### A Pluto.jl notebook ###
# v0.15.1

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing
        el
    end
end

# ╔═╡ c028100e-f47b-11eb-2547-df779889e265
using WaterLily, StaticArrays, PlutoUI, Interpolations, Plots, Images

# ╔═╡ b0c8df66-1e63-456e-be32-d469e6972e00
md"""
## Simulation of a swimming dogfish shark

**Gabriel D Weymouth** [![Twitter URL](https://img.shields.io/twitter/url/https/twitter.com/gabrielweymouth.svg?style=social&label=)](https://twitter.com/gabrielweymouth) | 11 Aug 2021

---

This post will demonstrate how to set up and simulate a model of a swimming dogfish shark using [Julia](https://julialang.org/) and [WaterLily.jl](https://www.youtube.com/watch?v=YsPkfZqbNSQ). The post is actually a [Pluto notebook](https://www.youtube.com/watch?v=IAF8DjrQSSk) and can be run interactively by clicking on the top right icon.

We'll use a simple model of the shark based on Lighthill's pioneering [paper on the swimming of slender fish](https://doi.org/10.1017/S0022112060001110). It focuses on the "backbone" of the fish; idealizing the shape as a thickness distribution on either side of the center, and the motion as a lateral ("side-to-side") traveling wave. Amazingly, this simple approach provides insight across a huge range of a swimming animals as illustrated in the image below. [Image credit: Gazzola et al, _Nature_ 2014](https://www.nature.com/articles/nphys3078)
"""

# ╔═╡ e9384124-9708-44eb-b5e7-75abc5177a69
begin
	url = "https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fnphys3078/MediaObjects/41567_2014_Article_BFnphys3078_Fig1_HTML.jpg"
	filename = download(url)
	lighthill = load(filename)
end


# ╔═╡ c8293b7b-0585-45b3-8763-8cfa4671b155
md"""

### Modeling the shark's thickness and motion

We'll define the thickness and motion distributions using `interpolate` to fit splines through a few points.
"""

# ╔═╡ a48ac92c-5cb0-4e51-a08f-b34bc8b06d04
fit = y-> scale(interpolate(y, BSpline(Quadratic(Line(OnGrid())))), range(0,1,length=length(y)))

# ╔═╡ ad96bcbe-4482-4486-b817-e788604d1d96
md"""
Here is an image of the dogfish shark we will model. [Image credit: Shark Trust](file:///C:/Users/admin/Downloads/Spiny%20Dogfish%20ST%20Factsheet.PDF)
"""

# ╔═╡ 47eb9e3f-5617-417a-8924-5a38d90390e8
# Fish shape
begin
	url2 = "https://pterosaurheresies.files.wordpress.com/2020/01/squalus-acanthias-invivo588.jpg"
	filename2 = download(url2)
	dogfish = load(filename2)
end

# ╔═╡ 8752bf4d-50e6-42ea-8d43-c41699769f77
md"""
The bottom view shows the outline we're interested in, and adding a few points along the length defines the thickness distribution function `thk`.
"""

# ╔═╡ 71c9bc8f-921d-4df4-acc2-7321ce6648e0
begin
	plot(dogfish)
	nose,len = (30,224),500
	width = [0.02,0.07,0.06,0.048,0.03,0.019,0.01]
	scatter!(nose[1].+len.*range(0,1,length=length(width)), 
		nose[2].-len.*width,color=:blue,legend=false)
	thk = fit(width)
	x = 0:0.01:1
	plot!(nose[1].+len.*x, [nose[2].-len.*thk.(x),nose[2].+len.*thk.(x)],color=:blue)
end

# ╔═╡ d830e69e-f085-495e-be2c-8ea44582fa20
md"""
Looking at [videos of swimming dogfish](https://youtu.be/nMa6lD2CQVI?t=200), we can see a couple general features
 - The motion of the front half of the body has a small amplitude (around 20% of the tail). This sets the amplitude evelope for the traveling wave.
"""

# ╔═╡ c2b7a082-c05d-41bb-af4c-eb23f6a331bf
begin
	envelope = [0.2,0.21,0.23,0.4,0.88,1.0]
	amp = fit(envelope)
end

# ╔═╡ ea51f472-7ffc-4062-a45c-1410d26b1c3e
md"""
- The wavelength of the traveling wave is a bit longer than the body length. 

