Tensor meshes

Tensor meshes are the most basic class of meshes that can be created with discretize. They belong to the class (TensorMesh). Tensor meshes can be defined in 1, 2 or 3 dimensions. Here we demonstrate:

• How to create basic tensor meshes

• How to include padding cells

• How to plot tensor meshes

• How to extract properties from meshes

Import Packages

Here we import the packages required for this tutorial.

from discretize import TensorMesh
import matplotlib.pyplot as plt
import numpy as np

# sphinx_gallery_thumbnail_number = 3

Basic Example

The easiest way to define a tensor mesh is to define the cell widths in x, y and z as 1D numpy arrays. And to provide the position of the bottom southwest corner of the mesh. We demonstrate this here for a 2D mesh (thus we do not need to consider the z-dimension).

ncx = 10  # number of core mesh cells in x
ncy = 15  # number of core mesh cells in y
dx = 15  # base cell width x
dy = 10  # base cell width y
hx = dx * np.ones(ncx)
hy = dy * np.ones(ncy)

x0 = 0
y0 = -150

mesh = TensorMesh([hx, hy], x0=[x0, y0])

mesh.plot_grid()

<Axes: xlabel='x1', ylabel='x2'>

Padding Cells and Plotting

For practical purposes, the user may want to define a region where the cell widths are increasing/decreasing in size. For example, padding is often used to define a large domain while reducing the total number of mesh cells. Here we demonstrate how to create tensor meshes that have padding cells.

ncx = 10  # number of core mesh cells in x
ncy = 15  # number of core mesh cells in y
dx = 15  # base cell width x
dy = 10  # base cell width y
npad_x = 4  # number of padding cells in x
npad_y = 4  # number of padding cells in y
exp_x = 1.25  # expansion rate of padding cells in x
exp_y = 1.25  # expansion rate of padding cells in y

# Use a list of tuples to define cell widths in each direction. Each tuple
# contains the cell width, number of cells and the expansion factor. A
# negative sign is used to indicate an interval where cells widths go
# from largest to smallest.
hx = [(dx, npad_x, -exp_x), (dx, ncx), (dx, npad_x, exp_x)]
hy = [(dy, npad_y, -exp_y), (dy, ncy), (dy, npad_y, exp_y)]

# We can use flags 'C', '0' and 'N' to shift the xyz position of the mesh
# relative to the origin
mesh = TensorMesh([hx, hy], x0="CN")

# We can apply the plot_grid method and output to a specified axes object
fig = plt.figure(figsize=(6, 6))
ax = fig.add_subplot(111)
mesh.plot_grid(ax=ax)
ax.set_xbound(mesh.x0[0], mesh.x0[0] + np.sum(mesh.h[0]))
ax.set_ybound(mesh.x0[1], mesh.x0[1] + np.sum(mesh.h[1]))
ax.set_title("Tensor Mesh")

Text(0.5, 1.0, 'Tensor Mesh')

Extracting Mesh Properties

Once the mesh is created, you may want to extract certain properties. Here, we show some properties that can be extracted from 2D meshes.

ncx = 10  # number of core mesh cells in x
ncy = 15  # number of core mesh cells in y
dx = 15  # base cell width x
dy = 10  # base cell width y
npad_x = 4  # number of padding cells in x
npad_y = 4  # number of padding cells in y
exp_x = 1.25  # expansion rate of padding cells in x
exp_y = 1.25  # expansion rate of padding cells in y

hx = [(dx, npad_x, -exp_x), (dx, ncx), (dx, npad_x, exp_x)]
hy = [(dy, npad_y, -exp_y), (dy, ncy), (dy, npad_y, exp_y)]

mesh = TensorMesh([hx, hy], x0="C0")

# The bottom west corner
x0 = mesh.x0

# The total number of cells
nC = mesh.nC

# An (nC, 2) array containing the cell-center locations
cc = mesh.gridCC

# A boolean array specifying which cells lie on the boundary
bInd = mesh.cell_boundary_indices

# Plot the cell areas (2D "volume")
s = mesh.cell_volumes

fig = plt.figure(figsize=(6, 6))
ax = fig.add_subplot(111)
mesh.plot_image(s, grid=True, ax=ax)
ax.set_xbound(mesh.x0[0], mesh.x0[0] + np.sum(mesh.h[0]))
ax.set_ybound(mesh.x0[1], mesh.x0[1] + np.sum(mesh.h[1]))
ax.set_title("Cell Areas")

Text(0.5, 1.0, 'Cell Areas')

3D Example

Here we show how the same approach can be used to create and extract properties from a 3D tensor mesh.

nc = 10  # number of core mesh cells in x, y and z
dh = 10  # base cell width in x, y and z
npad = 5  # number of padding cells
exp = 1.25  # expansion rate of padding cells

h = [(dh, npad, -exp), (dh, nc), (dh, npad, exp)]
mesh = TensorMesh([h, h, h], x0="C00")

# The bottom southwest corner
x0 = mesh.x0

# The total number of cells
nC = mesh.nC

# An (nC, 3) array containing the cell-center locations
cc = mesh.gridCC

# A boolean array specifying which cells lie on the boundary
bInd = mesh.cell_boundary_indices

# The cell volumes
v = mesh.cell_volumes

# Plot all cells volumes or plot cell volumes for a particular horizontal slice
fig = plt.figure(figsize=(9, 4))
ax1 = fig.add_subplot(121)
ax2 = fig.add_subplot(122)

mesh.plot_image(np.log10(v), grid=True, ax=ax1)
ax1.set_title("All Cell Log-Volumes")

cplot = mesh.plot_slice(np.log10(v), grid=True, ax=ax2, normal="Z", ind=2)
cplot[0].set_clim(np.min(np.log10(v)), np.max(np.log10(v)))
ax2.set_title("Cell Log-Volumes #2")

Text(0.5, 1.0, 'Cell Log-Volumes #2')