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Nongradient Based MPP Search MethodThe limit-state function must be second-order differentiable in order to implement the MPP search algorithms previously discussed. All these are basically gradient-based optimization algorithms that require accurate gradient evaluations of the limit-state function. In other words, for these methods to be effective, the limit-state function and its gradient should be smooth and continuous. However, for some problems, the limit-state function is not second-order differentiable as the examples shown in the following two figures. Such classes of problems may be found in the field of structural dynamics. Often, artificial difficulties originate from poor numerical methods that result in insufficient precision for gradient calculation. In this case, the derivatives are not available because limit-state functions are evaluated by an external software package that does not or cannot provide enough accurate or significant digits in the output. The round off error is amplified through the finite difference limit state function gradient. Consequently, the correct MPP cannot be recovered and the search algorithms fail to converge.
Figure 1 Example of Non-Second-Order Differentiable Limit-State Function
Figure 2 Example of Discontinuous Limit-State Function
The MPP simulation search method implemented in UNIPASS™ is a non-gradient-based method. It can identify the MPP for discrete limit-state functions and for problems that are discontinuous in the first-order derivative of the limit-state function. This method requires more limit-state function calculations than the previous MPP identification methods. However, it is still more affordable when compared to the computation effort required by conventional simulation methods such as Monte Carlo and Directional simulations.
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Last Updated 11/12/08
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