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//
// Copyright (C) 1998 - 2016 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__tensor_h
#define dealii__tensor_h
#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/table_indices.h>
#include <deal.II/base/tensor_accessors.h>
#include <deal.II/base/template_constraints.h>
#include <deal.II/base/utilities.h>
#include <cmath>
#include <ostream>
#include <vector>
DEAL_II_NAMESPACE_OPEN
// Forward declarations:
template <int dim, typename Number> class Point;
template <int rank_, int dim, typename Number = double> class Tensor;
#ifndef DOXYGEN
// Overload invalid tensor types of negative rank that come up during
// overload resolution of operator* and related contraction variants.
template <int dim, typename Number>
class Tensor<-2, dim, Number>
{
};
template <int dim, typename Number>
class Tensor<-1, dim, Number>
{
};
#endif /* DOXYGEN */
/**
* This class is a specialized version of the <tt>Tensor<rank,dim,Number></tt>
* class. It handles tensors of rank zero, i.e. scalars. The second template
* argument @p dim is ignored.
*
* This class exists because in some cases we want to construct objects of
* type Tensor@<spacedim-dim,dim,Number@>, which should expand to scalars,
* vectors, matrices, etc, depending on the values of the template arguments
* @p dim and @p spacedim. We therefore need a class that acts as a scalar
* (i.e. @p Number) for all purposes but is part of the Tensor template
* family.
*
* @tparam dim An integer that denotes the dimension of the space in which
* this tensor operates. This of course equals the number of coordinates that
* identify a point and rank-1 tensor. Since the current object is a rank-0
* tensor (a scalar), this template argument has no meaning for this class.
*
* @tparam Number The data type in which the tensor elements are to be stored.
* This will, in almost all cases, simply be the default @p double, but there
* are cases where one may want to store elements in a different (and always
* scalar) type. It can be used to base tensors on @p float or @p complex
* numbers or any other data type that implements basic arithmetic operations.
* Another example would be a type that allows for Automatic Differentiation
* (see, for example, the Sacado type used in step-33) and thereby can
* generate analytic (spatial) derivatives of a function that takes a tensor
* as argument.
*
* @ingroup geomprimitives
* @author Wolfgang Bangerth, 2009, Matthias Maier, 2015
*/
template <int dim, typename Number>
class Tensor<0,dim,Number>
{
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;
/**
* Publish the rank of this tensor to the outside world.
*/
static const unsigned int rank = 0;
/**
* Number of independent components of a tensor of rank 0.
*/
static const unsigned int n_independent_components = 1;
/**
* Declare a type that has holds real-valued numbers with the same precision
* as the template argument to this class. For std::complex<number>, this
* corresponds to type number, and it is equal to Number for all other
* cases. See also the respective field in Vector<Number>.
*
* This typedef is used to represent the return type of norms.
*/
typedef typename numbers::NumberTraits<Number>::real_type real_type;
/**
* Type of objects encapsulated by this container and returned by
* operator[](). This is a scalar number type for a rank 0 tensor.
*/
typedef Number value_type;
/**
* Declare an array type which can be used to initialize an object of this
* type statically. In case of a a tensor of rank 0 this is just the scalar
* number type Number.
*/
typedef Number array_type;
/**
* Constructor. Set to zero.
*/
Tensor ();
/**
* Copy constructor.
*/
Tensor (const Tensor<0,dim,Number> &initializer);
/**
* Constructor from tensors with different underlying scalar type. This
* obviously requires that the @p OtherNumber type is convertible to @p
* Number.
*/
template <typename OtherNumber>
Tensor (const Tensor<0,dim,OtherNumber> &initializer);
/**
* Constructor, where the data is copied from a C-style array.
*/
template <typename OtherNumber>
Tensor (const OtherNumber initializer);
/**
* Return a reference to the encapsulated Number object. Since rank-0
* tensors are scalars, this is a natural operation.
*
* This is the non-const conversion operator that returns a writable
* reference.
*/
operator Number &();
/**
* Return a reference to the encapsulated Number object. Since rank-0
* tensors are scalars, this is a natural operation.
*
* This is the const conversion operator that returns a read-only reference.
*/
operator const Number &() const;
/**
* Copy assignment operator.
*/
Tensor<0,dim,Number> &operator = (const Tensor<0,dim,Number> &rhs);
/**
* Assignment from tensors with different underlying scalar type. This
* obviously requires that the @p OtherNumber type is convertible to @p
* Number.
*/
template <typename OtherNumber>
Tensor<0,dim,Number> &operator = (const Tensor<0,dim,OtherNumber> &rhs);
/**
* Test for equality of two tensors.
*/
template<typename OtherNumber>
bool operator == (const Tensor<0,dim,OtherNumber> &rhs) const;
/**
* Test for inequality of two tensors.
*/
template<typename OtherNumber>
bool operator != (const Tensor<0,dim,OtherNumber> &rhs) const;
/**
* Add another scalar
*/
template<typename OtherNumber>
Tensor<0,dim,Number> &operator += (const Tensor<0,dim,OtherNumber> &rhs);
/**
* Subtract another scalar.
*/
template<typename OtherNumber>
Tensor<0,dim,Number> &operator -= (const Tensor<0,dim,OtherNumber> &rhs);
/**
* Multiply the scalar with a <tt>factor</tt>.
*/
template<typename OtherNumber>
Tensor<0,dim,Number> &operator *= (const OtherNumber factor);
/**
* Divide the scalar by <tt>factor</tt>.
*/
template<typename OtherNumber>
Tensor<0,dim,Number> &operator /= (const OtherNumber factor);
/**
* Tensor with inverted entries.
*/
Tensor<0,dim,Number> operator - () const;
/**
* Reset all values to zero.
*
* Note that this is partly inconsistent with the semantics of the @p
* 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.
*/
void clear ();
/**
* Return the Frobenius-norm of a tensor, i.e. the square root of the sum of
* the absolute squares of all entries. For the present case of rank-1
* tensors, this equals the usual <tt>l<sub>2</sub></tt> norm of the vector.
*/
real_type norm () const;
/**
* Return the square of the Frobenius-norm of a tensor, i.e. the sum of the
* absolute squares of all entries.
*/
real_type norm_square () const;
/**
* Read or write the data of this object to or from a stream for the purpose
* of serialization
*/
template <class Archive>
void serialize(Archive &ar, const unsigned int version);
/**
* Internal type declaration that is used to specialize the return type of
* operator[]() for Tensor<1,dim,Number>
*/
typedef Number tensor_type;
private:
/**
* The value of this scalar object.
*/
Number value;
/**
* Internal helper function for unroll.
*/
template <typename OtherNumber>
void unroll_recursion(Vector<OtherNumber> &result,
unsigned int &start_index) const;
/**
* Allow an arbitrary Tensor to access the underlying values.
*/
template <int, int, typename> friend class Tensor;
};
/**
* A general tensor class with an arbitrary rank, i.e. with an arbitrary
* number of indices. The Tensor class provides an indexing operator and a bit
* of infrastructure, but most functionality is recursively handed down to
* tensors of rank 1 or put into external templated functions, e.g. the
* <tt>contract</tt> family.
*
* Using this tensor class for objects 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 also
* makes the code easier to read because of the semantic difference between a
* tensor (an object that relates to a coordinate system and has
* transformation properties with regard to coordinate rotations and
* transforms) and matrices (which we consider as operators on arbitrary
* vector spaces related to linear algebra things).
*
* @tparam rank_ An integer that denotes the rank of this tensor. A rank-0
* tensor is a scalar, a rank-1 tensor is a vector with @p dim components, a
* rank-2 tensor is a matrix with dim-by-dim components, etc. There are
* specializations of this class for rank-0 and rank-1 tensors. There is also
* a related class SymmetricTensor for tensors of even rank whose elements are
* symmetric.
*
* @tparam dim An integer that denotes the dimension of the space in which
* this tensor operates. This of course equals the number of coordinates that
* identify a point and rank-1 tensor.
*
* @tparam Number The data type in which the tensor elements are to be stored.
* This will, in almost all cases, simply be the default @p double, but there
* are cases where one may want to store elements in a different (and always
* scalar) type. It can be used to base tensors on @p float or @p complex
* numbers or any other data type that implements basic arithmetic operations.
* Another example would be a type that allows for Automatic Differentiation
* (see, for example, the Sacado type used in step-33) and thereby can
* generate analytic (spatial) derivatives of a function that takes a tensor
* as argument.
*
* @ingroup geomprimitives
* @author Wolfgang Bangerth, 1998-2005, Matthias Maier, 2015
*/
template <int rank_, int dim, typename Number>
class Tensor
{
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;
/**
* Publish the rank of this tensor to the outside world.
*/
static const unsigned int rank = rank_;
/**
* Number of independent components of a tensor of current rank. This is dim
* times the number of independent components of each sub-tensor.
*/
static const unsigned int
n_independent_components = Tensor<rank_-1,dim>::n_independent_components *dim;
/**
* Type of objects encapsulated by this container and returned by
* operator[](). This is a tensor of lower rank for a general tensor, and a
* scalar number type for Tensor<1,dim,Number>.
*/
typedef typename Tensor<rank_-1,dim,Number>::tensor_type value_type;
/**
* Declare an array type which can be used to initialize an object of this
* type statically.
*/
typedef typename Tensor<rank_-1,dim,Number>::array_type
array_type[(dim != 0) ? dim : 1];
// ... avoid a compiler warning in case of dim == 0 and ensure that the
// array always has positive size.
/**
* Constructor. Initialize all entries to zero.
*/
Tensor ();
/**
* Copy constructor.
*/
Tensor (const Tensor<rank_,dim,Number> &initializer);
/**
* Constructor, where the data is copied from a C-style array.
*/
Tensor (const array_type &initializer);
/**
* Constructor from tensors with different underlying scalar type. This
* obviously requires that the @p OtherNumber type is convertible to @p
* Number.
*/
template <typename OtherNumber>
Tensor (const Tensor<rank_,dim,OtherNumber> &initializer);
/**
* Constructor that converts from a "tensor of tensors".
*/
template <typename OtherNumber>
Tensor (const Tensor<1,dim,Tensor<rank_-1,dim,OtherNumber> > &initializer);
/**
* Conversion operator to tensor of tensors.
*/
template <typename OtherNumber>
operator Tensor<1,dim,Tensor<rank_-1,dim,OtherNumber> > () const;
/**
* Read-Write access operator.
*/
value_type &operator [] (const unsigned int i);
/**
* Read-only access operator.
*/
const value_type &operator[](const unsigned int i) const;
/**
* Read access using TableIndices <tt>indices</tt>
*/
const Number &operator [] (const TableIndices<rank_> &indices) const;
/**
* Read and write access using TableIndices <tt>indices</tt>
*/
Number &operator [] (const TableIndices<rank_> &indices);
/**
* Copy assignment operator.
*/
Tensor &operator = (const Tensor<rank_,dim,Number> &rhs);
/**
* Assignment operator from tensors with different underlying scalar type.
* This obviously requires that the @p OtherNumber type is convertible to @p
* Number.
*/
template <typename OtherNumber>
Tensor &operator = (const Tensor<rank_,dim,OtherNumber> &rhs);
/**
* 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.
*/
Tensor<rank_,dim,Number> &operator = (const Number d);
/**
* Test for equality of two tensors.
*/
template <typename OtherNumber>
bool operator == (const Tensor<rank_,dim,OtherNumber> &) const;
/**
* Test for inequality of two tensors.
*/
template <typename OtherNumber>
bool operator != (const Tensor<rank_,dim,OtherNumber> &) const;
/**
* Add another tensor.
*/
template <typename OtherNumber>
Tensor<rank_,dim,Number> &operator += (const Tensor<rank_,dim,OtherNumber> &);
/**
* Subtract another tensor.
*/
template <typename OtherNumber>
Tensor<rank_,dim,Number> &operator -= (const Tensor<rank_,dim,OtherNumber> &);
/**
* Scale the tensor by <tt>factor</tt>, i.e. multiply all components by
* <tt>factor</tt>.
*/
template <typename OtherNumber>
Tensor<rank_,dim,Number> &operator *= (const OtherNumber factor);
/**
* Scale the vector by <tt>1/factor</tt>.
*/
template <typename OtherNumber>
Tensor<rank_,dim,Number> &operator /= (const OtherNumber factor);
/**
* Unary minus operator. Negate all entries of a tensor.
*/
Tensor<rank_,dim,Number> operator - () const;
/**
* Reset all values to zero.
*
* Note that this is partly inconsistent with the semantics of the @p
* 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.
