Reference documentation for deal.II version 8.5.0

Namespaces  
Contravariant  
Covariant  
Piola  
Rotations  
Functions  
Special operations  
template<int dim, typename Number >  
Tensor< 1, dim, Number >  nansons_formula (const Tensor< 1, dim, Number > &N, const Tensor< 2, dim, Number > &F) 
A collection of operations to assist in the transformation of tensor quantities from the reference to spatial configuration, and vice versa. These types of transformation are typically used to reexpress quantities measured or computed in one configuration in terms of a second configuration.
We will use the same notation for the coordinates \(\mathbf{X}, \mathbf{x}\), transformations \(\varphi\), differential operator \(\nabla_{0}\) and deformation gradient \(\mathbf{F}\) as discussed for namespace Physics::Elasticity.
As a further point on notation, we will follow Holzapfel (2007) and denote the push forward transformation as \(\chi\left(\bullet\right)\) and the pull back transformation as \(\chi^{1}\left(\bullet\right)\). We will also use the annotation \(\left(\bullet\right)^{\sharp}\) to indicate that a tensor \(\left(\bullet\right)\) is a contravariant tensor, and \(\left(\bullet\right)^{\flat}\) that it is covariant. In other words, these indices do not actually change the tensor, they just indicate the kind of object a particular tensor is.
Tensor<1,dim,Number> Physics::Transformations::nansons_formula  (  const Tensor< 1, dim, Number > &  N, 
const Tensor< 2, dim, Number > &  F  
) 
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.
\[ \mathbf{n} \frac{da}{dA} := \textrm{det} \mathbf{F} \, \mathbf{F}^{T} \cdot \mathbf{N} = \textrm{cof} \mathbf{F} \cdot \mathbf{N} \, . \]
[in]  N  The referential normal unit vector \(\mathbf{N}\) 
[in]  F  The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\) 