build_gsys: Build a smooth system limit-state function (g_sys)

In TesiproV: Reliability Analysis Methods for Structural Engineering

Build a smooth system limit-state function (g_sys)

Description

Constructs a **global system limit-state function** from a list of individual limit-state functions ('SYS_LSF' objects). The resulting function aggregates all component limit states using a **smooth approximation** of the minimum or maximum operator.

Usage

build_gsys(
  sys_input,
  sys_structure = "serial",
  smooth_kappa = 20,
  debug.level = 0
)

Arguments

sys_input	List of SYS_LSF objects defining the component limit-state functions of the system.
sys_structure	System aggregation structure. Supported options are "serial" System fails if any component fails. "parallel" System fails only if all components fail. function(g_vals) Custom aggregation function applied to the vector of component limit-state values.
smooth_kappa	Positive numeric value controlling the smoothness of the aggregation. Larger values approach the exact min or max operator.
debug.level	Optional verbosity level for diagnostic output.

Details

The smooth aggregation improves numerical stability and differentiability, which is beneficial for gradient-based reliability methods such as FORM, SORM, and importance sampling algorithms.

For a serial system the aggregation is

g_{\kappa}(x) = -\frac{1}{\kappa}\log \left( \sum_{i=1}^{m} \exp(-\kappa g_i(x)) \right)

which approximates

\min_i g_i(x).

For a parallel system the aggregation is

g_{\kappa}(x) = \frac{1}{\kappa}\log \left( \sum_{i=1}^{m} \exp(\kappa g_i(x)) \right)

which approximates

\max_i g_i(x).

The approximation becomes exact for \kappa \rightarrow \infty.

In addition to the system function, this helper also constructs a **flattened list of marginal distributions** and a **variable index map** that links the global variable vector to the variables of each individual limit-state function.

This function is primarily used internally by reliability algorithms that operate on a **single limit-state function**, such as importance sampling or crude Monte Carlo simulation.

By constructing a smooth global limit-state function, the system reliability problem can be treated as a standard single-LSF problem, enabling the reuse of algorithms originally developed for component reliability.

Value

A list containing:

g_sys

Callable system limit-state function function(x) operating on the global variable vector.

distr_flat

Flattened list of marginal distribution objects corresponding to all variables in the system.

var_map

List mapping each limit-state function to the indices of its variables in the global vector.

n_vars

Total number of basic variables in the system.

n_lsfs

Number of limit-state functions.

Author(s)

(C) 2021-2026 K. Nille-Hauf, T. Feiri, M. Ricker, T. Lux - Hochschule Biberach (until 2022), TU Dortmund University - Chair of Structural Concrete (since 2023)

See Also

smooth_min, smooth_max, MC_IS, SYS_LSF

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