biosspheres.formulations.mtf.reconstructions#

Functions

complex_spherical_harmonics_at_points(big_l, ...)

Returns the complex spherical harmonics of degree <= big_l, the basis of the Helmholtz routines, evaluated at the points with the given spherical coordinates.

rf_helmholtz_n_spheres(points, n, radii, ...)

Evaluates with the representation formula the solution of a Helmholtz transmission problem solved with the MTF.

rf_laplace_n_spheres(vector, n, radii, ...)

rf_laplace_n_spheres_call(n, radii, ...)

rf_laplace_n_spheres_plus_ex_function(...)

rf_laplace_n_spheres_plus_ex_function_call(n, ...)

rf_laplace_one_sphere_azimuthal_symmetry(...)

Reconstruction of the function u using the representation formula and the coefficients from solving the Laplace transmission problem with the mtf formulation.
biosspheres.formulations.mtf.reconstructions.complex_spherical_harmonics_at_points(big_l: int, cos_theta: ndarray, phi: ndarray) → ndarray[source]#

Returns the complex spherical harmonics of degree <= big_l, the basis of the Helmholtz routines, evaluated at the points with the given spherical coordinates.

Returns:

spherical_harmonics – Shape ((big_l + 1)**2, len(phi)). spherical_harmonics[l(l+1) + m] = Y_{l,m}, with Y_{l,-m} = (-1)**m conj(Y_{l,m}).

Return type:

np.ndarray

biosspheres.formulations.mtf.reconstructions.rf_helmholtz_n_spheres(points: ndarray, n: int, radii: ndarray, positions: list[ndarray], kii: ndarray, big_l: int, coefficients: ndarray) → ndarray[source]#

Evaluates with the representation formula the solution of a Helmholtz transmission problem solved with the MTF.

Outside the spheres it returns the scattered field u_0; inside the sphere j it returns u_j.

Parameters:

  • points (np.ndarray) – Shape (number of points, 3).

  • n (int) – >= 1, number of spheres.

  • radii (np.ndarray) – Length n.

  • positions (list[np.ndarray]) – Centers of the spheres.

  • kii (np.ndarray) – Length n + 1. kii[0] is the exterior wave number, kii[j] the one of sphere j.

  • big_l (int) – >= 0, max degree.

  • coefficients (np.ndarray) – Solution of the MTF system, length 4 * n * (big_l + 1)**2, in the order of mtf.mtf_n_matrix.

Returns:

u – Complex, length number of points.

Return type:

np.ndarray

biosspheres.formulations.mtf.reconstructions.rf_laplace_one_sphere_azimuthal_symmetry(vector: ndarray, rs: ndarray, ps: list[ndarray], big_l: int, coefficients: ndarray) → float[source]#

Reconstruction of the function u using the representation formula and the coefficients from solving the Laplace transmission problem with the mtf formulation.

Parameters:

  • vector (np.ndarray) – Length 3, representing the x, y and z coordinates in the Cartesian coordinate system.

  • rs (np.ndarray) – with the radii of the sphere. Should be of length one.

  • ps (list) – of np.ndarray

  • big_l (int)

  • coefficients (np.ndarray) – traces of the function.

Returns:

u – evaluation of the function u in the given vector.

Return type:

float