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// $Id: symmetric_tensor.h 21326 2010-06-24 14:55:56Z bangerth $
// Version: $Name$
//
// Copyright (C) 2005, 2006, 2008, 2009, 2010 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//---------------------------------------------------------------------------
#ifndef __deal2__symmetric_tensor_h
#define __deal2__symmetric_tensor_h
#include <base/tensor.h>
#include <base/table_indices.h>
DEAL_II_NAMESPACE_OPEN
template <int rank, int dim> class SymmetricTensor;
template <int dim> SymmetricTensor<2,dim> unit_symmetric_tensor ();
template <int dim> SymmetricTensor<4,dim> deviator_tensor ();
template <int dim> SymmetricTensor<4,dim> identity_tensor ();
template <int dim> SymmetricTensor<4,dim> invert (const SymmetricTensor<4,dim> &);
template <int dim2> double trace (const SymmetricTensor<2,dim2> &);
template <int dim> SymmetricTensor<2,dim>
deviator (const SymmetricTensor<2,dim> &);
template <int dim> double
determinant (const SymmetricTensor<2,dim> &);
namespace internal
{
/**
* A namespace for classes that are
* internal to how the SymmetricTensor
* class works.
*/
namespace SymmetricTensorAccessors
{
/**
* Create a TableIndices<2>
* object where the first entries
* up to <tt>position-1</tt> are
* taken from previous_indices,
* and new_index is put at
* position
* <tt>position</tt>. The
* remaining indices remain in
* invalid state.
*/
inline
TableIndices<2> merge (const TableIndices<2> &previous_indices,
const unsigned int new_index,
const unsigned int position)
{
Assert (position < 2, ExcIndexRange (position, 0, 2));
if (position == 0)
return TableIndices<2>(new_index);
else
return TableIndices<2>(previous_indices[0], new_index);
}
/**
* Create a TableIndices<4>
* object where the first entries
* up to <tt>position-1</tt> are
* taken from previous_indices,
* and new_index is put at
* position
* <tt>position</tt>. The
* remaining indices remain in
* invalid state.
*/
inline
TableIndices<4> merge (const TableIndices<4> &previous_indices,
const unsigned int new_index,
const unsigned int position)
{
Assert (position < 4, ExcIndexRange (position, 0, 4));
switch (position)
{
case 0:
return TableIndices<4>(new_index);
case 1:
return TableIndices<4>(previous_indices[0],
new_index);
case 2:
return TableIndices<4>(previous_indices[0],
previous_indices[1],
new_index);
case 3:
return TableIndices<4>(previous_indices[0],
previous_indices[1],
previous_indices[2],
new_index);
}
Assert (false, ExcInternalError());
return TableIndices<4>();
}
/**
* Typedef template magic
* denoting the result of a
* double contraction between two
* tensors or ranks rank1 and
* rank2. In general, this is a
* tensor of rank
* <tt>rank1+rank2-4</tt>, but if
* this is zero it is a single
* scalar double. For this case,
* we have a specialization.
*
* @author Wolfgang Bangerth, 2005
*/
template <int rank1, int rank2, int dim>
struct double_contraction_result
{
typedef ::dealii::SymmetricTensor<rank1+rank2-4,dim> type;
};
/**
* Typedef template magic
* denoting the result of a
* double contraction between two
* tensors or ranks rank1 and
* rank2. In general, this is a
* tensor of rank
* <tt>rank1+rank2-4</tt>, but if
* this is zero it is a single
* scalar double. For this case,
* we have a specialization.
*
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
struct double_contraction_result<2,2,dim>
{
typedef double type;
};
/**
* Declaration of typedefs for the type
* of data structures which are used to
* store symmetric tensors. For example,
* for rank-2 symmetric tensors, we use a
* flat vector to store all the
* elements. On the other hand, symmetric
* rank-4 tensors are mappings from
* symmetric rank-2 tensors into
* symmetric rank-2 tensors, so they can
* be represented as matrices, etc.
*
* This information is probably of little
* interest to all except the accessor
* classes that need it. In particular,
* you shouldn't make any assumptions
* about the storage format in your
* application programs.
*/
template <int rank, int dim>
struct StorageType;
/**
* Specialization of StorageType for
* rank-2 tensors.
*/
template <int dim>
struct StorageType<2,dim>
{
/**
* Number of independent components of a
* symmetric tensor of rank 2. We store
* only the upper right half of it.
*/
static const unsigned int
n_independent_components = (dim*dim + dim)/2;
/**
* Declare the type in which we actually
* store the data.
*/
typedef Tensor<1,n_independent_components> base_tensor_type;
};
/**
* Specialization of StorageType for
* rank-4 tensors.
*/
template <int dim>
struct StorageType<4,dim>
{
/**
* Number of independent components
* of a symmetric tensor of rank
* 2. Since rank-4 tensors are
* mappings between such objects, we
* need this information.
*/
static const unsigned int
n_rank2_components = (dim*dim + dim)/2;
/**
* Number of independent components of a
* symmetric tensor of rank 4.
*/
static const unsigned int
n_independent_components = (n_rank2_components *
StorageType<2,dim>::n_independent_components);
/**
* Declare the type in which we
* actually store the data. Symmetric
* rank-4 tensors are mappings
* between symmetric rank-2 tensors,
* so we can represent the data as a
* matrix if we represent the rank-2
* tensors as vectors.
*/
typedef Tensor<2,n_rank2_components> base_tensor_type;
};
/**
* Switch type to select a tensor of
* rank 2 and dimension <tt>dim</tt>,
* switching on whether the tensor
* should be constant or not.
*/
template <int rank, int dim, bool constness>
struct AccessorTypes;
/**
* Switch type to select a tensor of
* rank 2 and dimension <tt>dim</tt>,
* switching on whether the tensor
* should be constant or not.
*
* Specialization for constant tensors.
*/
template <int rank, int dim>
struct AccessorTypes<rank, dim,true>
{
typedef const ::dealii::SymmetricTensor<rank,dim> tensor_type;
typedef double reference;
};
/**
* Switch type to select a tensor of
* rank 2 and dimension <tt>dim</tt>,
* switching on whether the tensor
* should be constant or not.
*
* Specialization for non-constant
* tensors.
*/
template <int rank, int dim>
struct AccessorTypes<rank,dim,false>
{
typedef ::dealii::SymmetricTensor<rank,dim> tensor_type;
typedef double &reference;
};
/**
* @internal
*
* Class that acts as accessor to elements of type
* SymmetricTensor. The template parameter <tt>C</tt> may be either
* true or false, and indicates whether the objects worked on are
* constant or not (i.e. write access is only allowed if the value is
* false).
*
* Since with <tt>N</tt> indices, the effect of applying
* <tt>operator[]</tt> is getting access to something we <tt>N-1</tt>
* indices, we have to implement these accessor classes recursively,
* with stopping when we have only one index left. For the latter
* case, a specialization of this class is declared below, where
* calling <tt>operator[]</tt> gives you access to the objects
* actually stored by the tensor; the tensor class also makes sure
* that only those elements are actually accessed which we actually
* store, i.e. it reorders indices if necessary. The template
* parameter <tt>P</tt> indicates how many remaining indices there
* are. For a rank-2 tensor, <tt>P</tt> may be two, and when using
* <tt>operator[]</tt>, an object with <tt>P=1</tt> emerges.
*
* As stated for the entire namespace, you will not usually have to do
* with these classes directly, and should not try to use their
* interface directly as it may change without notice. In fact, since
* the constructors are made private, you will not even be able to
* generate objects of this class, as they are only thought as
* temporaries for access to elements of the table class, not for
* passing them around as arguments of functions, etc.
*
* This class is an adaptation of a similar class used for the Table
* class.
*
* @author Wolfgang Bangerth, 2002, 2005
*/
template <int rank, int dim, bool constness, int P>
class Accessor
{
public:
/**
* Import two typedefs from the
* switch class above.
*/
typedef typename AccessorTypes<rank,dim,constness>::reference reference;
typedef typename AccessorTypes<rank,dim,constness>::tensor_type tensor_type;
private:
/**
* Constructor. Take a
* reference to the tensor
* object which we will
* access.
