Reference documentation for deal.II version Git f93876b 20170818 07:46:40 +0200

#include <deal.II/base/symmetric_tensor.h>
Public Member Functions  
SymmetricTensor ()  
SymmetricTensor (const Tensor< 2, dim, Number > &t)  
SymmetricTensor (const Number(&array)[n_independent_components])  
template<typename OtherNumber >  
SymmetricTensor (const SymmetricTensor< rank, dim, OtherNumber > &initializer)  
template<typename OtherNumber >  
SymmetricTensor &  operator= (const SymmetricTensor< rank, dim, OtherNumber > &rhs) 
SymmetricTensor &  operator= (const Number &d) 
operator Tensor< rank, dim, Number > () const  
bool  operator== (const SymmetricTensor &) const 
bool  operator!= (const SymmetricTensor &) const 
template<typename OtherNumber >  
SymmetricTensor &  operator+= (const SymmetricTensor< rank, dim, OtherNumber > &) 
template<typename OtherNumber >  
SymmetricTensor &  operator= (const SymmetricTensor< rank, dim, OtherNumber > &) 
template<typename OtherNumber >  
SymmetricTensor &  operator*= (const OtherNumber &factor) 
template<typename OtherNumber >  
SymmetricTensor &  operator/= (const OtherNumber &factor) 
SymmetricTensor  operator () const 
internal::SymmetricTensorAccessors::double_contraction_result< rank, 2, dim, Number >::type  operator* (const SymmetricTensor< 2, dim, Number > &s) const 
internal::SymmetricTensorAccessors::double_contraction_result< rank, 4, dim, Number >::type  operator* (const SymmetricTensor< 4, dim, Number > &s) const 
Number &  operator() (const TableIndices< rank > &indices) 
Number  operator() (const TableIndices< rank > &indices) const 
internal::SymmetricTensorAccessors::Accessor< rank, dim, true, rank1, Number >  operator[] (const unsigned int row) const 
internal::SymmetricTensorAccessors::Accessor< rank, dim, false, rank1, Number >  operator[] (const unsigned int row) 
Number  operator[] (const TableIndices< rank > &indices) const 
Number &  operator[] (const TableIndices< rank > &indices) 
Number  access_raw_entry (const unsigned int unrolled_index) const 
Number &  access_raw_entry (const unsigned int unrolled_index) 
numbers::NumberTraits< Number >::real_type  norm () const 
void  clear () 
template<class Archive >  
void  serialize (Archive &ar, const unsigned int version) 
Static Public Member Functions  
static unsigned int  component_to_unrolled_index (const TableIndices< rank > &indices) 
static TableIndices< rank >  unrolled_to_component_indices (const unsigned int i) 
static std::size_t  memory_consumption () 
Static Public Attributes  
static const unsigned int  dimension = dim 
static const unsigned int  n_independent_components 
Private Types  
typedef internal::SymmetricTensorAccessors::StorageType< rank, dim, Number >  base_tensor_descriptor 
typedef base_tensor_descriptor::base_tensor_type  base_tensor_type 
Private Attributes  
base_tensor_type  data 
Friends  
template<int , int , typename >  
class  SymmetricTensor 
template<int dim2, typename Number2 >  
Number2  trace (const SymmetricTensor< 2, dim2, Number2 > &d) 
template<int dim2, typename Number2 >  
SymmetricTensor< 2, dim2, Number2 >  unit_symmetric_tensor () 
template<int dim2, typename Number2 >  
SymmetricTensor< 4, dim2, Number2 >  deviator_tensor () 
template<int dim2, typename Number2 >  
SymmetricTensor< 4, dim2, Number2 >  identity_tensor () 
Related Functions  
(Note that these are not member functions.)  
template<int rank, int dim, typename Number >  
SymmetricTensor< rank, dim, Number >  sum (const SymmetricTensor< rank, dim, Number > &local, const MPI_Comm &mpi_communicator) 
template<int rank, int dim, typename Number , typename OtherNumber >  
SymmetricTensor< rank, dim, typename ProductType< Number, OtherNumber >::type >  operator+ (const SymmetricTensor< rank, dim, Number > &left, const SymmetricTensor< rank, dim, OtherNumber > &right) 
template<int rank, int dim, typename Number , typename OtherNumber >  
SymmetricTensor< rank, dim, typename ProductType< Number, OtherNumber >::type >  operator (const SymmetricTensor< rank, dim, Number > &left, const SymmetricTensor< rank, dim, OtherNumber > &right) 
template<int rank, int dim, typename Number , typename OtherNumber >  
Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type >  operator+ (const SymmetricTensor< rank, dim, Number > &left, const Tensor< rank, dim, OtherNumber > &right) 
template<int rank, int dim, typename Number , typename OtherNumber >  
Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type >  operator+ (const Tensor< rank, dim, Number > &left, const SymmetricTensor< rank, dim, OtherNumber > &right) 
template<int rank, int dim, typename Number , typename OtherNumber >  
Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type >  operator (const SymmetricTensor< rank, dim, Number > &left, const Tensor< rank, dim, OtherNumber > &right) 
template<int rank, int dim, typename Number , typename OtherNumber >  
Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type >  operator (const Tensor< rank, dim, Number > &left, const SymmetricTensor< rank, dim, OtherNumber > &right) 
template<int dim, typename Number >  
Number  determinant (const SymmetricTensor< 2, dim, Number > &t) 
template<int dim, typename Number >  
Number  third_invariant (const SymmetricTensor< 2, dim, Number > &t) 
template<int dim, typename Number >  
Number  trace (const SymmetricTensor< 2, dim, Number > &d) 
template<int dim, typename Number >  
Number  first_invariant (const SymmetricTensor< 2, dim, Number > &t) 
template<typename Number >  
Number  second_invariant (const SymmetricTensor< 2, 1, Number > &) 
template<typename Number >  
Number  second_invariant (const SymmetricTensor< 2, 2, Number > &t) 
template<typename Number >  
Number  second_invariant (const SymmetricTensor< 2, 3, Number > &t) 
template<typename Number >  
std::array< Number, 1 >  eigenvalues (const SymmetricTensor< 2, 1, Number > &T) 
template<typename Number >  
std::array< Number, 2 >  eigenvalues (const SymmetricTensor< 2, 2, Number > &T) 
template<typename Number >  
std::array< Number, 3 >  eigenvalues (const SymmetricTensor< 2, 3, Number > &T) 
template<typename Number >  
std::array< std::pair< Number, Tensor< 1, 1, Number > >, 1 >  eigenvectors (const SymmetricTensor< 2, 1, Number > &T, const enum SymmetricTensorEigenvectorMethod=SymmetricTensorEigenvectorMethod::ql_implicit_shifts) 
template<int dim, typename Number >  
std::array< std::pair< Number, Tensor< 1, dim, Number > >, dim >  eigenvectors (const SymmetricTensor< 2, dim, Number > &T, const enum SymmetricTensorEigenvectorMethod method=SymmetricTensorEigenvectorMethod::ql_implicit_shifts) 
template<int rank, int dim, typename Number >  
SymmetricTensor< rank, dim, Number >  transpose (const SymmetricTensor< rank, dim, Number > &t) 
template<int dim, typename Number >  
SymmetricTensor< 2, dim, Number >  deviator (const SymmetricTensor< 2, dim, Number > &t) 
template<int dim, typename Number >  
SymmetricTensor< 2, dim, Number >  invert (const SymmetricTensor< 2, dim, Number > &) 
template<int dim, typename Number >  
SymmetricTensor< 4, dim, Number >  invert (const SymmetricTensor< 4, dim, Number > &t) 
template<>  
SymmetricTensor< 4, 3, double >  invert (const SymmetricTensor< 4, 3, double > &t) 
template<int dim, typename Number >  
SymmetricTensor< 4, dim, Number >  outer_product (const SymmetricTensor< 2, dim, Number > &t1, const SymmetricTensor< 2, dim, Number > &t2) 
template<int dim, typename Number >  
SymmetricTensor< 2, dim, Number >  symmetrize (const Tensor< 2, dim, Number > &t) 
template<int rank, int dim, typename Number >  
SymmetricTensor< rank, dim, Number >  operator* (const SymmetricTensor< rank, dim, Number > &t, const Number factor) 
template<int rank, int dim, typename Number >  
SymmetricTensor< rank, dim, Number >  operator* (const Number factor, const SymmetricTensor< rank, dim, Number > &t) 
template<int rank, int dim, typename Number >  
SymmetricTensor< rank, dim, Number >  operator/ (const SymmetricTensor< rank, dim, Number > &t, const Number factor) 
template<int rank, int dim>  
SymmetricTensor< rank, dim >  operator* (const SymmetricTensor< rank, dim > &t, const double factor) 
template<int rank, int dim>  
SymmetricTensor< rank, dim >  operator* (const double factor, const SymmetricTensor< rank, dim > &t) 
template<int rank, int dim>  
SymmetricTensor< rank, dim >  operator/ (const SymmetricTensor< rank, dim > &t, const double factor) 
template<int dim, typename Number >  
Number  scalar_product (const SymmetricTensor< 2, dim, Number > &t1, const SymmetricTensor< 2, dim, Number > &t2) 
template<int dim, typename Number >  
Number  scalar_product (const SymmetricTensor< 2, dim, Number > &t1, const Tensor< 2, dim, Number > &t2) 
template<int dim, typename Number >  
Number  scalar_product (const Tensor< 2, dim, Number > &t1, const SymmetricTensor< 2, dim, Number > &t2) 
template<typename Number >  
void  double_contract (SymmetricTensor< 2, 1, Number > &tmp, const SymmetricTensor< 4, 1, Number > &t, const SymmetricTensor< 2, 1, Number > &s) 
template<typename Number >  
void  double_contract (SymmetricTensor< 2, 1, Number > &tmp, const SymmetricTensor< 2, 1, Number > &s, const SymmetricTensor< 4, 1, Number > &t) 
template<typename Number >  
void  double_contract (SymmetricTensor< 2, 2, Number > &tmp, const SymmetricTensor< 4, 2, Number > &t, const SymmetricTensor< 2, 2, Number > &s) 
template<typename Number >  
