### A Pluto.jl notebook ###
# v0.17.2

using Markdown
using InteractiveUtils

# ╔═╡ 191b2740-c20f-11eb-1bc3-4554934741e5
using Plots, Swalbe

# ╔═╡ 993b8d98-0f13-4ac5-80d8-3ebd1b64f59b
md"# Droplet coalescence

A fairly simple show case is the dynamics of two coalescing sessile droplets.
The simulation is performed within the quasi two-dimensional (fast prototyping)  system.
Everything, besides the visualization with **Plots.jl**, is contained in the **Swalbe.jl** package.
However for convenience we define two more functions.
The first one contains the simulation, so we can run it with different parameters just by switching input arguments.
The second one is for data analysis.

## Experiment and analysis functions

- The fluid dynamics simulation `run_drop_coal()`
- Computation of the bridge height `brdige_height()`

One minor addition, for convenience we use a `recipe` that works quite similar to *matplotlibs* `markevery` function.

"

# ╔═╡ 645c6e07-c565-4f44-b29a-d9737fde9222
@recipe function f(::Type{Val{:samplemarkers}}, x, y, z; step = 10)
    n = length(y)
    sx, sy = x[1:step:n], y[1:step:n]
    # add an empty series with the correct type for legend markers
    @series begin
        seriestype := :path
        markershape --> :auto
        x := [Inf]
        y := [Inf]
    end
    # add a series for the line
    @series begin
        primary := false # no legend entry
        markershape := :none # ensure no markers
        seriestype := :path
        seriescolor := get(plotattributes, :seriescolor, :auto)
        x := x
        y := y
    end
    # return  a series for the sampled markers
    primary := false
    seriestype := :scatter
    markershape --> :auto
    x := sx
    y := sy
end

# ╔═╡ 301aecbc-a5fc-4cea-8a1e-7ca130377e93
"""
    run_drop_coal()

Simulation for droplet coalesence in quasi two dimensions.
"""
function run_drop_coal(
    sys::Swalbe.SysConst_1D;
    r₁=115,
    r₂=115, 
    θ₀=1/9,  
    verbos=true, 
    dump = 100, 
    fluid=zeros(sys.Tmax÷dump, sys.L)
)
    println("Simulating droplet coalecense")
    state = Swalbe.Sys(sys)
    drop_cent = (sys.L/3, 2*sys.L/3)
    state.height .= Swalbe.two_droplets(sys, r₁=r₁, r₂=r₂, θ₁=θ₀, θ₂=θ₀, center=drop_cent)
    Swalbe.equilibrium!(state)
    println("Starting the lattice Boltzmann time loop")
    for t in 1:sys.Tmax
        if t % sys.tdump == 0
            mass = 0.0
            mass = sum(state.height)
            if verbos
                println("Time step $t bridge height is $(round(state.height[Int(sys.L/2)], digits=3))")
            end
        end
        Swalbe.filmpressure!(state, sys)
        Swalbe.h∇p!(state)
        Swalbe.slippage!(state, sys)
        state.F .= -state.h∇p .- state.slip
        Swalbe.equilibrium!(state)
        Swalbe.BGKandStream!(state)
        Swalbe.moments!(state)
        
        Swalbe.snapshot!(fluid, state.height, t, dumping = dump)
    end

    return fluid
    
end

# ╔═╡ 518b7204-1301-4a54-8332-ead4f8619cb0
"""
	bridge_height()

Function that tracks the height or thickness at single well defined lattice point.
"""
function bridge_height(data; t1=100, t_dur=size(data)[1], pos=size(data)[2]÷2)
    bridge_h = []
    for i in 1:t_dur
        push!(bridge_h, data[i, pos])
    end
    return bridge_h
end

# ╔═╡ 69f24d80-6181-476c-bd01-a336edf6b8a5
# Need this for plotting purposes
begin
	os = true
	if Sys.iswindows()
		os = false
	elseif Sys.islinux()
		os = true
	end
end

# ╔═╡ b417f463-46e5-421d-8d3b-974532274c3d
md"## Initial condition

The plot below describes the initial condition of our numerical experiment.
Two droplets that are barely touching with an contact angle θ of 20° (or π/9).