*
* The second argument
* denotes the values of
* previous indices into the
* tensor. For example, for a
* rank-4 tensor, if P=2,
* then we will already have
* had two successive element
* selections (e.g. through
* <tt>tensor[1][2]</tt>),
* and the two index values
* have to be stored
* somewhere. This class
* therefore only makes use
* of the first rank-P
* elements of this array,
* but passes it on to the
* next level with P-1 which
* fills the next entry, and
* so on.
*
* The constructor is made
* private in order to prevent
* you having such objects
* around. The only way to
* create such objects is via
* the <tt>Table</tt> class, which
* only generates them as
* temporary objects. This
* guarantees that the accessor
* objects go out of scope
* earlier than the mother
* object, avoid problems with
* data consistency.
*/
Accessor (tensor_type &tensor,
const TableIndices<rank> &previous_indices);
/**
* Default constructor. Not
* needed, and invisible, so
* private.
*/
Accessor ();
/**
* Copy constructor. Not
* needed, and invisible, so
* private.
*/
Accessor (const Accessor &a);
public:
/**
* Index operator.
*/
Accessor<rank,dim,constness,P-1> operator [] (const unsigned int i);
private:
/**
* Store the data given to the
* constructor.
*/
tensor_type &tensor;
const TableIndices<rank> previous_indices;
// declare some other classes
// as friends. make sure to
// work around bugs in some
// compilers
#ifndef DEAL_II_NAMESP_TEMPL_FRIEND_BUG
template <int,int> friend class SymmetricTensor;
template <int,int,bool,int>
friend class Accessor;
# ifndef DEAL_II_TEMPL_SPEC_FRIEND_BUG
friend class ::dealii::SymmetricTensor<rank,dim>;
friend class Accessor<rank,dim,constness,P+1>;
# endif
#else
friend class SymmetricTensor<rank,dim>;
friend class Accessor<rank,dim,constness,P+1>;
#endif
};
/**
* @internal
* Accessor class for SymmetricTensor. This is the specialization for the last
* index, which actually allows access to the elements of the table,
* rather than recursively returning access objects for further
* subsets. The same holds for this specialization as for the general
* template; see there for more information.
*
* @author Wolfgang Bangerth, 2002, 2005
*/
template <int rank, int dim, bool constness>
class Accessor<rank,dim,constness,1>
{
public:
/**
* Import two typedefs from the
* switch class above.
*/
typedef typename AccessorTypes<rank,dim,constness>::reference reference;
typedef typename AccessorTypes<rank,dim,constness>::tensor_type tensor_type;
private:
/**
* Constructor. Take a
* reference to the tensor
* object which we will
* access.
*
* The second argument
* denotes the values of
* previous indices into the
* tensor. For example, for a
* rank-4 tensor, if P=2,
* then we will already have
* had two successive element
* selections (e.g. through
* <tt>tensor[1][2]</tt>),
* and the two index values
* have to be stored
* somewhere. This class
* therefore only makes use
* of the first rank-P
* elements of this array,
* but passes it on to the
* next level with P-1 which
* fills the next entry, and
* so on.
*
* For this particular
* specialization, i.e. for
* P==1, all but the last
* index are already filled.
*
* The constructor is made
* private in order to prevent
* you having such objects
* around. The only way to
* create such objects is via
* the <tt>Table</tt> class, which
* only generates them as
* temporary objects. This
* guarantees that the accessor
* objects go out of scope
* earlier than the mother
* object, avoid problems with
* data consistency.
*/
Accessor (tensor_type &tensor,
const TableIndices<rank> &previous_indices);
/**
* Default constructor. Not
* needed, and invisible, so
* private.
*/
Accessor ();
/**
* Copy constructor. Not
* needed, and invisible, so
* private.
*/
Accessor (const Accessor &a);
public:
/**
* Index operator.
*/
reference operator [] (const unsigned int);
private:
/**
* Store the data given to the
* constructor.
*/
tensor_type &tensor;
const TableIndices<rank> previous_indices;
// declare some other classes
// as friends. make sure to
// work around bugs in some
// compilers
#ifndef DEAL_II_NAMESP_TEMPL_FRIEND_BUG
template <int,int> friend class SymmetricTensor;
template <int,int,bool,int>
friend class SymmetricTensorAccessors::Accessor;
# ifndef DEAL_II_TEMPL_SPEC_FRIEND_BUG
friend class ::dealii::SymmetricTensor<rank,dim>;
friend class SymmetricTensorAccessors::Accessor<rank,dim,constness,2>;
# endif
#else
friend class SymmetricTensor<rank,dim>;
friend class Accessor<rank,dim,constness,2>;
#endif
};
}
}
/**
* Provide a class that stores symmetric tensors of rank 2,4,... efficiently,
* i.e. only store those off-diagonal 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 runtime-dependent dimension. It is
* also more efficient than using the more general <tt>Tensor</tt> class,
* since less 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 rank-2 tensor is obvious,
* tensors of rank 4 are considered symmetric if they are operators
* mapping symmetric rank-2 tensors onto symmetric rank-2
* tensors. This entails certain symmetry properties on the elements
* in their 4-dimensional index space, in particular that
* <tt>C<sub>ijkl</sub>=C<sub>jikl</sub>=C<sub>ijlk</sub></tt>. However,
* it does not imply the relation
* <tt>C<sub>ijkl</sub>=C<sub>klij</sub></tt>. 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 <tt>a:C:b=b:C:a</tt>
* or even a positive (semi-)definite form <tt>a:C:a</tt>, where
* <tt>a,b</tt> are symmetric rank-2 tensors and the colon indicates
* the common double-index 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
* stress-strain relationship is given by a symmetric rank-4 tensor.
*
* Note that symmetric tensors only exist with even numbers of indices. In
* other words, the only objects that you can use are
* <tt>SymmetricTensor<2,dim></tt>, <tt>SymmetricTensor<4,dim></tt>, etc, but
* <tt>SymmetricTensor<1,dim></tt> and <tt>SymmetricTensor<3,dim></tt> do not
* exist and their use will most likely lead to compiler errors.
*
*
* <h3>Accessing elements</h3>
*
* The elements of a tensor <tt>t</tt> can be accessed using the
* bracket operator, i.e. for a tensor of rank 4,
* <tt>t[0][1][0][1]</tt> accesses the element
* <tt>t<sub>0101</sub></tt>. This access can be used for both reading
* and writing (if the tensor is non-constant 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 rank-2 tensor that is assumed to
* be zero at the beginning, writing <tt>t[0][1]+=1; t[1][0]+=1;</tt>
* will lead to the same element being increased by one
* <em>twice</em>, 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.
*
* @ingroup geomprimitives
* @author Wolfgang Bangerth, 2005
*/
template <int rank, int dim>
class SymmetricTensor
{
public:
/**
* 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 <tt>dimension()</tt>
* 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.
*/
static const unsigned int dimension = dim;
/**
* 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.
*/
static const unsigned int n_independent_components
= internal::SymmetricTensorAccessors::StorageType<rank,dim>::
n_independent_components;
/**
* Default constructor. Creates a zero
* tensor.
*/
SymmetricTensor ();
/**
* Constructor. Generate a symmetric
* tensor from a general one. Assumes
* that @p 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 round-off 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
* round-off, then you may want to call
* the <tt>symmetrize</tt> function
* first. If you aren't sure, it is good
* practice to check before calling
* <tt>symmetrize</tt>.
*/
SymmetricTensor (const Tensor<2,dim> &t);
/**
* 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.
*/
SymmetricTensor (const double (&array) [internal::SymmetricTensorAccessors::StorageType<rank,dim>::n_independent_components]);
/**
* Assignment operator.
*/
SymmetricTensor & operator = (const SymmetricTensor &);
/**
* 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 <tt>d</tt>,
* allowing the intuitive
* notation <tt>t=0</tt> to reset
* all elements of the tensor to
* zero.
*/
SymmetricTensor & operator = (const double d);
/**
* Convert the present symmetric tensor
* into a full tensor with the same
* elements, but using the different
* storage scheme of full tensors.
*/
operator Tensor<rank,dim> () const;
/**
* Test for equality of two tensors.
*/
bool operator == (const SymmetricTensor &) const;
/**
* Test for inequality of two tensors.