*/
void clear ();
/**
* Return the Frobenius-norm of a tensor, i.e. the square root of the sum of
* the absolute squares of all entries. For the present case of rank-1
* tensors, this equals the usual <tt>l<sub>2</sub></tt> norm of the vector.
*/
typename numbers::NumberTraits<Number>::real_type norm() const;
/**
* Return the square of the Frobenius-norm of a tensor, i.e. the sum of the
* absolute squares of all entries.
*/
typename numbers::NumberTraits<Number>::real_type norm_square() const;
/**
* Fill a vector with all tensor elements.
*
* This function unrolls all tensor entries into a single, linearly numbered
* vector. As usual in C++, the rightmost index of the tensor marches
* fastest.
*/
template <typename OtherNumber>
void unroll (Vector<OtherNumber> &result) const;
/**
* Returns an unrolled index in the range [0,dim^rank-1] for the element of
* the tensor indexed by the argument to the function.
*/
static
unsigned int
component_to_unrolled_index(const TableIndices<rank_> &indices);
/**
* Opposite of component_to_unrolled_index: For an index in the range
* [0,dim^rank-1], return which set of indices it would correspond to.
*/
static
TableIndices<rank_> unrolled_to_component_indices(const unsigned int i);
/**
* Determine an estimate for the memory consumption (in bytes) of this
* object.
*/
static std::size_t memory_consumption ();
/**
* Read or write the data of this object to or from a stream for the purpose
* of serialization
*/
template <class Archive>
void serialize(Archive &ar, const unsigned int version);
/**
* Internal type declaration that is used to specialize the return type of
* operator[]() for Tensor<1,dim,Number>
*/
typedef Tensor<rank_, dim, Number> tensor_type;
private:
/**
* Array of tensors holding the subelements.
*/
Tensor<rank_-1, dim, Number> values[(dim != 0) ? dim : 1];
// ... avoid a compiler warning in case of dim == 0 and ensure that the
// array always has positive size.
/**
* Internal helper function for unroll.
*/
template <typename OtherNumber>
void unroll_recursion(Vector<OtherNumber> &result,
unsigned int &start_index) const;
/**
* Allow an arbitrary Tensor to access the underlying values.
*/
template <int, int, typename> friend class Tensor;
/**
* Point is allowed access to the coordinates. This is supposed to improve
* speed.
*/
friend class Point<dim,Number>;
};
/*---------------------- Inline functions: Tensor<0,dim> ---------------------*/
template <int dim,typename Number>
inline
Tensor<0,dim,Number>::Tensor ()
: value()
{
}
template <int dim, typename Number>
inline
Tensor<0,dim,Number>::Tensor (const Tensor<0,dim,Number> &p)
{
value = p.value;
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number>::Tensor (const OtherNumber initializer)
{
value = initializer;
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number>::Tensor (const Tensor<0,dim,OtherNumber> &p)
{
value = p.value;
}
template <int dim, typename Number>
inline
Tensor<0,dim,Number>::operator Number &()
{
Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
return value;
}
template <int dim, typename Number>
inline
Tensor<0,dim,Number>::operator const Number &() const
{
Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
return value;
}
template <int dim, typename Number>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator = (const Tensor<0,dim,Number> &p)
{
value = p.value;
return *this;
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator = (const Tensor<0,dim,OtherNumber> &p)
{
value = p.value;
return *this;
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
bool Tensor<0,dim,Number>::operator == (const Tensor<0,dim,OtherNumber> &p) const
{
return (value == p.value);
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
bool Tensor<0,dim,Number>::operator != (const Tensor<0,dim,OtherNumber> &p) const
{
return !((*this) == p);
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator += (const Tensor<0,dim,OtherNumber> &p)
{
value += p.value;
return *this;
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator -= (const Tensor<0,dim,OtherNumber> &p)
{
value -= p.value;
return *this;
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator *= (const OtherNumber s)
{
value *= s;
return *this;
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator /= (const OtherNumber s)
{
value /= s;
return *this;
}
template <int dim, typename Number>
inline
Tensor<0,dim,Number> Tensor<0,dim,Number>::operator - () const
{
return -value;
}
template <int dim, typename Number>
inline
typename Tensor<0,dim,Number>::real_type
Tensor<0,dim,Number>::norm () const
{
Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
return numbers::NumberTraits<Number>::abs (value);
}
template <int dim, typename Number>
inline
typename Tensor<0,dim,Number>::real_type
Tensor<0,dim,Number>::norm_square () const
{
Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
return numbers::NumberTraits<Number>::abs_square (value);
}
template <int dim, typename Number>
template <typename OtherNumber>
inline
void
Tensor<0, dim, Number>::unroll_recursion (Vector<OtherNumber> &result,
unsigned int &index) const
{
Assert(dim != 0, ExcMessage("Cannot unroll an object of type Tensor<0,0,Number>"));
result[index] = value;
++index;
}
template <int dim, typename Number>
inline
void Tensor<0,dim,Number>::clear ()
{
value = value_type();
}
template <int dim, typename Number>
template <class Archive>
inline
void Tensor<0,dim,Number>::serialize(Archive &ar, const unsigned int)
{
ar &value;
}
/*-------------------- Inline functions: Tensor<rank,dim> --------------------*/
template <int rank_, int dim, typename Number>
inline
Tensor<rank_,dim,Number>::Tensor ()
{
// All members of the c-style array values are already default initialized
// and thus all values are already set to zero recursively.
}
template <int rank_, int dim, typename Number>
inline
Tensor<rank_,dim,Number>::Tensor (const Tensor<rank_,dim,Number> &initializer)
{
if (dim > 0)
std::copy (&initializer[0], &initializer[0]+dim, &values[0]);
}
template <int rank_, int dim, typename Number>
inline
Tensor<rank_,dim,Number>::Tensor (const array_type &initializer)
{
for (unsigned int i=0; i<dim; ++i)
values[i] = initializer[i];
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number>::Tensor (const Tensor<rank_,dim,OtherNumber> &initializer)
{
for (unsigned int i=0; i!=dim; ++i)
values[i] = initializer[i];
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number>::Tensor
(const Tensor<1,dim,Tensor<rank_-1,dim,OtherNumber> > &initializer)
{
for (unsigned int i=0; i<dim; ++i)
values[i] = initializer[i];
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number>::
operator Tensor<1,dim,Tensor<rank_-1,dim,OtherNumber> > () const
{
return Tensor<1,dim,Tensor<rank_-1,dim,Number> > (values);
}
namespace internal
{
namespace TensorSubscriptor
{
template <typename ArrayElementType, int dim>
inline DEAL_II_ALWAYS_INLINE
ArrayElementType &
subscript (ArrayElementType *values,
const unsigned int i,
dealii::internal::int2type<dim>)
{
Assert (i<dim, ExcIndexRange(i, 0, dim));
return values[i];
}
template <typename ArrayElementType>
ArrayElementType &
subscript (ArrayElementType *,
const unsigned int,
dealii::internal::int2type<0>)
{
Assert(false, ExcMessage("Cannot access elements of an object of type Tensor<rank,0,Number>."));
static ArrayElementType t;
return t;
}
}
}
template <int rank_, int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
typename Tensor<rank_,dim,Number>::value_type &
Tensor<rank_,dim,Number>::operator[] (const unsigned int i)
{
return dealii::internal::TensorSubscriptor::subscript(values, i, dealii::internal::int2type<dim>());
}
template <int rank_, int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
const typename Tensor<rank_,dim,Number>::value_type &
Tensor<rank_,dim,Number>::operator[] (const unsigned int i) const
{
return dealii::internal::TensorSubscriptor::subscript(values, i, dealii::internal::int2type<dim>());
}
template <int rank_, int dim, typename Number>
inline
const Number &
Tensor<rank_,dim,Number>::operator[] (const TableIndices<rank_> &indices) const
{
Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<rank_,0,Number>"));
return TensorAccessors::extract<rank_>(*this, indices);
}
template <int rank_, int dim, typename Number>
inline
Number &
Tensor<rank_,dim,Number>::operator[] (const TableIndices<rank_> &indices)
{
Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<rank_,0,Number>"));
return TensorAccessors::extract<rank_>(*this, indices);
}
template <int rank_, int dim, typename Number>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator = (const Tensor<rank_,dim,Number> &t)
{
if (dim > 0)
std::copy (&t.values[0], &t.values[0]+dim, &values[0]);
return *this;
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator = (const Tensor<rank_,dim,OtherNumber> &t)
{
if (dim > 0)
std::copy (&t.values[0], &t.values[0]+dim, &values[0]);
return *this;
}
template <int rank_, int dim, typename Number>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator = (const Number d)
{
Assert (d == Number(), ExcMessage ("Only assignment with zero is allowed"));
(void) d;
for (unsigned int i=0; i<dim; ++i)
values[i] = Number();
return *this;
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
bool
Tensor<rank_,dim,Number>::operator == (const Tensor<rank_,dim,OtherNumber> &p) const
{
for (unsigned int i=0; i<dim; ++i)
if (values[i] != p.values[i])
return false;
return true;
}
// At some places in the library, we have Point<0> for formal reasons
// (e.g., we sometimes have Quadrature<dim-1> for faces, so we have
// Quadrature<0> for dim=1, and then we have Point<0>). To avoid warnings
// in the above function that the loop end check always fails, we
// implement this function here
template <>
template <>
inline
bool Tensor<1,0,double>::operator == (const Tensor<1,0,double> &) const
{
return true;
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
bool
Tensor<rank_,dim,Number>::operator != (const Tensor<rank_,dim,OtherNumber> &p) const
{
return !((*this) == p);
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator += (const Tensor<rank_,dim,OtherNumber> &p)
{
for (unsigned int i=0; i<dim; ++i)
values[i] += p.values[i];
return *this;
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator -= (const Tensor<rank_,dim,OtherNumber> &p)
{
for (unsigned int i=0; i<dim; ++i)
values[i] -= p.values[i];
return *this;
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator *= (const OtherNumber s)
{
for (unsigned int i=0; i<dim; ++i)
values[i] *= s;
return *this;
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator /= (const OtherNumber s)
{
for (unsigned int i=0; i<dim; ++i)
values[i] /= s;
return *this;
}
template <int rank_, int dim, typename Number>
inline
Tensor<rank_,dim,Number>
Tensor<rank_,dim,Number>::operator - () const
{
Tensor<rank_,dim,Number> tmp;
for (unsigned int i=0; i<dim; ++i)
tmp.values[i] = -values[i];
return tmp;
}
template <int rank_, int dim, typename Number>
inline
typename numbers::NumberTraits<Number>::real_type
Tensor<rank_,dim,Number>::norm () const
{
return std::sqrt (norm_square());
}
template <int rank_, int dim, typename Number>
inline
typename numbers::NumberTraits<Number>::real_type
Tensor<rank_,dim,Number>::norm_square () const
{
typename numbers::NumberTraits<Number>::real_type s = typename numbers::NumberTraits<Number>::real_type();
for (unsigned int i=0; i<dim; ++i)
s += values[i].norm_square();
return s;
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
void
Tensor<rank_, dim, Number>::unroll (Vector<OtherNumber> &result) const
{
AssertDimension (result.size(),(Utilities::fixed_power<rank_, unsigned int>(dim)));
unsigned int index = 0;
unroll_recursion (result, index);
}
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
void
Tensor<rank_, dim, Number>::unroll_recursion (Vector<OtherNumber> &result,
unsigned int &index) const
{
for (unsigned int i=0; i<dim; ++i)
values[i].unroll_recursion(result, index);
}
template <int rank_, int dim, typename Number>
inline
unsigned int
Tensor<rank_, dim, Number>::component_to_unrolled_index(const TableIndices<rank_> &indices)
{
unsigned int index = 0;
for (int r = 0; r < rank_; ++r)
index = index * dim + indices[r];
return index;
}
template <int rank_, int dim, typename Number>
inline
TableIndices<rank_>
Tensor<rank_, dim, Number>::unrolled_to_component_indices(const unsigned int i)
{
Assert (i < n_independent_components,
ExcIndexRange (i, 0, n_independent_components));
TableIndices<rank_> indices;
unsigned int remainder = i;
for (int r=rank_-1; r>=0; --r)
{
indices[r] = (remainder % dim);
remainder /= dim;
}
Assert (remainder == 0, ExcInternalError());
return indices;
}
template <int rank_, int dim, typename Number>
inline
void Tensor<rank_,dim,Number>::clear ()
{
for (unsigned int i=0; i<dim; ++i)
values[i] = value_type();
}
template <int rank_, int dim, typename Number>
inline
std::size_t
Tensor<rank_,dim,Number>::memory_consumption ()
{
return sizeof(Tensor<rank_,dim,Number>);
}
template <int rank_, int dim, typename Number>
template <class Archive>
inline
void
Tensor<rank_,dim,Number>::serialize(Archive &ar, const unsigned int)
{
ar &values;
}
/* ----------------- Non-member functions operating on tensors. ------------ */
/**
* @name Output functions for Tensor objects
*/
//@{
/**
* Output operator for tensors. Print the elements consecutively, with a space
* in between, two spaces between rank 1 subtensors, three between rank 2 and
* so on.
*
* @relates Tensor
*/
template <int rank_, int dim, typename Number>
inline
std::ostream &operator << (std::ostream &out, const Tensor<rank_,dim,Number> &p)
{
for (unsigned int i = 0; i < dim; ++i)
{
out << p[i];
if (i != dim - 1)
out << ' ';
}
return out;
}
/**
* Output operator for tensors of rank 0. Since such tensors are scalars, we
* simply print this one value.
*
* @relates Tensor<0,dim,Number>
*/
template <int dim, typename Number>
inline
std::ostream &operator << (std::ostream &out, const Tensor<0,dim,Number> &p)
{
out << static_cast<const Number &>(p);
return out;
}
//@}
/**
* @name Vector space operations on Tensor objects:
*/
//@{
#ifndef DEAL_II_WITH_CXX11
template <typename T, typename U, int rank, int dim>
struct ProductType<T,Tensor<rank,dim,U> >
{
typedef Tensor<rank,dim,typename ProductType<T,U>::type> type;
};
template <typename T, typename U, int rank, int dim>
struct ProductType<Tensor<rank,dim,T>,U>
{
typedef Tensor<rank,dim,typename ProductType<T,U>::type> type;
};
#endif
/**
* Scalar multiplication of a tensor of rank 0 with an object from the left.
*
* This function unwraps the underlying @p Number stored in the Tensor and
* multiplies @p object with it.
*
* @relates Tensor<0,dim,Number>
*/
template <int dim, typename Number, typename Other>
inline
typename ProductType<Other, Number>::type
operator * (const Other object,
const Tensor<0,dim,Number> &t)
{
return object * static_cast<const Number &>(t);
}
/**
* Scalar multiplication of a tensor of rank 0 with an object from the right.
*
* This function unwraps the underlying @p Number stored in the Tensor and
* multiplies @p object with it.
*
* @relates Tensor<0,dim,Number>
*/
template <int dim, typename Number, typename Other>
inline
typename ProductType<Number, Other>::type
operator * (const Tensor<0,dim,Number> &t,
const Other object)
{
return static_cast<const Number &>(t) * object;
}
/**
* Scalar multiplication of two tensors of rank 0.
*
* This function unwraps the underlying objects of type @p Number and @p
* OtherNumber that are stored within the Tensor and multiplies them. It
* returns an unwrapped number of product type.
*
* @relates Tensor<0,dim,Number>
*/
template <int dim, typename Number, typename OtherNumber>
inline
typename ProductType<Number, OtherNumber>::type
operator * (const Tensor<0, dim, Number> &src1,
const Tensor<0, dim, OtherNumber> &src2)
{
return static_cast<const Number &>(src1) *
static_cast<const OtherNumber &>(src2);
}
/**
* Division of a tensor of rank 0 by a scalar number.
*
* @relates Tensor<0,dim,Number>
*/
template <int dim, typename Number, typename OtherNumber>
inline
Tensor<0,dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator / (const Tensor<0,dim,Number> &t,
const OtherNumber factor)
{
return static_cast<Number>(t) / factor;
}
/**
* Add two tensors of rank 0.
*
* @relates Tensor<0,dim,Number>
*/
template <int dim, typename Number, typename OtherNumber>
inline
Tensor<0, dim, typename ProductType<Number, OtherNumber>::type>
operator+ (const Tensor<0,dim,Number> &p, const Tensor<0,dim,OtherNumber> &q)
{
return static_cast<const Number &>(p) + static_cast<const OtherNumber &>(q);
}
/**
* Subtract two tensors of rank 0.
*
* @relates Tensor<0,dim,Number>
*/
template <int dim, typename Number, typename OtherNumber>
inline
Tensor<0, dim, typename ProductType<Number, OtherNumber>::type>
operator- (const Tensor<0,dim,Number> &p, const Tensor<0,dim,OtherNumber> &q)
{
return static_cast<const Number &>(p) - static_cast<const OtherNumber &>(q);
}
/**
* Multiplication of a tensor of general rank with a scalar number from the
* right.
*
* Only multiplication with a scalar number type (i.e., a floating point
* number, a complex floating point number, etc.) is allowed, see the
* documentation of EnableIfScalar for details.
*
* @relates Tensor
*/
template <int rank, int dim,
typename Number,
typename OtherNumber>
inline
Tensor<rank,dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator * (const Tensor<rank,dim,Number> &t,
const OtherNumber factor)
{
// recurse over the base objects
Tensor<rank,dim,typename ProductType<Number,OtherNumber>::type> tt;
for (unsigned int d=0; d<dim; ++d)
tt[d] = t[d] * factor;
return tt;
}
/**
* Multiplication of a tensor of general rank with a scalar number from the
* left.
*
* Only multiplication with a scalar number type (i.e., a floating point
* number, a complex floating point number, etc.) is allowed, see the
* documentation of EnableIfScalar for details.
*
* @relates Tensor
*/
template <int rank, int dim,
typename Number,
typename OtherNumber>
inline
Tensor<rank,dim,typename ProductType<typename EnableIfScalar<Number>::type, OtherNumber>::type>
operator * (const Number factor,
const Tensor<rank,dim,OtherNumber> &t)
{
// simply forward to the operator above
return t * factor;
}
/**
* Division of a tensor of general rank with a scalar number. See the
* discussion on operator*() above for more information about template
* arguments and the return type.
*
* @relates Tensor
*/
template <int rank, int dim,
typename Number,
typename OtherNumber>
inline
Tensor<rank,dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator / (const Tensor<rank,dim,Number> &t,
const OtherNumber factor)
{
// recurse over the base objects
Tensor<rank,dim,typename ProductType<Number,OtherNumber>::type> tt;
for (unsigned int d=0; d<dim; ++d)
tt[d] = t[d] / factor;
return tt;
}
/**
* Addition of two tensors of general rank.
