Rheolef  7.2
an efficient C++ finite element environment
 
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<tt>field</tt>

piecewise polynomial finite element function

Description

This class represents a piecewise polynomial finite element function. Since this function spans onto a finite dimensional basis, it simply stores its coefficients on this basis: these coefficients are called the degrees of freedom (see space).

Interpolation

For any function u, its interpolation on the finite element space Xh as a field uh in Xh expresses simply via the interpolate function:

    Float u (const point& x) { return exp(x[0]*x[1]); }
    ...
    field uh = interpolate (Xh, u);

Linear algebra

Linear algebra, such as uh+vh, uh-vh and lambda*uh + mu*vh, where lambda and mu are of type Float, are supported. The duality product between two fields lh and uh writes simply dual(lh,uh). As we consider finite dimensional spaces, this duality product coincides with the usual Euclidean dot product in IR^dim(Xh). The application of the a bilinear form writes a(uh,vh) and is equivalent to dual(m*uh,vh).

Common accessors

For convenience, uh.max(), uh.min() and uh.max_abs() returns respectively the maximum, minimum and maximum of the absolute value of the degrees of freedom.

Non-linear algebra

Non-linear operations, such as sqrt(uh) or 1/uh are also available. Note that non-linear operations do not returns in general piecewise polynomials: the value returned by sqrt(uh) may be filtered by interpolate function:

    field vh = interpolate (Xh, sqrt(uh));

Also, the multiplication uh*vh and the division uh/vh returns a result that is not in the same discrete finite element space: its result also may be filtered by interpolate`:

    field wh = interpolate(Xh, uh*vh);

All standard unary and binary math functions abs, cos, sin... are extended to scalar fields. Also sqr(uh), the square of a field, and min(uh,vh), max(uh,vh) are provided. Binary functions can be used also with a scalar, as in

    field vh = interpolate (Xh, max (abs(uh), 0));
    field wh = interpolate (Xh, pow (abs(uh), 1./3));

For applying a user-provided function to a field, please see the compose function.

Access by domain

The restriction of a field to a geometric domain, says "boundary" writes uh["boundary"]: it represents the trace of the field on the boundary:

    space Xh (omega, "P1");
    uh["boundary"] = 0;

A possible alternative uses a geo domain as index:

    geo boundary = omega["boundary"];
    uh[boundary] = 0;

Multi-valued fields

A vector-valued field contains several components, as:

    space Xh (omega, "P2", "vector");
    field uh (Xh);

Conversely, for a tensor-valued field:

    space Th (omega, "P1d", "tensor");
    field sigma_h (Xh);

General space product interface

A general multi-component field writes:

    space Th (omega, "P1d", "tensor");
    space Vh (omega, "P2", "vector");
    space Qh (omega, "P1");
    space Xh = Th*Vh*Qh;
    field xh (Xh);
    field tau_h = xh[0]; // tensor-valued
    field uh    = xh[1]; // vector-valued
    field qh    = xh[2]; // scalar

Remark the hierarchical multi-component field structure: the first-component is tensor-valued and the second-one is vector-valued. There is no limitation upon the hierarchical number of levels in use: the hierarchy is not flattened.

The xh.size() returns the number of field components. When the field is scalar, it returns zero by convention, and xh[0] is undefined.

Direct access to degrees-of-freedom

The field class provides a STL-like container interface for accessing the degrees-of-freedom (dofs) of a finite element field uh. The number of dofs is uh.ndof() and any dof can be accessed via uh.dof(idof). In a distributed memory environment, a non-local dof at the partition interface can be obtain via uh.dis_dof(dis_idof) where dis_idof is the (global) distributed index assoiated to the distribution uh.ownership(). See distributor.

For better performances, a STL-like iterator interface is available, with uh.begin_dof() and uh.end_dof() returns iterators to the dofs on the current processor.

Representation

The degrees of freedom (see space) are splited between unknowns and blocked, i.e. uh=[uh.u,uh.b] for any field uh in Xh. Access to these vectors is allowed via some accessors: a read-only one, as uh.u() and uh.b(), and a read-and-write one, as uh.set_u() and uh.set_b(), see vec.

Note that blocked and unknown degrees of freedom could also be elegantly set by using a domain name indexation (see geo):

    geo omega ("circle");
    space Xh (omega, "P1");
    Xh.block ("boundary");
    field uh (Xh);
    uh ["boundary"] = 0;

Implementation

This documentation has been generated from file main/lib/field.h

The field class is simply an alias to the field_basic class

typedef field_basic<Float> field;
see the field page for the full documentation

The field_basic class provides an interface to a vector data container:

