* rotation angles and axes of rotation.
*
* @author Jean-Paul Pelteret, 2017
- */
+ */
namespace Rotations
{
/**
//@{
/**
- * Returns the rotation matrix for 2-d Euclidean space, namely
- * @f[
- * \mathbf{R} := \left[ \begin{array}{cc}
- * cos(\theta) & sin(\theta) \\
- * -sin(\theta) & cos(\theta)
- * \end{array}\right]
- * @f]
- * where $\theta$ is the rotation angle given in radians.
- * In particular, this describes the counter-clockwise rotation of a vector
- * relative to a
- * <a href="http://mathworld.wolfram.com/RotationMatrix.html">fixed set of right-handed axes</a>.
- *
- * @param[in] angle The rotation angle (about the z-axis) in radians
- */
+ * Returns the rotation matrix for 2-d Euclidean space, namely
+ * @f[
+ * \mathbf{R} := \left[ \begin{array}{cc}
+ * cos(\theta) & sin(\theta) \\
+ * -sin(\theta) & cos(\theta)
+ * \end{array}\right]
+ * @f]
+ * where $\theta$ is the rotation angle given in radians. In particular,
+ * this describes the counter-clockwise rotation of a vector relative to
+ * a <a href="http://mathworld.wolfram.com/RotationMatrix.html">fixed
+ * set of right-handed axes</a>.
+ *
+ * @param[in] angle The rotation angle (about the z-axis) in radians
+ */
template<typename Number>
Tensor<2,2,Number>
rotation_matrix_2d (const Number &angle);
/**
- * Returns the rotation matrix for 3-d Euclidean space.
- * Most concisely stated using the Rodrigues' rotation formula, this
- * function returns the equivalent of
- * @f[
- * \mathbf{R} := cos(\theta)\mathbf{I} + sin(\theta)\mathbf{W}
- * + (1-cos(\theta))\mathbf{u}\otimes\mathbf{u}
- * @f]
- * where $\mathbf{u}$ is the axial vector (an axial vector) and $\theta$
- * is the rotation angle given in radians, $\mathbf{I}$ is the identity
- * tensor and $\mathbf{W}$ is the skew symmetric tensor of $\mathbf{u}$.
- *
- * @dealiiWriggersA{374,9.194}
- * This presents Rodrigues' rotation formula, but the implementation used
- * in this function is described in this
- * <a href="https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle">wikipedia link</a>.
- * In particular, this describes the counter-clockwise rotation of a vector
- * <a href="http://mathworld.wolfram.com/RotationMatrix.html">in a plane with its normal</a>.
- * defined by the @p axis of rotation.
- * An alternative implementation is discussed at
- * <a href="https://www.gamedev.net/resources/_/technical/math-and-physics/do-we-really-need-quaternions-r1199">this link</a>,
- * but is inconsistent (sign-wise) with the Rodrigues' rotation formula as
- * it describes the rotation of a coordinate system.
- *
- * @param[in] axis A unit vector that defines the axis of rotation
- * @param[in] angle The rotation angle in radians
- */
+ * Returns the rotation matrix for 3-d Euclidean space. Most concisely
+ * stated using the Rodrigues' rotation formula, this function returns
+ * the equivalent of
+ * @f[
+ * \mathbf{R} := cos(\theta)\mathbf{I} + sin(\theta)\mathbf{W}
+ * + (1-cos(\theta))\mathbf{u}\otimes\mathbf{u}
+ * @f]
+ * where $\mathbf{u}$ is the axial vector (an axial vector) and $\theta$
+ * is the rotation angle given in radians, $\mathbf{I}$ is the identity
+ * tensor and $\mathbf{W}$ is the skew symmetric tensor of $\mathbf{u}$.
