NeutronDiffusion.jl

### A Pluto.jl notebook ###
# v0.17.7

using Markdown
using InteractiveUtils

# ╔═╡ 1addcc10-5304-11eb-06d7-e1c69bd44c54
using Plots 	# plotting backend: takes a sec for first compile

# ╔═╡ 54f41f50-530c-11eb-1a8e-0b08dda08164
md"# Neutron Diffusion

*Modeling the diffusion of a U-235 sample*

Under what conditions will a fission chain reaction be sustained in a uranium 235 sample? Let's assume that the neutron density, $n(\vec{r}, t)$ obeys ordinary diffusion theory:

$\frac{\partial}{\partial t} n(\vec{r}, t)=D \nabla ^{2}n+Cn$

Where $D$ is the diffusion constant and $C$ is the creation rate for neutrons. For $U ^{235}$, $D \approx 10 ^{5} m ^{2}/s$ and $C \approx 10 ^{8} s ^{-1}$. To simplify the analysis, consider a one-dimensional system of length $L$, so the diffusion equation is

$\frac{\partial}{\partial t} n(x, t)=D \frac{\partial ^{2}n}{\partial x ^{2}}+Cn$

with $-L/2≤x ≤L/2$. If all neutrons that reach the boundary escape from the system, then

$n(x=-L/2,t)=0; \hspace{1em} n(x=L/2,t)=0$

are the (Dirchlet) boundary conditions.

---"

# ╔═╡ 9a0fece0-530c-11eb-15bd-29ebb4adb032
md"The struct `Args` contains our parameters:"

# ╔═╡ c3554ecf-b07b-4b21-9485-1a5e37745a66
Base.@kwdef struct Args
	# Physical parameters
	D = 1e5 	# Diffusion constant for U-235 [m²/s]
	C = 1e8 	# Creation rate for U-235 [1/s]
	L = 1.0 	# Lenth of our 1D space [m]
	# Parameters for our simulation
	N = 100 	# Number of grid points
	τ = 1e-10 	# Time step [s]
end

# ╔═╡ b539f410-5308-11eb-16eb-5354e7f4bae2
md"Check stability of FTCS algorithm:"

# ╔═╡ db60f84b-2d63-41fd-8b29-99d2fbd39e6c
let args = Args(), h = args.L / (args.N - 1) 	# h → grid spacing
	args.τ ≤ h^2 / 2args.D 
end

# ╔═╡ 115a22b0-52ff-11eb-288b-f1f22602c21b
md"The reaction will grow exponentially if the critical length, $L_c$, is less than the length of our 1-D geometry. This relation establishes the *critical mass* of our 1-D geometry: if $L<L_c$, then the flux of neutrons escaping at the boundary will damp out the neutron density."

# ╔═╡ fcf86f10-52ff-11eb-30b7-bb29ca63f98d
let args = Args()
	(L_critical = π * √(args.D / args.C)) < args.L
end

# ╔═╡ 65f6b782-530d-11eb-1105-c1ede0ba670a
md"## FTCS algorithm

This function advances the state of the neutron density vector using a Forward in Time, Centered in Space algorithm (basically averaging neighboring points to solve the diffusion equation iteratively)"

# ╔═╡ 895801fb-e8ca-4290-9cb2-e0b0ee59a265
function step!(ρ::Vector, args::Args)
	""" FTCS step - see Garcia's Numerical Methods for Physics """
	h = args.L / (args.N - 1) # grid spacing
	# Dirchlet Boundary Conditions are n[1] = n[end] = 0
	ρ[2:end-1] = ρ[2:end-1] .+ 
		( args.D * args.τ / h^2 ) * ( ρ[3:end] .+ ρ[1:end-2] .- 2ρ[2:end-1] ) .+ 
		args.C * args.τ * ρ[2:end-1]
end

# ╔═╡ 6a57f8ac-cb6e-4e0d-bba0-e7c3e5e04e73
md"We'll make another convenience struct for position and neutron density arrays. This one will be mutable so we can update the density element."

# ╔═╡ 8d6b2ab3-94a0-42d6-9137-6ca786237a26
mutable struct NeutronDiffuser
	x::Vector 	# X positions
	ρ::Vector 	# Neutron Density
end

# ╔═╡ 37f699c3-9af7-4cef-9cdd-0f1ef241dc63
md"We'll also make another helper function `neutrondiffuser` to make sure we initialize our `NeutronDiffuser` struct correctly with respect to our `Args` and some initial condition."

