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Hi Arnold,<br>
<br>
I looked into this issue some when I was trying to calculate the
curvature of some of my temperature isosurfaces. Aleks Obabko
suggested that I tried smoothing the derivatives across the element
boundaries. So basically, I tried doing this:<br>
<br>
! Calculate the gradient of the temperature field<br>
! This derivative will be continuous within the element, but
there will be jumps across the element boundaries<br>
call gradm1(dTdx, dTdy, dTdz, t) <br>
<br>
! Smooth the derivative (dT/dx) so that it is continuous across the
element boundaries.<br>
call col2(dTdx,bm1,n) ! collocate with the mass
matrix<br>
call dssum(dTdx,nx1,ny1,nz1) ! smooth<br>
call col2(dTdx,bintm1,n) ! collocate with the inverse
mass matrix<br>
<br>
(Note: dssum just returns the sum of the shared GLL points across
the element boundaries.)<br>
<br>
I ran a test case, and I found that, in that case, this procedure
does generate good first and second derivatives. The test case was
basically just an artificial sphere that I set up in Nek5000. The
temperature varied with the radius of the sphere, and my goal was to
calculate smooth first and second derivatives of the temperature so
that I could accurately calculate the curvature of the sphere. I
looked at some comparisons between the calculated first and second
derivatives and the analytical solution. Generally, the accuracy
was good given the pretty low resolution that I was using.
(However, in order to find the curvature, I needed to multiply,
raise to powers, and divide the derivatives, so that ultimately the
relatively small derivative errors were magnified into unacceptably
large curvature errors.) <br>
<br>
So, ultimately, I think that this smoothing procedure should
definitely work well enough for most (but not all) purposes. If
you just want smooth derivatives for visualization, or you just want
to measure just the derivative itself, this will probably work.<br>
<br>
In your case of looking at the derivatives of the vorticity, you
could try doing this two ways and see which one works better.<br>
<br>
1) Calculate the vorticity, smooth the vorticity, take the
derivative, then smooth the derivative <br>
<br>
2) Calculate the vorticity, take the derivative, smooth the
derivative<br>
<br>
The first option has more smoothing steps, but the second option
preserves the representation of the vorticity within the element as
you take the derivative. Aleks generally recommends the second
option. <br>
<br>
What do you need the derivatives of the vorticity for exactly?
Depending on your end goal, one of the methods Paul suggested may
work better than this one.<br>
<br>
Elizabeth<br>
<br>
<br>
<div class="moz-cite-prefix">On 03/03/2017 06:32 AM,
<a class="moz-txt-link-abbreviated" href="mailto:nek5000-users@lists.mcs.anl.gov">nek5000-users@lists.mcs.anl.gov</a> wrote:<br>
</div>
<blockquote
cite="mid:mailman.1538.1488544332.2967.nek5000-users@lists.mcs.anl.gov"
type="cite">
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Hi Arnold,
<div><br>
</div>
<div>The velocity is only in C^0, which means it is continuous.
The first derivative</div>
<div>is defined everywhere, but is discontinuous, meaning that
the vorticity is</div>
<div>discontinuous.</div>
<div><br>
</div>
<div>The derivative of the vorticity involves the second
derivative of velocity, and this</div>
<div>is not defined.</div>
<div><br>
</div>
<div>All that being said, the vorticity will converge to a
continuous function exponentially</div>
<div>fast with increasing polynomial order. So if you want to
differentiate the vorticity</div>
<div>you have the option of increasing the polynomial order. </div>
<div><br>
</div>
<div>Alternatively, you can try to regularize the vorticity
prior to differentiating it.</div>
<div>A common strategy is to call dsavg( . ) for each component
of vorticity -- which</div>
<div>makes it continuous, but this wouldn't necessarily make the
derivatives better.</div>
<div>Other alternatives would be to use a combination of dsavg()
and some type of</div>
<div>low-pass filtering, e.g., by solving a Helmholtz problem of
the form (I-L^2 \nabla^2) vbar = v,</div>
<div>where L is the filter length. Another option is to look
at the context in which</div>
<div>you want to use your differentiated vorticity. Perhaps
this can be recast in a weak</div>
<div>form, in which one of the derivatives is moved onto the
(continuous and differentiable)</div>
<div>test functions, etc.</div>
<div><br>
</div>
<div>hth,</div>
<div><br>
</div>
<div>Paul</div>
<div><br>
<div style="font-family: Times New Roman; color: #000000;
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<div id="divRpF849118" style="direction: ltr;"><font
color="#000000" face="Tahoma" size="2"><b>From:</b>
<a class="moz-txt-link-abbreviated" href="mailto:nek5000-users-bounces@lists.mcs.anl.gov">nek5000-users-bounces@lists.mcs.anl.gov</a>
[<a class="moz-txt-link-abbreviated" href="mailto:nek5000-users-bounces@lists.mcs.anl.gov">nek5000-users-bounces@lists.mcs.anl.gov</a>] on behalf of
<a class="moz-txt-link-abbreviated" href="mailto:nek5000-users@lists.mcs.anl.gov">nek5000-users@lists.mcs.anl.gov</a>
[<a class="moz-txt-link-abbreviated" href="mailto:nek5000-users@lists.mcs.anl.gov">nek5000-users@lists.mcs.anl.gov</a>]<br>
<b>Sent:</b> Friday, March 03, 2017 6:02 AM<br>
<b>To:</b> <a class="moz-txt-link-abbreviated" href="mailto:nek5000-users@lists.mcs.anl.gov">nek5000-users@lists.mcs.anl.gov</a><br>
<b>Subject:</b> [Nek5000-users] Problem with Spatial
Derivatives of field<br>
</font><br>
</div>
<div>
<p>Hi Neks,</p>
<p>I'm having some problems with computing the gradients
of vorticity. <br>
</p>
<p>So what I do in the usercheck is something like : <br>
</p>
<p><br>
</p>
<p><br>
</p>
<p>call call comp_vort3(vort,work1,work2,vxp,vyp,vzp) !
Computing vorticity of perturbations
<br>
</p>
<p>call gradm1(derivvortx,derivvorty,derivvortz,vort(1,1))
<br>
</p>
<p><br>
</p>
<p><br>
</p>
<p>However, by doing this I do observe some abnormal
results when I plot the fields : derivvortx, derivvorty,
derivvortz.</p>
<p>Attached you will find a snapshot of the magnitude of
derivvortx and derivvorty (obtained using paraview)</p>
<p>I am wondering why I am having extremely high values a
the boudary of the elements. Has anyone noticed or faced
such a problem?
<br>
</p>
<p><br>
</p>
<p>Even when I increase lx1, reduce dt and reduce the mesh
size, the problem remains..</p>
<p><br>
</p>
<p>Thank you for your time,</p>
<p>Best wishes,</p>
<p><br>
</p>
<p>Arnold<br>
</p>
<p><br>
</p>
<p><br>
</p>
<p><br>
</p>
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