 |
The idea of probabilistic design is that there is a
fundamental difference between a analyzing a single deterministic
analysis scenario on one hand and simulating the entire range of
things that can happen in real life on the other hand.
Before I go into the details let me give you an
illustration of this difference I am talking about.
I guess most of you have a cell phone. I bet all of you
who have dropped you your cell phone you all dropped it at a height of
1.3yards and oriented by an inclination angle of 20 degrees and then
turned left by 30 degrees, didn’t you??
What I just described is a single deterministic
analysis scenario and that is what I have analyzed in this animation.
In reality of course you very probably have not dropped
your cell phone like that, because in reality you could drop a cell
phone at any arbitrary orientation and at any reasonable height. Any
of these possible dropping scenarios has its own probability of
occurrence. Probability of occurrence means for example that it is
very likely that would drop your cell phone from a height of 1 to 2
yards.
That you actually throw away your cell phone leading to
a very large dropping height - that is of course very unlikely.
Although, if you have just received the recent phone bill from your
wireless phone company then some people might actually think of
throwing the cell phone away, but this is of course a completely
different story.
The fact that in real life some scenarios are more
likely to occur then others and the fact there are things that can
happen and very likely can happen can happen that are never covered in
a single deterministic analysis run - this brings me to the
probabilistic design system.
|
 |
If you analyze a product, then
the product and how it is operated is described by certain input
parameters such as material, geometry and boundary conditions. As a
result of the analysis we get certain results like stresses, strains
and so on.
In real life all of these
input parameters are subjected to scatter - all of them are uncertain.
If you measure material properties you will directly see the scatter.
Also the geometrical extensions of a component can only be reproduced
within certain tolerances. In many cases the boundary conditions are
highly uncertain - think of the orientation of the cell phone.
( As a direct and unavoidable
consequence of scatter on the input side is the fact that the results
are uncertain as well.
|
 |
In a situation like that
probabilistic methods can be used to answer the following questions:
If we have scatter on the input
side how wide is the scatter on the output side? Is it negligible?? Is
it large??
If we do have scatter on
the output side by chance some the possible scenarios might no longer
fulfil certain design criteria that are defined in terms of output
parameters. This is then referred to as a failure probability.
If we do have such a
failure probability then of course we want to do something about it
and reduce it. In this case we ask what are the most important and
sensitive drivers on the input side are so we can tackle them.
|
 |
Another example where
uncertainties can be important, but are usually ignored is the
calculation of thermal stresses. The thermal stress is proportional to
the Young’s modulus, the thermal expansion coefficient and
temperature difference.
In the deterministic approach
the scatter of the two material parameter is typically ignored and
only the mean values E,mean and Alpha,mean are used to calculate the
expected thermal stress.
In a probabilistic approach we
can first take into account that young’s modulus is subjected to
scatter, for example with a standard deviation of ±5%. As a result of
the probabilistic analysis we will find that in about 1 out of 6 cases
the thermal stresses are at least 5% higher than the deterministically
calculated “expected” thermal stress. Still in 1 out of 40 cases
the thermal stresses are even more than 10% higher than the
“expected” thermal stress.
The situation get even worse if
we also include the scatter of the thermal expansion coefficient. If
both material are subjected to scatter (which is what they are!) then
we have even 1 out of 5 cases where the thermal stress exceeds the
deterministic result by 5% and 1 out of 12 cases where the thermal
stresses are in excess of 10% higher than the deterministic result.
It should be noted, thermal
stress that are 5% or 10% higher than expected don’t seem an awful
lot, but if we use these stress results in a low cycle fatigue
lifetime calculation then these differences can easily amount to a
factor of in the resulting lifetime.
|
 |
In the general case we analyze
components and products according to a much more complex process. For
example, the geometry is taken from a CAD model, which is then used in
a fluid dynamic analysis (CFD) and also a thermal as well as a
structural analysis is performed. Eventually we are also interested in
calculating the lifetime based on the temperature and stress results.
