New subject: Cell center on extruded mesh

24 Sep 2015

      Ah, my first instinct was that the FE way would be exactly equivalent, but
of course that's not true.

Anyway, lowest-order FE is similar to FV, so here's the FE approach I was
thinking of:

The space (mesh, "NCF", 1) has one degree of freedom per facet (
http://femtable.org/, 3rd column, bottom panel).  We want to find a
function in this space which is somehow the gradient of your p in DG0.

It'd be nice if we could just do
u = TrialFunction(W)
w = TestFunction(W)
a = dot(w, u)*dx
L = dot(w, grad(p))*dx
out = Function(W)
solve(a == L, out)

However, this clearly won't give the result you want, since p is in DG0, so
grad(p) is 0 everywhere, so the right hand side is 0.

Instead, integrate the right hand side by parts to obtain L = p_bc*dot(w,
n)*ds - div(w)*p*dx, where n = FacetNormal(mesh), and p_bc is some function
with the values of p that you want on the boundary (this can be just p, or
you might choose to put p_bc in a continuous space, for example).

Complete code for the 1D version of this is

---
from firedrake import *

mesh = UnitIntervalMesh(10)

V = FunctionSpace(mesh, "DG", 0)
f = Function(V).interpolate(Expression("2*x[0] - 1.0"))
f_bc = Function(FunctionSpace(mesh, "CG",
1)).interpolate(Expression("2*x[0] - 1.0"))

W = FunctionSpace(mesh, "CG", 1)
u = TrialFunction(W)
v = TestFunction(W)
a = u*v*dx
n = FacetNormal(mesh)

L = f_bc*v*n[0]*ds - v.dx(0)*f*dx

g = Function(W)
solve(a == L, g)

print g.dat.data
---

Indeed, this verifies that the FE 'weak gradient' of 2x-1 is 2 everywhere.
For the 3D version, W needs to be FunctionSpace(mesh, "NCF", 1), and L
needs to be as given above.

Andrew

On 24 September 2015 at 07:15, Buesing, Henrik <
HBuesing@eonerc.rwth-aachen.de> wrote:
...
--
Dipl.-Math. Henrik Büsing
Institute for Applied Geophysics and Geothermal Energy
E.ON Energy Research Center
RWTH Aachen University
------------------------------------------------------
Mathieustr. 10            |    Tel +49 (0)241 80 49907
52074 Aachen, Germany     |    Fax +49 (0)241 80 49889
------------------------------------------------------
http://www.eonerc.rwth-aachen.de/GGE
hbuesing@eonerc.rwth-aachen.de
------------------------------------------------------
*Von:* firedrake-bounces@imperial.ac.uk [mailto:
firedrake-bounces@imperial.ac.uk] *Im Auftrag von *Andrew McRae
*Gesendet:* 22 September 2015 20:42
*An:* firedrake@imperial.ac.uk
*Betreff:* Re: [firedrake] Cell center on extruded mesh
For the distances between cell centres, yes.
It feels like you're performing some operations that aren't particularly
finite element-y?  Would you be using the approximate gradient in a
variational form immediately, or are there further calculations you want to
carry out?
In the former case, I think there's a nice FE way to get what you want.
*[Buesing, Henrik] @Andrew: So, what would be the FE way of doing this?*
* Henrik*
In the latter case, we can probably hack something up if we're smart
enough...
On 22 September 2015 at 18:42, Buesing, Henrik <
HBuesing@eonerc.rwth-aachen.de> wrote:
------------------------------
*Von:* firedrake-bounces@imperial.ac.uk [firedrake-bounces@imperial.ac.uk]"
im Auftrag von "Andrew McRae [A.T.T.McRae@bath.ac.uk]
*Gesendet:* Dienstag, 22. September 2015 18:59
*An:* firedrake@imperial.ac.uk
*Betreff:* Re: [firedrake] Cell center on extruded mesh
...
...
Can you say a bit more about how you plan to use this?
I wonder to what end you would like these values?
I would like to approximate \grad(pw) on a DG0 Function space by
(p('+') - p('-'))/|x('+')-x('-')|, where x are the cell centers. On a
hexahedral mesh this should just give \Delta x, \Delta y, \Delta z, right?
Henrik
On 22 September 2015 at 17:52, Buesing, Henrik <
HBuesing@eonerc.rwth-aachen.de> wrote:
Dear Firedrakers,
on an extruded mesh:
*meshbase = RectangleMesh(Nx, Ny, Lx, Ly, quadrilateral=True) mesh =
ExtrudedMesh(meshbase, Nz, Delta_z) horiz_elt = FiniteElement("DG",
quadrilateral, 0) vert_elt = FiniteElement("DG", interval, 0) elt =
OuterProductElement(horiz_elt, vert_elt) *How can I access the element
center and the center of the element faces?
Thank you!
Henrik
--
Dipl.-Math. Henrik Büsing
Applied Geophysics and Geothermal Energy
E.ON Energy Research Center
RWTH Aachen University
-----------------------------------------------
Mathieustr. 10        | Tel +49 (0)241 80 49907
52074 Aachen, Germany | Fax +49 (0)241 80 49889
-----------------------------------------------
http://www.eonerc.rwth-aachen.de/GGE
hbuesing@eonerc.rwth-aachen.de
-----------------------------------------------

Re: [firedrake] Cell center on extruded mesh

Andrew McRae

Buesing, Henrik

tags

participants (2)