void  double_contract (SymmetricTensor< 2, 2, Number > &tmp, const SymmetricTensor< 2, 2, Number > &s, const SymmetricTensor< 4, 2, Number > &t) 
template<typename Number >  
void  double_contract (SymmetricTensor< 2, 3, Number > &tmp, const SymmetricTensor< 4, 3, Number > &t, const SymmetricTensor< 2, 3, Number > &s) 
template<typename Number >  
void  double_contract (SymmetricTensor< 2, 3, Number > &tmp, const SymmetricTensor< 2, 3, Number > &s, const SymmetricTensor< 4, 3, Number > &t) 
template<int dim, typename Number >  
Tensor< 1, dim, Number >  operator* (const SymmetricTensor< 2, dim, Number > &src1, const Tensor< 1, dim, Number > &src2) 
template<int dim, typename Number >  
Tensor< 1, dim, Number >  operator* (const Tensor< 1, dim, Number > &src1, const SymmetricTensor< 2, dim, Number > &src2) 
template<int dim, typename Number >  
std::ostream &  operator<< (std::ostream &out, const SymmetricTensor< 2, dim, Number > &t) 
template<int dim, typename Number >  
std::ostream &  operator<< (std::ostream &out, const SymmetricTensor< 4, dim, Number > &t) 
template<int dim, typename Number , typename T >  
Sacado::Fad::DFad< T >  scalar_product (const SymmetricTensor< 2, dim, Sacado::Fad::DFad< T > > &t1, const Tensor< 2, dim, Number > &t2) 
template<int dim, typename Number , typename T >  
Sacado::Fad::DFad< T >  scalar_product (const Tensor< 2, dim, Number > &t1, const SymmetricTensor< 2, dim, Sacado::Fad::DFad< T > > &t2) 
template<int dim, typename Number , typename T >  
Sacado::Fad::DFad< T >  scalar_product (const SymmetricTensor< 2, dim, Number > &t1, const Tensor< 2, dim, Sacado::Fad::DFad< T > > &t2) 
template<int dim, typename Number , typename T >  
Sacado::Fad::DFad< T >  scalar_product (const Tensor< 2, dim, Sacado::Fad::DFad< T > > &t1, const SymmetricTensor< 2, dim, Number > &t2) 
Provide a class that stores symmetric tensors of rank 2,4,... efficiently, i.e. only store those offdiagonal elements of the full tensor that are not redundant. For example, for symmetric 2x2 tensors, this would be the elements 11, 22, and 12, while the element 21 is equal to the 12 element.
Using this class for symmetric tensors of rank 2 has advantages over matrices in many cases since the dimension is known to the compiler as well as the location of the data. It is therefore possible to produce far more efficient code than for matrices with runtimedependent dimension. It is also more efficient than using the more general Tensor
class, since fewer elements are stored, and the class automatically makes sure that the tensor represents a symmetric object.
For tensors of higher rank, the savings in storage are even higher. For example for the 3x3x3x3 tensors of rank 4, only 36 instead of the full 81 entries have to be stored.
While the definition of a symmetric rank2 tensor is obvious, tensors of rank 4 are considered symmetric if they are operators mapping symmetric rank2 tensors onto symmetric rank2 tensors. This entails certain symmetry properties on the elements in their 4dimensional index space, in particular that C_{ijkl}=C_{jikl}=C_{ijlk}
. However, it does not imply the relation C_{ijkl}=C_{klij}
. Consequently, symmetric tensors of rank 4 as understood here are only tensors that map symmetric tensors onto symmetric tensors, but they do not necessarily induce a symmetric scalar product a:C:b=b:C:a
or even a positive (semi)definite form a:C:a
, where a,b
are symmetric rank2 tensors and the colon indicates the common doubleindex contraction that acts as a product for symmetric tensors.
Symmetric tensors are most often used in structural and fluid mechanics, where strains and stresses are usually symmetric tensors, and the stressstrain relationship is given by a symmetric rank4 tensor.
SymmetricTensor<2,dim>
, SymmetricTensor<4,dim>
, etc, but SymmetricTensor<1,dim>
and SymmetricTensor<3,dim>
do not exist and their use will most likely lead to compiler errors.The elements of a tensor t
can be accessed using the bracket operator, i.e. for a tensor of rank 4, t[0][1][0][1]
accesses the element t_{0101}
. This access can be used for both reading and writing (if the tensor is nonconstant at least). You may also perform other operations on it, although that may lead to confusing situations because several elements of the tensor are stored at the same location. For example, for a rank2 tensor that is assumed to be zero at the beginning, writing t[0][1]+=1; t[1][0]+=1;
will lead to the same element being increased by one twice, because even though the accesses use different indices, the elements that are accessed are symmetric and therefore stored at the same location. It may therefore be useful in application programs to restrict operations on individual elements to simple reads or writes.