*/
bool operator != (const SymmetricTensor &) const;
/**
* Add another tensor.
*/
SymmetricTensor & operator += (const SymmetricTensor &);
/**
* Subtract another tensor.
*/
SymmetricTensor & operator -= (const SymmetricTensor &);
/**
* Scale the tensor by <tt>factor</tt>,
* i.e. multiply all components by
* <tt>factor</tt>.
*/
SymmetricTensor & operator *= (const double factor);
/**
* Scale the vector by
* <tt>1/factor</tt>.
*/
SymmetricTensor & operator /= (const double factor);
/**
* Add two tensors. If possible, you
* should use <tt>operator +=</tt>
* instead since this does not need the
* creation of a temporary.
*/
SymmetricTensor operator + (const SymmetricTensor &s) const;
/**
* Subtract two tensors. If possible,
* you should use <tt>operator -=</tt>
* instead since this does not need the
* creation of a temporary.
*/
SymmetricTensor operator - (const SymmetricTensor &s) const;
/**
* Unary minus operator. Negate all
* entries of a tensor.
*/
SymmetricTensor operator - () const;
/**
* Product between the present
* symmetric tensor and a tensor
* of rank 2. For example, if the
* present object is also a
* rank-2 tensor, then this is
* the scalar-product double
* contraction
* <tt>a<sub>ij</sub>b<sub>ij</sub></tt>
* over all indices
* <tt>i,j</tt>. 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
* rank-4 tensor, the the result
* is a rank-2 tensor, 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 symmetrict
* 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
* <tt>Tensor</tt> objects. For
* symmetric tensors it therefore
* acts in a way that is commonly
* denoted by a "colon
* multiplication" in the
* mathematica literature.
*
* There are global functions
* <tt>double_contract</tt> 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.
*/
typename internal::SymmetricTensorAccessors::double_contraction_result<rank,2,dim>::type
operator * (const SymmetricTensor<2,dim> &s) const;
/**
* Contraction over two indices
* of the present object with the
* rank-4 symmetric tensor given
* as argument.
*/
typename internal::SymmetricTensorAccessors::double_contraction_result<rank,4,dim>::type
operator * (const SymmetricTensor<4,dim> &s) const;
/**
* Return a read-write reference
* to the indicated element.
*/
double & operator() (const TableIndices<rank> &indices);
/**
* Return the value of the
* indicated element as a
* read-only reference.
*
* We return the requested value
* as a constant reference rather
* than by value since this
* object may hold data types
* that may be large, and we
* don't know here whether
* copying is expensive or not.
*/
double operator() (const TableIndices<rank> &indices) const;
/**
* Access the elements of a row of this
* symmetric tensor. This function is
* called for constant tensors.
*/
internal::SymmetricTensorAccessors::Accessor<rank,dim,true,rank-1>
operator [] (const unsigned int row) const;
/**
* Access the elements of a row of this
* symmetric tensor. This function is
* called for non-constant tensors.
*/
internal::SymmetricTensorAccessors::Accessor<rank,dim,false,rank-1>
operator [] (const unsigned int row);
/**
* Access to an element where you
* specify the entire set of
* indices.
*/
double
operator [] (const TableIndices<rank> &indices) const;
/**
* Access to an element where you
* specify the entire set of
* indices.
*/
double &
operator [] (const TableIndices<rank> &indices);
/**
* Return the Frobenius-norm 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 <i>all</i> entries of
* the tensor, counting symmetry, not
* only the unique ones (for example, for
* rank-2 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).
*/
double norm () const;
/**
* 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
* <code>[0,n_independent_components)</code>
* the given entry in a symmetric
* tensor has.
*/
static
unsigned int
component_to_unrolled_index (const TableIndices<rank> &indices);
/**
* 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 rank-2
* tensors) or $(k,l,m,n)$ (for
* rank-4 tensors) corresponds to
* it.
*/
static
TableIndices<rank>
unrolled_to_component_indices (const unsigned int i);
/**
* Reset all values to zero.
*
* Note that this is partly inconsistent
* with the semantics of the @p clear()
* member functions of the STL 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.
*/
void clear ();
/**
* Determine an estimate for
* the memory consumption (in
* bytes) of this
* object.
*/
static unsigned int memory_consumption ();
private:
/**
* A structure that describes
* properties of the base tensor.
*/
typedef
internal::SymmetricTensorAccessors::StorageType<rank,dim>
base_tensor_descriptor;
/**
* Data storage type for a
* symmetric tensor.
*/
typedef typename base_tensor_descriptor::base_tensor_type base_tensor_type;
/**
* The place where we store the
* data of the tensor.
*/
base_tensor_type data;
/**
* Make all other symmetric tensors friends.
*/
template <int, int> friend class SymmetricTensor;
/**
* Make a few more functions friends.
*/
template <int dim2>
friend double trace (const SymmetricTensor<2,dim2> &d);
template <int dim2>
friend double determinant (const SymmetricTensor<2,dim2> &t);
template <int dim2>
friend SymmetricTensor<2,dim2>
deviator (const SymmetricTensor<2,dim2> &t);
template <int dim2>
friend SymmetricTensor<2,dim2> unit_symmetric_tensor ();
template <int dim2>
friend SymmetricTensor<4,dim2> deviator_tensor ();
template <int dim2>
friend SymmetricTensor<4,dim2> identity_tensor ();
template <int dim2>
friend SymmetricTensor<4,dim2> invert (const SymmetricTensor<4,dim2> &);
};
// ------------------------- inline functions ------------------------
#ifndef DOXYGEN
namespace internal
{
namespace SymmetricTensorAccessors
{
template <int rank, int dim, bool constness, int P>
Accessor<rank,dim,constness,P>::
Accessor (tensor_type &tensor,
const TableIndices<rank> &previous_indices)
:
tensor (tensor),
previous_indices (previous_indices)
{}
template <int rank, int dim, bool constness, int P>
Accessor<rank,dim,constness,P-1>
Accessor<rank,dim,constness,P>::operator[] (const unsigned int i)
{
return Accessor<rank,dim,constness,P-1> (tensor,
merge (previous_indices, i, rank-P));
}
template <int rank, int dim, bool constness>
Accessor<rank,dim,constness,1>::
Accessor (tensor_type &tensor,
const TableIndices<rank> &previous_indices)
:
tensor (tensor),
previous_indices (previous_indices)
{}
template <int rank, int dim, bool constness>
typename Accessor<rank,dim,constness,1>::reference
Accessor<rank,dim,constness,1>::operator[] (const unsigned int i)
{
return tensor(merge (previous_indices, i, rank-1));
}
}
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>::SymmetricTensor ()
{}
template <>
inline
SymmetricTensor<2,2>::SymmetricTensor (const Tensor<2,2> &t)
{
Assert (t[0][1] == t[1][0], ExcInternalError());
data[0] = t[0][0];
data[1] = t[1][1];
data[2] = t[0][1];
}
template <>
inline
SymmetricTensor<2,3>::SymmetricTensor (const Tensor<2,3> &t)
{
Assert (t[0][1] == t[1][0], ExcInternalError());
Assert (t[0][2] == t[2][0], ExcInternalError());
Assert (t[1][2] == t[2][1], ExcInternalError());
data[0] = t[0][0];
data[1] = t[1][1];
data[2] = t[2][2];
data[3] = t[0][1];
data[4] = t[0][2];
data[5] = t[1][2];
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>::SymmetricTensor (const double (&array) [internal::SymmetricTensorAccessors::StorageType<rank,dim>::n_independent_components])
:
data (array)
{}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim> &
SymmetricTensor<rank,dim>::operator = (const SymmetricTensor<rank,dim> &t)
{
data = t.data;
return *this;
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim> &
