This package can also be used for testing physics-informed neural networks, deepOnets and Fourier Neural operators

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

NodeLab

NodeLab is a simple MATLAB-package for unstructured node-generation and refinement for meshfree modeling in 2D arbitrary domains. The final output can be seen in the 'workspace'. Following are the description of Input/Output variables of NodeLab:

Input:

• box - a vector defining the bounding box of the domain.
• hbdy - the density of the boundary poit-cloud
• ptol - minimum distance between two nodes
• ctps - control-points for density variation
• radius - distance function for node density metric ### Output
• xy — an array containing interior nodes.
• bdy— an array containing boundary nodes.

Installation

• Download the package on your PC.
• Open MATLAB
• Go to the directory 'NodeLab'
• run the script 'setup.m'. This will add different directories of the NodeLab on your MATLAB path — only for the current session. Now you are ready to run the demos and generate nodes for your own project. Note: make sure you have MATLAB's statistics toolbox.

Tutorials

1.1

Generate nodes inside a circle within a bounding box (-1,1)^2 with no control points. matlab clear varibale; close all; clc box = [-1,-1; 1,1]; hbdy = 0.02; ptol = 0.001; [b] = draw_circ(0,0,1,2/hbdy); ctps = []; %ctps = [linspace(-0.5, 0.5,10); zeros(1,10)]'; radius = @(p,ctps) 0.05; % for fixed node-density %radius = @(p,ctps) 0.005+0.08*(min(pdist2(ctps, p))); % for variable node-density [xy] = NodeLab2D(b.sdf,box,ctps,ptol, radius); [bdy] = b.xy; clear box hbdy ptol ctps radius b %------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',12) hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize',12) axis('square'); set(gca,'visible','off')

1.2

Generate nodes inside a circle within a bounding box (-1,1)^2 but keep a varibale-density being highest at the single control point (0,0). matlab clear varibale; close all; clc box = [-1,-1; 1,1]; hbdy = 0.02; ptol = 0.001; [b] = draw_circ(0,0,1,2/hbdy); ctps = [0, 0]; radius = @(p,ctps) 0.005+0.08*(min(pdist2(ctps, p))); % for variable node-density [xy] = NodeLab2D(b.sdf,box,ctps,ptol, radius); [bdy] = b.xy; clear box hbdy ptol ctps radius b %------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',12) hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize',12) axis('square'); set(gca,'visible','off')

1.3

Generate nodes inside a circle within a bounding box (-1,1)^2 but keep a varibale-density being highest along a line between (-0.75,0) and (0.75,0). matlab %----------------------------------- clear varibale; close all; clc box = [-1,-1; 1,1]; hbdy = 0.02; ptol = 0.001; [b] = draw_circ(0,0,1,2/hbdy); %ctps = [0, 0]; ctps = [linspace(-0.5, 0.5,10); zeros(1,10)]'; %radius = @(p,ctps) 0.05; % for fixed node-density radius = @(p,ctps) 0.005+0.08*(min(pdist2(ctps, p))); % for variable node-density [xy] = NodeLab2D(b.sdf,box,ctps,ptol, radius); [bdy] = b.xy; clear box hbdy ptol ctps radius b %------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',12) hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize',12) axis('square'); set(gca,'visible','off')

1.4

Generate nodes inside a circle within a bounding box (-1,1)^2 but keep a varibale-density being highest along two seperate points: (-0.75,0), and (0.75,0). matlab %----------------------------------- clear varibale; close all; clc box = [-1,-1; 1,1]; hbdy = 0.02; ptol = 0.001; [b] = draw_circ(0,0,1,2/hbdy); ctps = [-0.75,0; 0.75,0]; radius = @(p,ctps) 0.005+0.08*(min(pdist2(ctps, p))); [xy] = NodeLab2D(b.sdf,box,ctps,ptol, radius); [bdy] = b.xy; clear box hbdy ptol ctps radius b %------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',12) hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize',12) axis('square'); set(gca,'visible','off')

