von Karman's question
Pavel Solin
let me share a funny thing. Many people
(including ourselves) present computer simulations
of non-symmetric von Karman vortices using symmetrical
problems, such as flow past symmetrically-placed
round or square obstacles. See, for example,
http://www.youtube.com/watch?v=mCh809uHBSE
(you have to wait until half of the video for the
nonsymmetry to occur).
However, most likely the solution has to stay symmetric
and the chaotic effects are caused by non-symmetry of
the mesh and finite computer arithmetic only. For the last
few days I occupied myself with many numerical experiments
and it seems that adaptivity with dynamical meshes does
greatly reduce these artifacts, see
http://hpfem.math.unr.edu/people/pavel/work/roznov.pdf
(still unfinished).
There is no doubt that von Karman vortices exist, of
course, but imho one has to introduce a physical nonsymmetry
to the problem. This is much worth further investigation,
because many people perform these and various
fluid-structure interaction simulations, including airplane
wing flutter, in *completely symmetric settings* and they
obtain nonsymmetric results. I am almost sure that what
they are showing are just numerical mistakes!
Best,
Pavel
--
Pavel Solin
University of Nevada, Reno
http://hpfem.math.unr.edu/people/pavel/
Hermes project: http://hpfem.org/
FemHub project: http://femhub.org/
Brian Granger
correct. Anytime you have a symmetry like this the solutions will
know about the symmetry.
I can imagine though what people will give as an excuse though... I
think they would say something like:
"In reality, there are small physical perturbations that make the real
physical system non-symmetric, so we are fine."
I don't agree with this, because physical asymmetry can be quite
different from asymmetric numerical artifacts. But, there is possibly
a very subtle dynamical symmetry breaking that happens in these
systems. The best thing to do to really understand this physics is to
introduce an actual asymmetry into the problem that is characterized
by some strength parameter (call it "a"). Then study your solutions
as a goes from 0 to + or - some small value. By doing this you can
make sure that the solutions follow the physical asymmetry (induced by
a) rather than any asymmetry from any numerical artifacts.
This reminds me of My Ph.D. advisor, Chris Greene, who used to always
ask us this question" "are you actually solving the differential
equation you claim to be?"
If the supposed solutions don't respect the symmetry of the original
differential equation, then they can't really be solutions.
This is really cool!
Cheers,
Brian
Ondrej Certik
>
> I am not familiar with this physics, but I think you *must* be
> correct. Anytime you have a symmetry like this the solutions will
> know about the symmetry.
There is no doubt that if you refine the mesh and the solution changes
dramatically like shown in Pavel's presentation, then the solution is
wrong. That part is clear. (That's also the reason why I like the
adaptive approach to FEM, because without it one doesn't really know
anything.)
What is not clear to me if the Karman's vertex is the solution of the
symmetric problem or not.
Pavel, I really like Brian's suggestion: what about introducing a
physical asymmetry by putting a "rod" of a length "a" nearby the
column (maybe even attaching it to it?), then solve it adaptively,
that should introduce the vertex and then go a->0. If you go "a->0",
e.g. a=0.000001, is the solution still the Karman's vertex? It may be
the case, that the solution for a=0 is symmetric, but for a=0.00001,
you always get the Karman's vertex. Well, I don't understand it, what
the solution actually is.
If I was at the lecture, this would be my question.
Ondrej
Pavel Solin
thanks for taking the time to read this and for your comments!
"In reality, there are small physical perturbations that make the real
physical system non-symmetric, so we are fine."
of computational mathematics. A numerical method is supposed
to solve the idealized model, it does not solve the physical reality.
Overall, computational fluid dynamics is much less understood
than many other computational sciences. There is a huge lack
of theoretical results (existence, uniqueness, etc.). That's one of
the reasons why people often check the numerical results against
"physical reality" instead of checking against exact solution of
the model. It will take many years before this paradigm is changed.
The example with the artificial oscillations is a very good tool for
this.
Of course no numerical method can really eliminate the problem
since there always is a nonzero approximation error. But one should
know when one can trust the results and when not. I am now playing
with higher Reynolds numbers and other symmetric geometries of
the obstacles, to gain more intuition.
Best,
Pavel
Pavel Solin
Brian's suggestion is really useful. The problem is deeper
than I first thought, I'll need some time to figure out
a strategy that could lead to a significant result. What I
have shown is numerical evidence, not a mathematical
proof. Perfect would be to prove mathematically that
the solution of a symmetric problem must be symmetric,
but I do not know if this is possible at all. I know people who
study theoretical aspects of the N-S equations so let me
talk to them.
> by putting a "rod" of a length "a" nearby the
Since the circular obstacle is defined using four identical
NURBS curves, it appears much easier to me to just change
a parameter of one of them.
> If you go "a->0", e.g. a=0.000001, is the solution still
> the Karman's vertex?
here is that the solution in the idealized symmetric case is
unstable - even an extremely subtle non- symmetry throws
it off-balance.
