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deal.II version GIT relicensing-6809-ge913b9bb34 2026-09-25 17:20:01+00:00
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deal.II version GIT relicensing-6809-ge913b9bb34 2026-09-25 17:20:01+00:00
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#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 > |
Functions | |
| 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) |
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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 457 of file derivative_form.h.
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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 497 of file derivative_form.h.
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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 522 of file derivative_form.h.
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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 548 of file derivative_form.h.
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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 575 of file derivative_form.h.
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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 608 of file derivative_form.h.
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Transpose of a rectangular DerivativeForm DF, mostly for compatibility reasons.
Definition at line 629 of file derivative_form.h.
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Specialization of apply_transformation() for a diagonal DerivativeForm.
Definition at line 643 of file derivative_form.h.
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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 668 of file derivative_form.h.
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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 696 of file derivative_form.h.
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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 726 of file derivative_form.h.
1.9.8