def create_transformation_matrix(coors):
r"""
Create a transposed coordinate transformation matrix, that transforms 3D
coordinates of quadrilateral cell vertices so that the transformed vertices
of a plane cell are in the :math:`x-y` plane. The rotation is performed
w.r.t. the centres of quadrilaterals.
Parameters
----------
coors : array
The coordinates of cell vertices, shape `(n_el, 4, 3)`.
Returns
-------
mtx_t : array
The transposed transformation matrix :math:`T`, i.e.
:math:`X_{inplane} = T^T X_{3D}`.
Notes
-----
:math:`T = [t_1, t_2, n]`, where :math:`t_1`, :math:`t_2`, are unit
in-plane (column) vectors and :math:`n` is the unit normal vector,
all mutually orthonormal.
"""
# Local coordinate system.
t1 = coors[:, 1, :] + coors[:, 2, :] - coors[:, 0, :] - coors[:, 3, :]
t2 = coors[:, 2, :] + coors[:, 3, :] - coors[:, 0, :] - coors[:, 1, :]
n = nm.cross(t1, t2)
t2 = nm.cross(n, t1)
t1 = t1 / norm(t1)[:, None]
t2 = t2 / norm(t2)[:, None]
n = n / norm(n)[:, None]
# Coordinate transformation matrix (transposed!).
mtx_t = nm.concatenate((t1[:, :, None],
t2[:, :, None],
n[:, :, None]), axis=2)
return mtx_t
def create_transformation_matrix(coors):
"""
Create a transposed coordinate transformation matrix, that
transforms 3D coordinates of element face nodes so that the
transformed nodes are in the `x-y` plane. The rotation is performed
w.r.t. the first node of each face.
Parameters
----------
coors : array
The coordinates of element nodes, shape `(n_el, n_ep, dim)`.
Returns
-------
mtx_t : array
The transposed transformation matrix :math:`T`, i.e.
:math:`X_{inplane} = T^T X_{3D}`.
Notes
-----
:math:`T = [t_1, t_2, n]`, where :math:`t_1`, :math:`t_2`, are unit
in-plane (column) vectors and :math:`n` is the unit normal vector,
all mutually orthonormal.
"""
# Local coordinate system.
t1 = coors[:, 1, :] - coors[:, 0, :]
t2 = coors[:, -1, :] - coors[:, 0, :]
n = nm.cross(t1, t2)
t2 = nm.cross(n, t1)
t1 = t1 / norm(t1)[:, None]
t2 = t2 / norm(t2)[:, None]
n = n / norm(n)[:, None]
# Coordinate transformation matrix (transposed!).
mtx_t = nm.concatenate((t1[:, :, None],
t2[:, :, None],
n[:, :, None]), axis=2)
return mtx_t