derivative_form.h File Reference

#include <deal.II/base/config.h>
#include <deal.II/base/tensor.h>

Go to the source code of this file.

Classes
class	DerivativeForm< order, dim, spacedim, Number >

Functions
template<int spacedim, int dim, typename Number1 , typename Number2 >
Tensor< 1, spacedim, typename ProductType< Number1, Number2 >::type >	apply_transformation (const DerivativeForm< 1, dim, spacedim, Number1 > &grad_F, const Tensor< 1, dim, Number2 > &d_x)
template<int spacedim, int dim, typename Number1 , typename Number2 >
DerivativeForm< 1, spacedim, dim, typename ProductType< Number1, Number2 >::type >	apply_transformation (const DerivativeForm< 1, dim, spacedim, Number1 > &grad_F, const Tensor< 2, dim, Number2 > &D_X)
template<int dim, typename Number1 , typename Number2 >
Tensor< 2, dim, typename ProductType< Number1, Number2 >::type >	apply_transformation (const DerivativeForm< 1, dim, dim, Number1 > &grad_F, const Tensor< 2, dim, Number2 > &D_X)
template<int spacedim, int dim, int n_components, typename Number1 , typename Number2 >
Tensor< 1, n_components, Tensor< 1, spacedim, typename ProductType< Number1, Number2 >::type > >	apply_transformation (const DerivativeForm< 1, dim, spacedim, Number1 > &grad_F, const Tensor< 1, n_components, Tensor< 1, dim, Number2 > > &D_X)
template<int spacedim, int dim, typename Number1 , typename Number2 >
Tensor< 2, spacedim, typename ProductType< Number1, Number2 >::type >	apply_transformation (const DerivativeForm< 1, dim, spacedim, Number1 > &DF1, const DerivativeForm< 1, dim, spacedim, Number2 > &DF2)
template<int dim, int spacedim, typename Number >
DerivativeForm< 1, spacedim, dim, Number >	transpose (const DerivativeForm< 1, dim, spacedim, Number > &DF)

Function Documentation

apply_transformation() [1/5]

template<int spacedim, int dim, typename Number1 , typename Number2 >

Tensor< 1, spacedim, typename ProductType< Number1, Number2 >::type > apply_transformation ( const DerivativeForm< 1, dim, spacedim, Number1 > &  grad_F, const Tensor< 1, dim, Number2 > &  d_x  )

inline

One of the uses of DerivativeForm is to apply it as a linear transformation. This function returns \(\nabla \mathbf F(\mathbf x) \Delta \mathbf x\), which approximates the change in \(\mathbf F(\mathbf x)\) when \(\mathbf x\) is changed by the amount \(\Delta \mathbf x\)

\[ \nabla \mathbf F(\mathbf x) \; \Delta \mathbf x \approx \mathbf F(\mathbf x + \Delta \mathbf x) - \mathbf F(\mathbf x). \]

The transformation corresponds to

\[ [\text{result}]_{i_1,\dots,i_k} = i\sum_{j} \left[\nabla \mathbf F(\mathbf x)\right]_{i_1,\dots,i_k, j} \Delta x_j \]

in index notation and corresponds to \([\Delta \mathbf x] [\nabla \mathbf F(\mathbf x)]^T\) in matrix notation.

Definition at line 454 of file derivative_form.h.

apply_transformation() [2/5]

template<int spacedim, int dim, typename Number1 , typename Number2 >

DerivativeForm< 1, spacedim, dim, typename ProductType< Number1, Number2 >::type > apply_transformation ( const DerivativeForm< 1, dim, spacedim, Number1 > &  grad_F, const Tensor< 2, dim, Number2 > &  D_X  )

inline

Similar to the previous apply_transformation(). Each row of the result corresponds to one of the rows of D_X transformed by grad_F, equivalent to \(\mathrm{D\_X} \, \mathrm{grad\_F}^T\) in matrix notation.

Definition at line 479 of file derivative_form.h.

apply_transformation() [3/5]

template<int dim, typename Number1 , typename Number2 >

Tensor< 2, dim, typename ProductType< Number1, Number2 >::type > apply_transformation ( const DerivativeForm< 1, dim, dim, Number1 > &  grad_F, const Tensor< 2, dim, Number2 > &  D_X  )

inline

Similar to the previous apply_transformation(), specialized for the case dim == spacedim where we can return a rank-2 tensor instead of the more general DerivativeForm. Each row of the result corresponds to one of the rows of D_X transformed by grad_F, equivalent to \(\mathrm{D\_X} \, \mathrm{grad\_F}^T\) in matrix notation.

Definition at line 505 of file derivative_form.h.

apply_transformation() [4/5]

template<int spacedim, int dim, int n_components, typename Number1 , typename Number2 >

Tensor< 1, n_components, Tensor< 1, spacedim, typename ProductType< Number1, Number2 >::type > > apply_transformation ( const DerivativeForm< 1, dim, spacedim, Number1 > &  grad_F, const Tensor< 1, n_components, Tensor< 1, dim, Number2 > > &  D_X  )

inline

Similar to the previous apply_transformation(). Each row of the result corresponds to one of the rows of D_X transformed by grad_F.

Definition at line 532 of file derivative_form.h.

apply_transformation() [5/5]

template<int spacedim, int dim, typename Number1 , typename Number2 >

Tensor< 2, spacedim, typename ProductType< Number1, Number2 >::type > apply_transformation ( const DerivativeForm< 1, dim, spacedim, Number1 > &  DF1, const DerivativeForm< 1, dim, spacedim, Number2 > &  DF2  )

inline

Similar to the previous apply_transformation(). In matrix notation, it computes \(DF2 \, DF1^{T}\). Moreover, the result of this operation \(\mathbf A\) can be interpreted as a metric tensor in \({\mathbb R}^\text{spacedim}\) which corresponds to the Euclidean metric tensor in \({\mathbb R}^\text{dim}\). For every pair of vectors \(\mathbf u, \mathbf v \in {\mathbb R}^\text{spacedim}\), we have:

\[ \mathbf u \cdot \mathbf A \mathbf v = \text{DF2}^{-1}(\mathbf u) \cdot \text{DF1}^{-1}(\mathbf v) \]

Definition at line 565 of file derivative_form.h.

transpose()

template<int dim, int spacedim, typename Number >

DerivativeForm< 1, spacedim, dim, Number > transpose ( const DerivativeForm< 1, dim, spacedim, Number > &  DF )

inline

Transpose of a rectangular DerivativeForm DF, mostly for compatibility reasons.

Definition at line 586 of file derivative_form.h.