private 
A structure that describes properties of the base tensor.
Definition at line 799 of file symmetric_tensor.h.

private 
Data storage type for a symmetric tensor.
Definition at line 804 of file symmetric_tensor.h.
SymmetricTensor< rank, dim, Number >::SymmetricTensor  (  ) 
Default constructor. Creates a tensor with all entries equal to zero.
SymmetricTensor< rank, dim, Number >::SymmetricTensor  (  const Tensor< 2, dim, Number > &  t  ) 
Constructor. Generate a symmetric tensor from a general one. Assumes that t
is already symmetric, and in debug mode this is in fact checked. Note that no provision is made to assure that the tensor is symmetric only up to roundoff error: if the incoming tensor is not exactly symmetric, then an exception is thrown. If you know that incoming tensor is symmetric only up to roundoff, then you may want to call the symmetrize
function first. If you aren't sure, it is good practice to check before calling symmetrize
.
SymmetricTensor< rank, dim, Number >::SymmetricTensor  (  const Number(&)  array[n_independent_components]  ) 
A constructor that creates a symmetric tensor from an array holding its independent elements. Using this constructor assumes that the caller knows the order in which elements are stored in symmetric tensors; its use is therefore discouraged, but if you think you want to use it anyway you can query the order of elements using the unrolled_index() function.
This constructor is currently only implemented for symmetric tensors of rank 2.
The size of the array passed is equal to SymmetricTensor<rank,dim>::n_independent_component; the reason for using the object from the internal namespace is to work around bugs in some older compilers.

explicit 
Copy constructor from tensors with different underlying scalar type. This obviously requires that the OtherNumber
type is convertible to Number
.
SymmetricTensor& SymmetricTensor< rank, dim, Number >::operator=  (  const SymmetricTensor< rank, dim, OtherNumber > &  rhs  ) 
Assignment operator from symmetric tensors with different underlying scalar type. This obviously requires that the OtherNumber
type is convertible to Number
.
SymmetricTensor& SymmetricTensor< rank, dim, Number >::operator=  (  const Number &  d  ) 
This operator assigns a scalar to a tensor. To avoid confusion with what exactly it means to assign a scalar value to a tensor, zero is the only value allowed for d
, allowing the intuitive notation t=0
to reset all elements of the tensor to zero.
SymmetricTensor< rank, dim, Number >::operator Tensor< rank, dim, Number >  (  )  const 
Convert the present symmetric tensor into a full tensor with the same elements, but using the different storage scheme of full tensors.
bool SymmetricTensor< rank, dim, Number >::operator==  (  const SymmetricTensor< rank, dim, Number > &  )  const 
Test for equality of two tensors.
bool SymmetricTensor< rank, dim, Number >::operator!=  (  const SymmetricTensor< rank, dim, Number > &  )  const 
Test for inequality of two tensors.
SymmetricTensor& SymmetricTensor< rank, dim, Number >::operator+=  (  const SymmetricTensor< rank, dim, OtherNumber > &  ) 
Add another tensor.
SymmetricTensor& SymmetricTensor< rank, dim, Number >::operator=  (  const SymmetricTensor< rank, dim, OtherNumber > &  ) 
Subtract another tensor.
SymmetricTensor& SymmetricTensor< rank, dim, Number >::operator*=  (  const OtherNumber &  factor  ) 
Scale the tensor by factor
, i.e. multiply all components by factor
.
SymmetricTensor& SymmetricTensor< rank, dim, Number >::operator/=  (  const OtherNumber &  factor  ) 
Scale the tensor by 1/factor
.
SymmetricTensor SymmetricTensor< rank, dim, Number >::operator  (  )  const 
Unary minus operator. Negate all entries of a tensor.
internal::SymmetricTensorAccessors::double_contraction_result<rank,2,dim,Number>::type SymmetricTensor< rank, dim, Number >::operator*  (  const SymmetricTensor< 2, dim, Number > &  s  )  const 
Product between the present symmetric tensor and a tensor of rank 2. For example, if the present object is also a rank2 tensor, then this is the scalarproduct double contraction a_{ij}b_{ij}
over all indices i,j
. In this case, the return value evaluates to a single scalar. While it is possible to define other scalar product (and associated induced norms), this one seems to be the most appropriate one.
If the present object is a rank4 tensor, then the result is a rank2 tensor, i.e., the operation contracts over the last two indices of the present object and the indices of the argument, and the result is a tensor of rank 2.
Note that the multiplication operator for symmetric tensors is defined to be a double contraction over two indices, while it is defined as a single contraction over only one index for regular Tensor
objects. For symmetric tensors it therefore acts in a way that is commonly denoted by a "colon multiplication" in the mathematical literature.
There are global functions double_contract
that do the same work as this operator, but rather than returning the result as a return value, they write it into the first argument to the function.
internal::SymmetricTensorAccessors::double_contraction_result<rank,4,dim,Number>::type SymmetricTensor< rank, dim, Number >::operator*  (  const SymmetricTensor< 4, dim, Number > &  s  )  const 
Contraction over two indices of the present object with the rank4 symmetric tensor given as argument.
Number& SymmetricTensor< rank, dim, Number >::operator()  (  const TableIndices< rank > &  indices  ) 
Return a readwrite reference to the indicated element.
Number SymmetricTensor< rank, dim, Number >::operator()  (  const TableIndices< rank > &  indices  )  const 
Return an element by value.
internal::SymmetricTensorAccessors::Accessor<rank,dim,true,rank1,Number> SymmetricTensor< rank, dim, Number >::operator[]  (  const unsigned int  row  )  const 
Access the elements of a row of this symmetric tensor. This function is called for constant tensors.
internal::SymmetricTensorAccessors::Accessor<rank,dim,false,rank1,Number> SymmetricTensor< rank, dim, Number >::operator[]  (  const unsigned int  row  ) 
Access the elements of a row of this symmetric tensor. This function is called for nonconstant tensors.
Number SymmetricTensor< rank, dim, Number >::operator[]  (  const TableIndices< rank > &  indices  )  const 
Return an element by value.
Exactly the same as operator().
Number& SymmetricTensor< rank, dim, Number >::operator[]  (  const TableIndices< rank > &  indices  ) 
Return a readwrite reference to the indicated element.
Exactly the same as operator().
Number SymmetricTensor< rank, dim, Number >::access_raw_entry  (  const unsigned int  unrolled_index  )  const 
Access to an element according to unrolled index. The function s.access_raw_entry(unrolled_index)
does the same as s[s.unrolled_to_component_indices(i)]
, but more efficiently.
Number& SymmetricTensor< rank, dim, Number >::access_raw_entry  (  const unsigned int  unrolled_index  ) 
Access to an element according to unrolled index. The function s.access_raw_entry(unrolled_index)
does the same as s[s.unrolled_to_component_indices(i)]
, but more efficiently.
numbers::NumberTraits<Number>::real_type SymmetricTensor< rank, dim, Number >::norm  (  )  const 
Return the Frobeniusnorm of a tensor, i.e. the square root of the sum of squares of all entries. This norm is induced by the scalar product defined above for two symmetric tensors. Note that it includes all entries of the tensor, counting symmetry, not only the unique ones (for example, for rank2 tensors, this norm includes adding up the squares of upper right as well as lower left entries, not just one of them, although they are equal for symmetric tensors).