Due to surface tension (γ) we expect that these two droplets will coalesce into a single one.
That is because a single droplet does have less surface area than two smaller ones and therefore it is an energetically favorable state."

# ╔═╡ 31fb4367-50e9-46d9-ad41-e0fc309549b4
begin
	# The initial configuration of the numerical experiment
	rad = 500
	h = Swalbe.two_droplets(Swalbe.SysConst_1D(L=1024), r₁=rad, r₂=rad)
	x_ = 1:1024
	p0 = plot(x_ , h, w=3, aspect_ratio=7, label="Initial conf.", xlabel="x [Δx]", ylabel="h(x)")
	ylims!(0,40)
end

# ╔═╡ 684e5cd8-aeae-411b-81e1-342df09d1bd8
md"## Experiment

We start the numerical experiment or simulation by simply calling the function that we defined before: `run_drop_coal()`.

The data will be saved to an array the contains n-time steps and the whole height field.
Something that is only possible with numerical simulations is to change parameters at will.
We do this by changing the surface tension γ of our test liquid.
Of course changing the contact angle θ or the kinematic viscosity ν would as well be possible."

# ╔═╡ 5f2202d7-f217-4fec-8669-b34bfcbf5008
begin
	# Sphere radius
	sphere_rad = 500
	# Different surface tension values
	γs = [0.00008, 0.0003, 0.0006, 0.0008]
	# Array that contains the simulation data
	data_merge = zeros(20000, 1024, length(γs))
	# Loop different surface tension values
	for γ in enumerate(γs)
		# System parameter, δ=50 can still considered small to medium slippage
		sys = Swalbe.SysConst_1D(L=1024, Tmax=2000000, δ=50.0, γ=γ[2])
		# The experiment
		data_merge[:,:,γ[1]] = run_drop_coal(sys, r₁=sphere_rad, r₂=sphere_rad)
	end
end

# ╔═╡ 03a2f622-202a-4847-bf20-2a072eb9f592
md"### Data

Below is an example of the data that was generated during the simulation.
We saved all `Tmax` time steps of all `L` lattice points in `data_merge[:,:,γ]` where we can specifying the desired `γ` value in the last column of the array.
Displayed is the data of the run with γ=0.00008, or simply the first run.

Here a short word of caution.
[The lattice Boltzmann method](https://link.springer.com/book/10.1007/978-3-319-44649-3) is not perfect, it is easy to programm and scales well.
One of the downsides however is the fact that this method is derived from the [Boltzmann equation](https://en.wikipedia.org/wiki/Boltzmann_equation).
Therefore trying to make sense of these surface tension values is not too helpful.
Interpreting results in terms of **SI** units is often not the best way.
Personally I like to work with dimensionless numbers and relevant length and time scales.

Instead of adding **SI** units to the plots we normalize with characteristic quantities of the experiment."

# ╔═╡ 056f47ed-2be2-4937-aa0c-6a9e353cfa92
# Data containing the γ=0.00008 simulation
data_merge[:,:,1] 

# ╔═╡ b35ee645-1220-4132-a9da-3097ade07dcd
md"Since a plot says more than several columns of the dataset here a plot of three different time steps.
The initial kink between the two droplets is quickly smoothed out and the bridge height h₀ is growing."

# ╔═╡ f5112422-d08a-4566-9ba7-ee3eed12cc9c
begin
	# Time evolution for a single surface tension
	lf = 8
	p01 = plot(p0)
	plot!(x_ ,data_merge[1000,:,1], w=3, alpha=0.6, label="t=1000Δt")
	plot!(x_ ,data_merge[5000,:,1], w=3, alpha=0.6, label="t=5000Δt")
	plot!(x_ ,data_merge[15000,:,1], w=3, alpha=0.6, label="t=15000Δt",
		  xlims=(150,1024),
		  # legend=:bottomright,
		  legendfontsize = lf,			# legend font size
          tickfont = (14),				# tick font and size
          guidefont = (15),				# label font and size
		  grid=:none)
end

# ╔═╡ 5cbbb115-47d8-4cb6-9d68-77c391425e86
md"### Under the bridge

The next point on the list is to measure the evolution of the bridge height h₀.
We expect that the height will grow according to some power law, in fact $h_0(t) = t^{\alpha}$ with $\alpha = 2/3$.

