Added double contraction operator :

benchmark/TensorAlgebraBenchmarks/TensorAlgebraBenchmarks.jl

@@ -49,3 +49,5 @@ SUITE["Tensor algebra"]["δδ_μ_2d"] = @benchmarkable δᵢₖδⱼₗ2D + δ
 SUITE["Tensor algebra"]["δδ_λ_2d"] = @benchmarkable 1.0 * δᵢⱼδₖₗ2D
 SUITE["Tensor algebra"]["Cofactor"] = @benchmarkable cof(A)
 SUITE["Tensor algebra"]["Det(A)Inv(A')"] = @benchmarkable det(A)*inv(A')
+SUITE["Tensor algebra"]["A:A"] = @benchmarkable A:A
+SUITE["Tensor algebra"]["tr(A'·A)"] = @benchmarkable tr(A'·A)

src/TensorAlgebra/Functions.jl

@@ -39,6 +39,7 @@ end
 
 
 function trAA(A::TensorValue{3, 3, T, N}) where {T, N}
+  @warn "The function trAA is deprecated and will be removed. Use the double contraction operator : instead." maxlog=1
   return sum(A.data[i]*A.data[i] for i in 1:N)
 end
 

src/TensorAlgebra/Operations.jl

@@ -10,6 +10,14 @@ function (*)(Ten1::TensorValue, Ten2::TensorValue)
 end
 
 
+"""
+Double contraction of second order tensors, which is equivalent to tr(a'·b)
+"""
+function (:)(a::TensorValue{D1,D2}, b::TensorValue{D1,D2}) where {D1,D2}
+  sum(a.data .* b.data)
+end
+
+
 @generated function (+)(A::TensorValue{D,D,Float64}, B::TensorValue{D,D,Float64}) where {D}
   str = ""
   for i in 1:D*D

src/TensorAlgebra/TensorAlgebra.jl

@@ -4,11 +4,8 @@ using Gridap
 using Gridap.TensorValues
 using StaticArrays
 using LinearAlgebra
-import Base: *
-import Base: +
-import Base: sqrt
+import Base: *, +, (:), sqrt
 
-export (*)
 export (×ᵢ⁴)
 export (+)
 export (⊗₁₂³)

test/TestTensorAlgebra/TensorAlgebraTests.jl

@@ -101,6 +101,13 @@ end
 end
 
 
+@testset "double contraction" begin
+  A = TensorValue(1.:9...)
+  @test A:A == tr(A'·A)
+  @test A:A == sum(i -> i^2, 1:9)
+end
+
+
 @testset "sqrt" begin
   A = TensorValue(1.:9...)
   A = A'*A + I3