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Internal

Public APIs

TestParticle.BiMaxwellian — Method
BiMaxwellian(B::Vector{U}, u0::Vector{T}, ppar::T, pperp::T, n; m=mᵢ)

Construct a BiMaxwellian distribution with magnetic field B, bulk velocity u0, parallel thermal pressure ppar, perpendicular thermal pressure pperp, and number density n in SI units. The default particle is proton.

source
TestParticle.Maxwellian — Method
Maxwellian(u0::Vector{T}, p::T, n; m=mᵢ)

Construct a Maxwellian distribution with bulk velocity u0, thermal pressure p, and number density n in SI units. The default particle is proton.

source
TestParticle.prepare — Method
prepare(grid::CartesianGrid, E, B; kwargs...) -> (q2m, E, B)

Return a tuple consists of particle charge-mass ratio for a prescribed species and interpolated EM field functions.

keywords

  • order::Int=1: order of interpolation in [1,2,3].

  • bc::Int=1: type of boundary conditions, 1 -> NaN, 2 -> periodic.

  • species::Species=Proton: particle species.

  • q::AbstractFloat=1.0: particle charge. Only works when Species=User.

  • m::AbstractFloat=1.0: particle mass. Only works when Species=User.

    prepare(grid::CartesianGrid, E, B, F; species=Proton, q=1.0, m=1.0) -> (q, m, E, B, F)

Return a tuple consists of particle charge, mass for a prescribed species of charge q and mass m, interpolated EM field functions, and external force F.

prepare(x::AbstractRange, y::AbstractRange, z::AbstractRange, E, B; kwargs...) -> (q2m, E, B)
+prepare(x, y, E, B; kwargs...) -> (q2m, E, B)
+prepare(x::AbstractRange, E, B; kwargs...) -> (q2m, E, B)

Direct range input for uniform grid in 2/3D is also accepted.

prepare(E, B; kwargs...) -> (q2m, E, B)

Return a tuple consists of particle charge-mass ratio for a prescribed species of charge q and mass m and analytic EM field functions. Prescribed species are Electron and Proton; other species can be manually specified with species=Ion/User, q and m.

prepare(E, B, F; kwargs...) -> (q, m, E, B, F)

Return a tuple consists of particle charge, mass for a prescribed species of charge q and mass m, analytic EM field functions, and external force F.

source
TestParticle.sample — Method
sample(vdf::Maxwellian)

Sample a 3D velocity from a Maxwellian distribution vdf using the Box-Muller method.

sample(vdf::BiMaxwellian)

Sample a 3D velocity from a BiMaxwellian distribution vdf using the Box-Muller method.

source
TestParticle.trace! — Method
trace!(dy, y, p::TPTuple, t)
+trace!(dy, y, p::FullTPTuple, t)

ODE equations for charged particle moving in static EM field with in-place form.

ODE equations for charged particle moving in static EM field and external force field with in-place form.

source
TestParticle.trace — Method
trace(y, p::TPTuple, t) -> SVector{6, Float64}
+trace(y, p::FullTPTuple, t) -> SVector{6, Float64}

ODE equations for charged particle moving in static EM field with out-of-place form.

ODE equations for charged particle moving in static EM field and external force field with out-of-place form.

source
TestParticle.trace_normalized! — Method
trace_normalized!(dy, y, p::TPNormalizedTuple, t)

Normalized ODE equations for charged particle moving in static EM field with in-place form. If the field is in 2D X-Y plane, periodic boundary should be applied for the field in z via the extrapolation function provided by Interpolations.jl.

source
TestParticle.trace_relativistic — Method
trace_relativistic(y, p::TPTuple, t) -> SVector{6, Float64}

ODE equations for relativistic charged particle moving in static EM field with out-of-place form.

source

Private types and methods

TestParticle.Field — Type
Field{itd, F} <: AbstractField{itd}

A representation of a field function f, defined by:

time-independent field

\[\mathbf{F} = F(\mathbf{x}),\]

time-dependent field

\[\mathbf{F} = F(\mathbf{x}, t).\]

Arguments

  • field_function::Function: the function of field.
  • itd::Bool: whether the field function is time dependent.
  • F: the type of field_function.
source
TestParticle.dipole_fieldline — Function
dipole_fieldline(L, ϕ, nP)

Creates nP points on one field line of the magnetic field from a dipole. In a centered dipole magnetic field model, the path along a given L shell can be described as r = L*cos²λ, where r is the radial distance (in planetary radii) to a point on the line, λ is its co-latitude, and L is the L-shell of interest.

source
TestParticle.getB_CS_harris — Method
getB_CS_harris(B₀, L)

Return the magnetic field at location r near a current sheet with magnetic strength B₀ and sheet length L.

source
TestParticle.getB_bottle — Method
getB_bottle(x, y, z, distance, a, b, I1, I2) -> Vector{Float}

Get magnetic field from a magnetic bottle. Reference: https://en.wikipedia.org/wiki/Magneticmirror#Magneticbottles

Arguments

  • x,y,z::Float: particle coordinates in [m].
  • distance::Float: distance between solenoids in [m].
  • a::Float: radius of each side coil in [m].
  • b::Float: radius of central coil in [m].
  • I1::Float: current in the solenoid times number of windings in side coils.
  • I2::Float: current in the central solenoid times number of windings in the

central loop.

source
TestParticle.getB_mirror — Method
getB_mirror(x, y, z, distance, a, I1) -> Vector{Float}

Get magnetic field from a magnetic mirror generated from two coils.

Arguments

  • x,y,z::Float: particle coordinates in [m].
  • distance::Float: distance between solenoids in [m].
  • a::Float: radius of each side coil in [m].
  • I1::Float: current in the solenoid times number of windings in side coils.
source
TestParticle.getB_tokamak_coil — Method
getB_tokamak_coil(x, y, z, a, b, ICoils, IPlasma)

Get the magnetic field from a Tokamak topology consists of 16 coils. Original: https://github.com/BoschSamuel/Simulation-of-a-Tokamak-Fusion-Reactor/blob/master/Simulation2.m

Arguments

  • x,y,z::Float: location in [m].
  • a::Float: radius of each coil in [m].
  • b::Float: radius of central region in [m].
  • ICoil::Float: current in the coil times number of windings.
  • IPlasma::Float: current of the plasma?
source
TestParticle.getB_tokamak_profile — Method
getB_tokamak_profile(x, y, z, q_profile, a, R₀, Bζ0)

Reconstruct the magnetic field distribution from a safe factor(q) profile. The formulations are from the book "Tokamak 4th Edition" by John Wesson.

Arguments

  • x,y,z::Float: location in [m].
  • q_profile::Function: profile of q. The variable of this function must be the normalized radius.
  • a::Float: minor radius [m].
  • R₀::Float: major radius [m].
  • Bζ0::Float: toroidal magnetic field on axis [T].
source
TestParticle.get_gc — Method
get_gc(param::Union{TPTuple, FullTPTuple})

Get three functions for plotting the orbit of guiding center.

For example:

param = prepare(E, B; species=Proton)
+gc = get_gc(params)
+# The definitions of stateinit, tspan, E and B are ignored.
+prob = ODEProblem(trace!, stateinit, tspan, param)
+sol = solve(prob, Vern7(); dt=2e-11)
+
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1], aspect = :data)
+gc_plot(x,y,z,vx,vy,vz) = (gc([x,y,z,vx,vy,vz])...,)
+lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6))
source
TestParticle.get_rotation_matrix — Method
get_rotation_matrix(axis::AbstractVector, angle::Real) --> SMatrix{3,3}

Create a rotation matrix for rotating a 3D vector around a unit axis by an angle in radians. Reference: Rotation matrix from axis and angle

Example

using LinearAlgebra
+v = [-0.5, 1.0, 1.0]
+v̂ = normalize(v)
+θ = deg2rad(-74)
+R = get_rotation_matrix(v̂, θ)
source
TestParticle.getchargemass — Method
getchargemass(species::Species, q::AbstractFloat, m::AbstractFloat)

Return charge and mass for species. if species = User, input q and m are returned.

source
TestParticle.getinterp — Function
getinterp(A, gridx, gridy, gridz, order::Int=1, bc::Int=1)
+getinterp(A, gridx, gridy, order::Int=1, bc::Int=1)

Return a function for interpolating array A on the grid given by gridx, gridy, and gridz.

Arguments

  • order::Int=1: order of interpolation in [1,2,3].
  • bc::Int=1: type of boundary conditions, 1 -> NaN, 2 -> periodic, 3 -> Flat.
source
TestParticle.guiding_center — Method
guiding_center(xu, param::Union{TPTuple, FullTPTuple})

Calculate the coordinates of the guiding center according to the phase space coordinates of a particle. Reference: https://en.wikipedia.org/wiki/Guiding_center

A simple definition:

\[\mathbf{X}=\mathbf{x}-m\frac{\mathbf{v}\times\mathbf{B}}{qB}\]

source
TestParticle.set_axes_equal — Method
set_axes_equal(ax)

Set 3D plot axes to equal scale for Matplotlib. Make axes of 3D plot have equal scale so that spheres appear as spheres and cubes as cubes. Required since ax.axis('equal') and ax.set_aspect('equal') don't work on 3D.

source
TestParticle.solve — Function
solve(prob::TraceProblem; trajectories::Int=1,
+   savestepinterval::Int=1, isoutofdomain::Function=ODE_DEFAULT_ISOUTOFDOMAIN)

Trace particles using the Boris method with specified prob.

keywords

  • trajectories::Int: number of trajectories to trace.
  • savestepinterval::Int: saving output interval.
  • isoutofdomain::Function: a function with input of position and velocity vector xv that determines whether to stop tracing.
source
TestParticle.update_velocity! — Method
update_velocity!(xv, paramBoris, param, dt)

Update velocity using the Boris method, Birdsall, Plasma Physics via Computer Simulation. Reference: https://apps.dtic.mil/sti/citations/ADA023511

source
diff --git a/v0.10.1/assets/documenter.js b/v0.10.1/assets/documenter.js new file mode 100644 index 00000000..c6562b55 --- /dev/null +++ b/v0.10.1/assets/documenter.js @@ -0,0 +1,1050 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.10.1/assets/themes/documenter-dark.css b/v0.10.1/assets/themes/documenter-dark.css new file mode 100644 index 00000000..53889fb9 --- /dev/null +++ b/v0.10.1/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .file-cta,html.theme--documenter-dark .file-name,html.theme--documenter-dark .select select,html.theme--documenter-dark .textarea,html.theme--documenter-dark .input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark .button{-moz-appearance:none;-webkit-appearance:none;align-items:center;border:1px solid transparent;border-radius:.4em;box-shadow:none;display:inline-flex;font-size:1rem;height:2.5em;justify-content:flex-start;line-height:1.5;padding-bottom:calc(0.5em - 1px);padding-left:calc(0.75em - 1px);padding-right:calc(0.75em - 1px);padding-top:calc(0.5em - 1px);position:relative;vertical-align:top}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark 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kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--documenter-dark 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h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content 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html.theme--documenter-dark .navbar-dropdown,html.theme--documenter-dark .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--documenter-dark .navbar-dropdown.is-right{left:auto;right:0}html.theme--documenter-dark .navbar-divider{display:block}html.theme--documenter-dark .navbar>.container .navbar-brand,html.theme--documenter-dark .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--documenter-dark .navbar>.container .navbar-menu,html.theme--documenter-dark .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop,html.theme--documenter-dark .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark 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a.navbar-item.is-active:not(:focus):not(:hover),html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--documenter-dark .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--documenter-dark .pagination{font-size:1rem;margin:-.25rem}html.theme--documenter-dark .pagination.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--documenter-dark .pagination.is-medium{font-size:1.25rem}html.theme--documenter-dark .pagination.is-large{font-size:1.5rem}html.theme--documenter-dark .pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar 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.hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/v0.10.1/assets/themeswap.js b/v0.10.1/assets/themeswap.js new file mode 100644 index 00000000..9f5eebe6 --- /dev/null +++ b/v0.10.1/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/v0.10.1/assets/warner.js b/v0.10.1/assets/warner.js new file mode 100644 index 00000000..3f6f5d00 --- /dev/null +++ b/v0.10.1/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/v0.10.1/democards/nocoverlisttheme.css b/v0.10.1/democards/nocoverlisttheme.css new file mode 100644 index 00000000..82c452d5 --- /dev/null +++ b/v0.10.1/democards/nocoverlisttheme.css @@ -0,0 +1,24 @@ +.list-card:hover{ + box-shadow: 0 4px 8px 0 rgba(0, 0, 0, 0.4), 0 6px 20px 0 rgba(0, 0, 0, 0.1); +} + +.list-card { + width: 800px; + max-height: 250px; + margin: 25px 10px; + box-shadow: 0 4px 8px 0 rgba(0,0,0,0.2); + transition: 0.3s; + border-radius: 5px; +} + +.list-card-text { + padding: 10px 5px; + font-style: oblique 40deg; + font-size: large; +} + +.list-card-description { + width: 550px; + max-height: 250px; + line-height: 1.5; +} diff --git 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zhNS39-{RORLP>Gl{Yt9|`l*~JD25zTfKIR+=(Z$8;~#Ij>WY3LdC%2mAGYXhxNHEf z3u;x{c_eS{U$&sBwlI{9--FIvD0yr9mkYoU-8s^t90_6Z3nStX#^vV zAfD*6r%#W_nnCW5c(IR9J1h#Nhl%>~O8|VL^L#^&Q9Q4{<7eG$j~x%ODb}A}_L;OG z6aTy679Bx&E{{~ z&e7FdEcvCQ!)eZ65Lp;_!CgbdjO;_fqK}0@B5R$uc;)Dul2idufvki-sI>WZQL^QP z^0IB0A32&HvvqY4mk$>ArmSD;9D=dggNiy$3@xLldgk)pmd1i zMpU@rJBnv0Rr%*flrasIh;axe|MmuU>y;{5$R!%oeq!prE2ak- z7QVXMd-s2yzl`T^uwQ43W`iO^jggtIBHA)E@etpNDAQr)nP=$BWn~OASc3{bQ7j{B z2&F$2r@bGBM+otv1fyP}QdWlW7^2ILzfB)~_uUaJ1mj3_$@xQ rout || abs(u[2]) > rout || abs(u[3]) > rout + return true + else + return false + end +end + +"Set initial conditions." +function prob_func(prob, i, repeat) + # initial particle energy + Ek = 5e3 # [eV] + # initial velocity, [m/s] + v₀ = sph2cart(c*sqrt(1-1/(1+Ek*qᵢ/(mᵢ*c^2))^2), 0.0, π/4) + # initial position, [m] + r₀ = sph2cart(13*Rₑ, π*i, π/2) + + prob = remake(prob; u0 = [r₀..., v₀...]) +end + +# obtain field +param = prepare(getE_dipole, getB_superposition) +stateinit = zeros(6) # particle position and velocity to be modified +tspan = (0.0, 2000.0) +trajectories = 2 + +prob = ODEProblem(trace!, stateinit, tspan, param) +ensemble_prob = EnsembleProblem(prob; prob_func, safetycopy=false) + +# See https://docs.sciml.ai/DiffEqDocs/stable/basics/common_solver_opts/ +# for the solver options +sols = solve(ensemble_prob, Vern9(), EnsembleSerial(); reltol=1e-5, + trajectories, isoutofdomain, dense=true, save_on=true) + +### Visualization + +f = Figure(fontsize=18) +ax = Axis3(f[1, 1], + title = "5 keV Protons in a vacuum superposition magnetosphere", + xlabel = "x [Re]", + ylabel = "y [Re]", + zlabel = "z [Re]", + aspect = :data, + limits = (-14, 14, -14, 14, -5, 5) +) + +invRE = 1 / Rₑ + +for (i, sol) in enumerate(sols) + l = lines!(ax, sol, idxs=(1, 2, 3)) + ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed! + scale!(ax.scene.plots[9+2*i-1], invRE, invRE, invRE) + ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i] +end + +# Field lines +function get_numerical_field(x, y, z) + bx = zeros(length(x), length(y), length(z)) + by = similar(bx) + bz = similar(bx) + + for i in CartesianIndices(bx) + pos = [x[i[1]], y[i[2]], z[i[3]]] + bx[i], by[i], bz[i] = getB_superposition(pos) + end + + bx, by, bz +end + +function trace_field!(ax, x, y, z, unitscale) + bx, by, bz = get_numerical_field(x, y, z) + + zs = 0.0 + nr, nϕ = 8, 4 + dϕ = 2π / nϕ + for r in range(8Rₑ, 16Rₑ, length=nr), ϕ in range(0, 2π-dϕ, length=nϕ) + xs = r * cos(ϕ) + ys = r * sin(ϕ) + + x1, y1, z1 = FieldTracer.trace(bx, by, bz, xs, ys, zs, x, y, z; ds=0.1, maxstep=10000) + + lines!(ax, x1.*unitscale, y1.*unitscale, z1.*unitscale, color=:gray) + end +end + +x = range(-18Rₑ, 18Rₑ, length=50) +y = range(-18Rₑ, 18Rₑ, length=50) +z = range(-18Rₑ, 18Rₑ, length=50) + +trace_field!(ax, x, y, z, invRE) + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_analytic_magnetosphere/index.html b/v0.10.1/examples/advanced/demo_analytic_magnetosphere/index.html new file mode 100644 index 00000000..65f18d21 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_analytic_magnetosphere/index.html @@ -0,0 +1,106 @@ + +Analytical magnetosphere · TestParticle.jl

Analytical magnetosphere

Source code compat Author Update time

This demo shows how to trace particles in a vacuum superposition of a dipolar magnetic field $\mathbf{B}_D$ with a uniform background magnetic field $\mathbf{B}_\mathrm{IMF}$. In this slightly modified dipole field, magnetic null points appear near 14 Earth's radii, and the particle orbits are also distorted from the idealized motions in Demo: magnetic dipole.

using TestParticle
+using TestParticle: getB_dipole, getE_dipole, sph2cart, mᵢ, qᵢ, c, Rₑ
+using OrdinaryDiffEq
+using StaticArrays
+using FieldTracer
+using CairoMakie
+
+function getB_superposition(xu)
+   getB_dipole(xu) + SA[0.0, 0.0, -10e-9]
+end
+
+"Boundary condition check."
+function isoutofdomain(u, p, t)
+   rout = 18Rₑ
+   if hypot(u[1], u[2], u[3]) < 1.1Rₑ ||
+      abs(u[1]) > rout || abs(u[2]) > rout || abs(u[3]) > rout
+      return true
+   else
+      return false
+   end
+end
+
+"Set initial conditions."
+function prob_func(prob, i, repeat)
+   # initial particle energy
+   Ek = 5e3 # [eV]
+   # initial velocity, [m/s]
+   v₀ = sph2cart(c*sqrt(1-1/(1+Ek*qᵢ/(mᵢ*c^2))^2), 0.0, π/4)
+   # initial position, [m]
+   r₀ = sph2cart(13*Rₑ, π*i, π/2)
+
+   prob = remake(prob; u0 = [r₀..., v₀...])
+end
+
+# obtain field
+param = prepare(getE_dipole, getB_superposition)
+stateinit = zeros(6) # particle position and velocity to be modified
+tspan = (0.0, 2000.0)
+trajectories = 2
+
+prob = ODEProblem(trace!, stateinit, tspan, param)
+ensemble_prob = EnsembleProblem(prob; prob_func, safetycopy=false)
+
+# See https://docs.sciml.ai/DiffEqDocs/stable/basics/common_solver_opts/
+# for the solver options
+sols = solve(ensemble_prob, Vern9(), EnsembleSerial(); reltol=1e-5,
+   trajectories, isoutofdomain, dense=true, save_on=true)
+
+### Visualization
+
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "5 keV Protons in a vacuum superposition magnetosphere",
+   xlabel = "x [Re]",
+   ylabel = "y [Re]",
+   zlabel = "z [Re]",
+   aspect = :data,
+   limits = (-14, 14, -14, 14, -5, 5)
+)
+
+invRE = 1 / Rₑ
+
+for (i, sol) in enumerate(sols)
+   l = lines!(ax, sol, idxs=(1, 2, 3))
+   ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+   scale!(ax.scene.plots[9+2*i-1], invRE, invRE, invRE)
+   ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i]
+end
+
+# Field lines
+function get_numerical_field(x, y, z)
+   bx = zeros(length(x), length(y), length(z))
+   by = similar(bx)
+   bz = similar(bx)
+
+   for i in CartesianIndices(bx)
+      pos = [x[i[1]], y[i[2]], z[i[3]]]
+      bx[i], by[i], bz[i] = getB_superposition(pos)
+   end
+
+   bx, by, bz
+end
+
+function trace_field!(ax, x, y, z, unitscale)
+   bx, by, bz = get_numerical_field(x, y, z)
+
+   zs = 0.0
+   nr, nϕ = 8, 4
+   dϕ = 2π / nϕ
+   for r in range(8Rₑ, 16Rₑ, length=nr), ϕ in range(0, 2π-dϕ, length=nϕ)
+      xs = r * cos(ϕ)
+      ys = r * sin(ϕ)
+
+      x1, y1, z1 = FieldTracer.trace(bx, by, bz, xs, ys, zs, x, y, z; ds=0.1, maxstep=10000)
+
+      lines!(ax, x1.*unitscale, y1.*unitscale, z1.*unitscale, color=:gray)
+   end
+end
+
+x = range(-18Rₑ, 18Rₑ, length=50)
+y = range(-18Rₑ, 18Rₑ, length=50)
+z = range(-18Rₑ, 18Rₑ, length=50)
+
+trace_field!(ax, x, y, z, invRE)
+

This page was generated using DemoCards.jl and Literate.jl.

diff --git a/v0.10.1/examples/advanced/demo_batsrus_3dstructured/index.html b/v0.10.1/examples/advanced/demo_batsrus_3dstructured/index.html new file mode 100644 index 00000000..9e764069 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_batsrus_3dstructured/index.html @@ -0,0 +1,108 @@ + +Tracing particle from PIC · TestParticle.jl

Tracing particle from PIC

Author Update time

This example shows how to trace charged particles in the structured SWMF outputs from an MHD-EPIC simulation of Ganymede. For more details about the field file format, please checkout Batsrus.jl.

