Volume Weighted Weibull

Initialize a variable with randomly generated numbers following a volume-weighted Weibull distribution

The VolumeWeightedWeibull class generates a spatially randomized distribution of a variable following a Weibull distribution, but weighted by the element volume to account for the fact that larger volumes are more likely to contain defects, and would thus have a reduced strength. This class follows the approach documented in Strack et al. (2015), and using it to describe local strength for fracture models minimizes mesh size dependence.

The randomized value of a given variable used to characterize a strength can be expressed as: (1) where is the median value of the strength variable, is the reference volume, which is the volume of a test specimen that has a median strength equal to , is a uniform random number on the interval from 0 to 1, and is the Weibull modulus.

This has two important differences from using the standard Weibull distribution, which can also be used to define a randomized strength in MOOSE using a combination of the WeibullDistribution object in the stochastic_tools module and the MOOSE RandomIC object:

1. This Weibull distribution is defined by two parameters: the Weibull modulus, , and the median value of the randomized variable, . This in contrast to the standard version, which has three parameters: the Weibull modulus, , the shape parameter, , and the location parameter, . The distribution generated by this Weibull distribution implicitly assumes that . The value for for the standard set of parameters in a Weibull distribution can be computed from and as:

2. The value of the samples from the standard Weibull distribution is scaled by the factor: As a result of this scaling, elements larger than the reference volume typically have decreased strength, and elements smaller than the reference volume have increased strength (because is typically greater than 1).

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 100
  ny = 100
[]

[Problem]
  solve = false
[]

[AuxVariables]
  [u_vww]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[ICs]
  [u_vww]
    type = VolumeWeightedWeibull
    variable = u_vww
    reference_volume = 0.0001 #This is the volume of an element for a 100x100 mesh
    weibull_modulus = 15.0
    median = 1.0
# When the reference_volume is equal to the element volume (so that there is no volume correction),
# this combination of Weibull modulus and median gives the same distribution that you would get with
# the following parameters in a WeibullDistribution:
#    weibull_modulus = 15.0
#    location = 0
#    scale = 1.024735156 #median * (-1/log(0.5))^(1/weibull_modulus)
  []
[]

[VectorPostprocessors]
  [./histo]
    type = VolumeHistogram
    variable = u_vww
    min_value = 0
    max_value = 2
    bin_number = 100
    execute_on = initial
    outputs = initial
  [../]
[]

[Executioner]
  type = Steady
[]

[Outputs]
  [./initial]
    type = CSV
    execute_on = initial
  [../]
[]