property SimplexMesh.average_cell_to_face#

Averaging operator from cell centers to faces (scalar quantities).

This property constructs an averaging operator that maps scalar quantities from cell centers to face. This averaging operator is used when a discrete scalar quantity defined cell centers must be projected to faces. Once constructed, the operator is stored permanently as a property of the mesh. See notes.

(n_faces, n_cells) scipy.sparse.csr_matrix

The scalar averaging operator from cell centers to faces


Let \(\boldsymbol{\phi_c}\) be a discrete scalar quantity that lives at cell centers. average_cell_to_face constructs a discrete linear operator \(\mathbf{A_{cf}}\) that projects \(\boldsymbol{\phi_c}\) to faces, i.e.:

\[\boldsymbol{\phi_f} = \mathbf{A_{cf}} \, \boldsymbol{\phi_c}\]

where \(\boldsymbol{\phi_f}\) approximates the value of the scalar quantity at the faces. For each face, we are performing a weighted average between the values at adjacent cell centers. In 1D, where adjacent cells \(i\) and \(i+1\) have widths \(h_i\) and \(h_{i+1}\), \(\phi\) on face is approximated by:

\[\phi_{i \! + \! 1/2} \approx \frac{h_{i+1} \phi_i + h_i \phi_{i+1}}{h_i + h_{i+1}}\]

On boundary faces, nearest neighbour is used to extrapolate the value from the nearest cell center. Once the operator is construct, the averaging is implemented as a matrix vector product, i.e.:

phi_f = Acf @ phi_c


Here we compute the values of a scalar function at cell centers. We then create an averaging operator to approximate the function on the faces. We choose to define a scalar function that is strongly discontinuous in some places to demonstrate how the averaging operator will smooth out discontinuities.

We start by importing the necessary packages and defining a mesh.

>>> from discretize import TensorMesh
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> h = np.ones(40)
>>> mesh = TensorMesh([h, h], x0="CC")

Create a scalar variable at cell centers

>>> phi_c = np.zeros(mesh.nC)
>>> xy = mesh.cell_centers
>>> phi_c[(xy[:, 1] > 0)] = 25.0
>>> phi_c[(xy[:, 1] < -10.0) & (xy[:, 0] > -10.0) & (xy[:, 0] < 10.0)] = 50.0

Next, we construct the averaging operator and apply it to the discrete scalar quantity to approximate the value at the faces.

>>> Acf = mesh.average_cell_to_face
>>> phi_f = Acf @ phi_c

Plot the results

>>> fig = plt.figure(figsize=(11, 5))
>>> ax1 = fig.add_subplot(121)
>>> mesh.plot_image(phi_c, ax=ax1, v_type="CC")
>>> ax1.set_title("Variable at cell centers", fontsize=16)
>>> ax2 = fig.add_subplot(122)
>>> mesh.plot_image(phi_f, ax=ax2, v_type="F")
>>> ax2.set_title("Averaged to faces", fontsize=16)
>>> plt.show()

(Source code, png, pdf)


Below, we show a spy plot illustrating the sparsity and mapping of the operator.

>>> fig = plt.figure(figsize=(9, 9))
>>> ax1 = fig.add_subplot(111)
>>> ax1.spy(Acf, ms=1)
>>> ax1.set_title("Cell Index", fontsize=12, pad=5)
>>> ax1.set_ylabel("Face Index", fontsize=12)
>>> plt.show()

(png, pdf)