discretize.base.BaseRegularMesh.average_edge_to_cell#
- property BaseRegularMesh.average_edge_to_cell#
Averaging operator from edges to cell centers (scalar quantities).
This property constructs a 2nd order averaging operator that maps scalar quantities from edges to cell centers. This averaging operator is used when a discrete scalar quantity defined on mesh edges 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_edges
)scipy.sparse.csr_matrix
The scalar averaging operator from edges to cell centers
- (
Notes
Let \(\boldsymbol{\phi_e}\) be a discrete scalar quantity that lives on mesh edges. average_edge_to_cell constructs a discrete linear operator \(\mathbf{A_{ec}}\) that projects \(\boldsymbol{\phi_e}\) to cell centers, i.e.:
\[\boldsymbol{\phi_c} = \mathbf{A_{ec}} \, \boldsymbol{\phi_e}\]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 edges. The operation is implemented as a matrix vector product, i.e.:
phi_c = Aec @ phi_e
Examples
Here we compute the values of a scalar function on the edges. 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 edges,
>>> phi_e = np.zeros(mesh.nE) >>> xy = mesh.edges >>> phi_e[(xy[:, 1] > 0)] = 25.0 >>> phi_e[(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.
>>> Aec = mesh.average_edge_to_cell >>> phi_c = Aec @ phi_e
And plot the results:
>>> fig = plt.figure(figsize=(11, 5)) >>> ax1 = fig.add_subplot(121) >>> mesh.plot_image(phi_e, ax=ax1, v_type="E") >>> ax1.set_title("Variable at edges", fontsize=16) >>> ax2 = fig.add_subplot(122) >>> mesh.plot_image(phi_c, ax=ax2, v_type="CC") >>> ax2.set_title("Averaged to cell centers", 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(Aec, ms=1) >>> ax1.set_title("Edge Index", fontsize=12, pad=5) >>> ax1.set_ylabel("Cell Index", fontsize=12) >>> plt.show()