shgpy.core.utilities module

shgpy.core.utilities.make_tensor_complex(tensor, prefix='real_', 'imag_', suffix='', '')

Substitute e.g. x in sympy expression with real_x+1j*imag_x.

In sympy, variables initalized by x = sympy.symbols('x') are by default assumed to be complex. In order to make this more explicit (e.g. for generate_contracted_fourier_transforms()), we replace x by real_x + 1j*imag_x.

Parameters

tensorndarray

prefixtuple of str, optional

The prefixes of the newly-created real and imaginary variables. Defaults to ('real_', 'imag_').

suffixtuple of str, optional

The prefixes of the newly-created real and imaginary variables. Defaults to ('', '').

Returns

complex_tensorndarray

shgpy.core.utilities.make_tensor_real(tensor)

Substitute e.g. x in sympy expression with its real counterpart.

In sympy, variables initalized by x = sympy.symbols('x') are by default assumed to be complex. In order to make this more explicit (e.g. for generate_contracted_fourier_transforms()), we replace sympy.Symbol('x') by sympy.Symbol('x', real=True).

Parameters

tensorndarray

Returns

real_tensorndarray

shgpy.core.utilities.map_to_real(sym)

Map a sympy.Symbol to its real counterpart.

shgpy.core.utilities.particularize(tensor, exclude=[], permute_all_indices=False)

Particularize tensor (e.g. enforce chi_ijk = chi_ikj).

Because (e.g.) P_i = chi_ijk E_j E_k, chi_ijk needs to be symmetric in its last two indices. This symmetry is not documented in SHG tables, so it needs to be implemented manually. Functions using the solve function of sympy.solvers.

Parameters

tensorndarray

Tensor to be particularized.

excludelist of sympy.Symbol objects

Variables to exlude from the solver. Defaults to [].

permute_all_indicesbool, optional

Whether to permute all indices, not just the last two. Useful for materials where Kleinman symmetry is preserved. Default is False.

Returns

particularized_tensorndarray

shgpy.core.utilities.rotation_matrix3(n, t)

Returns the 3x3 matrix which rotates by t about n

Parameters

nndarray

Axis of rotation (ndim = 3).

tint, float, …

Angle (in radians) to rotate by.

Returns

rotation_matrix3ndarray of float

shgpy.core.utilities.rotation_matrix3symb(n, t, ndigits=16)

Duplicate of rotation_matrix3(), but accepts sympy.Symbol angle argument.

Parameters

nndarray

Axis of rotation (ndim = 3).

tsympy.Symbol, …

Angle to rotate by.

Returns

rotation_matrix3symbndarray of sympy.Expr

shgpy.core.utilities.rotation_matrix_from_two_vectors(v_initial, v_final, accuracy=0.01, ndigits=16)

Return rotation matrix which takes v_initial -> v_final.

Parameters

v_initialndarray

v_finalndarray

accuracyfloat, optional

Precision to which v_final is achieved. Defaults to 0.1.

ndigitsint, optional

Number of digits to round off to. Defaults to 16.

Returns

rotation_matrix3ndarray of float

shgpy.core.utilities.tensor_contract(tensor, index_pairs)

Contract a tensor against to pairs of indices.

Parameters

tensorndarray

index_pairsarray_like of array_like of int

List of pairs of indices, e.g. [[1, 3], [2, 4]].

shgpy.core.utilities.tensor_product(*tensors)

Tensor product of multiple tensors.

shgpy.core.utilities.transform(tensor, operation)

Transform a tensor by a given operation.

Parameters

tensorndarray

operationndarray

Operation to transform tensor by. Should be rank 2.

Returns

transformed_tensorndarray

shgpy.core.utilities.union(*arrays)

Union of multiple arrays.