deal.II version GIT relicensing-2289-g1e5549a87a 2024-12-21 21:30:00+00:00
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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) |
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 \(\mathbf{n}\).
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.
\[ \chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \]
[in] | V | The (referential) vector to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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.
\[ \chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T} \]
[in] | T | The (referential) rank-2 tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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.
\[ \chi\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T} \]
[in] | T | The (referential) rank-2 symmetric tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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):
\[ \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl} \dealcoloneq F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL} \]
[in] | H | The (referential) rank-4 tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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):
\[ \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl} \dealcoloneq F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL} \]
[in] | H | The (referential) rank-4 symmetric tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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.
\[ \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \]
[in] | v | The (spatial) vector to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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.
\[ \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T} \]
[in] | t | The (spatial) tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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.
\[ \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T} \]
[in] | t | The (spatial) symmetric tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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):
\[ \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL} \dealcoloneq F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll} \]
[in] | h | The (spatial) tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |
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):
\[ \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL} \dealcoloneq F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll} \]
[in] | h | The (spatial) symmetric tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |