1. FORM - Linear function - Two Random variables

In the second example we have the limit state to be a linear function of two (\(d=2\)) independent Gaussian random variables

\(g(U) = -\frac{1}{\sqrt{d}}\sum_{i=1}^{d} u_i + \beta\)

The probability of failure in this case is \(P(F) ≈ 10^{−3}\) for \(β = 3.0902\)

Initially we have to import the necessary modules.

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


dist1 = Normal(loc=0., scale=1.)
dist2 = Normal(loc=0., scale=1.)

model = PythonModel(model_script='local_pfn.py', model_object_name="example2")
RunModelObject2 = RunModel(model=model)

Z = FORM(distributions=[dist1, dist2], runmodel_object=RunModelObject2)
Z.run()

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