+ *
+ * @dealiiWriggersA{374,9.194} This presents Rodrigues' rotation
+ * formula, but the implementation used in this function is described in
+ * this <a
+ * href="https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle">wikipedia
+ * link</a>. In particular, this describes the counter-clockwise
+ * rotation of a vector <a
+ * href="http://mathworld.wolfram.com/RotationMatrix.html">in a plane
+ * with its normal</a>. defined by the @p axis of rotation. An
+ * alternative implementation is discussed at <a
+ * href="https://www.gamedev.net/resources/_/technical/math-and-physics/do-we-really-need-quaternions-r1199">this
+ * link</a>, but is inconsistent (sign-wise) with the Rodrigues'
+ * rotation formula as it describes the rotation of a coordinate system.
+ *
+ * @param[in] axis A unit vector that defines the axis of rotation
+ * @param[in] angle The rotation angle in radians
+ */
template<typename Number>
Tensor<2,3,Number>
rotation_matrix_3d (const Point<3,Number> &axis,
/**
* Transformation of tensors that are defined in terms of a set of
* contravariant bases. Rank-1 and rank-2 contravariant tensors
- * $\left(\bullet\right)^{\sharp} = \mathbf{T}$ (and its spatial counterpart
- * $\mathbf{t}$) typically satisfy the relation
+ * $\left(\bullet\right)^{\sharp} = \mathbf{T}$ (and its spatial
+ * counterpart $\mathbf{t}$) typically satisfy the relation
* @f[
* \int_{V_{0}} \nabla_{0} \cdot \mathbf{T} \; dV
* = \int_{\partial V_{0}} \mathbf{T} \cdot \mathbf{N} \; dA
* = \int_{V_{t}} \nabla \cdot \mathbf{t} \; dv
* @f]
* where $V_{0}$ and $V_{t}$ are respectively control volumes in the
- * reference and spatial configurations, and their surfaces $\partial V_{0}$
- * and $\partial V_{t}$ have the outwards facing normals $\mathbf{N}$ and
- * $\mathbf{n}$.
+ * reference and spatial configurations, and their surfaces $\partial
+ * V_{0}$ and $\partial V_{t}$ have the outwards facing normals
+ * $\mathbf{N}$ and $\mathbf{n}$.
*
* @author Jean-Paul Pelteret, Andrew McBride, 2016
- */
+ */
namespace Contravariant
{
//@{
/**
- * Returns the result of the push forward transformation on a
- * contravariant vector, i.e.
- * @f[
- * \chi\left(\bullet\right)^{\sharp}
- * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp}
- * @f]
- *
- * @param[in] V The (referential) vector to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{V} \right)$
- */
+ * Returns the result of the push forward transformation on a
+ * contravariant vector, i.e.
+ * @f[
+ * \chi\left(\bullet\right)^{\sharp}
+ * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp}
+ * @f]
+ *
+ * @param[in] V The (referential) vector to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{V} \right)$
+ */
template <int dim, typename Number>
Tensor<1,dim,Number>
push_forward (const Tensor<1,dim,Number> &V,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-2
- * contravariant tensor, i.e.
- * @f[
- * \chi\left(\bullet\right)^{\sharp}
- * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
- * @f]
- *
- * @param[in] T The (referential) rank-2 tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{T} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-2
+ * contravariant tensor, i.e.
+ * @f[
+ * \chi\left(\bullet\right)^{\sharp}
+ * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
+ * @f]
+ *
+ * @param[in] T The (referential) rank-2 tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{T} \right)$
+ */
template <int dim, typename Number>
Tensor<2,dim,Number>
push_forward (const Tensor<2,dim,Number> &T,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-2
- * contravariant symmetric tensor, i.e.
- * @f[
- * \chi\left(\bullet\right)^{\sharp}
- * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
- * @f]
- *
- * @param[in] T The (referential) rank-2 symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{T} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-2
+ * contravariant symmetric tensor, i.e.