using Statistics: mean
+using Batsrus
+using TestParticle
+using OrdinaryDiffEq
+using FieldTracer
+using PyPlot
+
+## Utility functions
+
+function initial_conditions(i)
+   j = i - 1
+   [x[xc_], y[yc_+j], z[zc_], Uix[xc_,yc_+j,zc_], Uiy[xc_,yc_+j,zc_], Uiz[xc_,yc_+j,zc_]]
+end
+
+"Set initial conditions."
+function prob_func(prob, i, repeat)
+   remake(prob, u0=initial_conditions(i))
+end
+
+## Data processing
+
+filename = "3d_var_region0_0_t00002500_n00106468.out"
+data = readdata(filename)
+
+var = getvars(data, ["Bx", "By", "Bz", "Ex", "Ey", "Ez", "uxs0", "uys0", "uzs0", "uxs1", "uys1", "uzs1"])
+
+const RG = 2634e3 # [m]
+
+x = range(extrema(data.x[:,:,:,1])..., length=size(data.x, 1)) .* RG
+y = range(extrema(data.x[:,:,:,2])..., length=size(data.x, 2)) .* RG
+z = range(extrema(data.x[:,:,:,3])..., length=size(data.x, 3)) .* RG
+B = zeros(Float32, 3, length(x), length(y), length(z)) # [T]
+E = zeros(Float32, 3, length(x), length(y), length(z)) # [V/m]
+
+# Convert into SI units
+B[1,:,:,:] .= var["Bx"] .* 1e-9
+B[2,:,:,:] .= var["By"] .* 1e-9
+B[3,:,:,:] .= var["Bz"] .* 1e-9
+E[1,:,:,:] .= var["Ex"] .* 1e-6
+E[2,:,:,:] .= var["Ey"] .* 1e-6
+E[3,:,:,:] .= var["Ez"] .* 1e-6
+
+## Initial conditions
+
+Uex, Uey, Uez = var["uxs0"] .* 1e3, var["uys0"] .* 1e3, var["uzs0"] .* 1e3
+Uix, Uiy, Uiz = var["uxs1"] .* 1e3, var["uys1"] .* 1e3, var["uzs1"] .* 1e3
+
+xc_ = floor(Int, length(x)/2) + 1
+yc_ = floor(Int, length(y)/2) + 1
+zc_ = floor(Int, length(z)/2) + 1
+
+stateinit_e = [x[xc_], y[yc_], z[zc_], Uex[xc_,yc_,zc_], Uey[xc_,yc_,zc_], Uez[xc_,yc_,zc_]]
+stateinit_p = [x[xc_], y[yc_], z[zc_], Uix[xc_,yc_,zc_], Uiy[xc_,yc_,zc_], Uiz[xc_,yc_,zc_]]
+
+param_electron = prepare(x, y, z, E, B, species=Electron)
+tspan_electron = (0.0, 0.1)
+
+param_proton = prepare(x, y, z, E, B, species=Proton)
+tspan_proton = (0.0, 10.0)
+trajectories = 5
+
+prob_p = ODEProblem(trace!, stateinit_p, tspan_proton, param_proton)
+ensemble_prob = EnsembleProblem(prob_p, prob_func=prob_func)
+sol_p = solve(ensemble_prob, Vern9(), EnsembleThreads(); trajectories)
+
+## Visualization
+
+using3D()
+fig = plt.figure(figsize=(10,6))
+ax = fig.gca(projection="3d")
+
+## Field tracing
+
+for i in 1:10:length(x)
+   xs, ys, zs = x[i], 0.0, 0.0
+   x1, y1, z1 = trace(B[1,:,:,:], B[2,:,:,:], B[3,:,:,:], xs, ys, zs, x, y, z, ds=0.2, maxstep=1000)
+   line = ax.plot(x1 ./ RG, y1 ./RG, z1 ./ RG, "k-", alpha=0.3)
+end
+
+for i in 1:10:length(y)
+   xs, ys, zs = x[xc_], y[i], z[zc_]
+   x1, y1, z1 = trace(B[1,:,:,:], B[2,:,:,:], B[3,:,:,:], xs, ys, zs, x, y, z, ds=0.2, maxstep=1000)
+   line = ax.plot(x1 ./ RG, y1 ./RG, z1 ./ RG, "k-", alpha=0.3)
+end
+
+n = 200 # number of timepoints
+
+ts = range(0, stop=tspan_proton[2], length=n)
+for i = 1:trajectories
+   if sol_p[i].t[end] < tspan_proton[2]
+      ts⁺ = range(0, stop=sol_p[i].t[end], length=n)
+      ax.plot(sol_p[i](ts⁺,idxs=1) ./ RG, sol_p[i](ts⁺,idxs=2) ./ RG, sol_p[i](ts⁺,idxs=3) ./ RG, label="proton $i", lw=1.5)
+   else
+      ax.plot(sol_p[i](ts,idxs=1) ./ RG, sol_p[i](ts,idxs=2) ./ RG, sol_p[i](ts,idxs=3) ./ RG, label="proton $i", lw=1.5)
+   end
+end
+
+#ax.plot(sol_p[1,:], sol_p[2,:], sol_p[3,:], label="proton")
+
+ax.legend()
+title("Particle trajectory near Ganymede's magnetopause from PIC")
+xlabel("x [Rg]")
+ylabel("y [Rg]")
+zlabel("z [Rg]")
+
+ax.set_box_aspect([1.17,4,4])
+TestParticle.set_axes_equal(ax)

This page was generated using DemoCards.jl.

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HcmV?d00001 diff --git a/v0.10.1/examples/advanced/demo_boris_outofdomain.jl b/v0.10.1/examples/advanced/demo_boris_outofdomain.jl new file mode 100644 index 00000000..ea4e4028 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_boris_outofdomain.jl @@ -0,0 +1,79 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: qᵢ, mᵢ +using StaticArrays +using OrdinaryDiffEq +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +uniform_B(x) = SA[0.0, 0.0, 0.01] +uniform_E(x) = SA[0.0, 0.0, 0.0] + +"Set initial states." +function prob_func(prob, i, repeat) + prob = @views remake(prob; u0 = [prob.u0[1:3]..., 10.0 - i*2.0, prob.u0[5:6]...]) +end + +function isoutofdomain(xv, p, t) + if isnan(xv[1]) + return true + else + return false + end +end + +# Number of cells for the field along each dimension +nx, ny = 4, 6 +# Unit conversion factors between SI and dimensionless units +B₀ = 10e-9 # [T] +Ω = abs(qᵢ) * B₀ / mᵢ # [1/s] +t₀ = 1 / Ω # [s] +U₀ = 1.0 # [m/s] +l₀ = U₀ * t₀ # [m] +E₀ = U₀*B₀ # [V/m] + +x = range(0, 11, length=nx) # [l₀] +y = range(-21, 0, length=ny) # [l₀] + +B = fill(0.0, 3, nx, ny) # [B₀] +B[3,:,:] .= 1.0 + +E(x) = SA[0.0, 0.0, 0.0] # [E₀] + +# If bc == 1, we set a NaN value outside the domain (default); +# If bc == 2, we set periodic boundary conditions. +param = prepare(x, y, E, B; species=User, bc=1); + +# Initial conditions to be modified in prob_func +x0 = [0.0, 0.0, 0.0] # initial position [l₀] +u0 = [0.0, 0.0, 0.0] # initial velocity [v₀] +stateinit = [x0..., u0...] +tspan = (0.0, 1.5π) # 3/4 gyroperiod + +dt = 0.1 +savestepinterval = 1 +trajectories = 2 +prob = TraceProblem(stateinit, tspan, param; prob_func) + +sols = TestParticle.solve(prob; dt, savestepinterval, isoutofdomain, trajectories) + +f = Figure(fontsize = 18) +ax = Axis(f[1, 1], + title = "Proton trajectory", + xlabel = "X", + ylabel = "Y", + limits = (-10.1, 10.1, -20.1, 0.1), + aspect = DataAspect() +) + +for i in eachindex(sols) + lines!(ax, sols[i]; idxs=(1, 2), label=string(i)) + ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed! + ax.scene.plots[2*i-1].color = Makie.wong_colors()[i] +end + +axislegend(position=:lt, framevisible=false) + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_boris_outofdomain/index.html b/v0.10.1/examples/advanced/demo_boris_outofdomain/index.html new file mode 100644 index 00000000..c506b3f0 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_boris_outofdomain/index.html @@ -0,0 +1,73 @@ + +Advanced Boris tracing · TestParticle.jl

Advanced Boris tracing

Source code compat Author Update time

This example shows how to trace charged particles using the Boris method in dimensionless units with additionally boundary check. If the particles travel out of the domain specified by the field, the tracing will stop. Check Demo: Dimensionless Units for explaining the unit conversion, and Demo: Boris Method for introducing the Boris method.

using TestParticle
+using TestParticle: qᵢ, mᵢ
+using StaticArrays
+using OrdinaryDiffEq
+using CairoMakie
+
+uniform_B(x) = SA[0.0, 0.0, 0.01]
+uniform_E(x) = SA[0.0, 0.0, 0.0]
+
+"Set initial states."
+function prob_func(prob, i, repeat)
+   prob = @views remake(prob; u0 = [prob.u0[1:3]..., 10.0 - i*2.0, prob.u0[5:6]...])
+end
+
+function isoutofdomain(xv, p, t)
+   if isnan(xv[1])
+      return true
+   else
+      return false
+   end
+end
+
+# Number of cells for the field along each dimension
+nx, ny = 4, 6
+# Unit conversion factors between SI and dimensionless units
+B₀ = 10e-9            # [T]
+Ω = abs(qᵢ) * B₀ / mᵢ # [1/s]
+t₀ = 1 / Ω            # [s]
+U₀ = 1.0              # [m/s]
+l₀ = U₀ * t₀          # [m]
+E₀ = U₀*B₀            # [V/m]
+
+x = range(0, 11, length=nx) # [l₀]
+y = range(-21, 0, length=ny) # [l₀]
+
+B = fill(0.0, 3, nx, ny) # [B₀]
+B[3,:,:] .= 1.0
+
+E(x) = SA[0.0, 0.0, 0.0] # [E₀]
+
+# If bc == 1, we set a NaN value outside the domain (default);
+# If bc == 2, we set periodic boundary conditions.
+param = prepare(x, y, E, B; species=User, bc=1);

Note that we set a radius of 10, so the trajectory extent from -20 to 0 in y, and -10 to 10 in x. After half a cycle, the particle will move into the region where is field is not defined. The tracing will stop with the final step being all NaNs.

# Initial conditions to be modified in prob_func
+x0 = [0.0, 0.0, 0.0] # initial position [l₀]
+u0 = [0.0, 0.0, 0.0] # initial velocity [v₀]
+stateinit = [x0..., u0...]
+tspan = (0.0, 1.5π) # 3/4 gyroperiod
+
+dt = 0.1
+savestepinterval = 1
+trajectories = 2
+prob = TraceProblem(stateinit, tspan, param; prob_func)
+
+sols = TestParticle.solve(prob; dt, savestepinterval, isoutofdomain, trajectories)
+
+f = Figure(fontsize = 18)
+ax = Axis(f[1, 1],
+   title = "Proton trajectory",
+   xlabel = "X",
+   ylabel = "Y",
+   limits = (-10.1, 10.1, -20.1, 0.1),
+   aspect = DataAspect()
+)
+
+for i in eachindex(sols)
+   lines!(ax, sols[i]; idxs=(1, 2), label=string(i))
+   ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+   ax.scene.plots[2*i-1].color = Makie.wong_colors()[i]
+end
+
+axislegend(position=:lt, framevisible=false)
+

This page was generated using DemoCards.jl and Literate.jl.

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/dev/null +++ b/v0.10.1/examples/advanced/demo_currentsheet.jl @@ -0,0 +1,90 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: getB_CS_harris +using OrdinaryDiffEq +using StaticArrays +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +### Obtain field + +# Harris current sheet parameters in SI units +const B₀, L = 20e-9, 0.4TestParticle.Rₑ + +function getB(xu) + SVector{3}(getB_CS_harris(xu[1:3], B₀, L)) +end + +function getE(xu) + SA[0.0, 0.0, 0.0] +end + +### Initialize particles + +m = TestParticle.mᵢ +q = TestParticle.qᵢ +c = TestParticle.c +Rₑ = TestParticle.Rₑ +# Initial condition +stateinit = let + # initial particle energy, [eV] + Ek = 5e7 + # initial velocity, [m/s] + v₀ = [c*√(1-1/(1+Ek*q/(m*c^2))^2), 0.0, 0.0] + # initial position, [m] + r₀ = [-5.0Rₑ, 0.0, 0.0] + + [r₀..., v₀...] +end + +param = prepare(getE, getB) +tspan = (0.0, 10.0) + +prob = ODEProblem(trace!, stateinit, tspan, param) + +sol = solve(prob, Tsit5(); save_idxs=[1,2,3]) + +### Visualization + +f = Figure(fontsize=18) +ax = Axis3(f[1, 1], + title = "Particle trajectory near the Harris current sheet", + xlabel = "x [Re]", + ylabel = "y [Re]", + zlabel = "z [Re]", + aspect = :data, +) + +n = 100 # number of timepoints +ts = range(tspan..., length=n) +x = sol(ts, idxs=1)./Rₑ |> Vector +y = sol(ts, idxs=2)./Rₑ |> Vector +z = sol(ts, idxs=3)./Rₑ |> Vector + +l = lines!(ax, x, y, z, label="50 MeV proton, B0 = 20 nT") +axislegend() + +X, Y, Z = let xrange = range(-8, 8, length=20) + X = collect(Float32, xrange) + Y = zeros(Float32, size(X)...) + Z = zeros(Float32, size(X)...) + X, Y, Z +end + +B = zeros(Float32, 3, size(X)...) + +i = 1 +for (x,y) in zip(X, Y) + B[1+3*(i-1):3*i] = getB_CS_harris([x,0.0,0.0], 4e-2, 1.0) + global i += 1 +end + +for s = 1:3 + quiver!(ax, X, Y, Z, vec(B[1,:,:]), vec(B[2,:,:]), vec(B[3,:,:]), + color=:black, alpha=0.6, lengthscale=100.0) + @. Y -= 15 +end + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_currentsheet/index.html b/v0.10.1/examples/advanced/demo_currentsheet/index.html new file mode 100644 index 00000000..27fc4184 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_currentsheet/index.html @@ -0,0 +1,86 @@ + +Current sheet · TestParticle.jl

Current sheet

Source code compat Author Update time

This example shows how to trace protons in a stationary magnetic field that corresponds to the 1D Harris current sheet defined by a reference strength and width. Reference: https://en.wikipedia.org/wiki/Current_sheet

using TestParticle
+using TestParticle: getB_CS_harris
+using OrdinaryDiffEq
+using StaticArrays
+using CairoMakie
+
+### Obtain field
+
+# Harris current sheet parameters in SI units
+const B₀, L = 20e-9, 0.4TestParticle.Rₑ
+
+function getB(xu)
+   SVector{3}(getB_CS_harris(xu[1:3], B₀, L))
+end
+
+function getE(xu)
+   SA[0.0, 0.0, 0.0]
+end
+
+### Initialize particles
+
+m = TestParticle.mᵢ
+q = TestParticle.qᵢ
+c = TestParticle.c
+Rₑ = TestParticle.Rₑ
+# Initial condition
+stateinit = let
+   # initial particle energy, [eV]
+   Ek = 5e7
+   # initial velocity, [m/s]
+   v₀ = [c*√(1-1/(1+Ek*q/(m*c^2))^2), 0.0, 0.0]
+   # initial position, [m]
+   r₀ = [-5.0Rₑ, 0.0, 0.0]
+
+   [r₀..., v₀...]
+end
+
+param = prepare(getE, getB)
+tspan = (0.0, 10.0)
+
+prob = ODEProblem(trace!, stateinit, tspan, param)
+
+sol = solve(prob, Tsit5(); save_idxs=[1,2,3])
+
+### Visualization
+
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "Particle trajectory near the Harris current sheet",
+   xlabel = "x [Re]",
+   ylabel = "y [Re]",
+   zlabel = "z [Re]",
+   aspect = :data,
+)
+
+n = 100 # number of timepoints
+ts = range(tspan..., length=n)
+x = sol(ts, idxs=1)./Rₑ |> Vector
+y = sol(ts, idxs=2)./Rₑ |> Vector
+z = sol(ts, idxs=3)./Rₑ |> Vector
+
+l = lines!(ax, x, y, z, label="50 MeV proton, B0 = 20 nT")
+axislegend()
+
+X, Y, Z = let xrange = range(-8, 8, length=20)
+   X = collect(Float32, xrange)
+   Y = zeros(Float32, size(X)...)
+   Z = zeros(Float32, size(X)...)
+   X, Y, Z
+end
+
+B = zeros(Float32, 3, size(X)...)
+
+i = 1
+for (x,y) in zip(X, Y)
+   B[1+3*(i-1):3*i] = getB_CS_harris([x,0.0,0.0], 4e-2, 1.0)
+   global i += 1
+end
+
+for s = 1:3
+   quiver!(ax, X, Y, Z, vec(B[1,:,:]), vec(B[2,:,:]), vec(B[3,:,:]),
+      color=:black, alpha=0.6, lengthscale=100.0)
+   @. Y -= 15
+end
+

This page was generated using DemoCards.jl and Literate.jl.

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zMV-^D{&+6otXl_G<6trVmDU6`!xpfGo15&YMh_;t=pLoxA5eYHk6e7%)6q(uu?Gv(UHLlYXJ${u+xDtMJ{8Geu`N7gQSW{Kj5voZ&2NXiDF${U36p7^D!s&Ez`J&vsN2L`0$zbgKzp6VM_!= +Ensemble tracing · TestParticle.jl

Ensemble tracing

Source code compat Author Update time

This example demonstrates tracing multiple electrons in an analytic EM field and how to take advantage of the multithreading support in the ODE solver. A multiproc version is also available. Check the official documentation of DifferentialEquations.jl for details. In performing test particle tracing, we want to share the field information for all particles. This can be achieved in the ensemble problem with safetycopy=false.

using TestParticle
+using OrdinaryDiffEq
+using StaticArrays
+using CairoMakie
+
+"Set initial state for EnsembleProblem."
+function prob_func(prob, i, repeat)
+   prob = @views remake(prob, u0=[prob.u0[1:3]..., i/3, 0.0, 0.0])
+end
+
+# Initialization
+
+B(x) = SA[0, 0, 1e-11]
+E(x) = SA[0, 0, 1e-13]
+
+x0 = [0.0, 0.0, 0.0] # initial position, [m]
+u0 = [1.0, 0.0, 0.0] # initial velocity, [m/s]
+stateinit = [x0..., u0...]
+
+param = prepare(E, B, species=Electron)
+tspan = (0.0, 10.0)
+
+trajectories = 3
+
+# Solve for the trajectories
+
+prob = ODEProblem(trace!, stateinit, tspan, param)
+ensemble_prob = EnsembleProblem(prob; prob_func, safetycopy=false)
+sols = solve(ensemble_prob, Tsit5(), EnsembleThreads(); trajectories)
+
+# Visualization
+
+f = Figure(fontsize = 18)
+ax = Axis3(f[1, 1],
+   title = "Electron trajectories",
+   xlabel = "X",
+   ylabel = "Y",
+   zlabel = "Z",
+   aspect = :data,
+)
+
+for i in eachindex(sols)
+   lines!(ax, sols[i], idxs=(1,2,3), label="$i")
+   ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+   ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i]
+end
+

We can also solve this problem with the native Boris pusher. Note that the Boris pusher requires a additional parameters: a fixed timestep, and an output save interval.

dt = 0.1
+savestepinterval = 1
+
+# Solve for the trajectories
+
+prob = TraceProblem(stateinit, tspan, param; prob_func)
+trajs = TestParticle.solve(prob; dt, trajectories, savestepinterval)
+
+# Visualization
+
+f = Figure(fontsize = 18)
+ax = Axis3(f[1, 1],
+   title = "Electron trajectories",
+   xlabel = "X",
+   ylabel = "Y",
+   zlabel = "Z",
+   aspect = :data,
+)
+
+for i in eachindex(trajs)
+   lines!(ax, trajs[i]; idxs=(1,2,3), label="$i")
+   ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+   ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i]
+end
+

Note that by default linear interpolation is applied when plotting the trajectories from the Boris method.

This page was generated using DemoCards.jl and Literate.jl.