The slider below controls the wavelength of the traveling wave λ, which you can adjust to see the impact it has on the backbone over the motion cycle.
"""

# ╔═╡ 454f7849-2d9b-4250-94bb-9aed4c2f0d54
md"λ: $@bind λ Slider(0.5:0.1:2, show_value=true, default = 1.1)"

# ╔═╡ 67f29fd6-11ed-415f-abc0-2ed3c84fe60f
begin
	scatter(0:0.2:1, envelope)
	colors = palette(:cyclic_wrwbw_40_90_c42_n256)
	for t in 1/12:1/12:1
		plot!(x,amp.(x).*sin.(2π/λ*x.-2π*t),color=colors[floor(Int,t*256)])
	end
	plot!(ylim=(-1.4,1.4),legend=false)
end

# ╔═╡ db1ed38c-a1aa-4173-bb43-d81afedbfab9
md"""

### Setting up the simulation

Now the thickness and motion are defined, but how will we apply these to a fluid simulation? `WaterLily`uses an [immersed boundary method](https://eprints.soton.ac.uk/369635/) and [automatic differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation) to embed a body into the flow. The upshot is that we don't need to do any meshing; all we need is a signed distance function (SDF) to the surface.

Let's start by defining the SDF to the "backbone", which is a line segment from $x=0\ldots 1$. [See this great video from Inigo Quilez for a derivation of this sdf.](https://www.youtube.com/watch?v=PMltMdi1Wzg) The plot below shows the sdf and the zero contour, which is ... just a line segment. 

Simple adjustments to the SDF give us more control of the shape and position. By shifting the y offset as `y = y-shift`, we can move the body laterally. And by subtracting a thickness from the distance as `sdf = sdf-thickness`, we can give the line some width. This is all we need to model the shark.
"""

# ╔═╡ bfc73784-8c27-4cdb-a8e4-fbcbf9ddb3c2
md"shift: $@bind shift Slider(-1:0.5:2, show_value=true, default=0.5)"

# ╔═╡ 76dc572f-3cd2-4f1a-9296-2065c877a6ef
md"thickness: $@bind T Slider(0.001:0.25:2, show_value=true)"

# ╔═╡ 839db88c-a477-4718-88bb-796ab0cd6591
begin
	function segment_sdf(x,y) 
		s = clamp(x,0,1)         # distance along the segment
		y = y-shift              # shift laterally
		sdf = √sum(abs2,(x-s,y)) # line segment SDF
		return sdf-T*thk(s)      # subtract thickness
	end
	grid = -1:0.05:2
	contourf(grid,grid,segment_sdf,clim=(-1,2),linewidth=0)
	contour!(grid,grid,segment_sdf,levels=[0],color=:black) # zero contour
end

# ╔═╡ f33b9315-b2d1-4fab-91c9-ed3b63fb4d41
md"""
With the basic SDF tested out, we are ready to set up the WaterLily simulation using the function `fish` defined below:
 - The functions `thk` is passed in to create the `sdf` and the function `amp` is passed in to create the traveling wave `map`.
 - The only numerical parameter passed into `fish` is the length of the fish `L` measured in computational cells. This sets the resolution of the simulation and the size of the fluid arrays.
 - The other parameters are the tail amplitude `A` as a fraction of the length, the [Stouhal number](https://en.wikipedia.org/wiki/Strouhal_number) which sets the motion frequency `ω`, and the [Reynolds number](https://en.wikipedia.org/wiki/Reynolds_number) which sets the fluid viscosity `ν`.
"""

# ╔═╡ 11702be3-d125-4ea4-b7e9-74d34943a164
begin
	function fish(thk,amp,k=5.3;L=2^6,A=0.1,St=0.3,Re=1e4)
		# fraction along fish length
		s(x) = clamp(x[1]/L,0,1)
		
