Get triangular meshes for any shape

I would like to get triangular meshes for arbitrary shapes, with the most regular triangles possible and the ability to adjust the size of the triangles.

I have an example which is close but unsatisfactory because the triangles are very irregular. The example creates a toroid, then tries to get an alphashape and then tries to convert that into a triangular mesh. It's the only way I found without importing stl files.

I only need the vertices and faces. The rest of the pdemodels stuff is nice, but not necessary.

Examples of the kinds of other shapes I need this for: Cubes, prisms, regular N-gons, spheroid, part-torroids, cones, bowls, twists, etc (with holes possible in each). Nothing is spiky or furry, or rough. All surfaces are smooth.

My emphasis on triangular meshes is just because I need to exactly specify each face, and triagular meshes are efficient way of doing that. Other regular meshes are fine.

R=5; % outer radius of torus
r=2; % inner tube radius
th=linspace(0,2*pi,4*36); % e.g. 36 partitions along perimeter of the tube 
phi=linspace(0,2*pi,4*18); % e.g. 18 partitions along azimuth of torus
% we convert our vectors phi and th to [n x n] matrices with meshgrid command:
[Phi,Th]=meshgrid(phi,th); 
% now we generate n x n matrices for x,y,z according to eqn of torus
x=(R+r.*cos(Th)).*cos(Phi);
y=(R+r.*cos(Th)).*sin(Phi);
z=r.*sin(Th);

shp=alphaShape(x(:),y(:),z(:));
shp.Alpha=2.5;
[elements,nodes] = boundaryFacets(shp);
nodes = nodes';
elements = elements';
model = createpde();
mesh=geometryFromMesh(model,nodes,elements);
generateMesh(model)

pdeplot3D(model)

[John D'Errico]

You want an automatic scheme that will create a perfectly regular triangular mesh on any shape. Good luck in that.

[James Johnson]

@John D'Errico Thanks for showing interest. I'll add some helpful context later. That's not quite what I want. Not "any shapes": All shapes are either N-gons, created through extrusion, or created through revolution (except the holes, which are cut in). Hence the shapes are parameteric, this should make it easier. "perfect" also isn't quite right. I just want "not-garbage" (see the garbage toroid above) blender does this, as does MATLAB's own "vrworld". Any importing from an stl file does this. I just can't do extrusions and revolutions in MATLAB's "vrworld". I'll update my question with exactly why tomorrow.

Answer 1

[Nipun]

Hi James,

I understand that you want to generate triangular meshes for arbitrary shapes with regular triangles and adjustable triangle sizes. Here is a method to create a triangular mesh for a toroid using the alphaShape and generateMesh functions:

R = 5; % outer radius of torus
r = 2; % inner tube radius
th = linspace(0,2*pi,144); % partitions along perimeter of the tube 
phi = linspace(0,2*pi,72); % partitions along azimuth of torus
[Phi,Th] = meshgrid(phi,th); 
x = (R + r.*cos(Th)).*cos(Phi);
y = (R + r.*cos(Th)).*sin(Phi);
z = r.*sin(Th);

shp = alphaShape(x(:), y(:), z(:));
shp.Alpha = 2.5;
[elements, nodes] = boundaryFacets(shp);
nodes = nodes';
elements = elements';

model = createpde();
geometryFromMesh(model, nodes, elements);
generateMesh(model, 'GeometricOrder', 'linear', 'Hmax', 0.5);

pdeplot3D(model);

For more information on "alphaShape", "boundaryFacets" and "geometryFromMesh" functions, refer to the following MathWorks documentation:

1. alphaShape function in MATLAB: https://www.mathworks.com/help/matlab/ref/alphashape.html
2. boundaryFacets function in MATLAB: https://www.mathworks.com/help/matlab/ref/alphashape.boundaryfacets.html
3. geometryFromMesh function in MATLAB: https://www.mathworks.com/help/matlab/ref/alphashape.boundaryfacets.html

Hope this helps.

Regards,

Nipun