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zGo)f}0MNmvzbz>lhlwPk0Ux#j#?qecv*;rw>J{L9chXbY4%u5FKG)KxJaViC*W)dv z>vR=~MyQsqn_@n=^R~8cc*^BXaB@1bP$YU+$UlCFu2rVK_0blI z7Iv-L{cAb-t7Q*O4ys6p8ZD71MV&W4l{e2Xqx2szfOP1~UKXI*eb^rhX literal 0 HcmV?d00001 diff --git a/v0.10.1/examples/advanced/demo_magneticbottle.jl b/v0.10.1/examples/advanced/demo_magneticbottle.jl new file mode 100644 index 00000000..1916a7d8 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_magneticbottle.jl @@ -0,0 +1,130 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: getB_bottle +using OrdinaryDiffEq +using StaticArrays +using Printf +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +### Obtain field + +# Magnetic bottle parameters in SI units +const I1 = 20. # current in the solenoid +const N1 = 45 # number of windings +const I2 = 20. # current in the central solenoid +const N2 = 45 # number of windings +const distance = 10. # distance between solenoids +const a = 4.0 # radius of each coil +const b = 8.0 # radius of central coil + +function getB(xu) + SVector{3}(getB_bottle(xu[1], xu[2], xu[3], distance, a, b, I1*N1, I2*N2)) +end + +function getE(xu) + SA[0.0, 0.0, 0.0] +end + +### Initialize particles +m = TestParticle.mₑ +q = TestParticle.qₑ +c = TestParticle.c + +# initial velocity, [m/s] +v₀ = [2.75, 2.5, 1.3] .* 0.03c # confined +# initial position, [m] +r₀ = [0.8, 0.8, 0.0] +stateinit = [r₀..., v₀...] + +# Theoretically we can take advantage of the fact that magnetic field does not +# accelerate particles, so that γ remains constant. However, we are not doing +# that here since it is not generally true in the EM field. + +param = prepare(getE, getB; species=Electron) +tspan = (0.0, 1e-5) + +prob_rel = ODEProblem(trace_relativistic!, stateinit, tspan, param) +prob_non = ODEProblem(trace!, stateinit, tspan, param) + +vratio = √(v₀[1]^2+v₀[2]^2+v₀[3]^2)/c +E = (1 / √(1 - vratio^2) - 1)*m*c^2/abs(q)/1e6 + +v_str = @sprintf "V = %4.2f %s" vratio*100 "% c" +e_str = @sprintf "E = %6.4f MeV" E + +println(v_str) +println(e_str) + +# Default Tsit5() and many solvers does not work in this case! +sol_rel = solve(prob_rel, AB4(); dt=3e-9) +sol_non = solve(prob_non, AB4(); dt=3e-9) + +### Visualization + +f = Figure(fontsize=18) +ax1 = Axis3(f[1, 1]; + aspect = :data, + title = "Relativistic e⁻, \n"*v_str*", "*e_str + ) +ax2 = Axis3(f[1, 2]; + aspect = :data, + title = "Non-relativistic e⁻, \n"*v_str*", "*e_str + ) + +lines!(ax1, sol_rel, idxs=(1, 2, 3)) +lines!(ax2, sol_non, idxs=(1, 2, 3)) + +# Plot coils +θ = range(0, 2π, length=100) +x = a.*cos.(θ) +y = a.*sin.(θ) +z = fill(distance/2, size(x)) +for ax in (f[1,1], f[1,2]) + lines!(ax, x, y, z, color=(:red, 0.7)) +end +z = fill(-distance/2, size(x)) +for ax in (f[1,1], f[1,2]) + lines!(ax, x, y, z, color=(:red, 0.7)) +end + +x = b.*cos.(θ) +y = b.*sin.(θ) +z = fill(0.0, size(x)) +for ax in (f[1,1], f[1,2]) + lines!(ax, x, y, z, color=(:red, 0.7)) +end + +using FieldTracer + +xrange = range(-4, 4, length=20) +yrange = range(-4, 4, length=20) +zrange = range(-10, 10, length=20) + +Bx, By, Bz = let x=xrange, y=yrange, z=zrange + Bx = zeros(length(x), length(y), length(z)) + By = zeros(length(x), length(y), length(z)) + Bz = zeros(length(x), length(y), length(z)) + for k in eachindex(z), j in eachindex(y), i in eachindex(x) + Bx[i,j,k], By[i,j,k], Bz[i,j,k] = getB([x[i], y[j], z[k]]) + end + + Bx, By, Bz +end + +for i in 0:8 + if i == 0 + xs, ys, zs = 0.0, 0.0, 0.0 + else + xs, ys, zs = 3*cos(2π*(i-1)/8), 3*sin(2π*(i-1)/8), 0.0 + end + x1, y1, z1 = FieldTracer.trace(Bx, By, Bz, xs, ys, zs, xrange, yrange, zrange; + ds=0.1, maxstep=1000) + for ax in (f[1,1], f[1,2]) + lines!(ax, x1, y1, z1, color=:black) + end +end + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_magneticbottle/index.html b/v0.10.1/examples/advanced/demo_magneticbottle/index.html new file mode 100644 index 00000000..f5b8ac4e --- /dev/null +++ b/v0.10.1/examples/advanced/demo_magneticbottle/index.html @@ -0,0 +1,126 @@ + +Electron in a magnetic bottle · TestParticle.jl diff --git a/v0.10.1/examples/advanced/demo_magneticmirror-4.png b/v0.10.1/examples/advanced/demo_magneticmirror-4.png new file mode 100644 index 0000000000000000000000000000000000000000..f2c994948bd491f4537b7e0e6eea90ce0eb0bb65 GIT binary patch literal 253216 zcmeFZ^+|xP!7`Ft#o(8p}V_7TDrUL;;TN-`+n}9 zaPRg9L2&Q2*P1crm}4GaX-VN1NVrIFaBwd^iU`QU!NDWI!96ZQcnJRGlZUT7`0J^T 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00000000..c2273dea --- /dev/null +++ b/v0.10.1/examples/advanced/demo_magneticmirror.jl @@ -0,0 +1,110 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: getB_mirror +using OrdinaryDiffEq +using StaticArrays +using LinearAlgebra: normalize, norm, ⋅ +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +### Obtain field + +# Magnetic mirror parameters in SI units +const I = 20. # current in the solenoid +const N = 45 # number of windings +const distance = 10. # distance between solenoids +const a = 4.0 # radius of each coil + +function getB(xu) + SVector{3}(getB_mirror(xu[1], xu[2], xu[3], distance, a, I*N)) +end + +getE(xu) = SA[0.0, 0.0, 0.0] + +# velocity in the direction perpendicular to the magnetic field +function v_perp(t, x, y, z, vx, vy, vz) + xu = SA[x, y, z, vx, vy, vz] + vu = @view xu[4:6] + B = getB(xu) + b = normalize(B) + v_pa = (vu ⋅ b) .* b + + (t, norm(vu - v_pa)) +end + +# magnetic field +absB(t, x, y, z) = (t, hypot(getB(SA[x,y,z])...)) + +# μ, magnetic moment +function mu(t, x, y, z, vx, vy, vz) + xu = SA[x, y, z, vx, vy, vz] + + (t, v_perp(t, x, y, z, vx, vy, vz)[2]^2 / hypot(getB(xu)...) ) +end + +Et(xu) = hypot(xu[4:6]...) + +### Initialize particles +m = TestParticle.mₑ +q = TestParticle.qₑ +c = TestParticle.c + +# initial velocity, [m/s] +v₀ = [2.75, 2.5, 1.3] .* 0.001c # confined +##v₀ = [0.25, 0.25, 5.9595] .* 0.01c # escaped +# initial position, [m] +r₀ = [0.8, 0.8, 0.0] # confined +##r₀ = [1.5, 1.5, 2.4] # escaped +stateinit = [r₀..., v₀...] + +param = prepare(getE, getB; species=Electron) +tspan = (0.0, 1e-4) + +prob = ODEProblem(trace!, stateinit, tspan, param) + +# Default Tsit5() and many solvers does not work in this case! +# AB4() has better performance in maintaining magnetic moment conservation compared to AB3(). +sol_non = solve(prob, AB4(); dt=3e-9) + +### Visualization +f = Figure(size=(900, 600), fontsize=18) +ax1 = Axis3(f[1:3, 1], + title = "Magnetic Mirror", + xlabel = "x [m]", + ylabel = "y [m]", + zlabel = "z [m]", + aspect = :data, + azimuth = 0.9π, + elevation = 0.1π +) +ax2 = Axis(f[1,2], xlabel = "time [s]", ylabel = "B [T]") +ax3 = Axis(f[2,2], xlabel = "time [s]", ylabel = "v_perp [m/s]") +ax4 = Axis(f[3,2], xlabel = "time [s]", ylabel = "mu") + +lines!(ax1, sol_non, idxs=(1, 2, 3)) +lines!(ax2, sol_non, idxs=(absB, 0, 1, 2, 3)) +lines!(ax3, sol_non, idxs=(v_perp, 0, 1, 2, 3, 4, 5, 6)) +lines!(ax4, sol_non, idxs=(mu, 0, 1, 2, 3, 4, 5, 6)) + +# Plot coils +θ = range(0, 2π, length=100) +x = a.*cos.(θ) +y = a.*sin.(θ) +z = fill(distance/2, size(x)) +lines!(ax1, x, y, z, color=:red) +z = fill(-distance/2, size(x)) +lines!(ax1, x, y, z, color=:red) + +# # The distribution of magnetic field along the z-axis or x-axis +# Bz(z) = hypot(getB(SA[0.0, 0.0, z])...) +# Bx(x) = hypot(getB(SA[x, 0.0, 0.5*distance])...) +# z = collect(-10:0.01:10) +# x = collect(-0.99*a:0.01:0.99*a) +# # Ba = Bz.(z) +# Ba = Bx.(x) +# # lines(z, Ba, color=:red) +# lines(x, Ba, color=:red) + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_magneticmirror/index.html b/v0.10.1/examples/advanced/demo_magneticmirror/index.html new file mode 100644 index 00000000..cc56d160 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_magneticmirror/index.html @@ -0,0 +1,106 @@ + +Magnetic mirror · TestParticle.jl diff --git a/v0.10.1/examples/advanced/demo_output_func-9.png b/v0.10.1/examples/advanced/demo_output_func-9.png new file mode 100644 index 0000000000000000000000000000000000000000..70118ab60703d315eabba534a34eb2d8d14321bd GIT binary patch literal 89940 zcmeFZWmuK%)-}Au00jY+kPZ<+Qo2EqQ0bNwL`q7!K>?AH5|Az>M7p~~L?o1MxarPC z$2TwTy`STG-+$k)@6UJb@HvfDN zU}R*}OTa^+D&D~+cGI?oQYG`fPFuF*BbS$vk(uvIjEsnw>r50h?M~T^B`XOnHtEXK zuOHtH>oC59&*QMv%XpMdKtSLNpPpBLU(su*9xEF>!5J$%9w(b~&|1AgPELhFnPMYn z4DUWOv-=5ppjH?g`S5%<%_WrSmjQh*-TP6)l;)+wX=!P(?%56gj*YJ^`P|ghR`pnV0hqcp^~4 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b/v0.10.1/examples/advanced/demo_output_func.jl new file mode 100644 index 00000000..6c3c1204 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_output_func.jl @@ -0,0 +1,115 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: qᵢ, mᵢ +using OrdinaryDiffEq +using StaticArrays +using Statistics +using LinearAlgebra: normalize, ×, ⋅ +using Random +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +seed = 1 # seed for random number +Random.seed!(seed) + +"Set initial state for EnsembleProblem." +function prob_func(prob, i, repeat) + B0 = prob.p[3](prob.u0) + B0 = normalize(B0) + + Bperp1 = SA[0.0, -B0[3], B0[2]] |> normalize + Bperp2 = B0 × Bperp1 |> normalize + + # initial azimuthal angle + ϕ = 2π*rand() + # initial pitch angle + θ = acos(0.5) + + sinϕ, cosϕ = sincos(ϕ) + u = @. (B0*cos(θ) + Bperp1*(sin(θ)*cosϕ) + Bperp2*(sin(θ)*sinϕ)) * U₀ + + prob = @views remake(prob; u0=[prob.u0[1:3]..., u...]) +end + +# Number of cells for the field along each dimension +nx, ny, nz = 4, 6, 8 +# Spatial coordinates given in customized units +x = range(0, 1, length=nx) +y = range(0, 1, length=ny) +z = range(0, 1, length=nz) +# Numerical magnetic field given in customized units +B = Array{Float32, 4}(undef, 3, nx, ny, nz) + +B[1,:,:,:] .= 0.0 +B[2,:,:,:] .= 0.0 +B[3,:,:,:] .= 2.0 + +# Reference values for unit conversions between the customized and dimensionless units +const B₀ = let Bmag = @views hypot.(B[1,:,:,:], B[2,:,:,:], B[3,:,:,:]) + sqrt(mean(vec(Bmag) .^ 2)) +end +const U₀ = 1.0 +const l₀ = 2*nx +const t₀ = l₀ / U₀ +const E₀ = U₀ * B₀; + +# Dimensionless spatial extents [l₀] +x /= l₀ +y /= l₀ +z /= l₀ +# For full EM problems, the normalization of E and B should be done separately. +B ./= B₀ +E(x) = SA[0.0/E₀, 0.0/E₀, 0.0/E₀] + +# By default User type assumes q=1, m=1 +# bc=2 uses periodic boundary conditions +param = prepare(x, y, z, E, B; species=User, bc=2) + +# Initial condition +stateinit = let + x0 = [0.0, 0.0, 0.0] # initial position [l₀] + u0 = [1.0, 0.0, 0.0] # initial velocity [v₀] + [x0..., u0...] +end +# Time span +tspan = (0.0, 2π) # one averaged gyroperiod based on B₀ + +saveat = tspan[2] / 40 # save interval + +prob = ODEProblem(trace_normalized!, stateinit, tspan, param) + +"Set customized outputs for the ensemble problem." +function output_func(sol, i) + Bfunc = sol.prob.p[3] + b = Bfunc.(sol.u) + + μ = [@views b[i] ⋅ sol[4:6, i] / hypot(b[i]...) for i in eachindex(sol)] + + (sol.u, b, μ), false +end + +# Number of trajectories +trajectories = 2 + +ensemble_prob = EnsembleProblem(prob; prob_func, output_func, safetycopy=false) +sols = solve(ensemble_prob, Vern9(), EnsembleThreads(); trajectories, saveat); + +f = Figure(fontsize = 18) +ax = Axis3(f[1, 1], + title = "Proton trajectories", + xlabel = "X", + ylabel = "Y", + zlabel = "Z", + aspect = :data, +) + +for i in eachindex(sols) + xp = [s[1] for s in sols[i][1]] + yp = [s[2] for s in sols[i][1]] + zp = [s[3] for s in sols[i][1]] + lines!(ax, xp, yp, zp, label="$i") +end + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_output_func/index.html b/v0.10.1/examples/advanced/demo_output_func/index.html new file mode 100644 index 00000000..76035e26 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_output_func/index.html @@ -0,0 +1,107 @@ + +Ensemble tracing with extra saving · TestParticle.jl

Ensemble tracing with extra saving

Source code compat Author Update time

This example demonstrates tracing multiple protons in an analytic E field and numerical B field. See Demo: single tracing with additional diagnostics for explaining the unit conversion. Also check Demo: Ensemble for basic usages of the ensemble problem.

The output_func argument can be used to change saving outputs. It works as a reduction function, but here we demonstrate how to add additional outputs. Besides the regular outputs, we also save the magnetic field along the trajectory, together with the parallel velocity.

using TestParticle
+using TestParticle: qᵢ, mᵢ
+using OrdinaryDiffEq
+using StaticArrays
+using Statistics
+using LinearAlgebra: normalize, ×, ⋅
+using Random
+using CairoMakie
+
+seed = 1 # seed for random number
+Random.seed!(seed)
+
+"Set initial state for EnsembleProblem."
+function prob_func(prob, i, repeat)
+   B0 = prob.p[3](prob.u0)
+   B0 = normalize(B0)
+
+   Bperp1 = SA[0.0, -B0[3], B0[2]] |> normalize
+   Bperp2 = B0 × Bperp1 |> normalize
+
+   # initial azimuthal angle
+   ϕ = 2π*rand()
+   # initial pitch angle
+   θ = acos(0.5)
+
+   sinϕ, cosϕ = sincos(ϕ)
+   u = @. (B0*cos(θ) + Bperp1*(sin(θ)*cosϕ) + Bperp2*(sin(θ)*sinϕ)) * U₀
+
+   prob = @views remake(prob; u0=[prob.u0[1:3]..., u...])
+end
+
+# Number of cells for the field along each dimension
+nx, ny, nz = 4, 6, 8
+# Spatial coordinates given in customized units
+x = range(0, 1, length=nx)
+y = range(0, 1, length=ny)
+z = range(0, 1, length=nz)
+# Numerical magnetic field given in customized units
+B = Array{Float32, 4}(undef, 3, nx, ny, nz)
+
+B[1,:,:,:] .= 0.0
+B[2,:,:,:] .= 0.0
+B[3,:,:,:] .= 2.0
+
+# Reference values for unit conversions between the customized and dimensionless units
+const B₀ = let Bmag = @views hypot.(B[1,:,:,:], B[2,:,:,:], B[3,:,:,:])
+   sqrt(mean(vec(Bmag) .^ 2))
+end
+const U₀ = 1.0
+const l₀ = 2*nx
+const t₀ = l₀ / U₀
+const E₀ = U₀ * B₀;

Convert from customized to default dimensionless units

# Dimensionless spatial extents [l₀]
+x /= l₀
+y /= l₀
+z /= l₀
+# For full EM problems, the normalization of E and B should be done separately.
+B ./= B₀
+E(x) = SA[0.0/E₀, 0.0/E₀, 0.0/E₀]
+
+# By default User type assumes q=1, m=1
+# bc=2 uses periodic boundary conditions
+param = prepare(x, y, z, E, B; species=User, bc=2)
+
+# Initial condition
+stateinit = let
+   x0 = [0.0, 0.0, 0.0] # initial position [l₀]
+   u0 = [1.0, 0.0, 0.0] # initial velocity [v₀]
+   [x0..., u0...]
+end
+# Time span
+tspan = (0.0, 2π) # one averaged gyroperiod based on B₀
+
+saveat = tspan[2] / 40 # save interval
+
+prob = ODEProblem(trace_normalized!, stateinit, tspan, param)
+
+"Set customized outputs for the ensemble problem."
+function output_func(sol, i)
+   Bfunc = sol.prob.p[3]
+   b = Bfunc.(sol.u)
+
+   μ = [@views b[i] ⋅ sol[4:6, i] / hypot(b[i]...) for i in eachindex(sol)]
+
+   (sol.u, b, μ), false
+end
+
+# Number of trajectories
+trajectories = 2
+
+ensemble_prob = EnsembleProblem(prob; prob_func, output_func, safetycopy=false)
+sols = solve(ensemble_prob, Vern9(), EnsembleThreads(); trajectories, saveat);

Visualization

f = Figure(fontsize = 18)
+ax = Axis3(f[1, 1],
+   title = "Proton trajectories",
+   xlabel = "X",
+   ylabel = "Y",
+   zlabel = "Z",
+   aspect = :data,
+)
+
+for i in eachindex(sols)
+   xp = [s[1] for s in sols[i][1]]
+   yp = [s[2] for s in sols[i][1]]
+   zp = [s[3] for s in sols[i][1]]
+   lines!(ax, xp, yp, zp, label="$i")
+end
+

This page was generated using DemoCards.jl and Literate.jl.

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z?IXL;kUtYnUK&-(RTF>J+pB>B`0X8Z*(lU}=9`4K6~qz?q z3rK6^kHl?oH2AM}c6Sazux0D(?A%Wf>sBCskUzl;vIC6#`G0-k|Nmm->iyq6jQ_u8 d|K~NE&{SA-oOxPPC4u}~eO)7+T>6o7{{shiF6{sS literal 0 HcmV?d00001 diff --git a/v0.10.1/examples/advanced/demo_proton_dipole.jl b/v0.10.1/examples/advanced/demo_proton_dipole.jl new file mode 100644 index 00000000..e15048f0 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_proton_dipole.jl @@ -0,0 +1,96 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: getB_dipole, getE_dipole, sph2cart, dipole_fieldline, mᵢ, qᵢ, c, Rₑ +using OrdinaryDiffEq +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +# Initial condition +stateinit = let + # Initial particle energy + Ek = 5e7 # [eV] + # initial velocity, [m/s] + v₀ = sph2cart(c*sqrt(1-1/(1+Ek*qᵢ/(mᵢ*c^2))^2), 0.0, π/4) + # initial position, [m] + r₀ = sph2cart(2.5*Rₑ, 0.0, π/2) + [r₀..., v₀...] +end +# obtain field +param = prepare(getE_dipole, getB_dipole) +tspan = (0.0, 10.0) + +prob = ODEProblem(trace!, stateinit, tspan, param) + +sol = solve(prob, Vern9()) + +### Visualization + +f = Figure(fontsize=18) +ax = Axis3(f[1, 1], + title = "50 MeV Proton trajectory in Earth's dipole field", + xlabel = "x [Re]", + ylabel = "y [Re]", + zlabel = "z [Re]", + aspect = :data, + limits = (-2.5, 2.5, -2.5, 2.5, -1, 1) +) + +invRE = 1 / Rₑ +l = lines!(ax, sol, idxs=(1, 2, 3)) +##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed! +scale!(ax.scene.plots[9+1], invRE, invRE, invRE) + +for ϕ in range(0, stop=2*π, length=10) + lines!(dipole_fieldline(ϕ)..., color=:tomato, alpha=0.3) +end + +f = DisplayAs.PNG(f) #hide + +function get_energy_ratio(sol) + vx = getindex.(sol.u, 4) + vy = getindex.(sol.u, 5) + vz = getindex.(sol.u, 6) + + Einit = vx[1]^2 + vy[1]^2 + vz[1]^2 + Eend = vx[end]^2 + vy[end]^2 + vz[end]^2 + + (Eend - Einit) / Einit +end + +# `ImplicitMidpoint()` requires a fixed time step. +sol = solve(prob, ImplicitMidpoint(); dt=1e-3) +get_energy_ratio(sol) + +sol = solve(prob, ImplicitMidpoint(); dt=1e-4) +get_energy_ratio(sol) + +sol = solve(prob, Vern9()) +get_energy_ratio(sol) + +sol = solve(prob, Trapezoid()) +get_energy_ratio(sol) + +sol = solve(prob, Vern6()) +get_energy_ratio(sol) + +sol = solve(prob, Tsit5()) +get_energy_ratio(sol) + +# This is roughly equivalent in accuracy and performance with Vern9() and `reltol=1e-3` (default) +sol = solve(prob, Tsit5(); reltol=1e-4); + +using DiffEqCallbacks + +# p = (charge_mass_ratio, E, B) +dtFE(u, p, t) = 2π / (abs(p[1]) * hypot(p[3](u, t)...)) +cb = StepsizeLimiter(dtFE; safety_factor=1 // 10, max_step=true) + +sol = solve(prob, Vern9(); callback=cb, dt=0.1) # dt=0.1 is a dummy value +get_energy_ratio(sol) + +dt = 1e-4 +prob = TraceProblem(stateinit, tspan, param) +sol = TestParticle.solve(prob; dt)[1] +get_energy_ratio(sol) + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_proton_dipole/index.html b/v0.10.1/examples/advanced/demo_proton_dipole/index.html new file mode 100644 index 00000000..420045a8 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_proton_dipole/index.html @@ -0,0 +1,74 @@ + +Magnetic dipole · TestParticle.jl diff --git a/v0.10.1/examples/advanced/demo_savingcallback.jl b/v0.10.1/examples/advanced/demo_savingcallback.jl new file mode 100644 index 00000000..d9b8eb0a --- /dev/null +++ b/v0.10.1/examples/advanced/demo_savingcallback.jl @@ -0,0 +1,81 @@ +using TestParticle +using TestParticle: qᵢ, mᵢ +using OrdinaryDiffEq +using StaticArrays +using Statistics +using LinearAlgebra: normalize, ×, ⋅ +using DiffEqCallbacks + +# Number of cells for the field along each dimension +nx, ny, nz = 4, 6, 8 +# Spatial coordinates given in customized units +x = range(-0.5, 0.5, length=nx) +y = range(-0.5, 0.5, length=ny) +z = range(-0.5, 0.5, length=nz) +# Numerical magnetic field given in customized units +B = Array{Float32, 4}(undef, 3, nx, ny, nz) + +B[1,:,:,:] .= 0.0 +B[2,:,:,:] .= 0.0 +B[3,:,:,:] .= 2.0 + +# Reference values for unit conversions between the customized and dimensionless units +const B₀ = let Bmag = @views hypot.(B[1,:,:,:], B[2,:,:,:], B[3,:,:,:]) + sqrt(mean(vec(Bmag) .^ 2)) +end +const U₀ = 1.0 +const l₀ = 4*nx +const t₀ = l₀ / U₀ +const E₀ = U₀ * B₀ + +### Convert from customized to default dimensionless units +# Dimensionless spatial extents [l₀] +x /= l₀ +y /= l₀ +z /= l₀ +# For full EM problems, the normalization of E and B should be done separately. +B ./= B₀ +E(x) = SA[0.0/E₀, 0.0/E₀, 0.0/E₀] + +# By default User type assumes q=1, m=1; bc=2 uses periodic boundary conditions +param = prepare(x, y, z, E, B; species=User, bc=2) + +tspan = (0.0, π) # half averaged gyroperiod based on B₀ + +# Dummy initial state; positions have units l₀; velocities have units U₀ +stateinit = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0] + +prob = ODEProblem(trace_normalized!, stateinit, tspan, param) + +prob.u0[4:6] = let + B0 = prob.p[3](prob.u0) + B0 = normalize(B0) + + Bperp1 = SA[0.0, -B0[3], B0[2]] |> normalize + Bperp2 = B0 × Bperp1 |> normalize + + # initial velocity azimuthal angle + ϕ = 2π*0 + # initial velocity pitch angle w.r.t. B + θ = acos(0.0) + + sinϕ, cosϕ = sincos(ϕ) + @. (B0*cos(θ) + Bperp1*(sin(θ)*cosϕ) + Bperp2*(sin(θ)*sinϕ)) * U₀ +end + +saved_values = SavedValues(Float64, Tuple{SVector{3, Float64}, Float64}) + +function save_B_mu(u, t, integrator) + b = integrator.p[3](u) + μ = @views b ⋅ u[4:6] / hypot(b...) + + b, μ +end + +cb = SavingCallback(save_B_mu, saved_values) + +sol = solve(prob, Vern9(); callback=cb); + +saved_values + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_savingcallback/index.html b/v0.10.1/examples/advanced/demo_savingcallback/index.html new file mode 100644 index 00000000..dd51d43e --- /dev/null +++ b/v0.10.1/examples/advanced/demo_savingcallback/index.html @@ -0,0 +1,82 @@ + +Single tracing with additional diagnostics · TestParticle.jl

Single tracing with additional diagnostics

Source code compat Author Update time

This example demonstrates tracing one proton in an analytic E field and numerical B field. It also combines one type of normalization using a reference velocity U₀, a reference magnetic field B₀, and a reference time 1/Ω, where Ω is the gyrofrequency. This indicates that in the dimensionless units, a proton with initial perpendicular velocity 1 under magnetic field magnitude 1 will possess a gyro-radius of 1. In the dimensionless spatial coordinates, we can zoom in/out the EM field to control the number of discrete points encountered in a gyroperiod. For example, if dx=dy=dz=1, it means that a particle with perpendicular velocity 1 will "see" one discrete point along a certain direction oriented from the gyro-center within the gyro-radius; if dx=dy=dz=0.5, then the particle will "see" two discrete points. MHD models, for instance, are dimensionless by nature. There will be customized (dimensionless) units for (x,y,z,E,B) that we needs to convert the dimensionless units for computing. If we simulate a turbulence field with MHD, we want to include more discrete points within a gyro-radius for the effect of small scale perturbations to take place. (Otherwise within one gyro-period all you will see is a nice-looking helix!) However, we cannot simply shrink the spatial coordinates as we wish, otherwise we will quickly encounter the boundary of our simulation.

