# Generate Meshes#

FElupe provides a simple mesh generation module mesh. A `Mesh`

instance contains essentially two arrays: one with `points`

and another one containing the cell connectivities, called `cells`

. Only a single `cell_type`

is supported per `Mesh`

. Optionally the `cell_type`

is specified which is used if the mesh is saved as a VTK or a XDMF file. These cell types are identical to cell types used in meshio (VTK types): `line`

, `quad`

and `hexahedron`

for linear lagrange elements or `triangle`

and `tetra`

for 2- and 3-simplices or `VTK_LAGRANGE_HEXAHEDRON`

for 3d lagrange-cells with polynomial shape functions of arbitrary order.

```
import numpy as np
import felupe as fem
points = np.array([
[ 0, 0], # point 1
[ 1, 0], # point 2
[ 0, 1], # point 3
[ 1, 1], # point 4
], dtype=float)
cells = np.array([
[ 0, 1, 3, 2], # point-connectivity of first cell
])
mesh = fem.Mesh(points, cells, cell_type="quad")
# if needed, convert a FElupe mesh to a meshio-mesh
mesh_meshio = mesh.as_meshio()
# view the mesh in an interactive window
mesh.plot().show()
```

```
# take a screenshot of an off-screen view
img = mesh.screenshot(
filename="mesh.png",
transparent_background=True,
)
```

```
# take a screenshot of an off-screen view
img = mesh.screenshot(
filename="mesh.png",
transparent_background=True,
)
```

## A cube by hand#

First let’s start with the generation of a point at `x=0`

, expanded to a line from `x=1`

to `x=3`

with `n=7`

points. Next, the line is expanded into a rectangle. The `z`

argument of `expand()`

represents the total expansion. Again, an expansion of our rectangle leads to a hexahedron. Several other useful functions are available beside `expand()`

: `rotate()`

, `revolve()`

and `merge_duplicate_points()`

. With these simple tools at hand, rectangles, cubes or cylinders may be constructed with ease.

```
vert = fem.Point(a=1)
line = vert.expand(n=7, z=2)
rect = line.expand(n=5, z=5)
cube = rect.expand(n=6, z=3)
cube.plot().show()
```

Alternatively, these mesh-related tools are also provided as methods of a `Mesh`

.

```
cube = fem.mesh.Line(a=1, b=3, n=7).expand(n=5, z=5).expand(n=6, z=3)
cube.plot().show()
```

## Elementary Shapes#

Lines, rectangles, cubes, circles and triangles do not have to be constructed manually each time. Instead, some easier to use classes are povided by FElupe like `Line`

, `Rectangle`

or `Cube`

. For non equi-distant points per axis use `Grid`

.

```
cube = fem.Cube(a=(1, 0, 0), b=(3, 5, 3), n=(7, 5, 6))
cube.plot().show()
```

For circles, there is `Circle`

for the creation of a quad-mesh for a circle.

```
circle = fem.Circle(radius=1.5, centerpoint=[1, 2], n=6, sections=[0, 90, 180, 270])
circle.plot().show()
```

For triangles, there is `Triangle`

for the creation of a quad-mesh for a triangle. For positive cell volumes, the coordinates of `a`

, `b`

and `c`

must be sorted counter-clockwise around the center point.

```
triangle = fem.mesh.Triangle(a=(0, 0), b=(1, 0), c=(0, 1), n=5)
triangle.plot().show()
```

## Corner Modifications#

For a regular `Rectangle`

or a `Cube`

, corners may be modified by `modify_corners()`

. This is sometimes beneficial for compressive states of deformation.

```
rectangle = fem.mesh.Rectangle(n=6).modify_corners()
rectangle.plot().show()
```

## Cylinders#

Cylinders are created by a revolution of a rectangle.

```
r = 25
R = 50
H = 100
rect = fem.Rectangle(a=(-r, 0), b=(-R, H), n=(11, 41))
cylinder = rect.revolve(n=19, phi=-180, axis=1)
cylinder.plot().show()
```

## Fill between boundaries#

Meshed boundaries may be used to fill the area or volume in between for line and quad meshes. A plate with a hole is initiated by a line mesh, which is copied two times for the boundaries. The points arrays are updated for the hole and the upper edge. The face is filled by a quad mesh.

```
n = (11, 9)
phi = np.linspace(1, 0.5, n[0]) * np.pi / 2
line = fem.mesh.Line(n=n[0])
bottom = line.copy(points=0.5 * np.vstack([np.cos(phi), np.sin(phi)]).T)
top = line.copy(
points=np.vstack([np.linspace(0, 1, n[0]), np.linspace(1, 1, n[0])]).T
)
face = bottom.fill_between(top, n=n[1])
plate_with_hole = fem.mesh.concatenate(
[face, face.mirror(normal=[-1, 1, 0])]
).merge_duplicate_points()
plate_with_hole.plot().show()
```

Connect two quad-meshed faces by hexahedrons:

```
x = np.linspace(0, 1, 11)
y = np.linspace(0, 1, 11)
xg, yg = np.meshgrid(x, y, indexing="ij")
zg = (
0.5 + 0.3 * xg**2 + 0.5 * yg**2 - 0.7 * yg ** 3 + np.random.rand(11, 11) / 50
)
grid = fem.Grid(x, y)
top = grid.copy(points=np.hstack([grid.points, zg.reshape(-1, 1)]))
bottom = grid.copy(points=np.hstack([grid.points, 0 * zg.reshape(-1, 1)]))
bottom.points += [0.2, 0.1, 0]
bottom.points *= 0.75
mesh = bottom.fill_between(top, n=6)
mesh.plot().show()
```