		# fish geometry: thickened line SDF
		sdf(x,t) = √sum(abs2,x-L*SVector(s(x),0.))-L*thk(s(x))

		# fish motion: travelling wave
		U=1
		ω = 2π*St*U/(2A*L)
		function map(x,t)
			xc = x.-L # shift origin
			return xc-SVector(0.,A*L*amp(s(xc))*sin(k*s(xc)-ω*t))
		end
		
		# make the fish simulation
		return Simulation((4L+2,2L+2),[U,0.],L;
							ν=U*L/Re,body=AutoBody(sdf,map))
	end
	
	# Create the swimming shark
	L,A,St = 3*2^5,0.1,0.3
	swimmer = fish(thk,amp;L,A,St);
	
	# Save a time span for one swimming cycle
	period = 2A/St
	cycle = range(0,23/24*period,length=24)
end

# ╔═╡ d844c02d-41c8-43a8-95dd-742ea6c24f86
md"""
We can test our geometry by plotting the immersed boundary function `μ₀`; which equals 1 in the fluid and 0 in the body. 
"""

# ╔═╡ e618b2d9-abe8-4e97-aab2-e87c6278378f
@gif for t ∈ cycle
	measure!(swimmer,t*swimmer.L/swimmer.U);
	contour(swimmer.flow.μ₀[:,:,1]',
		aspect_ratio=:equal,legend=false,border=:none)
end

# ╔═╡ 7329a0ba-c57c-4d77-99da-2f8f66d5a94e
md"""

### Running visualizing and measuring the simulation

That animation of the motion looks great, so we are ready to run the flow simulator! 

The `sim_step!(sim,t,remeasure=true)` function runs the simulator up to time `t`, remeasuring the body position every time step. (`remeasure=false` by default since it takes a little extra computational time and isn't needed for statics geometries.)
"""

# ╔═╡ 845cbd96-c01b-49f7-94df-15836a68ebba
# run the simulation a few cycles (this takes few seconds)
begin 
	sim_step!(swimmer,3,remeasure=true)
	sim_time(swimmer)
end

# ╔═╡ d5c82801-8387-4ef0-9f34-3279a54603c5
md"""
The simulation has now run forward in time, but there are no visualizations or measurements by default. 

To see what is going on, lets make a gif of the vorticity `ω=curl(u)` to visualize the vortices in the wake of the shark. This requires simulating a cycle of motion, and computing the `curl` at all the points `@inside` the simulation. 
"""

# ╔═╡ a5c9b186-332f-4812-bb07-0bbe288f5091
begin
	# plot the vorcity ω=curl(u) scaled by the body length L and flow speed U
	function plot_vorticity(sim)
		@inside sim.flow.σ[I] = WaterLily.curl(3,I,sim.flow.u)*sim.L/sim.U
		contourf(sim.flow.σ', 
			color=palette(:BuGn), clims=(-10,10),linewidth=0,
			aspect_ratio=:equal,legend=false,border=:none)
	end

	# make a gif over a swimming cycle
	@gif for t ∈ sim_time(swimmer).+cycle
		sim_step!(swimmer,t,remeasure=true)
		plot_vorticity(swimmer)
	end
end

# ╔═╡ 85c555bb-f584-4064-9da7-f7f1e4fd5320
md"""
This is pretty (CFD does stand for _Colorful Fluid Dynamics_ after all), but also tells us something important about the flow. Notice that there are no eddies coming off the body anywhere other than the tail! This is a sign of efficiency since energy is only used to create those trailing vortices.

