def _staggered_curl_3d(self):
"""
Calculates the curl operator on a staggered three-dimensional field.
The resulting vector field is a staggered grid.
If the velocities of the vector potential were sampled at the lower faces of a cube, the resulting velocities
are sampled at the centers of the upper edges.
:param vector_potential: three-dimensional vector potential
:return: three-dimensional staggered vector field
"""
kernel = np.zeros((2, 2, 2, 3, 3), np.float32)
derivative = np.array([-1, 1])
# x-component: dz/dy - dy/dz
kernel[0, :, 0, 2, 0] = derivative
kernel[:, 0, 0, 1, 0] = -derivative
# y-component: dx/dz - dz/dx
kernel[:, 0, 0, 0, 1] = derivative
kernel[0, 0, :, 2, 1] = -derivative
# z-component: dy/dx - dx/dy
kernel[0, 0, :, 1, 2] = derivative
kernel[0, :, 0, 0, 2] = -derivative
vector_potential = math.pad(self.staggered,
[[0, 0], [0, 1], [0, 1], [0, 1], [0, 0]],
"SYMMETRIC")
vector_field = math.conv(vector_potential, kernel, padding="VALID")
return StaggeredGrid(vector_field)
def staggered_curl_2d(grid, pad_width=(1, 2)):
assert isinstance(grid, CenteredGrid)
kernel = math.zeros((3, 3, 1, 2))
kernel[1, :, 0, 0] = [0, 1, -1] # y-component: - dz/dx
kernel[:, 1, 0, 1] = [0, -1, 1] # x-component: dz/dy
scalar_potential = grid.padded([pad_width, pad_width]).data
vector_field = math.conv(scalar_potential, kernel, padding='valid')
return StaggeredGrid(vector_field, box=grid.box)
def _staggered_curl_2d(self):
kernel = np.zeros((2, 2, 1, 2), np.float32)
derivative = np.array([-1, 1])
# x-component: dz/dy
kernel[:, 0, 0, 0] = derivative
# y-component: - dz/dx
kernel[0, :, 0, 1] = -derivative
scalar_potential = math.pad(self.staggered,
[[0, 0], [0, 1], [0, 1], [0, 0]],
"SYMMETRIC")
vector_field = math.conv(scalar_potential, kernel, padding="VALID")
return StaggeredGrid(vector_field)
def _conv_laplace_3d(tensor):
kernel = np.zeros((3, 3, 3, 1, 1), np.float32)
kernel[1, 1, 1, 0, 0] = -6
kernel[(0, 1, 1, 1, 1, 2), (1, 0, 2, 1, 1, 1), (1, 1, 1, 0, 2, 1), 0,
0] = 1
return math.conv(tensor, kernel, padding="VALID")