Reference documentation for deal.II version GIT relicensing-1262-g480f56a2fa 2024-07-23 19:40:02+00:00
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Physics::Transformations::Contravariant 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 bases. Rank-1 and rank-2 contravariant tensors $$\left(\bullet\right)^{\sharp} = \mathbf{T}$$ (and its spatial counterpart $$\mathbf{t}$$) typically satisfy the relation

$\int_{V_{0}} \nabla_{0} \cdot \mathbf{T} \; dV = \int_{\partial V_{0}} \mathbf{T} \cdot \mathbf{N} \; dA = \int_{\partial V_{t}} \mathbf{T} \cdot \mathbf{n} \; da = \int_{V_{t}} \nabla \cdot \mathbf{t} \; dv$

where $$V_{0}$$ and $$V_{t}$$ are respectively control volumes in the reference and spatial configurations, and their surfaces $$\partial V_{0}$$ and $$\partial V_{t}$$ have the outwards facing normals $$\mathbf{N}$$ and $$\mathbf{n}$$.

◆ push_forward() [1/5]

template<int dim, typename Number >
 Tensor< 1, dim, Number > Physics::Transformations::Contravariant::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.

$\chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp}$

Parameters
 [in] V The (referential) vector to be operated on [in] F The 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::Contravariant::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.

$\chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}$

Parameters
 [in] T The (referential) rank-2 tensor to be operated on [in] F The 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::Contravariant::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.

$\chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}$

Parameters
 [in] T The (referential) rank-2 symmetric tensor to be operated on [in] F The 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::Contravariant::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):

$\left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl} \dealcoloneq F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}$

Parameters
 [in] H The (referential) rank-4 tensor to be operated on [in] F The 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::Contravariant::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):

$\left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl} \dealcoloneq F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}$

Parameters
 [in] H The (referential) rank-4 symmetric tensor to be operated on [in] F The 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::Contravariant::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.

$\chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}$

Parameters
 [in] v The (spatial) vector to be operated on [in] F The 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::Contravariant::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.

$\chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}$

Parameters
 [in] t The (spatial) tensor to be operated on [in] F The 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::Contravariant::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.

$\chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}$

Parameters
 [in] t The (spatial) symmetric tensor to be operated on [in] F The 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::Contravariant::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):

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

Parameters
 [in] h The (spatial) tensor to be operated on [in] F The 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::Contravariant::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):

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

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