The SavingCallback from DiffEqCallbacks.jl can be used to save additional outputs for diagnosis. Here we save the magnetic field along the trajectory, together with the parallel velocity. Note that SavingCallback is currently not compatible with ensemble problems; for multiple particle tracing with customized outputs, see Demo: ensemble tracing with extra saving.

using TestParticle
+using TestParticle: qᵢ, mᵢ
+using OrdinaryDiffEq
+using StaticArrays
+using Statistics
+using LinearAlgebra: normalize, ×, ⋅
+using DiffEqCallbacks
+
+# Number of cells for the field along each dimension
+nx, ny, nz = 4, 6, 8
+# Spatial coordinates given in customized units
+x = range(-0.5, 0.5, length=nx)
+y = range(-0.5, 0.5, length=ny)
+z = range(-0.5, 0.5, length=nz)
+# Numerical magnetic field given in customized units
+B = Array{Float32, 4}(undef, 3, nx, ny, nz)
+
+B[1,:,:,:] .= 0.0
+B[2,:,:,:] .= 0.0
+B[3,:,:,:] .= 2.0
+
+# Reference values for unit conversions between the customized and dimensionless units
+const B₀ = let Bmag = @views hypot.(B[1,:,:,:], B[2,:,:,:], B[3,:,:,:])
+   sqrt(mean(vec(Bmag) .^ 2))
+end
+const U₀ = 1.0
+const l₀ = 4*nx
+const t₀ = l₀ / U₀
+const E₀ = U₀ * B₀
+
+### Convert from customized to default dimensionless units
+# Dimensionless spatial extents [l₀]
+x /= l₀
+y /= l₀
+z /= l₀
+# For full EM problems, the normalization of E and B should be done separately.
+B ./= B₀
+E(x) = SA[0.0/E₀, 0.0/E₀, 0.0/E₀]
+
+# By default User type assumes q=1, m=1; bc=2 uses periodic boundary conditions
+param = prepare(x, y, z, E, B; species=User, bc=2)
+
+tspan = (0.0, π) # half averaged gyroperiod based on B₀
+
+# Dummy initial state; positions have units l₀; velocities have units U₀
+stateinit = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
+
+prob = ODEProblem(trace_normalized!, stateinit, tspan, param)
+
+prob.u0[4:6] = let
+   B0 = prob.p[3](prob.u0)
+   B0 = normalize(B0)
+
+   Bperp1 = SA[0.0, -B0[3], B0[2]] |> normalize
+   Bperp2 = B0 × Bperp1 |> normalize
+
+   # initial velocity azimuthal angle
+   ϕ = 2π*0
+   # initial velocity pitch angle w.r.t. B
+   θ = acos(0.0)
+
+   sinϕ, cosϕ = sincos(ϕ)
+   @. (B0*cos(θ) + Bperp1*(sin(θ)*cosϕ) + Bperp2*(sin(θ)*sinϕ)) * U₀
+end
+
+saved_values = SavedValues(Float64, Tuple{SVector{3, Float64}, Float64})
+
+function save_B_mu(u, t, integrator)
+   b = integrator.p[3](u)
+   μ = @views b ⋅ u[4:6] / hypot(b...)
+
+   b, μ
+end
+
+cb = SavingCallback(save_B_mu, saved_values)
+
+sol = solve(prob, Vern9(); callback=cb);

The extra values are saved in saved_values:

saved_values
SavedValues{tType=Float64, savevalType=Tuple{StaticArraysCore.SVector{3, Float64}, Float64}}
+t:
+[0.0, 0.0007064003808057418, 0.006151342961409062, 0.03990852562861322, 0.24729664360869114, 0.8655117969873891, 1.8106352974346311, 2.900697481526176, 3.141592653589793]
+saveval:
+Tuple{StaticArraysCore.SVector{3, Float64}, Float64}[([0.0, 0.0, 1.0], 6.123233995736766e-17), ([0.0, 0.0, 1.0], 6.123233995736766e-17), ([0.0, 0.0, 1.0], 6.123233995736766e-17), ([0.0, 0.0, 1.0], 6.123233995736766e-17), ([0.0, 0.0, 1.0], 6.123233995736766e-17), ([0.0, 0.0, 1.0], 6.123233995736766e-17), ([0.0, 0.0, 1.0], 6.123233995736766e-17), ([0.0, 0.0, 1.0], 6.123233995736766e-17), ([0.0, 0.0, 1.0], 6.123233995736766e-17)]

This page was generated using DemoCards.jl and Literate.jl.

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TestParticle: mᵢ, kB +using LinearAlgebra +using Statistics: mean, std +using Printf +using Random +using CairoMakie, PairPlots +CairoMakie.activate!(type = "png") #hide + +# For reproducible results +Random.seed!(1234) + +"Set initial conditions." +function prob_func(prob, i, repeat) + v₀ = sample(vdf₁) + r₀ = [5_000e3, 0.0, 0.0] + + prob = remake(prob; u0 = [r₀..., v₀...]) +end + +# MHD states in SI units +n₁ = 1.0e6 +T₁ = 720471.8506664868 +Pi₁ = 0.0049735919716217296 * 1e-9 +Pe₁ = 0.0049735919716217296 * 1e-9 +Pth₁ = Pi₁ + Pe₁ +V₁ = [-545.1484121835928, 0.0, 0.0] .* 1e3 +B₁ = [0.0, 0.0, 5.0] .* 1e-9 + +n₂ = 3.1662479035540008e6 +T₂ = 5.957947703036288e6 +Pi₂ = 0.13022511153415584 * 1e-9 +Pe₂ = 0.13022511153415584 * 1e-9 +Pth₂ = Pi₂ + Pe₂ +V₂ = [-172.17489874108816, 0.0, 0.0] .* 1e3 +B₂ = [0.0, 0.0, 15.831239517770003] .* 1e-9; + +E₁ = B₁ × V₁ +E₂ = B₂ × V₂ + +# Shock normal direction range +x = range(-5_000e3, 5_000e3, length=100) +B = repeat(B₁, 1, length(x)) +E = repeat(E₁, 1, length(x)) +# Index for the shock location +mid_ = length(x) ÷ 2 + +B[:, 1:mid_] .= B₂ +E[:, 1:mid_] .= E₂ + +const vdf₁ = Maxwellian(V₁, Pth₁, n₁; m=mᵢ) +vdf₂ = Maxwellian(V₂, Pth₂, n₂; m=mᵢ) + +trajectories = 400 +weight₁ = n₁ / trajectories # relation between test particle and real particles + +prob = let + # BC type 3 is Flat + param = prepare(x, E, B; species=Proton, bc=3); + stateinit = zeros(6) # particle position and velocity to be modified + tspan = (0.0, 30.0) + ODEProblem(trace!, stateinit, tspan, param) +end +ensemble_prob = EnsembleProblem(prob; prob_func, safetycopy=false) + +sols = solve(ensemble_prob, Vern9(), EnsembleSerial(); trajectories); + +function plot_traj(sols; azimuth=1.275pi, elevation=pi/8) + f = Figure(fontsize=18) + ax = Axis3(f[1, 1]; + title = "Particles across MHD shock", + xlabel = "x [km]", + ylabel = "y [km]", + zlabel = "z [km]", + aspect = :data, + azimuth, elevation, + ) + + invL = 1 / 1e3 + + for i in eachindex(sols) + lines!(ax, sols[i], idxs=(1,2,3), label="$i") + ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed! + ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[mod(i-1, 7)+1] + scale!(ax.scene.plots[9+2*i-1], invL, invL, invL) + end + + # Represent the shock front + p1 = Point3f(0.0, -2e2, -2e2) + p2 = Point3f(0.0, 2e2, -2e2) + p3 = Point3f(0.0, 2e2, 2e2) + p4 = Point3f(0.0, -2e2, 2e2) + + mesh!(ax, [p1, p2, p3], color = (:gray, 0.1), shading = Makie.automatic) + mesh!(ax, [p1, p4, p3], color = (:gray, 0.1), shading = Makie.automatic) + + f +end + +f = plot_traj(sols[1:4]) +f = DisplayAs.PNG(f) #hide + +function plot_dist(x, sols; nxchunks::Int=2, ntchunks::Int=20) + trange = range(sols[1].prob.tspan..., length=ntchunks) + + xrange = range(x[1], x[end], length=nxchunks+1) + dx = (x[end] - x[1]) / nxchunks + xmid = range(x[1] + 0.5dx, x[end] - 0.5dx, length=nxchunks) ./ 1e3 + + vx = [Float64[] for _ in 1:nxchunks] + + for sol in sols + for t in trange + xv = sol(t) + for i in 1:nxchunks + if xrange[i] < xv[1] ≤ xrange[i+1] + push!(vx[i], xv[4] / 1e3) + end + end + end + end + + for i in eachindex(vx) + if isempty(vx[i]) + push!(vx[i], 0.0) + end + end + + f = Figure(size = (1200, 600), fontsize=18) + ax = Axis(f[1, 1], + limits = (nothing, nothing, -750, 400), + title = "Phase space distributions at different spatial locations", + xlabel = "Location [km]", + ylabel = "Vx [km/s]", + xminorticksvisible = true) + + for i in 1:nxchunks + hist!(ax, vx[i], normalization = :pdf, bins = 50, + scale_to=-5000/nxchunks, offset=xmid[i], direction=:x) + end + + v̄x = mean.(vx) + vth = [std(vx[i]; corrected=false, mean=v̄x[i]) for i in 1:nxchunks] + means_str = [@sprintf "Vx: %d [km/s]" v̄x[i] for i in eachindex(v̄x)] + std_str = [@sprintf "Vth: %d [km/s]" vth[i] for i in eachindex(vth)] + text!(Point.(xmid.+400, 300.0), text = means_str, align = (:right, :center), + offset = (-60, 0), color = :black, fontsize=24) + text!(Point.(xmid.+400, -700.0), text = std_str, align = (:right, :center), + offset = (-60, 0), color = :black, fontsize=24) + + f +end + +f = plot_dist(x, sols; nxchunks=4, ntchunks=100) +f = DisplayAs.PNG(f) #hide + +println("Vth₁ = ", round(vdf₁.vth / 1e3, digits=2), " km/s") #hide +println("Vth₂ = ", round(vdf₂.vth / 1e3, digits=2), " km/s") #hide + +function plot_dist_pairplot(x, sols; ntchunks::Int=20) + nxchunks = 2 + trange = range(sols[1].prob.tspan..., length=ntchunks) + + xrange = range(x[1], x[end], length=nxchunks+1) + + table = [(; + vx = Float64[], + vy = Float64[], + vz = Float64[], + ) for _ in 1:nxchunks] + + for sol in sols + for t in trange + xv = sol(t) + for i in 1:nxchunks + if xrange[i] < xv[1] ≤ xrange[i+1] + push!(table[i].vx, xv[4] / 1e3) + push!(table[i].vy, xv[5] / 1e3) + push!(table[i].vz, xv[6] / 1e3) + end + end + end + end + + for i in eachindex(table) + if isempty(table[i].vx) + push!(table[i].vx, 0.0) + push!(table[i].vy, 0.0) + push!(table[i].vz, 0.0) + end + end + + f = Figure(size = (1000, 600), fontsize=18) + + c1 = Makie.wong_colors(0.5)[1] + c2 = Makie.wong_colors(0.5)[2] + + l1 = @sprintf "x: [%d, %d] km downstream" xrange[1]/1e3 xrange[2]/1e3 + l2 = @sprintf "x: [%d, %d] km upstream" xrange[2]/1e3 xrange[3]/1e3 + + pairplot(f[1,1], + PairPlots.Series(table[1], label=l1, color=c1, strokecolor=c1), + PairPlots.Series(table[2], label=l2, color=c2, strokecolor=c2), + bodyaxis=(; xgridvisible=true, ygridvisible=true), + diagaxis=(; xgridvisible=true, ygridvisible=true) + ) + + f +end + +f = plot_dist_pairplot(x, sols; ntchunks=20) +f = DisplayAs.PNG(f) #hide + +# MHD states in SI units +n₁ = 1.0e6 +T₁ = 720471.8506664868 +Pi₁ = 0.0049735919716217296 * 1e-9 +Pe₁ = 0.0049735919716217296 * 1e-9 +Pth₁ = Pi₁ + Pe₁ +V₁ = [-545.1484121835928, 0.0, 0.0] .* 1e3 +B₁ = [5.0, 0.0, 0.0] .* 1e-9 + +n₂ = 3.6363636363636362e6 +T₂ = 7.380333520264822e6 +Pi₂ = 0.18526630094290936 * 1e-9 +Pe₂ = 0.18526630094290936 * 1e-9 +Pth₂ = Pi₂ + Pe₂ +V₂ = [-149.91581335048804, 0.0, 0.0] .* 1e3 +B₂ = [5.0, 0.0, 0.0] .* 1e-9; + +E₁ = B₁ × V₁ +E₂ = B₂ × V₂ + +# Shock normal direction range +x = range(-5_000e3, 5_000e3, length=100) +B = repeat(B₁, 1, length(x)) +E = repeat(E₁, 1, length(x)) +# Index for the shock location +mid_ = length(x) ÷ 2 + +B[:, 1:mid_] .= B₂ +E[:, 1:mid_] .= E₂ + +vdf₁ = Maxwellian(V₁, Pth₁, n₁; m=mᵢ) +vdf₂ = Maxwellian(V₂, Pth₂, n₂; m=mᵢ) + +trajectories = 2 +weight₁ = n₁ / trajectories + +prob = let + # BC type 3 is Flat + param = prepare(x, E, B; species=Proton, bc=3); + stateinit = zeros(6) # particle position and velocity to be modified + tspan = (0.0, 14.0) + ODEProblem(trace!, stateinit, tspan, param) +end +ensemble_prob = EnsembleProblem(prob; prob_func, safetycopy=false) + +sols = solve(ensemble_prob, Vern9(), EnsembleSerial(); trajectories); + +f = plot_traj(sols; azimuth=1.08π, elevation=pi/16) +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_shock/index.html b/v0.10.1/examples/advanced/demo_shock/index.html new file mode 100644 index 00000000..03aec9ee --- /dev/null +++ b/v0.10.1/examples/advanced/demo_shock/index.html @@ -0,0 +1,244 @@ + +Shock · TestParticle.jl

Shock

Source code compat Author Update time

This example shows how to trace protons of a certain energy in MHD plane shocks.

using TestParticle, OrdinaryDiffEq
+using TestParticle: mᵢ, kB
+using LinearAlgebra
+using Statistics: mean, std
+using Printf
+using Random
+using CairoMakie, PairPlots
+
+# For reproducible results
+Random.seed!(1234)
+
+"Set initial conditions."
+function prob_func(prob, i, repeat)
+   v₀ = sample(vdf₁)
+   r₀ = [5_000e3, 0.0, 0.0]
+
+   prob = remake(prob; u0 = [r₀..., v₀...])
+end
Main.var"##55416".prob_func

Perpendicular shock is a special shock in which both the upstream and downstream plasma flows are perpendicular to the magnetic field, as well as the shock front.

# MHD states in SI units
+n₁ = 1.0e6
+T₁ = 720471.8506664868
+Pi₁ = 0.0049735919716217296 * 1e-9
+Pe₁ = 0.0049735919716217296 * 1e-9
+Pth₁ = Pi₁ + Pe₁
+V₁ = [-545.1484121835928, 0.0, 0.0] .* 1e3
+B₁ = [0.0, 0.0, 5.0] .* 1e-9
+
+n₂ = 3.1662479035540008e6
+T₂ = 5.957947703036288e6
+Pi₂ = 0.13022511153415584 * 1e-9
+Pe₂ = 0.13022511153415584 * 1e-9
+Pth₂ = Pi₂ + Pe₂
+V₂ = [-172.17489874108816, 0.0, 0.0] .* 1e3
+B₂ = [0.0, 0.0, 15.831239517770003] .* 1e-9;

In ideal MHD, the electric field simply contains the convection term. For perpendicular shocks, the electric field across the shock is continuous. In the upstream, particles follow straight lines because it's purely an ExB drift; across the shock, ExB drift changes to the downstream bulk velocity.

E₁ = B₁ × V₁
+E₂ = B₂ × V₂
+
+# Shock normal direction range
+x = range(-5_000e3, 5_000e3, length=100)
+B = repeat(B₁, 1, length(x))
+E = repeat(E₁, 1, length(x))
+# Index for the shock location
+mid_ = length(x) ÷ 2
+
+B[:, 1:mid_] .= B₂
+E[:, 1:mid_] .= E₂
+
+const vdf₁ = Maxwellian(V₁, Pth₁, n₁; m=mᵢ)
+vdf₂ = Maxwellian(V₂, Pth₂, n₂; m=mᵢ)
+
+trajectories = 400
+weight₁ = n₁ / trajectories # relation between test particle and real particles
+
+prob = let
+   # BC type 3 is Flat
+   param = prepare(x, E, B; species=Proton, bc=3);
+   stateinit = zeros(6) # particle position and velocity to be modified
+   tspan = (0.0, 30.0)
+   ODEProblem(trace!, stateinit, tspan, param)
+end
+ensemble_prob = EnsembleProblem(prob; prob_func, safetycopy=false)
+
+sols = solve(ensemble_prob, Vern9(), EnsembleSerial(); trajectories);

Sample particle trajectories

function plot_traj(sols; azimuth=1.275pi, elevation=pi/8)
+   f = Figure(fontsize=18)
+   ax = Axis3(f[1, 1];
+      title = "Particles across MHD shock",
+      xlabel = "x [km]",
+      ylabel = "y [km]",
+      zlabel = "z [km]",
+      aspect = :data,
+      azimuth, elevation,
+   )
+
+   invL = 1 / 1e3
+
+   for i in eachindex(sols)
+      lines!(ax, sols[i], idxs=(1,2,3), label="$i")
+      ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+      ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[mod(i-1, 7)+1]
+      scale!(ax.scene.plots[9+2*i-1], invL, invL, invL)
+   end
+
+   # Represent the shock front
+   p1 = Point3f(0.0, -2e2, -2e2)
+   p2 = Point3f(0.0, 2e2, -2e2)
+   p3 = Point3f(0.0, 2e2, 2e2)
+   p4 = Point3f(0.0, -2e2, 2e2)
+
+   mesh!(ax, [p1, p2, p3], color = (:gray, 0.1), shading = Makie.automatic)
+   mesh!(ax, [p1, p4, p3], color = (:gray, 0.1), shading = Makie.automatic)
+
+   f
+end
+
+f = plot_traj(sols[1:4])

Phase space distributions

function plot_dist(x, sols; nxchunks::Int=2, ntchunks::Int=20)
+   trange = range(sols[1].prob.tspan..., length=ntchunks)
+
+   xrange = range(x[1], x[end], length=nxchunks+1)
+   dx = (x[end] - x[1]) / nxchunks
+   xmid = range(x[1] + 0.5dx, x[end] - 0.5dx, length=nxchunks) ./ 1e3
+
+   vx = [Float64[] for _ in 1:nxchunks]
+
+   for sol in sols
+      for t in trange
+         xv = sol(t)
+         for i in 1:nxchunks
+            if xrange[i] < xv[1] ≤ xrange[i+1]
+               push!(vx[i], xv[4] / 1e3)
+            end
+         end
+      end
+   end
+
+   for i in eachindex(vx)
+      if isempty(vx[i])
+         push!(vx[i], 0.0)
+      end
+   end
+
+   f = Figure(size = (1200, 600), fontsize=18)
+   ax = Axis(f[1, 1],
+      limits = (nothing, nothing, -750, 400),
+      title = "Phase space distributions at different spatial locations",
+      xlabel = "Location [km]",
+      ylabel = "Vx [km/s]",
+      xminorticksvisible = true)
+
+   for i in 1:nxchunks
+      hist!(ax, vx[i], normalization = :pdf, bins = 50,
+         scale_to=-5000/nxchunks, offset=xmid[i], direction=:x)
+   end
+
+   v̄x = mean.(vx)
+   vth = [std(vx[i]; corrected=false, mean=v̄x[i]) for i in 1:nxchunks]
+   means_str = [@sprintf "Vx: %d [km/s]" v̄x[i] for i in eachindex(v̄x)]
+   std_str = [@sprintf "Vth: %d [km/s]" vth[i] for i in eachindex(vth)]
+   text!(Point.(xmid.+400, 300.0), text = means_str, align = (:right, :center),
+      offset = (-60, 0), color = :black, fontsize=24)
+   text!(Point.(xmid.+400, -700.0), text = std_str, align = (:right, :center),
+      offset = (-60, 0), color = :black, fontsize=24)
+
+   f
+end
+
+f = plot_dist(x, sols; nxchunks=4, ntchunks=100)

Even with 400 particles, we are still able to statistically approximate the velocity moment downstream of the perpendicular shock. While the upstream thermal speed is close to what we set, the downstream thermal speed is higher than our precalculated value. (Why?)

