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DanielVandH committed Jun 27, 2024
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Expand Up @@ -36,7 +36,7 @@ DelaunayTriangulation.jl has already seen use in several areas. DelaunayTriangul

# Examples

We give two examples of how the package can be used. Many more examples are given in the [documentation](https://juliageometry.github.io/DelaunayTriangulation.jl/stable/), including [tutorials](https://juliageometry.github.io/DelaunayTriangulation.jl/stable/tutorials/overview/) and [fully detailed applications](https://juliageometry.github.io/DelaunayTriangulation.jl/stable/applications/overview/) such as cell simulations. To fully demonstrate the utility of the package, our examples follow realistic applications. We omit code used for plotting with Makie.jl [@danisch2021makie] in the examples below for space reasons. The complete code is available [here].
We give two examples of how the package can be used. Many more examples are given in the [documentation](https://juliageometry.github.io/DelaunayTriangulation.jl/stable/), including [tutorials](https://juliageometry.github.io/DelaunayTriangulation.jl/stable/tutorials/overview/) and [fully detailed applications](https://juliageometry.github.io/DelaunayTriangulation.jl/stable/applications/overview/) such as cell simulations. To fully demonstrate the utility of the package, our examples follow realistic applications. We omit code used for plotting with Makie.jl [@danisch2021makie] in the examples below for space reasons. The complete code is available [here](https://github.com/JuliaGeometry/DelaunayTriangulation.jl/blob/paper/paper/paper.jl).

For our first example, we consider a domain motivated by mean exit time. In particular, consider the problem
```math
Expand All @@ -53,7 +53,7 @@ Here, $T(x, y)$ denotes the mean exit time of a particle exiting $\Omega$ with d

![The domain $\Omega$. The red part of the boundary defines the absorbing boundary $\Gamma_a$, and the blue part defines the reflecting boundary $\Gamma_r$.\label{fig0}](figure0.png)

The code to generate a mesh of the domain in Figure \ref{fig0} is given below. We use curves to define the boundary so that curve-bounded refinement can be applied [@gosselin2009delaunay]. The resulting mesh is shown in Figure \ref{fig1}, together with a solution of the mean exit time problem with $D = 6.25 \times 10^{-4}$; FiniteVolumeMethod.jl [@vandenheuvel2024finite] is used to solve this problem, and the code for this can be found [here]. Makie.jl [@danisch2021makie] is used for the plotting.
The code to generate a mesh of the domain in Figure \ref{fig0} is given below. We use curves to define the boundary so that curve-bounded refinement can be applied [@gosselin2009delaunay]. The resulting mesh is shown in Figure \ref{fig1}, together with a solution of the mean exit time problem with $D = 6.25 \times 10^{-4}$; FiniteVolumeMethod.jl [@vandenheuvel2024finite] is used to solve this problem, and the code for this can be found [here](https://github.com/JuliaGeometry/DelaunayTriangulation.jl/blob/paper/paper/paper.jl).

```julia
# The outer circle
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