What you see here is a rough
guess of what the uncertainties are associated with the individual
input variables of the calculation tools and steps. The higher number
are rather applicable for the MEMS area (micro-electromechanical
systems) and the lower number are applicable rather for all other
cases. Please note that the uncertainties apply to multiple input
variables in the relevant databases.
As a result of of these
uncertainties on the input side we can really ask ourselves what the
accuracy and the scatter of the overall lifetime result will be. To
address this question this is what probabilistic design is all about.
|
 |
This slide shows a comparison
between the deterministic and probabilistic approaches:
• Deterministic
analyses can only provide a binary information derived from the single
point solution (the analyzed component is OK or not). Probabilistic
methods provide also a probability for the solutions, e.g. a “design
for reliability” is possible.
• In
deterministic analyses uncertainties are taken into account by safety
margins that are stacked up blindly. I.e. we are designing for a
component with the weakest geometry and worst material
properties subjected to highest
load. Such a component is literally not existing. This leads to
an expensive over-design. In PDS the uncertainties are taken into
account in the way they appear in nature.
This way an over-design is avoided, which can safe a lot of
money.
|
 |
This slide shows a comparison
between the deterministic and probabilistic approaches:
• Deterministic
Methods only assess one specific design, whereas PDS
takes the deviations from that design into account that could
lead to a tolerance stack-up. Taking this into account enables a
smarter design-for-manufacturability, which also can safe a lot of
money.
• Deterministic
models have no concept of
the deviations that are possible apart from the “as planned”
design. In PDS this is inherently taken into account.
• Also
in the deterministic approach sensitivities can be evaluated. But this
is done by applying a plus and minus Delta-value to an input parameter
while keeping all others constant. This way interactions cannot be
covered. Interactions lead to the fact that a variation of two
parameters at the same time can have a much larger effect then the
combined effect of the variation of the two individual
parameters if varied one at a time. According to experience
these interaction are important for about 20% of all applications.
This is inherently taken into account in probabilistic sensitivities.
|
 |
I don’t want go into very
deeply into the details of probabilistic methods.
I just want to mention that we
have implemented two methods, namely Monte Carlo Simulations and
Response Surface Methods, which are most commonly used and widely
accepted. Each of these two methods have their own different sampling
techniques or sub-types if you will.
What makes these two types of
methods even more interesting is that they are most suited for
parallel, distributed processing. If one of these methods require say
a 100 analysis runs and you have enough computers available, then you
can run all these 100 runs in parallel, because they are completely
independent.
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
Speaking of parallel distributed
processing I think you all know how it works so I can go quickly over
this slide.
Each job is sent from a client
to a different server machine. The job is execute and the results
parameters are shipped back to the client.
It is quite a challenge to
implement a fully heterogeneous and cross-platform solution. But when
ANSYS 5.7 will be delivered by the end of the year then will
have the probabilistic design system in it and it will be based on a
fully heterogeneous parallel distribution solution
|
 |
As you know ANSYS is widely used
in the industry and here you see only a few of our customers who gave
us permission to show their company logo.
Our probabilistic design system
is already being tested by 35 companies worldwide and some of
them are illustrated with the green circles. The ANSYS Probabilistic
Design System will be an integral part of ANSYS 5.7.
|
 |
This application example is a
Micro-Eletromechanical device.
This is the package and inside
is the chip and on the chip is the device.
To give you an impression about
the size - this is an American "one cent" coin with a
diameter of 19mm. The device has a size of about 0.2mm.
|
 |
The device is widely used a
shock sensor, for example to deploy the airbag in case of a crash. But
I want to show the use of the device as an electromechanical filter.
As you see here the device is
electro-statically charged at the static comb and then due to
electro-static forces the moving comb gets pulled into the static
comb. The moving comb is supported by a structure that acts as a
bending spring and the spring is fixed at two points with pins in the
center. Obviously, the underlying problem involves a multi-physics
approach, because the solution is ultimately an equilibrium of the
electromagnetic forces and the elastic spring back forces of the
structure.
