Tutorial

Installation

MeshIntegrals.jl is registered in the official Julia General registry, enabling it to be easily installed using Julia on an internet-connected computer. MeshIntegrals.jl should usually be installed in conjunction with Meshes.jl and Unitful.jl which provide support for geometries and physical units, respectively.

MeshIntegrals.jl can be installed from the Julia REPL by entering pkg mode with the ] key and adding it by name

pkg> add MeshIntegrals

or by calling the built-in Pkg.jl library.

import Pkg
Pkg.add("MeshIntegrals")

Basic Usage

Usage of MeshIntegrals.jl typically also involves using Meshes.jl and Unitful.jl, so all three packages will be used in this example.

using Meshes
using MeshIntegrals
using Unitful

Define a Geometry to Use as an Integration Domain

Here, representing the domain being integrated over, we will define a Bezier curve whose path approximates a sine-wave on the xy-plane.

N = 361  # number of control points
xs = range(-π, π, length=N)  # x will be bounded in [-π, π] meters
curve = Meshes.BezierCurve([Point(x * u"m", sin(x) * u"m", 0.0u"m") for x in xs])

BezierCurve
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Define an Integrand Function

Here, we will define an integrand function of the following form, where all coordinates are defined in a physical space with meter units and the result is a quantity in units of Ohms per meter.

\[f(x, y, z) = \frac{1}{\sqrt{1 + \cos^2(x/\text{m})}} ~ \Omega/\text{m}\]

Integrand functions are expected to provide a method that takes a single Meshes.Point argument, so this can be written in Julia as

function f(p::Meshes.Point)
    x, y, z = Meshes.to(p)
    return (1 / sqrt(1 + cos(x / u"m")^2)) * u"Ω/m"
end

f (generic function with 1 method)

Alternatively, this could be written in the user-friendly notation

f(x, y, z) = (1 / sqrt(1 + cos(x / u"m")^2)) * u"Ω/m"
f(p::Meshes.Point) = f(Meshes.to(p)...)

f (generic function with 2 methods)

where the required f(Meshes.Point) method simply maps to the method f(x, y, z).

Integrating

This function can be integrated using recommended defaults simply by calling

integral(f, curve)  # -> Approximately 2π Ω

6.264795483127288 Ω

The alias function lineintegral works for this geometry since it has one parametric dimension. However, the aliases surfaceintegral and volumeintegral will throw an ArgumentError since the geometry is not a surface or volume.

lineintegral(f, curve)
# -> Approximately 2π Ω

surfaceintegral(f, curve)
# -> ERROR: ArgumentError: Performing a surface integral on a geometry with 1 parametric dimensions not supported.

volumeintegral(f, curve)
# -> ERROR: ArgumentError: Performing a volume integral on a geometry with 1 parametric dimensions not supported.

An IntegrationRule with settings can also be manually specified. The following example uses the adaptive Gauss-Kronrod method with a loosened absolute tolerance setting of $10^{-4}~\Omega$, which speeds up integration by sacrificing some accuracy.

integral(f, curve, GaussKronrod(atol = 1e-4u"Ω")) # -> Approximately (2π ± 1e-4) Ω

6.264795504552421 Ω

The integral function and its aliases also support Julia's do-syntax, which provides a code block to define a single-use anonymous function and then injects it as a first argument to the preceding call. This can be useful if the integrand function will not be used outside this integration call.

integral(curve) do p
    x, y, z = Meshes.to(p)
    (1 / sqrt(1 + cos(x / u"m")^2)) * u"Ω/m"
end # -> Approximately 2π Ω

6.264795483127288 Ω