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other_operators.md

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DocTestSetup = quote
    using Random
    Random.seed!(1234)
    using Tensors
end

Other operators

Pages = ["other_operators.md"]

Transpose-dot

The dot product between the transpose of a tensor with itself. Results in a symmetric tensor.

$$\mathbf{A} = \mathbf{B}^\text{T} \cdot \mathbf{B} \Leftrightarrow A_{ij} = B_{ki}^\text{T} B_{kj} = B_{ik} B_{kj}$$ $$\mathbf{A} = \mathbf{B} \cdot \mathbf{B}^\text{T} \Leftrightarrow A_{ij} = B_{ik} B_{jk}^\text{T} = B_{ik} B_{kj}$$ 
Tensors.tdot
Tensors.dott

Norm

The (2)-norm of a tensor is defined for a vector, second order tensor and fourth order tensor as

$$\|\mathbf{a}\| = \sqrt{\mathbf{a} \cdot \mathbf{a}} \Leftrightarrow \|a_i\| = \sqrt{a_i a_i},$$ $$\|\mathbf{A}\| = \sqrt{\mathbf{A} : \mathbf{A}} \Leftrightarrow \|A_{ij}\| = \sqrt{A_{ij} A_{ij}},$$ $$\|\mathsf{A}\| = \sqrt{\mathsf{A} :: \mathsf{A}} \Leftrightarrow \|A_{ijkl}\| = \sqrt{A_{ijkl} A_{ijkl}}.$$ 
Tensors.norm

Trace

The trace for a second order tensor is defined as the sum of the diagonal elements. This can be written as

$$\text{tr}(\mathbf{A}) = \mathbf{I} : \mathbf{A} \Leftrightarrow \text{tr}(A_{ij}) = A_{ii}.$$ 
Tensors.tr

Determinant

Determinant for a second order tensor.

Tensors.det

Inverse

Inverse of a second order tensor such that

$$\mathbf{A}^{-1} \cdot \mathbf{A} = \mathbf{I}$$

where \mathbf{I} is the second order identitiy tensor.

Tensors.inv

Transpose

Transpose of tensors is defined by changing the order of the tensor's "legs". The transpose of a vector/symmetric tensor is the vector/tensor itself. The transpose of a second order tensor can be written as:

$$A_{ij}^\text{T} = A_{ji}$$

and for a fourth order tensor the minor transpose can be written as

$$A_{ijkl}^\text{t} = A_{jilk}$$

and the major transpose as

$$A_{ijkl}^\text{T} = A_{klij}.$$ 
Tensors.transpose
Tensors.minortranspose
Tensors.majortranspose

Symmetric

The symmetric part of a second order tensor is defined by:

$$\mathbf{A}^\text{sym} = \frac{1}{2}(\mathbf{A} + \mathbf{A}^\text{T}) \Leftrightarrow A_{ij}^\text{sym} = \frac{1}{2}(A_{ij} + A_{ji}),$$

The major symmetric part of a fourth order tensor is defined by

$$\mathsf{A}^\text{majsym} = \frac{1}{2}(\mathsf{A} + \mathsf{A}^\text{T}) \Leftrightarrow A_{ijkl}^\text{majsym} = \frac{1}{2}(A_{ijkl} + A_{klij}).$$

The minor symmetric part of a fourth order tensor is defined by

$$A_{ijkl}^\text{minsym} = \frac{1}{4}(A_{ijkl} + A_{ijlk} + A_{jikl} + A_{jilk}).$$ 
Tensors.symmetric
Tensors.minorsymmetric
Tensors.majorsymmetric

Skew symmetric

The skew symmetric part of a second order tensor is defined by

$$\mathbf{A}^\text{skw} = \frac{1}{2}(\mathbf{A} - \mathbf{A}^\text{T}) \Leftrightarrow A^\text{skw}_{ij} = \frac{1}{2}(A_{ij} - A_{ji}).$$

The skew symmetric part of a symmetric tensor is zero.

