discretize.base.BaseTensorMesh.average_node_to_edge#

property BaseTensorMesh.average_node_to_edge#

Averaging operator from nodes to edges (scalar quantities).

This property constructs a 2nd order averaging operator that maps scalar quantities from nodes to edges; scalar at edges is organized in a 1D numpy.array of the form [x-edges, y-edges, z-edges]. This averaging operator is used when a discrete scalar quantity defined on mesh nodes must be projected to edges. Once constructed, the operator is stored permanently as a property of the mesh. See notes.

Returns:
(n_edges, n_nodes) scipy.sparse.csr_matrix

The scalar averaging operator from nodes to edges

Notes

Let ϕn be a discrete scalar quantity that lives on mesh nodes. average_node_to_edge constructs a discrete linear operator Ane that projects ϕn to edges, i.e.:

ϕe=Aneϕn

where ϕe approximates the value of the scalar quantity at edges. For each edge, we are simply averaging the values defined on the nodes it connects. The operation is implemented as a matrix vector product, i.e.:

phi_e = Ane @ 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 edges. 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 edges.

>>> Ane = mesh.average_node_to_edge
>>> phi_e = Ane @ 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_e, ax=ax2, v_type="E")
>>> ax2.set_title("Averaged to edges")
>>> plt.show()

(Source code, png, pdf)

../../_images/discretize-base-BaseTensorMesh-average_node_to_edge-1_00_00.png

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(Ane, ms=1)
>>> ax1.set_title("Node Index", fontsize=12, pad=5)
>>> ax1.set_ylabel("Edge Index", fontsize=12)
>>> plt.show()

(png, pdf)

../../_images/discretize-base-BaseTensorMesh-average_node_to_edge-1_01_00.png