# ╔═╡ 5b4c4cb3-fa31-43c0-ad47-d09dd32f7a29
function neutrondiffuser(args::Args; ic=:delta)
	x = range(-args.L/2, args.L/2; length=args.N)
	ρ = zeros(args.N)
	# Our Initial Condition is a delta spike in the middle
	ic == :delta && (ρ[args.N÷2] = args.N / args.L)
	
	NeutronDiffuser(x, ρ)
end

# ╔═╡ 41758b00-e8a6-4ad4-80d0-9a57460bb739
md"We can iterate over a given number of steps, producing a given number of data and writing that data to a $N×M$ matrix for neutron density, and a vector of length $M$ for time, where $M$ is the total number of samples we'll record (which is less than the total number of steps we'll take; perhaps a rate of one recording to every 5-10 FTCS steps) and $N$ is the number of grid points from our `Args` struct, i.e. `args.N`."

# ╔═╡ b3a20600-08dc-4bd3-95bf-e2fe7cc38b02
function iterate(nd::NeutronDiffuser, args::Args; steps=300, samples=50)
	logstep = steps ÷ samples 	# Iterations between logs
	ρ = zeros(args.N, samples) 	# Neutron density result matrix
	t = zeros(samples) 			# Time result vector

	logᵢ = 1
	for i ∈ 1:steps
		step!(nd.ρ, args)
		if i % logstep == 1 	# logging occasionally
			ρ[:, logᵢ] = copy(nd.ρ)
			t[logᵢ] = i*args.τ
			logᵢ += 1
		end
	end

	(ρ, t)
end

# ╔═╡ 5c911c31-629b-402b-8ed0-ad010f281542
md"## Plotting

We're all set up; let's go ahead and create our `Args` and `NeutronDiffuser` structs, then iterate using our `iterate` function."

# ╔═╡ de8e6fc3-029c-4d01-a003-ca9948f7b835
begin
	args = Args()
	diffuser = neutrondiffuser(args) 
	(ρ, t) = iterate(diffuser, args)
end;

# ╔═╡ 3b84522d-4878-422d-a298-3811dd9df281
contourf(t * 1e9, diffuser.x, ρ, title="Neutron Diffusion",
	 	 xlabel="Time (ns)", ylabel="Position (m)", c=:thermal)

# ╔═╡ fc5f0250-530c-11eb-15d9-bb6c5204bde9
surface(t*1e9, diffuser.x, ρ, c=:thermal, title="Neutron Diffusion",
	    xlabel="Time (ns)", ylabel="Position (m)", zlabel="Neutron Density")

# ╔═╡ f5f2acf3-a066-4cf0-8264-a6a0bc14bf7d
md"Let's make an animation just for funzies."

# ╔═╡ de6ac937-5d2d-45d7-b4c3-081b65bd5252
let x = range(-args.L/2, args.L/2; length=args.N), ρ = zeros(args.N)
ρ[args.N÷2] = args.N / args.L
steps = 300
totaltime = round(args.τ*steps * 10^9, digits=2)
@gif for i ∈ 1:steps
	step!(ρ, args)
	plot(x, ρ, lw=3, lab="$(round(args.τ*i * 10^9, digits=2)) ns")
	plot!(title="Neutron Diffusion Over $(totaltime) Nanoseconds")
	plot!(xlabel="Position (m)", ylabel="Neutron Density", ylims=(0, 100))
end every 6
end

# ╔═╡ ca6f707b-0743-4899-a86e-7f25834d82e3
md"## Source

Alejandro L. Garcia. [Numerical Methods for Physics (Python), Second Edition (Revised)](http://algarcia.org/nummeth/nummeth.html). Chapter 6.3: *Critical Mass* (156-159)."

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

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

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

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

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

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

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.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"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

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

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

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

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

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

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

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

[[deps.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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# ╠═895801fb-e8ca-4290-9cb2-e0b0ee59a265
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# ╠═1addcc10-5304-11eb-06d7-e1c69bd44c54
# ╠═de8e6fc3-029c-4d01-a003-ca9948f7b835
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# ╠═fc5f0250-530c-11eb-15d9-bb6c5204bde9
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# ╟─00000000-0000-0000-0000-000000000002