SymmetricTensor<rank,dim>::operator = (const double d)
{
Assert (d==0, ExcMessage ("Only assignment with zero is allowed"));
data = 0;
return *this;
}
template <>
inline
SymmetricTensor<2,1>::
operator Tensor<2,1> () const
{
const double t[1][1] = {{data[0]}};
return Tensor<2,1>(t);
}
template <>
inline
SymmetricTensor<2,2>::
operator Tensor<2,2> () const
{
const double t[2][2] = {{data[0], data[2]},
{data[2], data[1]}};
return Tensor<2,2>(t);
}
template <>
inline
SymmetricTensor<2,3>::
operator Tensor<2,3> () const
{
const double t[3][3] = {{data[0], data[3], data[4]},
{data[3], data[1], data[5]},
{data[4], data[5], data[2]}};
return Tensor<2,3>(t);
}
template <int rank, int dim>
inline
bool
SymmetricTensor<rank,dim>::operator == (const SymmetricTensor<rank,dim> &t) const
{
return data == t.data;
}
template <int rank, int dim>
inline
bool
SymmetricTensor<rank,dim>::operator != (const SymmetricTensor<rank,dim> &t) const
{
return data != t.data;
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim> &
SymmetricTensor<rank,dim>::operator += (const SymmetricTensor<rank,dim> &t)
{
data += t.data;
return *this;
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim> &
SymmetricTensor<rank,dim>::operator -= (const SymmetricTensor<rank,dim> &t)
{
data -= t.data;
return *this;
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim> &
SymmetricTensor<rank,dim>::operator *= (const double d)
{
data *= d;
return *this;
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim> &
SymmetricTensor<rank,dim>::operator /= (const double d)
{
data /= d;
return *this;
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>
SymmetricTensor<rank,dim>::operator + (const SymmetricTensor &t) const
{
SymmetricTensor tmp = *this;
tmp.data += t.data;
return tmp;
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>
SymmetricTensor<rank,dim>::operator - (const SymmetricTensor &t) const
{
SymmetricTensor tmp = *this;
tmp.data -= t.data;
return tmp;
}
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>
SymmetricTensor<rank,dim>::operator - () const
{
SymmetricTensor tmp = *this;
tmp.data = -tmp.data;
return tmp;
}
template <int rank, int dim>
inline
void
SymmetricTensor<rank,dim>::clear ()
{
data.clear ();
}
template <int rank, int dim>
inline
unsigned int
SymmetricTensor<rank,dim>::memory_consumption ()
{
return
internal::SymmetricTensorAccessors::StorageType<rank,dim>::memory_consumption ();
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<2,2,1>::type
SymmetricTensor<2,1>::operator * (const SymmetricTensor<2,1> &s) const
{
return data[0] * s.data[0];
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<2,2,2>::type
SymmetricTensor<2,2>::operator * (const SymmetricTensor<2,2> &s) const
{
return (data[0] * s.data[0] +
data[1] * s.data[1] +
2*data[2] * s.data[2]);
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<2,2,3>::type
SymmetricTensor<2,3>::operator * (const SymmetricTensor<2,3> &s) const
{
return (data[0] * s.data[0] +
data[1] * s.data[1] +
data[2] * s.data[2] +
2*data[3] * s.data[3] +
2*data[4] * s.data[4] +
2*data[5] * s.data[5]);
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<4,2,1>::type
SymmetricTensor<4,1>::
operator * (const SymmetricTensor<2,1> &s) const
{
const unsigned int dim = 1;
SymmetricTensor<2,dim> tmp;
tmp.data[0] = data[0][0] * s.data[0];
return tmp;
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<2,4,1>::type
SymmetricTensor<2,1>::
operator * (const SymmetricTensor<4,1> &s) const
{
const unsigned int dim = 1;
SymmetricTensor<2,dim> tmp;
tmp[0][0] = data[0] * s.data[0][0];
return tmp;
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<4,2,2>::type
SymmetricTensor<4,2>::
operator * (const SymmetricTensor<2,2> &s) const
{
const unsigned int dim = 2;
SymmetricTensor<2,dim> tmp;
for (unsigned int i=0; i<3; ++i)
tmp.data[i] = data[i][0] * s.data[0] +
data[i][1] * s.data[1] +
2 * data[i][2] * s.data[2];
return tmp;
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<2,4,2>::type
SymmetricTensor<2,2>::
operator * (const SymmetricTensor<4,2> &s) const
{
const unsigned int dim = 2;
SymmetricTensor<2,dim> tmp;
for (unsigned int i=0; i<3; ++i)
tmp.data[i] = data[0] * s.data[0][i] +
data[1] * s.data[1][i] +
2 * data[2] * s.data[2][i];
return tmp;
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<4,2,3>::type
SymmetricTensor<4,3>::
operator * (const SymmetricTensor<2,3> &s) const
{
const unsigned int dim = 3;
SymmetricTensor<2,dim> tmp;
for (unsigned int i=0; i<6; ++i)
tmp.data[i] = data[i][0] * s.data[0] +
data[i][1] * s.data[1] +
data[i][2] * s.data[2] +
2 * data[i][3] * s.data[3] +
2 * data[i][4] * s.data[4] +
2 * data[i][5] * s.data[5];
return tmp;
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<2,4,3>::type
SymmetricTensor<2,3>::
operator * (const SymmetricTensor<4,3> &s) const
{
const unsigned int dim = 3;
SymmetricTensor<2,dim> tmp;
for (unsigned int i=0; i<6; ++i)
tmp.data[i] = data[0] * s.data[0][i] +
data[1] * s.data[1][i] +
data[2] * s.data[2][i] +
2 * data[3] * s.data[3][i] +
2 * data[4] * s.data[4][i] +
2 * data[5] * s.data[5][i];
return tmp;
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<4,4,1>::type
SymmetricTensor<4,1>::
operator * (const SymmetricTensor<4,1> &s) const
{
const unsigned int dim = 1;
SymmetricTensor<4,dim> tmp;
tmp.data[0][0] = data[0][0] * s.data[0][0];
return tmp;
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<4,4,2>::type
SymmetricTensor<4,2>::
operator * (const SymmetricTensor<4,2> &s) const
{
const unsigned int dim = 2;
SymmetricTensor<4,dim> tmp;
for (unsigned int i=0; i<base_tensor_descriptor::n_rank2_components; ++i)
for (unsigned int j=0; j<base_tensor_descriptor::n_rank2_components; ++j)
tmp.data[i][j] = data[i][0] * s.data[0][j] +
data[i][1] * s.data[1][j] +
2*data[i][2] * s.data[2][j];
return tmp;
}
template <>
inline
internal::SymmetricTensorAccessors::double_contraction_result<4,4,3>::type
SymmetricTensor<4,3>::
operator * (const SymmetricTensor<4,3> &s) const
{
const unsigned int dim = 3;
SymmetricTensor<4,dim> tmp;
for (unsigned int i=0; i<base_tensor_descriptor::n_rank2_components; ++i)
for (unsigned int j=0; j<base_tensor_descriptor::n_rank2_components; ++j)
tmp.data[i][j] = data[i][0] * s.data[0][j] +
data[i][1] * s.data[1][j] +
data[i][2] * s.data[2][j] +
2*data[i][3] * s.data[3][j] +
2*data[i][4] * s.data[4][j] +
2*data[i][5] * s.data[5][j];
return tmp;
}
template <>
inline
double &
SymmetricTensor<2,1>::operator () (const TableIndices<2> &indices)
{
const unsigned int rank = 2;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < 1, ExcIndexRange (indices[r], 0, 1));
return data[0];
}
template <>
inline
double
SymmetricTensor<2,1>::operator () (const TableIndices<2> &indices) const
{
const unsigned int rank = 2;
const unsigned int dim = 1;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
return data[0];
}
template <>
inline
double &
SymmetricTensor<2,2>::operator () (const TableIndices<2> &indices)
{
const unsigned int rank = 2;
const unsigned int dim = 2;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
// first treat the main diagonal
// elements, which are stored
// consecutively at the beginning
if (indices[0] == indices[1])
return data[indices[0]];
// the rest is messier and requires a few
// switches. at least for the 2x2 case it
// is reasonably simple
Assert (((indices[0]==1) && (indices[1]==0)) ||
((indices[0]==0) && (indices[1]==1)),
ExcInternalError());
return data[2];
}
template <>
inline
double
SymmetricTensor<2,2>::operator () (const TableIndices<2> &indices) const
{
const unsigned int rank = 2;
const unsigned int dim = 2;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
// first treat the main diagonal