static 
Tensors can be unrolled by simply pasting all elements into one long vector, but for this an order of elements has to be defined. For symmetric tensors, this function returns which index within the range [0,n_independent_components)
the given entry in a symmetric tensor has.

static 
The opposite of the previous function: given an index \(i\) in the unrolled form of the tensor, return what set of indices \((k,l)\) (for rank2 tensors) or \((k,l,m,n)\) (for rank4 tensors) corresponds to it.
void SymmetricTensor< rank, dim, Number >::clear  (  ) 
Reset all values to zero.
Note that this is partly inconsistent with the semantics of the clear()
member functions of the standard library containers and of several other classes within deal.II, which not only reset the values of stored elements to zero, but release all memory and return the object into a virginial state. However, since the size of objects of the present type is determined by its template parameters, resizing is not an option, and indeed the state where all elements have a zero value is the state right after construction of such an object.

static 
Determine an estimate for the memory consumption (in bytes) of this object.
void SymmetricTensor< rank, dim, Number >::serialize  (  Archive &  ar, 
const unsigned int  version  
) 
Read or write the data of this object to or from a stream for the purpose of serialization

friend 
Make all other symmetric tensors friends.
Definition at line 814 of file symmetric_tensor.h.

friend 
Make a few more functions friends.

friend 
Return a unit symmetric tensor of rank 2, i.e., the dimbydim identity matrix.
Return a unit symmetric tensor of rank 2, i.e., the dimbydim identity matrix. This specialization of the function uses double
as the data type for the elements.
Definition at line 2864 of file symmetric_tensor.h.

friend 
Return the tensor of rank 4 that, when multiplied by a symmetric rank 2 tensor t
returns the deviator \(\textrm{dev}\ t\). It is the operator representation of the linear deviator operator.
For every tensor t
, there holds the identity deviator(t)==deviator_tensor<dim>()*t
, up to numerical roundoff. The reason this operator representation is provided is that one sometimes needs to invert operators like identity_tensor<dim>() + delta_t*deviator_tensor<dim>()
or similar.
Return the tensor of rank 4 that, when multiplied by a symmetric rank 2 tensor t
returns the deviator dev t
. It is the operator representation of the linear deviator operator.
For every tensor t
, there holds the identity deviator(t)==deviator_tensor<dim>()*t
, up to numerical roundoff. The reason this operator representation is provided is that one sometimes needs to invert operators like identity_tensor<dim>() + delta_t*deviator_tensor<dim>()
or similar.
Definition at line 2924 of file symmetric_tensor.h.

friend 
Return the fourthorder symmetric identity tensor which maps symmetric secondorder tensors to themselves.
Note that this tensor, even though it is the identity, has a somewhat funny form, and in particular does not only consist of zeros and ones. For example, for dim=2
, the identity tensor has all zero entries except for id[0][0][0][0]=id[1][1][1][1]=1
and id[0][1][0][1]=id[0][1][1][0]=id[1][0][0][1]=id[1][0][1][0]=1/2
. To see why this factor of 1/2 is necessary, consider computing A=Id : B
. For the element a_01
we have a_01=id_0100 b_00 + id_0111 b_11 + id_0101 b_01 + id_0110 b_10
. On the other hand, we need to have a_01=b_01
, and symmetry implies b_01=b_10
, leading to a_01=(id_0101+id_0110) b_01
, or, again by symmetry, id_0101=id_0110=1/2
. Similar considerations hold for the three dimensional case.
This issue is also explained in the introduction to step44.
Return the tensor of rank 4 that, when multiplied by a symmetric rank 2 tensor t
returns the deviator dev t
. It is the operator representation of the linear deviator operator.
Note that this tensor, even though it is the identity, has a somewhat funny form, and in particular does not only consist of zeros and ones. For example, for dim=2
, the identity tensor has all zero entries except for id[0][0][0][0]=id[1][1][1][1]=1
and id[0][1][0][1]=id[0][1][1][0]=id[1][0][0][1]=id[1][0][1][0]=1/2
. To see why this factor of 1/2 is necessary, consider computing A=Id . B
. For the element a_01
we have a_01=id_0100 b_00 + id_0111 b_11 + id_0101 b_01 + id_0110 b_10
. On the other hand, we need to have a_01=b_01
, and symmetry implies b_01=b_10
, leading to a_01=(id_0101+id_0110) b_01
, or, again by symmetry, id_0101=id_0110=1/2
. Similar considerations hold for the three dimensional case.
Definition at line 2997 of file symmetric_tensor.h.

