Meshing and Connectivity for FEM in 2D¶

Copyright (C) 2010-2020 Luke Olson
Copyright (C) 2020 Andreas Kloeckner

MIT License Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
In [7]:
import numpy as np
import scipy.linalg as la
import scipy.sparse as sparse

import matplotlib.pyplot as plt

Mesh the Domain¶

This uses meshpy, which under the hood uses Triangle.

pip install meshpy to install.

NB: Triangle is not open-source software. If you are looking for a quality mesher that is open-source (but a bit more complex to use), look at Gmsh.

In [4]:
import meshpy.triangle as triangle

def round_trip_connect(start, end):
    return [(i, i+1) for i in range(start, end)] + [(end, start)]

def make_mesh():
    points = [(-1, -1), (1, -1), (1, 1), (-1, 1)]
    facets = round_trip_connect(0, len(points)-1)

    circ_start = len(points)
    points.extend(
            (0.25 * np.cos(angle), 0.25 * np.sin(angle))
            for angle in np.linspace(0, 2*np.pi, 30, endpoint=False))

    facets.extend(round_trip_connect(circ_start, len(points)-1))

    def needs_refinement(vertices, area):
        bary = np.sum(np.array(vertices), axis=0)/3
        max_area = 0.01 + la.norm(bary, np.inf)*0.01
        return bool(area > max_area)

    info = triangle.MeshInfo()
    info.set_points(points)
    info.set_facets(facets)

    built_mesh = triangle.build(info, refinement_func=needs_refinement)
    return np.array(built_mesh.points), np.array(built_mesh.elements)

V, E = make_mesh()
In [5]:
nv = len(V)
ne = len(E)
print(V.shape)
print(E.shape)
print(E.max())
X, Y = V[:, 0], V[:, 1]

(237, 2)
(438, 3)
236
In [8]:
plt.figure(figsize=(7,7))
plt.gca().set_aspect("equal")
plt.triplot(X, Y, E)
Out[8]:
[<matplotlib.lines.Line2D at 0x7fb9a984f010>,
 <matplotlib.lines.Line2D at 0x7fb9a984f250>]

Explore Connectivity¶

Compute the vertex-to-edge connections as V2E.

In [ ]:
print('V shape: ', V.shape)
print('E shape: ', E.shape)
In [10]:
element_ids = np.empty((ne, 3), dtype=np.intp)
element_ids[:] = np.arange(ne).reshape(-1, 1)

V2E = sparse.coo_matrix(
    (np.ones((ne*3,), dtype=np.intp), 
     (E.ravel(), 
      element_ids.ravel(),)))
print('V2E shape: ', V2E.shape)

V2E shape:  (237, 438)

Compute

  • the element-to-element connections E2E, and
  • the vertex-to-vertex connections V2V.
In [12]:
E2E = V2E.T @ V2E
V2V = V2E @ V2E.T
In [13]:
print('V2V shape: ', V2V.shape)
print('E2E shape: ', E2E.shape)

V2V shape:  (237, 237)
E2E shape:  (438, 438)

Plot the vertex degrees.

In [14]:
plt.scatter(X, Y, c=V2V.diagonal(), clip_on=False)
plt.colorbar()
plt.show()

Explain this:

In [15]:
E2E.diagonal()
Out[15]:
array([3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3])
In [16]:
E2E.data[:] = 1
num_neighbors = np.array(E2E.sum(axis=0)).ravel()
plt.tripcolor(X, Y, triangles=E, facecolors=num_neighbors)
plt.colorbar()
plt.show()
In [ ]: