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 $\mathbf{c}$:

\[\mathbf{a} = \mathbf{B} \cdot \mathbf{c} \Leftrightarrow a_i = B_{ij} c_j\]
Base.LinAlg.dot — Function.
dot(::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.

Example:

julia> A = rand(Tensor{2, 2})
2×2 Tensors.Tensor{2,2,Float64,4}:
 0.590845  0.566237
 0.766797  0.460085

julia> B = rand(Tensor{1, 2})
2-element Tensors.Tensor{1,2,Float64,2}:
 0.794026
 0.854147

julia> dot(A, B)
2-element Tensors.Tensor{1,2,Float64,2}:
 0.952796
 1.00184

julia> A ⋅ B
2-element Tensors.Tensor{1,2,Float64,2}:
 0.952796
 1.00184
source

Double 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 — Function.
dcontract(::SecondOrderTensor, ::SecondOrderTensor)
dcontract(::SecondOrderTensor, ::FourthOrderTensor)
dcontract(::FourthOrderTensor, ::SecondOrderTensor)
dcontract(::FourthOrderTensor, ::FourthOrderTensor)

Computes 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.

Example:

julia> A = rand(SymmetricTensor{2, 2});

julia> B = rand(SymmetricTensor{2, 2});

julia> dcontract(A,B)
1.9732018397544984

julia> A ⊡ B
1.9732018397544984
source

Tensor 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 — Function.
otimes(::Vec, ::Vec)
otimes(::SecondOrderTensor, ::SecondOrderTensor)

Computes the open product between two tensors. The symbol ⊗, written \otimes, is overloaded for tensor products.

Example:

julia> A = rand(SymmetricTensor{2, 2});

julia> B = rand(SymmetricTensor{2, 2});

julia> A ⊗ B
2×2×2×2 Tensors.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  0.48365
source