Create high-quality 3D meshes with ease.
pygalmesh is a Python frontend to CGAL's 3D mesh generation capabilities. pygalmesh makes it easy to create high-quality 3D volume meshes, periodic volume meshes, and surface meshes.
CGAL offers two different approaches for mesh generation:
- Meshes defined implicitly by level sets of functions.
- Meshes defined by a set of bounding planes.
pygalmesh provides a front-end to the first approach, which has the following advantages and disadvantages:
- All boundary points are guaranteed to be in the level set within any specified residual. This results in smooth curved surfaces.
- Sharp intersections of subdomains (e.g., in unions or differences of sets) need to be specified manually (via feature edges, see below), which can be tedious.
On the other hand, the bounding-plane approach (realized by mshr), has the following properties:
- Smooth, curved domains are approximated by a set of bounding planes, resulting in more of less visible edges.
- Intersections of domains can be computed automatically, so domain unions etc. have sharp edges where they belong.
See here for other mesh generation tools.
import pygalmesh
s = pygalmesh.Ball([0, 0, 0], 1.0)
mesh = pygalmesh.generate_mesh(s, cell_size=0.2)
# mesh.points, mesh.cells, ...
You can write the mesh using meshio, e.g.,
import meshio
meshio.write("out.vtk", mesh)
The mesh generation comes with many more options, described here. Try, for example,
mesh = pygalmesh.generate_mesh(
s,
cell_size=0.2,
edge_size=0.1,
odt=True,
lloyd=True,
verbose=False
)
pygalmesh provides out-of-the-box support for balls, cuboids, ellipsoids, tori, cones, cylinders, and tetrahedra. Try for example
import pygalmesh
s0 = pygalmesh.Tetrahedron(
[0.0, 0.0, 0.0],
[1.0, 0.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, 1.0]
)
mesh = pygalmesh.generate_mesh(s0, cell_size=0.1, edge_size=0.1)
Supported are unions, intersections, and differences of all domains. As mentioned above, however, the sharp intersections between two domains are not automatically handled. Try for example
import pygalmesh
radius = 1.0
displacement = 0.5
s0 = pygalmesh.Ball([displacement, 0, 0], radius)
s1 = pygalmesh.Ball([-displacement, 0, 0], radius)
u = pygalmesh.Difference(s0, s1)
To sharpen the intersection circle, add it as a feature edge polygon line, e.g.,
a = numpy.sqrt(radius**2 - displacement**2)
edge_size = 0.15
n = int(2*numpy.pi*a / edge_size)
circ = [
[
0.0,
a * numpy.cos(i * 2*numpy.pi / n),
a * numpy.sin(i * 2*numpy.pi / n)
] for i in range(n)
]
circ.append(circ[0])
mesh = pygalmesh.generate_mesh(
u,
feature_edges=[circ],
cell_size=0.15,
edge_size=edge_size,
facet_angle=25,
facet_size=0.15,
cell_radius_edge_ratio=2.0
)
Note that the length of the polygon legs are kept in sync with the edge_size
of the
mesh generation. This makes sure that it fits in nicely with the rest of the mesh.
You can of course translate, rotate, scale, and stretch any domain. Try, for example,
import pygalmesh
s = pygalmesh.Stretch(
pygalmesh.Ball([0, 0, 0], 1.0),
[1.0, 2.0, 0.0]
)
mesh = pygalmesh.generate_mesh(s, cell_size=0.1)
pygalmesh lets you extrude any polygon into a 3D body. It even supports rotation alongside!
import pygalmesh
p = pygalmesh.Polygon2D([[-0.5, -0.3], [0.5, -0.3], [0.0, 0.5]])
edge_size = 0.1
domain = pygalmesh.Extrude(
p,
[0.0, 0.0, 1.0],
0.5 * 3.14159265359,
edge_size
)
mesh = pygalmesh.generate_mesh(
domain,
cell_size=0.1,
edge_size=edge_size,
verbose=False
)
Feature edges are automatically preserved here, which is why an edge length needs to be
given to pygalmesh.Extrude
.
