discretize.utils.volume_tetrahedron#
- discretize.utils.volume_tetrahedron(xyz, A, B, C, D)[source]#
Return the tetrahedron volumes for a specified set of verticies.
Let xyz be an (n, 3) array denoting a set of vertex locations. Any 4 vertex locations a, b, c and d can be used to define a tetrahedron. For the set of tetrahedra whose verticies are indexed in vectors A, B, C and D, this function returns the corresponding volumes. See algorithm: https://en.wikipedia.org/wiki/Tetrahedron#Volume
\[vol = {1 \over 6} \big | ( \mathbf{a - d} ) \cdot ( ( \mathbf{b - d} ) \times ( \mathbf{c - d} ) ) \big |\]- Parameters:
- xyz(
n_pts
, 3)numpy.ndarray
x,y, and z locations for all verticies
- A(
n_tetra
)numpy.ndarray
of
int
Vector containing the indicies for the a vertex locations
- B(
n_tetra
)numpy.ndarray
of
int
Vector containing the indicies for the b vertex locations
- C(
n_tetra
)numpy.ndarray
of
int
Vector containing the indicies for the c vertex locations
- D(
n_tetra
)numpy.ndarray
of
int
Vector containing the indicies for the d vertex locations
- xyz(
- Returns:
- (
n_tetra
)numpy.ndarray
Volumes of the tetrahedra whose vertices are indexed by A, B, C and D.
- (
Examples
Here we define a small 3D tensor mesh. 4 nodes are chosen to be the verticies of a tetrahedron. We compute the volume of this tetrahedron. Note that xyz locations for the verticies can be scattered and do not require regular spacing.
>>> from discretize.utils import volume_tetrahedron >>> from discretize import TensorMesh >>> import numpy as np >>> import matplotlib.pyplot as plt >>> import matplotlib as mpl >>> mpl.rcParams.update({"font.size": 14})
Define corners of a uniform cube
>>> h = [1, 1] >>> mesh = TensorMesh([h, h, h]) >>> xyz = mesh.nodes
Specify the indicies of the corner points
>>> A = np.array([0]) >>> B = np.array([6]) >>> C = np.array([8]) >>> D = np.array([24])
Compute volume for all tetrahedra and the extract first one
>>> vol = volume_tetrahedron(xyz, A, B, C, D) >>> vol = vol[0] >>> vol array([1.33333333])
Plotting small mesh and tetrahedron
>>> fig = plt.figure(figsize=(7, 7)) >>> ax = fig.gca(projection='3d') >>> mesh.plot_grid(ax=ax) >>> k = [0, 6, 8, 0, 24, 6, 24, 8] >>> xyz_tetra = xyz[k, :] >>> ax.plot(xyz_tetra[:, 0], xyz_tetra[:, 1], xyz_tetra[:, 2], 'r') >>> ax.text(-0.25, 0., 3., 'Volume of the tetrahedron: {:g} $m^3$'.format(vol)) >>> plt.show()