CollocationPoints.jl

CollocationPoints.jl is a Julia package for generating collocation points inside various domains, with support for variable point density. The primary goal here is to use this for solving PDEs with physics-informed machine learning

Features

• Generate uniformly distributed points in circles and rectangles.
• Generate points with variable density, concentrating points around specific points, lines, or boundaries.
• Define complex domains from boundary coordinates.
• Smooth boundary representations.

Running Examples

The package includes a comprehensive set of examples in the examples/ directory. To run them:

1. From the main directory in your terminal.
2. Start Julia: julia --project=. example/run_2D.jl

This will generate a series of plots in the examples/figures/ directory, showcasing the different capabilities of the package.

Data Files for Demos

The examples for complex geometries (L-shape, lake, island) require data files that define the boundary coordinates. You need to place these files in the examples/demos/ directory.

• Lshape.txt
• lake.txt
• island.txt

Each file should contain a list of x, y coordinates, one pair per line, separated by spaces.

Citations

• Mishra, Pankaj K (2019). NodeLab: A MATLAB package for meshfree node-generation and adaptive refinement. Journal of Open Source Software, 4(40), 1173, https://doi.org/10.21105/joss.01173
• Fornberg, B. and Flyer, N., 2015. Fast generation of 2-D node distributions for mesh-free PDE discretizations. Computers & Mathematics with Applications, 69(7), pp.531-544.