FORM {mistral}

First-order reliability method

Description

The First-Order Reliability Method computes an estimation of the failure probability by approximating the limit-state function at the Most Probable Failure Point with a hyperplane.

Usage

FORM(
  dimension,
  lsf,
  u.dep = rep(0, dimension),
  N.calls = 100,
  eps = 1e-07,
  Method = "HLRF",
  IS = FALSE,
  IS.ratio = 0.5,
  plot = FALSE,
  plot.lsf = FALSE,
  plot.lab = c("x_1", "x_2")
)

Arguments

dimension

the dimension of the input space.

lsf

the limit-state function.

u.dep

the starting point for the MPFP search.

N.calls

the total number of calls for the whole algorithm.

eps

stopping criterion: distance of two points between two iterations.

Method

choice of the method to search the design point: "AR" for Abdo-Rackwitz and "HLRF" for Hasofer-Lindt-Rackwitz-Fiessler.

IS

"TRUE" for using importance Sampling method with an standard Gaussian importance density centred at the MPFP.

IS.ratio

ratio of N.calls for the search of the design point by FORM. Default = 0.5. 1-IS.ratio = the remaining ratio to be used for importance sampling.

plot

to plot the generated samples.

plot.lsf

a boolean indicating if the lsf should be added to the plot. This requires the evaluation of the lsf over a grid and consequently should be used only for illustation purposes.

plot.lab

the x and y labels for the plot.

Details

The FORM method has to be used in the standard Gaussian input space. It is designed to estimate probability of the form P[g(\mathbf{X}) < 0] with g the limit-state function. This function has to be modified accordingly to fit into this framework

Furthermore, it should be able to handle matrix input of column vectors. See the mistral vignette for more info about lsf definition

Value

A list containing the following objects

p

Failure probability

indice.reliab

Reliability index

Ncall

Number of calls to f

Design.Point

Coordinates of the design point

fact.imp

Importance factors

variance

Standard error of the probability estimator (if IS = TRUE)

Interval.conf

Confidence interval of the estimator at 0.95 (if IS = TRUE)

DOE

List which contains the design of experiments

Author(s)

Vincent MOUTOUSSAMY and Clement WALTER [e-mail]

References

• O. Ditlevsen and H.O. Madsen. Structural reliability methods, Wiley, 1996
  

• M. Lemaire, A. Chateauneuf and J. Mitteau. Structural reliability, Wiley Online Library, 2009.
  

Examples

## Not run: 
# u.dep is a starting point for the research of the Most Probable Failing Point
# N.calls is a total number of calls
form <- mistral::FORM(dimension = 2, mistral::kiureghian, N.calls = 1000,
                     u.dep = c(0,0))
form$p

# use IS=TRUE to use an Importance Sampling scheme with a Gaussian standard
# proposal distribution centred at the MPFP
form.IS <- mistral::FORM(dimension = 2, mistral::kiureghian, N.calls = 1000,
                        u.dep = c(0,0),
                        IS = TRUE)
form.IS$p

## End(Not run)