// elements, which are stored
// consecutively at the beginning
if (indices[0] == indices[1])
return data[indices[0]];
// the rest is messier and requires a few
// switches. at least for the 2x2 case it
// is reasonably simple
Assert (((indices[0]==1) && (indices[1]==0)) ||
((indices[0]==0) && (indices[1]==1)),
ExcInternalError());
return data[2];
}
template <>
inline
double &
SymmetricTensor<2,3>::operator () (const TableIndices<2> &indices)
{
const unsigned int rank = 2;
const unsigned int dim = 3;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
// first treat the main diagonal
// elements, which are stored
// consecutively at the beginning
if (indices[0] == indices[1])
return data[indices[0]];
// the rest is messier and requires a few
// switches, but simpler if we just sort
// our indices
TableIndices<2> sorted_indices (indices);
sorted_indices.sort ();
if ((sorted_indices[0]==0) && (sorted_indices[1]==1))
return data[3];
else if ((sorted_indices[0]==0) && (sorted_indices[1]==2))
return data[4];
else if ((sorted_indices[0]==1) && (sorted_indices[1]==2))
return data[5];
else
Assert (false, ExcInternalError());
static double dummy_but_referenceable = 0;
return dummy_but_referenceable;
}
template <>
inline
double
SymmetricTensor<2,3>::operator () (const TableIndices<2> &indices) const
{
const unsigned int rank = 2;
const unsigned int dim = 3;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
// first treat the main diagonal
// elements, which are stored
// consecutively at the beginning
if (indices[0] == indices[1])
return data[indices[0]];
// the rest is messier and requires a few
// switches, but simpler if we just sort
// our indices
TableIndices<2> sorted_indices (indices);
sorted_indices.sort ();
if ((sorted_indices[0]==0) && (sorted_indices[1]==1))
return data[3];
else if ((sorted_indices[0]==0) && (sorted_indices[1]==2))
return data[4];
else if ((sorted_indices[0]==1) && (sorted_indices[1]==2))
return data[5];
else
Assert (false, ExcInternalError());
static double dummy_but_referenceable = 0;
return dummy_but_referenceable;
}
template <>
inline
double &
SymmetricTensor<4,1>::operator () (const TableIndices<4> &indices)
{
const unsigned int rank = 4;
const unsigned int dim = 1;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
return data[0][0];
}
template <>
inline
double
SymmetricTensor<4,1>::operator () (const TableIndices<4> &indices) const
{
const unsigned int rank = 4;
const unsigned int dim = 1;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
return data[0][0];
}
template <>
inline
double &
SymmetricTensor<4,2>::operator () (const TableIndices<4> &indices)
{
const unsigned int rank = 4;
const unsigned int dim = 2;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
// each entry of the tensor can be
// thought of as an entry in a
// matrix that maps the rolled-out
// rank-2 tensors into rolled-out
// rank-2 tensors. this is the
// format in which we store rank-4
// tensors. determine which
// position the present entry is
// stored in
unsigned int base_index[2] ;
if ((indices[0] == 0) && (indices[1] == 0))
base_index[0] = 0;
else if ((indices[0] == 1) && (indices[1] == 1))
base_index[0] = 1;
else
base_index[0] = 2;
if ((indices[2] == 0) && (indices[3] == 0))
base_index[1] = 0;
else if ((indices[2] == 1) && (indices[3] == 1))
base_index[1] = 1;
else
base_index[1] = 2;
return data[base_index[0]][base_index[1]];
}
template <>
inline
double
SymmetricTensor<4,2>::operator () (const TableIndices<4> &indices) const
{
const unsigned int rank = 4;
const unsigned int dim = 2;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
// each entry of the tensor can be
// thought of as an entry in a
// matrix that maps the rolled-out
// rank-2 tensors into rolled-out
// rank-2 tensors. this is the
// format in which we store rank-4
// tensors. determine which
// position the present entry is
// stored in
unsigned int base_index[2] ;
if ((indices[0] == 0) && (indices[1] == 0))
base_index[0] = 0;
else if ((indices[0] == 1) && (indices[1] == 1))
base_index[0] = 1;
else
base_index[0] = 2;
if ((indices[2] == 0) && (indices[3] == 0))
base_index[1] = 0;
else if ((indices[2] == 1) && (indices[3] == 1))
base_index[1] = 1;
else
base_index[1] = 2;
return data[base_index[0]][base_index[1]];
}
template <>
inline
double &
SymmetricTensor<4,3>::operator () (const TableIndices<4> &indices)
{
const unsigned int rank = 4;
const unsigned int dim = 3;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
// each entry of the tensor can be
// thought of as an entry in a
// matrix that maps the rolled-out
// rank-2 tensors into rolled-out
// rank-2 tensors. this is the
// format in which we store rank-4
// tensors. determine which
// position the present entry is
// stored in
unsigned int base_index[2] ;
if ((indices[0] == 0) && (indices[1] == 0))
base_index[0] = 0;
else if ((indices[0] == 1) && (indices[1] == 1))
base_index[0] = 1;
else if ((indices[0] == 2) && (indices[1] == 2))
base_index[0] = 2;
else if (((indices[0] == 0) && (indices[1] == 1)) ||
((indices[0] == 1) && (indices[1] == 0)))
base_index[0] = 3;
else if (((indices[0] == 0) && (indices[1] == 2)) ||
((indices[0] == 2) && (indices[1] == 0)))
base_index[0] = 4;
else
{
Assert (((indices[0] == 1) && (indices[1] == 2)) ||
((indices[0] == 2) && (indices[1] == 1)),
ExcInternalError());
base_index[0] = 5;
}
if ((indices[2] == 0) && (indices[3] == 0))
base_index[1] = 0;
else if ((indices[2] == 1) && (indices[3] == 1))
base_index[1] = 1;
else if ((indices[2] == 2) && (indices[3] == 2))
base_index[1] = 2;
else if (((indices[2] == 0) && (indices[3] == 1)) ||
((indices[2] == 1) && (indices[3] == 0)))
base_index[1] = 3;
else if (((indices[2] == 0) && (indices[3] == 2)) ||
((indices[2] == 2) && (indices[3] == 0)))
base_index[1] = 4;
else
{
Assert (((indices[2] == 1) && (indices[3] == 2)) ||
((indices[2] == 2) && (indices[3] == 1)),
ExcInternalError());
base_index[1] = 5;
}
return data[base_index[0]][base_index[1]];
}
template <>
inline
double
SymmetricTensor<4,3>::operator () (const TableIndices<4> &indices) const
{
const unsigned int rank = 4;
const unsigned int dim = 3;
for (unsigned int r=0; r<rank; ++r)
Assert (indices[r] < dim, ExcIndexRange (indices[r], 0, dim));
// each entry of the tensor can be
// thought of as an entry in a
// matrix that maps the rolled-out
// rank-2 tensors into rolled-out
// rank-2 tensors. this is the
// format in which we store rank-4
// tensors. determine which
// position the present entry is
// stored in
unsigned int base_index[2] ;
if ((indices[0] == 0) && (indices[1] == 0))
base_index[0] = 0;
else if ((indices[0] == 1) && (indices[1] == 1))
base_index[0] = 1;
else if ((indices[0] == 2) && (indices[1] == 2))
base_index[0] = 2;
else if (((indices[0] == 0) && (indices[1] == 1)) ||
((indices[0] == 1) && (indices[1] == 0)))
base_index[0] = 3;
else if (((indices[0] == 0) && (indices[1] == 2)) ||
((indices[0] == 2) && (indices[1] == 0)))
base_index[0] = 4;
else
{
Assert (((indices[0] == 1) && (indices[1] == 2)) ||
((indices[0] == 2) && (indices[1] == 1)),
ExcInternalError());
base_index[0] = 5;
}
if ((indices[2] == 0) && (indices[3] == 0))
base_index[1] = 0;
else if ((indices[2] == 1) && (indices[3] == 1))
base_index[1] = 1;
else if ((indices[2] == 2) && (indices[3] == 2))
base_index[1] = 2;
else if (((indices[2] == 0) && (indices[3] == 1)) ||
((indices[2] == 1) && (indices[3] == 0)))
base_index[1] = 3;
else if (((indices[2] == 0) && (indices[3] == 2)) ||
((indices[2] == 2) && (indices[3] == 0)))
base_index[1] = 4;
else
{
Assert (((indices[2] == 1) && (indices[3] == 2)) ||
((indices[2] == 2) && (indices[3] == 1)),
ExcInternalError());
