<div dir="ltr"><div>Hi,</div><div><br></div>This is a very nice book about SEM. It could be a good start, if you already know something about finite element method and spectral method. <div><br></div><div><a href="http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198528692.001.0001/acprof-9780198528692">http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198528692.001.0001/acprof-9780198528692</a><br>
</div><div><br></div><div><br></div><div>The second book is by CHQZ, a classic of spectral method, but also talks about SEM. </div><div><a href="http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-3-540-30727-3">http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-3-540-30727-3</a><br>
</div><div><br></div><div>Good lecture!</div><div><br></div><div>Tubois</div></div><div class="gmail_extra"><br><br><div class="gmail_quote">On Sat, Jan 11, 2014 at 11:59 AM, <span dir="ltr"><<a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.gov</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Dear Paul,<br>
Thank you for the information. I am a bit unfamiliar with the basic knowledge in SEM. Could you recommend me a nice short introduction for the theory? especially that deals with the issues below.<br>
<br>
Many thanks,<br>
Barak<div class="HOEnZb"><div class="h5"><br>
<br>
On 01/10/2014 05:07 PM, <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
Hi Barak,<br>
<br>
If the elements are of uniform size in a given direction then, to leading order, the<br>
resolution is<br>
<br>
dx ~ Lx / (N*Ex)<br>
<br>
where Lx = length in x, Ex=# of elements in x-direction, and N is polynomial order.<br>
<br>
Technically, the resolution is a bit less -- like (2/pi).<br>
<br>
There are two errors of primary concern --- approximation error and dispersion.<br>
<br>
If n=number of points in a given direction (n=N*ex the SEM case), then you have dx=L/n,<br>
and Nyquist says you can't resolve wavelengths shorter than 2*dx. This restriction applies<br>
to all methods, including Fourier spectral and SEM.<br>
<br>
For Fourier, all wavenumbers are transported with zero phase error (because of the periodic<br>
BCs + fact that they are eigenfunctions of the operators, etc.).<br>
<br>
For SEM, only 2/pi have diminishing error as N increases, and it is in fact not as small<br>
as machine epsilon (as is the case with Fourier spectral, only). However, the SEM is<br>
far superior to low-order approaches, and that's where it's strength lies. The primary<br>
reason to use SEM over Fourier is when you don't have homogeneous geometries or<br>
constant coefficient operators.<br>
<br>
Paul<br>
<br>
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Sent: Friday, January 10, 2014 8:30 AM<br>
To: <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a><br>
Subject: Re: [Nek5000-users] turbulent 3d box<br>
<br>
Hi Paul,<br>
Thanks for your help till now.<br>
Regarding the proper discretization of the 3d turbulent box setup. When<br>
simulating using a standard spectral method the smallest scale defines<br>
the highest wavenumber kmax. What is its equivalent in SEM?<br>
Happy new year 2014,<br>
Barak<br>
<br>
On 01/04/2014 03:40 AM, <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
Barak,<br>
<br>
I suggest comparing the SIZE files for various examples. From there you should<br>
be able to see how to properly modify SIZE to change the resolution.<br>
<br>
Of course, you'll need to reduce dt to maintain CFL ~ 0.5<br>
<br>
Paul<br>
<br>
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Sent: Sunday, December 22, 2013 4:25 PM<br>
To: <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a><br>
Subject: Re: [Nek5000-users] turbulent 3d box<br>
<br>
It didn't work (I tried both options). Do you have working example?<br>
(see my previous email and my attachment).<br>
Barak<br>
<br>
On 12/22/2013 11:20 PM, <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
Hi Paul,<br>
On 12/22/2013 11:14 PM, <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
Hi Barak,<br>
<br>
You can either increase lx1,ly1, lz1 in SIZE and re-make, or you<br>
can change nelx,nely,nelz in the .box file.<br>
<br>
</blockquote>
Thanks.<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Did you resolve the issue with the initial condition?<br>
<br>
</blockquote>
See my previous email. I have inside further questions.<br>
Best regards,<br>
Barak<br>
<br>
<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Paul<br>
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To: <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a><br>
Subject: Re: [Nek5000-users] turbulent 3d box<br>
<br>
Hi Paul,<br>
Before I run problems with smaller viscosity: how I increase the<br>
resolution? and remesh the box?<br>
Barak<br>
<br>
On 12/22/2013 10:24 PM, <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
Hi Barak,<br>
<br>
When I compute the L2 norm of the function, given as:<br>
<br>
v2 = [ sum_i vx_i b_i vx_i + vy_i b_i vy_i + vz_i b_i vz_i<br>
/ sum_i b_i ]^(1/2)<br>
<br>
it comes out to ~ 2.23<br>
<br>
Note that there are some places where ux==2, uy==0, uz==1,<br>
which would give an L2 norm of<br>
<br>
sqrt (2*2 + 0*0 + 1*1) = sqrt(5) > sqrt(3)<br>
<br>
?<br>
<br>
Paul<br>
<br>
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Sent: Sunday, December 22, 2013 11:04 AM<br>
To: <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a><br>
Subject: Re: [Nek5000-users] turbulent 3d box<br>
