Reference documentation for deal.II version 9.2.0
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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.
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.
\[ \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp} \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \]
[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::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.
\[ \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp} \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T} \]
[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::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.
\[ \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp} \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T} \]
[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::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)
\[ \textrm{det} \mathbf{F}^{-1} \; \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl} \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL} \]
[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::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)
\[ \textrm{det} \mathbf{F}^{-1} \; \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl} \dealcoloneq \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL} \]
[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::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.
\[ \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \]
[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::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.
\[ \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T} \]
[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::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.
\[ \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp} \dealcoloneq \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T} \]
[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::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)
\[ \textrm{det} \mathbf{F} \; \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL} \dealcoloneq \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll} \]
[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::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)
\[ \textrm{det} \mathbf{F} \; \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL} \dealcoloneq \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll} \]
[in] | h | The (spatial) symmetric tensor to be operated on |
[in] | F | The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) |