+ * @f[
+ * \chi\left(\bullet\right)^{\sharp}
+ * := \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
+ * @f]
+ *
+ * @param[in] T The (referential) rank-2 symmetric tensor to be operated
+ * on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{T} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<2,dim,Number>
push_forward (const SymmetricTensor<2,dim,Number> &T,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-4
- * contravariant tensor, i.e. (in index notation)
- * @f[
- * \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
- * := F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
- * @f]
- *
- * @param[in] H The (referential) rank-4 tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{H} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-4
+ * contravariant tensor, i.e. (in index notation)
+ * @f[
+ * \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
+ * := F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
+ * @f]
+ *
+ * @param[in] H The (referential) rank-4 tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{H} \right)$
+ */
template <int dim, typename Number>
Tensor<4,dim,Number>
push_forward (const Tensor<4,dim,Number> &H,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-4
- * contravariant symmetric tensor, i.e. (in index notation)
- * @f[
- * \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
- * := F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
- * @f]
- *
- * @param[in] H The (referential) rank-4 symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{H} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-4
+ * contravariant symmetric tensor, i.e. (in index notation)
+ * @f[
+ * \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
+ * := F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
+ * @f]
+ *
+ * @param[in] H The (referential) rank-4 symmetric tensor to be operated
+ * on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{H} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<4,dim,Number>
push_forward (const SymmetricTensor<4,dim,Number> &H,
//@{
/**
- * Returns the result of the pull back transformation on a
- * contravariant vector, i.e.
- * @f[
- * \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}
- * @f]
- *
- * @param[in] v The (spatial) vector to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{v} \right)$
- */
+ * Returns the result of the pull back transformation on a contravariant
+ * vector, i.e.
+ * @f[
+ * \chi^{-1}\left(\bullet\right)^{\sharp}
+ * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}
+ * @f]
+ *
+ * @param[in] v The (spatial) vector to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{v} \right)$
+ */
template <int dim, typename Number>
Tensor<1,dim,Number>
pull_back (const Tensor<1,dim,Number> &v,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-2
- * contravariant tensor, i.e.
- * @f[
- * \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
- * @f]
- *
- * @param[in] t The (spatial) tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{t} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-2
+ * contravariant tensor, i.e.
+ * @f[
+ * \chi^{-1}\left(\bullet\right)^{\sharp}
+ * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
+ * @f]
+ *
+ * @param[in] t The (spatial) tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{t} \right)$
+ */
template <int dim, typename Number>
Tensor<2,dim,Number>
pull_back (const Tensor<2,dim,Number> &t,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-2
- * contravariant symmetric tensor, i.e.
- * @f[
- * \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
- * @f]
- *
- * @param[in] t The (spatial) symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{t} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-2
+ * contravariant symmetric tensor, i.e.
+ * @f[
+ * \chi^{-1}\left(\bullet\right)^{\sharp}
+ * := \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
+ * @f]
+ *
+ * @param[in] t The (spatial) symmetric tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{t} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<2,dim,Number>
pull_back (const SymmetricTensor<2,dim,Number> &t,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-4
- * contravariant tensor, i.e. (in index notation)
- * @f[
- * \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
- * := F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
- * @f]
- *