2.1

Generate nodes in a rectangle within a bounding box (-1,1)^2. matlab clear variables; close all; clc %------------------------------------------------ box = [-1,-1; 1,1]; hbdy = 0.05; ptol = 0.01; ctps = []; radius = @(p, ctps) 0.05; [b] = draw_rect(-1,-1,1,1,2/hbdy); [xy] = NodeLab2D(b.sdf,box,ctps,ptol,radius); [bdy] = b.xy; clear box hbdy ptol ctps radius b %--------------------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',12);hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize',12); axis('square') set(gca,'visible','off')

2.2

Generate nodes in a rectangle within a bounding box (-1,1)^2. Make varibale density at two point-source. matlab % unit circle clear variables; close all; clc %------------------------------------------------ box = [-1,-1; 1,1]; hbdy = 0.02; ptol = 0.001; ctps = [-0.5,1; 0.5, 1]; radius = @(p,ctps) 0.005+0.05*(min(pdist2(ctps, p))); [b] = draw_rect(-1,-1,1,1,2/hbdy); [xy] = NodeLab2D(b.sdf,box,ctps,ptol,radius); [bdy] = b.xy; clear box hbdy ptol ctps radius b %--------------------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',12);hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize',12); axis('square') set(gca,'visible','off')

2.3

Generate nodes in a L-shape domain for point-singularity problems. matlab clear variables; close all; clc %---------------------------------------------------- box = [-1, -1; 1, 1 ]; hbdy = 0.025; ptol = 0.001; [b] = make_domain('Lshape.txt'); ctps = [0, 0]; radius = @(p,ctps) 0.005+0.05*(min(pdist2(ctps, p))); [xy] = NodeLab2D(b.sdf,box,ctps,ptol,radius); bdy = bsmooth(b.xy, hbdy); clear b box hbdy ctps radius ptol %----------------------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',12); hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize', 12); axis('square') set(gca,'visible','off')

3.1

Generate nodes in a Island model.
matlab clear variables; close all; clc %-------------------------------------------- box = [100.0, 145.0; 634.0, 799.0 ]; hbdy = 5; ptol = 1; [b] = make_domain('lake.txt'); % process lake points as boundary [bdy] = bsmooth(b.xy, hbdy); ctps = bdy; radius = @(p,ctps) 6 ; [xy] = NodeLab2D(b.sdf,box,ctps,ptol,radius); %-------------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',10); hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize', 10); axis('square') set(gca,'visible','off')

3.2

Generate nodes in a Island model. Variable-density being maximum at the boundary.
matlab clear variables; close all; clc %-------------------------------------------- box = [100.0, 145.0; 634.0, 799.0 ]; hbdy = 5; ptol = 1; [b] = make_domain('lake.txt'); % process lake points as boundary [bdy] = bsmooth(b.xy, hbdy); ctps = bdy; radius = @(p,ctps) 6 ; [xy] = NodeLab2D(b.sdf,box,ctps,ptol,radius); %-------------------------------------------- plot(xy(:,1), xy(:,2),'.k','MarkerSize',10); hold on plot(bdy(:,1), bdy(:,2), '.k','MarkerSize', 10); axis('square') set(gca,'visible','off')

3.3

Generate nodes in a Island model having a small lake inside it. Varibale-density being maximum at each boundaries. matlab % unit circle clear variables; close all; clc %----------------------------------------- box = [100.0, 145.0; 634.0, 799.0 ]; hbdy = 3; ptol = 1; [b1] = make_domain('lake.txt'); % process lake points as boundary [b2] = make_domain('island.txt'); b.sdf = @(p) max(b1.sdf(p), -b2.sdf(p)) ; b1.xy = bsmooth(b1.xy, hbdy); b2.xy = bsmooth(b2.xy, hbdy); b.xy = [b1.xy; b2.xy]; ctps = b.xy; radius = @(p,ctps) 2 + 0.2*(min(pdist2(ctps, p))); [x] = NodeLab2D(b.sdf,box,ctps,ptol,radius); %----------------------------------------- plot(x(:,1), x(:,2),'.k','MarkerSize',8); hold on plot(b.xy(:,1), b.xy(:,2), '.k','MarkerSize', 8); axis('square'); set(gca,'visible','off')