Reference documentation for deal.II version 8.5.1
Public Member Functions | Static Public Member Functions | Private Member Functions | Private Attributes | Related Functions | List of all members
DerivativeForm< order, dim, spacedim, Number > Class Template Reference

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

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

Public Member Functions

 DerivativeForm ()
 
 DerivativeForm (const Tensor< order+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< 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
 
double 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 Number >
Tensor< 1, spacedim, Number > apply_transformation (const DerivativeForm< 1, dim, spacedim, Number > &DF, const Tensor< 1, dim, Number > &T)
 
template<int spacedim, int dim, typename Number >
DerivativeForm< 1, spacedim, dim > apply_transformation (const DerivativeForm< 1, dim, spacedim, Number > &DF, const Tensor< 2, dim, Number > &T)
 
template<int spacedim, int dim, typename Number >
Tensor< 2, spacedim, Number > apply_transformation (const DerivativeForm< 1, dim, spacedim, Number > &DF1, const DerivativeForm< 1, dim, spacedim, Number > &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 $ 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, $\nabla f(\mathbf x)$ 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 f(\mathbf x) \mathbf d = \lim_{\varepsilon\rightarrow 0} \frac{f(\mathbf x + \varepsilon \mathbf d) - f(\mathbf x)}{\varepsilon}, \end{align*}

i.e., one needs to be able to multiply the matrix $\nabla f(\mathbf x)$ 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.

Author
Sebastian Pauletti, 2011, Luca Heltai, 2015

Definition at line 56 of file derivative_form.h.

Constructor & Destructor Documentation

◆ DerivativeForm() [1/2]

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

Constructor. Initialize all entries to zero.

◆ DerivativeForm() [2/2]

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.

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=() [1/2]

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=() [2/2]

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> to Tensor<order+1,dim,Number>. In particular, if order==1 and the derivative is the Jacobian of F, then Tensor[i] = grad(F^i).

◆ 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> 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, that is to say 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 $\sqrt{\sum_{ij} |DF_{ij}|^2}$.

◆ determinant()

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

Compute the volume element associated with the jacobian of the transformation 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 $F$, then the current function computes the covariant form of the derivative, namely $(\nabla F)G^{-1}$, where $G = (\nabla F)^{T}*(\nabla F)$. If $\nabla F$ is a square matrix (i.e., $F: {\mathbb R}^n \mapsto {\mathbb R}^n$), then this function simplifies to computing $\nabla F^{-T}$.

◆ 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 (*this) * $T^{T}$

Friends And Related Function Documentation

◆ apply_transformation() [1/3]

template<int spacedim, int dim, typename Number >
Tensor< 1, spacedim, Number > apply_transformation ( const DerivativeForm< 1, dim, spacedim, Number > &  DF,
const Tensor< 1, dim, Number > &  T 
)
related

One of the uses of DerivativeForm is to apply it as a transformation. This is what this function does. If T is DerivativeForm<1,dim,1> it computes $DF * T$, if T is DerivativeForm<1,dim,rank> it computes $T*DF^{T}$.

Author
Sebastian Pauletti, 2011

Definition at line 401 of file derivative_form.h.

◆ apply_transformation() [2/3]

template<int spacedim, int dim, typename Number >
DerivativeForm< 1, spacedim, dim > apply_transformation ( const DerivativeForm< 1, dim, spacedim, Number > &  DF,
const Tensor< 2, dim, Number > &  T 
)
related

Similar to previous apply_transformation. It computes $T*DF^{T}$.

Author
Sebastian Pauletti, 2011

Definition at line 422 of file derivative_form.h.

◆ apply_transformation() [3/3]

template<int spacedim, int dim, typename Number >
Tensor< 2, spacedim, Number > apply_transformation ( const DerivativeForm< 1, dim, spacedim, Number > &  DF1,
const DerivativeForm< 1, dim, spacedim, Number > &  DF2 
)
related

Similar to previous apply_transformation. It computes $DF2*DF1^{T}$

Author
Sebastian Pauletti, 2011

Definition at line 442 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.

Author
Sebastian Pauletti, 2011

Definition at line 464 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 154 of file derivative_form.h.

The documentation for this class was generated from the following file:
Generated by   doxygen 1.8.13