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

Model Input Uncertainties

Validated Deterministic Model

Tissue Properties

External Loads

Device Placement

Outcome Probabilities &

Sensitivities

Response and Failure Prediction

Pro

bab

ility

Performance Metric

Location Radius Load S-N Scatter0

0.2

0.4

0.6

0.8

1

Pro

ba

bili

stic

Se

ns

itiv

ity F

ac

tor

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

P

b

hL = 2400 mm

Random variables,normally distributed

h = 400 ± 20 mmb = 100 ± 5 mmP = 1000 ± 50 NE = 200 ± 10 GPa

Standard Normal Distribution

PDF CDF

0

0.1

0.2

0.3

0.4

0.5

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0x

f(x)

0

0.2

0.4

0.6

0.8

1

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0x

F(x

)

m = 0s = 1

Standard Normal Distribution

‣Normal (m=0, s=1)

‣Standard normal variate – (Note: Halder uses S)

‣All normal distributions can be simply transformed to the standard normal distribution

x

xxz

))a(z())b(z(dss21

exp)bxa(P 2)b(z

)a(z

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 - SensitivitySensitivity 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(si

2)) = 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