Vth₁ = 77.1 km/s
+Vth₂ = 221.7 km/s
+

A nice way to present the 3d distributions in 2d is via the pair plots:

function plot_dist_pairplot(x, sols; ntchunks::Int=20)
+   nxchunks = 2
+   trange = range(sols[1].prob.tspan..., length=ntchunks)
+
+   xrange = range(x[1], x[end], length=nxchunks+1)
+
+   table = [(;
+   vx = Float64[],
+   vy = Float64[],
+   vz = Float64[],
+   ) for _ in 1:nxchunks]
+
+   for sol in sols
+      for t in trange
+         xv = sol(t)
+         for i in 1:nxchunks
+            if xrange[i] < xv[1] ≤ xrange[i+1]
+               push!(table[i].vx, xv[4] / 1e3)
+               push!(table[i].vy, xv[5] / 1e3)
+               push!(table[i].vz, xv[6] / 1e3)
+            end
+         end
+      end
+   end
+
+   for i in eachindex(table)
+      if isempty(table[i].vx)
+         push!(table[i].vx, 0.0)
+         push!(table[i].vy, 0.0)
+         push!(table[i].vz, 0.0)
+      end
+   end
+
+   f = Figure(size = (1000, 600), fontsize=18)
+
+   c1 = Makie.wong_colors(0.5)[1]
+   c2 = Makie.wong_colors(0.5)[2]
+
+   l1 = @sprintf "x: [%d, %d] km downstream" xrange[1]/1e3 xrange[2]/1e3
+   l2 = @sprintf "x: [%d, %d] km upstream" xrange[2]/1e3 xrange[3]/1e3
+
+   pairplot(f[1,1],
+       PairPlots.Series(table[1], label=l1, color=c1, strokecolor=c1),
+       PairPlots.Series(table[2], label=l2, color=c2, strokecolor=c2),
+       bodyaxis=(; xgridvisible=true, ygridvisible=true),
+       diagaxis=(; xgridvisible=true, ygridvisible=true)
+   )
+
+   f
+end
+
+f = plot_dist_pairplot(x, sols; ntchunks=20)

For parallel shocks, we have a different scenario. Parallel shock is another special shock in which both the upstream and downstream plasma flows are parallel to the magnetic field, as well as perpendicular to the shock front.

# MHD states in SI units
+n₁ = 1.0e6
+T₁ = 720471.8506664868
+Pi₁ = 0.0049735919716217296 * 1e-9
+Pe₁ = 0.0049735919716217296 * 1e-9
+Pth₁ = Pi₁ + Pe₁
+V₁ = [-545.1484121835928, 0.0, 0.0] .* 1e3
+B₁ = [5.0, 0.0, 0.0] .* 1e-9
+
+n₂ = 3.6363636363636362e6
+T₂ = 7.380333520264822e6
+Pi₂ = 0.18526630094290936 * 1e-9
+Pe₂ = 0.18526630094290936 * 1e-9
+Pth₂ = Pi₂ + Pe₂
+V₂ = [-149.91581335048804, 0.0, 0.0] .* 1e3
+B₂ = [5.0, 0.0, 0.0] .* 1e-9;

There is no convection electric field in the parallel shock:

E₁ = B₁ × V₁
+E₂ = B₂ × V₂
3-element Vector{Float64}:
+  0.0
+ -0.0
+  0.0

Therefore, when we trace particles, there is no deceleration across the shock:

# Shock normal direction range
+x = range(-5_000e3, 5_000e3, length=100)
+B = repeat(B₁, 1, length(x))
+E = repeat(E₁, 1, length(x))
+# Index for the shock location
+mid_ = length(x) ÷ 2
+
+B[:, 1:mid_] .= B₂
+E[:, 1:mid_] .= E₂
+
+vdf₁ = Maxwellian(V₁, Pth₁, n₁; m=mᵢ)
+vdf₂ = Maxwellian(V₂, Pth₂, n₂; m=mᵢ)
+
+trajectories = 2
+weight₁ = n₁ / trajectories
+
+prob = let
+   # BC type 3 is Flat
+   param = prepare(x, E, B; species=Proton, bc=3);
+   stateinit = zeros(6) # particle position and velocity to be modified
+   tspan = (0.0, 14.0)
+   ODEProblem(trace!, stateinit, tspan, param)
+end
+ensemble_prob = EnsembleProblem(prob; prob_func, safetycopy=false)
+
+sols = solve(ensemble_prob, Vern9(), EnsembleSerial(); trajectories);
+
+f = plot_traj(sols; azimuth=1.08π, elevation=pi/16)

Clearly, test particle tracing in MHD parallel shocks fails to recover physics. MHD parallel shocks are essentially hydrodynamic shocks where magnetic field plays no role. Due to the lack of collision and other diffusion processes, we are unable to capture the correct microscopic scenario here.

This page was generated using DemoCards.jl and Literate.jl.

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Coil Tokamak

Source code compat Author Update time

This example shows how to trace protons in a stationary magnetic field that corresponds to a Tokamak represented by a circle of coils. A excellent introduction video to Tokamak can be found here in Mandarin.

using TestParticle
+using TestParticle: getB_tokamak_coil
+using OrdinaryDiffEq
+using StaticArrays
+using Statistics: mean
+using Printf
+using CairoMakie
+
+### Obtain field
+
+# Magnetic bottle parameters in SI units
+const ICoil = 80. # current in the coil
+const N = 15000 # number of windings
+const IPlasma = 1e6 # current in the plasma
+const a = 1.5 # radius of each coil
+const b = 0.8 # radius of central region
+
+function getB(xu)
+   SVector{3}(getB_tokamak_coil(xu[1], xu[2], xu[3], a, b, ICoil*N, IPlasma))
+end
+
+function getE(xu)
+   SA[0.0, 0.0, 0.0]
+end
+
+### Initialize particles
+m = TestParticle.mᵢ
+q = TestParticle.qᵢ
+c = TestParticle.c
+
+# initial velocity, [m/s]
+v₀ = [-0.1, -0.15, 0.0] .* c
+# initial position, [m]
+r₀ = [2.3, 0.0, 0.0]
+stateinit = [r₀..., v₀...]
+
+param = prepare(getE, getB; species=Proton)
+tspan = (0.0, 1e-6)
+
+prob = ODEProblem(trace!, stateinit, tspan, param)
+
+@printf "Speed = %6.4f %s\n" √(v₀[1]^2+v₀[2]^2+v₀[3]^2)/c*100 "% speed of light"
+@printf "Energy = %6.4f MeV\n" (1/√(1-(v₀[1]/c)^2-(v₀[2]/c)^2-(v₀[3]/c)^2)-1)*m*c^2/abs(q)/1e6
+
+# Default Tsit5() alone does not work in this case! You need to set a maximum
+# timestep to maintain stability, or choose a different algorithm as well.
+# The sample figure in the gallery is generated with AB3() and dt=2e-11.
+sol = solve(prob, Tsit5(); dt=2e-11, save_idxs=[1,2,3])
+
+### Visualization
+
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "Particle trajectory in Tokamak",
+   xlabel = "x [m]",
+   ylabel = "y [m]",
+   zlabel = "z [m]",
+   aspect = :data,
+)
+
+plot!(sol, label="proton")
+
+# Plot coils
+θ = range(0, 2π, length=201)
+y = a * cos.(θ)
+z = a * sin.(θ)
+for i in 0:17
+   ϕ = i*π/9
+   plot!(y*sin(ϕ) .+ (a+b)*sin(ϕ), y*cos(ϕ) .+ (a+b)*cos(ϕ), z, color=(:red, 0.3))
+end
+
+# Plot Tokamak
+u = range(0, 2π, length=100)
+v = range(0, 2π, length=100)
+
+U = [y for _ in u, y in v]
+V = [x for x in u, _ in v]
+
+X = @. (a + b + (a - 0.05)*cos(U)) * cos(V)
+Y = @. (a + b + (a - 0.05)*cos(U)) * sin(V)
+Z = @. (a - 0.05) * sin(U)
+
+wireframe!(ax, X, Y, Z, color=(:blue, 0.1), linewidth=0.5, transparency=true)
+

This page was generated using DemoCards.jl and Literate.jl.

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dzwOhXWh!;62Ev@ZP01_Sr*%;C4%PPT{{zx%KLY>& literal 0 HcmV?d00001 diff --git a/v0.10.1/examples/basics/demo_ExB_drift.jl b/v0.10.1/examples/basics/demo_ExB_drift.jl new file mode 100644 index 00000000..cff9c8b2 --- /dev/null +++ b/v0.10.1/examples/basics/demo_ExB_drift.jl @@ -0,0 +1,66 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: get_gc +using OrdinaryDiffEq +using StaticArrays +using LinearAlgebra: ⋅, ×, normalize +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +# Analytic EM fields +uniform_B(x) = SA[0, 0, 1e-8] +uniform_E(x) = SA[1e-9, 0, 0] + +# Trace the orbit of the guiding center +function trace_gc!(dx, x, p, t) + _, E, B, sol = p + xu = sol(t) + Bv = B(x) + b = normalize(Bv) + v_par = @views (xu[4:6] ⋅ b) .* b + B2 = sum(Bv.^2) + dx[1:3] = (E(x) × Bv) / B2 + v_par +end +# Initial condition +stateinit = let x0 = [1.0, 0.0, 0.0], v0 = [0.0, 1.0, 0.1] + [x0..., v0...] +end +# Time span +tspan = (0, 20) + +# Trace particle +param = prepare(uniform_E, uniform_B, species=Proton) +prob = ODEProblem(trace!, stateinit, tspan, param) +sol = solve(prob, Vern9()) + +# Functions for obtaining the guiding center from actual trajectory +gc = get_gc(param) +gc_x0 = gc(stateinit) +prob_gc = ODEProblem(trace_gc!, gc_x0, tspan, (param..., sol)) +sol_gc = solve(prob_gc, Vern9(); save_idxs=[1,2,3]); + +# Numeric and analytic results +f = Figure(fontsize=18) +ax = Axis3(f[1, 1], + title = "ExB Drift", + xlabel = "x [m]", + ylabel = "y [m]", + zlabel = "z [m]", + aspect = :data, + azimuth = 0.3π, +) + +gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,) + +lines!(ax, sol, idxs=(1, 2, 3)) +lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6)) +lines!(ax, sol_gc, idxs=(1, 2, 3)) + +for i in 1:3 + ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed! + ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i] +end + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/basics/demo_ExB_drift/index.html b/v0.10.1/examples/basics/demo_ExB_drift/index.html new file mode 100644 index 00000000..70d1d9a1 --- /dev/null +++ b/v0.10.1/examples/basics/demo_ExB_drift/index.html @@ -0,0 +1,62 @@ + +E×B drift · TestParticle.jl

E×B drift

Source code compat Author Author Update time

This example demonstrates a single proton motion under uniform E and B fields. The electric field is parallel to the magnetic field in the z-direction, so the motion consists of a cyclotron gyration and an acceleration along z. On top of that, particles also exhibit an ExB drift in the direction perpendicular to both E and B field. Note that in this simple ExB drift case, the analytic and numeric guiding centers overlaps. More theoretical details can be found in ExB Drift.

using TestParticle
+using TestParticle: get_gc
+using OrdinaryDiffEq
+using StaticArrays
+using LinearAlgebra: ⋅, ×, normalize
+using CairoMakie
+
+# Analytic EM fields
+uniform_B(x) = SA[0, 0, 1e-8]
+uniform_E(x) = SA[1e-9, 0, 0]
+
+# Trace the orbit of the guiding center
+function trace_gc!(dx, x, p, t)
+    _, E, B, sol = p
+    xu = sol(t)
+    Bv = B(x)
+    b = normalize(Bv)
+    v_par = @views (xu[4:6] ⋅ b) .* b
+    B2 = sum(Bv.^2)
+    dx[1:3] = (E(x) × Bv) / B2 + v_par
+end
+# Initial condition
+stateinit = let x0 = [1.0, 0.0, 0.0], v0 = [0.0, 1.0, 0.1]
+    [x0..., v0...]
+end
+# Time span
+tspan = (0, 20)
+
+# Trace particle
+param = prepare(uniform_E, uniform_B, species=Proton)
+prob = ODEProblem(trace!, stateinit, tspan, param)
+sol = solve(prob, Vern9())
+
+# Functions for obtaining the guiding center from actual trajectory
+gc = get_gc(param)
+gc_x0 = gc(stateinit)
+prob_gc = ODEProblem(trace_gc!, gc_x0, tspan, (param..., sol))
+sol_gc = solve(prob_gc, Vern9(); save_idxs=[1,2,3]);
+
+# Numeric and analytic results
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "ExB Drift",
+   xlabel = "x [m]",
+   ylabel = "y [m]",
+   zlabel = "z [m]",
+   aspect = :data,
+   azimuth = 0.3π,
+)
+
+gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,)
+
+lines!(ax, sol, idxs=(1, 2, 3))
+lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6))
+lines!(ax, sol_gc, idxs=(1, 2, 3))
+
+for i in 1:3
+    ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+    ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i]
+end
+

This page was generated using DemoCards.jl and Literate.jl.

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HcmV?d00001 diff --git a/v0.10.1/examples/basics/demo_FLR.jl b/v0.10.1/examples/basics/demo_FLR.jl new file mode 100644 index 00000000..a5fef9bc --- /dev/null +++ b/v0.10.1/examples/basics/demo_FLR.jl @@ -0,0 +1,78 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: get_gc +using OrdinaryDiffEq +using StaticArrays +using LinearAlgebra: ×, ⋅, norm, normalize +using Tensors: laplace +import Tensors: Vec as Vec3 +# using SpecialFunctions +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +function uniform_B(x) + return SA[0, 0, 1e-8] +end + +function nonuniform_E(x) + return SA[1e-9*cos(0.3*x[1]), 0, 0] +end + +# trace the orbit of the guiding center +function trace_gc!(dx, x, p, t) + q2m, E, B, sol = p + xu = sol(t) + xp = @view xu[1:3] + Bv = B(xp) + b = normalize(Bv) + v_par = (xu[4:6] ⋅ b) .* b # (v ⋅ b)b + v_perp = xu[4:6] - v_par + r4 = (norm(v_perp) / q2m / norm(Bv))^2 / 4 + EB(x) = (E(x) × B(x)) / norm(B(x))^2 + # dx[1:3] = EB(xp) + v_par + dx[1:3] = EB(x) + r4*laplace.(EB, Vec3(x...)) + v_par + + # more accurate + # dx[1:3] = besselj0(0.3*norm(v_perp)/q2m/norm(Bv))*EB(x) + v_par +end + +# Initial condition +stateinit = let x0 = [1.0, 0, 0], v0 = [0.0, 1.0, 0.1] + [x0..., v0...] +end +# Time span +tspan = (0, 20) +param = prepare(nonuniform_E, uniform_B, species=Proton) +prob = ODEProblem(trace!, stateinit, tspan, param) +sol = solve(prob, Vern9()) + +gc = get_gc(param) +gc_x0 = gc(stateinit) +prob_gc = ODEProblem(trace_gc!, gc_x0, tspan, (param..., sol)) +sol_gc = solve(prob_gc, Tsit5(); save_idxs=[1,2,3]) + +# numeric result and analytic result +f = Figure(fontsize=18) +ax = Axis3(f[1, 1], + title = "Finite Larmor Radius Effect", + xlabel = "x [m]", + ylabel = "y [m]", + zlabel = "z [m]", + aspect = :data, + azimuth = 0.3π, +) + +gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,) + +lines!(ax, sol, idxs=(1, 2, 3)) +lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6)) +lines!(ax, sol_gc, idxs=(1, 2, 3)) + +for i in 1:3 + ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed! + ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i] +end + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/basics/demo_FLR/index.html b/v0.10.1/examples/basics/demo_FLR/index.html new file mode 100644 index 00000000..e0666451 --- /dev/null +++ b/v0.10.1/examples/basics/demo_FLR/index.html @@ -0,0 +1,74 @@ + +Finite-Larmor-Radius effect · TestParticle.jl

Finite-Larmor-Radius effect

Source code compat Author Update time

More theoretical details can be found in Non-uniform E Field.

using TestParticle
+using TestParticle: get_gc
+using OrdinaryDiffEq
+using StaticArrays
+using LinearAlgebra: ×, ⋅, norm, normalize
+using Tensors: laplace
+import Tensors: Vec as Vec3
+# using SpecialFunctions
+using CairoMakie
+
+function uniform_B(x)
+    return SA[0, 0, 1e-8]
+end
+
+function nonuniform_E(x)
+    return SA[1e-9*cos(0.3*x[1]), 0, 0]
+end
+
+# trace the orbit of the guiding center
+function trace_gc!(dx, x, p, t)
+    q2m, E, B, sol = p
+    xu = sol(t)
+    xp = @view xu[1:3]
+    Bv = B(xp)
+    b = normalize(Bv)
+    v_par = (xu[4:6] ⋅ b) .* b # (v ⋅ b)b
+    v_perp = xu[4:6] - v_par
+    r4 = (norm(v_perp) / q2m / norm(Bv))^2 / 4
+    EB(x) = (E(x) × B(x)) / norm(B(x))^2
+    # dx[1:3] = EB(xp) + v_par
+    dx[1:3] = EB(x) + r4*laplace.(EB, Vec3(x...)) + v_par
+
+    # more accurate
+    # dx[1:3] = besselj0(0.3*norm(v_perp)/q2m/norm(Bv))*EB(x) + v_par
+end
+
+# Initial condition
+stateinit = let x0 = [1.0, 0, 0], v0 = [0.0, 1.0, 0.1]
+    [x0..., v0...]
+end
+# Time span
+tspan = (0, 20)
+param = prepare(nonuniform_E, uniform_B, species=Proton)
+prob = ODEProblem(trace!, stateinit, tspan, param)
+sol = solve(prob, Vern9())
+
+gc = get_gc(param)
+gc_x0 = gc(stateinit)
+prob_gc = ODEProblem(trace_gc!, gc_x0, tspan, (param..., sol))
+sol_gc = solve(prob_gc, Tsit5(); save_idxs=[1,2,3])
+
+# numeric result and analytic result
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "Finite Larmor Radius Effect",
+   xlabel = "x [m]",
+   ylabel = "y [m]",
+   zlabel = "z [m]",
+   aspect = :data,
+   azimuth = 0.3π,
+)
+
+gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,)
+
+lines!(ax, sol, idxs=(1, 2, 3))
+lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6))
+lines!(ax, sol_gc, idxs=(1, 2, 3))
+
+for i in 1:3
+    ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+    ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i]
+end
+

This page was generated using DemoCards.jl and Literate.jl.