- * @param[in] h The (spatial) tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{h} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-4
+ * contravariant tensor, i.e. (in index notation)
+ * @f[
+ * \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
+ * := F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
+ * @f]
+ *
+ * @param[in] h The (spatial) tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{h} \right)$
+ */
template <int dim, typename Number>
Tensor<4,dim,Number>
pull_back (const Tensor<4,dim,Number> &h,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-4
- * contravariant symmetric tensor, i.e. (in index notation)
- * @f[
- * \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
- * := F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
- * @f]
- *
- * @param[in] h The (spatial) symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{h} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-4
+ * contravariant symmetric tensor, i.e. (in index notation)
+ * @f[
+ * \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
+ * := F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
+ * @f]
+ *
+ * @param[in] h The (spatial) symmetric tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{h} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<4,dim,Number>
pull_back (const SymmetricTensor<4,dim,Number> &h,
/**
* Transformation of tensors that are defined in terms of a set of
* covariant basis vectors. Rank-1 and rank-2 covariant tensors
- * $\left(\bullet\right)^{\flat} = \mathbf{T}$ (and its spatial counterpart
- * $\mathbf{t}$) typically satisfy the relation
+ * $\left(\bullet\right)^{\flat} = \mathbf{T}$ (and its spatial
+ * counterpart $\mathbf{t}$) typically satisfy the relation
* @f[
* \int_{\partial V_{0}} \left[ \nabla_{0} \times \mathbf{T} \right] \cdot \mathbf{N} \; dA
* = \oint_{\partial A_{0}} \mathbf{T} \cdot \mathbf{L} \; dL
* = \int_{\partial V_{t}} \left[ \nabla \times \mathbf{t} \right] \cdot \mathbf{n} \; da
* @f]
* where the control surfaces $\partial V_{0}$ and $\partial V_{t}$ with
- * outwards facing normals $\mathbf{N}$ and $\mathbf{n}$ are bounded by the
- * curves $\partial A_{0}$ and $\partial A_{0}$ that are, respectively,
- * associated with the line directors $\mathbf{L}$ and $\mathbf{l}$.
+ * outwards facing normals $\mathbf{N}$ and $\mathbf{n}$ are bounded by
+ * the curves $\partial A_{0}$ and $\partial A_{0}$ that are,
+ * respectively, associated with the line directors $\mathbf{L}$ and
+ * $\mathbf{l}$.
*
* @author Jean-Paul Pelteret, Andrew McBride, 2016
- */
+ */
namespace Covariant
{
//@{
/**
- * Returns the result of the push forward transformation on a covariant
- * vector, i.e.
- * @f[
- * \chi\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat}
- * @f]
- *
- * @param[in] V The (referential) vector to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{V} \right)$
- */
+ * Returns the result of the push forward transformation on a covariant
+ * vector, i.e.
+ * @f[
+ * \chi\left(\bullet\right)^{\flat}
+ * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat}
+ * @f]
+ *
+ * @param[in] V The (referential) vector to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{V} \right)$
+ */
template <int dim, typename Number>
Tensor<1,dim,Number>
push_forward (const Tensor<1,dim,Number> &V,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-2
- * covariant tensor, i.e.
- * @f[
- * \chi\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}^{-1}
- * @f]
- *
- * @param[in] T The (referential) rank-2 tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{T} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-2
+ * covariant tensor, i.e.
+ * @f[
+ * \chi\left(\bullet\right)^{\flat}
+ * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}^{-1}
+ * @f]
+ *
+ * @param[in] T The (referential) rank-2 tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{T} \right)$
+ */
template <int dim, typename Number>
Tensor<2,dim,Number>
push_forward (const Tensor<2,dim,Number> &T,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-2
- * covariant symmetric tensor, i.e.
- * @f[
- * \chi\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}^{-1}
- * @f]
- *
- * @param[in] T The (referential) rank-2 symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{T} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-2
+ * covariant symmetric tensor, i.e.
