biosspheres.formulations.massmatrices#

This module has implementation for routines that return mass matrices for one or several spheres when using spherical harmonics basis as test and trial functions in a boundary integral formulation setting. All routines return the diagonal of the matrix (the rest of the entries are zero).

Routine listings#

j_block two_j_blocks n_j_blocks n_two_j_blocks

Functions

`j_block`(big_l, r[, azimuthal])	Returns a numpy array with the boundary integral operator corresponding to the identity evaluated and tested with spherical harmonics of order 0 in a sphere of radius r.
`n_j_blocks`(big_l, radii[, azimuthal])	Returns a numpy array with the result concatenated of the j_block(big_l, r, azimuthal) for each value r in radii.
`n_two_j_blocks`(big_l, radii[, azimuthal])	Returns a numpy array with the result concatenated of the two_j_blocks(big_l, r, azimuthal) for each value r in radii.
`two_j_blocks`(big_l, r[, azimuthal])	Returns a numpy array with the result of j_block(big_l, r, azimuthal) concatenated with itself.

biosspheres.formulations.massmatrices.j_block(big_l: int, r: float, azimuthal: bool = True) → ndarray[source]#

Returns a numpy array with the boundary integral operator corresponding to the identity evaluated and tested with spherical harmonics of order 0 in a sphere of radius r. (Only returns the diagonal).

Notes

mass_matrix[l] = ( I Y_l,0 ; Y_l,0 )_L^2(surface sphere radius r). = r**2 for each l such that 0 <= l <= big_l, and with ( ., .)_L^2(surface sphere radius r): inner product where indicated.

Parameters:

big_l (int) – >= 0, max degree. For big_l over 3000 a warning will be given.
r (float) – > 0, radius.
azimuthal (bool) – Default True.

Returns:

mass_matrix – with real values. If azimuthal = True

Shape ((big_l+1), (big_l+1))

Else: Shape ((big_l+1)**2, (big_l+1)**2)

Return type:

np.ndarray

biosspheres.formulations.massmatrices.n_j_blocks(big_l: int, radii: ndarray, azimuthal: bool = True) → ndarray[source]#

Returns a numpy array with the result concatenated of the j_block(big_l, r, azimuthal) for each value r in radii.

Parameters:

big_l (int) – >= 0, max degree. For big_l over 3000 a warning will be given.
radii (np.ndarray) – Array with the radii of the spheres.
azimuthal (bool) – Default True.

Returns:

mass_matrix – with real values. If azimuthal = True

Shape (len(radii) * (big_l+1), len(radii) * (big_l+1))

Else: Shape (len(radii) * (big_l+1)**2, len(radii) * (big_l+1)**2)

Return type:

np.ndarray

See also

j_block, j_blocks, n_two_j_blocks

biosspheres.formulations.massmatrices.n_two_j_blocks(big_l: int, radii: ndarray, azimuthal: bool = True) → ndarray[source]#

Returns a numpy array with the result concatenated of the two_j_blocks(big_l, r, azimuthal) for each value r in radii.

Parameters:

big_l (int) – >= 0, max degree. For big_l over 3000 a warning will be given.
radii (np.ndarray) – Array with the radii of the spheres. Should be one dimensional.
azimuthal (bool) – Default True.

Returns:

mass_matrix – with real values. If azimuthal = True

Shape (2 * len(radii) * (big_l+1),
2 * len(radii) * (big_l+1))

Else

Shape (2 * len(radii) * (big_l+1)**2,: 2 * len(radii) * (big_l+1)**2)

Return type:

np.ndarray

See also

j_block, j_blocks, n_j_blocks

biosspheres.formulations.massmatrices.two_j_blocks(big_l: int, r: float, azimuthal: bool = True) → ndarray[source]#

Returns a numpy array with the result of j_block(big_l, r, azimuthal) concatenated with itself.

Parameters:

big_l (int) – >= 0, max degree. For big_l over 3000 a warning will be given.
r (float) – > 0, radius.
azimuthal (bool) – Default True.

Returns:

mass_matrix – with real values. If azimuthal = True

Shape (2 * (big_l+1), 2 * (big_l+1))

Else: Shape (2 * (big_l+1)**2, 2 * (big_l+1)**2)

Return type:

np.ndarray