jaxmat.tensors.mappings module

jaxmat.tensors.mappings module#

full_rank2_map(d)[source]#

Construct an index map for a full (non-symmetric) rank-2 tensor.

The flat array ordering is:

\[\{\boldsymbol{A}\} = [A_{11}, A_{22}, A_{33}, A_{12}, A_{21}, A_{13}, A_{31}, A_{23}, A_{32}]\]

for \(d=3\).

Parameters:

d (int) – Spatial dimension.

Return type:

tuple[tuple[Array, Array], Array]

Returns:

  • (I, J) (tuple[jax.Array, jax.Array]) – Index arrays of shape (d*d,) such that:

    tensor[..., I[k], J[k]] <-> array[..., k]

  • W (jax.Array) – Weight array of shape (d*d,), filled with ones. (Provided for compatibility with weighted indexed tensor backends.)

kelvin_rank2_map(d)[source]#

Construct an index map for a symmetric rank-2 tensor in Kelvin-Mandel notation.

The Kelvin vector is defined as:

$${boldsymbol{A}} = [A_{11}, A_{22}, A_{33},

sqrt{2} A_{12}, sqrt{2} A_{13}, sqrt{2} A_{23}]$$

for \(d=3\). This scaling makes the Kelvin basis orthonormal, ensuring:

\[\boldsymbol{A} : \boldsymbol{B} = \{\boldsymbol{A}\} \cdot \{\boldsymbol{B}\}\]

Parameters:

d (int) – Spatial dimension.

Return type:

tuple[tuple[Array, Array], Array]

Returns:

  • (I, J) (tuple[jax.Array, jax.Array]) – Index arrays of shape (d(d+1)/2,) mapping each Kelvin component to a tensor slot.

  • W (jax.Array) – Weight array of shape (d(d+1)/2,) containing \(1\) for diagonal components, \(\sqrt{2}\) for off-diagonal components.

full_rank4_map(d)[source]#

Construct an index map for a full (non-symmetric) rank-4 tensor (Tensor4).

The mapping is defined as:

\[C_{\alpha\beta} = C_{ijkl}\]

where \(\alpha\) and \(\beta\) are the Kelvin coefficients of the \((i,j)\) and \((k,l)\) index pairs.

For \(d = 3\), this produces a \(9\times 9\) representation of a 4th-rank tensor

Parameters:

d (int) – Spatial dimension.

Return type:

tuple[tuple[Array, Array, Array, Array], Array]

Returns:

  • (I, J, K, L) (tuple[jax.Array, jax.Array, jax.Array, jax.Array]) – Index arrays of shape (N, N) where N = d*d, mapping:

    tensor[..., I[a,b], J[a,b], K[a,b], L[a,b]] <-> array[..., a, b]

  • W (jax.Array) – Weight array of shape (N, N), filled with ones.

kelvin_rank4_map(d)[source]#

Construct an index map for a symmetric rank-4 tensor in Kelvin-Mandel notation.

The 4th-rank Kelvin tensor is defined as:

\[C_{\alpha\beta} = s_{\alpha} s_{\beta} C_{ijkl}\]

where \(\alpha\) and \(\beta\) are the Kelvin coefficients (weights \(s_{\alpha},s_{\beta}\) are 1 and \(\sqrt{2}\) for shear components) of the \((i,j)\) and \((k,l)\) index pairs.

For \(d = 3\), this produces a \(6\times 6\) representation of a 4th-rank tensor with minor symmetries.

Parameters:

d (int) – Spatial dimension.

Return type:

tuple[tuple[Array, Array, Array, Array], Array]

Returns:

  • (I, J, K, L) (tuple[jax.Array, jax.Array, jax.Array, jax.Array]) – Index arrays of shape (N, N) where N = d(d+1)/2, mapping:

    tensor[..., I[a,b], J[a,b], K[a,b], L[a,b]] <-> kelvin[..., a, b]

  • W (jax.Array) – Weight array of shape (N, N) with Kelvin scaling:

    W[a,b] = S[a] * S[b]

Notes

This mapping enforces minor symmetry (ij) and (kl) by construction.

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