Luckily this is a fairly easy task, since by definition the touching point of the droplets is at the center of the substrate."

# ╔═╡ ddcf4747-a3d5-4eaf-8614-c620f37724fc
begin
	bridges = zeros(20000, length(γs))
	for i in 1:length(γs)
		bridges[:, i] = bridge_height(data_merge[:,:,i])
	end
end

# ╔═╡ f7db6fb5-eaa0-4954-8730-89647cc73629
md"Now there are two data sets
- Film thicknesses: $h(t,x)$
- bridge heights: $h_0(t,γ)$ 

Especially the bridge heights should likely collapse to a single curve upon rescaling with a characteristic time scale $\tau$ which we define as

$$\tau = \frac{\mu R}{\gamma},$$

where $R$ is the radius of the sphere from which the spherical cap has been cut, $\gamma$ is the surface tension and $\mu$ is the liquids viscosity.
This time scale can be understood as a capillary time scale, see [Zitz et al.](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.100.033313)."

# ╔═╡ b32505e2-5fc8-48d1-8381-8050a8d292b5
"""
	τ(γ, R, μ)

Computes a characteristic time scale.
"""
function τ(γ; R=500, μ=1/6)
	time_c = 0.0
	time_c = R*μ/γ
	return time_c
end

# ╔═╡ e4cc9ddc-cf63-47ba-8b13-ddf9265c8338
md"## Results
In the following we show the results collected.
First being the evolution of the bridge height h₀(t).
This information may not be too insightful, because the data is given in terms of lattice Boltzmann units (l.b.u.).
To overcome this issue we have two natural scales.
On the one hand a time scale defined by **τ** and on the other hand a length scale given by the spheres radii **R**.

- τ ... time
- R ... length

In the plot below we see that *pure* data spans four different curves, based on their respective surface tensions.
"

# ╔═╡ 842006de-fb79-41be-8af1-459cb7bdde2e
begin
	labels_gamma = ["γ=0.00008" "γ=0.0003" "γ=0.0006" "γ=0.0008"]
	γ_h = [0.00008, 0.0003, 0.0006, 0.0008]
	time_lbm = 100:100:2000000
	p1 = plot(time_lbm, 
		      bridges, 
		      xlabel="t [Δt]", 
		      ylabel="h₀", 
		      label=labels_gamma, 
			  xticks=([0:1000000:2000000;], ["0", "1×10⁶", "2×10⁶"]),
		      xlims=(0,2100000),
			  ylims=(0, 40),
		      w = 3, 							# line width
		      st = :samplemarkers, 				# some recipy stuff
		      step = 1800, 						# density of markers
		      marker = (8, :auto, 0.6),			# marker size 
		      legend=:bottomright,
			  legendfontsize = lf,			# legend font size
              tickfont = (14),	# tick font and size
              guidefont = (15),	# label font and size
			  grid=:none
	         )
end

# ╔═╡ 96b773b7-e158-47c4-b25c-75e4fef655b8
md"In the box below the lattice Boltzmann time steps are normalized and nondimensionalized with τ."

# ╔═╡ 0db816bf-9dc2-43f3-a315-887c9ca68ff4
begin
	# Normalize the simulation time step with the viscous time
	time_norm = zeros(length(time_lbm), length(γs))
	for i in enumerate(γs)
		time_norm[:, i[1]] .= time_lbm ./ τ(i[2])
	end
end

# ╔═╡ 6451c7dd-344f-41d1-a869-7a733ad5b2b1
md"With the normalized time we can plot the data of the bridge height h₀ again.
Interestingly now the data *collapses* to a single master curve, independent of our parameters.

