deal.II version 9.7.0
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derivative_form.h File Reference
#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/numbers.h>
#include <deal.II/base/template_constraints.h>
#include <deal.II/base/tensor.h>
#include <cstddef>

Go to the source code of this file.

Classes

class  DerivativeForm< order, dim, spacedim, Number >

Function Documentation

◆ operator<<()

template<int order, int dim, int spacedim, typename Number>
std::ostream & operator<< ( std::ostream & out,
const DerivativeForm< order, dim, spacedim, Number > & df )
related

Output operator for DerivativeForm. Print the elements consecutively, with a space in between, two spaces between rank 1 subtensors, three between rank 2 and so on.

Definition at line 459 of file derivative_form.h.

◆ 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 )
related

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 499 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 )
related

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 524 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 )
related

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 550 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 )
related

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 577 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 )
related

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 610 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)
related

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

Definition at line 631 of file derivative_form.h.

◆ apply_diagonal_transformation() [1/4]

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

Specialization of apply_transformation() for a diagonal DerivativeForm.

Definition at line 645 of file derivative_form.h.

◆ apply_diagonal_transformation() [2/4]

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

Similar to the previous apply_diagonal_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 670 of file derivative_form.h.

◆ apply_diagonal_transformation() [3/4]

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_diagonal_transformation ( const DerivativeForm< 1, dim, spacedim, Number1 > & grad_F,
const Tensor< 1, n_components, Tensor< 1, dim, Number2 > > & D_X )
related

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

Definition at line 698 of file derivative_form.h.

◆ apply_diagonal_transformation() [4/4]

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

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 728 of file derivative_form.h.