+ * @f[
+ * \chi\left(\bullet\right)^{\flat}
+ * := \mathbf{F}^{-T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}^{-1}
+ * @f]
+ *
+ * @param[in] T The (referential) rank-2 symmetric tensor to be operated
+ * on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{T} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<2,dim,Number>
push_forward (const SymmetricTensor<2,dim,Number> &T,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-4
- * covariant tensor, i.e. (in index notation)
- * @f[
- * \left[ \chi\left(\bullet\right)^{\flat} \right]_{ijkl}
- * := F^{-T}_{iI} F^{-T}_{jJ} \left(\bullet\right)^{\flat}_{IJKL} F^{-T}_{kK} F^{-T}_{lL}
- * @f]
- *
- * @param[in] H The (referential) rank-4 tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{H} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-4
+ * covariant tensor, i.e. (in index notation)
+ * @f[
+ * \left[ \chi\left(\bullet\right)^{\flat} \right]_{ijkl}
+ * := F^{-T}_{iI} F^{-T}_{jJ} \left(\bullet\right)^{\flat}_{IJKL} F^{-T}_{kK} F^{-T}_{lL}
+ * @f]
+ *
+ * @param[in] H The (referential) rank-4 tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{H} \right)$
+ */
template <int dim, typename Number>
Tensor<4,dim,Number>
push_forward (const Tensor<4,dim,Number> &H,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-4
- * covariant symmetric tensor, i.e. (in index notation)
- * @f[
- * \left[ \chi\left(\bullet\right)^{\flat} \right]_{ijkl}
- * := F^{-T}_{iI} F^{-T}_{jJ} \left(\bullet\right)^{\flat}_{IJKL} F^{-T}_{kK} F^{-T}_{lL}
- * @f]
- *
- * @param[in] H The (referential) rank-4 symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi\left( \mathbf{H} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-4
+ * covariant symmetric tensor, i.e. (in index notation)
+ * @f[
+ * \left[ \chi\left(\bullet\right)^{\flat} \right]_{ijkl}
+ * := F^{-T}_{iI} F^{-T}_{jJ} \left(\bullet\right)^{\flat}_{IJKL} F^{-T}_{kK} F^{-T}_{lL}
+ * @f]
+ *
+ * @param[in] H The (referential) rank-4 symmetric tensor to be operated
+ * on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi\left( \mathbf{H} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<4,dim,Number>
push_forward (const SymmetricTensor<4,dim,Number> &H,
//@{
/**
- * Returns the result of the pull back transformation on a
- * covariant vector, i.e.
- * @f[
- * \chi^{-1}\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat}
- * @f]
- *
- * @param[in] v The (spatial) vector to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{v} \right)$
- */
+ * Returns the result of the pull back transformation on a covariant
+ * vector, i.e.
+ * @f[
+ * \chi^{-1}\left(\bullet\right)^{\flat}
+ * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat}
+ * @f]
+ *
+ * @param[in] v The (spatial) vector to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{v} \right)$
+ */
template <int dim, typename Number>
Tensor<1,dim,Number>
pull_back (const Tensor<1,dim,Number> &v,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-2
- * covariant tensor, i.e.
- * @f[
- * \chi^{-1}\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}
- * @f]
- *
- * @param[in] t The (spatial) tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{t} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-2
+ * covariant tensor, i.e.
+ * @f[
+ * \chi^{-1}\left(\bullet\right)^{\flat}
+ * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}
+ * @f]
+ *
+ * @param[in] t The (spatial) tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{t} \right)$
+ */
template <int dim, typename Number>
Tensor<2,dim,Number>
pull_back (const Tensor<2,dim,Number> &t,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-2
- * covariant symmetric tensor, i.e.
- * @f[
- * \chi^{-1}\left(\bullet\right)^{\flat}
- * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}
- * @f]
- *
- * @param[in] t The (spatial) symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{t} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-2
+ * covariant symmetric tensor, i.e.
