discretize.base.BaseRegularMesh.average_face_to_cell#
- property BaseRegularMesh.average_face_to_cell#
Averaging operator from faces to cell centers (scalar quantities).
This property constructs a 2nd order averaging operator that maps scalar quantities from faces to cell centers. This averaging operator is used when a discrete scalar quantity defined on mesh faces must be projected to cell centers. Once constructed, the operator is stored permanently as a property of the mesh. See notes.
- Returns
- (
n_cells
,n_faces
)scipy.sparse.csr_matrix
The scalar averaging operator from faces to cell centers
- (
Notes
Let \(\boldsymbol{\phi_f}\) be a discrete scalar quantity that lives on mesh faces. average_face_to_cell constructs a discrete linear operator \(\mathbf{A_{fc}}\) that projects \(\boldsymbol{\phi_f}\) to cell centers, i.e.:
\[\boldsymbol{\phi_c} = \mathbf{A_{fc}} \, \boldsymbol{\phi_f}\]where \(\boldsymbol{\phi_c}\) approximates the value of the scalar quantity at cell centers. For each cell, we are simply averaging the values defined on its faces. The operation is implemented as a matrix vector product, i.e.:
phi_c = Afc @ phi_f
Examples
Here we compute the values of a scalar function on the faces. We then create an averaging operator to approximate the function at cell centers. 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")
Then we create a scalar variable on faces
>>> phi_f = np.zeros(mesh.nF) >>> xy = mesh.faces >>> phi_f[(xy[:, 1] > 0)] = 25.0 >>> phi_f[(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 cell centers.
>>> Afc = mesh.average_face_to_cell >>> phi_c = Afc @ phi_f
And finally plot the results:
(Source code, png, pdf)
Below, we show a spy plot illustrating the sparsity and mapping of the operator