Overview

Estimation of first-passage probability under stochastic wind excitations

The Wind module estimates the probability that structural response exceeds a prescribed threshold under stochastic wind excitation. Its practical goal is to produce accurate first-passage probability curves with far fewer dynamic simulations by combining surrogate modeling and adaptive learning.

Primary objective

Quantify threshold exceedance risk for wind-excited structures without relying on exhaustive Monte Carlo simulation.

Core engine

A heteroscedastic Gaussian-process surrogate captures both the trend and the changing uncertainty of the maximum response.

Main outputs

Estimated first-passage probability curves together with surrogate predictions that describe the adaptive learning process.

Method

How it works

The workflow reduces a high-dimensional stochastic problem into a tractable conditional-surrogate estimation task.

The Wind algorithm targets the first-passage probability, meaning the probability that the maximum structural response within a time interval exceeds a threshold:

$$p_f(u_0,\tau)=P\left(\max_{0<t\le\tau}|u(\mathbf{X},\mathbf{Z},t)| \ge u_0\right)$$

Here, $\mathbf{X}$ contains time-invariant uncertain parameters such as structural properties and wind-model parameters, while $\mathbf{Z}$ represents the stochastic wind-excitation sequence. Direct evaluation is expensive because the excitation process makes the problem very high-dimensional.

To reduce that difficulty, the method reformulates the problem in terms of the conditional distribution of the maximum response given the time-invariant variables:

$$p_f=\int p_{f|X}(\mathbf{x}^*) f_X(\mathbf{x}^*)\,d\mathbf{x}^*$$

The conditional first-passage probability is then approximated using a lognormal model for the maximum response:

$$p_{f|X}(\mathbf{x}^*) = 1-\Phi\left(\frac{\ln u_0-\lambda(\mathbf{x}^*)}{\zeta(\mathbf{x}^*)}\right)$$

Once $\lambda(\mathbf{x})$ and $\zeta(\mathbf{x})$ are known, the total failure probability can be estimated efficiently by sampling only over the lower-dimensional variable space $\mathbf{X}$.

Those distribution parameters are estimated with a heteroscedastic Gaussian process:

$$\ln M = y(\mathbf{x}) + \varepsilon(\mathbf{x}), \qquad \varepsilon(\mathbf{x}) \sim N(0,r(\mathbf{x}))$$

This matters because the variability caused by stochastic wind loads is not uniform across the input space. The surrogate therefore models both the response trend and the location-dependent uncertainty.

From this model, the algorithm predicts

$$\hat{\lambda}_{GP}(\mathbf{x}^*) = E_p[\ln M^*], \qquad \hat{\zeta}_{GP}(\mathbf{x}^*) = \sqrt{\mathrm{Var}_p[\ln M^*]}$$

and substitutes them into the conditional failure expression. Adaptive learning then adds the most informative simulation points for the target threshold:

$$\mathbf{x}_{best} = \arg\max_{\mathbf{x}_k} \, p_{f|X}(\mathbf{x}_k) = \arg\min_{\mathbf{x}_k} \Phi\left(\frac{\ln u_0-\hat{\lambda}_{GP}(\mathbf{x}_k)}{\hat{\zeta}_{GP}(\mathbf{x}_k)}\right)$$

This strategy focuses new simulations near influential threshold-crossing regions, so the module can estimate wind-induced first-passage probability accurately with many fewer structural simulations than a direct Monte Carlo approach.

Inputs

What you need

One Python configuration file defines the structural model, wind environment, geometry, thresholds, and simulation controls.

The required root sections are St, wind, Geo, samp, and u_o_exact.
St defines the structural model, including numStory, dimensions, density, damping ratio vector zeta, and stiffness vector k.
wind defines the wind environment, including V_refer, Gust_factor, drag, rho_air, kappa, optional EC, and the nested coherence decay parameters in CohDecay.
Geo defines the spatial grid for the wind field, and samp defines the sampling frequency and duration.
u_o_exact is the threshold vector used for probability evaluation.
Main control values include numb_sim, adap_sim, n_sample, iter_hyp, x_test_min, and x_test_max, with optional values such as numb_rvs, u_id_p, and target_n.

Implementation note: all numeric fields must be valid numbers, vector lengths in the structural model must match the number of stories, and the file is evaluated directly as a Python-style configuration.