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 | / OmegaThe 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");