discretize.base.BaseRegularMesh.average_node_to_face#
- property BaseRegularMesh.average_node_to_face#
Averaging operator from nodes to faces (scalar quantities).
This property constructs a 2nd order averaging operator that maps scalar quantities from nodes to edges; scalar at faces is organized in a 1D numpy.array of the form [x-faces, y-faces, z-faces]. This averaging operator is used when a discrete scalar quantity defined on mesh nodes must be projected to faces. Once constructed, the operator is stored permanently as a property of the mesh. See notes.
- Returns:
- (
n_faces
,n_nodes
)scipy.sparse.csr_matrix
The scalar averaging operator from nodes to faces
- (
Notes
Let \(\boldsymbol{\phi_n}\) be a discrete scalar quantity that lives on mesh nodes. average_node_to_face constructs a discrete linear operator \(\mathbf{A_{nf}}\) that projects \(\boldsymbol{\phi_n}\) to faces, i.e.:
\[\boldsymbol{\phi_f} = \mathbf{A_{nf}} \, \boldsymbol{\phi_n}\]where \(\boldsymbol{\phi_f}\) approximates the value of the scalar quantity at faces. For each face, we are simply averaging the values at the nodes which outline the face. The operation is implemented as a matrix vector product, i.e.:
phi_f = Anf @ phi_n
Examples
Here we compute the values of a scalar function on the nodes. We then create an averaging operator to approximate the function at 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")
Then we, create a scalar variable on nodes
>>> phi_n = np.zeros(mesh.nN) >>> xy = mesh.nodes >>> phi_n[(xy[:, 1] > 0)] = 25.0 >>> phi_n[(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 on the faces.
>>> Anf = mesh.average_node_to_face >>> phi_f = Anf @ phi_n
Plot the results,
>>> fig = plt.figure(figsize=(11, 5)) >>> ax1 = fig.add_subplot(121) >>> mesh.plot_image(phi_n, ax=ax1, v_type="N") >>> ax1.set_title("Variable at nodes") >>> ax2 = fig.add_subplot(122) >>> mesh.plot_image(phi_f, ax=ax2, v_type="F") >>> ax2.set_title("Averaged to faces") >>> 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(Anf, ms=1) >>> ax1.set_title("Node Index", fontsize=12, pad=5) >>> ax1.set_ylabel("Face Index", fontsize=12) >>> plt.show()