*
* @tparam rank The rank of both tensors.
*
* @relates Tensor
*/
template <int rank, int dim, typename Number, typename OtherNumber>
inline
Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
operator+ (const Tensor<rank,dim,Number> &p, const Tensor<rank,dim,OtherNumber> &q)
{
Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tmp (p);
for (unsigned int i=0; i<dim; ++i)
tmp[i] += q[i];
return tmp;
}
/**
* Subtraction of two tensors of general rank.
*
* @tparam rank The rank of both tensors.
*
* @relates Tensor
*/
template <int rank, int dim, typename Number, typename OtherNumber>
inline
Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
operator- (const Tensor<rank,dim,Number> &p, const Tensor<rank,dim,OtherNumber> &q)
{
Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tmp (p);
for (unsigned int i=0; i<dim; ++i)
tmp[i] -= q[i];
return tmp;
}
//@}
/**
* @name Contraction operations and the outer product for tensor objects
*/
//@{
/**
* The dot product (single contraction) for tensors: Return a tensor of rank
* $(\text{rank}_1 + \text{rank}_2 - 2)$ that is the contraction of the last
* index of a tensor @p src1 of rank @p rank_1 with the first index of a
* tensor @p src2 of rank @p rank_2:
* @f[
* \text{result}_{i_1,..,i_{r1},j_1,..,j_{r2}}
* = \sum_{k}
* \text{left}_{i_1,..,i_{r1}, k}
* \text{right}_{k, j_1,..,j_{r2}}
* @f]
*
* @note For the Tensor class, the multiplication operator only performs a
* contraction over a single pair of indices. This is in contrast to the
* multiplication operator for SymmetricTensor, which does the double
* contraction.
*
* @note In case the contraction yields a tensor of rank 0 the scalar number
* is returned as an unwrapped number type.
*
* @relates Tensor
* @author Matthias Maier, 2015
*/
template <int rank_1, int rank_2, int dim,
typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
typename Tensor<rank_1 + rank_2 - 2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
operator * (const Tensor<rank_1, dim, Number> &src1,
const Tensor<rank_2, dim, OtherNumber> &src2)
{
typename Tensor<rank_1 + rank_2 - 2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type result;
TensorAccessors::internal::ReorderedIndexView<0, rank_2, const Tensor<rank_2, dim, OtherNumber> >
reordered = TensorAccessors::reordered_index_view<0, rank_2>(src2);
TensorAccessors::contract<1, rank_1, rank_2, dim>(result, src1, reordered);
return result;
}
/**
* Generic contraction of a pair of indices of two tensors of arbitrary rank:
* Return a tensor of rank $(\text{rank}_1 + \text{rank}_2 - 2)$ that is the
* contraction of index @p index_1 of a tensor @p src1 of rank @p rank_1 with
* the index @p index_2 of a tensor @p src2 of rank @p rank_2:
* @f[
* \text{result}_{i_1,..,i_{r1},j_1,..,j_{r2}}
* = \sum_{k}
* \text{left}_{i_1,..,k,..,i_{r1}}
* \text{right}_{j_1,..,k,..,j_{r2}}
* @f]
*
* If for example the first index (<code>index_1==0</code>) of a tensor
* <code>t1</code> shall be contracted with the third index
* (<code>index_2==2</code>) of a tensor <code>t2</code>, the invocation of
* this function is
* @code
* contract<0, 2>(t1, t2);
* @endcode
*
* @note The position of the index is counted from 0, i.e.,
* $0\le\text{index}_i<\text{range}_i$.
*
* @note In case the contraction yields a tensor of rank 0 the scalar number
* is returned as an unwrapped number type.
*
* @relates Tensor
* @author Matthias Maier, 2015
*/
template <int index_1, int index_2,
int rank_1, int rank_2, int dim,
typename Number, typename OtherNumber>
inline
typename Tensor<rank_1 + rank_2 - 2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
contract (const Tensor<rank_1, dim, Number> &src1,
const Tensor<rank_2, dim, OtherNumber> &src2)
{
Assert(0 <= index_1 && index_1 < rank_1,
ExcMessage("The specified index_1 must lie within the range [0,rank_1)"));
Assert(0 <= index_2 && index_2 < rank_2,
ExcMessage("The specified index_2 must lie within the range [0,rank_2)"));
using namespace TensorAccessors;
using namespace TensorAccessors::internal;
// Reorder index_1 to the end of src1:
ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number> >
reord_01 = reordered_index_view<index_1, rank_1>(src1);
// Reorder index_2 to the end of src2:
ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber> >
reord_02 = reordered_index_view<index_2, rank_2>(src2);
typename Tensor<rank_1 + rank_2 - 2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
result;
TensorAccessors::contract<1, rank_1, rank_2, dim>(result, reord_01, reord_02);
return result;
}
/**
* Generic contraction of two pairs of indices of two tensors of arbitrary
* rank: Return a tensor of rank $(\text{rank}_1 + \text{rank}_2 - 4)$ that is
* the contraction of index @p index_1 with index @p index_2, and index @p
* index_3 with index @p index_4 of a tensor @p src1 of rank @p rank_1 and a
* tensor @p src2 of rank @p rank_2:
* @f[
* \text{result}_{i_1,..,i_{r1},j_1,..,j_{r2}}
* = \sum_{k, l}
* \text{left}_{i_1,..,k,..,l,..,i_{r1}}
* \text{right}_{j_1,..,k,..,l..,j_{r2}}
* @f]
*
* If for example the first index (<code>index_1==0</code>) shall be
* contracted with the third index (<code>index_2==2</code>), and the second
* index (<code>index_3==1</code>) with the first index
* (<code>index_4==0</code>) the invocation of this function is this function
* is
* @code
* contract<0, 2, 1, 0>(t1, t2);
* @endcode
*
* @note The position of the index is counted from 0, i.e.,
* $0\le\text{index}_i<\text{range}_i$.
*
* @note In case the contraction yields a tensor of rank 0 the scalar number
* is returned as an unwrapped number type.
*
* @relates Tensor
* @author Matthias Maier, 2015
*/
template <int index_1, int index_2, int index_3, int index_4,
int rank_1, int rank_2, int dim,
typename Number, typename OtherNumber>
inline
typename Tensor<rank_1 + rank_2 - 4, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
double_contract (const Tensor<rank_1, dim, Number> &src1,
const Tensor<rank_2, dim, OtherNumber> &src2)
{
Assert(0 <= index_1 && index_1 < rank_1,
ExcMessage("The specified index_1 must lie within the range [0,rank_1)"));
Assert(0 <= index_3 && index_3 < rank_1,
ExcMessage("The specified index_3 must lie within the range [0,rank_1)"));
Assert(index_1 != index_3,
ExcMessage("index_1 and index_3 must not be the same"));
Assert(0 <= index_2 && index_2 < rank_2,
ExcMessage("The specified index_2 must lie within the range [0,rank_2)"));
Assert(0 <= index_4 && index_4 < rank_2,
ExcMessage("The specified index_4 must lie within the range [0,rank_2)"));
Assert(index_2 != index_4,
ExcMessage("index_2 and index_4 must not be the same"));
using namespace TensorAccessors;
using namespace TensorAccessors::internal;
// Reorder index_1 to the end of src1:
ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number> >
reord_1 = TensorAccessors::reordered_index_view<index_1, rank_1>(src1);
// Reorder index_2 to the end of src2:
ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber> >
reord_2 = TensorAccessors::reordered_index_view<index_2, rank_2>(src2);
// Now, reorder index_3 to the end of src1. We have to make sure to
// preserve the orginial ordering: index_1 has been removed. If
// index_3 > index_1, we have to use (index_3 - 1) instead:
ReorderedIndexView<(index_3 < index_1 ? index_3 : index_3 - 1), rank_1, ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number> > >
reord_3 = TensorAccessors::reordered_index_view<index_3 < index_1 ? index_3 : index_3 - 1, rank_1>(reord_1);
// Now, reorder index_4 to the end of src2. We have to make sure to
// preserve the orginial ordering: index_2 has been removed. If
// index_4 > index_2, we have to use (index_4 - 1) instead:
ReorderedIndexView<(index_4 < index_2 ? index_4 : index_4 - 1), rank_2, ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber> > >
reord_4 = TensorAccessors::reordered_index_view<index_4 < index_2 ? index_4 : index_4 - 1, rank_2>(reord_2);
typename Tensor<rank_1 + rank_2 - 4, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
result;
TensorAccessors::contract<2, rank_1, rank_2, dim>(result, reord_3, reord_4);
return result;
}
/**
* The scalar product, or (generalized) Frobenius inner product of two tensors
* of equal rank: Return a scalar number that is the result of a full
* contraction of a tensor @p left and @p right:
* @f[
* \sum_{i_1,..,i_r}
* \text{left}_{i_1,..,i_r}
* \text{right}_{i_1,..,i_r}
* @f]
*
* @relates Tensor
* @author Matthias Maier, 2015
*/
template <int rank, int dim, typename Number, typename OtherNumber>
inline
typename ProductType<Number, OtherNumber>::type
scalar_product (const Tensor<rank, dim, Number> &left,
const Tensor<rank, dim, OtherNumber> &right)
{
typename ProductType<Number, OtherNumber>::type result;
TensorAccessors::contract<rank, rank, rank, dim>(result, left, right);
return result;
}
/**
* Full contraction of three tensors: Return a scalar number that is the
* result of a full contraction of a tensor @p left of rank @p rank_1, a
* tensor @p middle of rank $(\text{rank}_1+\text{rank}_2)$ and a tensor @p
* right of rank @p rank_2:
* @f[
* \sum_{i_1,..,i_{r1},j_1,..,j_{r2}}
* \text{left}_{i_1,..,i_{r1}}
* \text{middle}_{i_1,..,i_{r1},j_1,..,j_{r2}}
* \text{right}_{j_1,..,j_{r2}}
* @f]
*
* @relates Tensor
* @author Matthias Maier, 2015
*/
template <int rank_1, int rank_2, int dim,
typename T1, typename T2, typename T3>
typename ProductType<T1, typename ProductType<T2, T3>::type>::type
contract3 (const Tensor<rank_1, dim, T1> &left,
const Tensor<rank_1 + rank_2, dim, T2> &middle,
const Tensor<rank_2, dim, T3> &right)
{
typedef typename ProductType<T1, typename ProductType<T2, T3>::type>::type
return_type;
return TensorAccessors::contract3<rank_1, rank_2, dim, return_type>(
left, middle, right);
}
/**
* The outer product of two tensors of @p rank_1 and @p rank_2: Returns a
* tensor of rank $(\text{rank}_1 + \text{rank}_2)$:
* @f[
* \text{result}_{i_1,..,i_{r1},j_1,..,j_{r2}}
* = \text{left}_{i_1,..,i_{r1}}\,\text{right}_{j_1,..,j_{r2}.}
* @f]
*
* @relates Tensor
* @author Matthias Maier, 2015
*/
template <int rank_1, int rank_2, int dim,
typename Number, typename OtherNumber>
inline
Tensor<rank_1 + rank_2, dim, typename ProductType<Number, OtherNumber>::type>
outer_product(const Tensor<rank_1, dim, Number> &src1,
const Tensor<rank_2, dim, OtherNumber> &src2)
{
typename Tensor<rank_1 + rank_2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type result;
TensorAccessors::contract<0, rank_1, rank_2, dim>(result, src1, src2);
return result;
}
//@}
/**
* @name Special operations on tensors of rank 1
*/
//@{
/**
* Returns the cross product in 2d. This is just a rotation by 90 degrees
* clockwise to compute the outer normal from a tangential vector. This
* function is defined for all space dimensions to allow for dimension
* independent programming (e.g. within switches over the space dimension),
* but may only be called if the actual dimension of the arguments is two
* (e.g. from the <tt>dim==2</tt> case in the switch).
*
* @relates Tensor
* @author Guido Kanschat, 2001
*/
template <int dim, typename Number>
inline
Tensor<1,dim,Number>
cross_product_2d (const Tensor<1,dim,Number> &src)
{
Assert (dim==2, ExcInternalError());
Tensor<1, dim, Number> result;
result[0] = src[1];
result[1] = -src[0];
return result;
}
/**
* Returns the cross product of 2 vectors in 3d. This function is defined for
* all space dimensions to allow for dimension independent programming (e.g.
* within switches over the space dimension), but may only be called if the
* actual dimension of the arguments is three (e.g. from the <tt>dim==3</tt>
* case in the switch).