base_index[1] = 5;
}
return data[base_index[0]][base_index[1]];
}
template <int rank, int dim>
internal::SymmetricTensorAccessors::Accessor<rank,dim,true,rank-1>
SymmetricTensor<rank,dim>::operator [] (const unsigned int row) const
{
return
internal::SymmetricTensorAccessors::
Accessor<rank,dim,true,rank-1> (*this, TableIndices<rank> (row));
}
template <int rank, int dim>
internal::SymmetricTensorAccessors::Accessor<rank,dim,false,rank-1>
SymmetricTensor<rank,dim>::operator [] (const unsigned int row)
{
return
internal::SymmetricTensorAccessors::
Accessor<rank,dim,false,rank-1> (*this, TableIndices<rank> (row));
}
template <int rank, int dim>
inline
double
SymmetricTensor<rank,dim>::operator [] (const TableIndices<rank> &indices) const
{
return data[component_to_unrolled_index(indices)];
}
template <int rank, int dim>
inline
double &
SymmetricTensor<rank,dim>::operator [] (const TableIndices<rank> &indices)
{
return data[component_to_unrolled_index(indices)];
}
template <>
inline
double
SymmetricTensor<2,1>::norm () const
{
return std::fabs(data[0]);
}
template <>
inline
double
SymmetricTensor<2,2>::norm () const
{
return std::sqrt(data[0]*data[0] + data[1]*data[1] + 2*data[2]*data[2]);
}
template <>
inline
double
SymmetricTensor<2,3>::norm () const
{
return std::sqrt(data[0]*data[0] + data[1]*data[1] + data[2]*data[2] +
2*data[3]*data[3] + 2*data[4]*data[4] + 2*data[5]*data[5]);
}
template <>
inline
double
SymmetricTensor<4,1>::norm () const
{
return std::fabs(data[0][0]);
}
template <>
inline
double
SymmetricTensor<4,2>::norm () const
{
const unsigned int dim = 2;
// this is not really efficient and
// could be improved by counting
// how often each tensor entry is
// accessed, but this isn't a
// really frequent operation anyway
double t = 0;
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=0; j<dim; ++j)
for (unsigned int k=0; k<dim; ++k)
for (unsigned int l=0; l<dim; ++l)
{
const double a = (*this)[i][j][k][l];
t += a * a;
}
return std::sqrt(t);
}
template <>
inline
double
SymmetricTensor<4,3>::norm () const
{
const unsigned int dim = 3;
// this is not really efficient and
// could be improved by counting
// how often each tensor entry is
// accessed, but this isn't a
// really frequent operation anyway
double t = 0;
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=0; j<dim; ++j)
for (unsigned int k=0; k<dim; ++k)
for (unsigned int l=0; l<dim; ++l)
{
const double a = (*this)[i][j][k][l];
t += a * a;
}
return std::sqrt(t);
}
template <>
inline
unsigned int
SymmetricTensor<2,1>::component_to_unrolled_index (const TableIndices<2> &indices)
{
const unsigned int dim = 1;
Assert (indices[0] < dim, ExcIndexRange(indices[0], 0, dim));
Assert (indices[1] < dim, ExcIndexRange(indices[1], 0, dim));
return 0;
}
template <>
inline
unsigned int
SymmetricTensor<2,2>::component_to_unrolled_index (const TableIndices<2> &indices)
{
const unsigned int dim = 2;
Assert (indices[0] < dim, ExcIndexRange(indices[0], 0, dim));
Assert (indices[1] < dim, ExcIndexRange(indices[1], 0, dim));
static const unsigned int table[dim][dim] = {{0, 2},
{2, 1}};
return table[indices[0]][indices[1]];
}
template <>
inline
unsigned int
SymmetricTensor<2,3>::component_to_unrolled_index (const TableIndices<2> &indices)
{
const unsigned int dim = 3;
Assert (indices[0] < dim, ExcIndexRange(indices[0], 0, dim));
Assert (indices[1] < dim, ExcIndexRange(indices[1], 0, dim));
static const unsigned int table[dim][dim] = {{0, 3, 4},
{3, 1, 5},
{4, 5, 2}};
return table[indices[0]][indices[1]];
}
template <>
inline
unsigned int
SymmetricTensor<2,4>::component_to_unrolled_index (const TableIndices<2> &indices)
{
const unsigned int dim = 4;
Assert (indices[0] < dim, ExcIndexRange(indices[0], 0, dim));
Assert (indices[1] < dim, ExcIndexRange(indices[1], 0, dim));
static const unsigned int table[dim][dim] = {{0, 4, 5, 6},
{4, 1, 7, 8},
{5, 7, 2, 9},
{6, 8, 9, 3}};
return table[indices[0]][indices[1]];
}
template <>
inline
TableIndices<2>
SymmetricTensor<2,1>::unrolled_to_component_indices (const unsigned int i)
{
Assert (i < n_independent_components, ExcIndexRange(i, 0, n_independent_components));
return TableIndices<2>(0,0);
}
template <>
inline
TableIndices<2>
SymmetricTensor<2,2>::unrolled_to_component_indices (const unsigned int i)
{
Assert (i < n_independent_components, ExcIndexRange(i, 0, n_independent_components));
static const TableIndices<2> table[n_independent_components] =
{ TableIndices<2> (0,0),
TableIndices<2> (1,1),
TableIndices<2> (0,1) };
return table[i];
}
template <>
inline
TableIndices<2>
SymmetricTensor<2,3>::unrolled_to_component_indices (const unsigned int i)
{
Assert (i < n_independent_components, ExcIndexRange(i, 0, n_independent_components));
static const TableIndices<2> table[n_independent_components] =
{ TableIndices<2> (0,0),
TableIndices<2> (1,1),
TableIndices<2> (2,2),
TableIndices<2> (0,1),
TableIndices<2> (0,2),
TableIndices<2> (1,2) };
return table[i];
}
#endif // DOXYGEN
/* ----------------- Non-member functions operating on tensors. ------------ */
/**
* Compute the determinant of a tensor or rank 2. The determinant is
* also commonly referred to as the third invariant of rank-2 tensors.
*
* For the present case of one-dimensional tensors, the determinant
* equals the only element and is therefore equivalent to the trace.
*
* For greater notational simplicity, there is also a <tt>third_invariant</tt>
* function that returns the determinant of a tensor.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
inline
double determinant (const SymmetricTensor<2,1> &t)
{
return t.data[0];
}
/**
* Compute the determinant of a tensor or rank 2. The determinant is
* also commonly referred to as the third invariant of rank-2 tensors.
*
* For greater notational simplicity, there is also a <tt>third_invariant</tt>
* function that returns the determinant of a tensor.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
inline
double determinant (const SymmetricTensor<2,2> &t)
{
return (t.data[0] * t.data[1] - t.data[2]*t.data[2]);
}
/**
* Compute the determinant of a tensor or rank 2. The determinant is
* also commonly referred to as the third invariant of rank-2 tensors.
*
* For greater notational simplicity, there is also a <tt>third_invariant</tt>
* function that returns the determinant of a tensor.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
inline
double determinant (const SymmetricTensor<2,3> &t)
{
// in analogy to general tensors, but
// there's something to be simplified for
// the present case
return ( t.data[0]*t.data[1]*t.data[2]
-t.data[0]*t.data[5]*t.data[5]
-t.data[1]*t.data[4]*t.data[4]
-t.data[2]*t.data[3]*t.data[3]
+2*t.data[3]*t.data[4]*t.data[5] );
}
/**
* Compute the determinant of a tensor or rank 2. This function therefore
* computes the same value as the <tt>determinant()</tt> functions and is only
* provided for greater notational simplicity (since there are also functions
* <tt>first_invariant</tt> and <tt>second_invariant</tt>).
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
inline
double third_invariant (const SymmetricTensor<2,dim> &t)
{
return determinant (t);
}
/**
* 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 rank-2
* tensor.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
double trace (const SymmetricTensor<2,dim> &d)
{
double t=0;
for (unsigned int i=0; i<dim; ++i)
t += d.data[i];
return t;
}
/**
* Compute the trace of a tensor or rank 2. This function therefore
* computes the same value as the <tt>trace()</tt> functions and is only
* provided for greater notational simplicity (since there are also functions
* <tt>second_invariant</tt> and <tt>third_invariant</tt>).