Pavel
Brian Granger
> than I first thought, I'll need some time to figure out
> a strategy that could lead to a significant result. What I
> have shown is numerical evidence, not a mathematical
> proof. Perfect would be to prove mathematically that
> the solution of a symmetric problem must be symmetric,
> but I do not know if this is possible at all. I know people who
> study theoretical aspects of the N-S equations so let me
> talk to them.
The relevant thing to prove would be this:
If the boundary conditions are symmetric about x=0 and the initial
condition is also symmetric, then the solutions will also be symmetric
at t>0.
What is the exact form of the PDE you are solving?
>> by putting a "rod" of a length "a" nearby the
>
> This would require complicated remeshing for every new 'a'.
> Since the circular obstacle is defined using four identical
> NURBS curves, it appears much easier to me to just change
> a parameter of one of them.
Yes, I think this makes the most sense rather than introducing
something entirely new.
Pavel Solin
The relevant thing to prove would be this:
> Brian's suggestion is really useful. The problem is deeper
> than I first thought, I'll need some time to figure out
> a strategy that could lead to a significant result. What I
> have shown is numerical evidence, not a mathematical
> proof. Perfect would be to prove mathematically that
> the solution of a symmetric problem must be symmetric,
> but I do not know if this is possible at all. I know people who
> study theoretical aspects of the N-S equations so let me
> talk to them.
If the boundary conditions are symmetric about x=0 and the initial
condition is also symmetric, then the solutions will also be symmetric
at t>0.
What is the exact form of the PDE you are solving?
Standard incompressible Navier-Stokes equations.
Yes, I think this makes the most sense rather than introducing
>> by putting a "rod" of a length "a" nearby the
>
> This would require complicated remeshing for every new 'a'.
> Since the circular obstacle is defined using four identical
> NURBS curves, it appears much easier to me to just change
> a parameter of one of them.
something entirely new.
Robert Cimrman
Pavel Solin wrote:
> Hi Brian,
> thanks for taking the time to read this and for your comments!
>
> "In reality, there are small physical perturbations that make the real
> physical system non-symmetric, so we are fine."
>
> You are absolutely right, and this is a profound misunderstanding
> of computational mathematics. A numerical method is supposed
> to solve the idealized model, it does not solve the physical reality.
> Overall, computational fluid dynamics is much less understood
> than many other computational sciences. There is a huge lack
> of theoretical results (existence, uniqueness, etc.). That's one of
> the reasons why people often check the numerical results against
> "physical reality" instead of checking against exact solution of
> the model. It will take many years before this paradigm is changed.
> The example with the artificial oscillations is a very good tool for
> this.
Playing the devil's advocate: in practice, _all_ you care about is the
experimental verification, not some abstract theoretical one.
Consider:
reality -(1)- idealized model -(2)- numerical solution
\----------------(3)-----------------------/
People often shortcut by "joining" the verification steps (1) and (2) by
checking (3), and when it works, it's ok in a pragmatic way.
But you are the mathematician, so I am looking forward to what you come
about - the question at hand has some deep philosophical background
(modelling reality by continuous models, why is it possible at all?,
nature of chaotic behaviour, ...), so I stay tuned :)
r.
Pavel Solin
Hi Pavel,
Playing the devil's advocate: in practice, _all_ you care about is the
Pavel Solin wrote:
> Hi Brian,
> thanks for taking the time to read this and for your comments!
>
> "In reality, there are small physical perturbations that make the real
> physical system non-symmetric, so we are fine."
>
> You are absolutely right, and this is a profound misunderstanding
> of computational mathematics. A numerical method is supposed
> to solve the idealized model, it does not solve the physical reality.
> Overall, computational fluid dynamics is much less understood
> than many other computational sciences. There is a huge lack
> of theoretical results (existence, uniqueness, etc.). That's one of
> the reasons why people often check the numerical results against
> "physical reality" instead of checking against exact solution of
> the model. It will take many years before this paradigm is changed.
> The example with the artificial oscillations is a very good tool for
> this.
experimental verification, not some abstract theoretical one.
a very loud NOT_TRUE here :)
Consider:
reality -(1)- idealized model -(2)- numerical solution
\----------------(3)-----------------------/
People often shortcut by "joining" the verification steps (1) and (2) by
checking (3), and when it works, it's ok in a pragmatic way.
This is not OK. Using your diagram, you need to check two things
which are mutually independent of each other: (1) that the model
describes the reality well. Babuska calls this "validation". (2) That
your numerical methods solves the model well. This is called
"verification". Again Babuska's terminology and very famous
these days. If you merge these two things, you may commit serious
crimes which may work out one time but kill you later. Basically,
you can cancel one mistake with another one.