We can dig in and get some quantitative measurements from the simulation as well. The function `∮nds` takes a integral over the body surface. By passing in the pressure `p`, we can measure the thrust force and side force generated by the shark!
"""

# ╔═╡ c4c3fa5c-79bc-4d55-827f-c952b129ab5f
begin
	function get_force(sim,t)
		sim_step!(sim,t,remeasure=true)
		return WaterLily.∮nds(sim.flow.p,sim.body,t*sim.L/sim.U)./(0.5*sim.L*sim.U^2)
	end
	forces = [get_force(swimmer,t) for t ∈ sim_time(swimmer).+cycle]
	"got forces"
end

# ╔═╡ 9fc14582-108f-4434-83da-f7b32b676bd3
scatter(cycle./period,[first.(forces),last.(forces)], 
	labels=permutedims(["thrust","side"]), 
	xlabel="scaled time", ylabel="scaled force")

# ╔═╡ 9c67282b-e26a-42bf-836e-fc464b8ca63d
md"""
We can learn a lot from this simple plot. For example, the side-to-side force has the same frequency as the swimming motion itself while the thrust force has double the frequency, with a peak every time the tail passes through the centerline. 

### Next steps

This simple model is a great start and it opens up a ton of avenues for improving the shark simulation and suggesting research questions:
 - The instantaneous net forces should be zero in a free swimming body! We could add reaction motions to our model to achieve this. Would the model shark swim in a straight line if we did this, or is a control-loop needed?
 - Real sharks are 3D (gasp!). While we could easily extend this approach using 2D splines, it will take much longer to simulate. Is there a way use GPUs to accelerate the simulations without completely changing the solver?
 - If we were going to make a bio-inspired robot of this shark, we will have constraints on the shape and motion and powering available. Can we use this framework to help optimize our robotic within it's contraints?

Below you can find links to all the packages used in this notebook. Happy Simulating!

1. [Interpolations.jl](https://juliamath.github.io/Interpolations.jl/stable/)
1. [JuliaDiff](https://juliadiff.org/)
1. [JuliaImages](https://juliaimages.org/stable/)
1. [Plots.jl](http://docs.juliaplots.org/latest/)
1. [Pluto.jl](https://github.com/fonsp/Pluto.jl) and [PlutoUI.jl](https://github.com/fonsp/PlutoUI.jl)
1. [StaticArrays.jl](https://github.com/JuliaArrays/StaticArrays.jl)
1. [WaterLily.jl](https://github.com/weymouth/WaterLily.jl)
"""

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[[Distributed]]
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[[DocStringExtensions]]
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[[Documenter]]
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[[Downloads]]
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[[EarCut_jll]]
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[[Expat_jll]]
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[[Fontconfig_jll]]
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[[ForwardDiff]]
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[[FreeType2_jll]]
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[[FriBidi_jll]]
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[[GLFW_jll]]
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[[GR]]
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[[GR_jll]]
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[[GeometryBasics]]
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[[Gettext_jll]]
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[[Glib_jll]]
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[[Grisu]]
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[[HTTP]]
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[[IOCapture]]
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[[IfElse]]
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[[ImageMagick]]
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[[ImageMagick_jll]]
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[[ImageQualityIndexes]]
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[[ImageShow]]
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[[Images]]
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[[IniFile]]
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[[IntelOpenMP_jll]]
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[[InteractiveUtils]]
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[[Interpolations]]
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[[IntervalSets]]
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[[JLLWrappers]]
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[[JSON]]
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[[JpegTurbo_jll]]
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[[LAME_jll]]
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[[LZO_jll]]
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[[LaTeXStrings]]
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[[Latexify]]
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[[LazyArtifacts]]
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[[LibCURL]]
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[[LibCURL_jll]]
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[[LibGit2]]
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[[LibSSH2_jll]]
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[[LibVPX_jll]]
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[[Libdl]]
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[[Libffi_jll]]
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[[Libgcrypt_jll]]
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[[Libglvnd_jll]]
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[[Libgpg_error_jll]]
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[[Libiconv_jll]]
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[[Libmount_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
version = "2.35.0+0"

[[Libtiff_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "340e257aada13f95f98ee352d316c3bed37c8ab9"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.3.0+0"

[[Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066"
uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700"
version = "2.36.0+0"

[[LinearAlgebra]]
deps = ["Libdl"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[LogExpFunctions]]
deps = ["DocStringExtensions", "LinearAlgebra"]
git-tree-sha1 = "7bd5f6565d80b6bf753738d2bc40a5dfea072070"
uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
version = "0.2.5"

