deal.II version 9.7.1
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DerivativeForm< order, dim, spacedim, Number > Class Template Reference

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

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 63 of file derivative_form.h.

Inheritance diagram for DerivativeForm< order, dim, spacedim, Number >:
Inheritance graph
[legend]

Public Member Functions

 DerivativeForm ()=default
 DerivativeForm (const Tensor< order+1, dim, Number > &)
 DerivativeForm (const Tensor< 1, spacedim, Tensor< order, 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 > &)
template<typename OtherNumber>
DerivativeForm & operator= (const DerivativeForm< order, dim, spacedim, OtherNumber > &df)
 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
Tensor< 2, dim, Number > first_fundamental_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]

(Note that these are not member symbols.)

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

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< 1, spacedim, Tensor< order, dim, Number > > & )

Constructor. This constructor initializes the data stored by this object from a spacedim x dim^order array that is represented by a Tensor with spacedim components each of which is a Tensor of rank order and size dim.

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=() [1/4]

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=() [2/4]

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=() [3/4]

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=() [4/4]

template<int order, int dim, int spacedim, typename Number = double>
template<typename OtherNumber>
DerivativeForm & DerivativeForm< order, dim, spacedim, Number >::operator= ( const DerivativeForm< order, dim, spacedim, OtherNumber > & df)

Number conversion 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 $\nabla {\mathbf F}^{-T}$.

◆ first_fundamental_form()

template<int order, int dim, int spacedim, typename Number = double>
Tensor< 2, dim, Number > DerivativeForm< order, dim, spacedim, Number >::first_fundamental_form ( ) const

Compute the first fundamental form. This is the tensor of dot products of the columns of the current object.

Note
This function is only defined for order == 1.

◆ memory_consumption()

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

◆ 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.

Member Data Documentation

◆ tensor

template<int order, int dim, int spacedim, typename Number = double>
Tensor<order, dim, Number> DerivativeForm< order, dim, spacedim, Number >::tensor[spacedim]
private

Array of tensors holding the subelements.

Definition at line 204 of file derivative_form.h.

The documentation for this class was generated from the following file: