tensor4

d-dimensional physical fourth-order tensor

Description

The tensor4 class defines a d^4 array with floating coefficients. This class is suitable for defining fourth-order tensors, i.e. field with d^4 matrix values at each physical position.

It is represented as a fourth-dimensional array of coordinates. The coordinate indexes start at zero and finishes at d-1, e.g. a(0,0,0,0), a(0,0,0,1), ..., a(2,2,2,2).

The default constructor set all components to zero:

    tensor4 a;

The standard linear algebra is supported.

Implementation

This documentation has been generated from file fem/geo_element/tensor4.h

The tensor4 class is simply an alias to the tensor4_basic class

typedef tensor4_basic<Float> tensor4;

The tensor4_basic class is a template class with the floating type as parameter:

template<class T>
class tensor4_basic {
public:
 
  typedef size_t size_type;
  typedef T      element_type;
  typedef T      float_type;
 
// allocators:
 
  tensor4_basic ();
  explicit tensor4_basic (const T& init_val);
  tensor4_basic (const tensor4_basic<T>& a);
  static tensor4_basic<T> eye (size_type d = 3);
 
  tensor4_basic (const std::initializer_list<std::initializer_list<
                       std::initializer_list<std::initializer_list<T> > > >& il);
 
// affectation:
 
  tensor4_basic<T>& operator= (const tensor4_basic<T>& a);
  tensor4_basic<T>& operator= (const T& val);
 
// accessors:
 
  T&       operator()(size_type i, size_type j, size_type k, size_type l);
  const T& operator()(size_type i, size_type j, size_type k, size_type l) const;
 
  tensor_basic<T>&  operator()(size_type i, size_type j);
  const tensor_basic<T>& operator()(size_type i, size_type j) const;
 
// algebra:
 
  tensor4_basic<T>  operator* (const T& k) const;
  tensor4_basic<T>  operator/ (const T& k) const;
  tensor4_basic<T>  operator+ (const tensor4_basic<T>& b) const;
  tensor4_basic<T>  operator- (const tensor4_basic<T>& b) const;
  tensor4_basic<T>& operator+= (const tensor4_basic<T>&);
  tensor4_basic<T>& operator-= (const tensor4_basic<T>&);
  tensor4_basic<T>& operator*= (const T& k);
  tensor4_basic<T>& operator/= (const T& k) { return operator*= (1./k); }
 
// io:
  std::ostream& put (std::ostream& out, size_type d=3) const;

};

The norm and contracted product with a second-order tensor is provided, together with the dexp fuinction, that represents the derivative of the tensor matrix function.

template <class T>
T norm (const tensor4_basic<T>& a) { return sqrt(norm2(a)); }
 
template <class T>
T norm2 (const tensor4_basic<T>&);
 
template <class T>
tensor_basic<T> ddot (const tensor4_basic<T>&, const tensor_basic<T>&);
 
template <class T>
tensor_basic<T> ddot (const tensor_basic<T>&, const tensor4_basic<T>&);
 
template <class T>
tensor4_basic<T> dexp (const tensor_basic<T>& a, size_t d = 3);