4. Built-in meshes

This demo is implemented in a single Python file, demo_built-in.py, and demonstrates use of the built-in meshes in DOLFIN.

This demo illustrates:

• How to define some of the different built-in meshes in DOLFIN

4.1. Problem definition

The demo focuses on the built-in meshes. We will look at the following meshes:

• UnitIntervalMesh
• UnitSquareMesh
• RectangleMesh
• UnitCubeMesh
• BoxMesh

4.2. Implementation

First, the dolfin module is imported:

from dolfin import *

The first mesh we make is a mesh over the unit interval \((0,1)\). UnitIntervalMesh takes the number of intervals \((n_x)\) as input argument, and the total number of vertices is therefore \((n_x+1)\).

mesh = UnitIntervalMesh(10)
print("Plotting a UnitIntervalMesh")
plot(mesh, title="Unit interval")

This produces a mesh looking as follows:

We then make our first version of a mesh on the unit square \([0,1] \times [0,1]\). We must give the number of cells in the horizontal and vertical directions as the first two arguments to UnitSquareMesh. There is a third optional argument that indicates the direction of the diagonals. This can be set to “left”, “right”, “right/left”, “left/right”, or “crossed”. We can also omit this argument and thereby use the default direction “right”.

mesh = UnitSquareMesh(10, 10)
print("Plotting a UnitSquareMesh")
plot(mesh, title="Unit square")

Our second version of a mesh on the unit square has diagonals to the left, the third version has crossed diagonals and our final version has diagonals to both left and right:

mesh = UnitSquareMesh(10, 10, "left")
print("Plotting a UnitSquareMesh")
plot(mesh, title="Unit square (left)")

mesh = UnitSquareMesh(10, 10, "crossed")
print("Plotting a UnitSquareMesh")
plot(mesh, title="Unit square (crossed)")

mesh = UnitSquareMesh(10, 10, "right/left")
print("Plotting a UnitSquareMesh")
plot(mesh, title="Unit square (right/left)")

The class RectangleMesh (\(x_0,y_0,x_1,y_1,n_x,n_y\), direction) creates a mesh on a rectangle with one corner in \((x_0,y_0)\) and the opposite corner in \((x_1,y_1)\). \(n_x\) and \(n_y\) specify the number of cells in the \(x\)- and \(y\)-directions, and as above the direction of the diagonals is given as a final optional argument (“left”, “right”, “left/right”, or “crossed”). In the first mesh we use the default direction (“right”) of the diagonal, and in the second mesh we use diagonals to both left and right.

mesh = RectangleMesh(0.0, 0.0, 10.0, 4.0, 10, 10)
print("Plotting a RectangleMesh")
plot(mesh, title="Rectangle")

mesh = RectangleMesh(-3.0, 2.0, 7.0, 6.0, 10, 10, "right/left")
print("Plotting a RectangleMesh")
plot(mesh, title="Rectangle (right/left)")

To make a mesh of the 3D unit cube \([0,1] \times [0,1] \times [0,1]\), we use UnitCubeMesh. UnitCubeMesh takes the number of cells in the \(x\)-, \(y\)- and \(z\)-direction as the only three arguments.

mesh = UnitCubeMesh(10, 10, 10)
print("Plotting a UnitCubeMesh")
plot(mesh, title="Unit cube")

Finally we will demonstrate a mesh on a rectangular prism in 3D. BoxMesh (\(x_0,y_0,z_0,x_1,y_1,z_1,x_n,y_n,z_n\)) takes the coordinates of the first corner(\(x_0,y_0,z_0\)) as the three first arguments, the coordinates of the opposite corner (\(x_1,y_1,z_1\)) as the next three arguments, while the last three arguments specify the number of points in the \(x\)-, \(y\)- and \(z\)-direction.

Meshes for more complex geometries may be created using the mshr library, which functions as a plugin to DOLFIN, providing support for Constructive Solid Geometry (CSG) and mesh generation. For more details, refer to the mshr documentation.

mesh = BoxMesh(0.0, 0.0, 0.0, 10.0, 4.0, 2.0, 10, 10, 10)
print("Plotting a BoxMesh")
plot(mesh, title="Box")

By calling interactive we are allowed to resize, move and rotate the plots.

interactive()

4.3. Complete code

from __future__ import print_function
from dolfin import *

mesh = UnitIntervalMesh(10)
print("Plotting a UnitIntervalMesh")
plot(mesh, title="Unit interval")

mesh = UnitSquareMesh(10, 10)
print("Plotting a UnitSquareMesh")
plot(mesh, title="Unit square")

mesh = UnitSquareMesh(10, 10, "left")
print("Plotting a UnitSquareMesh")
plot(mesh, title="Unit square (left)")

mesh = UnitSquareMesh(10, 10, "crossed")
print("Plotting a UnitSquareMesh")
plot(mesh, title="Unit square (crossed)")

mesh = UnitSquareMesh(10, 10, "right/left")
print("Plotting a UnitSquareMesh")
plot(mesh, title="Unit square (right/left)")

mesh = RectangleMesh(0.0, 0.0, 10.0, 4.0, 10, 10)
print("Plotting a RectangleMesh")
plot(mesh, title="Rectangle")

mesh = RectangleMesh(-3.0, 2.0, 7.0, 6.0, 10, 10, "right/left")
print("Plotting a RectangleMesh")
plot(mesh, title="Rectangle (right/left)")

mesh = UnitCubeMesh(10, 10, 10)
print("Plotting a UnitCubeMesh")
plot(mesh, title="Unit cube")

mesh = BoxMesh(0.0, 0.0, 0.0, 10.0, 4.0, 2.0, 10, 10, 10)
print("Plotting a BoxMesh")
plot(mesh, title="Box")

interactive()