Operations

cross(x::Vec, y::Vec)
x × y

Compute the cross product between two vectors. The infix operation x × y (where × can be typed by \times<tab>) is a synonym for cross(x, y).

Examples

julia> x = rand(Vec{3})
3-element Vec{3, Float64}:
 0.32597672886359486
 0.5490511363155669
 0.21858665481883066

julia> y = rand(Vec{3})
3-element Vec{3, Float64}:
 0.8942454282009883
 0.35311164439921205
 0.39425536741585077

julia> x × y
3-element Vec{3, Float64}:
  0.13928086435138393
  0.0669520417303531
 -0.37588028973385323

norm(::AbstractTensor)

Compute norm of a tensor.

Examples

julia> x = rand(Mat{3, 3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> norm(x)
1.8223398556552728

normalize(x)

Compute x / norm(x).

tr(A)

Compute the trace of a square tensor A.

Examples

julia> A = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> tr(A)
1.1733382401532275

inv(A)

Compute the inverse of a tensor A.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> inv(x) * x ≈ one(x)
true

julia> A = rand(SymmetricFourthOrderTensor{3});

julia> A ⊡₂ inv(A) ≈ one(A)
true

contract(x, y, ::Val{N})

Conduct contraction of N inner indices. For example, N=2 contraction for third-order tensors $A_{ij} = B_{ikl} C_{klj}$ can be computed as follows:

Examples

julia> B = rand(Tensor{Tuple{3,3,3}});

julia> C = rand(Tensor{Tuple{3,3,3}});

julia> A = contract(B, C, Val(2))
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 3.70978  2.47156  3.91807
 2.90966  2.30881  3.25965
 1.78391  1.38714  2.2079

The following infix operators are also available for specific contractions:

• x ⊡ y (where ⊡ can be typed by \boxdot<tab> ): contract(x, y, Val(1))
• x ⊡₂ y (where ⊡₂ can be typed by \boxdot<tab>\_2<tab> ): contract(x, y, Val(2))
• x ⊗ y (where ⊗ can be typed by \otimes<tab> ): contract(x, y, Val(0))

contract(x, y, Val(xdims), Val(ydims))

Perform contraction over the given dimensions.

Examples

julia> A = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> B = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.748415  0.00744801  0.682533
 0.578232  0.199377    0.956741
 0.727935  0.439243    0.647855

julia> contract(A, B, Val(1), Val(2)) ≈ @einsum C[i,j] := A[k,i] * B[j,k]
true

julia> contract(A, B, Val((1,2)), Val((2,1))) ≈ @einsum c := A[i,j] * B[j,i]
true

tensor(x::AbstractTensor, y::AbstractTensor)
x ⊗ y

Compute tensor product such as $A_{ij} = x_i y_j$. x ⊗ y (where ⊗ can be typed by \otimes<tab>) is a synonym for tensor(x, y).

Examples

julia> x = rand(Vec{3})
3-element Vec{3, Float64}:
 0.32597672886359486
 0.5490511363155669
 0.21858665481883066

julia> y = rand(Vec{3})
3-element Vec{3, Float64}:
 0.8942454282009883
 0.35311164439921205
 0.39425536741585077

julia> A = x ⊗ y
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.291503  0.115106   0.128518
 0.490986  0.193876   0.216466
 0.19547   0.0771855  0.086179

x^⊗(n)

n-fold tensor product of a tensor x.

Examples

julia> x = rand(Vec{2})
2-element Vec{2, Float64}:
 0.32597672886359486
 0.5490511363155669

julia> x^⊗(3)
2×2×2 Tensor{Tuple{Symmetry{Tuple{2, 2, 2}}}, Float64, 3, 4}:
[:, :, 1] =
 0.0346386  0.0583426
 0.0583426  0.098268

[:, :, 2] =
 0.0583426  0.098268
 0.098268   0.165515

@einsum [TensorType] expr

Performs tensor computations using the Einstein summation convention. Since @einsum cannot fully infer tensor symmetries, it is possible to annotate the returned tensor type (though this is not checked for correctness). This can help eliminate the computation of the symmetric part, improving performance.

