#include <deal.II/base/derivative_form.h>

Inheritance diagram for DerivativeForm< order, dim, spacedim, Number >:

Public Member Functions
	DerivativeForm ()=default

	DerivativeForm (const Tensor< order+1, dim, Number > &)

	DerivativeForm (const Tensor< order, spacedim, Tensor< 1, dim, Number > > &)

Tensor< order, dim, Number > &	operator[] (const unsigned int i)

const Tensor< order, dim, Number > &	operator[] (const unsigned int i) const

DerivativeForm &	operator= (const Tensor< order+1, dim, Number > &)

DerivativeForm &	operator= (const Tensor< order, spacedim, Tensor< 1, dim, Number > > &)

DerivativeForm &	operator= (const Tensor< 1, dim, Number > &)

	operator Tensor< order+1, dim, Number > () const

	operator Tensor< 1, dim, Number > () const

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

numbers::NumberTraits< Number >::real_type	norm () const

Number	determinant () const

DerivativeForm< 1, dim, spacedim, Number >	covariant_form () const

Static Public Member Functions
static std::size_t	memory_consumption ()

static ::ExceptionBase &	ExcInvalidTensorIndex (int arg1)

Private Member Functions
DerivativeForm< 1, dim, spacedim, Number >	times_T_t (const Tensor< 2, dim, Number > &T) const

Private Attributes
Tensor< order, dim, Number >	tensor [spacedim]

Related Functions
(Note that these are not member 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 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)

Detailed Description

template<int order, int dim, int spacedim, typename Number = double>
class DerivativeForm< order, dim, spacedim, Number >

This class represents the (tangential) derivatives of a function $\mathbf F: {\mathbb R}^{\text{dim}} \rightarrow {\mathbb R}^{\text{spacedim}}$ . Such functions are always used to map the reference dim-dimensional cell into spacedim-dimensional space. For such objects, the first derivative of the function is a linear map from ${\mathbb R}^{\text{dim}}$ to ${\mathbb R}^{\text{spacedim}}$ , i.e., it can be represented as a matrix in ${\mathbb R}^{\text{spacedim}\times \text{dim}}$ . This makes sense since one would represent the first derivative, $\nabla \mathbf F(\mathbf x)$ with $\mathbf x\in {\mathbb R}^{\text{dim}}$ , in such a way that the directional derivative in direction $\mathbf d\in {\mathbb R}^{\text{dim}}$ so that

$\begin{align*} \nabla \mathbf F(\mathbf x) \mathbf d = \lim_{\varepsilon\rightarrow 0} \frac{\mathbf F(\mathbf x + \varepsilon \mathbf d) - \mathbf F(\mathbf x)}{\varepsilon}, \end{align*}$

i.e., one needs to be able to multiply the matrix $\nabla \mathbf F(\mathbf x)$ by a vector in ${\mathbb R}^{\text{dim}}$ , and the result is a difference of function values, which are in ${\mathbb R}^{\text{spacedim}}$ . Consequently, the matrix must be of size $\text{spacedim}\times\text{dim}$ .

Similarly, the second derivative is a bilinear map from ${\mathbb R}^{\text{dim}} \times {\mathbb R}^{\text{dim}}$ to ${\mathbb R}^{\text{spacedim}}$ , which one can think of a rank-3 object of size $\text{spacedim}\times\text{dim}\times\text{dim}$ .

In deal.II we represent these derivatives using objects of type DerivativeForm<1,dim,spacedim,Number>, DerivativeForm<2,dim,spacedim,Number> and so on.

Definition at line 58 of file derivative_form.h.

Constructor & Destructor Documentation

◆ DerivativeForm() [1/3]

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm< order, dim, spacedim, Number >::DerivativeForm ( )

default

Constructor. Initialize all entries to zero.

◆ DerivativeForm() [2/3]

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm< order, dim, spacedim, Number >::DerivativeForm ( const Tensor< order+1, dim, Number > & )

Constructor from a tensor.

◆ DerivativeForm() [3/3]

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm< order, dim, spacedim, Number >::DerivativeForm ( const Tensor< order, spacedim, Tensor< 1, dim, Number > > & )

Constructor from a tensor.

Member Function Documentation

◆ operator[]() [1/2]

template<int order, int dim, int spacedim, typename Number = double>

Tensor< order, dim, Number > & DerivativeForm< order, dim, spacedim, Number >::operator[] ( const unsigned int i )

Read-Write access operator.