related 
Perform an MPI sum of the entries of a symmetric tensor.

related 
Addition of two symmetric tensors of equal rank. The result is another SymmetricTensor that has a number type that is compatible with the operation.
If possible (e.g. when Number
and OtherNumber
are of the same type, or if the result of Number() + OtherNumber()
is another Number
), you should use operator +=
instead since this does not require the creation of a temporary variable.
Definition at line 2117 of file symmetric_tensor.h.

related 
Subtraction of two symmetric tensors of equal rank. The result is another SymmetricTensor that has a number type that is compatible with the operation.
If possible (e.g. when Number
and OtherNumber
are of the same type, or if the result of Number() + OtherNumber()
is another Number
), you should use operator +=
instead since this does not require the creation of a temporary variable.
Definition at line 2141 of file symmetric_tensor.h.

related 
Addition of a SymmetricTensor and a general Tensor of equal rank. The result is a general Tensor that has a number type that is compatible with the operation.
Definition at line 2160 of file symmetric_tensor.h.

related 
Addition of a general Tensor with a SymmetricTensor of equal rank. The result is a general Tensor that has a number type that is compatible with the operation.
Definition at line 2177 of file symmetric_tensor.h.

related 
Subtraction of a general Tensor from a SymmetricTensor of equal rank. The result is a general Tensor that has a number type that is compatible with the operation.
Definition at line 2194 of file symmetric_tensor.h.

related 
Subtraction of a SymmetricTensor from a general Tensor of equal rank. The result is a general Tensor that has a number type that is compatible with the operation.
Definition at line 2211 of file symmetric_tensor.h.

related 
Compute the determinant of a tensor or rank 2. The determinant is also commonly referred to as the third invariant of rank2 tensors.
For a onedimensional tensor, the determinant equals the only element and is therefore equivalent to the trace.
For greater notational simplicity, there is also a third_invariant
function that returns the determinant of a tensor.
Definition at line 2234 of file symmetric_tensor.h.

related 
Compute the determinant of a tensor or rank 2. This function therefore computes the same value as the determinant()
functions and is only provided for greater notational simplicity (since there are also functions first_invariant() and second_invariant()).
Definition at line 2273 of file symmetric_tensor.h.

related 
Compute and return the trace of a tensor of rank 2, i.e. the sum of its diagonal entries. The trace is the first invariant of a rank2 tensor.
Definition at line 2288 of file symmetric_tensor.h.

related 
Compute the trace of a tensor or rank 2. This function therefore computes the same value as the trace()
functions and is only provided for greater notational simplicity (since there are also functions second_invariant() and third_invariant()).
Definition at line 2308 of file symmetric_tensor.h.

related 
Compute the second invariant of a tensor of rank 2. The second invariant of a tensor \(\sigma\) is defined as \(II(\sigma) = \frac 12 \left[ (\text{trace}\ \sigma)^2 \text{trace}\ (\sigma^2) \right]\).
For the kind of arguments to this function, i.e., a rank2 tensor of size 1, the result is simply zero.
Definition at line 2328 of file symmetric_tensor.h.

related 
Compute the second invariant of a tensor of rank 2. The second invariant of a tensor \(\sigma\) is defined as \(II(\sigma) = \frac 12 \left[ (\text{trace}\ \sigma)^2 \text{trace}\ (\sigma^2) \right]\).
For the kind of arguments to this function, i.e., a rank2 tensor of size 2, the result is (counting indices starting at one) \(II(\sigma) = \frac 12 \left[ (\sigma_{11} + \sigma_{22})^2 (\sigma_{11}^2+2\sigma_{12}^2+\sigma_{22}^2) \right] = \sigma_{11}\sigma_{22}  \sigma_{12}^2\). As expected, for the \(2\times 2\) symmetric tensors this function handles, this equals the determinant of the tensor. (This is so because for \(2\times 2\) symmetric tensors, there really are only two invariants, so the second and third invariant are the same; the determinant is the third invariant.)
Definition at line 2357 of file symmetric_tensor.h.

related 
Compute the second invariant of a tensor of rank 2. The second invariant of a tensor \(\sigma\) is defined as \(II(\sigma) = \frac 12 \left[ (\text{trace}\ \sigma)^2 \text{trace}\ (\sigma^2) \right]\).
Definition at line 2375 of file symmetric_tensor.h.

related 
Return the eigenvalues of a symmetric 1x1 tensor of rank 2. The (single) entry of the tensor is, of course, equal to the (single) eigenvalue.