To give a concrete example, my friends climatologists use
compressible Euler equations (i.e. inviscid flow) with
a no-slip boundary condition for the velocity at height h = 0.
Since they use finite differences, they see a boundary layer
which makes them happy. They believe that their model
works correctly because of this. Once they refine their
mesh in the "boundary layer", however, they will see it
disappear, since there are no boundary layers in inviscid
flow, and eventually the smile will disappear from their faces :)
But you are the mathematician, so I am looking forward to what you come
about - the question at hand has some deep philosophical background
(modelling reality by continuous models, why is it possible at all?,
nature of chaotic behaviour, ...), so I stay tuned :)
Best,
Pavel
r.
Ondrej Certik
I generally agree, only I would add, that as long as you are sure that
you are solving your idealised model correctly, it doesn't make sense
to solve it more precisely than the precision of the idealised model
itself. In particular, it absolutely makes no sense to solve
eigenvalues of schroediger equation for the Pb atom to 6 decimal
digits, when we know, that even the second decimal digit is already
physically wrong (not taking into account relativistic corrections).
E.g. all I want is to solve to 2 decimal digits, however, I want to be
sure that it's correct. So to be sure, I solve to 6 decimal digits,
refine the mesh, do my best, etc (e.g. I do all of this only to be
reasonable sure that my 2 digits are correct). At least that's how I
understand it.
Ondrej
Pavel Solin
You are right. The two things (verification and validation)
are independent. Thus it might be necessary to solve the Pb
atom to six digits in order to determine whether your method
is good or not. After you know that it solves the model correctly,
you can only use as many digits as you find appropriate, that's
a different question.
Pavel
E.g. all I want is to solve to 2 decimal digits, however, I want to be
sure that it's correct. So to be sure, I solve to 6 decimal digits,
refine the mesh, do my best, etc (e.g. I do all of this only to be
reasonable sure that my 2 digits are correct). At least that's how I
understand it.
Ondrej
Robert Cimrman
Pavel Solin wrote:
>>> You are absolutely right, and this is a profound misunderstanding
>>> of computational mathematics. A numerical method is supposed
>>> to solve the idealized model, it does not solve the physical reality.
>>> Overall, computational fluid dynamics is much less understood
>>> than many other computational sciences. There is a huge lack
>>> of theoretical results (existence, uniqueness, etc.). That's one of
>>> the reasons why people often check the numerical results against
>>> "physical reality" instead of checking against exact solution of
>>> the model. It will take many years before this paradigm is changed.
>>> The example with the artificial oscillations is a very good tool for
>>> this.
>> Playing the devil's advocate: in practice, _all_ you care about is the
>> experimental verification, not some abstract theoretical one.
>
>
> a very loud NOT_TRUE here :)
True, it's wrong, but still, people do that. You are 100% right that it
may and sooner or later will backlash them. But often the practical
engineers doing the computations have enough experience to judge a
solution "better" than an automatic scheme agnostic to the underlying
physics. I mean, when an actual engine _is_ more efficient just as a
(wrong) computation predicted, then who cares how they arrived at the
design. I am not one of those, though, so it is great you work on the
adaptivity :)
>>
>> Consider:
>>
>> reality -(1)- idealized model -(2)- numerical solution
>> \----------------(3)-----------------------/
>>
>> People often shortcut by "joining" the verification steps (1) and (2) by
>> checking (3), and when it works, it's ok in a pragmatic way.
>
>
> This is not OK. Using your diagram, you need to check two things
> which are mutually independent of each other: (1) that the model
> describes the reality well. Babuska calls this "validation". (2) That
> your numerical methods solves the model well. This is called
> "verification". Again Babuska's terminology and very famous
> these days. If you merge these two things, you may commit serious
> crimes which may work out one time but kill you later. Basically,
> you can cancel one mistake with another one.
> To give a concrete example, my friends climatologists use
> compressible Euler equations (i.e. inviscid flow) with
> a no-slip boundary condition for the velocity at height h = 0.
> Since they use finite differences, they see a boundary layer
> which makes them happy. They believe that their model
> works correctly because of this. Once they refine their
> mesh in the "boundary layer", however, they will see it
> disappear, since there are no boundary layers in inviscid
> flow, and eventually the smile will disappear from their faces :)
I am not arguing with you, you know :)
Anyway, weather forecasts for more than one day suck in unstable
conditions, so in climatology, I guess, measurements are even more the key.
r.
Pavel Solin
thanks for the discussion. We could talk about this
forever, of course. I really think that we should not
make today's standards our own. When I am 60
and see that I went wrong direction, I will not
be sorry.
Pavel
Robert Cimrman
> Hi Robert,
> thanks for the discussion. We could talk about this
> forever, of course. I really think that we should not
> make today's standards our own. When I am 60
> and see that I went wrong direction, I will not
> be sorry.
I will buy you a beer in Roznov :)
r.
Ondrej Certik
I will have a beer today.
Ondrej