Reference documentation for deal.II version 9.6.1
 
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Physics::Transformations::Piola 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 contravariant basis vectors and scale with the inverse of the volume change associated with the mapping.

Function Documentation

◆ push_forward() [1/5]

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

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

\[ \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
 \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
 \left(\bullet\right)^{\sharp}
\]

Parameters
[in]VThe (referential) vector to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left(
\mathbf{X} \right)$
Returns
$\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
\mathbf{V} \right)$

◆ push_forward() [2/5]

template<int dim, typename Number>
Tensor< 2, dim, Number > Physics::Transformations::Piola::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 contravariant tensor, i.e.

\[ \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
   \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
\left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
\]

Parameters
[in]TThe (referential) rank-2 tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left(
\mathbf{X} \right)$
Returns
$\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
\mathbf{T} \right)$

◆ push_forward() [3/5]

template<int dim, typename Number>
SymmetricTensor< 2, dim, Number > Physics::Transformations::Piola::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 contravariant symmetric tensor, i.e.

\[ \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
   \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot
\left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
\]

Parameters
[in]TThe (referential) rank-2 symmetric tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left(
\mathbf{X} \right)$
Returns
$\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
\mathbf{T} \right)$

◆ push_forward() [4/5]

template<int dim, typename Number>
Tensor< 4, dim, Number > Physics::Transformations::Piola::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 contravariant tensor, i.e. (in index notation):

\[ \textrm{det} \mathbf{F}^{-1} \; \left[
\chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
   \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ}
\left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
\]

Parameters
[in]HThe (referential) rank-4 tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left(
\mathbf{X} \right)$
Returns
$\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
\mathbf{H} \right)$

◆ push_forward() [5/5]

template<int dim, typename Number>
SymmetricTensor< 4, dim, Number > Physics::Transformations::Piola::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 contravariant symmetric tensor, i.e. (in index notation):

\[ \textrm{det} \mathbf{F}^{-1} \; \left[
\chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
   \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ}
\left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
\]

Parameters
[in]HThe (referential) rank-4 symmetric tensor to be operated on
[in]FThe deformation gradient tensor $\mathbf{F} \left(
\mathbf{X} \right)$
Returns
$\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
\mathbf{H} \right)$

◆ pull_back() [1/5]

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

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

\[ \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
   \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
\left(\bullet\right)^{\sharp}
\]

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

◆ pull_back() [2/5]

template<int dim, typename Number>
Tensor< 2, dim, Number > Physics::Transformations::Piola::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 contravariant tensor, i.e.

\[ \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
   \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
\left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
\]

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

◆ pull_back() [3/5]

template<int dim, typename Number>
SymmetricTensor< 2, dim, Number > Physics::Transformations::Piola::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 contravariant symmetric tensor, i.e.

\[ \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
   \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot
\left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
\]

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

◆ pull_back() [4/5]

template<int dim, typename Number>
Tensor< 4, dim, Number > Physics::Transformations::Piola::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):

\[ \textrm{det} \mathbf{F} \; \left[
\chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
   \dealcoloneq \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj}
\left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
\]

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

◆ pull_back() [5/5]

template<int dim, typename Number>
SymmetricTensor< 4, dim, Number > Physics::Transformations::Piola::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):

\[ \textrm{det} \mathbf{F} \; \left[
\chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
   \dealcoloneq \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj}
\left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
\]

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