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

Basics of Reliability Recall that we can find exact CDF of a system response variable if… All inputs are statistically independent normal variables and the functional relationship g(X) is a linear function of X i ’s All inputs are statistically independent lognormal variables and the functional relationship g(X) is a multiplicative function of the X i ’s Risk-based reliability employs the above concepts Option 1: limit state Z = R – S, then POF = P(Z ≤ 0) Option 2: limit state Y = R/S (safety factor), then POF = P(Y ≤ 1) These two options produce different results for POF (see Haldar, Example 7.3), which is undesirable Also, in practice we often want to consider different known or unknown limit states, different input distributions, and correlations among inputs

Monte Carlo Simulation Given: 1. input distributions 2. deterministic model (can be “black box” so g(X) is not known in closed form) Find: POF Method: 1.Randomly generate numbers (u i ) 0 to 1 2.Generate input values from CDF’s, x i = F -1 (u i ) 3.Input x i into deterministic model 4.Evaluate limit state g(X) 5.POF = # of failures / # of trials 6.Repeat until POF converges

Monte Carlo Simulation Accuracy is related to the number of trials Example: six sigma requires at least 1 million 30 seconds/trial  347 days Advantages Guaranteed to work every time Guaranteed to converge to the correct solution Disadvantages Inefficient Requires a large number of trials Results in large computation times

Reliability Index To address limitations of risk-based reliability with greater efficiency than MC, we introduce the safety index or reliability index,  Consider the familiar limit state, Z = R – S, where R and S are independent normal variables Then we can write, and POF = P(Z ≤ 0), or in terms of  …

Example: TIC 7 Recall that if we wish to consider a complex limit state that is a function of many inputs, then we can simplify the problem by estimating the limit state with a first-order Taylor series expansion… Furthermore, if we do not have a closed form expression for the limit state, then finite difference approximations can be used to estimate the partial derivatives Then we can easily find…

Example: TIC 7 The limit state for axial stress in a spinal rod was given as, which is a simple linear expression It was assumed that all inputs were uncorrelated variables with unknown distributions The form of the functional relationship was assumed unknown The mean and standard deviation of each input was given

Example: TIC 7 Forward difference

Example: TIC 7 Conclusions: MC provides excellent results MC requires many trials to provide accurate results MC cannot capture CDF tails without trials MV method more accurate than MC for few trials, much more efficient MV based on a linearization of the limit state about the mean values of the inputs – only accurate for linear or approximately linear functions

Advanced Mean Value Method (AMV) We would like to overcome linear limitation of MV method Try second (or higher) order Taylor series expansion…(SORM, Ch.8) There is a simpler way to do it… We would also like to overcome lack of invariance exhibited by basic risk-based reliability methods… Give different answers depending on form of limit state Give different answers depending on input distributions Let us consider the AMV method, which includes improvements on the MV method to address both of the above limitations To develop the AMV method, we must introduce the… Most Probable Point (MPP) of failure

Most Probable Point (MPP) Inputs must first be transformed to standard or “reduced” variates Computed in the same way as standard normal variates, but the variable need not be normal

Most Probable Point (MPP) Failure Safe

Most Probable Point (MPP) Closest to the origin → highest frequency, most likely

Most Probable Point (MPP) Consider the curve that defines a fixed value of the limit state, there are many combinations of inputs that represent points on that curve MPP is the most probable combination of the input variables that satisfy the limit state equation – that is the most probable combination of input values leading to a specified value of the limit state (e.g., in reliability theory, g = 0 defines failure) Values of the input variables can be put into a column vector, where the prime symbol indicates each variable has been transformed to a reduced or standard variate The MPP can then be written, where the star superscript identifies the MPP in the reduced or standard variate coordinate system

AMV Method (NESSUS) AFOSM Method uses MPP to compute the Hasofer-Lind reliability index,  HL, developed for a linear performance function (limit state) Similar to the AMV method in NESSUS Joint PDF

AMV Method (NESSUS) The AFOSM / AMV Method can compute the reliability index exactly in the case of a linear limit state AFOSM / AMV Method can also be used to compute reliability index for a non-linear limit state When limit state is non-linear, finding the reliability index,  HL (minimum distance to the origin), becomes an optimization problem… Minimize the distance, subject to the constraint, Note, const. is usually zero for finding POF, but one may also wish to find performance at certain P-levels, in which case const. ≠ 0

Recognize that the point on any curve or n-dimensional surface that is closest to the origin is the point at which the function gradient passes through the origin Geometry of MPP MPP→closest to the origin →highest likelihood on joint PDF in the reduced coord. space * gradient direction gradient → perpendicular to tangent direction

If we can compute/estimate the gradient, then we can look along that direction (from the origin) for an intersection with the limit state to estimate the MPP Geometry of MPP MPP→closest to the origin →highest likelihood on joint PDF in the reduced coord. space * gradient direction gradient → perpendicular to tangent direction

AMV Method (NESSUS) The optimization problem can be solved in closed form using the Lagrange multiplier method (see Haldar, p.201) From this we find the reliability index,  HL, which is the magnitude of the minimum distance from the origin to the limit state in the reduced coordinate system The MPP can be expressed as, where * alpha is a unit vector in the direction of the limit state gradient

AMV Method (NESSUS) Assuming one is seeking values of the performance function (limit state) at various P-levels, the steps in the AMV method are: 1.Define the limit state equation 2.Complete the MV method to estimate g(X) at each P-level of interest, if the limit state is non-linear these estimates will be poor 3.Assume an initial value of the MPP, usually the means 4.Compute the partial derivatives and find alpha (unit vector in direction of the function gradient) 5.Now, if you are seeking to find the performance (value of limit state) at various P-levels, then there will be a different value of the reliability index  HL at each P-level. It will be some known value and you can estimate the MPP for each P-level as…

AMV Method (NESSUS) The steps in the AMV method (continued): 6.Convert the MPP from reduced coordinates back to original coordinates 7.Obtain an updated estimate of g( ) for each P-level using the relevant MPP’s computed in step 6

AMV Example Consider the recumbent cycling analysis MATLAB: Fap = pedal_trial_nessus(rqk,rtx,rty,rhky,angle)

AMV Example A/P force at knee, F ap, is assumed deterministic Estimate bending stress at the hip with simple expression where,

AMV Example Let the limit state be defined as, Find the 90% performance level for the limit state Step 1: the limit state equation is defined above Step 2: complete the MV method to make a first order estimate of g( ) at the 90% level

AMV Example Step 2: MV method (continued)

AMV Example Step 3: assume the MPP is initially at, Step 4: compute the partial derivatives at the MPP in the reduced coordinate system, then find alpha

AMV Example Step 4: compute the partial derivatives at the MPP in the reduced coordinate system, then find alpha

AMV Example Step 4: the geometry…showing Linear and MV results with alpha in the reduced coordinate system Recall: generally, the true form of g( ) will not be known, so the solid curves in the plot would also not be known, but they are shown here for comparison

Step 5: estimate MPP at 90%:  HL = -  -1 (0.9) = AMV Example

Step 6: convert MPP from reduced coordinates back to original coordinates Step 7: Update estimate for g( )… AMV Example

AMV Example: Final Results

AMV Example Conclusions Improved accuracy compared to MV method for non-linear limit state functions Based on MPP in reduced coordinate space, which eliminates invariance problem Designed to work with uncorrelated normal variables We will explore later how to expand to non-normal variables, advanced methods are also available to consider correlated non-normal inputs (NESSUS can do this)