FORM: First-Order Reliability Method (FORM)
In TesiproV: Reliability Analysis Methods for Structural Engineering

View source: R/algo_FORM.R

FORM	R Documentation

First-Order Reliability Method (FORM)

Description

First-Order Reliability Method (FORM) for the approximation of failure probabilities in structural reliability analysis.

The FORM estimates the probability of failure by transforming the basic random variables into standard normal space and approximating the limit-state function by a first-order (linear) Taylor expansion at the design point.

The reliability index is defined as the minimum distance from the origin to the limit-state surface in standard normal space. The probability of failure is then approximated using the standard normal cumulative distribution function.

Failure is defined as g(x) < 0. The reliability index is positive if the origin of the standard normal space lies in the safe domain and negative otherwise. The probability of failure is p_f = \Phi(-\beta).

Solver selection: With an explicitly given optim_type exactly this algorithm is run and nothing else. If it does not converge, a warning is issued and the last iterate is returned with converged = FALSE. With optim_type = "auto" (default) a robust chain is used: SLSQP (nloptr), and iHLRF as a fallback. A warning is only issued if both fail.

Usage

FORM(
  lsf = NULL,
  lDistr,
  n_optim = 100L,
  loctol = 1e-05,
  optim_type = "auto",
  lsf_list = NULL,
  debug.level = 0
)

Arguments

`lsf`	Objective function representing the limit-state, e.g. `function(R,E){R-E}`. Supplied automatically by a SYS_ object - do not provide manually.
`lDistr`	List of distribution objects created by TesiproV. Supplied automatically by a SYS_ object - do not provide manually.
`n_optim`	Maximum number of iterations. For `"slsqp"` it is the maximum number of function evaluations of the optimizer. In `"auto"` SLSQP may use `max(200, 10 * n_optim)` evaluations.
`loctol`	Relative tolerance. Converged means `\|g(u)\| \le loctol \cdot g_{scale}` and the distance of `u` from the line through the origin along the gradient is `\le loctol \cdot \max(1, \\|u\\|)`. SLSQP results are accepted with at most `max(loctol, 1e-5)`.
`optim_type`	Solver: `"auto"` - (default) SLSQP, then iHLRF if needed `"ihlrf"` - improved HLRF with merit-function line-search (Zhang & Der Kiureghian 1997), no fallback `"rackfies"` - Rackwitz-Fiessler iteration (classic HLRF), no fallback `"slsqp"` - SQP with BFGS (Kraft 1988, nloptr), no fallback `"auglag"` - Augmented Lagrangian (nloptr), random starts
`lsf_list`	Reserved for a future system-FORM (not active).
`debug.level`	Verbosity level: 0 = silent, 1 = basic info, 2 = detailed output.

Value

A list (content depends on optim_type): * beta - Hasofer-Lind reliability index * pf - probability of failure \Phi(-\beta) * x_points - design point in physical space * u_points - design point in standard normal space * dy - gradient in standard normal space at the design point * alpha - direction cosines * n_sim - number of (i)HLRF iterations performed * converged - TRUE if the convergence criteria are met * solver - algorithm that delivered the result ("HLRF", "iHLRF", "SLSQP") * fallback - TRUE if optim_type = "auto" had to switch from SLSQP to iHLRF, otherwise FALSE * conv_info - named vector of final convergence metrics

Author(s)

References

Hasofer, A. M., & Lind, N. C. (1974). An exact and invariant first-order reliability format. Journal of the Engineering Mechanics Division, ASCE, 100(1), 111-121.

Rackwitz, R., & Fieszler, B. (1978). Structural reliability under combined random load sequences. Computers & Structures, 9(5), 489-494.

Zhang, Y., & Der Kiureghian, A. (1997). Finite element reliability methods for inelastic structures. Report UCB/SEMM-97/05, University of California, Berkeley.

Kraft, D. (1988). A software package for sequential quadratic programming. Technical Report DFVLR-FB 88-28, Institut fuer Dynamik der Flugsysteme, Oberpfaffenhofen.

TesiproV documentation built on Oct. 28, 2026, 5:13 p.m.