Using Stochastic Tools with Multiphysics Models

The purpose of this document is to present a multiphysics example using the Stochastic Tools Module. The intention is to showcase the capabilities of the module to produce statistically relevant results including uncertainty propagation and sensitivity, as well as the module's surrogate modeling infrastructure. The problem of interest is a thermomechanics model using a combination of the Heat Transfer and Solid Mechanics modules. The problem involves multiple uncertain material properties and multiple quantities of interest (QoI). Using both Monte Carlo sampling and polynomial chaos surrogate modeling, the effect of these properties' uncertainties are quantified with uncertainty propagation and global sensitivity analysis.

Problem Description

The problem of interest involves a steady-state thermomechanics model. The geometry is a 3-D finite hollow cylinder with two concentric layers of different material properties, seen in Figure 1. Due to symmetry, only 1/8 of the cylinder is represented by the mesh. The inner surface of the ring is exposed to a surface heat source that has a cosine shape along the axis of the cylinder with its peak being at the center. The end and outside of the cylinder have convective boundary conditions. The cylinder is free to displace in all directions due to the thermal expansion. The relevant material and geometric properties are listed in Table 1. This table also lists the "uncertain" parameters that will be described in the following section. For reference, the temperature and displacement profiles are shown in Figure 4 to Figure 8 where the uncertain properties are set to some arbitrary values. The full input file for this model is shown in Listing 1.

Figure 1: Model Problem Geometry

Table 1: Material Properties for Thermomechanics Cylinder

PropertySymbolValueUnits
Half Cylinder Height3m
Inner Radius1.0m
Inner Ring Width0.1m
Outer Ring Width0.1m
Outer Heat Transfer Coef.10W/mK
Outer Free Temperature300K
End Heat Transfer Coef.10W/mK
End Free Temperature300K
Heat Source MagnitudeUncertainW
Inner Thermal ConductivityUncertainW/mK
Outer Thermal ConductivityUncertainW/mK
Inner Young's ModulusUncertainPa
Outer Young's ModulusUncertainPa
Inner Poisson's RatioUncertain
Outer Poisson's RatioUncertain
Inner Thermal Expansion Coef.Uncertain1/K
Outer Thermal Expansion Coef.Uncertain1/K
At Rest Temperature300K

Figure 4: Temperature (K)

Figure 5: Displacement (m)

Figure 6: X-Displacement (m)

Figure 7: Y-Displacement (m)

Figure 8: Z-Displacement (m)

Listing 1: Thermomechanics model input file

# Generate 1/4 of a 2-ring disk and extrude it by half to obtain
# 1/8 of a 3D tube.  Mirror boundary conditions will exist on the
# cut portions.

[Mesh]
  [disk]
    type = ConcentricCircleMeshGenerator
    num_sectors = 10
    radii = '1.0 1.1 1.2'
    rings = '1 1 1'
    has_outer_square = false
    preserve_volumes = false
    portion = top_right
  []
  [ring]
    type = BlockDeletionGenerator
    input = disk
    block = 1
    new_boundary = 'inner'
  []
  [cylinder]
    type = MeshExtruderGenerator
    input = ring
    extrusion_vector = '0 0 1.5'
    num_layers = 15
    bottom_sideset = 'back'
    top_sideset = 'front'
  []
[]

[Variables]
  [T]
    initial_condition = 300
  []
  [disp_x]
  []
  [disp_y]
  []
  [disp_z]
  []
[]

[Kernels]
  [hc]
    type = HeatConduction
    variable = T
  []
  [TensorMechanics]
    displacements = 'disp_x disp_y disp_z'
  []
[]

[BCs]
  [temp_inner]
    type = FunctionNeumannBC
    variable = T
    boundary = 'inner'
    function = surface_source
  []
  [temp_front]
    type = ConvectiveHeatFluxBC
    variable = T
    boundary = 'front'
    T_infinity = 300
    heat_transfer_coefficient = 10
  []
  [temp_outer]
    type = ConvectiveHeatFluxBC
    variable = T
    boundary = 'outer'
    T_infinity = 300
    heat_transfer_coefficient = 10
  []
  # mirror boundary conditions.
  [disp_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'left'
    value = 0.0
  []
  [disp_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'bottom'
    value = 0.0
  []
  [disp_z]
    type = DirichletBC
    variable = disp_z
    boundary = 'back'
    value = 0.0
  []
[]