+ * @f[
+ * \chi^{-1}\left(\bullet\right)^{\flat}
+ * := \mathbf{F}^{T} \cdot \left(\bullet\right)^{\flat} \cdot \mathbf{F}
+ * @f]
+ *
+ * @param[in] t The (spatial) symmetric tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{t} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<2,dim,Number>
pull_back (const SymmetricTensor<2,dim,Number> &t,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-4
- * contravariant tensor, i.e. (in index notation)
- * @f[
- * \left[ \chi^{-1}\left(\bullet\right)^{\flat} \right]_{IJKL}
- * := F^{T}_{Ii} F^{T}_{Jj} \left(\bullet\right)^{\flat}_{ijkl} F^{T}_{Kk} F^{T}_{Ll}
- * @f]
- *
- * @param[in] h The (spatial) tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{h} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-4
+ * contravariant tensor, i.e. (in index notation)
+ * @f[
+ * \left[ \chi^{-1}\left(\bullet\right)^{\flat} \right]_{IJKL}
+ * := F^{T}_{Ii} F^{T}_{Jj} \left(\bullet\right)^{\flat}_{ijkl} F^{T}_{Kk} F^{T}_{Ll}
+ * @f]
+ *
+ * @param[in] h The (spatial) tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{h} \right)$
+ */
template <int dim, typename Number>
Tensor<4,dim,Number>
pull_back (const Tensor<4,dim,Number> &h,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-4
- * contravariant symmetric tensor, i.e. (in index notation)
- * @f[
- * \left[ \chi^{-1}\left(\bullet\right)^{\flat} \right]_{IJKL}
- * := F^{T}_{Ii} F^{T}_{Jj} \left(\bullet\right)^{\flat}_{ijkl} F^{T}_{Kk} F^{T}_{Ll}
- * @f]
- *
- * @param[in] h The (spatial) symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\chi^{-1}\left( \mathbf{h} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-4
+ * contravariant symmetric tensor, i.e. (in index notation)
+ * @f[
+ * \left[ \chi^{-1}\left(\bullet\right)^{\flat} \right]_{IJKL}
+ * := F^{T}_{Ii} F^{T}_{Jj} \left(\bullet\right)^{\flat}_{ijkl} F^{T}_{Kk} F^{T}_{Ll}
+ * @f]
+ *
+ * @param[in] h The (spatial) symmetric tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\chi^{-1}\left( \mathbf{h} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<4,dim,Number>
pull_back (const SymmetricTensor<4,dim,Number> &h,
* change associated with the mapping.
*
* @author Jean-Paul Pelteret, Andrew McBride, 2016
- */
+ */
namespace Piola
{
//@{
/**
- * Returns the result of the push forward transformation on a
- * contravariant vector, i.e.
- * @f[
- * \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp}
- * @f]
- *
- * @param[in] V The (referential) vector to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left( \mathbf{V} \right)$
- */
+ * Returns the result of the push forward transformation on a
+ * contravariant vector, i.e.
+ * @f[
+ * \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
+ * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp}
+ * @f]
+ *
+ * @param[in] V The (referential) vector to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
+ * \mathbf{V} \right)$
+ */
template <int dim, typename Number>
Tensor<1,dim,Number>
push_forward (const Tensor<1,dim,Number> &V,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-2
- * contravariant tensor, i.e.
- * @f[
- * \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
- * @f]
- *
- * @param[in] T The (referential) rank-2 tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left( \mathbf{T} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-2
+ * contravariant tensor, i.e.
+ * @f[
+ * \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
+ * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
+ * @f]
+ *
+ * @param[in] T The (referential) rank-2 tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
+ * \mathbf{T} \right)$
+ */
template <int dim, typename Number>
Tensor<2,dim,Number>
push_forward (const Tensor<2,dim,Number> &T,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-2
- * contravariant symmetric tensor, i.e.
- * @f[
- * \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
- * @f]
- *
- * @param[in] T The (referential) rank-2 symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left( \mathbf{T} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-2
+ * contravariant symmetric tensor, i.e.
