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<p>Thank you for the email, Zak,</p>
<p>I have not looked into continuation methods for this problem yet,
and I'd love to hear your thoughts on it!</p>
<p>The problem I have is two-fold:</p>
<ol>
<li>Obtaining the eigenvectors of the system <br>
K.v = lam*M.v<br>
where the K and M are stiffness and mass matrices coming from a
finite element model without any Dirichlet boundary conditions
(i.e., having 6 Rigid body modes) and with a few RBE3
constraints (introducing degrees of freedom with "zero mass" in
the mass matrix). So the system has, in addition to the 6 rigid
body modes, a few "spurious" null vectors coming from these RBE3
constraints (which are enforced using Lagrange multipliers). <br>
</li>
<li>Conducting a linear solve of the system:<br>
K.x = b<br>
where K is from above.</li>
</ol>
<p>What I'm trying to do with both of these is to conduct a
Hurty/Craig-Bampton Component Mode Synthesis, which is like a
Schur Condensation with a few "fixed interface modal" DoFs added
to the reduced system.</p>
<p>So both can be solved by obtaining the vectors that span the null
space of the system.<br>
</p>
<p>I've created two separate threads for the two problems because I
felt both are slightly different questions. But I understand now
that the whole thing boils down to obtaining the nullspaces.<br>
</p>
<p>Thank you,<br>
Nidish<br>
</p>
<div class="moz-cite-prefix">On 8/17/20 1:33 PM, zakaryah wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CABm8tB6kv74G1E4XSsa1RZ+cLnkK4zA0anfqeP8Y0Z9_ZT9dUQ@mail.gmail.com">
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<div class="gmail_default" style="font-size:small">Hi Nidish,</div>
<div class="gmail_default" style="font-size:small"><br>
</div>
<div class="gmail_default" style="font-size:small">I may not
fully understand your problem, but it sounds like you could
benefit from continuation methods. Have you looked into this?
If it's helpful, I have some experience with this and I can
discuss with you by email.</div>
<div class="gmail_default" style="font-size:small"><br>
</div>
<div class="gmail_default" style="font-size:small">Cheers, Zak<br>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Mon, Aug 17, 2020 at 2:31
PM Nidish <<a href="mailto:nb25@rice.edu"
moz-do-not-send="true">nb25@rice.edu</a>> wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0px 0px 0px
0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">Thankfully
for this step, the matrix is not dense. But thank you.<br>
<br>
Nidish<br>
<br>
On 8/17/20 8:52 AM, Jose E. Roman wrote:<br>
><br>
>> El 17 ago 2020, a las 14:27, Barry Smith <<a
href="mailto:bsmith@petsc.dev" target="_blank"
moz-do-not-send="true">bsmith@petsc.dev</a>> escribió:<br>
>><br>
>><br>
>> Nidish,<br>
>><br>
>> Your matrix is dense, correct? MUMPS is for
sparse matrices.<br>
>><br>
>> Then I guess you could use Scalapack <a
href="http://netlib.org/scalapack/slug/node48.html#SECTION04323200000000000000"
rel="noreferrer" target="_blank" moz-do-not-send="true">http://netlib.org/scalapack/slug/node48.html#SECTION04323200000000000000</a>
to do the SVD. The work is order N^3 and parallel efficiency
may not be great but it might help you solve your problem.<br>
>><br>
>> I don't know if SLEPc has an interface to
Scalapack for SVD or not.<br>
> Yes, SLEPc (master) has interfaces for ScaLAPACK and
Elemental for both SVD and (symmetric) eigenvalues.<br>
><br>
> Jose<br>
><br>
>> Barry<br>
>><br>
>><br>
>><br>
>><br>
>><br>
>><br>
>><br>
>>> On Aug 17, 2020, at 2:51 AM, Jose E. Roman <<a
href="mailto:jroman@dsic.upv.es" target="_blank"
moz-do-not-send="true">jroman@dsic.upv.es</a>> wrote:<br>
>>><br>
>>> You can use SLEPc's SVD to compute the nullspace,
but it has pitfalls: make sure you use an absolute convergence
test (not relative); for the particular case of zero singular
vectors, accuracy may not be very good and convergence may be
slow (with the corresponding high computational cost).<br>
>>><br>
>>> MUMPS has functionality to get a basis of the
nullspace, once you have computed the factorization. But I
don't know if this is easily accessible via PETSc.<br>
>>><br>
>>> Jose<br>
>>><br>
>>><br>
>>><br>
>>>> El 17 ago 2020, a las 3:10, Nidish <<a
href="mailto:nb25@rice.edu" target="_blank"
moz-do-not-send="true">nb25@rice.edu</a>> escribió:<br>
>>>><br>
>>>> Oh damn. Alright, I'll keep trying out the
different options.<br>
>>>><br>
>>>> Thank you,<br>
>>>> Nidish<br>
>>>><br>
>>>> On 8/16/20 8:05 PM, Barry Smith wrote:<br>
>>>>> SVD is enormously expensive, needs to be
done on a full dense matrix so completely impractical. You
need the best tuned iterative method, Jose is the by far the
most knowledgeable about that.<br>
>>>>><br>
>>>>> Barry<br>
>>>>><br>
>>>>><br>
>>>>>> On Aug 16, 2020, at 7:46 PM, Nidish
