deal.II version 9.7.1
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Physics::Transformations::Covariant Namespace Reference

Functions

Push forward operations
template<int dim, typename Number>
Tensor< 1, dim, Number > push_forward (const Tensor< 1, dim, Number > &V, const Tensor< 2, dim, Number > &F)
template<int dim, typename Number>
Tensor< 2, dim, Number > push_forward (const Tensor< 2, dim, Number > &T, const Tensor< 2, dim, Number > &F)
template<int dim, typename Number>
SymmetricTensor< 2, dim, Number > push_forward (const SymmetricTensor< 2, dim, Number > &T, const Tensor< 2, dim, Number > &F)
template<int dim, typename Number>
Tensor< 4, dim, Number > push_forward (const Tensor< 4, dim, Number > &H, const Tensor< 2, dim, Number > &F)
template<int dim, typename Number>
SymmetricTensor< 4, dim, Number > push_forward (const SymmetricTensor< 4, dim, Number > &H, const Tensor< 2, dim, Number > &F)
Pull back operations
template<int dim, typename Number>
Tensor< 1, dim, Number > pull_back (const Tensor< 1, dim, Number > &v, const Tensor< 2, dim, Number > &F)
template<int dim, typename Number>
Tensor< 2, dim, Number > pull_back (const Tensor< 2, dim, Number > &t, const Tensor< 2, dim, Number > &F)
template<int dim, typename Number>
SymmetricTensor< 2, dim, Number > pull_back (const SymmetricTensor< 2, dim, Number > &t, const Tensor< 2, dim, Number > &F)
template<int dim, typename Number>
Tensor< 4, dim, Number > pull_back (const Tensor< 4, dim, Number > &h, const Tensor< 2, dim, Number > &F)
template<int dim, typename Number>
SymmetricTensor< 4, dim, Number > pull_back (const SymmetricTensor< 4, dim, Number > &h, const Tensor< 2, dim, Number > &F)

Detailed Description

Transformation of tensors that are defined in terms of a set of covariant basis vectors. Rank-1 and rank-2 covariant tensors $\left(\bullet\right)^{\flat} = \mathbf{T}$ (and its spatial counterpart $\mathbf{t}$) typically satisfy the relation

\‍[ \int_{\partial V_{0}} \left[ \nabla_{0} \times \mathbf{T} \right] \cdot \mathbf{N} \; dA = \oint_{\partial A_{0}} \mathbf{T} \cdot \mathbf{L} \; dL = \oint_{\partial A_{t}} \mathbf{t} \cdot \mathbf{l} \; dl = \int_{\partial V_{t}} \left[ \nabla \times \mathbf{t} \right] \cdot \mathbf{n} \; da \‍]

where the control surfaces $\partial V_{0}$ and $\partial V_{t}$ with outwards facing normals $\mathbf{N}$ and $\mathbf{n}$ are bounded by the curves $\partial A_{0}$ and $\partial A_{t}$ that are, respectively, associated with the line directors $\mathbf{L}$ and $\mathbf{l}$.

Function Documentation

◆ push_forward() [1/5]

template<int dim, typename Number>
Tensor< 1, dim, Number > Physics::Transformations::Covariant::push_forward ( const Tensor< 1, dim, Number > & V,
const Tensor< 2, dim, Number > & F )

Return the result of the push forward transformation on a covariant vector, i.e.

\‍[ \chi\left(\bullet\right)^{\flat} \dealcoloneq \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \‍]

Parameters
[in]VThe (referential) vector to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi\left( \mathbf{V} \right)$

◆ push_forward() [2/5]

template<int dim, typename Number>
Tensor< 2, dim, Number > Physics::Transformations::Covariant::push_forward ( const Tensor< 2, dim, Number > & T,
const Tensor< 2, dim, Number > & F )

Return the result of the push forward transformation on a rank-2 covariant tensor, i.e.

\‍[ \chi\left(\bullet\right)^{\flat} \dealcoloneq \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}^{-1} \‍]

Parameters
[in]TThe (referential) rank-2 tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi\left( \mathbf{T} \right)$

◆ push_forward() [3/5]

template<int dim, typename Number>
SymmetricTensor< 2, dim, Number > Physics::Transformations::Covariant::push_forward ( const SymmetricTensor< 2, dim, Number > & T,
const Tensor< 2, dim, Number > & F )

Return the result of the push forward transformation on a rank-2 covariant symmetric tensor, i.e.