related 
Return the eigenvalues of a symmetric 2x2 tensor of rank 2. The array of eigenvalues is sorted in descending order.
For 2x2 tensors, the eigenvalues of tensor \(T\) are the roots of the characteristic polynomial \(0 = \lambda^{2}  \lambda\textrm{tr}(T) + \textrm{det}(T)\) as given by \(\lambda = \frac{\textrm{tr}(T) \pm \sqrt{[\textrm{tr}(T)]^{2}  4\textrm{det}(T)}}{2}\).

related 
Return the eigenvalues of a symmetric 3x3 tensor of rank 2. The array of eigenvalues is sorted in descending order.
For 3x3 tensors, the eigenvalues of tensor \(T\) are the roots of the characteristic polynomial \(0 = \lambda^{3}  \lambda^{2}\textrm{tr}(T)  \frac{1}{2} \lambda (\textrm{tr}(T^{2})  [\textrm{tr}(T)]^{2})  \textrm{det}(T)\).

related 
Return the eigenvalues and eigenvectors of a symmetric 1x1 tensor of rank 2.
Definition at line 2744 of file symmetric_tensor.h.

related 
Return the eigenvalues and eigenvectors of a realvalued rank2 symmetric tensor \(T\). The array of matched eigenvalue and eigenvector pairs is sorted in descending order (determined by the eigenvalues).
The specialized algorithms utilized in computing the eigenvectors are presented in
Definition at line 2781 of file symmetric_tensor.h.

related 
Return the transpose of the given symmetric tensor. Since we are working with symmetric objects, the transpose is of course the same as the original tensor. This function mainly exists for compatibility with the Tensor class.
Definition at line 2821 of file symmetric_tensor.h.

related 
Compute the deviator of a symmetric tensor, which is defined as dev[s] = s  1/dim*tr[s]*I
, where I
is the identity operator. This quantity equals the original tensor minus its contractive or dilative component and refers to the shear in, for example, elasticity.
Definition at line 2840 of file symmetric_tensor.h.

related 
Invert a symmetric rank2 tensor.
Definition at line 3064 of file symmetric_tensor.h.

related 
Invert a symmetric rank4 tensor. Since symmetric rank4 tensors are mappings from and to symmetric rank2 tensors, they can have an inverse. This function computes it, if it exists, for the case that the dimension equals either 1 or 2.
If a tensor is not invertible, then the result is unspecified, but will likely contain the results of a division by zero or a very small number at the very least.
Definition at line 3178 of file symmetric_tensor.h.

related 
Invert a symmetric rank4 tensor. Since symmetric rank4 tensors are mappings from and to symmetric rank2 tensors, they can have an inverse. This function computes it, if it exists, for the case that the dimension equals 3.
If a tensor is not invertible, then the result is unspecified, but will likely contain the results of a division by zero or a very small number at the very least.

related 
Return the tensor of rank 4 that is the outer product of the two tensors given as arguments, i.e. the result \(T=t1 \otimes t2\) satisfies T phi = t1 (t2 : phi)
for all symmetric tensors phi
.
For example, the deviator tensor can be computed as identity_tensor<dim,Number>()  1/d*outer_product(unit_symmetric_tensor<dim,Number>(), unit_symmetric_tensor<dim,Number>())
, since the (double) contraction with the unit tensor yields the trace of a symmetric tensor.
Definition at line 3294 of file symmetric_tensor.h.

related 
Return the symmetrized version of a full rank2 tensor, i.e. (t+transpose(t))/2, as a symmetric rank2 tensor. This is the version for general dimensions.
Definition at line 3322 of file symmetric_tensor.h.

related 
Multiplication of a symmetric tensor of general rank with a scalar from the right. This version of the operator is used if the scalar has the same data type as is used to store the elements of the symmetric tensor.
Definition at line 3345 of file symmetric_tensor.h.

related 
Multiplication of a symmetric tensor of general rank with a scalar from the left. This version of the operator is used if the scalar has the same data type as is used to store the elements of the symmetric tensor.
Definition at line 3365 of file symmetric_tensor.h.

related 
Division of a symmetric tensor of general rank by a scalar.
Definition at line 3455 of file symmetric_tensor.h.

related 
Multiplication of a symmetric tensor of general rank with a scalar from the right.
Definition at line 3474 of file symmetric_tensor.h.

related 
Multiplication of a symmetric tensor of general rank with a scalar from the left.
Definition at line 3493 of file symmetric_tensor.h.

related 
Division of a symmetric tensor of general rank by a scalar.
Definition at line 3511 of file symmetric_tensor.h.

related 
Compute the scalar product \(a:b=\sum_{i,j} a_{ij}b_{ij}\) between two tensors \(a,b\) of rank 2. In the current case where both arguments are symmetric tensors, this is equivalent to calling the expression t1*t2
which uses the overloaded operator*
between two symmetric tensors of rank 2.
Definition at line 3531 of file symmetric_tensor.h.

related 
Compute the scalar product \(a:b=\sum_{i,j} a_{ij}b_{ij}\) between two tensors \(a,b\) of rank 2. We don't use operator*
for this operation since the product between two tensors is usually assumed to be the contraction over the last index of the first tensor and the first index of the second tensor, for example \((a\cdot b)_{ij}=\sum_k a_{ik}b_{kj}\).
Definition at line 3550 of file symmetric_tensor.h.

