# 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:

## 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)")  Unstructured ellipsoid and ellipse meshes can be created if DOLFIN is configured with CGAL. Using CircleMesh For a circle centered at (0, 0) with radius 1 and cell size 0.2:

mesh = CircleMesh(Point(0.0, 0.0), 1.0, 0.2)
print "Plotting a CircleMesh"
plot(mesh, title="Circle (unstructured)")


Using EllipseMesh for an ellipse centered at (0, 0) with ‘radii’ of 3 and 1 in the $$x$$- and $$y$$ directions, respectively, and cell size 0.2:

mesh = EllipseMesh(Point(0.0, 0.0), [3.0, 1.0], 0.2)
print "Plotting an EllipseMesh"
plot(mesh, title="Ellipse mesh (unstructured)")


Using SphereMesh for a sphere centered at (0, 0, 0) with radius 1 and cell size 0.2:

mesh = SphereMesh(Point(0.0, 0.0, 0.0), 1.0, 0.2)
print "Plotting a SphereMesh"
plot(mesh, title="Sphere mesh (unstructured)")


Using EllipsoidMesh For an ellipsoid centered at (0, 0, 0.0), with ‘radii’ of 3, 1 and 2 in the $$x$$-, $$y$$ and :mathz-directions, respectively, and cell size 0.2:

mesh = EllipsoidMesh(Point(0.0, 0.0, 0.0), [3.0, 1.0, 2.0], 0.2)
print "Plotting an EllipsoidMesh"
plot(mesh, title="Ellipsoid mesh (unstructured)")


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.

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 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)")

if has_cgal():
mesh = CircleMesh(Point(0.0, 0.0), 1.0, 0.2)
print "Plotting a CircleMesh"
plot(mesh, title="Circle (unstructured)")

mesh = EllipseMesh(Point(0.0, 0.0), [3.0, 1.0], 0.2)
print "Plotting an EllipseMesh"
plot(mesh, title="Ellipse mesh (unstructured)")

mesh = SphereMesh(Point(0.0, 0.0, 0.0), 1.0, 0.2)
print "Plotting a SphereMesh"
plot(mesh, title="Sphere mesh (unstructured)")

mesh = EllipsoidMesh(Point(0.0, 0.0, 0.0), [3.0, 1.0, 2.0], 0.2)
print "Plotting an EllipsoidMesh"
plot(mesh, title="Ellipsoid mesh (unstructured)")

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()