Tensors

A tensor, parametrized by an order and a dimension, is in general defined by

an array or a set of condensed parameters (e.g. isotropic tensors),
a basis,
a set of variances (covariant :cov or contravariant :cont) useful if the basis is not orthonormal.

In practice, the type of basis conditions the type of tensor (TensCanonical, TensRotated, TensOrthogonal, Tens or even TensISO in case of isotropic tensor).

julia> ℬ = Basis(Sym[0 1 1; 1 0 1; 1 1 0])
Basis{3, Sym}
→ basis: 3×3 Matrix{Sym}:
 0  1  1
 1  0  1
 1  1  0
→ dual basis: 3×3 Matrix{Sym}:
 -1/2   1/2   1/2
  1/2  -1/2   1/2
  1/2   1/2  -1/2
→ covariant metric tensor: 3×3 Symmetric{Sym, Matrix{Sym}}:
 2  1  1
 1  2  1
 1  1  2
→ contravariant metric tensor: 3×3 Symmetric{Sym, Matrix{Sym}}:
  3/4  -1/4  -1/4
 -1/4   3/4  -1/4
 -1/4  -1/4   3/4

julia> V = Tens(Tensor{1,3}(i -> symbols("v$i", real = true)))
(v1)𝐞¹ + (v2)𝐞² + (v3)𝐞³

julia> components(V, ℬ, (:cont,))
3-element Vector{Sym}:
 -v1/2 + v2/2 + v3/2
  v1/2 - v2/2 + v3/2
  v1/2 + v2/2 - v3/2

julia> components(V, ℬ, (:cov,))
3-element Vector{Sym}:
 v₂ + v₃
 v₁ + v₃
 v₁ + v₂

julia> ℬ̄ = normalize(ℬ)
Basis{3, Sym}
→ basis: 3×3 Matrix{Sym}:
         0  sqrt(2)/2  sqrt(2)/2
 sqrt(2)/2          0  sqrt(2)/2
 sqrt(2)/2  sqrt(2)/2          0
→ dual basis: 3×3 Matrix{Sym}:
 -sqrt(2)/2   sqrt(2)/2   sqrt(2)/2
  sqrt(2)/2  -sqrt(2)/2   sqrt(2)/2
  sqrt(2)/2   sqrt(2)/2  -sqrt(2)/2
→ covariant metric tensor: 3×3 Symmetric{Sym, Matrix{Sym}}:
   1  1/2  1/2
 1/2    1  1/2
 1/2  1/2    1
→ contravariant metric tensor: 3×3 Symmetric{Sym, Matrix{Sym}}:
  3/2  -1/2  -1/2
 -1/2   3/2  -1/2
 -1/2  -1/2   3/2

julia> components(V, ℬ̄, (:cov,))
3-element Vector{Sym}:
 sqrt(2)*v2/2 + sqrt(2)*v3/2
 sqrt(2)*v1/2 + sqrt(2)*v3/2
 sqrt(2)*v1/2 + sqrt(2)*v2/2

julia> T = Tens(Tensor{2,3}((i, j) -> symbols("t$i$j", real = true)))
(t11)𝐞¹⊗𝐞¹ + (t21)𝐞²⊗𝐞¹ + (t31)𝐞³⊗𝐞¹ + (t12)𝐞¹⊗𝐞² + (t22)𝐞²⊗𝐞² + (t32)𝐞³⊗𝐞² + (t13)𝐞¹⊗𝐞³ + (t23)𝐞²⊗𝐞³ + (t33)𝐞³⊗𝐞³

julia> components(T, ℬ, (:cov, :cov))
3×3 Matrix{Sym}:
 t₂₂ + t₂₃ + t₃₂ + t₃₃  t₂₁ + t₂₃ + t₃₁ + t₃₃  t₂₁ + t₂₂ + t₃₁ + t₃₂
 t₁₂ + t₁₃ + t₃₂ + t₃₃  t₁₁ + t₁₃ + t₃₁ + t₃₃  t₁₁ + t₁₂ + t₃₁ + t₃₂
 t₁₂ + t₁₃ + t₂₂ + t₂₃  t₁₁ + t₁₃ + t₂₁ + t₂₃  t₁₁ + t₁₂ + t₂₁ + t₂₂

julia> factor(simplify(components(T, ℬ, (:cont, :cov))))
3×3 Matrix{Sym}:
 -(t12 + t13 - t22 - t23 - t32 - t33)/2  -(t11 + t13 - t21 - t23 - t31 - t33)/2  -(t11 + t12 - t21 - t22 - t31 - t32)/2
  (t12 + t13 - t22 - t23 + t32 + t33)/2   (t11 + t13 - t21 - t23 + t31 + t33)/2   (t11 + t12 - t21 - t22 + t31 + t32)/2
  (t12 + t13 + t22 + t23 - t32 - t33)/2   (t11 + t13 + t21 + t23 - t31 - t33)/2   (t11 + t12 + t21 + t22 - t31 - t32)/2