Another interesting obeservation is that the curve has a straight part in the **loglog** scale, which means that the data follows a powerlaw.
Luckily for us the exponent of the powerlaw α is exactly the one that we want to observe for a partially wetting substrate with small contact angle (θ < π/2).

$$\alpha = 2/3.$$

The black dashed line in the plot below.
"

# ╔═╡ 23fd2d46-22c8-4dae-8437-d3fd4628f6c3
begin
	p2 = plot(time_norm, 
	      bridges ./ sphere_rad, 
	      xlabel="t/τ", 
	      ylabel="h₀/R₀", 
	      label=labels_gamma, 
	      w = 3, 							# line width
		  alpha=0.6,
	      ylims=(2e-4, 2e-1),
		  xticks=([0.00001,0.001, 0.1, 10], ["10⁻⁵", "10⁻³", "10⁻¹", "10"]),
		  st = :samplemarkers, 				# some recipy stuff
		  step = 1000, 						# density of markers
		  marker = (8, :auto, 0.6),			# marker size 
		  legendfontsize = lf,			# legend font size
          tickfont = (14),	# tick font and size
          guidefont = (15),	# label font and size
	      axis=:log, 
		  grid=:none,
	      legend=:topleft)
	exponent = 2/3
	there = collect(1e-5:1e-5:10)
	plot!(there, 0.023 .* there.^(exponent), c=:black, w=4, l=:dash, label="t^(2/3)")
end

# ╔═╡ b1de636a-81c7-410c-8767-153a1b9a3079
md"## Conclusion

Two sessile droplets placed next to each other on a partially wetting substrate are coalescing into a single one. 
The dynamics of this process, the growth of the bridge height h₀, can be explained with a powerlaw.
Interestingly the data of the different surface tensions γ can be collapsed to a single master curve upon rescaling."

# ╔═╡ 88f037e3-1ec3-4360-84d3-a6c496731c6c
begin
	# Collecting results in a single plot
	l = @layout[a{0.6h}; b c]
	all_p = plot(p01, p1, p2, layout = l)
end

# ╔═╡ 85104c76-b2fc-4813-8205-34de12792e0c
#savefig(all_p, "..\\paper\\drop_coal.png")

# ╔═╡ 00000000-0000-0000-0000-000000000001
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[compat]
Plots = "~1.22.6"
Swalbe = "~1.0.0"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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[[Distributed]]
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[[DocStringExtensions]]
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[[Downloads]]
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[[EarCut_jll]]
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[[Expat_jll]]
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[[FFMPEG_jll]]
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[[MathOptInterface]]
deps = ["BenchmarkTools", "CodecBzip2", "CodecZlib", "JSON", "JSONSchema", "LinearAlgebra", "MutableArithmetics", "OrderedCollections", "SparseArrays", "Test", "Unicode"]
git-tree-sha1 = "575644e3c05b258250bb599e57cf73bbf1062901"
uuid = "b8f27783-ece8-5eb3-8dc8-9495eed66fee"
version = "0.9.22"

[[MathProgBase]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "9abbe463a1e9fc507f12a69e7f29346c2cdc472c"
uuid = "fdba3010-5040-5b88-9595-932c9decdf73"
version = "0.7.8"

[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"]
git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
version = "1.0.3"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Measures]]
git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f"
uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.2"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"

[[MutableArithmetics]]
deps = ["LinearAlgebra", "SparseArrays", "Test"]
git-tree-sha1 = "8d9496b2339095901106961f44718920732616bb"
uuid = "d8a4904e-b15c-11e9-3269-09a3773c0cb0"
version = "0.2.22"

[[NaNMath]]
git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.5"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+0"

[[OpenLibm_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "05823500-19ac-5b8b-9628-191a04bc5112"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.10+0"

[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
version = "0.5.5+0"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[PCRE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
version = "8.44.0+0"