\‍[ \chi\left(\bullet\right)^{\flat} \dealcoloneq \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}^{-1} \‍]

Parameters
[in]TThe (referential) rank-2 symmetric tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi\left( \mathbf{T} \right)$

◆ push_forward() [4/5]

template<int dim, typename Number>
Tensor< 4, dim, Number > Physics::Transformations::Covariant::push_forward ( const Tensor< 4, dim, Number > & H,
const Tensor< 2, dim, Number > & F )

Return the result of the push forward transformation on a rank-4 covariant tensor, i.e. (in index notation):

\‍[ \left[ \chi\left(\bullet\right)^{\flat} \right]_{ijkl} \dealcoloneq F^{-T}_{iI} F^{-T}_{jJ} \left(\bullet\right)^{\flat}_{IJKL} F^{-T}_{kK} F^{-T}_{lL} \‍]

Parameters
[in]HThe (referential) rank-4 tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi\left( \mathbf{H} \right)$

◆ push_forward() [5/5]

template<int dim, typename Number>
SymmetricTensor< 4, dim, Number > Physics::Transformations::Covariant::push_forward ( const SymmetricTensor< 4, dim, Number > & H,
const Tensor< 2, dim, Number > & F )

Return the result of the push forward transformation on a rank-4 covariant symmetric tensor, i.e. (in index notation):

\‍[ \left[ \chi\left(\bullet\right)^{\flat} \right]_{ijkl} \dealcoloneq F^{-T}_{iI} F^{-T}_{jJ} \left(\bullet\right)^{\flat}_{IJKL} F^{-T}_{kK} F^{-T}_{lL} \‍]

Parameters
[in]HThe (referential) rank-4 symmetric tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi\left( \mathbf{H} \right)$

◆ pull_back() [1/5]

template<int dim, typename Number>
Tensor< 1, dim, Number > Physics::Transformations::Covariant::pull_back ( const Tensor< 1, dim, Number > & v,
const Tensor< 2, dim, Number > & F )

Return the result of the pull back transformation on a covariant vector, i.e.

\‍[ \chi^{-1}\left(\bullet\right)^{\flat} \dealcoloneq \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \‍]

Parameters
[in]vThe (spatial) vector to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi^{-1}\left( \mathbf{v} \right)$

◆ pull_back() [2/5]

template<int dim, typename Number>
Tensor< 2, dim, Number > Physics::Transformations::Covariant::pull_back ( const Tensor< 2, dim, Number > & t,
const Tensor< 2, dim, Number > & F )

Return the result of the pull back transformation on a rank-2 covariant tensor, i.e.

\‍[ \chi^{-1}\left(\bullet\right)^{\flat} \dealcoloneq \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F} \‍]

Parameters
[in]tThe (spatial) tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi^{-1}\left( \mathbf{t} \right)$

◆ pull_back() [3/5]

template<int dim, typename Number>
SymmetricTensor< 2, dim, Number > Physics::Transformations::Covariant::pull_back ( const SymmetricTensor< 2, dim, Number > & t,
const Tensor< 2, dim, Number > & F )

Return the result of the pull back transformation on a rank-2 covariant symmetric tensor, i.e.

\‍[ \chi^{-1}\left(\bullet\right)^{\flat} \dealcoloneq \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F} \‍]

Parameters
[in]tThe (spatial) symmetric tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi^{-1}\left( \mathbf{t} \right)$

◆ pull_back() [4/5]

template<int dim, typename Number>
Tensor< 4, dim, Number > Physics::Transformations::Covariant::pull_back ( const Tensor< 4, dim, Number > & h,
const Tensor< 2, dim, Number > & F )

Return the result of the pull back transformation on a rank-4 contravariant tensor, i.e. (in index notation):

\‍[ \left[ \chi^{-1}\left(\bullet\right)^{\flat} \right]_{IJKL} \dealcoloneq F^{T}_{Ii} F^{T}_{Jj} \left(\bullet\right)^{\flat}_{ijkl} F^{T}_{Kk} F^{T}_{Ll} \‍]

Parameters
[in]hThe (spatial) tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi^{-1}\left( \mathbf{h} \right)$

◆ pull_back() [5/5]

template<int dim, typename Number>
SymmetricTensor< 4, dim, Number > Physics::Transformations::Covariant::pull_back ( const SymmetricTensor< 4, dim, Number > & h,
const Tensor< 2, dim, Number > & F )

Return the result of the pull back transformation on a rank-4 contravariant symmetric tensor, i.e. (in index notation):

\‍[ \left[ \chi^{-1}\left(\bullet\right)^{\flat} \right]_{IJKL} \dealcoloneq F^{T}_{Ii} F^{T}_{Jj} \left(\bullet\right)^{\flat}_{ijkl} F^{T}_{Kk} F^{T}_{Ll} \‍]

Parameters
[in]hThe (spatial) symmetric tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
Returns
$\chi^{-1}\left( \mathbf{h} \right)$