Tensors.skew

Deviatoric tensor

The deviatoric part of a second order tensor is defined by

$$\mathbf{A}^\text{dev} = \mathbf{A} - \frac{1}{3} \mathrm{tr}[\mathbf{A}] \mathbf{I} \Leftrightarrow A_{ij}^\text{dev} = A_{ij} - \frac{1}{3}A_{kk}\delta_{ij}.$$ 
Tensors.dev

Volumetric tensor

The volumetric part of a second order tensor is defined by

$$\mathbf{A}^\text{vol} = \frac{1}{3} \mathrm{tr}[\mathbf{A}] \mathbf{I} \Leftrightarrow A_{ij}^\text{vol} = \frac{1}{3}A_{kk}\delta_{ij}.$$ 
Tensors.vol

Cross product

The cross product between two vectors is defined as

$$\mathbf{a} = \mathbf{b} \times \mathbf{c} \Leftrightarrow a_i = \epsilon_{ijk} b_j c_k$$ 
Tensors.cross

Eigenvalues and eigenvectors

The eigenvalues and eigenvectors of a (symmetric) second order tensor, \mathbf{A} can be solved from the eigenvalue problem

$$\mathbf{A} \cdot \mathbf{v}_i = \lambda_i \mathbf{v}_i \qquad i = 1, \dots, \text{dim}$$

where \lambda_i are the eigenvalues and \mathbf{v}_i are the corresponding eigenvectors. For a symmetric fourth order tensor, \mathsf{A} the second order eigentensors and eigenvalues can be solved from

$$\mathsf{A} : \mathbf{V}_i = \lambda_i \mathbf{V}_i \qquad i = 1, \dots, \text{dim}$$

where \lambda_i are the eigenvalues and \mathbf{V}_i the corresponding eigentensors.

Tensors.eigen
Tensors.eigvals
Tensors.eigvecs

Tensor square root

Square root of a symmetric positive definite second order tensor S, defined such that

$$\sqrt{\mathbf{S}} \cdot \sqrt{\mathbf{S}} = S.$$ 
Tensors.sqrt

Rotations

Tensors.rotate
Tensors.rotation_tensor

Special operations

For computing a special dot product between two vectors \mathbf{a} and \mathbf{b} with a fourth order symmetric tensor \mathbf{C} such that a_k C_{ikjl} b_l there is dotdot(a, C, b). This function is useful because it is the expression for the tangent matrix in continuum mechanics when the displacements are approximated by scalar shape functions.

Tensors.dotdot

Voigt format

For some operations it is convenient to easily switch to the so called "Voigt"-format. For example when solving a local problem in a plasticity model. To simplify the conversion between tensors and Voigt format, see tovoigt, tovoigt! and fromvoigt documented below. Care must be exercised when combined with differentiation, see Differentiation of Voigt format further down.

Tensors.tovoigt
Tensors.tovoigt!
Tensors.fromvoigt

Differentiation of Voigt format

Differentiating with a Voigt representation of a symmetric tensor may lead to incorrect results when converted back to tensors. The tomandel, tomandel!, and frommandel versions of tovoigt, tovoigt!, and fromvoigt can then be used. As illustrated by the following example, this will give the correct result. In general, however, direct differentiation of Tensors is faster (see Automatic Differentiation).

julia> using Tensors, ForwardDiff

julia> fun(X::SymmetricTensor{2}) = X;

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

# Differentiation of a tensor directly (correct)
julia> tovoigt(gradient(fun, A))
3×3 Matrix{Float64}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  0.5

# Converting to Voigt format, perform differentiation, convert back (WRONG!)
julia> ForwardDiff.jacobian(
           v -> tovoigt(fun(fromvoigt(SymmetricTensor{2,2}, v))),
           tovoigt(A)
       )
3×3 Matrix{Float64}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

# Converting to Mandel format, perform differentiation, convert back (correct)
julia> tovoigt(
           frommandel(SymmetricTensor{4,2},
                ForwardDiff.jacobian(
                    v -> tomandel(fun(frommandel(SymmetricTensor{2,2}, v))),
                    tomandel(A)
                )
           )
       )
3×3 Matrix{Float64}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  0.5