demo16 of Im2mesh package

demo16 - How to add points or nodes to polygonal boundaries before meshing
Cite as: Ma, J., & Li, Y. (2025). Im2mesh: A MATLAB/Octave package for generating finite element mesh based on 2D multi-phase image (2.1.5). Zenodo. https://doi.org/10.5281/zenodo.14847059
Jiexian Ma, mjx0799@gmail.com. Project website.
Table of Contents
Note Setup Get boundries from image Coordinates of 4 corners Add midpoint to the bottom edge Add a point near the bottom edge Add multiple points to the upper edge Generate mesh Add points to a specific polyline Generate mesh

Note

I suggest familiarizing yourself with Im2mesh_GUI before learning Im2mesh package. With graphical user interface, Im2mesh_GUI will help you better understand the workflow and parameters of Im2mesh package.
Im2mesh_GUI: https://www.mathworks.com/matlabcentral/fileexchange/179684-im2mesh_gui-2d-image-to-finite-element-mesh
If you are using Im2mesh package in MATLAB, you need to install MATLAB Image Processing Toolbox and Mapping Toolbox because Im2mesh package use a few functions in these toolboxes.

Setup

Before we start, please set folder "Im2mesh_Matlab" as your current folder of MATLAB.
clearvars
Set default image size.
x = 250; y = 250; width = 350; height = 350;
set(groot, 'DefaultFigurePosition', [x,y,width,height])
% To reset:
% set(groot, 'DefaultFigurePosition', 'factory')
Function poly2mesh use a mesh generator called MESH2D (developed by Darren Engwirda). We can use the following command to add the folder 'mesh2d-master' to the path of MATLAB.
addpath(genpath('mesh2d-master'))

Get boundries from image

We have demostrated how to do this in demo04.
im = imread('Shape.tif');
if size(im,3) == 3; im = rgb2gray( im ); end
% image to polygon boundary
boundsRaw = im2Bounds( im );
boundsCtrlP = getCtrlPnts( boundsRaw, false, size(im) );
% smooth boundary
lambda = 0.5; mu = -0.5; iters = 100;
threshold_num_turning = 0; threshold_num_vert_Smo = 0;
boundsSmooth = smoothBounds( boundsCtrlP, lambda, mu, iters, ...
threshold_num_turning, threshold_num_vert_Smo );
% simplify polygon boundary
tolerance = 0.3; threshold_num_vert_Sim = 0;
boundsSimplified = simplifyBounds( boundsSmooth, tolerance, ...
threshold_num_vert_Sim );
boundsSimplified = delZeroAreaPoly( boundsSimplified );
% clear up redundant vertices
% only control points and turning points will remain
boundsClear = getCtrlPnts( boundsSimplified, false );
boundsClear = simplifyBounds( boundsClear, 0 );
% show all vertices
plotBounds( boundsClear, false, 'ko-' );

Coordinates of 4 corners

Before adding points to boundary, we want to know about the x y coordinates of 4 corners in polygonal boundaries .
The image size is
size(im)
ans = 1×2
80 92
The height and width of the input image is 80 and 92, respectively.
So the x y coordinates of 4 corners in polygonal boundaries will be
corners = [0.5, 0.5] + [ 0, 0;
size(im,2), 0;
size(im,2), size(im,1);
0, size(im,1) ];
corners
corners = 4×2
0.5000 0.5000 92.5000 0.5000 92.5000 80.5000 0.5000 80.5000
The midpoint in the bottom boundary is
xyMidBottom = 0.5*(corners(1,:) + corners(2,:));
We will use these points later.
% upper right corrner
xUpRight = corners(3,1);
yUpRight = corners(3,2);
% upper left corner
xUpLeft = corners(4,1);
yUpLeft = corners(4,2);

Add midpoint to the bottom edge

We can use function addPnt2Bound to add a point to polygonal boundaries.
tol_dist = 1e-2; % distance tolerance
boundsNew = addPnt2Bound( xyMidBottom, boundsClear, tol_dist );
% show all vertices
plotBounds( boundsNew, false, 'ko-' );
We saw that the midpoint of the bottom edge is added to the boundary.
When we generate finite element mesh, the mesh will have this point in the vertices. The midpoint maybe useful when defining boundary condition in some cases.