[[Parameters]]
deps = ["OrderedCollections", "UnPack"]
git-tree-sha1 = "34c0e9ad262e5f7fc75b10a9952ca7692cfc5fbe"
uuid = "d96e819e-fc66-5662-9728-84c9c7592b0a"
version = "0.12.3"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "98f59ff3639b3d9485a03a72f3ab35bab9465720"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.0.6"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlotThemes]]
deps = ["PlotUtils", "Requires", "Statistics"]
git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "2.0.1"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"]
git-tree-sha1 = "b084324b4af5a438cd63619fd006614b3b20b87b"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.15"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"]
git-tree-sha1 = "ba43b248a1f04a9667ca4a9f782321d9211aa68e"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.22.6"

[[PooledArrays]]
deps = ["DataAPI", "Future"]
git-tree-sha1 = "a193d6ad9c45ada72c14b731a318bedd3c2f00cf"
uuid = "2dfb63ee-cc39-5dd5-95bd-886bf059d720"
version = "1.3.0"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.2"

[[PrettyTables]]
deps = ["Crayons", "Formatting", "Markdown", "Reexport", "Tables"]
git-tree-sha1 = "d940010be611ee9d67064fe559edbb305f8cc0eb"
uuid = "08abe8d2-0d0c-5749-adfa-8a2ac140af0d"
version = "1.2.3"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[Profile]]
deps = ["Printf"]
uuid = "9abbd945-dff8-562f-b5e8-e1ebf5ef1b79"

[[Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+0"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[Random123]]
deps = ["Libdl", "Random", "RandomNumbers"]
git-tree-sha1 = "0e8b146557ad1c6deb1367655e052276690e71a3"
uuid = "74087812-796a-5b5d-8853-05524746bad3"
version = "1.4.2"

[[RandomNumbers]]
deps = ["Random", "Requires"]
git-tree-sha1 = "043da614cc7e95c703498a491e2c21f58a2b8111"
uuid = "e6cf234a-135c-5ec9-84dd-332b85af5143"
version = "1.5.3"

[[RecipesBase]]
git-tree-sha1 = "44a75aa7a527910ee3d1751d1f0e4148698add9e"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.1.2"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.4.1"

[[Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.1.3"

[[Revise]]
deps = ["CodeTracking", "Distributed", "FileWatching", "JuliaInterpreter", "LibGit2", "LoweredCodeUtils", "OrderedCollections", "Pkg", "REPL", "Requires", "UUIDs", "Unicode"]
git-tree-sha1 = "41deb3df28ecf75307b6e492a738821b031f8425"
uuid = "295af30f-e4ad-537b-8983-00126c2a3abe"
version = "3.1.20"

[[RoundingEmulator]]
git-tree-sha1 = "40b9edad2e5287e05bd413a38f61a8ff55b9557b"
uuid = "5eaf0fd0-dfba-4ccb-bf02-d820a40db705"
version = "0.2.1"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[SetRounding]]
git-tree-sha1 = "d7a25e439d07a17b7cdf97eecee504c50fedf5f6"
uuid = "3cc68bcd-71a2-5612-b932-767ffbe40ab0"
version = "0.2.1"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "2d57e14cd614083f132b6224874296287bfa3979"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "1.8.0"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.13"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "eb35dcc66558b2dda84079b9a1be17557d32091a"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.12"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.3"

[[Swalbe]]
deps = ["BSON", "CUDA", "DataFrames", "FileIO", "JLD2", "LazySets", "Parameters", "Random", "Revise", "Statistics"]
git-tree-sha1 = "b7cc26d63f598239153e748091d524f55b5ab574"
uuid = "1073fb09-a5e2-4e80-8bde-f3562efda53f"
version = "1.0.0"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "fed34d0e71b91734bf0a7e10eb1bb05296ddbcd0"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.6.0"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[TimerOutputs]]
deps = ["ExprTools", "Printf"]
git-tree-sha1 = "7cb456f358e8f9d102a8b25e8dfedf58fa5689bc"
uuid = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
version = "0.5.13"

[[TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "216b95ea110b5972db65aa90f88d8d89dcb8851c"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.6"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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