 |
Reference documentation for deal.II version 9.2.0
|
|
Reference documentation for deal.II version 9.2.0
|
#include <deal.II/base/derivative_form.h>
Public Member Functions | |
| DerivativeForm ()=default | |
| 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 |
| 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 Number > | |
| Tensor< 1, spacedim, Number > | apply_transformation (const DerivativeForm< 1, dim, spacedim, Number > &grad_F, const Tensor< 1, dim, Number > &d_x) |
| template<int spacedim, int dim, typename Number > | |
| DerivativeForm< 1, spacedim, dim, Number > | apply_transformation (const DerivativeForm< 1, dim, spacedim, Number > &grad_F, const Tensor< 2, dim, Number > &D_X) |
| 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) |
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 60 of file derivative_form.h.
|
default |
Constructor. Initialize all entries to zero.
| DerivativeForm< order, dim, spacedim, Number >::DerivativeForm | ( | const Tensor< order+1, dim, Number > & | ) |
Constructor from a tensor.
| Tensor<order, dim, Number>& DerivativeForm< order, dim, spacedim, Number >::operator[] | ( | const unsigned int | i | ) |
Read-Write access operator.
| const Tensor<order, dim, Number>& DerivativeForm< order, dim, spacedim, Number >::operator[] | ( | const unsigned int | i | ) | const |
Read-only access operator.
| DerivativeForm& DerivativeForm< order, dim, spacedim, Number >::operator= | ( | const Tensor< order+1, dim, Number > & | ) |
Assignment operator.
| DerivativeForm& DerivativeForm< order, dim, spacedim, Number >::operator= | ( | const Tensor< 1, dim, Number > & | ) |
Assignment operator.
| 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] = \(\nabla F_i(\mathbf x)\).
| DerivativeForm< order, dim, spacedim, Number >::operator Tensor< 1, dim, Number > | ( | ) | const |
Converts a DerivativeForm<1, dim, 1, Number> to Tensor<1, dim, Number>.
| 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.
| numbers::NumberTraits<Number>::real_type DerivativeForm< order, dim, spacedim, Number >::norm | ( | ) | const |
Compute the Frobenius norm of this form, i.e., the expression \(\sqrt{\sum_{ij} |DF_{ij}|^2} = \sqrt{\sum_{ij} |\frac{\partial F_i}{\partial x_j}|^2}\).
| 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 \(\sqrt{\det(DF^T \,DF)}\).
| 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 \(\nabla {\mathbf F}^{-T}\).
|
static |
Determine an estimate for the memory consumption (in bytes) of this object.
|
private |
Auxiliary function that computes \(A T^{T}\) where A represents the current object.
|
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 399 of file derivative_form.h.
|
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 \(\text{D\_X} \, \text{grad\_F}^T\) in matrix notation.
Definition at line 421 of file derivative_form.h.
|
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 449 of file derivative_form.h.
|
related |
Transpose of a rectangular DerivativeForm DF, mostly for compatibility reasons.
Definition at line 470 of file derivative_form.h.
|
private |
Array of tensors holding the subelements.
Definition at line 168 of file derivative_form.h.
1.8.17