NodeLab

NodeLab is a simple MATLAB-repository for node-generation and adaptive refinement for testing, and implementing various meshfree methods for

NodeLab is a simple MATLAB-repository for node-generation and adaptive refinement for testing, and implementing various meshfree methods for solving PDEs in arbitrary domains. The core-algorithm behind this package is the node-placing [1] approach because of its simplicity, computational speed and the quality of the distribution. The node-placing method has been used for creating initial node-distribution in the \emph{bounding-box} of the desired domain. To represent the geometry of the domain, NodeLab couples the Signed-distance function (SDF) [2], which can computed based on a priory information about the domain. As a results, NodeLab can take the following geometric objects as inputs:

1. simple shapes (like rectangles, and circles), required to create the desired geometry. For example, the SDF for a typical plate with a hole can be computed through a rectangle and a circle.
2. a level-set representation $D(x,y)=0$ of the domain, or
3. some discrete set of point cloud on the boundary, which need not to be uniformly sampled. The boundary can be smoothed through curve-interpolation according to the fill-distance near the boundary, which provides the flexibility to create the domain by manually digitizing of the geometry from a hand-drawing, digital-drawing, or a downloaded image.

The node-distribution in NodeLab can be refined non-uniformly by adapting the information provided through control-points, which are an input from the user. control-points provide spatial locations where the user needs relatively finer nodes.

Future developments in this package may include its extension to 3D, and surfaces, an optional graphical user interface, improvement in the adaptivity by using Machine Learning algorithms to decide the control-points, etc. NodeLab is intended to be an open-source and collaborative project, where developers and users could contribute to make (and keep) it --- state-of-the-art by incorporating the improvements as the research in this field grows with time.
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[1]. Bengt Fornberg, Natasha Flyer (2015), Fast generation of 2-D node distributions for mesh-free PDE discretizations, Computers & Mathematics with Applications, 69(7) 531-544,
[2]. Per-Olof Persson and Gilbert Strang, A Simple Mesh Generator in MATLAB, SIAM Review, 46(2) 2004 329-345.