◆ operator[]() [2/2]

template<int order, int dim, int spacedim, typename Number = double>

const Tensor< order, dim, Number > & DerivativeForm< order, dim, spacedim, Number >::operator[] ( const unsigned int i ) const

Read-only access operator.

◆ operator=() [1/3]

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm & DerivativeForm< order, dim, spacedim, Number >::operator= ( const Tensor< order+1, dim, Number > & )

Assignment operator.

◆ operator=() [2/3]

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm & DerivativeForm< order, dim, spacedim, Number >::operator= ( const Tensor< order, spacedim, Tensor< 1, dim, Number > > & )

Assignment operator.

◆ operator=() [3/3]

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm & DerivativeForm< order, dim, spacedim, Number >::operator= ( const Tensor< 1, dim, Number > & )

Assignment operator.

◆ operator Tensor< order+1, dim, Number >()

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm< order, dim, spacedim, Number >::operator Tensor< order+1, dim, Number > ( ) const

Converts a DerivativeForm <order, dim, dim, Number> to Tensor<order+1, dim, Number>. In particular, if order == 1 and the derivative is the Jacobian of $\mathbf F(\mathbf x)$ , then Tensor[i] = $\nabla F_i(\mathbf x)$ .

◆ operator Tensor< 1, dim, Number >()

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm< order, dim, spacedim, Number >::operator Tensor< 1, dim, Number > ( ) const

Converts a DerivativeForm<1, dim, 1, Number> to Tensor<1, dim, Number>.

◆ transpose()

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm< 1, spacedim, dim, Number > DerivativeForm< order, dim, spacedim, Number >::transpose ( ) const

Return the transpose of a rectangular DerivativeForm, viewed as a two dimensional matrix.

◆ norm()

template<int order, int dim, int spacedim, typename Number = double>

numbers::NumberTraits< Number >::real_type DerivativeForm< order, dim, spacedim, Number >::norm ( ) const

Compute the Frobenius norm of this form, i.e., the expression $\sqrt{\sum_{ij} |DF_{ij}|^2} = \sqrt{\sum_{ij} |\frac{\partial F_i}{\partial x_j}|^2}$ .

◆ determinant()

template<int order, int dim, int spacedim, typename Number = double>

Number DerivativeForm< order, dim, spacedim, Number >::determinant ( ) const

Compute the volume element associated with the jacobian of the transformation $\mathbf F$ . That is to say if $DF$ is square, it computes $\det(DF)$ , in case DF is not square returns $\sqrt{\det(DF^T \,DF)}$ .

◆ covariant_form()

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm< 1, dim, spacedim, Number > DerivativeForm< order, dim, spacedim, Number >::covariant_form ( ) const

Assuming that the current object stores the Jacobian of a mapping $\mathbf F$ , then the current function computes the covariant form of the derivative, namely $(\nabla \mathbf F) {\mathbf G}^{-1}$ , where $\mathbf G = (\nabla \mathbf F)^{T}(\nabla \mathbf F)$ . If $\nabla \mathbf F$ is a square matrix (i.e., $\mathbf F: {\mathbb R}^n \mapsto {\mathbb R}^n$ ), then this function simplifies to computing $\nabla {\mathbf F}^{-T}$ .

◆ memory_consumption()

template<int order, int dim, int spacedim, typename Number = double>

static std::size_t DerivativeForm< order, dim, spacedim, Number >::memory_consumption ( )

static

Determine an estimate for the memory consumption (in bytes) of this object.

◆ times_T_t()

template<int order, int dim, int spacedim, typename Number = double>

DerivativeForm< 1, dim, spacedim, Number > DerivativeForm< order, dim, spacedim, Number >::times_T_t ( const Tensor< 2, dim, Number > & T ) const

private

Auxiliary function that computes $A T^{T}$ where A represents the current object.

Friends And Related Function Documentation

◆ apply_transformation() [1/4]

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
	)

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 431 of file derivative_form.h.

◆ apply_transformation() [2/4]

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
	)

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 456 of file derivative_form.h.

◆ apply_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_transformation	(	const DerivativeForm< 1, dim, spacedim, Number1 > &	grad_F,
		const Tensor< 1, n_components, Tensor< 1, dim, Number2 > > &	D_X
	)

Definition at line 484 of file derivative_form.h.

◆ apply_transformation() [4/4]

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
	)

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: