s_grad_grad(3rheolef) [debian man page]

s_grad_grad(3rheolef)						    rheolef-6.1 					     s_grad_grad(3rheolef)

NAME

       s_grad_grad -- grad_grad-like operator for the Stokes stream function computation

SYNOPSIS

	     form(const space V, const space& V, "s_grad_grad");

DESCRIPTION

       Assembly  the  form associated to the -div(grad) variant operator on a finite element space V.  The V space may be a either P1 or P2 finite
       element space.  See also form(2) and space(2).  On  cartesian  coordinate  systems,  the  form  coincide  with  the  "grad_grad"  one  (see
       grad_grad(3)):

		      /
		      |
	     a(u,v) = |  grad(u).grad(v) dx
		      |
		      / Omega

       The stream function on tri-dimensionnal cartesian coordinate systems is such that

	      u = curl psi
	      div psi = 0

       where u is the velocity field. Taking the curl of the first relation, using the identity:

	       curl(curl(psi)) = -div(grad(psi)) + grad(div(psi))

       and using the div(psi)=0 relation leads to:

	       -div(grad(psi)) = curl(u)

       This relation leads to a variational formulation involving the the "grad_grad" and the "curl" forms (see grad_grad(3), curl(3)).

       In  the	axisymmetric  case, the stream function psi is scalar ans is defined from the velocity field u=(ur,uz) by (see Batchelor, 6th ed.,
       1967, p 543):

			d psi			    d psi
	     uz = (1/r) -----	 and   ur = - (1/r) -----
			 d r			     d r

       See also http://en.wikipedia.org/wiki/Stokes_stream_function .  Multiplying by rot(xi)=(d xi/dr, -d xi/dz), and integrating with r  dr  dz,
       we get a well-posed variationnal problem:

	       a(psi,xi) = b(xi,u)

       with

			 /
			 | (d psi d xi	 d psi d xi)
	     a(psi,xi) = | (----- ---- + ----- ----) dr dz
			 | ( d r  d r	  d z  d z )
			 / Omega

       and

		       /
		       | (d xi	    d xi   )
	     b(xi,u) = | (---- ur - ---- uz) r dr dz
		       | (d z	    d r    )
		       / Omega

       Notice  that  a	is  symmetric definite positive, but without the 'r' weight as is is usual for axisymmetric standard forms.  The b form is
       named "s_curl", for the Stokes curl variant of the "curl" operator (see s_curl(3)) as it is closely related to  the  "curl"  operator,  but
       differs by the r and 1/r factors, as:

			  (	  d (r xi)     d xi )
	       curl(xi) = ( (1/r) -------- ; - -----)
			  (	    d r        d z  )

       while

			  ( d xi       d xi )
	     s_curl(xi) = ( ----  ;  - ---- )
			  ( d r        d z  )

EXAMPLE

       The following piece of code build the form associated to the P1 approximation:

	       geo g("square");
	       space V(g, "P1");
	       form a(V, V, "s_grad_grad");

SEE ALSO

       form(2), space(2), grad_grad(3), grad_grad(3), curl(3), s_curl(3)

rheolef-6.1							    rheolef-6.1 					     s_grad_grad(3rheolef)

Check Out this Related Man Page

form_element(7rheolef)						    rheolef-6.1 					    form_element(7rheolef)

NAME

       form_element - bilinear form on a single element

SYNOPSYS

       The form_element class defines functions that compute a bilinear form defined between two polynomial basis on a single geometrical element.
       This bilinear form is represented by a matrix.

       The bilinear form is designated by a string, e.g. "mass", "grad_grad", ...  indicating the form. The form depends also of  the  geometrical
       element: triangle, square, tetrahedron (see geo_element(2)).

IMPLEMENTATION NOTE

       The  form_element class is managed by (see smart_pointer(2)).  This class uses a pointer on a pure virtual class form_element_rep while the
       effective code refers to the specific concrete derived classes: mass, grad_grad, etc.

IMPLEMENTATION

       template <class T, class M>
       class form_element : public smart_pointer<form_element_rep<T,M> > {
       public:

       // typedefs:

	   typedef form_element_rep<T,M>	 rep;
	   typedef smart_pointer<rep>		 base;
	   typedef typename rep::size_type	 size_type;
	   typedef typename rep::vertex_type	 vertex_type;
	   typedef typename rep::space_type	 space_type;
	   typedef typename rep::geo_type	 geo_type;
	   typedef typename rep::coordinate_type coordinate_type;

       // constructors:

	   form_element ();
	   form_element (
	       std::string	     name,
	       const space_type&     X,
	       const space_type&     Y,
	       const geo_type&	     omega,
	       const quadrature_option_type& qopt);

       // accessors & modifier:

	   void operator() (const geo_element& K, ublas::matrix<T>& m) const;
	   virtual bool is_symmetric () const;

       // for scalar-weighted forms:

	   void set_weight (const field_basic<T,M>& wh) const;
	   bool is_weighted() const;
	   const field_basic<T,M>& get_weight () const;

       // for banded level set method:

	   bool is_on_band() const;
	   const band_basic<T,M>& get_band() const;
	   void set_band (const band_basic<T,M>& bh) const;

       };

SEE ALSO

       geo_element(2), smart_pointer(2)

rheolef-6.1							    rheolef-6.1 					    form_element(7rheolef)

Linux and UNIX Man Pages

s_grad_grad(3rheolef) [debian man page]

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