Examples

julia> A = rand(Mat{3,3});

julia> B = rand(Mat{3,3});

julia> (@einsum C[i,j] := A[j,k] * B[k,i]) ≈ (A * B)'
true

julia> (@einsum c := A[i,j] * A[i,j]) ≈ A ⋅ A
true

julia> (@einsum SymmetricSecondOrderTensor{3} D[i,j] := A[k,i] * A[k,j]) ≈ A' * A
true

symmetric(::AbstractSecondOrderTensor)
symmetric(::AbstractSecondOrderTensor, uplo)

Compute the symmetric part of a second order tensor.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> symmetric(x)
3×3 SymmetricSecondOrderTensor{3, Float64, 6}:
 0.325977  0.721648  0.585856
 0.721648  0.353112  0.594901
 0.585856  0.594901  0.49425

julia> symmetric(x, :U)
3×3 SymmetricSecondOrderTensor{3, Float64, 6}:
 0.325977  0.894245  0.953125
 0.894245  0.353112  0.795547
 0.953125  0.795547  0.49425

skew(A)

Compute skew-symmetric (anti-symmetric) part of a second order tensor.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> symmetric(x) + skew(x) ≈ x
true

skew(ω::Vec{3})

Construct a skew-symmetric (anti-symmetric) tensor W from a vector ω as

\[\bm{\omega} = \begin{Bmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \end{Bmatrix}, \quad \bm{W} = \begin{bmatrix} 0 & -\omega_3 & \omega_2 \\ \omega_3 & 0 & -\omega_1 \\ -\omega_2 & \omega_1 & 0 \end{bmatrix}\]

Examples

julia> skew(Vec(1,2,3))
3×3 Tensor{Tuple{3, 3}, Int64, 2, 9}:
  0  -3   2
  3   0  -1
 -2   1   0

minorsymmetric(::AbstractFourthOrderTensor) -> SymmetricFourthOrderTensor

Compute the minor symmetric part of a fourth order tensor.

Examples

julia> x = rand(Tensor{Tuple{3,3,3,3}});

julia> minorsymmetric(x) ≈ @einsum y[i,j,k,l] := (x[i,j,k,l] + x[j,i,k,l] + x[i,j,l,k] + x[j,i,l,k]) / 4
true

rotmatx(θ::Number)

Construct rotation matrix around x axis.

\[\bm{R}_x = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos{\theta} & -\sin{\theta} \\ 0 & \sin{\theta} & \cos{\theta} \end{bmatrix}\]

rotmaty(θ::Number)

Construct rotation matrix around y axis.

\[\bm{R}_y = \begin{bmatrix} \cos{\theta} & 0 & \sin{\theta} \\ 0 & 1 & 0 \\ -\sin{\theta} & 0 & \cos{\theta} \end{bmatrix}\]

rotmatz(θ::Number)

Construct rotation matrix around z axis.

\[\bm{R}_z = \begin{bmatrix} \cos{\theta} & -\sin{\theta} & 0 \\ \sin{\theta} & \cos{\theta} & 0 \\ 0 & 0 & 1 \end{bmatrix}\]

rotmat(θ::Number)

Construct 2D rotation matrix.

\[\bm{R} = \begin{bmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{bmatrix}\]

Examples

julia> rotmat(deg2rad(30))
2×2 Tensor{Tuple{2, 2}, Float64, 2, 4}:
 0.866025  -0.5
 0.5        0.866025

rotmat(θ::Vec{3}; sequence::Symbol)

Convert Euler angles to rotation matrix. Use 3 characters belonging to the set (X, Y, Z) for intrinsic rotations, or (x, y, z) for extrinsic rotations.

Examples

julia> α, β, γ = map(deg2rad, rand(3));

julia> rotmat(Vec(α,β,γ), sequence = :XYZ) ≈ rotmatx(α) * rotmaty(β) * rotmatz(γ)
true

julia> rotmat(Vec(α,β,γ), sequence = :xyz) ≈ rotmatz(γ) * rotmaty(β) * rotmatx(α)
true

julia> rotmat(Vec(α,β,γ), sequence = :XYZ) ≈ rotmat(Vec(γ,β,α), sequence = :zyx)
true

rotmat(a => b)

Construct rotation matrix rotating vector a to b. The norms of two vectors must be the same.

Examples

julia> a = normalize(rand(Vec{3}))
3-element Vec{3, Float64}:
 0.4829957515506539
 0.8135223859352438
 0.3238771859304809

julia> b = normalize(rand(Vec{3}))
3-element Vec{3, Float64}:
 0.8605677447967596
 0.3398133016944055
 0.3794075336718636

julia> R = rotmat(a => b)
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 -0.00540771   0.853773   0.520617
  0.853773    -0.267108   0.446905
  0.520617     0.446905  -0.727485

julia> R * a ≈ b
true

rotmat(θ, n::Vec)

Construct rotation matrix from angle θ and axis n.

Examples

julia> x = Vec(1.0, 0.0, 0.0)
3-element Vec{3, Float64}:
 1.0
 0.0
 0.0

julia> n = Vec(0.0, 0.0, 1.0)
3-element Vec{3, Float64}:
 0.0
 0.0
 1.0

julia> rotmat(π/2, n) * x
3-element Vec{3, Float64}:
 6.123233995736766e-17
 1.0
 0.0

rotmat(::Quaternion)

Construct rotation matrix from quaternion.