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--- /dev/null +++ b/v0.10.1/examples/basics/demo_boris.jl @@ -0,0 +1,98 @@ +import DisplayAs #hide +using TestParticle +using StaticArrays +using OrdinaryDiffEq +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +function plot_trajectory(sol_boris, sol1, sol2) + f = Figure(size=(700, 600), fontsize=18) + ax = Axis(f[1, 1], aspect=1, limits = (-3, 1, -2, 2), + xlabel = "X", + ylabel = "Y") + idxs = (1, 2) + ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed! + l0 = lines!(ax, sol_boris[1]; idxs, linewidth=2, label="Boris") + l1 = lines!(ax, sol1; idxs, linewidth=2, linestyle=:dashdot, label="Tsit5 fixed") + l2 = linesegments!(ax, sol2; idxs, linewidth=2, linestyle=:dot, label="Tsit5 adaptive") + + ax.scene.plots[1].linewidth = 2 + ax.scene.plots[5].linewidth = 2 + + ax.scene.plots[3].color = Makie.wong_colors()[2] + ax.scene.plots[5].color = Makie.wong_colors()[3] + + scale!(ax.scene.plots[1], invrL, invrL) + scale!(ax.scene.plots[3], invrL, invrL) + scale!(ax.scene.plots[5], invrL, invrL) + + axislegend(position=:rt, framevisible=false) + + f +end + +const Bmag = 0.01 +uniform_B(x) = SA[0.0, 0.0, Bmag] +uniform_E(x) = SA[0.0, 0.0, 0.0] + +x0 = [0.0, 0.0, 0.0] +v0 = [0.0, 1e5, 0.0] +stateinit = [x0..., v0...] + +param = prepare(uniform_E, uniform_B, species=Electron) + +# Reference parameters +const tperiod = 2π / (abs(param[1]) * hypot(param[3]([0.0,0.0,0.0], 0.0)...)) +const rL = hypot(v0...) / (abs(param[1]) * Bmag) +const invrL = 1 / rL; + +tspan = (0.0, tperiod) +dt = tperiod / 4 + +prob = TraceProblem(stateinit, tspan, param) + +sol_boris = TestParticle.solve(prob; dt, savestepinterval=1); + +prob = ODEProblem(trace!, stateinit, tspan, param) +sol1 = solve(prob, Tsit5(); adaptive=false, dt, dense=false, saveat=dt); +sol2 = solve(prob, Tsit5()); + +# Visualization +f = plot_trajectory(sol_boris, sol1, sol2) +f = DisplayAs.PNG(f) #hide + +dt = tperiod / 8 + +prob = TraceProblem(stateinit, tspan, param) + +sol_boris = TestParticle.solve(prob; dt, savestepinterval=1); + +prob = ODEProblem(trace!, stateinit, tspan, param) +sol1 = solve(prob, Tsit5(); adaptive=false, dt, dense=false, saveat=dt); + +# Visualization +f = plot_trajectory(sol_boris, sol1, sol2) +f = DisplayAs.PNG(f) #hide + +tspan = (0.0, 200*tperiod) +dt = tperiod / 12 + +prob_boris = TraceProblem(stateinit, tspan, param) +prob = ODEProblem(trace!, stateinit, tspan, param) + +sol_boris = TestParticle.solve(prob_boris; dt, savestepinterval=10); +sol1 = solve(prob, Tsit5(); adaptive=false, dt, dense=false, saveat=dt); +sol2 = solve(prob, Tsit5()); + +# Visualization +f = plot_trajectory(sol_boris, sol1, sol2) +f = DisplayAs.PNG(f) #hide + +@time sol_boris = TestParticle.solve(prob_boris; dt, savestepinterval=10)[1]; +@time sol1 = solve(prob, Tsit5(); adaptive=false, dt, dense=false, saveat=dt); +@time sol2 = solve(prob, Tsit5()); + +t = tspan[2] / 2 +sol_boris(t) + +# This file was generated using Literate.jl, 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z?WJ@bk)1dr%b3R#r+L$YLg<(cv%~YYb6=xJuF#QjtfwuRk1;}kL_R_zp2k@0>AIa6 z`~C>$_!srsMm16#1gZE$5-&mN6!nCO9%M95ylWpfDRfr5c@Ay=dNV^58~c5rABMua z6-Kc}HN6wVfZV{7{Ppo579XD}3UUa#qkp8ohGeVNmKsch1w1=(?AWlt2cw}Ul$*f; zDKqdEtzNrU@8Mq!IeksW@fy9q{<1Fzkcz1 zjR`uEZ{L6ajsJ!{%9AsrJdhmX<8SZnknJu%JWs+BIvQH#Bo71&Mh@sH5-?T>g1C^!pnj z!%I@Cm%BSNL+Cg}usI$dU;OjfPii$cKkMSfn%de-cNCGBws8s4o@gKw*nuL9i+MY# zaB^E# ∂B_θ/∂θ = 0 + θ = atan(x[3]/(x[1]+3)) + r = hypot(x[1]+3, x[3]) + return SA[-1e-7*sin(θ)/r, 0, 1e-7*cos(θ)/r] +end + +function zero_E(x) + return SA[0, 0, 0] +end + +abs_B(x) = norm(curved_B(x)) # |B| + +# Trace the orbit of the guiding center using analytical drifts +function trace_gc!(dx, x, p, t) + q2m, E, B, sol = p + xu = sol(t) + gradient_B = gradient(abs_B, x) # ∇|B| + Bv = B(x) + b = normalize(Bv) + v_par = (xu[4:6] ⋅ b).*b # (v⋅b)b + v_perp = xu[4:6] - v_par + Ω = q2m*norm(Bv) + κ = jacobian(B, x)*Bv # B⋅∇B + # v⟂^2*(B×∇|B|)/(2*Ω*B^2) + v∥^2*(B×(B⋅∇B))/(Ω*B^3) + (E×B)/B^2 + v∥ + dx[1:3] = norm(v_perp)^2*(Bv × gradient_B)/(2*Ω*norm(Bv)^2) + + norm(v_par)^2*(Bv × κ)/Ω/norm(Bv)^3 + (E(x) × Bv)/norm(Bv)^2 + v_par +end + +# Initial conditions +stateinit = let x0 = [1.0, 0.0, 0.0], v0 = [0.0, 1.0, 0.1] + [x0..., v0...] +end +# Time span +tspan = (0, 40) +# Trace particle +param = prepare(zero_E, curved_B, species=Proton) +prob = ODEProblem(trace!, stateinit, tspan, param) +sol = solve(prob, Vern9()) +# Functions for obtaining the guiding center from actual trajectory +gc = get_gc(param) +gc_x0 = gc(stateinit) +prob_gc = ODEProblem(trace_gc!, gc_x0, tspan, (param..., sol)) +sol_gc = solve(prob_gc, Tsit5(); save_idxs=[1,2,3]) + +# Numeric and analytic results +f = Figure(fontsize=18) +ax = Axis3(f[1, 1], + title = "Curvature Drift", + xlabel = "x [m]", + ylabel = "y [m]", + zlabel = "z [m]", + aspect = :data, + azimuth = 0.3π, +) + +gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,) + +lines!(ax, sol, idxs=(1, 2, 3)) +lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6)) +lines!(ax, sol_gc, idxs=(1, 2, 3)) + +for i in 1:3 + ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed! + ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i] +end + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/basics/demo_curvature_B/index.html b/v0.10.1/examples/basics/demo_curvature_B/index.html new file mode 100644 index 00000000..a168f06a --- /dev/null +++ b/v0.10.1/examples/basics/demo_curvature_B/index.html @@ -0,0 +1,77 @@ + +Curl-B drift · TestParticle.jl diff --git a/v0.10.1/examples/basics/demo_dimensionless-6.png b/v0.10.1/examples/basics/demo_dimensionless-6.png new file mode 100644 index 0000000000000000000000000000000000000000..45a2fcba3ea6130614bf359cc2fa0259849e17c6 GIT binary patch literal 34848 zcmeFZ1y|JD+W`8*Kolt{X%G=4rI8i{R9d8lZWv%F=~fU00gIMKy1PM2q>+{e>F$QR zXU6lMbJzN>`w6aRorTWqc=prDJ0%4foEzjfAPB;F`sDF*2)dFFL6`loE`d)(@$_=Q 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+using TestParticle +using TestParticle: qᵢ, mᵢ +using OrdinaryDiffEq + +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +# Number of cells for the field along each dimension +nx, ny, nz = 4, 6, 8 +# Unit conversion factors between SI and dimensionless units +B₀ = 10e-9 # [T] +Ω = abs(qᵢ) * B₀ / mᵢ # [1/s] +t₀ = 1 / Ω # [s] +U₀ = 1.0 # [m/s] +l₀ = U₀ * t₀ # [m] +E₀ = U₀*B₀ # [V/m] +# All quantities are in dimensionless units +x = range(-10, 10, length=nx) # [l₀] +y = range(-10, 10, length=ny) # [l₀] +z = range(-10, 10, length=nz) # [l₀] + +B = fill(0.0, 3, nx, ny, nz) # [B₀] +B[3,:,:,:] .= 1.0 +E = fill(0.0, 3, nx, ny, nz) # [E₀] + +param = prepare(x, y, z, E, B; species=User) + +# Initial condition +stateinit = let + x0 = [0.0, 0.0, 0.0] # initial position [l₀] + u0 = [4.0, 0.0, 0.0] # initial velocity [v₀] + [x0..., u0...] +end +# Time span +tspan = (0.0, π) # half gyroperiod + +prob = ODEProblem(trace_normalized!, stateinit, tspan, param) +sol = solve(prob, Vern9()) + +### Visualization +f = Figure(fontsize = 18) +ax = Axis(f[1, 1], + title = "Proton trajectory", + xlabel = "X", + ylabel = "Y", + limits = (-4.1, 4.1, -8.1, 0.1), + aspect = DataAspect() +) + +lines!(ax, sol, vars=(1,2)) + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/basics/demo_dimensionless/index.html b/v0.10.1/examples/basics/demo_dimensionless/index.html new file mode 100644 index 00000000..a8eb4a38 --- /dev/null +++ b/v0.10.1/examples/basics/demo_dimensionless/index.html @@ -0,0 +1,51 @@ + +Dimensionless Units · TestParticle.jl

Dimensionless Units

Source code compat Author Update time

This example shows how to trace charged particles in dimensionless units. After normalization, $q=1, B=1, m=1$ so that the gyroradius $r_L = mv_\perp/qB = v_\perp$. All the quantities are given in dimensionless units. If the magnetic field is homogeneous and the initial perpendicular velocity is 4, then the gyroradius is 4. To convert them to the original units, $v_\perp = 4*U_0$ and $r_L = 4*l_0$. Check Demo: single tracing with additional diagnostics for explaining the unit conversion.

Tracing in dimensionless units is beneficial for many scenarios. For example, MHD simulations do not have intrinsic scales. Therefore, we can do dimensionless particle tracing in MHD fields, and then convert to any scale we would like.

Now let's demonstrate this with trace_normalized!.

using TestParticle
+using TestParticle: qᵢ, mᵢ
+using OrdinaryDiffEq
+
+using CairoMakie
+
+# Number of cells for the field along each dimension
+nx, ny, nz = 4, 6, 8
+# Unit conversion factors between SI and dimensionless units
+B₀ = 10e-9            # [T]
+Ω = abs(qᵢ) * B₀ / mᵢ # [1/s]
+t₀ = 1 / Ω            # [s]
+U₀ = 1.0              # [m/s]
+l₀ = U₀ * t₀          # [m]
+E₀ = U₀*B₀            # [V/m]
+# All quantities are in dimensionless units
+x = range(-10, 10, length=nx) # [l₀]
+y = range(-10, 10, length=ny) # [l₀]
+z = range(-10, 10, length=nz) # [l₀]
+
+B = fill(0.0, 3, nx, ny, nz) # [B₀]
+B[3,:,:,:] .= 1.0
+E = fill(0.0, 3, nx, ny, nz) # [E₀]
+
+param = prepare(x, y, z, E, B; species=User)
+
+# Initial condition
+stateinit = let
+   x0 = [0.0, 0.0, 0.0] # initial position [l₀]
+   u0 = [4.0, 0.0, 0.0] # initial velocity [v₀]
+   [x0..., u0...]
+end
+# Time span
+tspan = (0.0, π) # half gyroperiod
+
+prob = ODEProblem(trace_normalized!, stateinit, tspan, param)
+sol = solve(prob, Vern9())
+
+### Visualization
+f = Figure(fontsize = 18)
+ax = Axis(f[1, 1],
+   title = "Proton trajectory",
+   xlabel = "X",
+   ylabel = "Y",
+   limits = (-4.1, 4.1, -8.1, 0.1),
+   aspect = DataAspect()
+)
+
+lines!(ax, sol, vars=(1,2))
+

This page was generated using DemoCards.jl and Literate.jl.

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    Dimensionless and Dimensional Tracing

    Source code compat Author Update time

    This example shows how to trace charged particles in both dimensional and dimensionless units. We first solve the Lorentz equation in SI units, and then convert the quantities to normalized units and solve it again in dimensionless units.

    The Lorentz equation in SI units is written as

    \[\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \frac{q}{m}\left( \mathbf{v}\times\mathbf{B} + \mathbf{E} \right)\]

    It can be normalized to

    \[\frac{\mathrm{d}\mathbf{v}^\prime}{\mathrm{d}t^\prime} = \mathbf{v}^\prime\times\mathbf{B}^\prime + \mathbf{E}^\prime\]

    with the following transformation

    \[\begin{aligned} +\mathbf{v} &= \mathbf{v}^\prime V_0 \\ +t &= t^\prime t_0 = t^\prime \Omega^{-1} = t^\prime \frac{m}{qB_0} \\ +\mathbf{B} &= \mathbf{B}^\prime B_0 \\ +\mathbf{E} &= \mathbf{E}^\prime E_0 = \mathbf{E}^\prime V_0 B_0 +\end{aligned}\]

    where all the coefficients with subscript 0 are expressed in SI units. All the variables with a prime are written in the dimensionless units.

    using TestParticle
+using TestParticle: c, qᵢ, mᵢ
+using OrdinaryDiffEq
+using StaticArrays
+using CairoMakie
+
+# Unit conversion factors between SI and dimensionless units
+const B₀ = 1e-8             # [T]
+const U₀ = c                # [m/s]
+const E₀ = U₀*B₀            # [V/m]
+const Ω = abs(qᵢ) * B₀ / mᵢ # [1/s]
+const t₀ = 1 / Ω            # [s]
+const l₀ = U₀ * t₀          # [m]
+# Electric field magnitude in SI units
+const Emag = 1e-8           # [V/m]
+### Solving in SI units
+B(x) = SA[0, 0, B₀]
+E(x) = SA[Emag, 0.0, 0.0]
+
+# Initial conditions
+x0 = [0.0, 0.0, 0.0] # [m]
+v0 = [0.0, 0.01c, 0.0] # [m/s]
+stateinit1 = [x0..., v0...]
+tspan1 = (0, 2π*t₀) # [s]
+
+param1 = prepare(E, B, species=Proton)
+prob1 = ODEProblem(trace!, stateinit1, tspan1, param1)
+sol1 = solve(prob1, Vern9(); reltol=1e-4, abstol=1e-6)
+
+### Solving in dimensionless units
+B_normalize(x) = SA[0, 0, B₀/B₀]
+E_normalize(x) = SA[Emag/E₀, 0.0, 0.0]
+# For full EM problems, the normalization of E and B should be done separately.
+param2 = prepare(E_normalize, B_normalize; species=User)
+# Scale initial conditions by the conversion factors
+x0 ./= l₀
+v0 ./= U₀
+tspan2 = (0, 2π)
+stateinit2 = [x0..., v0...]
+
+prob2 = ODEProblem(trace_normalized!, stateinit2, tspan2, param2)
+sol2 = solve(prob2, Vern9(); reltol=1e-4, abstol=1e-6)
+
+### Visualization
+f = Figure(fontsize=18)
+ax = Axis(f[1, 1],
+    xlabel = "x",
+    ylabel = "y",
+    aspect = DataAspect(),
+)
+
+lines!(ax, sol1, idxs=(1, 2))
+# Interpolate dimensionless solutions and map back to SI units
+xp, yp = let trange = range(tspan2..., length=40)
+   sol2.(trange, idxs=1) .* l₀, sol2.(trange, idxs=2) .* l₀
+end
+lines!(ax, xp, yp, linestyle=:dashdot, linewidth=5)
+

    We see that the results are almost identical, with only floating point numerical errors. Tracing in dimensionless units usually allows larger timesteps, which leads to faster computation.

    This page was generated using DemoCards.jl and Literate.jl.

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literal 0 HcmV?d00001 diff --git a/v0.10.1/examples/basics/demo_dimensionless_periodic.jl b/v0.10.1/examples/basics/demo_dimensionless_periodic.jl new file mode 100644 index 00000000..39fdfadf --- /dev/null +++ b/v0.10.1/examples/basics/demo_dimensionless_periodic.jl @@ -0,0 +1,56 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: qᵢ, mᵢ +using OrdinaryDiffEq +using StaticArrays +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +# Number of cells for the field along each dimension +nx, ny = 4, 6 +# Unit conversion factors between SI and dimensionless units +B₀ = 10e-9 # [T] +Ω = abs(qᵢ) * B₀ / mᵢ # [1/s] +t₀ = 1 / Ω # [s] +U₀ = 1.0 # [m/s] +l₀ = U₀ * t₀ # [m] +E₀ = U₀*B₀ # [V/m] + +x = range(-10, 10, length=nx) # [l₀] +y = range(-10, 10, length=ny) # [l₀] + +B = fill(0.0, 3, nx, ny) # [B₀] +B[3,:,:] .= 1.0 + +E(x) = SA[0.0, 0.0, 0.0] # [E₀] + +# If bc == 1, we set a NaN value outside the domain (default); +# If bc == 2, we set periodic boundary conditions. +param = prepare(x, y, E, B; species=User, bc=2); + +# Initial conditions +stateinit = let + x0 = [0.0, 0.0, 0.0] # initial position [l₀] + u0 = [10.0, 0.0, 0.0] # initial velocity [v₀] + [x0..., u0...] +end +# Time span +tspan = (0.0, 1.5π) # 3/4 gyroperiod + +prob = ODEProblem(trace_normalized!, stateinit, tspan, param) +sol = solve(prob, Vern9()); + +f = Figure(fontsize = 18) +ax = Axis(f[1, 1], + title = "Proton trajectory", + xlabel = "X", + ylabel = "Y", + limits = (-10.1, 10.1, -20.1, 0.1), + aspect = DataAspect() +) + +lines!(ax, sol, vars=(1,2)) + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/basics/demo_dimensionless_periodic/index.html b/v0.10.1/examples/basics/demo_dimensionless_periodic/index.html new file mode 100644 index 00000000..b5be180e --- /dev/null +++ b/v0.10.1/examples/basics/demo_dimensionless_periodic/index.html @@ -0,0 +1,48 @@ + +Tracing with Dimensionless Units and Periodic Boundary · TestParticle.jl

    Tracing with Dimensionless Units and Periodic Boundary

    Source code compat Author Update time

    This example shows how to trace charged particles in dimensionless units and EM fields with periodic boundaries in a 2D spatial domain. For details about dimensionless units, please check Demo: dimensionless tracing.

    Now let's demonstrate this with trace_normalized!.

    using TestParticle
+using TestParticle: qᵢ, mᵢ
+using OrdinaryDiffEq
+using StaticArrays
+using CairoMakie
+
+# Number of cells for the field along each dimension
+nx, ny = 4, 6
+# Unit conversion factors between SI and dimensionless units
+B₀ = 10e-9            # [T]
+Ω = abs(qᵢ) * B₀ / mᵢ # [1/s]
+t₀ = 1 / Ω            # [s]
+U₀ = 1.0              # [m/s]
+l₀ = U₀ * t₀          # [m]
+E₀ = U₀*B₀            # [V/m]
+
+x = range(-10, 10, length=nx) # [l₀]
+y = range(-10, 10, length=ny) # [l₀]
+
+B = fill(0.0, 3, nx, ny) # [B₀]
+B[3,:,:] .= 1.0
+
+E(x) = SA[0.0, 0.0, 0.0] # [E₀]
+
+# If bc == 1, we set a NaN value outside the domain (default);
+# If bc == 2, we set periodic boundary conditions.
+param = prepare(x, y, E, B; species=User, bc=2);

    Note that we set a radius of 10, so the trajectory extent from -20 to 0 in y, which is beyond the original y range.

    # Initial conditions
+stateinit = let
+   x0 = [0.0, 0.0, 0.0] # initial position [l₀]
+   u0 = [10.0, 0.0, 0.0] # initial velocity [v₀]
+   [x0..., u0...]
+end
+# Time span
+tspan = (0.0, 1.5π) # 3/4 gyroperiod
+
+prob = ODEProblem(trace_normalized!, stateinit, tspan, param)
+sol = solve(prob, Vern9());

    Visualization

    f = Figure(fontsize = 18)
+ax = Axis(f[1, 1],
+   title = "Proton trajectory",
+   xlabel = "X",
+   ylabel = "Y",
+   limits = (-10.1, 10.1, -20.1, 0.1),
+   aspect = DataAspect()
+)
+
+lines!(ax, sol, vars=(1,2))
+

    This page was generated using DemoCards.jl and Literate.jl.

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zr!rplylFMT(XqCCc3y1b%*cZ(q%^t56Xy23ej0`b{(av9bXT|${Jloh#43IFTaO+J zY75t-`})w{|8ZV;Vp1bjmU*h@&^|rzRWmn#$HZAZDS%qsP7ixY7@W|I#>4kAJvgH4 byE?l(FWWxA%&J00co+-kE!B +Proton and electron in a static EM field · TestParticle.jl

    Proton and electron in a static EM field

    Source code compat Author Author Update time

    This example demonstrates tracing a single proton and electron motion under a uniform B field in real physical parameters. The E field is assumed to be zero such that there is no particle acceleration. Due to the fact that $m_p / m_e \doteq 1836$, the proton gyro-radius is 1800 times larger than the electron, if they start with the same velocity as in this case. In more common cases we would compare electrons and protons with the same energy, and their gyro-radii differ by a factor of $\sqrt{m_p/m_e} \sim 40$.

    using TestParticle
+using OrdinaryDiffEq
+using CairoMakie
+
+### Initialize grid and field
+x = range(-10, 10, length=15)
+y = range(-10, 10, length=20)
+z = range(-10, 10, length=25)
+
+B = fill(0.0, 3, length(x), length(y), length(z)) # [T]
+E = fill(0.0, 3, length(x), length(y), length(z)) # [V/m]
+B[3,:,:,:] .= 1e-11
+E[3,:,:,:] .= 5e-13
+
+### Initialize particles
+stateinit = let
+   x0 = [0.0, 0.0, 0.0] # initial position, [m]
+   u0 = [1.0, 0.0, 0.0] # initial velocity, [m/s]
+   [x0..., u0...]
+end
+param_electron = prepare(x, y, z, E, B, species=Electron)
+tspan_electron = (0.0, 15.0)
+
+param_proton = prepare(x, y, z, E, B, species=Proton)
+tspan_proton = (0.0, 10.0)
+
+### Solve for the trajectories
+prob_e = ODEProblem(trace!, stateinit, tspan_electron, param_electron)
+prob_p = ODEProblem(trace!, stateinit, tspan_proton, param_proton)
+
+sol_e = solve(prob_e, Vern9())
+sol_p = solve(prob_p, Vern9())
+
+### Visualization
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "Electron and Ion Trajectories",
+   xlabel = "x [m]",
+   ylabel = "y [m]",
+   zlabel = "z [m]",
+   aspect = :data,
+)
+plot!(sol_e, idxs=(1, 2, 3), color=:tomato, label="electron")
+plot!(sol_p, idxs=(1, 2, 3), color=:deepskyblue3, label="proton")
+
+##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+ax.scene.plots[9+2*1-1].color = :tomato
+ax.scene.plots[9+2*2-1].color = :deepskyblue3
+
+axislegend()
+

    This page was generated using DemoCards.jl and Literate.jl.

