pygalmesh 0.2.2

Python frontend to CGAL's 3D mesh generation capabilities

A Python frontend to CGAL’s 3D mesh generation capabilities.

pygalmesh makes it easy to create high-quality 3D volume and surface meshes.

Background

CGAL offers two different approaches for mesh generation:

1. Meshes defined implicitly by level sets of functions.

2. 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.

Other Python mesh generators are pygmsh (a frontend to gmsh) and MeshPy. meshzoo provides some basic canonical meshes.

Examples

A simple ball

Ball

import pygalmesh

s = pygalmesh.Ball([0, 0, 0], 1.0)
pygalmesh.generate_mesh(s, 'out.mesh', cell_size=0.2)

CGAL’s mesh generator returns Medit-files, which can be processed by, e.g., meshio.

import meshio
vertices, cells, _, _, _ = meshio.read('out.mesh')

The mesh generation comes with many more options, described here. Try, for example,

pygalmesh.generate_mesh(
    s,
    'out.mesh',
    cell_size=0.2,
    edge_size=0.1,
    odt=True,
    lloyd=True,
    verbose=False
    )

Other primitive shapes

Tetrahedron

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]
        )
pygalmesh.generate_mesh(
        s0, 'out.mesh', cell_size=0.1, edge_size=0.1
        )

Domain combinations

Balls difference

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])

pygalmesh.generate_mesh(
        u,
        'out.mesh',
        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.

Domain deformations

Egg

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]
        )

pygalmesh.generate_mesh(
        s,
        'out.mesh',
        cell_size=0.1
        )

Extrusion of 2D polygons

triangle rotated

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
        )
pygalmesh.generate_mesh(
        domain,
        'out.mesh',
        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.

Rotation bodies

triangle ring extruded

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.ring_extrude(p, edge_size)
pygalmesh.generate_mesh(
        domain,
        'out.mesh',
        cell_size=0.1,
        edge_size=edge_size,
        verbose=False
        )

Your own custom level set function

triangle ring extruded

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()
pygalmesh.generate_mesh(d, 'out.mesh', 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.

Surface meshes

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)
pygalmesh.generate_surface_mesh(
        s,
        'out.off',
        angle_bound=30,
        radius_bound=0.1,
        distance_bound=0.1
        )

The output format is OFF which again is handled by meshio.

Refer to CGAL’s documention for the options.

Meshes from OFF files

elephant

If you have an OFF file at hand (like elephant.off or these), pygalmesh generates the mesh via

import pygalmesh

pygalmesh.generate_from_off(
        'elephant.off',
        'out.mesh',
        facet_angle=25.0,
        facet_size=0.15,
        facet_distance=0.008,
        cell_radius_edge_ratio=3.0,
        verbose=False
        )

Installation

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

pip install -U pygalmesh

you can install/upgrade.

meshio (sudo -H pip install meshio) can be helpful in processing the meshes.

Manual installation

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 sudo python 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/ and make.

Testing

To run the pygalmesh unit tests, check out this repository and type

pytest

Distribution

To create a new release

1. bump the __version__ number (in setup.py and src/pygalmesh.i)

2. publish to PyPi and GitHub:

   make publish

License

pygalmesh is published under the MIT license.