[[Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[MKL_jll]]
deps = ["Artifacts", "IntelOpenMP_jll", "JLLWrappers", "LazyArtifacts", "Libdl", "Pkg"]
git-tree-sha1 = "c253236b0ed414624b083e6b72bfe891fbd2c7af"
uuid = "856f044c-d86e-5d09-b602-aeab76dc8ba7"
version = "2021.1.1+1"

[[MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "0fb723cd8c45858c22169b2e42269e53271a6df7"
uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
version = "0.5.7"

[[MappedArrays]]
git-tree-sha1 = "18d3584eebc861e311a552cbb67723af8edff5de"
uuid = "dbb5928d-eab1-5f90-85c2-b9b0edb7c900"
version = "0.4.0"

[[Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[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 = "4ea90bd5d3985ae1f9a908bd4500ae88921c5ce7"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.0"

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

[[MosaicViews]]
deps = ["MappedArrays", "OffsetArrays", "PaddedViews", "StackViews"]
git-tree-sha1 = "b34e3bc3ca7c94914418637cb10cc4d1d80d877d"
uuid = "e94cdb99-869f-56ef-bcf0-1ae2bcbe0389"
version = "0.3.3"

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

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

[[Netpbm]]
deps = ["ColorVectorSpace", "FileIO", "ImageCore"]
git-tree-sha1 = "09589171688f0039f13ebe0fdcc7288f50228b52"
uuid = "f09324ee-3d7c-5217-9330-fc30815ba969"
version = "1.0.1"

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

[[OffsetArrays]]
deps = ["Adapt"]
git-tree-sha1 = "5cc97a6f806ba1b36bac7078b866d4297ae8c463"
uuid = "6fe1bfb0-de20-5000-8ca7-80f57d26f881"
version = "1.10.4"

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

[[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"

[[PNGFiles]]
deps = ["Base64", "CEnum", "ImageCore", "IndirectArrays", "OffsetArrays", "libpng_jll"]
git-tree-sha1 = "520e28d4026d16dcf7b8c8140a3041f0e20a9ca8"
uuid = "f57f5aa1-a3ce-4bc8-8ab9-96f992907883"
version = "0.3.7"

[[PaddedViews]]
deps = ["OffsetArrays"]
git-tree-sha1 = "59925f4ae6861cddc2313a47514b93b6740f9b6f"
uuid = "5432bcbf-9aad-5242-b902-cca2824c8663"
version = "0.5.9"

[[Parameters]]
deps = ["OrderedCollections", "UnPack"]
git-tree-sha1 = "2276ac65f1e236e0a6ea70baff3f62ad4c625345"
uuid = "d96e819e-fc66-5662-9728-84c9c7592b0a"
version = "0.12.2"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "477bf42b4d1496b454c10cce46645bb5b8a0cf2c"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.0.2"

[[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"

[[PkgVersion]]
deps = ["Pkg"]
git-tree-sha1 = "a7a7e1a88853564e551e4eba8650f8c38df79b37"
uuid = "eebad327-c553-4316-9ea0-9fa01ccd7688"
version = "0.1.1"

[[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 = "501c20a63a34ac1d015d5304da0e645f42d91c9f"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.11"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "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 = "e39bea10478c6aff5495ab522517fae5134b40e3"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.20.0"

[[PlutoUI]]
deps = ["Base64", "Dates", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "Suppressor"]
git-tree-sha1 = "44e225d5837e2a2345e69a1d1e01ac2443ff9fcb"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.9"

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

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

[[ProgressMeter]]
deps = ["Distributed", "Printf"]
git-tree-sha1 = "afadeba63d90ff223a6a48d2009434ecee2ec9e8"
uuid = "92933f4c-e287-5a05-a399-4b506db050ca"
version = "1.7.1"

[[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"

[[RangeArrays]]
git-tree-sha1 = "b9039e93773ddcfc828f12aadf7115b4b4d225f5"
uuid = "b3c3ace0-ae52-54e7-9d0b-2c1406fd6b9d"
version = "0.3.2"