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
inline
double first_invariant (const SymmetricTensor<2,dim> &t)
{
return trace (t);
}
/**
* Compute the second invariant a tensor or rank 2. The second invariant is
* defined as <tt>I2 = 1/2[ (trace sigma)^2 - trace (sigma^2) ]</tt>.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
inline
double second_invariant (const SymmetricTensor<2,dim> &t)
{
return (trace(t)*trace(t) - t*t) / 2;
}
/**
* 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.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>
transpose (const SymmetricTensor<rank,dim> &t)
{
return t;
}
/**
* Compute the deviator of a symmetric tensor, which is defined as <tt>dev[s]
* = s - 1/dim*tr[s]*I</tt>, where <tt>I</tt> 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.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
inline
SymmetricTensor<2,dim>
deviator (const SymmetricTensor<2,dim> &t)
{
SymmetricTensor<2,dim> tmp = t;
// subtract scaled trace from the diagonal
const double tr = trace(t) / dim;
for (unsigned int i=0; i<dim; ++i)
tmp.data[i] -= tr;
return tmp;
}
/**
* Return a unit symmetric tensor of rank 2 and dimension 1.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
inline
SymmetricTensor<2,1>
unit_symmetric_tensor<1> ()
{
SymmetricTensor<2,1> tmp;
tmp.data[0] = 1;
return tmp;
}
/**
* Return a unit symmetric tensor of rank 2 and dimension 2.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
inline
SymmetricTensor<2,2>
unit_symmetric_tensor<2> ()
{
SymmetricTensor<2,2> tmp;
tmp.data[0] = tmp.data[1] = 1;
return tmp;
}
/**
* Return a unit symmetric tensor of rank 2 and dimension 3.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
inline
SymmetricTensor<2,3>
unit_symmetric_tensor<3> ()
{
SymmetricTensor<2,3> tmp;
tmp.data[0] = tmp.data[1] = tmp.data[2] = 1;
return tmp;
}
/**
* Return the tensor of rank 4 that, when multiplied by a symmetric rank 2
* tensor <tt>t</tt> returns the deviator <tt>dev t</tt>. It is the operator
* representation of the linear deviator operator.
*
* For every tensor <tt>t</tt>, there holds the identity
* <tt>deviator(t)==deviator_tensor<dim>()*t</tt>, up to numerical
* round-off. The reason this operator representation is provided is that one
* sometimes needs to invert operators like <tt>identity_tensor<dim>() +
* delta_t*deviator_tensor<dim>()</tt> or similar.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
inline
SymmetricTensor<4,dim>
deviator_tensor ()
{
SymmetricTensor<4,dim> tmp;
// fill the elements treating the diagonal
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=0; j<dim; ++j)
tmp.data[i][j] = (i==j ? 1 : 0) - 1./dim;
// then fill the ones that copy over the
// non-diagonal elements. note that during
// the double-contraction, we handle the
// off-diagonal elements twice, so simply
// copying requires a weight of 1/2
for (unsigned int i=dim;
i<internal::SymmetricTensorAccessors::StorageType<4,dim>::n_rank2_components;
++i)
tmp.data[i][i] = 0.5;
return tmp;
}
/**
* Return the tensor of rank 4 that, when multiplied by a symmetric rank 2
* tensor <tt>t</tt> returns the deviator <tt>dev t</tt>. 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 <tt>dim=2</tt>, the identity tensor has all zero entries
* except for <tt>id[0][0][0][0]=id[1][1][1][1]=1</tt> and
* <tt>id[0][1][0][1]=id[0][1][1][0]=id[1][0][0][1]=id[1][0][1][0]=1/2</tt>. To
* see why this factor of 1/2 is necessary, consider computing <tt>A=Id
* . B</tt>. For the element <tt>a_01</tt> we have <tt>a_01=id_0100 b_00 +
* id_0111 b_11 + id_0101 b_01 + id_0110 b_10</tt>. On the other hand, we need
* to have <tt>a_01=b_01</tt>, and symmetry implies <tt>b_01=b_10</tt>,
* leading to <tt>a_01=(id_0101+id_0110) b_01</tt>, or, again by symmetry,
* <tt>id_0101=id_0110=1/2</tt>. Similar considerations hold for the
* three-dimensional case.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
inline
SymmetricTensor<4,dim>
identity_tensor ()
{
SymmetricTensor<4,dim> tmp;
// fill the elements treating the diagonal
for (unsigned int i=0; i<dim; ++i)
tmp.data[i][i] = 1;
// then fill the ones that copy over the
// non-diagonal elements. note that during
// the double-contraction, we handle the
// off-diagonal elements twice, so simply
// copying requires a weight of 1/2
for (unsigned int i=dim;
i<internal::SymmetricTensorAccessors::StorageType<4,dim>::n_rank2_components;
++i)
tmp.data[i][i] = 0.5;
return tmp;
}
/**
* Invert a symmetric rank-4 tensor. Since symmetric rank-4 tensors are
* mappings from and to symmetric rank-2 tensors, they can have an
* inverse. This function computes it, if it exists, for the case that the
* dimension equals 1.
*
* 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.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
inline
SymmetricTensor<4,1>
invert (const SymmetricTensor<4,1> &t)
{
SymmetricTensor<4,1> tmp;
tmp.data[0][0] = 1./t.data[0][0];
return tmp;
}
/**
* Invert a symmetric rank-4 tensor. Since symmetric rank-4 tensors are
* mappings from and to symmetric rank-2 tensors, they can have an
* inverse. This function computes it, if it exists, for the case that the
* dimension equals 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.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
inline
SymmetricTensor<4,2>
invert (const SymmetricTensor<4,2> &t)
{
SymmetricTensor<4,2> tmp;
// inverting this tensor is a little more
// complicated than necessary, since we
// store the data of 't' as a 3x3 matrix
// t.data, but the product between a rank-4
// and a rank-2 tensor is really not the
// product between this matrix and the
// 3-vector of a rhs, but rather
//
// B.vec = t.data * mult * A.vec
//
// where mult is a 3x3 matrix with
// entries [[1,0,0],[0,1,0],[0,0,2]] to
// capture the fact that we need to add up
// both the c_ij12*a_12 and the c_ij21*a_21
// terms
//
// in addition, in this scheme, the
// identity tensor has the matrix
// representation mult^-1.
//
// the inverse of 't' therefore has the
// matrix representation
//
// inv.data = mult^-1 * t.data^-1 * mult^-1
//
// in order to compute it, let's first
// compute the inverse of t.data and put it
// into tmp.data; at the end of the
// function we then scale the last row and
// column of the inverse by 1/2,
// corresponding to the left and right
// multiplication with mult^-1
const double t4 = t.data[0][0]*t.data[1][1],
t6 = t.data[0][0]*t.data[1][2],
t8 = t.data[0][1]*t.data[1][0],
t00 = t.data[0][2]*t.data[1][0],
t01 = t.data[0][1]*t.data[2][0],
t04 = t.data[0][2]*t.data[2][0],
t07 = 1.0/(t4*t.data[2][2]-t6*t.data[2][1]-t8*t.data[2][2]+
t00*t.data[2][1]+t01*t.data[1][2]-t04*t.data[1][1]);
tmp.data[0][0] = (t.data[1][1]*t.data[2][2]-t.data[1][2]*t.data[2][1])*t07;
tmp.data[0][1] = -(t.data[0][1]*t.data[2][2]-t.data[0][2]*t.data[2][1])*t07;
tmp.data[0][2] = -(-t.data[0][1]*t.data[1][2]+t.data[0][2]*t.data[1][1])*t07;
tmp.data[1][0] = -(t.data[1][0]*t.data[2][2]-t.data[1][2]*t.data[2][0])*t07;
tmp.data[1][1] = (t.data[0][0]*t.data[2][2]-t04)*t07;
tmp.data[1][2] = -(t6-t00)*t07;
tmp.data[2][0] = -(-t.data[1][0]*t.data[2][1]+t.data[1][1]*t.data[2][0])*t07;
tmp.data[2][1] = -(t.data[0][0]*t.data[2][1]-t01)*t07;
tmp.data[2][2] = (t4-t8)*t07;
// scale last row and column as mentioned
// above
tmp.data[2][0] /= 2;
tmp.data[2][1] /= 2;
tmp.data[0][2] /= 2;
tmp.data[1][2] /= 2;
tmp.data[2][2] /= 4;
return tmp;
}
/**
* Invert a symmetric rank-4 tensor. Since symmetric rank-4 tensors are
* mappings from and to symmetric rank-2 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.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <>
SymmetricTensor<4,3>
invert (const SymmetricTensor<4,3> &t);
// this function is implemented in the .cc file
/**
* 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
* <tt>T phi = t1 (t2 : phi)</tt> for all symmetric tensors <tt>phi</tt>.