template <class T, class M = rheo_default_memory_model>
class field_basic: public details::field_wdof_base<field_basic<T,M>> {
public :
// typedefs:
using wdof_base = details::field_wdof_base<field_basic<T,M>>;
using rdof_base = details::field_rdof_base<field_basic<T,M>>;
using size_type = std::size_t;
using scalar_type = T;
using memory_type = M;
using geo_type = geo_basic <float_type,memory_type>;
using space_type = space_basic<float_type,memory_type>;
using dis_reference = typename vec<scalar_type,memory_type>::dis_reference;
using value_type = T; // TODO: run-time dependent, set it to undeterminated<T>
using result_type = T; // TODO: run-time dependent, set it to undeterminated<T>
class iterator;
class const_iterator;
// allocator/deallocator:
explicit field_basic (
const space_type& V,
const T& init_value = std::numeric_limits<T>::max());
void resize (
const space_type& V,
const T& init_value = std::numeric_limits<T>::max());
// expressions: field_expr or field_lazy are accepted here
// TODO: merge with FieldLazy piecewise_poly
template <class Expr, class Sfinae =
typename std::enable_if<
( details::is_field_expr_affine_homogeneous<Expr>::value
&& ! details::is_field<Expr>::value
)
|| details::is_field_lazy<Expr>::value
>::type>
field_basic (const Expr& expr);
field_basic (const std::initializer_list<details::field_concat_value<T,M> >& init_list);
// assignments:
field_basic<T,M>& operator= (const field_basic<T,M>&);
field_basic<T,M>& operator= (const std::initializer_list<details::field_concat_value<T,M> >& init_list);
// accessors:
const space_type& get_space() const { return _V; }
const geo_type& get_geo() const { return _V.get_geo(); }
std::string get_approx() const { return _V.get_approx(); }
valued_type valued_tag() const { return _V.valued_tag(); }
const std::string& valued() const { return _V.valued(); }
#ifdef TO_CLEAN
std::string name() const { return _V.name(); }
bool have_homogeneous_space (space_basic<T,M>& Xh) const { Xh = get_space(); return true; }
#endif // TO_CLEAN
// accessors & modifiers to unknown & blocked parts:
const vec<T,M>& u() const { return _u; }
const vec<T,M>& b() const { return _b; }
vec<T,M>& set_u() { dis_dof_indexes_requires_update(); return _u; }
vec<T,M>& set_b() { dis_dof_indexes_requires_update(); return _b; }
// accessors to extremas:
T min() const;
T max() const;
T max_abs() const;
T min_abs() const;
// accessors by degrees-of-freedom (dof):
const distributor& ownership() const { return get_space().ownership(); }
const communicator& comm() const { return ownership().comm(); }
size_type ndof() const { return ownership().size(); }
size_type dis_ndof() const { return ownership().dis_size(); }
T& dof (size_type idof);
const T& dof (size_type idof) const;
const T& dis_dof (size_type dis_idof) const;
// write access to non-local dis_idof changes to others procs
iterator begin_dof();
iterator end_dof();
const_iterator begin_dof() const;
const_iterator end_dof() const;
// input/output:
idiststream& get (idiststream& ips);
odiststream& put (odiststream& ops) const;
odiststream& put_field (odiststream& ops) const;
// evaluate uh(x) where x is given locally as hat_x in K:
T dis_evaluate (const point_basic<T>& x, size_type i_comp = 0) const;
T operator() (const point_basic<T>& x) const { return dis_evaluate (x,0); }
point_basic<T> dis_vector_evaluate (const point_basic<T>& x) const;
field::size_type size_type
Definition branch.cc:430
see the communicator page for the full documentation
size_type dis_size() const
global and local sizes
size_type size(size_type iproc) const
const communicator_type & comm() const
void dis_dof_indexes_requires_update() const
Definition field.h:725
T max_abs() const
Definition field.h:834
const T & dis_dof(size_type dis_idof) const
Definition field.cc:377
vec< T, M > _u
Definition field.h:477
vec< T, M > _b
Definition field.h:478
size_type dis_ndof() const
Definition field.h:299
space_type _V
Definition field.h:476
valued_type valued_tag() const
Definition field.h:273
std::string get_approx() const
Definition field.h:272
idiststream & get(idiststream &ips)
Definition field.cc:122
point_basic< T > dis_vector_evaluate(const point_basic< T > &x) const
Definition field.cc:469
T & dof(size_type idof)
Definition field.h:738
odiststream & put(odiststream &ops) const
Definition field.cc:367
space_constant::valued_type valued_type
Definition field.h:232
vec< T, M > & set_b()
Definition field.h:285
const vec< T, M > & b() const
Definition field.h:283
field_basic< T, M > & operator=(const field_basic< T, M > &)
Definition field.h:894
dis_reference dis_dof_entry(size_type dis_idof)
Definition field.cc:398
details::field_wdof_base< field_basic< T, M > > wdof_base
Definition field.h:223
details::field_rdof_base< field_basic< T, M > > rdof_base
Definition field.h:224
vec< T, M > & set_u()
Definition field.h:284
odiststream & put_field(odiststream &ops) const
Definition field.cc:303
typename vec< scalar_type, memory_type >::dis_reference dis_reference
Definition field.h:231
T operator()(const point_basic< T > &x) const
Definition field.h:320
iterator begin_dof()
Definition field.h:595
T min() const
Definition field.h:786
geo_basic< float_type, memory_type > geo_type
Definition field.h:229
T max() const
Definition field.h:802
T min_abs() const
Definition field.h:818
const geo_type & get_geo() const
Definition field.h:271
const space_type & get_space() const
Definition field.h:270
iterator end_dof()
Definition field.h:603
const vec< T, M > & u() const
Definition field.h:282
size_type ndof() const
Definition field.h:298
const std::string & valued() const
Definition field.h:274
typename float_traits< T >::type float_type
Definition field.h:228
void resize(const space_type &V, const T &init_value=std::numeric_limits< T >::max())
Definition field.cc:49
space_basic< float_type, memory_type > space_type
Definition field.h:230
const distributor & ownership() const
Definition field.h:296
const communicator & comm() const
Definition field.h:297
T dis_evaluate(const point_basic< T > &x, size_type i_comp=0) const
Definition field.cc:439
Expr1::float_type T
Definition field_expr.h:230
bool have_homogeneous_space(space_basic< scalar_type, memory_type > &Vh) const
Expr1::memory_type M
};
namespace details {
// concepts:
template<class T, class M>
struct is_field<field_basic<T,M>> : std::true_type {};
// field class is not an ordinary function/functor : for compose(field,args...) filtering
template <typename T, typename M> struct function_traits <field_basic<T,M> > {};
template<class T, class M>
struct field_traits<field_basic<T,M>> {
using size_type = std::size_t;
using scalar_type = T;
using memory_type = M;
};
} // namespace details