## Combinations of elementary shapes#

The elementary shapes are combined to create more complex shapes, e.g. a planar triangular shaped face connected to three arms with rounded ends.

```
rectangle = fem.Rectangle(a=(-1, 0), b=(1, 5), n=(13, 26))
circle = fem.Circle(radius=1, centerpoint=(0, 5), sections=(0, 90), n=4)
triangle = fem.mesh.Triangle(a=(-1, 0), b=(1, 0), c=(0, -np.sqrt(12) / 2), n=7)
arm = fem.mesh.concatenate([rectangle, circle])
center = triangle.points.mean(axis=0)
arms = [arm.rotate(phi, axis=2, center=center) for phi in [0, 120, 240]]
mesh = fem.mesh.concatenate([triangle, *arms]).merge_duplicate_points(decimals=8)
mesh.plot().show()
```

For quad- and hexahedron-meshes it is possible to extract the boundaries of the mesh by a boundary region. The boundary-mesh consists of the quad-cells which have their **first edge** located at the boundary. Hence, these are not the original cells connected to the boundary. The boundary line-mesh is available as a method. In FElupe, boundaries of cell (volumes) are considered as faces and hence, the line-mesh for the edges of a quad-mesh is obtained by a mesh-*face* method of the boundary region.

```
boundary = fem.RegionQuadBoundary(mesh)
boundary.mesh.plot().show()
```

```
boundary.mesh_faces().plot().show()
```

A three-dimensional example demonstrates a combination of two different expansions of a rectangle, fill-betweens of two lines and a circle.

```
circle = fem.Circle(radius=1, centerpoint=(0, 0), sections=(0, 90, 180, 270), n=6)
phi = np.linspace(1, 0.5, 6) * np.pi / 2
line = fem.mesh.Line(n=6)
curve = line.copy(points=1.0 * np.vstack([np.cos(phi), np.sin(phi)]).T)
top = line.copy(points=np.vstack([np.linspace(0, 1.5, 6), np.linspace(1.5, 1.5, 6)]).T)
transition = curve.fill_between(top, n=6)
transition = fem.mesh.concatenate([transition, transition.mirror(normal=[-1, 1, 0])])
rect = fem.Rectangle(a=(-1.5, 1.5), b=(1.5, 5.0), n=(11, 14))
rect.points[:, 0] *= 1 + (rect.points[:, 1] - 1.5) / 10
face = fem.mesh.concatenate([
transition,
transition.mirror(normal=[1, 0, 0]),
fem.mesh.Line(a=-1.5, b=-1, n=6).revolve(n=21, phi=180, axis=2).flip(),
rect
])
mesh = fem.mesh.concatenate([
face.expand(n=6, z=0.5),
circle.expand(n=11, z=1),
]).merge_duplicate_points(decimals=8)
mesh.plot().show()
```

The boundary mesh isn’t visualized correctly in PyVista and in ParaView because there are two duplicated cells at the edges. However, this is not a bug - it’s a feature. Each face on the surface has one attached cell - with the surface face as its first face. Hence, at edges, there are two overlapping cells with different point connectivity.

```
boundary = fem.RegionHexahedronBoundary(mesh)
boundary.mesh.plot().show()
```

## Boundary modification (runouts)#

Indentations (runouts) of the boundary edges or faces are defined by a centerpoint, an axis and their relative amounts (values) per axis. Optionally, the transformation of the point coordinates is restricted to a list of given points.

```
block = plate_with_hole.expand(z=0.5)
x, y, z = block.points.T
solid = block.add_runouts(
centerpoint=[0, 0, 0],
axis=2,
values=[0.07, 0.02],
exponent=5, # shape parameter
normalize=True,
mask=np.arange(block.npoints)[np.sqrt(x**2 + y**2) > 0.5]
)
solid.plot().show()
```

## Triangle and Tetrahedron meshes#

Any quad or tetrahedron mesh may be subdivided (triangulated) to meshes out of Triangles or Tetrahedrons by `triangulate()`

.

```
rectangle = fem.Rectangle(n=5).triangulate()
rectangle.plot().show()
```

```
cube = fem.Cube(n=5).triangulate()
cube.plot().show()
```

```
cube = fem.Cube(n=5).triangulate(mode=0)
cube.plot().show()
```

## Meshes with midpoints#

If a mesh with midpoints is required by a region, functions for edge, face and volume midpoint insertions are provided in `add_midpoints_edges()`

, `add_midpoints_faces()`

and `add_midpoints_volumes()`

. A low-order mesh, e.g. a mesh with cell-type `quad`

, can be converted to a quadratic mesh with `convert()`

. By default, only midpoints on edges are inserted. Hence, the resulting cell-type is `quad8`

. If midpoints on faces are also calculated, the resulting cell-type is `quad9`

.

```
rectangle_quad4 = fem.Rectangle(n=6)
rectangle_quad8 = rectangle_quad4.convert(order=2)
rectangle_quad9 = fem.mesh.convert(rectangle_quad4, order=2, calc_midfaces=True)
```

The same also applies on meshes with triangles.

```
rectangle_triangle3 = fem.Rectangle(n=6).triangulate()
rectangle_triangle6 = rectangle_triangle3.add_midpoints_edges()
```

While views on higher-order meshes are possible, it is suggested to use ParaView for the visualization of meshes with midpoints due to the improved representation of the cells.