Module tensor 

Tensor algebra operations for continuum mechanics.

Provides second-order tensors (Tensor2), fourth-order tensors (Tensor4), and a Voigt notation helper for symmetric second-order tensors.

Structs

CpDecomposition

CP (CANDECOMP/PARAFAC) decomposition result.

DenseTensor

A dense tensor of arbitrary rank stored in row-major (C-order) layout.

KelvinTensor

Kelvin (Mandel) notation utilities for symmetric fourth-order tensors.

MandelNotation

Mandel notation for symmetric second-order tensors.

Tensor2

Second-order tensor (3×3 matrix) stored as row-major [[f64; 3]; 3].

Tensor3

Rank-3 tensor with 27 components stored as data[i][j][k].

Tensor4

Fourth-order tensor with 81 components stored as [i][j][k][l].

TensorBasis

Tensor basis transformation utilities.

TensorTrain

A tensor in Tensor Train (TT / MPS) format.

TtCore

A single Tensor Train core of shape (r_left, n, r_right).

TuckerDecomposition

Tucker decomposition result for a 3-way tensor.

VoigtTensor

Voigt notation utilities for symmetric 3×3 tensors.

Functions

bilinear_form

Compute the bilinear form v^T A w for a Tensor2.

bulk_modulus

Compute the bulk modulus K from Lame parameters.

cauchy_stress_linear

Compute Cauchy stress from linear elasticity: sigma = C : epsilon.

compliance_from_stiffness

Compliance tensor from stiffness: S = C^{-1} in Voigt form (6×6 matrix inversion).

cp_als

Alternating Least Squares (ALS) for CP decomposition of a 3-way tensor.

cp_reconstruct

Reconstruct a DenseTensor from a CP decomposition.

cp_relative_error

Relative reconstruction error: ||T - T_approx||_F / ||T||_F.

einsum_2d

Einstein summation helper for common rank-2 tensor operations.

elasticity_stress

Compute the double inner product C :: T for a 4th-order elasticity tensor C and a symmetric second-order tensor T. Returns sigma_ij = C_ijkl T_kl.

eshelby_sphere

Build the Eshelby inclusion tensor for an isotropic matrix.

exp_tensor_pade

Approximate matrix exponential via Padé (2,2) approximation.

fourth_order_skew_identity

The 4th-order skew (antisymmetric) identity tensor: I^A_ijkl = (delta_ik delta_jl - delta_il delta_jk)/2.

fourth_order_symmetric_identity

The 4th-order symmetric identity tensor I^S_ijkl = (delta_ik delta_jl + delta_il delta_jk)/2.

general_einsum

Einstein summation over a pair of DenseTensors with explicit free and contracted indices.

hill_average

Hill average: arithmetic mean of Voigt and Reuss bounds.

invert_6x6

Invert a 6×6 matrix using Gauss–Jordan elimination.

is_symmetric_modes

Check whether a DenseTensor is symmetric under swap of two given modes.

lame_to_young_poisson

Compute the engineering Young’s modulus E and Poisson’s ratio nu from Lame parameters.

log_tensor_approx

Approximate matrix logarithm of a tensor close to identity: ln(I + X) ≈ X - X²/2 + X³/3.

neo_hookean_stiffness

Compute the 4th-order stiffness tensor for Neo-Hookean material at deformation F.

polar_decompose_right

Right polar decomposition of a non-singular 3×3 tensor F = R U.

quadratic_form

Compute the quadratic form v^T A v for a Tensor2.

reuss_average

Harmonic mean of two stiffness tensors (Reuss bound): C_Reuss = (C_a^{-1} + C_b^{-1})^{-1} / 2 (in Voigt/Kelvin form).

shear_modulus

Compute the shear modulus (= mu).

symmetrize_tensor

Build a fully symmetric DenseTensor by averaging over all permutations of indices (only for rank-2 and rank-3 tensors for now; rank-2 reduces to symmetrising a matrix).

tensor_contraction

Tensor contraction: contract modes mode_a of a and mode_b of b over their shared dimension.

tensor_outer

Compute the general outer product of two DenseTensors: result has rank = rank(a) + rank(b).

transversely_isotropic_stiffness

Build a transversely isotropic stiffness tensor given five elastic constants.

tucker_hosvd

Higher-order SVD (HOSVD) Tucker decomposition of a 3-way tensor.

tucker_reconstruct

Reconstruct a 3-way tensor from a Tucker decomposition.

vec_outer

Outer product of two 3-vectors: result_ij = a_i b_j (returns Tensor2).

voigt_average

Arithmetic mean of two stiffness tensors (Voigt bound): C_Voigt = (C_a + C_b) / 2.

young_poisson_to_lame

Compute Lame parameters from Young’s modulus E and Poisson’s ratio nu.