NESSUS Advanced Mean Value with Iterations (AMV+) [1]

NESSUS Advanced Mean Value with Iterations (AMV+) is a most probable point (MPP) based analytical method. It provides a significant improvement over MV, where first it uses approximation to estimate MPP and then performs additional iterations when locating the MPP in order to obtain a more accurate result. A new Taylor series approximation is used for each new MPP and the process is repeated until method converges on an MPP (i.e. no significant change in MPP over a few iterations) or 10 iterations are completed. AMV+ uses finite difference approximations to estimate the gradients. The step size can be set using the Finite Difference Step Size option available but care should be taken for non normal variables that the finite difference step does not move outside of the feasible range of the variable. The method does not support non-smooth responses and failed runs as it is a gradient based method. This method generally requires less than a couple hundred evaluations to compute a probability for less than 10 design variables. AMV+ should always be preferred to MV method. This method should not be used with discrete design variables or discrete response variables. Multiple local MPPs may induce error in the probability estimation by this method and that error tends to be most pronounced for large probabilities of failure.

Advanced Mean Value (AMV) improves the mean value method by using a correction for errors introduced from the truncation of a Taylor's series. The AMV model is defined as:

where H(Z_MV) is defined as the difference between the values of Z_MV and Z calculated at the Most Probable Point Locus (MPPL) of Z_MV. The MPPL is defined by connecting all the MPP's for different z values. The MPPL of Z_MV is generally nonlinear curve in the u-space. The key to the AMV method is the reduction of the truncation error by replacing higher-order terms H(X) by a simplified function H(Z_MV) dependent on Z_MV. Ideally H(Z_MV) function should be based on the exact MPPL of the Z-function but AMV simplifies the procedure by using MPPL of Z_MV. Thus optimization of truncation error is not optimum.

Analyzer does not provide AMV but it provides an even better method AMV+. This is an improvement over AMV by using an improved expansion point, which is done by an iteration procedure. Based initially on Z_MV and by keeping track of the MPPL, the exact MPP for a limit state Z = z can be computed to establish the AMV+ model defined as: 

where x* is the converged MPP for Z = z. The variables X are generally non-normal and dependent, thus AMV+ model which is linear is in X-space. The above equation provides an initial estimate of the curvature (about the MPP) in the u-space that can be used for approximate probability of failure analysis. The steps followed by AMV+ could be summarized as follows:

• Construct mean based response function based on the mean values of the design variables and use it as initial guess for MPP
• Use the AMV approximation to come up with a better MPP.
• Perform a curve fit operation and then we will have a new MPP
• The above steps are performed until the MPP converges or 10 iterations are completed.

Special caution should be used when using probability values in cases where the method has not been able to converge on an MPP. In such cases some experimentation can be carried on to vary the Finite Difference Step Size option.

Control Parameters

References:1. NESSUS Theoretical Manual, February 17, 2012, Section 8.2-8.3