How to design a solid Body for even Meshing in Matlab? (Hemisphere)

Hello dear Matlab community,

I currently try to:

• create a surface from x y z coordinates (easily done with meshgrid and surf)
• export the surface into FreeCad/Gmsh for creating an even Mesh

The result should look as follows (which was fully created in FreeCad):

Why I try doing all of this ?

• Because I know how to design exact surfaces in Matlab

(In this case I need to exactyl define the contact angle between hemisphere and surface)

• Because I only can create even Meshes in FreeCad

In Matlab I only get an uneven Mesh:

So once againg: Is there a workflow for creating a solid body in Matlab that can be evenly triangulated?

[Stefan Kerber]

Hi Niklas,

if I understand your prpblem correctly, it should be possible to define a regular net using the meshgid function and then interpolate your function to this regular grid. I guess a good staring point for this could be Interpolation for 2-D gridded data in meshgrid format - MATLAB interp2 (mathworks.com)

Does this help?

[Niklas Kurz]

Hi Stefan,

thank you for you suggestions. I tinkered for a while with it, but interp2 doesn't seem to work with the way I defined the hemisphere

R = 1;

n = 30; 

phi   = linspace(0,2*pi,n);

theta = linspace(0,pi/2,n);

[phigrid,thetagrid] = meshgrid(phi,theta);

xsphere = R*cos(phigrid).*sin(thetagrid);

ysphere = R*sin(phigrid).*sin(thetagrid);

zsphere = R*cos(thetagrid); 

My aim still persisted in extracting merely the surface of the hemisphere in order to perform proper (in the sense of 'even') meshing, but none of the solutions that I have browsed (like surf2stl or as you said interp2) seem to do the job - the wireframe is always uneven as you can observe in the Matlab plot above .

However, that's not as bad, since I managed to control the sphere parameters in FreeCad now.

It also may be done in Matlab, but of course Matlab was not designed for Modeling objects.

Answer 1

[Yatharth]

Hi Niklas,

I understand that you want to create an even mesh of a hemisphere in MATLAB.

You can create an even mesh using “generateMesh” function. You will be needing a STL file to generate the even mesh.

Here is an example for the same:

1. Creating a hemisphere and ploting it using Triangulation.

% Define hemisphere parameters
radius = 1;  % Radius of the hemisphere
resolution = 10;  % Number of points along the circumference

% Generate hemisphere coordinates
theta = linspace(0, pi/2, resolution);
phi = linspace(0, 2*pi, 2*resolution);
[theta, phi] = meshgrid(theta, phi);
x = radius * sin(theta) .* cos(phi);
y = radius * sin(theta) .* sin(phi);
z = radius * cos(theta);

points = [x(:), y(:), z(:)];

% Create the triangulation
TR = delaunayTriangulation(points);

Warning: Duplicate data points have been detected and removed.
The Triangulation indices are defined with respect to the unique set of points in delaunayTriangulation.

% Plot the triangulated hemisphere
tetramesh(TR);

2. You cannot directly use "stlwrite" function to make STL file as Tetrahedron Triangulation is not supported. STL file will eventually act as an input for the "generateMesh" function. . Therefore we will make an "alphaShape" and extract the facets of the "alphaShape" that represents the surface of the AlphaShape using the "boundaryFacets" function

% Making the alphaShape
x = x(:);
y= y(:);
z = z(:);
P = [x y z];
P = unique(P,'rows');
shp = alphaShape(P,1.5);
plot(shp)
axis equal

% Using the boundaryFacets function to make the surface of alphaShape
[tri, xyz] = boundaryFacets(shp);
trisurf(tri,xyz(:,1),xyz(:,2),xyz(:,3),...
    'FaceColor','cyan','FaceAlpha',0.3) 
axis equal

TR = triangulation(tri,xyz);

3. Now we can simply use the stlwrite function to generate the stl file and generate even mesh using "generateMesh" function

% Write the triangulation object to an STL file
stlwrite(TR,'hemisphere.stl');

% Read the STL file and generate an even mesh
model = createpde;
importGeometry(model, 'hemisphere.stl');
generateMesh(model, 'Hmax', 0.1);  % Set maximum element size for even meshing

% Plot the mesh
pdeplot3D(model);

Have a look at the respective documentations for the functions used:

1. delaunayTriangulation https://www.mathworks.com/help/matlab/ref/delaunaytriangulation.html
2. alphaShape https://www.mathworks.com/help/matlab/ref/alphashape.html
3. boundaryFacets https://www.mathworks.com/help/matlab/ref/alphashape.boundaryfacets.html
4. triangulation https://www.mathworks.com/help/matlab/ref/triangulation.html
5. stlwrite https://www.mathworks.com/help/matlab/ref/stlwrite.html
6. generateMesh https://www.mathworks.com/help/pde/ug/pde.pdemodel.generatemesh.html

I hope this helps!

[Niklas Kurz]

Awesome, exactly what I was hoping for! Matlab ist able to do it. (Of cousre it is). Also thanks for the really detailed answer, it is all clear to me right now. I finally hope that your answer will provide help for others who ran into the same problem (I feel like I was not the only one)

Cheers!