1. MEGN 537 – Probabilistic Biomechanics Ch
MEGN 537 – Probabilistic Biomechanics Ch.7 – First Order Reliability Methods
Anthony J Petrella, PhD

2. A Summary of AMV
We desire to find various probability levels of a certain outcome metric for a biomechanical system
We have a computational model for the system, but cannot write a closed-form expression for the response function (limit state)
In other words, we do not know g(X) for the response of interest
Begin with the MV method and sample the limit state, g(X), to build a linear model of the system using a first-order Taylor series expansion about the means, call this glinear(X)
The linear model gives us some idea of g(X) and allows us to estimate the first and second moments of g(X)…

3. A Summary of AMV
If we assume glinear(X) is normally distributed, then the first and second moments allow us to find values of the function at various probabilities
Values of glinear(X) are then taken as first order estimates for values of the actual limit state, g(X), at various probabilities…
If g(X) is linear, then glinear(X) is an exact representation and the probability levels will be accurate
If g(X) is non-linear, then glinear(X) will only be accurate near the expansion point (all inputs set to mean values) and probability levels other than 0.5 (mean for normal variable) will exhibit error

4. A Summary of AMV For example, consider the non-linear limit state,
where,

5. AMV Geometry
Starting with the MV method we can perturb/sample g(X) to develop the linear approximation, glinear(X)
We can then estimate the mean & standard deviation, and we can compute the value of glinear(X) at various probability levels (see table)
Then we can plot glinear(X) in the reduced variate space (see plot)

6. AMV Geometry
Starting with the MV method we can perturb/sample g(X) to develop the linear approximation, glinear(X)
We can then estimate the mean & standard deviation, and we can compute the value of glinear(X) at various probability levels (see table)
Then we can plot glinear(X) in the reduced variate space (see plot)
LOW STRESS
HIGH STRESS

7. AMV Geometry
In the reduced variate space probability levels are circles – think of the plot below as a top view of the joint PDF of all inputs
To find the value of g(X) corresponding to a specific probability level, one must find the g(X) curve tangent to the desired probability circle
If g(X) is non-linear, then for a given fixed value of the response g(X) may be a very different curve than glinear(X) (see plot)
Recall the origin in reduced variate space corresponds to 50% probability = means of the inputs, which were normal
Notice glinear_50% and g50% agree perfectly at the origin because the Taylor series expansion was centered there
Realize g(X) is normally not known!
LOW STRESS
HIGH STRESS

8. AMV Example: Final Results

9. AMV Geometry
Note that the function values glinear_90% = g90% are the same values
That is, the value of the response is identical, but the values of l_fem’ and h_hip’ are very different for each curve
The point where glinear_90% is tangent to the 90% prob circle is a good guess for the values of l_fem’ and h_hip’ that will give an accurate estimate of g90%
AMV involves finding that tangent point (l_fem’*,h_hip’*) and then recalculating g(X)

10. AMV Geometry
The red dot is (l_fem’*,h_hip’*), the tangency point for glinear_90%
When we recalculate g90% at (l_fem’*,h_hip’*) we obtain an updated value of g(X) and the curve naturally passes through (l_fem’*,h_hip’*)
Note however that the updated curve may not be exactly tangent to the 90% prob circle, so there may still be a small bit of error (see figure below)

11. AMV Example: Final Results

12. Homework 6 - AMV Your execution commands should look like this…
copy ..\..\pedal_data.mat
copy ..\..\pedal_trial_nessus.m
copy ..\..\unit.m
matlab –wait -r pedal_prob_nessus
del pedal_data.mat
del pedal_trial_nessus.m
del unit.m