<<a href="mailto:nb25@rice.edu" target="_blank"
moz-do-not-send="true">nb25@rice.edu</a>> wrote:<br>
>>>>>><br>
>>>>>> Thank you for the suggestions.<br>
>>>>>><br>
>>>>>> I'm getting a zero pivot error for
the LU in slepc while calculating the rest of the modes.<br>
>>>>>><br>
>>>>>> Would conducting an SVD for just the
stiffness matrix and then using the singular vectors as bases
for the nullspace work? I haven't tried this out just yet, but
I'm wondering if you could provide me insights into whether
this will.<br>
>>>>>><br>
>>>>>> Thanks,<br>
>>>>>> Nidish<br>
>>>>>><br>
>>>>>> On 8/16/20 2:50 PM, Barry Smith
wrote:<br>
>>>>>>> If you know part of your null
space explicitly (for example the rigid body modes) I would
recommend you always use that information explicitly since it
is extremely expensive numerically to obtain. Thus rather than
numerically computing the entire null space compute the part
orthogonal to the part you already know. Presumably SLEPc has
tools to help do this, naively I would just orthogonalized
against the know subspace during the computational process but
there are probably better ways.<br>
>>>>>>><br>
>>>>>>> Barry<br>
>>>>>>><br>
>>>>>>><br>
>>>>>>><br>
>>>>>>><br>
>>>>>>>> On Aug 16, 2020, at 11:26 AM,
Nidish <<a href="mailto:nb25@rice.edu" target="_blank"
moz-do-not-send="true">nb25@rice.edu</a>> wrote:<br>
>>>>>>>><br>
>>>>>>>> Well some of the zero
eigenvectors are rigid body modes, but there are some more
which are introduced by lagrange-multiplier based constraint
enforcement, which are non trivial.<br>
>>>>>>>><br>
>>>>>>>> My final application is for a
nonlinear simulation, so I don't mind the extra computational
effort initially. Could you have me the suggested solver
configurations to get this type of eigenvectors in slepc?<br>
>>>>>>>><br>
>>>>>>>> Nidish<br>
>>>>>>>> On Aug 16, 2020, at 00:17,
Jed Brown <<a href="mailto:jed@jedbrown.org"
target="_blank" moz-do-not-send="true">jed@jedbrown.org</a>>
wrote:<br>
>>>>>>>> It's possible to use this or
a similar algorithm in SLEPc, but keep in mind that it's more
expensive to compute these eigenvectors than to solve a linear
system. Do you have a sequence of systems with the same null
space?<br>
>>>>>>>><br>
>>>>>>>> You referred to the null
space as "rigid body modes". Why can't those be written
down? Note that PETSc has convenience routines for computing
rigid body modes from coordinates.<br>
>>>>>>>><br>
>>>>>>>> Nidish <<br>
>>>>>>>> <a
href="mailto:nb25@rice.edu" target="_blank"
moz-do-not-send="true">nb25@rice.edu</a><br>
>>>>>>>>> writes:<br>
>>>>>>>><br>
>>>>>>>> I just use the standard eigs
function (<a
href="https://www.mathworks.com/help/matlab/ref/eigs.html"
rel="noreferrer" target="_blank" moz-do-not-send="true">https://www.mathworks.com/help/matlab/ref/eigs.html</a><br>
>>>>>>>> ) as a black box. I think it
uses a lanczos type method under the hood.<br>
>>>>>>>><br>
>>>>>>>> Nidish<br>
>>>>>>>><br>
>>>>>>>> On Aug 15, 2020, 21:42, at
21:42, Barry Smith <<br>
>>>>>>>> <a
href="mailto:bsmith@petsc.dev" target="_blank"
moz-do-not-send="true">bsmith@petsc.dev</a><br>
>>>>>>>>> wrote:<br>
>>>>>>>><br>
>>>>>>>> Exactly what algorithm are
you using in Matlab to get the 10 smallest<br>
>>>>>>>> eigenvalues and their
corresponding eigenvectors?<br>
>>>>>>>><br>
>>>>>>>> Barry<br>
>>>>>>>><br>
>>>>>>>><br>
>>>>>>>><br>
>>>>>>>> On Aug 15, 2020, at 8:53 PM,
Nidish <<a href="mailto:nb25@rice.edu" target="_blank"
moz-do-not-send="true">nb25@rice.edu</a><br>
>>>>>>>>> wrote:<br>
>>>>>>>> The section on solving
singular systems in the manual starts with<br>
>>>>>>>><br>
>>>>>>>> assuming that the singular
eigenvectors are already known.<br>
>>>>>>>><br>
>>>>>>>><br>
>>>>>>>> I have a large system where
finding the singular eigenvectors is not<br>
>>>>>>>><br>
>>>>>>>> trivially written down. How
would you recommend I proceed with making<br>
>>>>>>>> initial estimates? In MATLAB
(with MUCH smaller matrices), I conduct an<br>
>>>>>>>> eigensolve for the first 10
smallest eigenvalues and take the<br>
>>>>>>>> eigenvectors corresponding to
the zero eigenvalues from this. This<br>
>>>>>>>> approach doesn't work here
since I'm unable to use SLEPc for solving<br>
>>>>>>>><br>
>>>>>>>><br>
>>>>>>>> K.v = lam*M.v<br>
>>>>>>>><br>
>>>>>>>> for cases where K is positive
semi-definite (contains a few "rigid<br>
>>>>>>>><br>
>>>>>>>> body modes") and M is
strictly positive definite.<br>
>>>>>>>><br>
>>>>>>>><br>
>>>>>>>> I'd appreciate any assistance
you may provide with this.<br>
>>>>>>>><br>
>>>>>>>> Thank you,<br>
>>>>>>>> Nidish<br>
>>>>>>>><br>
>>>>>> -- <br>
>>>>>> Nidish<br>
>>>> -- <br>
>>>> Nidish<br>
-- <br>
Nidish<br>
</blockquote>
</div>
</blockquote>
<div class="moz-signature">-- <br>
Nidish</div>
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