Special tensors are available

tensId2(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: second-order identity (𝟏ᵢⱼ = δᵢⱼ = 1 if i=j otherwise 0)
tensId4(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: fourth-order identity with minor symmetries (𝕀 = 𝟏 ⊠ˢ 𝟏 i.e. (𝕀)ᵢⱼₖₗ = (δᵢₖδⱼₗ+δᵢₗδⱼₖ)/2)
tensJ4(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: fourth-order spherical projector (𝕁 = (𝟏 ⊗ 𝟏) / dim i.e. (𝕁)ᵢⱼₖₗ = δᵢⱼδₖₗ/dim)
tensK4(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: fourth-order deviatoric projector (𝕂 = 𝕀 - 𝕁 i.e. (𝕂)ᵢⱼₖₗ = (δᵢₖδⱼₗ+δᵢₗδⱼₖ)/2 - δᵢⱼδₖₗ/dim)
ISO(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: returns 𝕀, 𝕁, 𝕂

The useful tensor products are the following:

⊗ tensor product
⊗ˢ symmetrized tensor product
⊠ modified tensor product
⊠ˢ symmetrized modified tensor product
⋅ contracted product
⊡ double contracted product
⊙ quadruple contracted product

NOTE: more information about modified tensor products can be found in Sébastien Brisard's blog.

julia> 𝟏 = tensId2(3, Sym)
(1) 𝟏

julia> 𝕀, 𝕁, 𝕂 = ISO(3, Sym) ;

julia> 𝕀 == 𝟏 ⊠ˢ 𝟏
true

julia> 𝕁 == (𝟏 ⊗ 𝟏)/3
true

julia> a = Tens(Vec{3}((i,) -> symbols("a$i", real = true))) ;

julia> b = Tens(Vec{3}((i,) -> symbols("b$i", real = true))) ;

julia> a ⊗ b
(a1*b1)𝐞¹⊗𝐞¹ + (a2*b1)𝐞²⊗𝐞¹ + (a3*b1)𝐞³⊗𝐞¹ + (a1*b2)𝐞¹⊗𝐞² + (a2*b2)𝐞²⊗𝐞² + (a3*b2)𝐞³⊗𝐞² + (a1*b3)𝐞¹⊗𝐞³ + (a2*b3)𝐞²⊗𝐞³ + (a3*b3)𝐞³⊗𝐞³

julia> a ⊗ˢ b
(a1*b1)𝐞¹⊗𝐞¹ + (a1*b2/2 + a2*b1/2)𝐞²⊗𝐞¹ + (a1*b3/2 + a3*b1/2)𝐞³⊗𝐞¹ + (a1*b2/2 + a2*b1/2)𝐞¹⊗𝐞² + (a2*b2)𝐞²⊗𝐞² + (a2*b3/2 + a3*b2/2)𝐞³⊗𝐞² + (a1*b3/2 + a3*b1/2)𝐞¹⊗𝐞³
 + (a2*b3/2 + a3*b2/2)𝐞²⊗𝐞³ + (a3*b3)𝐞³⊗𝐞³

julia> (θ, ϕ, r), (𝐞ᶿ, 𝐞ᵠ, 𝐞ʳ), ℬˢ = init_spherical()
((θ, ϕ, r), (Sym[1, 0, 0], Sym[0, 1, 0], Sym[0, 0, 1]), Sym[cos(θ)*cos(ϕ) -sin(ϕ) sin(θ)*cos(ϕ); sin(ϕ)*cos(θ) cos(ϕ) sin(θ)*sin(ϕ); -sin(θ) 0 cos(θ)])

julia> R = rot3(θ, ϕ)
3×3 RotZYZ{Sym} with indices SOneTo(3)×SOneTo(3)(ϕ, θ, 0):
 cos(θ)⋅cos(ϕ)  -sin(ϕ)  sin(θ)⋅cos(ϕ)
 sin(ϕ)⋅cos(θ)   cos(ϕ)  sin(θ)⋅sin(ϕ)
       -sin(θ)        0         cos(θ)

julia> A = Tens(R * a)
(a1*cos(θ)*cos(ϕ) - a2*sin(ϕ) + a3*sin(θ)*cos(ϕ))𝐞¹ + (a1*sin(ϕ)*cos(θ) + a2*cos(ϕ) + a3*sin(θ)*sin(ϕ))𝐞² + (-a1*sin(θ) + a3*cos(θ))𝐞³

julia> simplify(change_tens(A, ℬˢ))
(a1)𝐞¹ + (a2)𝐞² + (a3)𝐞³