Provided by: librheolef-dev_6.6-1build2_amd64 
NAME
field - piecewise polynomial finite element field
DESCRIPTION
Store degrees of freedom associated to a mesh and a piecewise polynomial approximation,
with respect to the numbering defined by the underlying space(2).
This class contains two vectors, namely unknown and blocked degrees of freedoms, and the
associated finite element space. Blocked and unknown degrees of freedom can be set by
using domain name indexation:
geo omega ("circle");
space Xh (omega, "P1");
Xh.block ("boundary");
field uh (Xh);
uh ["boundary"] = 0;
INTERPOLATION
Interpolation of a function u in a field uh with respect to the interpolation writes:
Float u (const point& x) { return 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 vh writes simply
dual(lh,vh): for discrete fields, it corresponds to a simple Euclidian dot product in
IR^n. The application of a bilinear form (see form(2)) writes m(uh,vh) and is equivalent
to dual(m*uh,vh).
NON-LINEAR ALGEBRA
Non-linear operations, such as sqrt(uh) or 1/uh are also available. Notice that non-
linear operations do not returns in general picewise polynomials: the value returned by
sqrt(uh) may be filtered by interpolate,
field vh = interpolate (Xh, sqrt(uh));
the Lagrange interpolant, to becomes a piecewise polynomial: All standard unary and binary
math functions abs, cos, sin... are available on 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, use the compose function:
field vh = interpolate(Xh, compose(f, uh));
field wh = interpolate(Xh, compose(f, uh, vh));
The composition supports also general unary and binary class-functions. 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 may be filtered by the interpolate operator:
field wh = interpolate(Xh, uh*vh);
Any function or class function can be used in nonlinear expressions: the function is
interpolated in the specified finite element space.
There is a special predefined class-function named normal that represents the outer unnit
normal vector on a boundary domain or surfacic mesh:
size_t k = omega.order();
string n_approx = "P" + itos(k-1) + "d";
space Nh (omega["boundary"], n_approx, "vector");
field nh = interpolate(Nh, normal());
The normal() function could appear in any nonlinear field expression: it is evaluated on
the fly, based on the current mesh. Notice that when using isoparametric elements, the
normal vector is no more constant along any face of the mesh. Also, on general curved
domains, the unit normal vector is discontinuous accross boundary element interfaces.
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;
Extraction of the trace as a field is also possible:
field wh = uh["boundary"];
The space associated to the trace writes wh.get_space() and is equivalent to
space(omega["boundary"], "P1"). See see space(2).
VECTOR VALUED FIELD
A vector-valued field contains several components, as:
space Xh (omega, "P2", "vector");
field uh (Xh);
field vh = uh[0] - uh[1];
field nh = norm (uh);
The norm function returns the euclidian norm of the vector-valuated field at each degree
of freedom: its result is a scalar field.
TENSOR VALUED FIELD
A tensor-valued field can be constructed and used as:
space Th (omega, "P1d", "tensor");
field sigma_h (Xh);
field trace_h = sigma(0,0) + sigma_h(1,1);
field nh = norm (sigma_h);
The norm function returns the euclidian norm of the tensor-valuated field at each degree
of freedom: its result is a scalar field. Notice that, as tensor-valued fields are
symmetric, extra-diagonals are counted twice.
GENERAL MULTI-COMPONENT 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.
For any field xh, the string xh.valued() returns "scalar" for a scalar field and "vector"
for a vector-valued one. Other possible valued are "tensor" and "other". The xh.size()
returns the number of field components. When the field is scalar, it returns zero by
convention, and xh[0] is undefined. A vector-valued field has d components, where
d=omega.dimension(). A tensor-valued field has d*(d+1)/2 components, where
d=omega.dimension().
BLOCKED AND UNBLOCKED ARRAYS
The field class contains two arrays of degrees-of-freedom (dof) associated respectively to
blocked and unknown dofs. Blocked dofs corresponds to Dirichlet boundary conditions, as
specified by space (See see space(2)). For simpliity, direct public access to these array
is allowed, as uh.b and uh.u: see see vec(2).
LOW-LEVEL DEGREE-OF-FREEDOM ACCESS
The field class provides a STL-like container interface for accessing the degrees-of-
freedom (dof) of a finite element field uh. The number of dofs is uh.ndof() and any dof
can be accessed via uh.dof(idof). A non-local dof at the partition interface can be
obtain via uh.dis_dof(dis_idof) where dis_idof is the (global) distribued index assoiated
to the distribution uh.ownership().
For performances, a STL-like iterator interface is available, with uh.begin_dof() and
uh.end_dof() returns iterators to the arrays of dofs on the current processor. See see
array(2) for more about distributed arrays.
For convenience, uh.max(), uh.min() and uh.max_abs() retuns respectively the maximum,
minimum and maximum of the absolute value of the degrees of freedom.
FILE FORMAT
TODO
IMPLEMENTATION NOTE
The field expression use the expression template technics in order to avoid temporaries
when evaluating complex expressions.
IMPLEMENTATION
template <class T, class M = rheo_default_memory_model>
class field_basic : public std::unary_function<point_basic<typename scalar_traits<T>::type>,T> {
public :
// typedefs:
typedef typename std::size_t size_type;
typedef M memory_type;
typedef T scalar_type;
typedef typename float_traits<T>::type float_type;
// typedef undeterminated_basic<T> value_type; // TODO
typedef T value_type; // TO_CLEAN
typedef space_constant::valued_type valued_type;
typedef geo_basic <float_type,M> geo_type;
typedef space_basic<float_type,M> space_type;
class iterator;
class const_iterator;
// allocator/deallocator:
field_basic();
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());
field_basic (const field_indirect<T,M>&);
field_basic<T, M>& operator= (const field_indirect<T,M>&);
field_basic (const field_indirect_const<T,M>&);
field_basic<T, M>& operator= (const field_indirect_const<T,M>&);
field_basic (const field_component<T,M>&);
field_basic<T, M>& operator= (const field_component<T,M>&);
field_basic (const field_component_const<T,M>&);
field_basic<T, M>& operator= (const field_component_const<T,M>&);
template <class Expr> field_basic (const field_expr<Expr>&);
template <class Expr> field_basic<T, M>& operator= (const field_expr<Expr>&);
field_basic<T, M>& operator= (const T&);
// initializer list (c++ 2011):
#ifdef _RHEOLEF_HAVE_STD_INITIALIZER_LIST
field_basic (const std::initializer_list<field_concat_value<T,M> >& init_list);
field_basic<T,M>& operator= (const std::initializer_list<field_concat_value<T,M> >& init_list);
#endif // _RHEOLEF_HAVE_STD_INITIALIZER_LIST
// accessors:
const space_type& get_space() const { return _V; }
const geo_type& get_geo() const { return _V.get_geo(); }
std::string stamp() const { return _V.stamp(); }
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(); }
// accessors & modifiers to unknown & blocked parts:
const vec<T,M>& u() const { dis_dof_update_needed(); return _u; }
const vec<T,M>& b() const { dis_dof_update_needed(); return _b; }
vec<T,M>& set_u() { return _u; }
vec<T,M>& set_b() { return _b; }
// accessors to extremas:
T min() const;
T max() const;
T max_abs() const;
T min_abs() const;
// accessors by domains:
field_indirect<T,M> operator[] (const geo_basic<T,M>& dom);
field_indirect_const<T,M> operator[] (const geo_basic<T,M>& dom) const;
field_indirect<T,M> operator[] (std::string dom_name);
field_indirect_const<T,M> operator[] (std::string dom_name) const;
// accessors by components:
size_type size() const { return _V.size(); }
field_component<T,M> operator[] (size_type i_comp);
field_component_const<T,M> operator[] (size_type i_comp) const;
field_component<T,M> operator() (size_type i_comp, size_type j_comp);
field_component_const<T,M> operator() (size_type i_comp, size_type j_comp) 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;
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;
// internals:
public:
// evaluate uh(x) where x is given locally as hat_x in K:
// requiers to call field::dis_dof_upgrade() before.
T evaluate (const geo_element& K, const point_basic<T>& hat_xq, size_type i_comp = 0) const;
// propagate changed values shared at partition boundaries to others procs
void dis_dof_update() const;
template <class Expr>
void assembly_internal (
const geo_basic<T,M>& dom,
const geo_basic<T,M>& band,
const band_basic<T,M>& gh,
const Expr& expr,
const quadrature_option_type& qopt,
bool is_on_band);
template <class Expr>
void assembly (
const geo_basic<T,M>& domain,
const Expr& expr,
const quadrature_option_type& qopt);
template <class Expr>
void assembly (
const band_basic<T,M>& gh,
const Expr& expr,
const quadrature_option_type& qopt);
protected:
void dis_dof_update_internal() const;
void dis_dof_update_needed() const;
// data:
space_type _V;
vec<T,M> _u;
vec<T,M> _b;
mutable bool _dis_dof_update_needed;
};
template <class T, class M>
idiststream& operator >> (odiststream& ips, field_basic<T,M>& u);
template <class T, class M>
odiststream& operator << (odiststream& ops, const field_basic<T,M>& uh);
typedef field_basic<Float> field;
typedef field_basic<Float,sequential> field_sequential;
SEE ALSO
space(2), form(2), space(2), space(2), vec(2), array(2)