band(2rheolef) [debian man page]

band(2rheolef)							    rheolef-6.1 						    band(2rheolef)

NAME

       band - compute the band arround a level set

DESCRIPTION

       Given  a  function  fh defined in a domain Lambda, compute the band of elements intersecting the level set defined by {x in Lambda, fh(x) =
       0}.  This class is used for solving problems defined on a surface described by a level set function (See level_set(4)).

ACCESSORS

       Each side in the surface mesh, as returned by the level_set member function, is included into an element of the band mesh, as  returned	by
       the band member function. Moreover, in the distributed memory environment, this correspondance is on the same process, so local indexes can
       be used for this correspondance: this is the sid_ie2bnd_ie member functions.

BAND TOPOLOGY AND DOMAINS

       For the direct resolution of systems posed on the band, the mesh returned by the band() provides some domains of vertices.  The "zero" ver-
       tex domain lists all vertices xi such that fh(xi)=0.  The "isolated" vertex domain lists all vertices xi such that fh(xi)!=0 and xi is con-
       tained by only one element K and all vertices xj!=xi of K satifies fh(xj)=0.  Others vertices of the band, separated by the zero  and  iso-
       lated  ones, are organizd by connected components: the n_connex_component member function returns its number.  Corresponding vertex domains
       of the band are named "cc<i>" where <i> should be replaced by any number between 0 and n_connex_component-1.

IMPLEMENTATION

       template <class T, class M = rheo_default_memory_model>
       class band_basic {
       public:

	 typedef typename geo_basic<T,M>::size_type size_type;

       // allocators:

	 band_basic();
	 band_basic(const field_basic<T,M>& fh,
	   const level_set_option_type& opt = level_set_option_type());

       /// accessors:

	 const geo_basic<T,M>& band() const { return _band; }
	 const geo_basic<T,M>& level_set() const { return _gamma; }
	 size_type sid_ie2bnd_ie (size_type sid_ie) const { return _sid_ie2bnd_ie [sid_ie]; }
	 size_type n_connected_component() const { return _ncc; }

       // data:
       protected:
	 geo_basic<T,M>     _gamma;
	 geo_basic<T,M>     _band;
	 array<size_type,M> _sid_ie2bnd_ie;
	 size_type	    _ncc;
       };
       typedef band_basic<Float> band;

SEE ALSO

       level_set(4)

rheolef-6.1							    rheolef-6.1 						    band(2rheolef)

form(2rheolef)							    rheolef-6.1 						    form(2rheolef)

NAME

       form - representation of a finite element bilinear form

DESCRIPTION

       The form class groups four sparse matrix, associated to a bilinear form on two finite element spaces:

	      a: U*V   ----> IR
		(u,v)  |---> a(u,v)

       The operator A associated to the bilinear form is defined by:

	      A: U  ----> V'
		 u  |---> A(u)

       where  u and v are fields (see field(2)), and A(u) is such that a(u,v)=<A(u),v> for all u in U and v in V and where <.,.> denotes the dual-
       ity product between V and V'.  Since V is a finite dimensional spaces, the duality product is the euclidian product in IR^dim(V).

       Since both U and V are finite dimensional spaces, the linear operator can be represented by a matrix.  The form	class  is  represented	by
       four  sparse  matrix in csr format (see csr(2)), associated to unknown and blocked degrees of freedom of origin and destination spaces (see
       space(2)).

EXAMPLE

       The operator A associated to a bilinear form a(.,.) by the relation (Au,v) = a(u,v) could be applied by using a sample matrix notation A*u,
       as shown by the following code:

	     geo omega("square");
	     space V (omega,"P1");
	     form a (V,V,"grad_grad");
	     field uh = interpolate (fct, V);
	     field vh = a*uh;
	     cout << v;

       The form-field vh=a*uh operation is equivalent to the following matrix-vector operations:

	    vh.set_u() = a.uu()*uh.u() + a.ub()*uh.b();
	    vh.set_b() = a.bu()*uh.u() + a.bb()*uh.b();

ALGABRA

	Forms, as matrices (see csr(2)), support linear algebra:
	Adding or substracting two forms writes a+b and a-b, respectively,
	and multiplying a form by a field uh writes a*uh.
	Thus, any linear combination of forms is available.

IMPLEMENTATION

       template<class T, class M>
       class form_basic {
       public :
       // typedefs:

	   typedef typename csr<T,M>::size_type    size_type;
	   typedef T				   value_type;
	   typedef typename scalar_traits<T>::type float_type;
	   typedef geo_basic<float_type,M>	   geo_type;
	   typedef space_basic<float_type,M>	   space_type;

       // allocator/deallocator:

	   form_basic ();
	   form_basic (const form_basic<T,M>&);

	   form_basic (const space_type& X, const space_type& Y,
	       const std::string& name = "",
	       quadrature_option_type  qopt = quadrature_option_type(quadrature_option_type::max_family,0));

	   form_basic (const space_type& X, const space_type& Y,
	       const std::string& name,
	       const geo_basic<T,M>& gamma,
	       quadrature_option_type  qopt = quadrature_option_type(quadrature_option_type::max_family,0));

	   form_basic (const space_type& X, const space_type& Y,
	       const std::string& name,
	       const field_basic<T,M>& weight,
	       quadrature_option_type  qopt = quadrature_option_type(quadrature_option_type::max_family,0));

	   form_basic (const space_type& X, const space_type& Y,
	       const std::string& name,
	       const band_basic<T,M>& bh,
	       quadrature_option_type  qopt = quadrature_option_type(quadrature_option_type::max_family,0));

       // allocators from initializer list (c++ 2011):

       #ifdef _RHEOLEF_HAVE_STD_INITIALIZER_LIST
	   form_basic (const std::initializer_list<form_concat_value<T,M> >& init_list);
	   form_basic (const std::initializer_list<form_concat_line <T,M> >& init_list);
       #endif // _RHEOLEF_HAVE_STD_INITIALIZER_LIST

       // accessors:

	   const space_type& get_first_space() const;
	   const space_type& get_second_space() const;
	   const geo_type&   get_geo() const;

	   const communicator& comm() const;

       // linear algebra:

	   form_basic<T,M>  operator+  (const form_basic<T,M>& b) const;
	   form_basic<T,M>  operator-  (const form_basic<T,M>& b) const;
	   form_basic<T,M>& operator*= (const T& lambda);
	   field_basic<T,M> operator*  (const field_basic<T,M>& xh) const;
       #ifdef TO_CLEAN
	   template <class Expr>
	   field_basic<T,M> operator*  (const field_expr<Expr>& xh) const;
       #endif // TO_CLEAN
	   field_basic<T,M> trans_mult (const field_basic<T,M>& yh) const;

	   float_type operator () (const field_basic<T,M>& uh, const field_basic<T,M>& vh) const;

       // io:

	   odiststream& put (odiststream& ops, bool show_partition = true) const;
	   void dump (std::string name) const;

       // accessors & modifiers to unknown & blocked parts:

	   const csr<T,M>&     uu() const { return _uu; }
	   const csr<T,M>&     ub() const { return _ub; }
	   const csr<T,M>&     bu() const { return _bu; }
	   const csr<T,M>&     bb() const { return _bb; }
		 csr<T,M>& set_uu()	  { return _uu; }
		 csr<T,M>& set_ub()	  { return _ub; }
		 csr<T,M>& set_bu()	  { return _bu; }
		 csr<T,M>& set_bb()	  { return _bb; }

       // data
       protected:
	   space_type  _X;
	   space_type  _Y;
	   csr<T,M>    _uu;
	   csr<T,M>    _ub;
	   csr<T,M>    _bu;
	   csr<T,M>    _bb;

       // internals:
	   void assembly (const form_element<T,M>&   form_e,
			  const geo_basic<T,M>&      X_geo,
			  const geo_basic<T,M>&      Y_geo,
			  bool	X_geo_is_background = true);
	   void form_init (
			  const std::string&	  name,
			  bool			  has_weight,
			  const field_basic<T,M>& weight,
			  quadrature_option_type  qopt);

       };
       template<class T, class M> form_basic<T,M> trans (const form_basic<T,M>& a);
       typedef form_basic<Float,rheo_default_memory_model> form;

SEE ALSO

       field(2), csr(2), space(2), csr(2)

rheolef-6.1							    rheolef-6.1 						    form(2rheolef)