<br>
Hi Paul,<br>
My definition for velocity magnitude is:<br>
|v|=sqrt(ux^2+uy^2+uz^2)<br>
so if you see, there are two components in the form:<br>
cos^2(x)+sin^2(x)<br>
etc in each direction<br>
so in fact<br>
|v|=sqrt(3) for any x,y,z.<br>
Isn't correct?<br>
Barak<br>
<br>
On 12/22/2013 05:05 PM, <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<br>
Hi Barak,<br>
<br>
If your initial condition is:<br>
<br>
ux=(cos(y)+sin(z))<br>
uy=(cos(z)+sin(x))<br>
uz=(cos(x)+sin(y))<br>
<br>
and your domain is [0,2pi]^3, how is it possible that your velocity<br>
magnitude<br>
is less than 2?<br>
<br>
<br>
What is your measure of velocity magnitude? When I compute the L2<br>
norm of this initial condition I'm seeing about 2.3+<br>
<br>
Paul<br>
<br>
------------------------------<u></u>------------------------------<u></u>------------<br>
<br>
*From:* <a href="mailto:nek5000-users-bounces@lists.mcs.anl.gov" target="_blank">nek5000-users-bounces@lists.<u></u>mcs.anl.gov</a><br>
[<a href="mailto:nek5000-users-bounces@lists.mcs.anl.gov" target="_blank">nek5000-users-bounces@lists.<u></u>mcs.anl.gov</a>] on behalf of<br>
<a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a> [<a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a>]<br>
*Sent:* Sunday, December 22, 2013 7:53 AM<br>
*To:* <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a><br>
*Subject:* Re: [Nek5000-users] turbulent 3d box<br>
<br>
Hi Paul,<br>
It seems that something does not work, comparing my simulations using a<br>
standard spectral code (pseudo-spectral). For nu=0.1, I used 16 points<br>
for each space direction. Now for this case, the nonlinear terms vanish<br>
(just cancel each other) and the solution is just the ABC flow if you<br>
set it as initial condition. Then since the forcing is f=nu*ABC the<br>
dissipation term (Laplacian) dissipation=-nu*ABC and therefore the<br>
initial value (ABC) remains as it is. If we decrease the viscosity<br>
above<br>
the critical value of 1/12 (if I remember correctly) there is a<br>
bifurcation (Hopf...) and we obtain a new periodic solution (in time),<br>
where the ABC modes are dominating combined with some high wave numbers<br>
that are oscillating.<br>
<br>
>From my first tests using NEK, I find that the velocity field (for<br>
nu>1/12) does not agree with the above characteristics. For example,<br>
the<br>
max magnitude of the velocity field should be for the ABC flow sqrt(3),<br>
whereas here I get a value close to 2.5.<br>
<br>
Do you have any idea what's wrong?<br>
If you wish to read some literature about the ABC flow as a forcing for<br>
Navier Stokes or as a prescribed velocity for generation of magnetic<br>
field (i.e., kinematic generation of magnetic field MHD):<br>
<br>
1. D. Galloway and U. Frisch. A numerical investigation of magnetic<br>
field generation in a flow with chaotic streamlines. Geophysical &<br>
Astrophysical Fluid Dynamics, 29(1):13–18, 1984.<br>
<br>
2. D. Galloway and U. Frisch. Dynamo action in a family of flows with<br>
chaotic streamlines.<br>
Geophysical & Astrophysical Fluid Dynamics, 36(1):53–83, 1986.<br>
<br>
3. Podvigina, Pouquet, On the non-linear stability of the 1:1:1 ABC<br>
flow,<br>
Phys. D, Vol. 75, 4, 1994.<br>
<br>
<br>
With my best wishes,<br>
Barak<br>
<br>
<br>
Quoting <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a><br>
<mailto:<a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.<u></u>mcs.anl.gov</a>>:<br>
<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Hi Barak,<br>
<br>
Thanks for the note about the ABC flow -- that's very useful.<br>
<br>
What is your target viscosity?<br>
If you want it to be 0.1, then you would set param 2 in the .rea file<br>
to be 0.1.<br>
Setting param 2 = -0.1 would imply that Re=0.1 or equivalently nu=10.<br>
Does this resolve the issue for you?<br>
<br>
Once it's working we could add it as a documented example in the<br>
example<br>
suite.<br>
<br>
Thanks!<br>
<br>
Paul<br>
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Subject: Re: [Nek5000-users] turbulent 3d box<br>
<br>
Hi Paul,<br>
Regarding the 3dbox:<br>
The forcing which I propose is known as the ABC flow<br>
(Arnold-Beltrami-Childress) and has steady solutions (in the form<br>
of ABC<br>
flow) with Re=1/nu (nu the viscosity) Re=<12 (in periodic box).<br>
The forcing should be in the form:<br>
f=Re*ABC in order to balance the dissipation term.<br>
This settings can serve as a benchmark for pure hydro/mhd problems.<br>
<br>
I tried to fix the example which you have sent me, but I don't succeed<br>
retreiving the expected values.<br>
I attach the relevant files( tarball file).<br>
Best regards,<br>
Barak<br>
<br>
<br>
<br>
On 12/18/2013 03:29 AM, <a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.mcs.anl.<u></u>gov</a><br>
<mailto:<a href="mailto:nek5000-users@lists.mcs.anl.gov" target="_blank">nek5000-users@lists.<u></u>mcs.anl.gov</a>> wrote:<br>
<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Hi Barak,<br>
<br>
I added a 3d periodic box under "3dbox" in the examples<br>
directory.<br>
<br>
Paul<br>
<br>
<br>
Dear nek's,<br>
I would like to run the following problem:<br>
a 3D periodic box, small viscosity and the following forcing:<br>
<br>
fx=cos(y)+sin(z)<br>
fy=cos(z)+sin(x)<br>
fz=cos(x)+sin(y)<br>
<br>
Do you have a similar example?<br>
<br>
Many thanks,<br>
Barak<br>
<br>
<br>
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