[[Ratios]]
git-tree-sha1 = "37d210f612d70f3f7d57d488cb3b6eff56ad4e41"
uuid = "c84ed2f1-dad5-54f0-aa8e-dbefe2724439"
version = "0.4.0"

[[RecipesBase]]
git-tree-sha1 = "b3fb709f3c97bfc6e948be68beeecb55a0b340ae"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.1.1"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "2a7a2469ed5d94a98dea0e85c46fa653d76be0cd"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.3.4"

[[Reexport]]
git-tree-sha1 = "5f6c21241f0f655da3952fd60aa18477cf96c220"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.1.0"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
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[[Rotations]]
deps = ["LinearAlgebra", "StaticArrays", "Statistics"]
git-tree-sha1 = "2ed8d8a16d703f900168822d83699b8c3c1a5cd8"
uuid = "6038ab10-8711-5258-84ad-4b1120ba62dc"
version = "1.0.2"

[[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"

[[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"

[[SimpleTraits]]
deps = ["InteractiveUtils", "MacroTools"]
git-tree-sha1 = "5d7e3f4e11935503d3ecaf7186eac40602e7d231"
uuid = "699a6c99-e7fa-54fc-8d76-47d257e15c1d"
version = "0.9.4"

[[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", "LogExpFunctions", "OpenSpecFun_jll"]
git-tree-sha1 = "508822dca004bf62e210609148511ad03ce8f1d8"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
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[[StackViews]]
deps = ["OffsetArrays"]
git-tree-sha1 = "46e589465204cd0c08b4bd97385e4fa79a0c770c"
uuid = "cae243ae-269e-4f55-b966-ac2d0dc13c15"
version = "0.1.1"

[[Static]]
deps = ["IfElse"]
git-tree-sha1 = "62701892d172a2fa41a1f829f66d2b0db94a9a63"
uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
version = "0.3.0"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "b28f39450421d07d89ab5d126fd15e5246350e8a"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.11"

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

[[StatsAPI]]
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uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "fed1ec1e65749c4d96fc20dd13bea72b55457e62"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.9"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "000e168f5cc9aded17b6999a560b7c11dda69095"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.0"

[[Suppressor]]
git-tree-sha1 = "a819d77f31f83e5792a76081eee1ea6342ab8787"
uuid = "fd094767-a336-5f1f-9728-57cf17d0bbfb"
version = "0.2.0"

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

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
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[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "d0c690d37c73aeb5ca063056283fde5585a41710"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.5.0"

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

[[TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
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version = "0.1.1"

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

[[TiffImages]]
deps = ["ColorTypes", "DocStringExtensions", "FileIO", "FixedPointNumbers", "IndirectArrays", "Inflate", "OffsetArrays", "OrderedCollections", "PkgVersion", "ProgressMeter"]
git-tree-sha1 = "03fb246ac6e6b7cb7abac3b3302447d55b43270e"
uuid = "731e570b-9d59-4bfa-96dc-6df516fadf69"
version = "0.4.1"

[[TiledIteration]]
deps = ["OffsetArrays"]
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uuid = "06e1c1a7-607b-532d-9fad-de7d9aa2abac"
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[[URIs]]
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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"

[[WaterLily]]
deps = ["Documenter", "ForwardDiff", "LinearAlgebra", "StaticArrays"]
git-tree-sha1 = "9d17cfa8a5e12d568bddd18f4332d1919316fc28"
uuid = "ed894a53-35f9-47f1-b17f-85db9237eebd"
version = "0.2.3"

[[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"

[[WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "59e2ad8fd1591ea019a5259bd012d7aee15f995c"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.3"

[[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", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "acc685bcf777b2202a904cdcb49ad34c2fa1880c"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.14.0+4"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7a5780a0d9c6864184b3a2eeeb833a0c871f00ab"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "0.1.6+4"

[[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 = "d713c1ce4deac133e3334ee12f4adff07f81778f"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2020.7.14+2"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "487da2f8f2f0c8ee0e83f39d13037d6bbf0a45ab"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.0.0+3"

[[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"
"""

# ╔═╡ Cell order:
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