Home  Blog Has Your Simulation Converged?
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Has Your Simulation Converged? |
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Everyone who has run any
numerical simulation has had to ask themselves this question at one
time or another. How do you tell if the simulation is complete
and converged? There are several ways to check for
convergence. The most general is to check the residuals for each
variable being solved, but other more specific checks can be made by
using the integrated summaries and even placing monitor points as a
tell-tale indication of when the solution is “done”. Each of
these approaches will be discussed in this note.
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Why is convergence important?
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The solution process is an
iterative process and by definition values are changing from one
iteration to the next. If the change is significant then that
means the results that you are relying upon for decision making are
therefore also changing significantly. So how can you make good
decisions if you are not confident that the solution is complete?
Sometimes it is obvious and an unconverged solution will show
unphysical results. This should not be a surprise and before
questioning those unphysical results we must satisfy ourselves that the
solver has completed it’s job and produced a converged solution.
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How much convergence is
required?
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Every numerical solution
contains errors. The key is to understand how big those errors
are and whether their level is acceptable in the particular
application. The acceptable level of error can vary
enormously. While a simulation comparing to minute experimental
details for purposes of replacing experimental tests may require high
level of convergence, a simulation to make a go / no-go decision for a
particular design concept may require a lower level of
convergence.
You will often hear terms like “orders of magnitude”, “percent error”,
or “significant digits”. All basically can be related as one
order of magnitude is similar to 10% error and similar to one
significant digit, two orders of magnitude is similar to 1% error and
two significant digits, etc. A general rule of thumb is that 3-5
orders of magnitude of convergence is a good idea.
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How is convergence measured?
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Residuals
– solver residuals
represent the absolute error in the solution of a particular variable.
CFD-ACE sums the absolute value of this error over all the cells in the
simulation and presents that information for each variable for each
iteration in the residual file (modelname.RSL). Because the error is
not normalized by cell count or solution value you will see that the
residual value itself is generally dependent on the scale of the
variable being solved. For example, values for velocity are often on
the order of 1-100 m/s while values for enthalpy may be on the order of
300000 J/kg. For this reason the “residual” for velocity may be
0.001 while the residual for enthalpy may be 1. Only upon looking
at the drop in residuals can we get an indication of the overall level
of convergence. The rule of thumb is to look at the first
residual or the maximum residual calculated and then look for 3-5
orders of magnitude drop from there.
When using the CFD-ACE-GUI residual plotter the Normalize button will
plot the residuals normalized by their largest value. In this case the
residual plots will be more closely spaced on the plot and their values
represent the order of magnitude fall from their peak values. Note that
if your initial guess values for a steady calculation are far away from
the final solution then the initial residuals will be quite large
relative to that of the converged solution. In that case, it may be
necessary to obtain additional orders of magnitude residual reduction
before full convergence is reached.
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Printed
Summaries – using the
printed summaries options you can get information like mass flow, heat
transfer, species flux, electric current, etc. integrated through all
boundaries in the system. A converged solution will show that the
total quantity of say mass coming into the system equals that going
out. The difference between incoming and outgoing is called the
imbalance and it should be zero ( or small relative to the incoming
value).
Monitor Points – monitor
points allow you to track a variable’s value
at a particular location. When the value stops changing you could
be near convergence. You should be aware, however, that variable values
will approach their final converged value at different rates depending
upon where they lie in the flow field. Generally, the smallest flow
structures are the slowest to converge.
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Is a converged solution
“correct”?
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A converged solution means that
the solver has done it’s job and produced the best solution it can for
the given problem. However, that does not guarantee that the
solution is correct. There are other factors that are involved,
most notably the influence of discretization (grid system) and problem
definition (user setup). A poor quality or overly coarse grid
system as well as user errors in setup could cause the solver to
produce inaccurate results.
If this topic interests you, then please read on to see a case study
which puts these ideas into practice.
If you have any questions about this feature or would like us to
discuss some other topic in the future, please let us know.
Regards,
Richard Thoms
Manager, ESI-Group CFD Customer Support
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Case Study: Laminar Flow over
a
Backward Facing Step
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A low Reynolds steady flow is
sent over a backward facing step. For this note we will only
discuss the convergence of the “U” equation (velocity in the
x-direction) but the same arguments and details could be shown for the
“V” and “P” equations. The simulation was run for 1000 iterations
and we will look at the level of convergence after 1, 10, 100, and 1000
iterations.
Residual File – a quick
look at the residual plot shows that
the residuals reached a level of 1E-17 after
approximately 500 iterations and remained flat thereafter. This
indicates that the solution reached “machine accuracy” and in fact
running anything after 500 iterations was not causing any change in the
solution.
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Iteration
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U
Residual
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Comment
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1
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5.08E-04
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initial
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2
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2.80E-02
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peak
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10
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4.15E-04
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2
orders from peak
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100
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1.98E-06
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4
orders from peak
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1000
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1.10E-017
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15
orders from peak
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So according to the guidelines
we consider this case converged four orders of magnitude after just 100
iterations.
Printed Summaries – for this case we
asked for the solver to write the
printed mass flow summary. After 1000 iterations that summary
looks like this: |
You can see that the inflow is
equal to the outflow to 6 significant digits and in fact the
imbalance/inflow indicates that we have 16 orders of magnitude of
convergence. Looking at selected snapshots of the mass flow summary
after 1, 10, 100, and 1000 iterations
shows the following:
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Guidelines indicate that the
mass flow summary shows 5 orders of convergence after just 10
iterations and if mass flow is all we were looking for then that may
have been good enough, but in this case we want to see the flow
recirculation and that may take longer to develop. This shows
that even if mass flow continuity is satisfied it may not be sufficient
to
determine if the simulation is converged.
Monitor Points – we put a monitor
point just 1 step height downstream
form the step in the center of the channel. The plot of that data
can be seen below and it is clear that after 100 iterations the U
velocity is not changing at that location anymore. Looking at
selected snapshots of the monitor point data around 1, 10, 100, and
1000 iterations shows the following:
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So according to the guidelines
we could say that the monitor point showed 5 orders of magnitude of
convergence after just 100 iterations.
Based upon all of the above
convergence tests (residuals, mass flow
summary, monitor points) we can safely say that this case was converged
to a suitable level of accuracy after just 100 iterations. Below
are contour plots of the x-direction velocity taken at various
iterations. It is clear that the solution at 100 iterations and
1000 iterations are quite similar.
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