01-25-2017
I tend to have this deployed to look at in general (if needed) :
InstallationApacheTomcat * psi-probe/psi-probe Wiki * GitHub
The software is gpl, has a relatively small footprint when deployed.
Of course, standard commands work as well (vmstat, iostat etc.), but this should help in investigation.
You can use various 'developers consoles' or firebugs to examine where
it's stuck from the client side using browser.
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LEARN ABOUT DEBIAN
s_grad_grad
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)