The manufacturing process of the
device kind of a photo process where the geometry of the device is
“burned” into the silicon chip. Since, the device is so tiny the
inaccuracy of this process results in a large amount of scatter in the
geometrical extensions of the members of the device. In this case I
have used 14 random input variables to describe the manufacturing
scatter of geometry parameters such as lengths and widths of the
fingers and the spring beams of the device. Also the scatter of two
material properties was taken into account.
As the relevant output parameter
I am looking at the deflection of how far the moving comb gets pulled
into the static part.
|
 |
For the entire sensor model I
performed a probabilistic analysis with all 14 random geometry and the
2 random material parameters. Here, the results of 400 Monte Carlo
Simulations and the Response Surface Method involving 49
Finite-element runs and 10’000 simulation runs using the derived
response surface are used. Without going into details I just want to
mention that the 10'000 simulations on the response surface took about
1 second to run, so it is the Finite-Element runs that are
"expensive" in terms of computations time, and here you
compare the 400 runs for Monte Carlo with the 49 for Response Surface
Methods.
From the histogram shown on the
right side we can see two important findings:
First there is very good
agreement between the Monte Carlo results and the response surface
results.
Secondly, the scatter range for
the maximum deflection of the moving comb has a factor of about 3
between the lowest and the largest value. Again this shows that the
uncertainties involved on the material and geometry side have a huge
influence on the result parameter.
|
 |
Shown in this slide is the
cumulative distribution curve of the maximum deflection.
Again the results of the
Response surface method show very good agreement with that of the
Monte Carlo simulation method.
You can use a curve like that to
design a product to achieve a certain required reliability. For
example, if the deflection of the moving comb must be between 0.0065
and 0.0105. The curve says that there is s probability of about 17% to
drop below the 0.0065 limit and a 7% (that is 100% minus 93%)
probability of exceeding the 0.0105 limit. Together, this makes up for
a 24% of probability of falling out side the required interval. In
other word in this example the device would have a reliability of 76%.
If this is not satisfactory then the sensitivities shown in the
following slide can be used to improve the design.
Generally speaking, if you
design a component for reliability then the quality control costs and
the warranty costs can be minimized.
Again, probabilistic methods can
help to make more qualified decisions about the efficient use of
valuable resources money.
|
 |
Again, also for the
sensitivities there is very good agreement between the results of the
two methods. The response surface method has a slight advantage here,
because the sensitivities here are based on 10’000 data points, so
it has better resolution to look into the details of smaller
sensitivities, where Monte Carlo with only 400 data points can no
longer safely detect the significance of these smaller sensitivities.
You can use the findings to more
efficiently guide the design process towards improving of the product
or to guide the quality control process toward looking into the right
and important parameters.
In addition, you can also use
this information to justify the spending of money at the right place
where it is most efficiently spent. For example the material database
is very often based on a rather small number of samples (specimen).
Then it certainly makes sense to spend money for more measurements to
get a more accurate understanding of the scatter of the Young’s
modulus. To spend more money to more accurately measure the
Poisson’s ratio would be a waste, because the Poisson’s ratio
turned out to be insignificant.
Again, probabilistic methods can
help to make more qualified decisions about the efficient use of
valuable resources such as time and money.
|
 |
As an example for the
application of probabilistic methods the analysis of a turbine blade
is shown here. The example is a cooled (hollow) rotating blade. The
probabilistic analysis includes the randomness of a total of 17 input
parameters. For example, the blades are manufactured by precision
casting. During casting of the blade a slight shift of the core that
makes up the hollow cavity can occur. This core shift makes the wall
of blade thinner on one side and thicker on the respective other. Also
there is an oxidation protection coating on the hotgas surface of the
blade. The thickness of the coating is not an exact value after it is
applied, but variations from the targeted thickness may appear.