Examples

julia> q = quaternion(π/4, Vec(0,0,1))
0.9238795325112867 + 0.0𝙞 + 0.0𝙟 + 0.3826834323650898𝙠

julia> rotmat(q)
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.707107  -0.707107  0.0
 0.707107   0.707107  0.0
 0.0        0.0       1.0

rotate(x, R::SecondOrderTensor)

Rotate x using the rotation matrix R. This function preserves the symmetry of the matrix.

Examples

julia> R = rotmatz(π/4)
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.707107  -0.707107  0.0
 0.707107   0.707107  0.0
 0.0        0.0       1.0

julia> rotate(Vec(1,0,0), R)
3-element Vec{3, Float64}:
 0.7071067811865476
 0.7071067811865475
 0.0

julia> A = rand(SymmetricSecondOrderTensor{3})
3×3 SymmetricSecondOrderTensor{3, Float64, 6}:
 0.325977  0.549051  0.218587
 0.549051  0.894245  0.353112
 0.218587  0.353112  0.394255

julia> rotate(A, R) ≈ R * A * R'
true

rotate(x::Vec, q::Quaternion)

Rotate x by quaternion q.

Examples

julia> v = Vec(1.0, 0.0, 0.0)
3-element Vec{3, Float64}:
 1.0
 0.0
 0.0

julia> rotate(v, quaternion(π/4, Vec(0,0,1)))
3-element Vec{3, Float64}:
 0.7071067811865475
 0.7071067811865476
 0.0

vol(A)

Compute the volumetric part of a square tensor A. This is only available in 3D.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> vol(x)
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.391113  0.0       0.0
 0.0       0.391113  0.0
 0.0       0.0       0.391113

julia> vol(x) + dev(x) ≈ x
true

dev(::AbstractSecondOrderTensor{3})
dev(::AbstractSymmetricSecondOrderTensor{3})

Compute the deviatoric part of a square tensor. This is only available in 3D.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> dev(x)
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 -0.065136   0.894245   0.953125
  0.549051  -0.0380011  0.795547
  0.218587   0.394255   0.103137

julia> tr(dev(x))
5.551115123125783e-17

vonmises(::AbstractSymmetricSecondOrderTensor{3})

Compute the von Mises stress.

\[q = \sqrt{\frac{3}{2} \mathrm{dev}(\bm{\sigma}) : \mathrm{dev}(\bm{\sigma})} = \sqrt{3J_2}\]

Examples

julia> σ = rand(SymmetricSecondOrderTensor{3})
3×3 SymmetricSecondOrderTensor{3, Float64, 6}:
 0.325977  0.549051  0.218587
 0.549051  0.894245  0.353112
 0.218587  0.353112  0.394255

julia> vonmises(σ)
1.3078860814690232

stress_invariants(::AbstractSecondOrderTensor{3})
stress_invariants(::AbstractSymmetricSecondOrderTensor{3})
stress_invariants(::Vec{3})

Return a tuple storing stress invariants.

\[\begin{aligned} I_1(\bm{\sigma}) &= \mathrm{tr}(\bm{\sigma}) \\ I_2(\bm{\sigma}) &= \frac{1}{2} (\mathrm{tr}(\bm{\sigma})^2 - \mathrm{tr}(\bm{\sigma}^2)) \\ I_3(\bm{\sigma}) &= \det(\bm{\sigma}) \end{aligned}\]

Examples

julia> σ = rand(SymmetricSecondOrderTensor{3})
3×3 SymmetricSecondOrderTensor{3, Float64, 6}:
 0.325977  0.549051  0.218587
 0.549051  0.894245  0.353112
 0.218587  0.353112  0.394255

julia> I₁, I₂, I₃ = stress_invariants(σ)
(1.6144775244804341, 0.2986572249840249, -0.0025393241133506677)

deviatoric_stress_invariants(::AbstractSecondOrderTensor{3})
deviatoric_stress_invariants(::AbstractSymmetricSecondOrderTensor{3})
deviatoric_stress_invariants(::Vec{3})

Return a tuple storing deviatoric stress invariants.

\[\begin{aligned} J_1(\bm{\sigma}) &= \mathrm{tr}(\mathrm{dev}(\bm{\sigma})) = 0 \\ J_2(\bm{\sigma}) &= \frac{1}{2} \mathrm{tr}(\mathrm{dev}(\bm{\sigma})^2) \\ J_3(\bm{\sigma}) &= \frac{1}{3} \mathrm{tr}(\mathrm{dev}(\bm{\sigma})^3) \end{aligned}\]

Examples

julia> σ = rand(SymmetricSecondOrderTensor{3})
3×3 SymmetricSecondOrderTensor{3, Float64, 6}:
 0.325977  0.549051  0.218587
 0.549051  0.894245  0.353112
 0.218587  0.353112  0.394255

julia> J₁, J₂, J₃ = deviatoric_stress_invariants(σ)
(0.0, 0.5701886673667987, 0.14845380911930367)