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
:covor 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$j", real = true)))
(t11)𝐞¹⊗𝐞¹ + (t21)𝐞²⊗𝐞¹ + (t31)𝐞³⊗𝐞¹ + (t12)𝐞¹⊗𝐞² + (t22)𝐞²⊗𝐞² + (t32)𝐞³⊗𝐞² + (t13)𝐞¹⊗𝐞³ + (t23)𝐞²⊗𝐞³ + (t33)𝐞³⊗𝐞³
julia> components(T, ℬ, (:cov, :cov))
3×3 Matrix{Sym}:
t₂₂ + t₂₃ + t₃₂ + t₃₃ t₂₁ + t₂₃ + t₃₁ + t₃₃ t₂₁ + t₂₂ + t₃₁ + t₃₂
t₁₂ + t₁₃ + t₃₂ + t₃₃ t₁₁ + t₁₃ + t₃₁ + t₃₃ t₁₁ + t₁₂ + t₃₁ + t₃₂
t₁₂ + t₁₃ + t₂₂ + t₂₃ t₁₁ + t₁₃ + t₂₁ + t₂₃ t₁₁ + t₁₂ + t₂₁ + t₂₂
julia> factor(simplify(components(T, ℬ, (:cont, :cov))))
3×3 Matrix{Sym}:
-(t12 + t13 - t22 - t23 - t32 - t33)/2 -(t11 + t13 - t21 - t23 - t31 - t33)/2 -(t11 + t12 - t21 - t22 - t31 - t32)/2
(t12 + t13 - t22 - t23 + t32 + t33)/2 (t11 + t13 - t21 - t23 + t31 + t33)/2 (t11 + t12 - t21 - t22 + t31 + t32)/2
(t12 + t13 + t22 + t23 - t32 - t33)/2 (t11 + t13 + t21 + t23 - t31 - t33)/2 (t11 + t12 + t21 + t22 - t31 - t32)/2Special tensors are available
tensId2(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: second-order identity (𝟏ᵢⱼ = δᵢⱼ = 1 if i=j otherwise 0)tensId4(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: fourth-order identity with minor symmetries (𝕀 = 𝟏 ⊠ˢ 𝟏i.e.(𝕀)ᵢⱼₖₗ = (δᵢₖδⱼₗ+δᵢₗδⱼₖ)/2)tensJ4(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: fourth-order spherical projector (𝕁 = (𝟏 ⊗ 𝟏) / dimi.e.(𝕁)ᵢⱼₖₗ = δᵢⱼδₖₗ/dim)tensK4(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: fourth-order deviatoric projector (𝕂 = 𝕀 - 𝕁i.e.(𝕂)ᵢⱼₖₗ = (δᵢₖδⱼₗ+δᵢₗδⱼₖ)/2 - δᵢⱼδₖₗ/dim)ISO(::Val{dim} = Val(3), ::Val{T} = Val(Sym)) where {dim,T<:Number}: returns𝕀, 𝕁, 𝕂
The useful tensor products are the following:
⊗tensor product⊗ˢsymmetrized tensor product⊠modified tensor product⊠ˢsymmetrized modified tensor product⋅contracted product⊡double contracted product⊙quadruple contracted product
NOTE: more information about modified tensor products can be found in Sébastien Brisard's blog.
julia> 𝟏 = tensId2(3, Sym)
(1) 𝟏
julia> 𝕀, 𝕁, 𝕂 = ISO(3, Sym) ;
julia> 𝕀 == 𝟏 ⊠ˢ 𝟏
true
julia> 𝕁 == (𝟏 ⊗ 𝟏)/3
true
julia> a = Tens(Vec{3}((i,) -> symbols("a$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