
 10.333782192844467
 10.33364429334216
fig, ax, sc = contourf(x, y, reshape(itp_vals, length(x), length(y)), colormap=:matter, levels=-10:2:40)
fig

The issue here is that the interpolant is trying to extrapolate inside the hole and outside of the annulus. To avoid this, you need to pass project=false.

itp_vals = itp(_x, _y; method=Farin(), project=false)
160000-element Vector{Float64}:
 Inf
 Inf
 Inf
 Inf
 Inf
  ⋮
 Inf
 Inf
 Inf
 Inf
 Inf
fig, ax, sc = contourf(x, y, reshape(itp_vals, length(x), length(y)), colormap=:matter, levels=-10:2:40)
fig

Just the code

An uncommented version of this example is given below. You can view the source code for this file here.

using DelaunayTriangulation, FiniteVolumeMethod, CairoMakie
R₁ = 0.2
R₂ = 1.0
θ = collect(LinRange(0, 2π, 100))
θ[end] = 0.0 # get the endpoints to match
x = [
    [R₂ .* cos.(θ)], # outer first
    [reverse(R₁ .* cos.(θ))] # then inner - reverse to get clockwise orientation
]
y = [
    [R₂ .* sin.(θ)], #
    [reverse(R₁ .* sin.(θ))]
]
boundary_nodes, points = convert_boundary_points_to_indices(x, y)
tri = triangulate(points; boundary_nodes)
A = get_total_area(tri)
refine!(tri; max_area=1e-4A)
triplot(tri)

mesh = FVMGeometry(tri)

fig = Figure()
ax = Axis(fig[1, 1])
outer = [get_point(tri, i) for i in get_neighbours(tri, -1)]
inner = [get_point(tri, i) for i in get_neighbours(tri, -2)]
triplot!(ax, tri)
scatter!(ax, outer, color=:red)
scatter!(ax, inner, color=:blue)
fig

outer_bc = (x, y, t, u, p) -> zero(u)
inner_bc = (x, y, t, u, p) -> oftype(u, 50(1 - exp(-t / 2)))
types = (Neumann, Dirichlet)
BCs = BoundaryConditions(mesh, (outer_bc, inner_bc), types)

initial_condition_f = (x, y) -> begin
    10 * exp(-25 * ((x + 0.5) * (x + 0.5) + (y + 0.5) * (y + 0.5))) - 5 * exp(-50 * ((x + 0.3) * (x + 0.3) + (y + 0.5) * (y + 0.5))) - 10 * exp(-45 * ((x - 0.5) * (x - 0.5) + (y - 0.5) * (y - 0.5)))
end
diffusion_function = (x, y, t, u, p) -> one(u)
initial_condition = [initial_condition_f(x, y) for (x, y) in each_point(tri)]
final_time = 2.0
prob = FVMProblem(mesh, BCs;
    diffusion_function,
    final_time,
    initial_condition)

using OrdinaryDiffEq, LinearSolve
sol = solve(prob, TRBDF2(linsolve=KLUFactorization()), saveat=0.2)

fig = Figure(fontsize=38)
for (i, j) in zip(1:3, (1, 6, 11))
    ax = Axis(fig[1, i], width=600, height=600,
        xlabel="x", ylabel="y",
        title="t = $(sol.t[j])",
        titlealign=:left)
    tricontourf!(ax, tri, sol.u[j], levels=-10:2:40, colormap=:matter)
    tightlimits!(ax)
end
resize_to_layout!(fig)
fig

add_ghost_triangles!(tri)

x = LinRange(-R₂, R₂, 400)
y = LinRange(-R₂, R₂, 400)
interp_vals = zeros(length(x), length(y))
u = sol.u[6]
last_triangle = Ref((1, 1, 1))
for (j, _y) in enumerate(y)
    for (i, _x) in enumerate(x)
        T = jump_and_march(tri, (_x, _y), try_points=last_triangle[])
        last_triangle[] = indices(T) # used to accelerate jump_and_march, since the points we're looking for are close to each other
        if DelaunayTriangulation.is_ghost_triangle(T) # don't extrapolate
            interp_vals[i, j] = NaN
        else
            interp_vals[i, j] = pl_interpolate(prob, T, sol.u[6], _x, _y)
        end
    end
end
fig, ax, sc = contourf(x, y, interp_vals, levels=-10:2:40, colormap=:matter)
fig

using NaturalNeighbours
_x = vec([x for x in x, y in y]) # NaturalNeighbours.jl needs vector data
_y = vec([y for x in x, y in y])
itp = interpolate(tri, u, derivatives=true)

itp_vals = itp(_x, _y; method=Farin())

fig, ax, sc = contourf(x, y, reshape(itp_vals, length(x), length(y)), colormap=:matter, levels=-10:2:40)
fig

itp_vals = itp(_x, _y; method=Farin(), project=false)

fig, ax, sc = contourf(x, y, reshape(itp_vals, length(x), length(y)), colormap=:matter, levels=-10:2:40)
fig

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