*
* @relates Tensor
* @author Guido Kanschat, 2001
*/
template <int dim, typename Number>
inline
Tensor<1,dim,Number>
cross_product_3d (const Tensor<1,dim,Number> &src1,
const Tensor<1,dim,Number> &src2)
{
Assert (dim==3, ExcInternalError());
Tensor<1, dim, Number> result;
result[0] = src1[1]*src2[2] - src1[2]*src2[1];
result[1] = src1[2]*src2[0] - src1[0]*src2[2];
result[2] = src1[0]*src2[1] - src1[1]*src2[0];
return result;
}
//@}
/**
* @name Special operations on tensors of rank 2
*/
//@{
/**
* Compute the determinant of a tensor or rank 2.
*
* @relates Tensor
* @author Wolfgang Bangerth, 2009
*/
template <int dim, typename Number>
inline
Number determinant (const Tensor<2,dim,Number> &t)
{
// Compute the determinant using the Laplace expansion of the
// determinant. We expand along the last row.
Number det = Number();
for (unsigned int k=0; k<dim; ++k)
{
Tensor<2,dim-1,Number> minor;
for (unsigned int i=0; i<dim-1; ++i)
for (unsigned int j=0; j<dim-1; ++j)
minor[i][j] = t[i][j<k ? j : j+1];
const Number cofactor = ((k % 2 == 0) ? -1. : 1.) * determinant(minor);
det += t[dim-1][k] * cofactor;
}
return ((dim % 2 == 0) ? 1. : -1.) * det;
}
/**
* Specialization for dim==1.
*
* @relates Tensor
*/
template <typename Number>
inline
Number determinant (const Tensor<2,1,Number> &t)
{
return t[0][0];
}
/**
* Compute and return the trace of a tensor of rank 2, i.e. the sum of its
* diagonal entries.
*
* @relates Tensor
* @author Wolfgang Bangerth, 2001
*/
template <int dim, typename Number>
Number trace (const Tensor<2,dim,Number> &d)
{
Number t=d[0][0];
for (unsigned int i=1; i<dim; ++i)
t += d[i][i];
return t;
}
/**
* Compute and return the inverse of the given tensor. Since the compiler can
* perform the return value optimization, and since the size of the return
* object is known, it is acceptable to return the result by value, rather
* than by reference as a parameter.
*
* @relates Tensor
* @author Wolfgang Bangerth, 2000
*/
template <int dim, typename Number>
inline
Tensor<2,dim,Number>
invert (const Tensor<2,dim,Number> &t)
{
Number return_tensor [dim][dim];
switch (dim)
{
case 1:
return_tensor[0][0] = 1.0/t[0][0];
break;
case 2:
// this is Maple output,
// thus a bit unstructured
{
const Number det = t[0][0]*t[1][1]-t[1][0]*t[0][1];
const Number t4 = 1.0/det;
return_tensor[0][0] = t[1][1]*t4;
return_tensor[0][1] = -t[0][1]*t4;
return_tensor[1][0] = -t[1][0]*t4;
return_tensor[1][1] = t[0][0]*t4;
break;
}
case 3:
{
const Number t4 = t[0][0]*t[1][1],
t6 = t[0][0]*t[1][2],
t8 = t[0][1]*t[1][0],
t00 = t[0][2]*t[1][0],
t01 = t[0][1]*t[2][0],
t04 = t[0][2]*t[2][0],
det = (t4*t[2][2]-t6*t[2][1]-t8*t[2][2]+
t00*t[2][1]+t01*t[1][2]-t04*t[1][1]),
t07 = 1.0/det;
return_tensor[0][0] = (t[1][1]*t[2][2]-t[1][2]*t[2][1])*t07;
return_tensor[0][1] = (t[0][2]*t[2][1]-t[0][1]*t[2][2])*t07;
return_tensor[0][2] = (t[0][1]*t[1][2]-t[0][2]*t[1][1])*t07;
return_tensor[1][0] = (t[1][2]*t[2][0]-t[1][0]*t[2][2])*t07;
return_tensor[1][1] = (t[0][0]*t[2][2]-t04)*t07;
return_tensor[1][2] = (t00-t6)*t07;
return_tensor[2][0] = (t[1][0]*t[2][1]-t[1][1]*t[2][0])*t07;
return_tensor[2][1] = (t01-t[0][0]*t[2][1])*t07;
return_tensor[2][2] = (t4-t8)*t07;
break;
}
// if desired, take over the
// inversion of a 4x4 tensor
// from the FullMatrix
default:
AssertThrow (false, ExcNotImplemented());
}
return Tensor<2,dim,Number>(return_tensor);
}
/**
* Return the transpose of the given tensor.
*
* @relates Tensor
* @author Wolfgang Bangerth, 2002
*/
template <int dim, typename Number>
inline
Tensor<2,dim,Number>
transpose (const Tensor<2,dim,Number> &t)
{
Tensor<2, dim, Number> tt;
for (unsigned int i=0; i<dim; ++i)
{
tt[i][i] = t[i][i];
for (unsigned int j=i+1; j<dim; ++j)
{
tt[i][j] = t[j][i];
tt[j][i] = t[i][j];
};
}
return tt;
}
/**
* Return the $l_1$ norm of the given rank-2 tensor, where $||t||_1 = \max_j
* \sum_i |t_{ij}|$ (maximum of the sums over columns).
*
* @relates Tensor
* @author Wolfgang Bangerth, 2012
*/
template <int dim, typename Number>
inline
double
l1_norm (const Tensor<2,dim,Number> &t)
{
double max = 0;
for (unsigned int j=0; j<dim; ++j)
{
double sum = 0;
for (unsigned int i=0; i<dim; ++i)
sum += std::fabs(t[i][j]);
if (sum > max)
max = sum;
}
return max;
}
/**
* Return the $l_\infty$ norm of the given rank-2 tensor, where $||t||_\infty
* = \max_i \sum_j |t_{ij}|$ (maximum of the sums over rows).
*
* @relates Tensor
* @author Wolfgang Bangerth, 2012
*/
template <int dim, typename Number>
inline
double
linfty_norm (const Tensor<2,dim,Number> &t)
{
double max = 0;
for (unsigned int i=0; i<dim; ++i)
{
double sum = 0;
for (unsigned int j=0; j<dim; ++j)
sum += std::fabs(t[i][j]);
if (sum > max)
max = sum;
}
return max;
}
//@}
DEAL_II_NAMESPACE_CLOSE
// include deprecated non-member functions operating on Tensor
#include <deal.II/base/tensor_deprecated.h>
#endif
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