A FEniCS Tutorial - FEniCS Project

# The diagonals can be right, left or crossed
mesh = UnitSquare(6, 10, ’left’)
mesh = UnitSquare(6, 10, ’crossed’)
# Domain [0,3]x[0,2] with 6x10 divisions and left diagonals
mesh = Rectangle(0, 0, 3, 2, 6, 10, ’left’)
# 6x10x5 boxes in the unit cube, each box gets 6 tetrahedra:
mesh = UnitCube(6, 10, 5)
# Domain [-1,1]x[-1,0]x[-1,2] with 6x10x5 divisions
mesh = Box(-1, -1, -1, 1, 0, 2, 6, 10, 5)
# 10 divisions in radial directions
mesh = UnitCircle(10)
mesh = UnitSphere(10)
4.2 Transforming Mesh Coordinates
A mesh that is denser toward a boundary is often desired to increase accuracy in
that region. Given a mesh with uniformly spaced coordinates x 0 ,...,x M−1 in
[a,b], the coordinate transformation ξ = (x−a)/(b−a) maps x onto ξ ∈ [0,1]. A
new mapping η = ξ s , for some s > 1, stretches the mesh toward ξ = 0 (x = a),
while η = ξ 1/s makes a stretching toward ξ = 1 (x = b). Mapping the η ∈ [0,1]
coordinates back to [a,b] gives new, stretched x coordinates,
( ) s x−a
¯x = a+(b−a)
(86)
b−a
toward x = a, or
¯x = a+(b−a)
( ) 1/s x−a
(87)
b−a
toward x = b
One way of creating more complex geometries is to transform the vertex
coordinates in a rectangular mesh according to some formula. Say we want to
create a part of a hollow cylinder of Θ degrees, with inner radius a and outer
radius b. A standard mapping from polar coordinates to Cartesian coordinates
can be used to generate the hollow cylinder. Given a rectangle in (¯x,ȳ) space
such that a ≤ ¯x ≤ b and 0 ≤ ȳ ≤ 1, the mapping
ˆx = ¯xcos(Θȳ), ŷ = ¯xsin(Θȳ),
takes a point in the rectangular (¯x,ȳ) geometry and maps it to a point (ˆx,ŷ) in
a hollow cylinder.
The corresponding Python code for first stretching the mesh and then mapping
it onto a hollow cylinder looks as follows:
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