DEAL_II_NAMESPACE_OPEN
/**
- * This class represents the (tangential) derivatives of a function $ f:
+ * 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, $\nabla f(\mathbf x)$ with $\mathbf x\in
+ * 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
* @f{align*}{
- * \nabla f(\mathbf x) \mathbf d
+ * \nabla \mathbf F(\mathbf x) \mathbf d
* = \lim_{\varepsilon\rightarrow 0}
- * \frac{f(\mathbf x + \varepsilon \mathbf d) - f(\mathbf x)}{\varepsilon},
+ * \frac{\mathbf F(\mathbf x + \varepsilon \mathbf d) - \mathbf F(\mathbf
+ * x)}{\varepsilon},
* @f}
- * 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 ${\mathbb R}^{\text{spacedim}}$.
+ * 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
operator=(const Tensor<1, dim, Number> &);
/**
- * 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).
+ * 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<order + 1, dim, Number>() const;
/**
- * Converts a DerivativeForm <1, dim, 1> to Tensor<1,dim,Number>.
+ * Converts a DerivativeForm<1, dim, 1, Number> to Tensor<1, dim, Number>.
*/
operator Tensor<1, dim, Number>() const;
/**
- * Return the transpose of a rectangular DerivativeForm, that is to say
+ * Return the transpose of a rectangular DerivativeForm,
* viewed as a two dimensional matrix.
*/
DerivativeForm<1, spacedim, dim, Number>
/**
* Compute the Frobenius norm of this form, i.e., the expression
- * $\sqrt{\sum_{ij} |DF_{ij}|^2}$.
+ * $\sqrt{\sum_{ij} |DF_{ij}|^2} =
+ * \sqrt{\sum_{ij} |\frac{\partial F_i}{\partial x_j}|}$.
*/
typename numbers::NumberTraits<Number>::real_type
norm() const;
/**
* Compute the volume element associated with the jacobian of the
- * transformation F. That is to say if $DF$ is square, it computes
+ * transformation $\mathbf F$. That is to say if $DF$ is square, it computes
* $\det(DF)$, in case DF is not square returns $\sqrt{\det(DF^T * DF)}$.
*/
Number
/**
* Assuming that the current object stores the Jacobian of a mapping
- * $F$, then the current function computes the <i>covariant</i> 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:
+ * $\mathbf F$, then the current function computes the <i>covariant</i> 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 $\nabla F^{-T}$.
+ * simplifies to computing $\nabla {\mathbf F}^{-T}$.
*/
DerivativeForm<1, dim, spacedim, Number>
covariant_form() const;
/**
- * One of the uses of DerivativeForm is to apply it as a transformation. This
- * is what this function does. If @p DF is DerivativeForm<1,dim,1> it computes
- * $DF * T$, if @p DF is DerivativeForm<1,dim,rank> it computes $T*DF^{T}$.
+ * 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 $\Delta \mathbf x$
+ * @f[
+ * \nabla \mathbf F(\mathbf x) \; \Delta \mathbf x
+ * \approx
+ * \mathbf F(\mathbf x + \Delta \mathbf x) - \mathbf F(\mathbf x).
+ * @f]
+ * The transformation corresponds to
+ * @f[
+ * [\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
+ * @f]
+ * in index notation and corresponds to
+ * $[\Delta \mathbf x] [\nabla \mathbf F(\mathbf x)]^T$ in matrix notation.
*
* @relatesalso DerivativeForm
- * @author Sebastian Pauletti, 2011
+ * @author Sebastian Pauletti, 2011, Reza Rastak, 2019
*/
template <int spacedim, int dim, typename Number>
inline Tensor<1, spacedim, Number>
-apply_transformation(const DerivativeForm<1, dim, spacedim, Number> &DF,
- const Tensor<1, dim, Number> & T)
+apply_transformation(const DerivativeForm<1, dim, spacedim, Number> &grad_F,
+ const Tensor<1, dim, Number> & d_x)
{
Tensor<1, spacedim, Number> dest;
for (unsigned int i = 0; i < spacedim; ++i)
- dest[i] = DF[i] * T;
+ dest[i] = grad_F[i] * d_x;
return dest;
}
/**
- * Similar to previous apply_transformation. It computes $T*DF^{T}$.
+ * Similar to the previous apply_transformation().
+ * Each row of the result corresponds to one of the rows of @p D_X transformed
+ * by @p grad_F, equivalent to $\text{D\_X} * \text{grad\_F}^T$ in matrix notation.
*
* @relatesalso DerivativeForm
- * @author Sebastian Pauletti, 2011
+ * @author Sebastian Pauletti, 2011, Reza Rastak, 2019
*/
// rank=2
template <int spacedim, int dim, typename Number>
inline DerivativeForm<1, spacedim, dim>
-apply_transformation(const DerivativeForm<1, dim, spacedim, Number> &DF,
- const Tensor<2, dim, Number> & T)
+apply_transformation(const DerivativeForm<1, dim, spacedim, Number> &grad_F,
+ const Tensor<2, dim, Number> & D_X)
{
DerivativeForm<1, spacedim, dim> dest;
for (unsigned int i = 0; i < dim; ++i)
- dest[i] = apply_transformation(DF, T[i]);
+ dest[i] = apply_transformation(grad_F, D_X[i]);
return dest;
}
/**
- * Similar to previous apply_transformation. It computes $DF2*DF1^{T}$
+ * 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:
+ * @f[
+ * \mathbf u \cdot \mathbf A \mathbf v =
+ * \text{DF2}^{-1}(\mathbf u) \cdot \text{DF1}^{-1}(\mathbf v)
+ * @f]
*
* @relatesalso DerivativeForm
- * @author Sebastian Pauletti, 2011
+ * @author Sebastian Pauletti, 2011, Reza Rastak, 2019
*/
template <int spacedim, int dim, typename Number>
inline Tensor<2, spacedim, Number>
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