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p::TestParticle.TPTuple, t) + dx .= v +end + +"f1" +function lorentz!(dv, v, x, p::TestParticle.TPTuple, t) + q2m, E, B = p + dv .= q2m*(E(x, t) + v × (B(x, t))) +end + +### Initialize field + +function uniform_B(x) + return SA[0, 0, B₀] +end + +function uniform_E(x) + return SA[E₀, 0.0, 0.0] +end + +function zero_B(x) + return SA[0.0, 0.0, 0.0] +end + +function zero_E(x) + return SA[0.0, 0.0, 0.0] +end + +"Check energy conservation." +E(dx, dy, dz) = 1 // 2 * (dx^2 + dy^2 + dz^2) + +### Initialize particles + +x0 = [0.0, 0, 0] +v0 = [0.0, 1e2, 0.0] +stateinit = [x0..., v0...] +tspan_proton = (0.0, 2000.0); + +param_proton = prepare(zero_E, uniform_B, species=Proton) + +### Solve for the trajectories + +prob_p = DynamicalODEProblem(lorentz!, location!, v0, x0, tspan_proton, param_proton) + +Ωᵢ = TestParticle.qᵢ * B₀ / TestParticle.mᵢ +Tᵢ = 2π / Ωᵢ +println("Number of gyrations: ", tspan_proton[2] / Tᵢ) + +sol = solve(prob_p, ImplicitMidpoint(), dt = Tᵢ/15) + +f = Figure(fontsize=18) +ax = Axis(f[1, 1], + title = "Proton in a uniform B field and zero E field", + xlabel = "x", + ylabel = "y", + aspect = 1, +) + +lines!(ax, sol, idxs=(1, 2)) + +f = DisplayAs.PNG(f) #hide + +param_proton = prepare(uniform_E, zero_B, species=Proton) + +# acceleration, [m/s²] +a = param_proton[1] * E₀ +# predicted final speed, [m/s] +v_final_predict = a * tspan_proton[2] +# predicted travel distance, [m/s] +d_final_predict = 0.5 * tspan_proton[2] * v_final_predict +# predicted energy gain, [eV] +E_predict = E₀ * d_final_predict + +prob_p = DynamicalODEProblem(lorentz!, location!, v0, x0, tspan_proton, param_proton) + +sol = solve(prob_p, Vern6()) + +energy = map(x -> E(x[1:3]...), sol.u) .* TestParticle.mᵢ; + +println("predicted final speed: $v_final_predict [m/s]") #hide + +println("simulated final speed: $(sol.u[end][1]) [m/s]") #hide + +println("predicted travel distance: $d_final_predict [m]") #hide + +println("simulated travel distance: $(sol.u[end][4]) [m]") #hide + +println("predicted energy gain: $E_predict [eV]") #hide + +println("simulated final energy: $(energy[end] / TestParticle.qᵢ) [eV]") #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/basics/demo_energy_conservation/index.html b/v0.10.1/examples/basics/demo_energy_conservation/index.html new file mode 100644 index 00000000..cc0bf912 --- /dev/null +++ b/v0.10.1/examples/basics/demo_energy_conservation/index.html @@ -0,0 +1,90 @@ + +Energy Conservation · TestParticle.jl

    Energy Conservation

    Source code compat Author Update time

    This example demonstrates the energy conservation of a single proton motion in two cases. The first one is under a uniform B field and zero E field. The second on is under a zero B field and uniform E field.

    using TestParticle
+using OrdinaryDiffEq
+using StaticArrays
+using LinearAlgebra: ×
+using CairoMakie
+
+const B₀ = 1e-8 # [T]
+const E₀ = 3e-2 # [V/m]
+
+"f2"
+function location!(dx, v, x, p::TestParticle.TPTuple, t)
+   dx .= v
+end
+
+"f1"
+function lorentz!(dv, v, x, p::TestParticle.TPTuple, t)
+   q2m, E, B = p
+   dv .= q2m*(E(x, t) + v × (B(x, t)))
+end
+
+### Initialize field
+
+function uniform_B(x)
+   return SA[0, 0, B₀]
+end
+
+function uniform_E(x)
+   return SA[E₀, 0.0, 0.0]
+end
+
+function zero_B(x)
+   return SA[0.0, 0.0, 0.0]
+end
+
+function zero_E(x)
+   return SA[0.0, 0.0, 0.0]
+end
+
+"Check energy conservation."
+E(dx, dy, dz) = 1 // 2 * (dx^2 + dy^2 + dz^2)
+
+### Initialize particles
+
+x0 = [0.0, 0, 0]
+v0 = [0.0, 1e2, 0.0]
+stateinit = [x0..., v0...]
+tspan_proton = (0.0, 2000.0);

    Uniform B field and zero E field

    param_proton = prepare(zero_E, uniform_B, species=Proton)
+
+### Solve for the trajectories
+
+prob_p = DynamicalODEProblem(lorentz!, location!, v0, x0, tspan_proton, param_proton)
+
+Ωᵢ = TestParticle.qᵢ * B₀ / TestParticle.mᵢ
+Tᵢ = 2π / Ωᵢ
+println("Number of gyrations: ", tspan_proton[2] / Tᵢ)
+
+sol = solve(prob_p, ImplicitMidpoint(), dt = Tᵢ/15)
+
+f = Figure(fontsize=18)
+ax = Axis(f[1, 1],
+    title = "Proton in a uniform B field and zero E field",
+    xlabel = "x",
+    ylabel = "y",
+    aspect = 1,
+)
+
+lines!(ax, sol, idxs=(1, 2))
+

    Zero B field and uniform E field

    param_proton = prepare(uniform_E, zero_B, species=Proton)
+
+# acceleration, [m/s²]
+a = param_proton[1] * E₀
+# predicted final speed, [m/s]
+v_final_predict = a * tspan_proton[2]
+# predicted travel distance, [m/s]
+d_final_predict = 0.5 * tspan_proton[2] * v_final_predict
+# predicted energy gain, [eV]
+E_predict = E₀ * d_final_predict
+
+prob_p = DynamicalODEProblem(lorentz!, location!, v0, x0, tspan_proton, param_proton)
+
+sol = solve(prob_p, Vern6())
+
+energy = map(x -> E(x[1:3]...), sol.u) .* TestParticle.mᵢ;

    Predicted final speed

    predicted final speed: 5.744086746808526e9 [m/s]
+

    Simulated final speed

    simulated final speed: 5.744086746808613e9 [m/s]
+

    Predicted travel distance

    predicted travel distance: 5.744086746808526e12 [m]
+

    Simulated travel distance

    simulated travel distance: 5.744086746768604e12 [m]
+

    Predicted final energy

    predicted energy gain: 1.723226024042558e11 [eV]
+

    Simulated final energy

    simulated final energy: 1.7232260240426102e11 [eV]
+

    This page was generated using DemoCards.jl and Literate.jl.

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    Grad-B drift

    Source code compat Author Author Update time

    This example demonstrates a single proton motion under a non-uniform B field with gradient ∇B ⊥ B. The orbit of guiding center includes some high order terms, it is different from the formula of magnetic field gradient drift of some textbooks which just preserves the first order term. It is more complex than the simpler ExB drift. More theoretical details can be found in Grad-B Drift, and Fundamentals of Plasma Physics by Paul Bellan.

    using TestParticle
+using TestParticle: get_gc
+using OrdinaryDiffEq
+using StaticArrays
+using LinearAlgebra: ×, ⋅, normalize, norm
+using ForwardDiff: gradient
+using CairoMakie
+
+function grad_B(x)
+    return SA[0, 0, 1e-8 + 1e-9*x[2]]
+end
+
+function uniform_E(x)
+    return SA[1e-9, 0, 0]
+end
+
+abs_B(x) = norm(grad_B(x))
+
+# Trace the orbit of the guiding center using analytical drifts
+function trace_gc!(dx, x, p, t)
+    q2m, E, B, sol = p
+    xu = sol(t)
+    gradient_B = gradient(abs_B, x)
+    Bv = B(x)
+    b = normalize(Bv)
+    v_par = (xu[4:6] ⋅ b) .* b
+    v_perp = xu[4:6] - v_par
+    dx[1:3] = norm(v_perp)^2*(Bv × gradient_B)/(2*q2m*norm(Bv)^3) +
+       (E(x) × Bv) / norm(Bv)^2 + v_par
+end
+
+# Initial condition
+stateinit = let x0 = [1.0, 0, 0], v0 = [0.0, 1.0, 0.1]
+    [x0..., v0...]
+end
+# Time span
+tspan = (0, 20)
+param = prepare(uniform_E, grad_B, species=Proton)
+prob = ODEProblem(trace!, stateinit, tspan, param)
+sol = solve(prob, Vern9())
+# Functions for obtaining the guiding center from actual trajectory
+gc = get_gc(param)
+gc_x0 = gc(stateinit)
+prob_gc = ODEProblem(trace_gc!, gc_x0, tspan, (param..., sol))
+sol_gc = solve(prob_gc, Tsit5(); save_idxs=[1,2,3])
+
+# Numeric and analytic results
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "Grad-B Drift",
+   xlabel = "x [m]",
+   ylabel = "y [m]",
+   zlabel = "z [m]",
+   aspect = :data,
+   azimuth = 0.3π,
+)
+
+gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,)
+
+lines!(ax, sol, idxs=(1, 2, 3))
+lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6))
+lines!(ax, sol_gc, idxs=(1, 2, 3))
+
+for i in 1:3
+    ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+    ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i]
+end
+

    Note that in this grad-B drift case, the analytic and numeric guiding centers have different trajectories.

    This page was generated using DemoCards.jl and Literate.jl.

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z%HVQ9b4d(7q^qdy$i2uax=@Q6#z*KCzng!NJN+pp{qOnrSlmBJbN{=cT9!xLA1Y9H Rzl971Az>jBkyaRY`5Oj3!4Uue literal 0 HcmV?d00001 diff --git a/v0.10.1/examples/basics/demo_gravity_drift.jl b/v0.10.1/examples/basics/demo_gravity_drift.jl new file mode 100644 index 00000000..07c4685d --- /dev/null +++ b/v0.10.1/examples/basics/demo_gravity_drift.jl @@ -0,0 +1,34 @@ +import DisplayAs #hide +using TestParticle +using TestParticle: get_gc +using OrdinaryDiffEq +using StaticArrays +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +# Analytic EM fields +B(x) = SA[0.0, 1e-8, 0.0] +E(x) = SA[0.0, 0.0, 0.0] + +# Earth's gravity +F(x) = SA[0.0, 0.0, -TestParticle.mᵢ*9.8] + +# Initial static particle +stateinit = let x0 = [1.0, 0.0, 0.0], v0 = [0.0, 0.0, 0.0] + [x0..., v0...] +end +# Time span +tspan = (0, 1.0) + +param = prepare(E, B, F, species=Proton) +prob = ODEProblem(trace!, stateinit, tspan, param) +sol = solve(prob, Vern9()) +# drift in x-direction + free fall in z-direction +f = lines(sol, idxs=(3,1); + figure = (; size = (800, 400), fontsize=18), + axis = (; title="ExF Drift", xlabel="Z [m]", ylabel="X [m]", aspect = DataAspect()) +) + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/basics/demo_gravity_drift/index.html b/v0.10.1/examples/basics/demo_gravity_drift/index.html new file mode 100644 index 00000000..41a1e1ef --- /dev/null +++ b/v0.10.1/examples/basics/demo_gravity_drift/index.html @@ -0,0 +1,30 @@ + +ExF drift · TestParticle.jl

    ExF drift

    Source code compat Author Author Update time

    This example demonstrates a single proton motion under uniform B and gravity fields.

    using TestParticle
+using TestParticle: get_gc
+using OrdinaryDiffEq
+using StaticArrays
+using CairoMakie
+
+# Analytic EM fields
+B(x) = SA[0.0, 1e-8, 0.0]
+E(x) = SA[0.0, 0.0, 0.0]
+
+# Earth's gravity
+F(x) = SA[0.0, 0.0, -TestParticle.mᵢ*9.8]
+
+# Initial static particle
+stateinit = let x0 = [1.0, 0.0, 0.0], v0 = [0.0, 0.0, 0.0]
+   [x0..., v0...]
+end
+# Time span
+tspan = (0, 1.0)
+
+param = prepare(E, B, F, species=Proton)
+prob = ODEProblem(trace!, stateinit, tspan, param)
+sol = solve(prob, Vern9())
+# drift in x-direction + free fall in z-direction
+f = lines(sol, idxs=(3,1);
+   figure = (; size = (800, 400), fontsize=18),
+   axis = (; title="ExF Drift", xlabel="Z [m]", ylabel="X [m]", aspect = DataAspect())
+)
+

    This page was generated using DemoCards.jl and Literate.jl.

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TestParticle.jl

    Polarization drift

    Source code compat Author Author Update time

    This example demonstrates a single proton motion under time-varying E field. More theoretical details can be found in Time-Varying E Drift, and Fundamentals of Plasma Physics by Paul Bellan.

    using TestParticle
+using TestParticle: get_gc
+using OrdinaryDiffEq
+using StaticArrays
+using CairoMakie
+
+function uniform_B(x)
+    return SA[0, 0, 1e-8]
+end
+
+function time_varying_E(x, t)
+    # return SA[0, 1e-9*cos(0.1*t), 0]
+    return SA[0, 1e-9*0.1*t, 0]
+end
+
+# Initial condition
+stateinit = let x0 = [1.0, 0, 0], v0 = [0.0, 1.0, 0.1]
+    [x0..., v0...]
+end
+# Time span
+tspan = (0, 100)
+param = prepare(time_varying_E, uniform_B, species=Proton)
+prob = ODEProblem(trace!, stateinit, tspan, param)
+sol = solve(prob, Vern9())
+# Functions for obtaining the guiding center from actual trajectory
+gc = get_gc(param)
+v_perp(xu) = hypot(xu[4], xu[5])
+
+# Visualization
+f = Figure(size=(800, 600), fontsize=18)
+ax1 = Axis3(f[1:3, 1],
+   title = "Polarization Drift",
+   xlabel = "x [m]",
+   ylabel = "y [m]",
+   zlabel = "z [m]",
+   aspect = :data,
+   azimuth = 0.9π,
+   elevation = 0.1π
+)
+ax2 = Axis(f[1,2], xlabel = "time [s]", ylabel = "v_perp [m/s]")
+ax3 = Axis(f[2,2], xlabel = "time [s]", ylabel = "y [m]")
+ax4 = Axis(f[3,2], xlabel = "time [s]", ylabel = "gc_y [m]")
+
+gc_y(t, x, y, z, vx, vy, vz) = (t, gc(SA[x, y, z, vx, vy, vz])[2])
+v_perp(t, vy, vz) = (t, sqrt(vy^2 + vz^2))
+
+lines!(ax1, sol, idxs=(1, 2, 3))
+lines!(ax2, sol, idxs=(v_perp, 0, 5, 6))
+lines!(ax3, sol, idxs=2)
+lines!(ax4, sol, idxs=(gc_y, 0, 1, 2, 3, 4, 5, 6))
+

    This page was generated using DemoCards.jl and Literate.jl.

    diff --git a/v0.10.1/examples/covers/democards_logo.svg b/v0.10.1/examples/covers/democards_logo.svg new file mode 100644 index 00000000..caaa936c --- /dev/null +++ b/v0.10.1/examples/covers/democards_logo.svg @@ -0,0 +1,47 @@ + + + + + Produced by OmniGraffle 7.7.1 + 2019-08-23 08:34:14 +0000 + + + square + + + diff --git a/v0.10.1/examples/index.html b/v0.10.1/examples/index.html new file mode 100644 index 00000000..59565c41 --- /dev/null +++ b/v0.10.1/examples/index.html @@ -0,0 +1,164 @@ + +Examples · TestParticle.jl

    Examples

    This section contains thorough examples for using TestParticle.jl. If you prefer to run the demos on your local computer with interactive support, we suggest switching from using CairoMakie (which is a Makie frontend for static 2D rendering) to using GLMakie (which is another Makie frontend for interactive 3D rendering).

    Basics

    + + +

    Energy Conservation

    +

    Demonstrate energy conservation in uniform fields.

    +
    +
    + + +

    Boris method

    +

    Simple electron trajectory under uniform B and zero E

    +
    +
    + + +

    Helix motion

    +

    Simple proton trajectory under uniform B and zero E

    +
    +
    + + +

    Dimensionless Units

    +

    Tracing charged particle in dimensionless units

    +
    +
    + + +

    Tracing with Dimensionless Units and Periodic Boundary

    +

    Tracing charged particle in dimensionless units and periodic boundary

    +
    +
    + + +

    Dimensionless and Dimensional Tracing

    +

    Tracing charged particle in both dimensional and dimensionless units.

    +
    +
    + + +

    Proton and electron in a static EM field

    +

    Simple particle trajectories under uniform B and zero E

    +
    +
    + + +

    Multiple particles

    +

    Tracing multiple charged particles in a static EM field

    +
    +
    + + +

    E×B drift

    +

    Simple ExB drift demonstration

    +
    +
    + + +

    ExF drift

    +

    Simple proton trajectory under uniform B and gravity

    +
    +
    + + +

    Grad-B drift

    +

    Simple magnetic field gradient drift demonstration

    +
    +
    + + +

    Curl-B drift

    +

    Simple magnetic field curvature drift demonstration

    +
    +
    + + +

    Finite-Larmor-Radius effect

    +

    Finite Larmor radius effect demonstration using Makie

    +
    +
    + + +

    Polarization drift

    +

    Simple polarization drift under time-varying E field

    +
    +

    Advanced

    + + +

    Advanced Boris tracing

    +

    Boris ensemble tracing with field out-of-domain check

    +
    +
    + + +

    Ensemble tracing

    +

    Tracing multiple charged particles in a static EM field

    +
    +
    + + +

    Single tracing with additional diagnostics

    +

    Tracing one charged particle with additional diagnostics

    +
    +
    + + +

    Ensemble tracing with extra saving

    +

    Tracing multiple charged particles in a static EM field

    +
    +
    + + +

    Current sheet

    +

    Tracing charged particle in the Harris current sheet

    +
    +
    + + +

    Magnetic mirror

    +

    Charged particle in the magnetic mirror

    +
    +
    + + +

    Electron in a magnetic bottle

    +

    Tracing charged particle in a magnetic bottle

    +
    +
    + + +

    Magnetic dipole

    +

    Tracing charged particle in a static analytic dipole magnetic field

    +
    +
    + + +

    Analytical magnetosphere

    +

    Tracing charged particle in an analytical magnetosphere

    +
    +
    + + +

    Shock

    +

    Tracing charged particle in an MHD plane shock

    +
    +
    + + +

    Coil Tokamak

    +

    Tracing protons in a Tokamak

    +
    +
    + + +

    Tokamak profile

    +

    Tracing passing and trapped proton in a Tokamak

    +
    +
    + + +

    Tracing particle from PIC

    +

    This demo shows how to trace particles from structured SWMF outputs

    +
    +
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    Multiple particles

    Source code compat Author Update time

    using TestParticle
+using OrdinaryDiffEq
+using Random
+using CairoMakie
+
+# For reproducible results
+Random.seed!(1234)
+
+function trace(x, y, z, E, B; trajectories::Int=10)
+   # Initialize particles
+   x0 = [0.0, 0.0, 0.0] # initial position, [m]
+   u0 = [1.0, 0.0, 0.0] # initial velocity, [m/s]
+   stateinit = [x0..., u0...]
+
+   param = prepare(x, y, z, E, B, species=Electron)
+   tspan = (0.0, 15.0)
+
+   prob = ODEProblem(trace!, stateinit, tspan, param)
+
+   sols = Vector{ODESolution}(undef, trajectories)
+   # Sample from a Maxwellian with bulk speed 0 and thermal speed 1.0
+   vdf = Maxwellian([0.0, 0.0, 0.0], 1.0)
+   v = [sample(vdf) for _ in 1:trajectories]
+
+   for i in 1:trajectories
+      #prob = remake(prob; u0=[x0..., v[:,i]...])
+      prob.u0[4:6] = v[i]
+
+      sol = solve(prob, Vern9())
+      sols[i] = sol
+   end
+
+   return sols
+end
+
+### Initialize grid and field
+x = range(-10, 10, length=15)
+y = range(-10, 10, length=20)
+z = range(-10, 10, length=25)
+
+B = fill(0.0, 3, length(x), length(y), length(z)) # [T]
+E = fill(0.0, 3, length(x), length(y), length(z)) # [V/m]
+B[3,:,:,:] .= 1e-11
+E[3,:,:,:] .= 5e-13
+
+trajectories = 4
+
+### Solve for the trajectories
+
+sols = trace(x, y, z, E, B; trajectories)
+
+### Visualization
+f = Figure(fontsize = 18)
+ax = Axis3(f[1, 1],
+   title = "Particle trajectories",
+   xlabel = "X [m]",
+   ylabel = "Y [m]",
+   zlabel = "Z [m]",
+   aspect = :data,
+)
+
+for i in eachindex(sols)
+   lines!(ax, sols[i], idxs=(1, 2, 3), label="$i")
+end
+

    This page was generated using DemoCards.jl and Literate.jl.

    • Curl-B drift

    Source code compat Author Author Update time

    This example demonstrates a single proton motion under a vacuum non-uniform B field with gradient and curvature. Similar to the magnetic field gradient drift, analytic calculation should include both of the gradient drift and the curvature drift. More theoretical details can be found in Curvature Drift and Computational Plasma Physics by Toshi Tajima.

    using TestParticle
+using TestParticle: get_gc
+using OrdinaryDiffEq
+using StaticArrays
+using LinearAlgebra: normalize, norm, ×, ⋅
+using ForwardDiff: gradient, jacobian
+using CairoMakie
+
+function curved_B(x)
+    # satisify ∇⋅B=0
+    # B_θ = 1/r => ∂B_θ/∂θ = 0
+    θ = atan(x[3]/(x[1]+3))
+    r = hypot(x[1]+3, x[3])
+    return SA[-1e-7*sin(θ)/r, 0, 1e-7*cos(θ)/r]
+end
+
+function zero_E(x)
+    return SA[0, 0, 0]
+end
+
+abs_B(x) = norm(curved_B(x))  # |B|
+
+# Trace the orbit of the guiding center using analytical drifts
+function trace_gc!(dx, x, p, t)
+    q2m, E, B, sol = p
+    xu = sol(t)
+    gradient_B = gradient(abs_B, x)  # ∇|B|
+    Bv = B(x)
+    b = normalize(Bv)
+    v_par = (xu[4:6] ⋅ b).*b  # (v⋅b)b
+    v_perp = xu[4:6] - v_par
+    Ω = q2m*norm(Bv)
+    κ = jacobian(B, x)*Bv  # B⋅∇B
+    # v⟂^2*(B×∇|B|)/(2*Ω*B^2) + v∥^2*(B×(B⋅∇B))/(Ω*B^3) + (E×B)/B^2 + v∥
+    dx[1:3] = norm(v_perp)^2*(Bv × gradient_B)/(2*Ω*norm(Bv)^2) +
+        norm(v_par)^2*(Bv × κ)/Ω/norm(Bv)^3 + (E(x) × Bv)/norm(Bv)^2 + v_par
+end
+
+# Initial conditions
+stateinit = let x0 = [1.0, 0.0, 0.0], v0 = [0.0, 1.0, 0.1]
+    [x0..., v0...]
+end
+# Time span
+tspan = (0, 40)
+# Trace particle
+param = prepare(zero_E, curved_B, species=Proton)
+prob = ODEProblem(trace!, stateinit, tspan, param)
+sol = solve(prob, Vern9())
+# Functions for obtaining the guiding center from actual trajectory
+gc = get_gc(param)
+gc_x0 = gc(stateinit)
+prob_gc = ODEProblem(trace_gc!, gc_x0, tspan, (param..., sol))
+sol_gc = solve(prob_gc, Tsit5(); save_idxs=[1,2,3])
+
+# Numeric and analytic results
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "Curvature Drift",
+   xlabel = "x [m]",
+   ylabel = "y [m]",
+   zlabel = "z [m]",
+   aspect = :data,
+   azimuth = 0.3π,
+)
+
+gc_plot(x, y, z, vx, vy, vz) = (gc(SA[x, y, z, vx, vy, vz])...,)
+
+lines!(ax, sol, idxs=(1, 2, 3))
+lines!(ax, sol, idxs=(gc_plot, 1, 2, 3, 4, 5, 6))
+lines!(ax, sol_gc, idxs=(1, 2, 3))
+
+for i in 1:3
+    ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+    ax.scene.plots[9+2*i-1].color = Makie.wong_colors()[i]
+end
+

    This page was generated using DemoCards.jl and Literate.jl.