+ * @f[
+ * \textrm{det} \mathbf{F}^{-1} \; \chi\left(\bullet\right)^{\sharp}
+ * := \frac{1}{\textrm{det} \mathbf{F}} \; \mathbf{F} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{T}
+ * @f]
+ *
+ * @param[in] T The (referential) rank-2 symmetric tensor to be operated
+ * on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
+ * \mathbf{T} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<2,dim,Number>
push_forward (const SymmetricTensor<2,dim,Number> &T,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-4
- * contravariant tensor, i.e. (in index notation)
- * @f[
- * \textrm{det} \mathbf{F}^{-1} \; \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
- * @f]
- *
- * @param[in] H The (referential) rank-4 tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left( \mathbf{H} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-4
+ * contravariant tensor, i.e. (in index notation)
+ * @f[
+ * \textrm{det} \mathbf{F}^{-1} \; \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
+ * := \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
+ * @f]
+ *
+ * @param[in] H The (referential) rank-4 tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
+ * \mathbf{H} \right)$
+ */
template <int dim, typename Number>
Tensor<4,dim,Number>
push_forward (const Tensor<4,dim,Number> &H,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the push forward transformation on a rank-4
- * contravariant symmetric tensor, i.e. (in index notation)
- * @f[
- * \textrm{det} \mathbf{F}^{-1} \; \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
- * := \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
- * @f]
- *
- * @param[in] H The (referential) rank-4 symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left( \mathbf{H} \right)$
- */
+ * Returns the result of the push forward transformation on a rank-4
+ * contravariant symmetric tensor, i.e. (in index notation)
+ * @f[
+ * \textrm{det} \mathbf{F}^{-1} \; \left[ \chi\left(\bullet\right)^{\sharp} \right]_{ijkl}
+ * := \frac{1}{\textrm{det} \mathbf{F}} \; F_{iI} F_{jJ} \left(\bullet\right)^{\sharp}_{IJKL} F_{kK} F_{lL}
+ * @f]
+ *
+ * @param[in] H The (referential) rank-4 symmetric tensor to be operated
+ * on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\frac{1}{\textrm{det} \mathbf{F}} \; \chi\left(
+ * \mathbf{H} \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<4,dim,Number>
push_forward (const SymmetricTensor<4,dim,Number> &H,
//@{
/**
- * Returns the result of the pull back transformation on a
- * contravariant vector, i.e.
- * @f[
- * \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}
- * @f]
- *
- * @param[in] v The (spatial) vector to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{v} \right)$
- */
+ * Returns the result of the pull back transformation on a contravariant
+ * vector, i.e.
+ * @f[
+ * \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
+ * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp}
+ * @f]
+ *
+ * @param[in] v The (spatial) vector to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{v}
+ * \right)$
+ */
template <int dim, typename Number>
Tensor<1,dim,Number>
pull_back (const Tensor<1,dim,Number> &v,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-2
- * contravariant tensor, i.e.
- * @f[
- * \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
- * @f]
- *
- * @param[in] t The (spatial) tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{t} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-2
+ * contravariant tensor, i.e.
+ * @f[
+ * \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
+ * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
+ * @f]
+ *
+ * @param[in] t The (spatial) tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{t}
+ * \right)$
+ */
template <int dim, typename Number>
Tensor<2,dim,Number>
pull_back (const Tensor<2,dim,Number> &t,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-2
- * contravariant symmetric tensor, i.e.
- * @f[
- * \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
- * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
- * @f]
- *
- * @param[in] t The (spatial) symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{t} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-2
+ * contravariant symmetric tensor, i.e.
