The ISPUD node performs Importance Sampling Procedure Using the Design Points (ISPUD) to compute the probability of failure of a single or multiple failure regions.
The method is a variant of Importance Sampling. Therein, a different joint probability distribution than the one specified by the parameters is used to generate the pseudo-random design point vectors. Similar to Monte Carlo, probability of failure is computed as the ratio of the number of samples associated with failure events to the total number of samples. The difference between the distribution function which is used for sampling in the simulation, and the actual distribution function of the design space is compensated for by associating weight factors to each failure event. The idea is to use a sampling simulation distribution function which optimally approximates the conditional distribution within the failure domain. Thus the variance of the estimator is reduced.
ISPUD uses a sampling distribution with a mean value that is identical to the design point. The design point is the point on the limit state which is closest to the origin in standard normal space. The covariance of the simulation distribution is identical to the original covariance (measured in standard normal space). Through the shift of the mean value, the number of samples in the failure domain will be strongly increased. In case of multiple failure regions, for each region an additional sampling density with the original covariance is considered, which would result in a joint multi-modal density function.
The design points may be either determined by FORM or can be specified as start designs.
ISPUD is specifically designed for small probabilities of failure. It can reliably treat problems where the distribution function within the failure domain can be represented by a copula model, for example where a multi-modal domain contributes to the probability of failure. Additional assumptions on convexity, differentiability and other properties of the problem are inherited from the procedure that finds the design point (for example FORM/NLPQL). Compared with FORM, ISPUD may provide an improvement to the accuracy of the FORM solution if the error due to the linearisation of the limit state by FORM is not negligible.
Further information about methods of reliability analysis used in optiSLang can be found here.
Initialization Options
To access the options shown in the following table, double-click the ISPUD system on the Scenery pane and switch to the ISPUD tab.
| Option | Description | ||
|---|---|---|---|
| Find design points by |
Selects the algorithm used to find the design point. You can select one of the following options:
| ||
| Prescribed sample size | If selected, a fixed number of samples is computed. | ||
| Prescribed accuracy | If selected, the number of samples is automatically increased until a given accuracy is reached. | ||
| Prescribed sample size | |||
| Total number of samples | Sets the number of samples to be computed. | ||
| Prescribed accuracy | |||
| Desired accuracy (C.O.V.) | Terminates the algorithm as soon as the coefficient of variation (C.O.V.) of the estimator for the probability of failure falls below this threshold. | ||
| Maximum number of samples | Sets the maximum allowed number of samples. This represents an upper bound that limits the computing time if the desired accuracy cannot be reached. | ||
| Number of samples per increment | Sets the amount of incremental increase of the sample size. After each increment the termination criteria is checked. | ||
| Vary discrete parameters in sampling | If selected, the discrete distribution types are transformed to the standard normal space similarly to the continuous parameters. In this case, different discrete values may be sampled in a single failure region. If cleared, the discrete parameters are kept fixed for each sampling density according to the center or design point found by FORM or given in the start designs. | ||
Additional Options
This algorithm allows Additional Options.