Add a point near the bottom edge

The point to be added does not have to lie exactly on the pristine boundary. The point can have some distance away from the pristine boundary. Check the following example.
Here we set distance tolerance 'tol_dist' to 8. If the shortest distance between the point and the pristine boundary is less than 'tol_dist', the point will be added to the boundary. Otherwise, the point will not be added.
point = [20,-5];
tol_dist = 8; % distance tolerance
boundsNew = addPnt2Bound( point, boundsNew, tol_dist );
% show all vertices
plotBounds( boundsNew, false, 'ko-' );

Add multiple points to the upper edge

We can add multiple point to the boundary.
Suppose we want to add 58 points between the upper left corner and the upper right corner. Let's define these points.
x = linspace( xUpLeft, xUpRight, 60 )';
y = linspace( yUpLeft, yUpRight, 60 )';
points = [x,y];
Add these points to the boundary.
tol_dist = 1e-2; % distance tolerance
boundsNew = addPnt2Bound( points, boundsNew, tol_dist );
% show all vertices
plotBounds( boundsNew, false, 'k.-' );

Generate mesh

We use function poly2mesh to generate mesh.
Function poly2mesh uses mesh generator MESH2D (developed by Darren Engwirda).
[ poly_node, poly_edge ] = getPolyNodeEdge( boundsNew );
hmax = 500;
mesh_kind = 'delaunay';
grad_limit = 0.25;
[ vert,tria,tnum ] = poly2mesh( poly_node, poly_edge, hmax, mesh_kind, grad_limit );
Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 5 156 208 10 170 428 13 173 468 Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 10 234 208 10 234 208 20 240 1431 30 243 1478 30 243 1478 Smooth triangulation... ------------------------------------------------------- |ITER.| |MOVE(X)| |DTRI(X)| ------------------------------------------------------- 4 427 1458 8 69 1458 12 0 1458
plotMeshes(vert,tria,tnum);
Awesome! We saw that the mesh near the upper boundary is denser due to the inserted nodes.

Add points to a specific polyline

We can function addPnt2Bound to add points to a specific polyline.
Suppose we are interested in polygon - boundsClear{1}{2}. Let's plot it.
poly = boundsClear{1}{2};
figure; hold on; axis equal
plot( poly(:,1), poly(:,2), 'ko-' );
% add numbering
for i = 1: length(poly)
text( poly(i,1), poly(i,2), num2str(i), ...
'HorizontalAlignment', 'left', ...
'VerticalAlignment', 'top', ...
'FontSize', 8, 'Color', 'red');
end
hold off
Suppose we are interested in the polyline between vertex 37 and vertex 40. Let's extract it.
polyline = poly(37:40,:);
plot( polyline(:,1), polyline(:,2), 'ko-' ); axis equal
Then, we can add more vertices to the interested polyline by repeatedly inserting midpoints to edges. We do this by function insertMidPnt.
polyline = insertMidPnt( polyline );
polyline = insertMidPnt( polyline );
polyline = insertMidPnt( polyline );
polyline = insertMidPnt( polyline );
plot( polyline(:,1), polyline(:,2), 'ko-' ); axis equal
Add the polyline to the bounday.
tol_dist = 1e-2; % distance tolerance
boundsNew = addPnt2Bound( polyline, boundsClear, tol_dist );
% show all vertices
plotBounds( boundsNew, false, 'ko-' );
We saw the added nodes.

Generate mesh

We use function poly2mesh to generate mesh.
[ poly_node, poly_edge ] = getPolyNodeEdge( boundsNew );
hmax = 500;
mesh_kind = 'delaunay';
grad_limit = 0.25;
[ vert,tria,tnum,vert2,tria2 ] = poly2mesh( poly_node, poly_edge, hmax, mesh_kind, grad_limit );
Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 5 148 238 10 166 456 16 167 588 Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 10 232 238 11 233 238 20 238 1801 23 238 1817 Smooth triangulation... ------------------------------------------------------- |ITER.| |MOVE(X)| |DTRI(X)| ------------------------------------------------------- 4 538 1793 8 128 1793 12 0 1793
plotMeshes(vert,tria,tnum);
Awesome! We saw that the mesh near the polyline is denser due to the inserted nodes.
% reset image size
set(groot, 'DefaultFigurePosition', 'factory')
% end of demo