Polygons in the x-z-plane can also be rotated around the z-axis to yield a rotation body.
import pygalmesh
p = pygalmesh.Polygon2D([[0.5, -0.3], [1.5, -0.3], [1.0, 0.5]])
edge_size = 0.1
domain = pygalmesh.RingExtrude(p, edge_size)
mesh = pygalmesh.generate_mesh(
domain,
cell_size=0.1,
edge_size=edge_size,
verbose=False
)
If all of the variety is not enough for you, you can define your own custom level set
function. You simply need to subclass pygalmesh.DomainBase
and specify a function,
e.g.,
import pygalmesh
class Heart(pygalmesh.DomainBase):
def __init__(self):
super(Heart, self).__init__()
return
def eval(self, x):
return (x[0]**2 + 9.0/4.0 * x[1]**2 + x[2]**2 - 1)**3 \
- x[0]**2 * x[2]**3 - 9.0/80.0 * x[1]**2 * x[2]**3
def get_bounding_sphere_squared_radius(self):
return 10.0
d = Heart()
mesh = pygalmesh.generate_mesh(d, cell_size=0.1)
Note that you need to specify the square of a bounding sphere radius, used as an input to CGAL's mesh generator.
If you want to have local refinement, you can use
generate_with_sizing_field
. It works just like generate_mesh
except that it takes a
SizingFieldBase
object as cell_size
.
# define a cell_size function
class Field(pygalmesh.SizingFieldBase):
def eval(self, x):
return abs(numpy.sqrt(numpy.dot(x, x)) - 0.5) / 5 + 0.025
mesh = pygalmesh.generate_with_sizing_field(
pygalmesh.Ball([0.0, 0.0, 0.0], 1.0),
facet_angle=30,
facet_size=0.1,
facet_distance=0.025,
cell_radius_edge_ratio=2,
cell_size=Field(),
)
If you're only after the surface of a body, pygalmesh has generate_surface_mesh
for
you. It offers fewer options (obviously, cell_size
is gone), but otherwise works the
same way:
import pygalmesh
s = pygalmesh.Ball([0, 0, 0], 1.0)
mesh = pygalmesh.generate_surface_mesh(
s,
angle_bound=30,
radius_bound=0.1,
distance_bound=0.1
)
Refer to CGAL's documention for the options.
pygalmesh also interfaces CGAL's 3D periodic mesh generation. Besides a domain, one needs to specify a bounding box, and optionally the number of copies in the output (1, 2, 4, or 8). Example:
import pygalmesh
class Schwarz(pygalmesh.DomainBase):
def __init__(self):
super(Schwarz, self).__init__()
return
def eval(self, x):
x2 = numpy.cos(x[0] * 2 * numpy.pi)
y2 = numpy.cos(x[1] * 2 * numpy.pi)
z2 = numpy.cos(x[2] * 2 * numpy.pi)
return x2 + y2 + z2
mesh = pygalmesh.generate_periodic_mesh(
Schwarz(),
[0, 0, 0, 1, 1, 1],
cell_size=0.05,
facet_angle=30,
facet_size=0.05,
facet_distance=0.025,
cell_radius_edge_ratio=2.0,
number_of_copies_in_output=4,
# odt=True,
# lloyd=True,
verbose=False
)
If you have a surface mesh at hand (like elephant.vtu), pygalmesh generates a volume mesh on the command line via
pygalmesh-volume-from-surface elephant.vtu out.vtk --cell-size 1.0 --odt
(See pygalmesh-volume-from-surface -h
for all options.)
In Python, do
import pygalmesh
mesh = pygalmesh.generate_volume_mesh_from_surface_mesh(
"elephant.vtu",
facet_angle=25.0,
facet_size=0.15,
facet_distance=0.008,
cell_radius_edge_ratio=3.0,
verbose=False
)
It is also possible to generate meshes from INR voxel files, e.g., liver.inr either on the command line
pygalmesh-from-inr liver.inr out.vtu --cell-size 5.0 --odt
(see pygalmesh-from-inr -h
for all options) or from Python
import pygalmesh
mesh = pygalmesh.generate_from_inr(
"liver.inr",
cell_size=5.0,
verbose=False,
)
For installation, pygalmesh needs CGAL and Eigen installed on your system. They are typically available on your Linux distribution, e.g., on Ubuntu
sudo apt install libcgal-dev libeigen3-dev
After that, pygalmesh can be installed from the Python Package Index, so with
pip3 install -U pygalmesh
you can install/upgrade.
meshio (pip3 install meshio
)
can be helpful in processing the meshes.
For manual installation (if you're a developer or just really keen on getting the bleeding edge version of pygalmesh), there are two possibilities:
- Get the sources, type
python3 setup.py install
. This does the trick most the time. - As a fallback, there's a CMake-based installation. Simply go
cmake /path/to/sources/
andmake
.
To run the pygalmesh unit tests, check out this repository and type
pytest
pygalmesh is published under the MIT license.