related 
Compute the scalar product \(a:b=\sum_{i,j} a_{ij}b_{ij}\) between two tensors \(a,b\) of rank 2. We don't use operator*
for this operation since the product between two tensors is usually assumed to be the contraction over the last index of the first tensor and the first index of the second tensor, for example \((a\cdot b)_{ij}=\sum_k a_{ik}b_{kj}\).
Definition at line 3573 of file symmetric_tensor.h.

related 
Double contraction between a rank4 and a rank2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrixvector multiplication.
This function does the same as the member operator* of the SymmetricTensor class. It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general tensor class.
Definition at line 3598 of file symmetric_tensor.h.

related 
Double contraction between a rank4 and a rank2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrixvector multiplication.
This function does the same as the member operator* of the SymmetricTensor class. It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general tensor class.
Definition at line 3625 of file symmetric_tensor.h.

related 
Double contraction between a rank4 and a rank2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrixvector multiplication.
This function does the same as the member operator* of the SymmetricTensor class. It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general tensor class.
Definition at line 3651 of file symmetric_tensor.h.

related 
Double contraction between a rank4 and a rank2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrixvector multiplication.
This function does the same as the member operator* of the SymmetricTensor class. It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general tensor class.
Definition at line 3684 of file symmetric_tensor.h.

related 
Double contraction between a rank4 and a rank2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrixvector multiplication.
This function does the same as the member operator* of the SymmetricTensor class. It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general tensor class.
Definition at line 3717 of file symmetric_tensor.h.

related 
Double contraction between a rank4 and a rank2 symmetric tensor, resulting in the symmetric tensor of rank 2 that is given as first argument to this function. This operation is the symmetric tensor analogon of a matrixvector multiplication.
This function does the same as the member operator* of the SymmetricTensor class. It should not be used, however, since the member operator has knowledge of the actual data storage format and is at least 2 orders of magnitude faster. This function mostly exists for compatibility purposes with the general tensor class.
Definition at line 3753 of file symmetric_tensor.h.

related 
Multiply a symmetric rank2 tensor (i.e., a matrix) by a rank1 tensor (i.e., a vector). The result is a rank1 tensor (i.e., a vector).
Definition at line 3780 of file symmetric_tensor.h.

related 
Multiply a rank1 tensor (i.e., a vector) by a symmetric rank2 tensor (i.e., a matrix). The result is a rank1 tensor (i.e., a vector).
Definition at line 3800 of file symmetric_tensor.h.

related 
Output operator for symmetric tensors of rank 2. Print the elements consecutively, with a space in between, two spaces between rank 1 subtensors, three between rank 2 and so on. No special amends are made to represents the symmetry in the output, for example by outputting only the unique entries.
Definition at line 3819 of file symmetric_tensor.h.

related 
Output operator for symmetric tensors of rank 4. Print the elements consecutively, with a space in between, two spaces between rank 1 subtensors, three between rank 2 and so on. No special amends are made to represents the symmetry in the output, for example by outputting only the unique entries.
Definition at line 3847 of file symmetric_tensor.h.

related 
Compute the scalar product \(a:b=\sum_{i,j} a_{ij}b_{ij}\) between two tensors \(a,b\) of rank 2. We don't use operator*
for this operation since the product between two tensors is usually assumed to be the contraction over the last index of the first tensor and the first index of the second tensor, for example \((a\cdot b)_{ij}=\sum_k a_{ik}b_{kj}\).
Definition at line 94 of file sacado_product_types.h.

related 
Compute the scalar product \(a:b=\sum_{i,j} a_{ij}b_{ij}\) between two tensors \(a,b\) of rank 2. We don't use operator*
for this operation since the product between two tensors is usually assumed to be the contraction over the last index of the first tensor and the first index of the second tensor, for example \((a\cdot b)_{ij}=\sum_k a_{ik}b_{kj}\).
Definition at line 117 of file sacado_product_types.h.

related 
Compute the scalar product \(a:b=\sum_{i,j} a_{ij}b_{ij}\) between two tensors \(a,b\) of rank 2. We don't use operator*
for this operation since the product between two tensors is usually assumed to be the contraction over the last index of the first tensor and the first index of the second tensor, for example \((a\cdot b)_{ij}=\sum_k a_{ik}b_{kj}\).
Definition at line 136 of file sacado_product_types.h.

related 
Compute the scalar product \(a:b=\sum_{i,j} a_{ij}b_{ij}\) between two tensors \(a,b\) of rank 2. We don't use operator*
for this operation since the product between two tensors is usually assumed to be the contraction over the last index of the first tensor and the first index of the second tensor, for example \((a\cdot b)_{ij}=\sum_k a_{ik}b_{kj}\).
Definition at line 159 of file sacado_product_types.h.

static 
Provide a way to get the dimension of an object without explicit knowledge of it's data type. Implementation is this way instead of providing a function dimension()
because now it is possible to get the dimension at compile time without the expansion and preevaluation of an inlined function; the compiler may therefore produce more efficient code and you may use this value to declare other data types.
Definition at line 528 of file symmetric_tensor.h.

static 
An integer denoting the number of independent components that fully describe a symmetric tensor. In \(d\) space dimensions, this number equals \(\frac 12 (d^2+d)\) for symmetric tensors of rank 2.
Definition at line 536 of file symmetric_tensor.h.

private 
The place where we store the data of the tensor.
Definition at line 809 of file symmetric_tensor.h.