+ * @f[
+ * \textrm{det} \mathbf{F} \; \chi^{-1}\left(\bullet\right)^{\sharp}
+ * := \textrm{det} \mathbf{F} \; \mathbf{F}^{-1} \cdot \left(\bullet\right)^{\sharp} \cdot \mathbf{F}^{-T}
+ * @f]
+ *
+ * @param[in] t The (spatial) symmetric tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{t}
+ * \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<2,dim,Number>
pull_back (const SymmetricTensor<2,dim,Number> &t,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-4
- * contravariant tensor, i.e. (in index notation)
- * @f[
- * \textrm{det} \mathbf{F} \; \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
- * := \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
- * @f]
- *
- * @param[in] h The (spatial) tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{h} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-4
+ * contravariant tensor, i.e. (in index notation)
+ * @f[
+ * \textrm{det} \mathbf{F} \; \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
+ * := \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
+ * @f]
+ *
+ * @param[in] h The (spatial) tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{h}
+ * \right)$
+ */
template <int dim, typename Number>
Tensor<4,dim,Number>
pull_back (const Tensor<4,dim,Number> &h,
const Tensor<2,dim,Number> &F);
/**
- * Returns the result of the pull back transformation on a rank-4
- * contravariant symmetric tensor, i.e. (in index notation)
- * @f[
- * \textrm{det} \mathbf{F} \; \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
- * := \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
- * @f]
- *
- * @param[in] h The (spatial) symmetric tensor to be operated on
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{h} \right)$
- */
+ * Returns the result of the pull back transformation on a rank-4
+ * contravariant symmetric tensor, i.e. (in index notation)
+ * @f[
+ * \textrm{det} \mathbf{F} \; \left[ \chi^{-1}\left(\bullet\right)^{\sharp} \right]_{IJKL}
+ * := \textrm{det} \mathbf{F} \; F^{-1}_{Ii} F^{-1}_{Jj} \left(\bullet\right)^{\sharp}_{ijkl} F^{-1}_{Kk} F^{-1}_{Ll}
+ * @f]
+ *
+ * @param[in] h The (spatial) symmetric tensor to be operated on
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return $\textrm{det} \mathbf{F} \; \chi^{-1}\left( \mathbf{h}
+ * \right)$
+ */
template <int dim, typename Number>
SymmetricTensor<4,dim,Number>
pull_back (const SymmetricTensor<4,dim,Number> &h,
//@{
/**
- * Returns the result of applying Nanson's formula for the transformation of
- * the material surface area element $d\mathbf{A}$ to the current surfaces
- * area element $d\mathbf{a}$ under the nonlinear transformation map
- * $\mathbf{x} = \boldsymbol{\varphi} \left( \mathbf{X} \right)$.
- *
- * The returned result is the spatial normal scaled by the ratio of areas
- * between the reference and spatial surface elements, i.e.
- * @f[
- * \mathbf{n} \frac{da}{dA}
- * := \textrm{det} \mathbf{F} \, \mathbf{F}^{-T} \cdot \mathbf{N}
- * = \textrm{cof} \mathbf{F} \cdot \mathbf{N} \, .
- * @f]
- *
- * @param[in] N The referential normal unit vector $\mathbf{N}$
- * @param[in] F The deformation gradient tensor $\mathbf{F} \left( \mathbf{X} \right)$
- * @return The scaled spatial normal vector $\mathbf{n} \frac{da}{dA}$
- *
- * @dealiiHolzapfelA{75,2.55}
- * @dealiiWriggersA{23,3.11}
- */
+ * Returns the result of applying Nanson's formula for the transformation
+ * of the material surface area element $d\mathbf{A}$ to the current
+ * surfaces area element $d\mathbf{a}$ under the nonlinear transformation
+ * map $\mathbf{x} = \boldsymbol{\varphi} \left( \mathbf{X} \right)$.
+ *
+ * The returned result is the spatial normal scaled by the ratio of areas
+ * between the reference and spatial surface elements, i.e.
+ * @f[
+ * \mathbf{n} \frac{da}{dA}
+ * := \textrm{det} \mathbf{F} \, \mathbf{F}^{-T} \cdot \mathbf{N}
+ * = \textrm{cof} \mathbf{F} \cdot \mathbf{N} \, .
+ * @f]
+ *
+ * @param[in] N The referential normal unit vector $\mathbf{N}$
+ * @param[in] F The deformation gradient tensor $\mathbf{F} \left(
+ * \mathbf{X} \right)$
+ * @return The scaled spatial normal vector $\mathbf{n}
+ * \frac{da}{dA}$
+ *
+ * @dealiiHolzapfelA{75,2.55} @dealiiWriggersA{23,3.11}
+ */
template<int dim, typename Number>
Tensor<1,dim,Number>
nansons_formula (const Tensor<1,dim,Number> &N,
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