1. Probabilistic Analysis using FEA A. Petrella

2. 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?

3. What is Probabilistic Analysis Model Input Uncertainties Validated Deterministic Model Tissue Properties External Loads Device Placement Outcome Probabilities & Sensitivities Response and Failure Prediction Probability Performance Metric Sensitivity Factors

4. 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

5. Prob… an example with Excel P b h L = 2400 mm Random variables, normally distributed h = 400 ± 20 mm b = 100 ± 5 mm P = 1000 ± 50 N E = 200 ± 10 GPa

6. Standard Normal Distribution PDF CDF  = 0  = 1

7. Standard Normal Distribution ‣ Normal (  =0,  =1) ‣ Standard normal variate – (Note: Halder uses S) ‣ All normal distributions can be simply transformed to the standard normal distribution

8. Generating Random Trials

9. Back to the Beam Example… 500 MC To get the 10% lower and 90% upper bounds… Use Excel functions: “large()” and “small()”

10. Beam Example in ANSYS ‣ ANSYS uses the term… “Sig Sigma Analysis” …this is most likely marketing since 6  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

11. Beam Example in ANSYS - Deflection

12. Beam Example in ANSYS - Stress

13. 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(s i 2 )) = 1 shsh sbsb sPsP sEsE shsh sbsb sPsP sEsE

14. 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