Transcript AMNIRS-II: Thesis proposal

Probabilistic Analysis using FEA
A. Petrella
What is Probabilistic Analysis
‣ All input parameters have some uncertainty
‣ What is the uncertainty in outcome metrics?
‣ How sensitive are outcomes to different inputs
‣ Which inputs are most important and how can we
design for a specific probability of performance?
What is Probabilistic Analysis
Outcome
Probabilities &
Sensitivities
Validated
Deterministic Model
Probability
Model Input
Uncertainties
Tissue Properties
Performance Metric
1
Device Placement
Probabilistic Sensitivity Factor
External Loads
Response and Failure
Prediction
0.8
0.6
0.4
0.2
0
Location
Radius
Load
S-N Scatter
Sensitivity Factors
Probabilistic Methods
‣ Monte Carlo (MC) is the simplest prob method… input
distributions randomly sampled to form trials
‣ MC is robust and will always converge, but this usually
requires many thousands of trials
‣ It may be impractical to perform 1000’s of trials with
an FE model that requires hours for one solution
‣ There are more advanced methods that require fewer
trials and many modern programs implement these
methods… e.g., ANSYS uses DOE + Response Surface
Prob… an example with Excel
Random variables,
normally distributed
P
h
L = 2400 mm
b
h = 400 ± 20 mm
b = 100 ± 5 mm
P = 1000 ± 50 N
E = 200 ± 10 GPa
Standard Normal Distribution
CDF
PDF
0.5
m=0
s=1
0.4
0.8
0.3
F(x)
f(x)
1
0.6
0.2
0.4
0.1
0.2
0
-6.0
-4.0
-2.0
0.0
x
2.0
4.0
6.0
0
-6.0
-4.0
-2.0
0.0
x
2.0
4.0
6.0
Standard Normal Distribution
‣ Normal (m=0, s=1)
‣ Standard normal variate
x  mx
z
sx
– (Note: Halder uses S)
‣ All normal distributions can be simply
transformed to the standard normal
distribution

z(b)
 1 2
P(a  x  b) 
exp  s ds  (z(b))  (z(a))
 2 
z(a)
Generating Random Trials
Back to the Beam Example… 500 MC
To get the 10% lower and 90% upper bounds…
Use Excel functions: “large()” and “small()”
Beam Example in ANSYS
‣ ANSYS uses the term…
“Sig Sigma Analysis”
…this is most likely marketing
since 6s is popular in industry
‣ Prob trials are taken from a
response surface (quadratic
polynomial regression) built on a
results from a DOE
‣ This is how ANSYS avoids 1000’s of
trials required for a brute force MC
Beam Example in ANSYS - Deflection
Beam Example in ANSYS - Stress
Beam Example in ANSYS - Sensitivity
Sensitivity factors are the
components of a unit
vector in the direction of
the function gradient…
(i.e., stress = f(h,b,P,E))
…then sqrt(sum(si2)) = 1
sh
sb
sP
sE
sh
sb
sP
sE
How does Prob Compare?
‣ Provides information on sensitivities similar to
DOE and Response Surface methods
‣ Prob provides insight into how uncertainty in your
input parameters will affect outcome metrics
‣ Allows you to design for probability of specific
outcomes… e.g., 90% upper bound