It is not necessary to explain
all random input parameters as listed here. Suffice it to say that the
random input parameters are from all categories, namely geometry,
material and loads. Also it should be emphasized that various
different statistical distribution function can be applied to describe
and quantify the randomness of the input parameter, such as the
Gaussian distribution, the uniform distribution or the lognormal
distribution (the ANSYS/PDS has many more)
The Finite-Element model has
about 60’000 elements and 180’000 nodes. One single analysis run
includes a thermal analysis to evaluate the temperature field (shown
here) and a structural analysis to evaluate the thermo-mechanical
stresses. Based on these result also the low cycle fatigue lifetime
(LCF), the creep lifetime and the time until the oxidation protection
layer has been eroded through is calculated. One such complete
analysis takes about 2.5 hours.
|
 |
Crucial in today’s business
environment is the development of reliable products. Only reliable
products keep the occurrence of premature failures (a failure that
happens before the end of the warranty period) at an acceptable level
or avoid such failures completely.
The most important measure for a
reliable product is a low failure probability. As a result of the
probabilistic analysis in this diagram the probability of a failure
due to one of the three failure modes (LCF, creep, oxidation) is
plotted versus the operation time in years. A particular failure
probability can be derived from this plot by choosing a value on the
X-axis for the operation time (i.e. the time how long the blade is
supposed to be in service) and then going up to the probability curves
related to the failure modes and reading the probability on the
Y-Axis.
In this diagram the results
calculated with the “response surface method” is compared with the
results gained from 500 Monte Carlo simulations. in this example, the
Monte Carlo Simulation results represent kind of a “true”
benchmark values which the “response surface method” result must
comply with for the failure probability ranging from 2% to 98%.
Obviously, there is a very good agreement between the results of the
two methods in this probability range.
Designing a product for a low
failure probability ultimately leads to “built-in-quality”, which
leads to reduced costs for the manufacturer and an increase in
customer satisfaction.
|
 |
Probabilistic methods also
automatically and inherently deliver probabilistic sensitivities.
Probabilistic sensitivities describe how much the scatter (or the
failure probability) of a particular random output parameter (shown
here is the LCF lifetime) is affected by the scatter of the individual
random input parameters. The probabilistic design system in ANSYS
sorts the input parameters into two groups - the significant and the
insignificant ones. Then the significant input parameters are ranked
by the importance and plotted.
These probabilistic
sensitivities proved highly values information in three ways.
1.) If the resulting failure
probability is not acceptable, i.e. too high (for example as derived
for the previous diagram) then we clearly need to improve the design
in order to achieve an acceptable level. The sensitivities clearly
indicate which input parameters are the drivers of the high failure
probability and therefore must be tackled in order to reduce it.
Hence, the input parameters must be tackled in the order of their
importance. There is no point in focusing on unimportant parameters.
2.) Sometimes the scatter of
some input parameters are just estimated based on no or very little
measurement data. If these parameters turn out to be very important
for the reliability of the design then this clearly indicates that for
example lab test must be done to collect more data about that input
parameter.
3.) If the current design is
sufficient, i.e. has an acceptably low failure probability, then there
is typically the need to save money without sacrificing the achieve
reliability. In this case the manufacturing requirements for the input
parameters can be relaxed an a possibly coarser and cheaper
manufacturing process can be chosen. Or the quality assurance
requirements for those parameters can be relaxed. This typically leads
to huge saving for manufacturing process related to geometry
parameters.
|
 |
With this I would like to
summarize my presentation.
I hope I managed to illustrate
that probabilistic methods are a very powerful tools to take the
randomness and the uncertainties into account as they occur everywhere
in real life.
In industrial applications Monte
Carlo Methods and Response Surface Methods are mostly used and
established and they are even getting much more accepted due to the
advances in computer technology and parallel distributed computing,
for which both of these methods are perfectly suited.
I illustrated that using
probabilistic methods you can design for more reliable products, a
higher quality of your products and more robust products. And all of
that by saving a lot of money at the same time.
I want to conclude the
presentation with a quote from the founder of our company Dr. John
Swanson: "A couple of years ago the engineering community had to
learn the world is not linear but is better described using non-linear
methods. We are now about to learn that the world is not deterministic
and is more realistically described by probabilistic methods."
And with this statement I want
to finish my presentation and I thank you for your attention.
|