MEGN 537 – Probabilistic Biomechanics Ch.7 – First Order Reliability Methods

1. MEGN 537 – Probabilistic BiomechanicsCh.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 thereduced variate space (see plot)

6. 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 agiven fixed value of the responseg(X) may be a very differentcurve than glinear(X) (see plot) • Recall the origin in reducedvariate space corresponds to50% probability = means ofthe inputs, which were normal • Notice glinear(X) and g(X) agreeperfectly at the origin becausethe Taylor series expansion wascentered there • Realize g(X) is normally not known!

7. AMV Geometry • Note that glinear_90% = g90% • 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% istangent to the 90% prob circle isa good guess for the values ofl_fem’ and h_hip’ that will producean accurate estimate of g90% • AMV involves finding that tangentpoint (l_fem’*,h_hip’*) and thenrecalculating g(X)

8. 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 updatedcurve may not be exactly tangentto the 90% prob circle, so theremay still be a small bit of error(see figure below)

9. Homework 6 - AMV • Your execution commands should look like this… copy ..\..\pedal_data.mat copy ..\..\pedal_trial_nessus.m copy ..\..\unit.m matlab -wait -nodesktop -nojvm -nosplash -minimize -r pedal_prob_nessus