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Reference documentation for deal.II version 9.3.3
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Namespaces | |
namespace | Contravariant |
namespace | Covariant |
namespace | Piola |
namespace | 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) |
Basis transformations | |
template<int dim, typename Number > | |
Tensor< 1, dim, Number > | basis_transformation (const Tensor< 1, dim, Number > &V, const Tensor< 2, dim, Number > &B) |
template<int dim, typename Number > | |
Tensor< 2, dim, Number > | basis_transformation (const Tensor< 2, dim, Number > &T, const Tensor< 2, dim, Number > &B) |
template<int dim, typename Number > | |
SymmetricTensor< 2, dim, Number > | basis_transformation (const SymmetricTensor< 2, dim, Number > &T, const Tensor< 2, dim, Number > &B) |
template<int dim, typename Number > | |
Tensor< 4, dim, Number > | basis_transformation (const Tensor< 4, dim, Number > &H, const Tensor< 2, dim, Number > &B) |
template<int dim, typename Number > | |
SymmetricTensor< 4, dim, Number > | basis_transformation (const SymmetricTensor< 4, dim, Number > &H, const Tensor< 2, dim, Number > &B) |
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 re-express quantities measured or computed in one configuration in terms of a second configuration.
We will use the same notation for the coordinates , transformations
, differential operator
and deformation gradient
as discussed for namespace Physics::Elasticity.
As a further point on notation, we will follow Holzapfel (2007) and denote the push forward transformation as and the pull back transformation as
. We will also use the annotation
to indicate that a tensor
is a contravariant tensor, and
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 | ||
) |
Return the result of applying Nanson's formula for the transformation of the material surface area element to the current surfaces area element
under the nonlinear transformation map
.
The returned result is the spatial normal scaled by the ratio of areas between the reference and spatial surface elements, i.e.
[in] | N | The referential normal unit vector ![]() |
[in] | F | The deformation gradient tensor ![]() |
Tensor< 1, dim, Number > Physics::Transformations::basis_transformation | ( | const Tensor< 1, dim, Number > & | V, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a vector with a changed basis, i.e.
[in] | V | The vector to be transformed ![]() |
[in] | B | The transformation matrix ![]() |
Tensor< 2, dim, Number > Physics::Transformations::basis_transformation | ( | const Tensor< 2, dim, Number > & | T, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a rank-2 tensor with a changed basis, i.e.
[in] | T | The tensor to be transformed ![]() |
[in] | B | The transformation matrix ![]() |
SymmetricTensor< 2, dim, Number > Physics::Transformations::basis_transformation | ( | const SymmetricTensor< 2, dim, Number > & | T, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a symmetric rank-2 tensor with a changed basis, i.e.
[in] | T | The tensor to be transformed ![]() |
[in] | B | The transformation matrix ![]() |
Tensor< 4, dim, Number > Physics::Transformations::basis_transformation | ( | const Tensor< 4, dim, Number > & | H, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a rank-4 tensor with a changed basis, i.e. (in index notation):
[in] | H | The tensor to be transformed ![]() |
[in] | B | The transformation matrix ![]() |
SymmetricTensor< 4, dim, Number > Physics::Transformations::basis_transformation | ( | const SymmetricTensor< 4, dim, Number > & | H, |
const Tensor< 2, dim, Number > & | B | ||
) |
Return a symmetric rank-4 tensor with a changed basis, i.e. (in index notation):
[in] | H | The tensor to be transformed ![]() |
[in] | B | The transformation matrix ![]() |