*
* For example, the deviator tensor can be computed as
* <tt>identity_tensor<dim>() -
* 1/d*outer_product(unit_symmetric_tensor<dim>(),
* unit_symmetric_tensor<dim>())</tt>, since the (double) contraction with the
* unit tensor yields the trace of a symmetric tensor.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
template <int dim>
inline
SymmetricTensor<4,dim>
outer_product (const SymmetricTensor<2,dim> &t1,
const SymmetricTensor<2,dim> &t2)
{
SymmetricTensor<4,dim> tmp;
// fill only the elements really needed
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=i; j<dim; ++j)
for (unsigned int k=0; k<dim; ++k)
for (unsigned int l=k; l<dim; ++l)
tmp[i][j][k][l] = t1[i][j] * t2[k][l];
return tmp;
}
/**
* Return the symmetrized version of a full rank-2 tensor,
* i.e. (t+transpose(t))/2, as a symmetric rank-2 tensor. This is the version
* for dim==1.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
inline
SymmetricTensor<2,1>
symmetrize (const Tensor<2,1> &t)
{
const double array[1]
= { t[0][0] };
return SymmetricTensor<2,1>(array);
}
/**
* Return the symmetrized version of a full rank-2 tensor,
* i.e. (t+transpose(t))/2, as a symmetric rank-2 tensor. This is the version
* for dim==2.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
inline
SymmetricTensor<2,2>
symmetrize (const Tensor<2,2> &t)
{
const double array[3]
= { t[0][0], t[1][1], (t[0][1] + t[1][0])/2 };
return SymmetricTensor<2,2>(array);
}
/**
* Return the symmetrized version of a full rank-2 tensor,
* i.e. (t+transpose(t))/2, as a symmetric rank-2 tensor. This is the version
* for dim==3.
*
* @relates SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
inline
SymmetricTensor<2,3>
symmetrize (const Tensor<2,3> &t)
{
const double array[6]
= { t[0][0], t[1][1], t[2][2],
(t[0][1] + t[1][0])/2,
(t[0][2] + t[2][0])/2,
(t[1][2] + t[2][1])/2 };
return SymmetricTensor<2,3>(array);
}
/**
* Multiplication of a symmetric tensor of general rank with a scalar double
* from the right.
*
* @relates SymmetricTensor
*/
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>
operator * (const SymmetricTensor<rank,dim> &t,
const double factor)
{
SymmetricTensor<rank,dim> tt = t;
tt *= factor;
return tt;
}
/**
* Multiplication of a symmetric tensor of general rank with a scalar double
* from the left.
*
* @relates SymmetricTensor
*/
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>
operator * (const double factor,
const SymmetricTensor<rank,dim> &t)
{
SymmetricTensor<rank,dim> tt = t;
tt *= factor;
return tt;
}
/**
* Division of a symmetric tensor of general rank by a scalar double.
*
* @relates SymmetricTensor
*/
template <int rank, int dim>
inline
SymmetricTensor<rank,dim>
operator / (const SymmetricTensor<rank,dim> &t,
const double factor)
{
SymmetricTensor<rank,dim> tt = t;
tt /= factor;
return tt;
}
/**
* Double contraction between a rank-4 and a rank-2 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 matrix-vector 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.
*
* @related SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
inline
void
double_contract (SymmetricTensor<2,1> &tmp,
const SymmetricTensor<4,1> &t,
const SymmetricTensor<2,1> &s)
{
tmp[0][0] = t[0][0][0][0] * s[0][0];
}
/**
* Double contraction between a rank-4 and a rank-2 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 matrix-vector 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.
*
* @related SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
inline
void
double_contract (SymmetricTensor<2,1> &tmp,
const SymmetricTensor<2,1> &s,
const SymmetricTensor<4,1> &t)
{
tmp[0][0] = t[0][0][0][0] * s[0][0];
}
/**
* Double contraction between a rank-4 and a rank-2 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 matrix-vector 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.
*
* @related SymmetricTensor @author Wolfgang Bangerth, 2005
*/
inline
void
double_contract (SymmetricTensor<2,2> &tmp,
const SymmetricTensor<4,2> &t,
const SymmetricTensor<2,2> &s)
{
const unsigned int dim = 2;
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=i; j<dim; ++j)
tmp[i][j] = t[i][j][0][0] * s[0][0] +
t[i][j][1][1] * s[1][1] +
2 * t[i][j][0][1] * s[0][1];
}
/**
* Double contraction between a rank-4 and a rank-2 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 matrix-vector 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.
*
* @related SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
inline
void
double_contract (SymmetricTensor<2,2> &tmp,
const SymmetricTensor<2,2> &s,
const SymmetricTensor<4,2> &t)
{
const unsigned int dim = 2;
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=i; j<dim; ++j)
tmp[i][j] = s[0][0] * t[0][0][i][j] * +
s[1][1] * t[1][1][i][j] +
2 * s[0][1] * t[0][1][i][j];
}
/**
* Double contraction between a rank-4 and a rank-2 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 matrix-vector 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.
*
* @related SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
inline
void
double_contract (SymmetricTensor<2,3> &tmp,
const SymmetricTensor<4,3> &t,
const SymmetricTensor<2,3> &s)
{
const unsigned int dim = 3;
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=i; j<dim; ++j)
tmp[i][j] = t[i][j][0][0] * s[0][0] +
t[i][j][1][1] * s[1][1] +
t[i][j][2][2] * s[2][2] +
2 * t[i][j][0][1] * s[0][1] +
2 * t[i][j][0][2] * s[0][2] +
2 * t[i][j][1][2] * s[1][2];
}
/**
* Double contraction between a rank-4 and a rank-2 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 matrix-vector 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.
*
* @related SymmetricTensor
* @author Wolfgang Bangerth, 2005
*/
inline
void
double_contract (SymmetricTensor<2,3> &tmp,
const SymmetricTensor<2,3> &s,
const SymmetricTensor<4,3> &t)
{
const unsigned int dim = 3;
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=i; j<dim; ++j)
tmp[i][j] = s[0][0] * t[0][0][i][j] +
s[1][1] * t[1][1][i][j] +
s[2][2] * t[2][2][i][j] +
2 * s[0][1] * t[0][1][i][j] +
2 * s[0][2] * t[0][2][i][j] +
2 * s[1][2] * t[1][2][i][j];
}
/**
* 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.
*
* @relates SymmetricTensor
*/
template <int dim>
inline
std::ostream & operator << (std::ostream &out,
const SymmetricTensor<2,dim> &t)
{
//make out lives a bit simpler by outputing
//the tensor through the operator for the
//general Tensor class
Tensor<2,dim> tt;
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=0; j<dim; ++j)
tt[i][j] = t[i][j];
return out << tt;
}
/**
* 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.
*
* @relates SymmetricTensor
*/
template <int dim>
inline
std::ostream & operator << (std::ostream &out,
const SymmetricTensor<4,dim> &t)
{
//make out lives a bit simpler by outputing
//the tensor through the operator for the
//general Tensor class
Tensor<4,dim> tt;
for (unsigned int i=0; i<dim; ++i)
for (unsigned int j=0; j<dim; ++j)
for (unsigned int k=0; k<dim; ++k)
for (unsigned int l=0; l<dim; ++l)
tt[i][j][k][l] = t[i][j][k][l];
return out << tt;
}
DEAL_II_NAMESPACE_CLOSE
#endif
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