2. FORM - Linear function - Three Random variables

In the third example we have the limit state to be a linear function of three (\(d=3\)) independent Gaussian random variables

\[g(X_1, X_2, X_3) = 6.2X_1 -X_2X_3^2\]

\[\mu_{X_1}=20, \mu_{X_2}=5, \mu_{X_3}=4\]

\[\sigma_{X_1}=3.5, \sigma_{X_2}=0.8, \sigma_{X_3}=0.4\]

The probability of failure in this case is \(P_f ≈ 0.079\) for \(\beta = 1.413\)

Initially we have to import the necessary modules.

from UQpy.run_model.RunModel import RunModel
from UQpy.run_model.model_execution.PythonModel import PythonModel
from UQpy.distributions import Normal
from UQpy.reliability import FORM

dist1 = Normal(loc=20., scale=3.5)
dist2 = Normal(loc=5., scale=0.8)
dist3 = Normal(loc=4., scale=0.4)

model = PythonModel(model_script='local_pfn.py', model_object_name="example3",)
run_model = RunModel(model=model)

form = FORM(distributions=[dist1, dist2, dist3], runmodel_object=run_model)
form.run()

print('Design point in standard normal space: %s' % form.design_point_u)
print('Design point in original space: %s' % form.design_point_x)
print('Hasofer-Lind reliability index: %s' % form.beta)
print('FORM probability of failure: %s' % form.failure_probability)