Remeshing 3D triangulated surface for better mesh quaility

Question

I am using marching cubes to extract a 2D surface from a volume. In this case a Gyroid.

import numpy as np
from numpy import sin, cos, pi
from skimage import measure
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

def gyroid(x, y, z, t):
    return cos(x)*sin(y) + cos(y)*sin(z) + cos(z)*sin(x) - t

lattice_param = 1.0
strut_param = 0.0
resolution = 31j

x, y, z = pi*np.mgrid[-1:1:resolution, -1:1:resolution, -1:1:resolution] * lattice_param
vol = gyroid(x, y, z, strut_param)

verts, faces = measure.marching_cubes(vol, 0, spacing=(0.1, 0.1, 0.1)) # , normals, values

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_trisurf(verts[:, 0], verts[:, 1], faces, verts[:, 2], cmap='ocean', lw=1)

This all works fine but the mesh quailty is appalling in a lot of places. I cant run any FEA on the meshes as many of the elements/faces have near zero area or are highly distorted.

Is there a way of either remeshing given the vertices & ensuring particular elementface/facet metrics (such as aspect ratio) or forcing marching cubes to do such a thing?

I am not bothered about moving vertices as long as the mesh is a fair approximation.

Answer 1

One option could be to 're-mesh' the marching cubes output using a surface meshing package. Essentially this means the marching cubes triangulation would serve as an initial surface definition to be re-triangulated.

There are many techniques one could employ to do this. A few options that may be useful (all C++ / C implementations):

• JIGSAW : a restricted , frontal-delaunay algorithm^^ that will generally build very high-quality surface Delaunay triangulations. For the type of object shown I'd expect it to work well. In the demos included (provided in MATLAB ) several examples address the re-meshing of marching cubes output.
• CGAL : a restricted , delaunay-refinement approach that can also build surface Delaunay triangulations, but uses a slightly different algorithm to JIGSAW and also includes CVT -type mesh optimisation schemes.
• MMG : a collection of re-meshing/optimisation strategies (as I understand it) that can be used to transform (and hence improve) an initial mesh via the iterative application of local modifications.

^^ I'm the author of JIGSAW , so, shameless promotion here basically.

Answer 2

Weird triangles come from weird data, not from the method used for triangulation.

I can say that a Delaunay triangulation achieves the best triangle-area/triangle-perimeter ratio (best theorethical ratio is with equilateral triangles). But you can't use if for your mesh because Delaunay triangulation outputs a convex mesh.

You have a tight task ahead. Some thoughts:

• Remove cuasi-duplicated points before meshing. This will avoid many near-zero-area triangles.
• Detect thin triangles. Subdivide them into more 'regular' triangles.
• You may create a dense regular grid. Each z can be found traversing the mesh until you find a triangle that contains x,y coordinates of the point in the grid and then interpolating the z . If you have some extra info, like neigbours or hierachy, the search can be done faster than inspecting every triangle. Triangulating the grid is quite simple.

Answer 3

Use can use pygalmesh for surface remeshing, too. (It's a easy-to-use interface to CGAL.)

import pygalmesh

# create verts, faces

meshio.write_points_cells("in.vtu", verts, [("triangle", faces)])

mesh = pygalmesh.remesh_surface(
    "in.vtu",
    edge_size=0.025,
    facet_angle=25,
    facet_size=0.1,
    facet_distance=0.001,
    verbose=False,
)
# mesh.points, mesh.cells

Answer 4

I suggest using 3d adaptive remeshing with smoothing. The Geogram programming library ( http://alice.loria.fr/software/geogram/doc/html/index.html ) provides a nice implementation. Please see: https://twitter.com/brunolevy01/status/1132343120690122752?lang=en