# pankajkmishra / NodeLab

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.
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## 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

• 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.

## 1.1

Generate nodes inside a circle within a bounding box (-1,1)^2 with no control points.

```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
[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).

```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
[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).

```%-----------------------------------
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
[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).

```%-----------------------------------
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];
[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.

```clear variables; close all; clc
%------------------------------------------------
box    = [-1,-1; 1,1];
hbdy   = 0.05;
ptol   = 0.01;
ctps   = [];
[b]    = draw_rect(-1,-1,1,1,2/hbdy);
[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.

```% 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];
[b]    = draw_rect(-1,-1,1,1,2/hbdy);
[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.

```clear variables; close all; clc
%----------------------------------------------------
box    = [-1, -1; 1, 1 ];
hbdy   = 0.025;
ptol   = 0.001;
[b]    = make_domain('Lshape.txt');
ctps   = [0, 0];
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.

```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;
%--------------------------------------------
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.

```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;
%--------------------------------------------
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.

```% 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)));