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

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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?

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Stefan Kerber 2023-11-16

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 2023-11-17

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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.

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Answer 1

Yatharth 2023-11-29

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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:

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

I hope this helps!

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Niklas Kurz 2023-11-29

编辑：Niklas Kurz 2023-11-29

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!

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