Backward step

The backward step is a mesh (or domain) in \(\mathbb{R}^n\), \(n\in\left\lbrace2,3\right\rbrace\). The domain is illustrated in the following figure.

Fig. 1 The illustration of the backward step domain.

When \(n=3\), The domain is extended to the thrid axis, \(z\)-axis, perpendicular to the plane. The parameters are

backward_step(x1=1, x2=1, y1=0.25, y2=0.25, z=None, periodic=False)

Parameters:

x1float, default=1

See the illustration.

x2float, default=1

See the illustration.

y1float, default=0.25

See the illustration.

y2float, default=0.25

See the illustration.

zfloat, None, default=None

When it is None, it gives a two-dimensional domain. Otherwise, \((z>0)\), it gives a three-dimensional one.

periodicbool, default=False

When the domain is 3d, whether it is periodic along the z-axis?

Boundary units

The domain is divided into three regions,

region 0	bottom-right region, \([x_1, (x_1+x_2)]\times[0, y_1]\)
region 1	top-right region , \([x_1, (x_1+x_2)]\times[y_1, (y_1+y_2)]\)
region 2	top-left region, \([0, x_1]\times[y_1, (y_1+y_2)]\)

Thus, for a 2-dimensional domain, it has 8 boundary units, i.e.,

>>> boundary_units_set = {
...     0: [1, 1, 1, 0],
...     1: [0, 1, 0, 1],
...     2: [1, 0, 1, 1],
... }

For example, for the bottom-right region 0, its left (\(x^-\)), right (\(x^+\)) and bottom (\(y^-\)) faces are boundary units, while its north (\(y^+\)) face is not. So we have 0: [1, 1, 1, 0] in the set.

And a 3-dimensional domain, it has 8 + 6 (\(2\times3\) \(z\)-direction) boundary units, i.e.,

>>> boundary_units_set = {
...     0: [1, 1, 1, 0, 1, 1],
...     1: [0, 1, 0, 1, 1, 1],
...     2: [1, 0, 1, 1, 1, 1],
... }

Examples

3d

Below codes will lead to a three-dimensional backward step mesh of \(3*5*5*5\) elements (because the domain is splited into 3 regions).

>>> ph.config.set_embedding_space_dim(3)
>>> manifold = ph.manifold(3)
>>> mesh = ph.mesh(manifold)
>>> msepy, obj = ph.fem.apply('msepy', locals())
>>> manifold = obj['manifold']
>>> mesh = obj['mesh']
>>> msepy.config(manifold)('backward_step')
>>> msepy.config(mesh)([5, 5, 5])
>>> mesh.visualize(saveto=None_or_custom_path_3)  
<Figure size ...

Fig. 2 The three-dimensional backward step mesh of \(3*5*5*5\) elements.

2d

To make a two-dimensional backward step mesh of \(3*24*6\) elements, just do

>>> ph.config.set_embedding_space_dim(2)
>>> manifold = ph.manifold(2)
>>> mesh = ph.mesh(manifold)
>>> msepy, obj = ph.fem.apply('msepy', locals())
>>> manifold = obj['manifold']
>>> mesh = obj['mesh']
>>> msepy.config(manifold)('backward_step')
>>> msepy.config(mesh)([24, 6])
>>> mesh.visualize(saveto=None_or_custom_path_2)  
<Figure size ...

Fig. 3 The two-dimensional backward step mesh of \(3*24*6\) elements.

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