First order second moment

# Import auxiliary libraries for demonstration

import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
from scipy import stats

plt.rcParams[ "figure.figsize" ] = [ 5, 4 ]

plt.rcParams[ "figure.dpi" ] = 80
plt.rcParams[ "font.family" ] = "Times New Roman"
plt.rcParams[ "font.size" ] = '14'

Mean value FOSM

FOSM with mean value algorithm can be expressed by the following equation,

\[\beta = \frac{g(\mu_1, \mu_2, \dots, \mu_n)}{\sqrt{ \sum_{i=1}^n \alpha_i \sigma_i}}\]

where \(g\) is the limit state function (LSF); \(\alpha_i\) is given by,

\[\alpha_i = \frac{\partial g}{ \partial X_i} \Bigr|_{\mu_i}\]

Function mvalFOSM implements the FOSM with mean value algorithm.

Reference: Nowak, A.S. and Collins, K.R., 2012. Reliability of structures. CRC press.

Function help

from ffpack.rrm import mvalFOSM
help( mvalFOSM )

Help on function mvalFOSM in module ffpack.rrm.firstOrderSecondMoment:

mvalFOSM(dim, g, dg, mus, sigmas, dx=1e-06)
    First order second moment method based on mean value algorithm.

    Parameters
    ----------
    dim: integer
        Space dimension ( number of random variables ).
    g: function
        Limit state function. It will be called like g( [ x1, x2, ... ] ).
    dg: array_like of function
        Gradient of the limit state function. It should be an array_like of function
        like dg = [ dg_dx1, dg_dx2, ... ]. To get the derivative of i-th random
        variable at ( x1*, x2*, ... ), dg[ i ]( x1*, x2*, ... ) will be called.
        dg can be None, see the following Notes.
    mus: 1d array
        Mean of the random variables.
    sigmas: 1d array
        Variance of the random variables.
    dx : scalar, optional
        Spacing for auto differentiation. Not required if dg is provided.

    Returns
    -------
    beta: scalar
        Reliability index.
    pf: scalar
        probability of failure.

    Raises
    ------
    ValueError
        If the dim is less than 1.
        If the dim does not match the length of mus and sigmas.

    Notes
    -----
    If dg is None, the numerical differentiation will be used. The tolerance of the
    numerical differentiation can be changed in globalConfig.

    Examples
    --------
    >>> from ffpack.rrm import mvalFOSM
    >>> dim = 2
    >>> g = lambda X: 3 * X[ 0 ] - 2 * X[ 1 ]
    >>> dg = [ lambda X: 3, lambda X: -2 ]
    >>> mus = [ 1, 1 ]
    >>> sigmas = [ 3, 4 ]
    >>> beta, pf = mvalFOSM( dim, g, dg, mus, sigmas)

Example with explicit derivative of LSF

# Define the dimension for the FOSM problem
dim = 2

# Define the limit state function (LSF) g
g = lambda X: 3 * X[ 0 ] - 2 * X[ 1 ]

# Explicit derivative of LSF
# dg is a list in which each element is a partial derivative function of g w.r.t. X
# dg[0] = partial g / partial X[0]
# dg[1] = partial g / partial X[1]
dg = [ lambda X: 3, lambda X: -2 ]

# Mean and standard deviation of the random variables
mus = [ 1, 1 ]
sigmas = [ 3, 4 ]

# Use mean value algorithm to get results
beta, pf = mvalFOSM( dim, g, dg, mus, sigmas)

print( "Reliability index: " )
print( beta )
print()
print( "Failure probability: " )
print( pf )

Reliability index:
0.08304547985373997

Failure probability:
0.46690768839408386

Example with automatic differentiation of LSF

# Define the dimension for the FOSM problem
dim = 2

# Define the limit state function (LSF) g
g = lambda X: 3 * X[ 0 ] - 2 * X[ 1 ]

# If dg is None, the internal automatic differentiation algorithm will be used
dg = None

# Mean and standard deviation of the random variables
mus = [ 1, 1 ]
sigmas = [ 3, 4 ]

# Use mean value algorithm to get results
beta, pf = mvalFOSM( dim, g, dg, mus, sigmas)

print( "Reliability index: " )
print( beta )
print()
print( "Failure probability: " )
print( pf )

Reliability index:
0.08304547985853711

Failure probability:
0.46690768839217667