    Boris method

    Source code compat Author Update time

    This example demonstrates a single electron motion under a uniform B field. The E field is assumed to be zero such that there is no particle acceleration. We use the Boris method for phase space conservation under a fixed time step. This is compared against other ODE general algorithms for performance and accuracy.

    using TestParticle
+using StaticArrays
+using OrdinaryDiffEq
+using CairoMakie
+
+function plot_trajectory(sol_boris, sol1, sol2)
+   f = Figure(size=(700, 600), fontsize=18)
+   ax = Axis(f[1, 1], aspect=1, limits = (-3, 1, -2, 2),
+      xlabel = "X",
+      ylabel = "Y")
+   idxs = (1, 2)
+   ##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+   l0 = lines!(ax, sol_boris[1]; idxs, linewidth=2, label="Boris")
+   l1 = lines!(ax, sol1; idxs, linewidth=2, linestyle=:dashdot, label="Tsit5 fixed")
+   l2 = linesegments!(ax, sol2; idxs, linewidth=2, linestyle=:dot, label="Tsit5 adaptive")
+
+   ax.scene.plots[1].linewidth = 2
+   ax.scene.plots[5].linewidth = 2
+
+   ax.scene.plots[3].color = Makie.wong_colors()[2]
+   ax.scene.plots[5].color = Makie.wong_colors()[3]
+
+   scale!(ax.scene.plots[1], invrL, invrL)
+   scale!(ax.scene.plots[3], invrL, invrL)
+   scale!(ax.scene.plots[5], invrL, invrL)
+
+   axislegend(position=:rt, framevisible=false)
+
+   f
+end
+
+const Bmag = 0.01
+uniform_B(x) = SA[0.0, 0.0, Bmag]
+uniform_E(x) = SA[0.0, 0.0, 0.0]
+
+x0 = [0.0, 0.0, 0.0]
+v0 = [0.0, 1e5, 0.0]
+stateinit = [x0..., v0...]
+
+param = prepare(uniform_E, uniform_B, species=Electron)
+
+# Reference parameters
+const tperiod = 2π / (abs(param[1]) * hypot(param[3]([0.0,0.0,0.0], 0.0)...))
+const rL = hypot(v0...) / (abs(param[1]) * Bmag)
+const invrL = 1 / rL;

    We first trace the particle for one period with a discrete time step of a quarter period.

    tspan = (0.0, tperiod)
+dt = tperiod / 4
+
+prob = TraceProblem(stateinit, tspan, param)
+
+sol_boris = TestParticle.solve(prob; dt, savestepinterval=1);

    Let's compare against the default ODE solver Tsit5 from DifferentialEquations.jl, in both fixed time step mode and adaptive mode:

    prob = ODEProblem(trace!, stateinit, tspan, param)
+sol1 = solve(prob, Tsit5(); adaptive=false, dt, dense=false, saveat=dt);
+sol2 = solve(prob, Tsit5());
+
+# Visualization
+f = plot_trajectory(sol_boris, sol1, sol2)

    It is clear that the Boris method comes with larger phase errors ($\mathcal{O}(\Delta t)$) compared with Tsit5. The phase error gets smaller using a smaller dt:

    dt = tperiod / 8
+
+prob = TraceProblem(stateinit, tspan, param)
+
+sol_boris = TestParticle.solve(prob; dt, savestepinterval=1);
+
+prob = ODEProblem(trace!, stateinit, tspan, param)
+sol1 = solve(prob, Tsit5(); adaptive=false, dt, dense=false, saveat=dt);
+
+# Visualization
+f = plot_trajectory(sol_boris, sol1, sol2)

    The Boris pusher shines when we do long time tracing, which is fast and conserves energy:

    tspan = (0.0, 200*tperiod)
+dt = tperiod / 12
+
+prob_boris = TraceProblem(stateinit, tspan, param)
+prob = ODEProblem(trace!, stateinit, tspan, param)
+
+sol_boris = TestParticle.solve(prob_boris; dt, savestepinterval=10);
+sol1 = solve(prob, Tsit5(); adaptive=false, dt, dense=false, saveat=dt);
+sol2 = solve(prob, Tsit5());
+
+# Visualization
+f = plot_trajectory(sol_boris, sol1, sol2)

    Fixed time step Tsit5() is ok, but adaptive Tsit5() is pretty bad for long time evolutions. The change in radius indicates change in energy, which is sometimes known as numerical heating.

    Another aspect to compare is performance:

    @time sol_boris = TestParticle.solve(prob_boris; dt, savestepinterval=10)[1];
+@time sol1 = solve(prob, Tsit5(); adaptive=false, dt, dense=false, saveat=dt);
+@time sol2 = solve(prob, Tsit5());
      0.000109 seconds (257 allocations: 21.609 KiB)
+  0.000785 seconds (2.46 k allocations: 377.031 KiB)
+  0.000747 seconds (11.86 k allocations: 1.329 MiB)
+

    We can extract the solution (x, y, z, vx, vy, vz) at any given time by performing a linear interpolation:

    t = tspan[2] / 2
+sol_boris(t)
    6-element StaticArraysCore.MVector{6, Float64} with indices SOneTo(6):
+    -3.911568318655031e-5
+    -5.5137270837814596e-5
+     0.0
+ 99639.26547015733
+  8486.269885387323
+     0.0

    The Boris method is faster and consumes less memory. However, in practice, it is pretty hard to find an optimal algorithm. When calling OrdinaryDiffEq.jl, we recommend using Vern9() as a starting point instead of Tsit5(), especially combined with adaptive timestepping.

    This page was generated using DemoCards.jl and Literate.jl.

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OrdinaryDiffEq +using StaticArrays +using Statistics: mean +using Printf +using CairoMakie +CairoMakie.activate!(type = "png") #hide + +### Obtain field + +# Magnetic bottle parameters in SI units +const ICoil = 80. # current in the coil +const N = 15000 # number of windings +const IPlasma = 1e6 # current in the plasma +const a = 1.5 # radius of each coil +const b = 0.8 # radius of central region + +function getB(xu) + SVector{3}(getB_tokamak_coil(xu[1], xu[2], xu[3], a, b, ICoil*N, IPlasma)) +end + +function getE(xu) + SA[0.0, 0.0, 0.0] +end + +### Initialize particles +m = TestParticle.mᵢ +q = TestParticle.qᵢ +c = TestParticle.c + +# initial velocity, [m/s] +v₀ = [-0.1, -0.15, 0.0] .* c +# initial position, [m] +r₀ = [2.3, 0.0, 0.0] +stateinit = [r₀..., v₀...] + +param = prepare(getE, getB; species=Proton) +tspan = (0.0, 1e-6) + +prob = ODEProblem(trace!, stateinit, tspan, param) + +@printf "Speed = %6.4f %s\n" √(v₀[1]^2+v₀[2]^2+v₀[3]^2)/c*100 "% speed of light" +@printf "Energy = %6.4f MeV\n" (1/√(1-(v₀[1]/c)^2-(v₀[2]/c)^2-(v₀[3]/c)^2)-1)*m*c^2/abs(q)/1e6 + +# Default Tsit5() alone does not work in this case! You need to set a maximum +# timestep to maintain stability, or choose a different algorithm as well. +# The sample figure in the gallery is generated with AB3() and dt=2e-11. +sol = solve(prob, Tsit5(); dt=2e-11, save_idxs=[1,2,3]) + +### Visualization + +f = Figure(fontsize=18) +ax = Axis3(f[1, 1], + title = "Particle trajectory in Tokamak", + xlabel = "x [m]", + ylabel = "y [m]", + zlabel = "z [m]", + aspect = :data, +) + +plot!(sol, label="proton") + +# Plot coils +θ = range(0, 2π, length=201) +y = a * cos.(θ) +z = a * sin.(θ) +for i in 0:17 + ϕ = i*π/9 + plot!(y*sin(ϕ) .+ (a+b)*sin(ϕ), y*cos(ϕ) .+ (a+b)*cos(ϕ), z, color=(:red, 0.3)) +end + +# Plot Tokamak +u = range(0, 2π, length=100) +v = range(0, 2π, length=100) + +U = [y for _ in u, y in v] +V = [x for x in u, _ in v] + +X = @. (a + b + (a - 0.05)*cos(U)) * cos(V) +Y = @. (a + b + (a - 0.05)*cos(U)) * sin(V) +Z = @. (a - 0.05) * sin(U) + +wireframe!(ax, X, Y, Z, color=(:blue, 0.1), linewidth=0.5, transparency=true) + +f = DisplayAs.PNG(f) #hide + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/v0.10.1/examples/advanced/demo_tokamak_coil/index.html b/v0.10.1/examples/advanced/demo_tokamak_coil/index.html new file mode 100644 index 00000000..508a37c8 --- /dev/null +++ b/v0.10.1/examples/advanced/demo_tokamak_coil/index.html @@ -0,0 +1,85 @@ + +Coil Tokamak · TestParticle.jl

    Magnetic dipole

    Source code compat Author Update time

    This example shows how to trace protons of a certain energy in a analytic Earth-like magnetic dipole field. There is a combination of grad-B drift, curvature drift, and the bounce motion between mirror points. It demonstrates the motions corresponding to the three adiabatic invariants.

    using TestParticle
+using TestParticle: getB_dipole, getE_dipole, sph2cart, dipole_fieldline, mᵢ, qᵢ, c, Rₑ
+using OrdinaryDiffEq
+using CairoMakie
+
+# Initial condition
+stateinit = let
+   # Initial particle energy
+   Ek = 5e7 # [eV]
+   # initial velocity, [m/s]
+   v₀ = sph2cart(c*sqrt(1-1/(1+Ek*qᵢ/(mᵢ*c^2))^2), 0.0, π/4)
+   # initial position, [m]
+   r₀ = sph2cart(2.5*Rₑ, 0.0, π/2)
+   [r₀..., v₀...]
+end
+# obtain field
+param = prepare(getE_dipole, getB_dipole)
+tspan = (0.0, 10.0)
+
+prob = ODEProblem(trace!, stateinit, tspan, param)
+
+sol = solve(prob, Vern9())
+
+### Visualization
+
+f = Figure(fontsize=18)
+ax = Axis3(f[1, 1],
+   title = "50 MeV Proton trajectory in Earth's dipole field",
+   xlabel = "x [Re]",
+   ylabel = "y [Re]",
+   zlabel = "z [Re]",
+   aspect = :data,
+   limits = (-2.5, 2.5, -2.5, 2.5, -1, 1)
+)
+
+invRE = 1 / Rₑ
+l = lines!(ax, sol, idxs=(1, 2, 3))
+##TODO: wait for https://github.com/MakieOrg/Makie.jl/issues/3623 to be fixed!
+scale!(ax.scene.plots[9+1], invRE, invRE, invRE)
+
+for ϕ in range(0, stop=2*π, length=10)
+   lines!(dipole_fieldline(ϕ)..., color=:tomato, alpha=0.3)
+end
+

    Solver algorithm matters in terms of energy conservation. In the above we used Verner's “Most Efficient” 9/8 Runge-Kutta method. Let's check other algorithms.

    function get_energy_ratio(sol)
+   vx = getindex.(sol.u, 4)
+   vy = getindex.(sol.u, 5)
+   vz = getindex.(sol.u, 6)
+
+   Einit = vx[1]^2 + vy[1]^2 + vz[1]^2
+   Eend = vx[end]^2 + vy[end]^2 + vz[end]^2
+
+   (Eend - Einit) / Einit
+end
+
+# `ImplicitMidpoint()` requires a fixed time step.
+sol = solve(prob, ImplicitMidpoint(); dt=1e-3)
+get_energy_ratio(sol)
    -0.00045982272113011914
    sol = solve(prob, ImplicitMidpoint(); dt=1e-4)
+get_energy_ratio(sol)
    -4.940132338997509e-9
    sol = solve(prob, Vern9())
+get_energy_ratio(sol)
    -0.0070445259008460786
    sol = solve(prob, Trapezoid())
+get_energy_ratio(sol)
    -0.011643134931582444
    sol = solve(prob, Vern6())
+get_energy_ratio(sol)
    -0.06589639870599219
    sol = solve(prob, Tsit5())
+get_energy_ratio(sol)
    0.5351974995171843

    Default stepsize settings may not be enough for our problem. By using a smaller abstol and reltol, we can guarantee much better conservation at a higher cost:

    # This is roughly equivalent in accuracy and performance with Vern9() and `reltol=1e-3` (default)
+sol = solve(prob, Tsit5(); reltol=1e-4);

    Or, for adaptive time step algorithms like Vern9(), with the help of callbacks, we can enforce a largest time step smaller than 1/10 of the local gyroperiod:

    using DiffEqCallbacks
+
+# p = (charge_mass_ratio, E, B)
+dtFE(u, p, t) = 2π / (abs(p[1]) * hypot(p[3](u, t)...))
+cb = StepsizeLimiter(dtFE; safety_factor=1 // 10, max_step=true)
+
+sol = solve(prob, Vern9(); callback=cb, dt=0.1) # dt=0.1 is a dummy value
+get_energy_ratio(sol)
    -5.399366794067525e-7

    This is much more accurate, at the cost of more iterations. In terms of accuracy, this is roughly equivalent to solve(prob, Vern9(); reltol=1e-7); in terms of performance, it is 2x slower (0.04s v.s. 0.02s) and consumes about the same amount of memory 42 MiB. We can also use the classical Boris method implemented within the package:

    dt = 1e-4
+prob = TraceProblem(stateinit, tspan, param)
+sol = TestParticle.solve(prob; dt)[1]
+get_energy_ratio(sol)
    -1.0158504769154868e-15

    The Boris method requires a fixed time step. It takes about 0.05s and consumes 53 MiB memory. In this specific case, the time step is determined empirically. If we increase the time step to 1e-2 seconds, the trajectory becomes completely off (but the energy is still conserved). Therefore, as a rule of thumb, we should not use the default Tsit5() scheme without decreasing reltol. Use adaptive Vern9() for an unfamiliar field configuration, then switch to more accurate schemes if needed. A more thorough test can be found here.

    This page was generated using DemoCards.jl and Literate.jl.

    Magnetic mirror

    Source code compat Author Author Update time

    This example demonstrates the particle motion trajectory in a magnetic mirror and also illustrates the conservation of magnetic moment. From the third figure on the right, it can be seen that the zero-order quantity of magnetic moment is conserved, but its high-order part is not conserved under this definition of magnetic moment and oscillates rapidly. We can observe from this the oscillation characteristics of magnetic moments at different levels.

    This example is based on demo_magneticbottle.jl.

    using TestParticle
+using TestParticle: getB_mirror
+using OrdinaryDiffEq
+using StaticArrays
+using LinearAlgebra: normalize, norm, ⋅
+using CairoMakie
+
+### Obtain field
+
+# Magnetic mirror parameters in SI units
+const I = 20. # current in the solenoid
+const N = 45 # number of windings
+const distance = 10. # distance between solenoids
+const a = 4.0 # radius of each coil
+
+function getB(xu)
+   SVector{3}(getB_mirror(xu[1], xu[2], xu[3], distance, a, I*N))
+end
+
+getE(xu) = SA[0.0, 0.0, 0.0]
+
+# velocity in the direction perpendicular to the magnetic field
+function v_perp(t, x, y, z, vx, vy, vz)
+   xu = SA[x, y, z, vx, vy, vz]
+   vu = @view xu[4:6]
+   B = getB(xu)
+   b = normalize(B)
+   v_pa = (vu ⋅ b) .* b
+
+   (t, norm(vu - v_pa))
+end
+
+# magnetic field
+absB(t, x, y, z) = (t, hypot(getB(SA[x,y,z])...))
+
+# μ, magnetic moment
+function mu(t, x, y, z, vx, vy, vz)
+   xu = SA[x, y, z, vx, vy, vz]
+
+   (t, v_perp(t, x, y, z, vx, vy, vz)[2]^2 / hypot(getB(xu)...) )
+end
+
+Et(xu) = hypot(xu[4:6]...)
+
+### Initialize particles
+m = TestParticle.mₑ
+q = TestParticle.qₑ
+c = TestParticle.c
+
+# initial velocity, [m/s]
+v₀ = [2.75, 2.5, 1.3] .* 0.001c  # confined
+##v₀ = [0.25, 0.25, 5.9595] .* 0.01c  # escaped
+# initial position, [m]
+r₀ = [0.8, 0.8, 0.0]  # confined
+##r₀ = [1.5, 1.5, 2.4]  # escaped
+stateinit = [r₀..., v₀...]
+
+param = prepare(getE, getB; species=Electron)
+tspan = (0.0, 1e-4)
+
+prob = ODEProblem(trace!, stateinit, tspan, param)
+
+# Default Tsit5() and many solvers does not work in this case!
+# AB4() has better performance in maintaining magnetic moment conservation compared to AB3().
+sol_non = solve(prob, AB4(); dt=3e-9)
+
+### Visualization
+f = Figure(size=(900, 600), fontsize=18)
+ax1 = Axis3(f[1:3, 1],
+   title = "Magnetic Mirror",
+   xlabel = "x [m]",
+   ylabel = "y [m]",
+   zlabel = "z [m]",
+   aspect = :data,
+   azimuth = 0.9π,
+   elevation = 0.1π
+)
+ax2 = Axis(f[1,2], xlabel = "time [s]", ylabel = "B [T]")
+ax3 = Axis(f[2,2], xlabel = "time [s]", ylabel = "v_perp [m/s]")
+ax4 = Axis(f[3,2], xlabel = "time [s]", ylabel = "mu")
+
+lines!(ax1, sol_non, idxs=(1, 2, 3))
+lines!(ax2, sol_non, idxs=(absB, 0, 1, 2, 3))
+lines!(ax3, sol_non, idxs=(v_perp, 0, 1, 2, 3, 4, 5, 6))
+lines!(ax4, sol_non, idxs=(mu, 0, 1, 2, 3, 4, 5, 6))
+
+# Plot coils
+θ = range(0, 2π, length=100)
+x = a.*cos.(θ)
+y = a.*sin.(θ)
+z = fill(distance/2, size(x))
+lines!(ax1, x, y, z, color=:red)
+z = fill(-distance/2, size(x))
+lines!(ax1, x, y, z, color=:red)
+
+# # The distribution of magnetic field along the z-axis or x-axis
+# Bz(z) = hypot(getB(SA[0.0, 0.0, z])...)
+# Bx(x) = hypot(getB(SA[x, 0.0, 0.5*distance])...)
+# z = collect(-10:0.01:10)
+# x = collect(-0.99*a:0.01:0.99*a)
+# # Ba = Bz.(z)
+# Ba = Bx.(x)
+# # lines(z, Ba, color=:red)
+# lines(x, Ba, color=:red)
+

    This page was generated using DemoCards.jl and Literate.jl.

    Electron in a magnetic bottle

    Source code compat Author Update time

    This example shows how to trace non-relativistic and relativistic electrons in a stationary magnetic field that corresponds to a magnetic bottle. Reference wiki

    using TestParticle
+using TestParticle: getB_bottle
+using OrdinaryDiffEq
+using StaticArrays
+using Printf
+using CairoMakie
+
+### Obtain field
+
+# Magnetic bottle parameters in SI units
+const I1 = 20. # current in the solenoid
+const N1 = 45 # number of windings
+const I2 = 20. # current in the central solenoid
+const N2 = 45 # number of windings
+const distance = 10. # distance between solenoids
+const a = 4.0 # radius of each coil
+const b = 8.0 # radius of central coil
+
+function getB(xu)
+   SVector{3}(getB_bottle(xu[1], xu[2], xu[3], distance, a, b, I1*N1, I2*N2))
+end
+
+function getE(xu)
+   SA[0.0, 0.0, 0.0]
+end
+
+### Initialize particles
+m = TestParticle.mₑ
+q = TestParticle.qₑ
+c = TestParticle.c
+
+# initial velocity, [m/s]
+v₀ = [2.75, 2.5, 1.3] .* 0.03c  # confined
+# initial position, [m]
+r₀ = [0.8, 0.8, 0.0]
+stateinit = [r₀..., v₀...]
+
+# Theoretically we can take advantage of the fact that magnetic field does not
+# accelerate particles, so that γ remains constant. However, we are not doing
+# that here since it is not generally true in the EM field.
+
+param = prepare(getE, getB; species=Electron)
+tspan = (0.0, 1e-5)
+
+prob_rel = ODEProblem(trace_relativistic!, stateinit, tspan, param)
+prob_non = ODEProblem(trace!, stateinit, tspan, param)
+
+vratio = √(v₀[1]^2+v₀[2]^2+v₀[3]^2)/c
+E = (1 / √(1 - vratio^2) - 1)*m*c^2/abs(q)/1e6
+
+v_str = @sprintf "V = %4.2f %s" vratio*100 "% c"
+e_str = @sprintf "E = %6.4f MeV" E
+
+println(v_str)
+println(e_str)
+
+# Default Tsit5() and many solvers does not work in this case!
+sol_rel = solve(prob_rel, AB4(); dt=3e-9)
+sol_non = solve(prob_non, AB4(); dt=3e-9)
+
+### Visualization
+
+f = Figure(fontsize=18)
+ax1 = Axis3(f[1, 1];
+   aspect = :data,
+   title = "Relativistic e⁻, \n"*v_str*", "*e_str
+   )
+ax2 = Axis3(f[1, 2];
+   aspect = :data,
+   title = "Non-relativistic e⁻, \n"*v_str*", "*e_str
+   )
+
+lines!(ax1, sol_rel, idxs=(1, 2, 3))
+lines!(ax2, sol_non, idxs=(1, 2, 3))
+
+# Plot coils
+θ = range(0, 2π, length=100)
+x = a.*cos.(θ)
+y = a.*sin.(θ)
+z = fill(distance/2, size(x))
+for ax in (f[1,1], f[1,2])
+   lines!(ax, x, y, z, color=(:red, 0.7))
+end
+z = fill(-distance/2, size(x))
+for ax in (f[1,1], f[1,2])
+   lines!(ax, x, y, z, color=(:red, 0.7))
+end
+
+x = b.*cos.(θ)
+y = b.*sin.(θ)
+z = fill(0.0, size(x))
+for ax in (f[1,1], f[1,2])
+   lines!(ax, x, y, z, color=(:red, 0.7))
+end
+
+using FieldTracer
+
+xrange = range(-4, 4, length=20)
+yrange = range(-4, 4, length=20)
+zrange = range(-10, 10, length=20)
+
+Bx, By, Bz = let x=xrange, y=yrange, z=zrange
+   Bx = zeros(length(x), length(y), length(z))
+   By = zeros(length(x), length(y), length(z))
+   Bz = zeros(length(x), length(y), length(z))
+   for k in eachindex(z), j in eachindex(y), i in eachindex(x)
+      Bx[i,j,k], By[i,j,k], Bz[i,j,k] = getB([x[i], y[j], z[k]])
+   end
+
+   Bx, By, Bz
+end
+
+for i in 0:8
+   if i == 0
+      xs, ys, zs = 0.0, 0.0, 0.0
+   else
+      xs, ys, zs = 3*cos(2π*(i-1)/8), 3*sin(2π*(i-1)/8), 0.0
+   end
+   x1, y1, z1 = FieldTracer.trace(Bx, By, Bz, xs, ys, zs, xrange, yrange, zrange;
+      ds=0.1, maxstep=1000)
+   for ax in (f[1,1], f[1,2])
+      lines!(ax, x1, y1, z1, color=:black)
+   end
+end
+

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