Create high-quality meshes with ease.

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pygalmesh is a Python frontend to CGAL's 2D and 3D mesh generation capabilities. pygalmesh makes it easy to create high-quality 2D, 3D volume meshes, periodic volume meshes, and surface meshes.

Examples

2D meshes

CGAL generates 2D meshes from linear constraints.

```python import numpy as np import pygalmesh

points = np.array([[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [0.0, 1.0]]) constraints = [[0, 1], [1, 2], [2, 3], [3, 0]]

mesh = pygalmesh.generate2d( points, constraints, maxedgesize=1.0e-1, numlloyd_steps=10, )

mesh.points, mesh.cells

```

The quality of the mesh isn't very good, but can be improved with optimesh.

A simple ball

```python import pygalmesh

s = pygalmesh.Ball([0, 0, 0], 1.0) mesh = pygalmesh.generatemesh(s, maxcell_circumradius=0.2)

mesh.points, mesh.cells, ...

```

You can write the mesh with

python mesh.write("out.vtk")

You can use any format supported by meshio.

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

python mesh = pygalmesh.generate_mesh( s, max_cell_circumradius=0.2, odt=True, lloyd=True, verbose=False )

Other primitive shapes

pygalmesh provides out-of-the-box support for balls, cuboids, ellipsoids, tori, cones, cylinders, and tetrahedra. Try for example

```python 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.generatemesh( s0, maxcellcircumradius=0.1, maxedgesizeatfeatureedges=0.1, ) ```

Domain combinations

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

```python 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.,

```python import numpy as np 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)

add circle

a = np.sqrt(radius2 - displacement2) maxedgesizeatfeatureedges = 0.15 n = int(2 * np.pi * a / maxedgesizeatfeatureedges) alpha = np.linspace(0.0, 2 * np.pi, n + 1) alpha[-1] = alpha[0] circ = a * np.column_stack([np.zeros(n + 1), np.cos(alpha), np.sin(alpha)])

mesh = pygalmesh.generatemesh( u, extrafeatureedges=[circ], maxcellcircumradius=0.15, maxedgesizeatfeatureedges=maxedgesizeatfeatureedges, minfacetangle=25, maxradiussurfacedelaunayball=0.15, maxcircumradiusedgeratio=2.0, ) ```

Note that the length of the polygon legs are kept in sync with max_edge_size_at_feature_edges of the mesh generation. This makes sure that it fits in nicely with the rest of the mesh.

Domain deformations

You can of course translate, rotate, scale, and stretch any domain. Try, for example,

```python import pygalmesh

s = pygalmesh.Stretch(pygalmesh.Ball([0, 0, 0], 1.0), [1.0, 2.0, 0.0])

mesh = pygalmesh.generatemesh(s, maxcell_circumradius=0.1) ```

Extrusion of 2D polygons

pygalmesh lets you extrude any polygon into a 3D body. It even supports rotation alongside!

```python import pygalmesh

p = pygalmesh.Polygon2D([[-0.5, -0.3], [0.5, -0.3], [0.0, 0.5]]) maxedgesizeatfeatureedges = 0.1 domain = pygalmesh.Extrude( p, [0.0, 0.0, 1.0], 0.5 * 3.14159265359, maxedgesizeatfeatureedges, ) mesh = pygalmesh.generatemesh( domain, maxcellcircumradius=0.1, maxedgesizeatfeatureedges=maxedgesizeatfeature_edges, verbose=False, ) ```

Feature edges are automatically preserved here, which is why an edge length needs to be given to pygalmesh.Extrude.

Rotation bodies

Polygons in the x-z-plane can also be rotated around the z-axis to yield a rotation body.

```python import pygalmesh

p = pygalmesh.Polygon2D([[0.5, -0.3], [1.5, -0.3], [1.0, 0.5]]) maxedgesizeatfeatureedges = 0.1 domain = pygalmesh.RingExtrude(p, maxedgesizeatfeatureedges) mesh = pygalmesh.generatemesh( domain, maxcellcircumradius=0.1, maxedgesizeatfeatureedges=maxedgesizeatfeature_edges, verbose=False, ) ```

Your own custom level set function

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.,

```python import pygalmesh

class Heart(pygalmesh.DomainBase): def init(self): super().init()

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.generatemesh(d, maxcell_circumradius=0.1) ```

Note that you need to specify the square of a bounding sphere radius, used as an input to CGAL's mesh generator.

Local refinement

Use generate_mesh with a function (regular or lambda) as max_cell_circumradius. The same goes for max_edge_size_at_feature_edges, max_radius_surface_delaunay_ball, and max_facet_distance.

