Binary Operations
Dot product (single contraction)
The dot product (or single contraction) between a tensor of order n and a tensor of order m is a tensor of order m + n - 2. For example, single contraction between two vectors $\mathbf{b}$ and $\mathbf{c}$ can be written as:
\[a = \mathbf{b} \cdot \mathbf{c} \Leftrightarrow a = b_i c_i\]
and single contraction between a second order tensor $\mathbf{B}$ and a vector $\mathbf{c}$:
\[\mathbf{a} = \mathbf{B} \cdot \mathbf{c} \Leftrightarrow a_i = B_{ij} c_j\]
LinearAlgebra.dot — Functiondot(::Vec, ::Vec)
dot(::Vec, ::SecondOrderTensor)
dot(::SecondOrderTensor, ::Vec)
dot(::SecondOrderTensor, ::SecondOrderTensor)Computes the dot product (single contraction) between two tensors. The symbol ⋅, written \cdot, is overloaded for single contraction.
Examples
julia> A = rand(Tensor{2, 2})
2×2 Tensor{2, 2, Float64, 4}:
0.590845 0.566237
0.766797 0.460085
julia> B = rand(Tensor{1, 2})
2-element Vec{2, Float64}:
0.7940257103317943
0.8541465903790502
julia> dot(A, B)
2-element Vec{2, Float64}:
0.9527955925660736
1.0018368881367576
julia> A ⋅ B
2-element Vec{2, Float64}:
0.9527955925660736
1.0018368881367576dot(::SymmetricTensor{2})Compute the dot product of a symmetric second order tensor with itself. Return a SymmetricTensor.
Examples
julia> A = rand(SymmetricTensor{2,3})
3×3 SymmetricTensor{2, 3, Float64, 6}:
0.590845 0.766797 0.566237
0.766797 0.460085 0.794026
0.566237 0.794026 0.854147
julia> dot(A)
3×3 SymmetricTensor{2, 3, Float64, 6}:
1.2577 1.25546 1.42706
1.25546 1.43013 1.47772
1.42706 1.47772 1.68067Double contraction
A double contraction between two tensors contracts the two most inner indices. The result of a double contraction between a tensor of order n and a tensor of order m is a tensor of order m + n - 4. For example, double contraction between two second order tensors $\mathbf{B}$ and $\mathbf{C}$ can be written as:
\[a = \mathbf{B} : \mathbf{C} \Leftrightarrow a = B_{ij} C_{ij}\]
and double contraction between a fourth order tensor $\mathsf{B}$ and a second order tensor $\mathbf{C}$:
\[\mathbf{A} = \mathsf{B} : \mathbf{C} \Leftrightarrow A_{ij} = B_{ijkl} C_{kl}\]
Tensors.dcontract — Functiondcontract(::SecondOrderTensor, ::SecondOrderTensor)
dcontract(::SecondOrderTensor, ::FourthOrderTensor)
dcontract(::FourthOrderTensor, ::SecondOrderTensor)
dcontract(::FourthOrderTensor, ::FourthOrderTensor)Compute the double contraction between two tensors. The symbol ⊡, written \boxdot, is overloaded for double contraction. The reason : is not used is because it does not have the same precedence as multiplication.
Examples
julia> A = rand(SymmetricTensor{2, 2});
julia> B = rand(SymmetricTensor{2, 2});
julia> dcontract(A,B)
1.9732018397544984
julia> A ⊡ B
1.9732018397544984Tensor product (open product)
The tensor product (or open product) between a tensor of order n and a tensor of order m is a tensor of order m + n. For example, open product between two vectors $\mathbf{b}$ and $\mathbf{c}$ can be written as:
\[\mathbf{A} = \mathbf{b} \otimes \mathbf{c} \Leftrightarrow A_{ij} = b_i c_j\]
and open product between two second order tensors $\mathbf{B}$ and $\mathbf{C}$:
\[\mathsf{A} = \mathbf{B} \otimes \mathbf{C} \Leftrightarrow A_{ijkl} = B_{ij} C_{kl}\]
Tensors.otimes — Functionotimes(::Vec, ::Vec)
otimes(::SecondOrderTensor, ::SecondOrderTensor)Compute the open product between two tensors. The symbol ⊗, written \otimes, is overloaded for tensor products.
Examples
julia> A = rand(SymmetricTensor{2, 2});
julia> B = rand(SymmetricTensor{2, 2});
julia> A ⊗ B
2×2×2×2 SymmetricTensor{4, 2, Float64, 9}:
[:, :, 1, 1] =
0.271839 0.352792
0.352792 0.260518
[:, :, 2, 1] =
0.469146 0.608857
0.608857 0.449607
[:, :, 1, 2] =
0.469146 0.608857
0.608857 0.449607
[:, :, 2, 2] =
0.504668 0.654957
0.654957 0.48365otimes(::Vec)Compute the open product of a vector with itself. Return a SymmetricTensor.
Examples
julia> A = rand(Vec{2})
2-element Vec{2, Float64}:
0.5908446386657102
0.7667970365022592
julia> otimes(A)
2×2 SymmetricTensor{2, 2, Float64, 3}:
0.349097 0.453058
0.453058 0.587978Permuted tensor products
Two commonly used permutations of the open product are the "upper" open product ($\bar{\otimes}$) and "lower" open product ($\underline{\otimes}$) defined between second order tensors $\mathbf{B}$ and $\mathbf{C}$ as
\[\mathsf{A} = \mathbf{B} \bar{\otimes} \mathbf{C} \Leftrightarrow A_{ijkl} = B_{ik} C_{jl}\\ \mathsf{A} = \mathbf{B} \underline{\otimes} \mathbf{C} \Leftrightarrow A_{ijkl} = B_{il} C_{jk}\]
Tensors.otimesu — Functionotimesu(::SecondOrderTensor, ::SecondOrderTensor)Compute the "upper" open product between two tensors.
Examples
julia> A = rand(SymmetricTensor{2, 2});
julia> B = rand(SymmetricTensor{2, 2});
julia> otimesu(A, B)
2×2×2×2 Tensor{4, 2, Float64, 16}:
[:, :, 1, 1] =
0.271839 0.469146
0.352792 0.608857
[:, :, 2, 1] =
0.352792 0.608857
0.260518 0.449607
[:, :, 1, 2] =
0.469146 0.504668
0.608857 0.654957
[:, :, 2, 2] =
0.608857 0.654957
0.449607 0.48365Tensors.otimesl — Functionotimesl(::SecondOrderTensor, ::SecondOrderTensor)Compute the "lower" open product between two tensors.
Examples
julia> A = rand(SymmetricTensor{2, 2});
julia> B = rand(SymmetricTensor{2, 2});
julia> otimesl(A, B)
2×2×2×2 Tensor{4, 2, Float64, 16}:
[:, :, 1, 1] =
0.271839 0.469146
0.352792 0.608857
[:, :, 2, 1] =
0.469146 0.504668
0.608857 0.654957
[:, :, 1, 2] =
0.352792 0.608857
0.260518 0.449607
[:, :, 2, 2] =
0.608857 0.654957
0.449607 0.48365