Physics::Transformations::Covariant Namespace Reference

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 \(\mathbf{l}\).

Author

Jean-Paul Pelteret, Andrew McBride, 2016

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]	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::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]	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::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]	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::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]	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::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]	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::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]	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::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]	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::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]	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::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]	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::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]	h	The (spatial) symmetric tensor to be operated on
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns

\(\chi^{-1}\left( \mathbf{h} \right)\)