```python import numpy as np import pygalmesh

mesh = pygalmesh.generatemesh( pygalmesh.Ball([0.0, 0.0, 0.0], 1.0), minfacetangle=30.0, maxradiussurfacedelaunayball=0.1, maxfacetdistance=0.025, maxcircumradiusedgeratio=2.0, maxcellcircumradius=lambda x: abs(np.sqrt(np.dot(x, x)) - 0.5) / 5 + 0.025, ) ```

Surface meshes

If you're only after the surface of a body, pygalmesh has generate_surface_mesh for you. It offers fewer options (obviously, max_cell_circumradius is gone), but otherwise works the same way:

```python import pygalmesh

s = pygalmesh.Ball([0, 0, 0], 1.0) mesh = pygalmesh.generatesurfacemesh( s, minfacetangle=30.0, maxradiussurfacedelaunayball=0.1, maxfacetdistance=0.1, ) ```

Refer to CGAL's documentation for the options.

Periodic volume meshes

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:

```python import numpy as np import pygalmesh

class Schwarz(pygalmesh.DomainBase): def init(self): super().init()

def eval(self, x):
    x2 = np.cos(x[0] * 2 * np.pi)
    y2 = np.cos(x[1] * 2 * np.pi)
    z2 = np.cos(x[2] * 2 * np.pi)
    return x2 + y2 + z2

mesh = pygalmesh.generateperiodicmesh( Schwarz(), [0, 0, 0, 1, 1, 1], maxcellcircumradius=0.05, minfacetangle=30, maxradiussurfacedelaunayball=0.05, maxfacetdistance=0.025, maxcircumradiusedgeratio=2.0, numberofcopiesin_output=4, # odt=True, # lloyd=True, verbose=False, ) ```

Volume meshes from surface meshes

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

```python import pygalmesh

mesh = pygalmesh.generatevolumemeshfromsurfacemesh( "elephant.vtu", minfacetangle=25.0, maxradiussurfacedelaunayball=0.15, maxfacetdistance=0.008, maxcircumradiusedgeratio=3.0, verbose=False, ) ```

Meshes from INR voxel files

It is also possible to generate meshes from INR voxel files, e.g., here either on the command line

pygalmesh from-inr skull_2.9.inr out.vtu --cell-size 5.0 --odt

(see pygalmesh from-inr -h for all options) or from Python

```python import pygalmesh

mesh = pygalmesh.generatefrominr( "skull2.9.inr", maxcell_circumradius=5.0, verbose=False, ) ```

Meshes from numpy arrays representing 3D images

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pygalmesh can help generating unstructed meshes from 3D numpy int arrays specifying the subdomains. Subdomains with key 0 are not meshed.

```python import pygalmesh import numpy as np

x_ = np.linspace(-1.0, 1.0, 50) y_ = np.linspace(-1.0, 1.0, 50) z_ = np.linspace(-1.0, 1.0, 50) x, y, z = np.meshgrid(x, y, z_)

vol = np.empty((50, 50, 50), dtype=np.uint8) idx = x2 + y2 + z2 < 0.52 vol[idx] = 1 vol[~idx] = 0

voxel_size = (0.1, 0.1, 0.1)

mesh = pygalmesh.generatefromarray( vol, voxelsize, maxfacetdistance=0.2, maxcell_circumradius=0.1 ) mesh.write("ball.vtk") ```

The code below creates a mesh from the 3D breast phantom from Lou et al available here. The phantom comprises four tissue types (background, fat, fibrograndular, skin, vascular tissues). The generated mesh conforms to tissues interfaces.

```python import pygalmesh import meshio

with open("MergedPhantom.DAT", "rb") as fid: vol = np.fromfile(fid, dtype=np.uint8)

vol = vol.reshape((722, 411, 284)) voxel_size = (0.2, 0.2, 0.2)

mesh = pygalmesh.generatefromarray( vol, voxelsize, maxfacetdistance=0.2, maxcell_circumradius=1.0 ) mesh.write("breast.vtk") ```

In addition, we can specify different mesh sizes for each tissue type. The code below sets the mesh size to 1 mm for the skin tissue (label 4), 0.5 mm for the vascular tissue (label 5), and 2 mm for all other tissues (default).

python mesh = pygalmesh.generate_from_array( vol, voxel_size, max_facet_distance=0.2, max_cell_circumradius={"default": 2.0, 4: 1.0, 5: 0.5}, ) mesh.write("breast_adapted.vtk")

Surface remeshing

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pygalmesh can help remeshing an existing surface mesh, e.g., lion-head.off. On the command line, use

pygalmesh remesh-surface lion-head.off out.vtu -e 0.025 -a 25 -s 0.1 -d 0.001

(see pygalmesh remesh-surface -h for all options) or from Python

```python import pygalmesh

mesh = pygalmesh.remeshsurface( "lion-head.off", maxedgesizeatfeatureedges=0.025, minfacetangle=25, maxradiussurfacedelaunayball=0.1, maxfacetdistance=0.001, 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

On MacOS with homebrew,

brew install cgal eigen

After that, pygalmesh can be installed from the Python Package Index, so with

pip install -U pygalmesh

you can install/upgrade.

Troubleshooting

If pygalmesh fails to build due to fatal error: 'Eigen/Dense' file not found you will need to create a symbolic link for Eigen to be detected, e.g.

cd /usr/local/include sudo ln -sf eigen3/Eigen Eigen

It's possible that eigen3 could be in /usr/include instead of /usr/local/install.

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 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/ and make.

Testing

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

pytest

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.

See here for other mesh generation tools.

License

This software is available under the GPLv3 license as well as a commercial license. If you'd like to use this software commercially, please contact the author.