[Materials]
  [cond_inner]
    type = GenericConstantMaterial
    block = 2
    prop_names = thermal_conductivity
    prop_values = 25
  []
  [cond_outer]
    type = GenericConstantMaterial
    block = 3
    prop_names = thermal_conductivity
    prop_values = 100
  []
  [elasticity_tensor_inner]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 2.1e5
    poissons_ratio = 0.3
    block = 2
  []
  [elasticity_tensor_outer]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 3.1e5
    poissons_ratio = 0.2
    block = 3
  []
  [thermal_strain_inner]
    type = ComputeThermalExpansionEigenstrain
    thermal_expansion_coeff = 2e-6
    temperature = T
    stress_free_temperature = 300
    eigenstrain_name = eigenstrain_inner
    block = 2
  []
  [thermal_strain_outer]
    type = ComputeThermalExpansionEigenstrain
    thermal_expansion_coeff = 1e-6
    temperature = T
    stress_free_temperature = 300
    eigenstrain_name = eigenstrain_outer
    block = 3
  []
  [strain_inner] #We use small deformation mechanics
    type = ComputeSmallStrain
    displacements = 'disp_x disp_y disp_z'
    eigenstrain_names = 'eigenstrain_inner'
    block = 2
  []
  [strain_outer] #We use small deformation mechanics
    type = ComputeSmallStrain
    displacements = 'disp_x disp_y disp_z'
    eigenstrain_names = 'eigenstrain_outer'
    block = 3
  []
  [stress] #We use linear elasticity
    type = ComputeLinearElasticStress
  []
[]

[Functions]
  [surface_source]
    type = ParsedFunction
    expression = 'Q_t*pi/2.0/3.0 * cos(pi/3.0*z)'
    symbol_names = 'Q_t'
    symbol_values = heat_source
  []
[]

[Executioner]
  type = Steady

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_gmres_restart'
  petsc_options_value = 'hypre boomeramg 101'

  l_max_its = 30
  nl_max_its = 100
  nl_abs_tol = 1e-9
  l_tol = 1e-04
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Controls]
  [stochastic]
    type = SamplerReceiver
  []
[]

[VectorPostprocessors]
  [temp_center]
    type = LineValueSampler
    variable = T
    start_point = '1 0 0'
    end_point = '1.2 0 0'
    num_points = 11
    sort_by = 'x'
  []
  [temp_end]
    type = LineValueSampler
    variable = T
    start_point = '1 0 1.5'
    end_point = '1.2 0 1.5'
    num_points = 11
    sort_by = 'x'
  []
  [dispx_center]
    type = LineValueSampler
    variable = disp_x
    start_point = '1 0 0'
    end_point = '1.2 0 0'
    num_points = 11
    sort_by = 'x'
  []
  [dispx_end]
    type = LineValueSampler
    variable = disp_x
    start_point = '1 0 1.5'
    end_point = '1.2 0 1.5'
    num_points = 11
    sort_by = 'x'
  []
  [dispz_end]
    type = LineValueSampler
    variable = disp_z
    start_point = '1 0 1.5'
    end_point = '1.2 0 1.5'
    num_points = 11
    sort_by = 'x'
  []
[]

[Postprocessors]
  [heat_source]
    type = FunctionValuePostprocessor
    function = 1
    scale_factor = 10000
    execute_on = linear
  []
  [temp_center_inner]
    type = PointValue
    variable = T
    point = '1 0 0'
  []
  [temp_center_outer]
    type = PointValue
    variable = T
    point = '1.2 0 0'
  []
  [temp_end_inner]
    type = PointValue
    variable = T
    point = '1 0 1.5'
  []
  [temp_end_outer]
    type = PointValue
    variable = T
    point = '1.2 0 1.5'
  []
  [dispx_center_inner]
    type = PointValue
    variable = disp_x
    point = '1 0 0'
  []
  [dispx_center_outer]
    type = PointValue
    variable = disp_x
    point = '1.2 0 0'
  []
  [dispx_end_inner]
    type = PointValue
    variable = disp_x
    point = '1 0 1.5'
  []
  [dispx_end_outer]
    type = PointValue
    variable = disp_x
    point = '1.2 0 1.5'
  []
  [dispz_inner]
    type = PointValue
    variable = disp_z
    point = '1 0 1.5'
  []
  [dispz_outer]
    type = PointValue
    variable = disp_z
    point = '1.2 0 1.5'
  []
[]

[Outputs]
  exodus = false
  csv = false
[]
(modules/combined/examples/stochastic/graphite_ring_thermomechanics.i)

Uncertain Parameters

A total of nine properties of the model are not known exactly, but with some known probability of values occurring. The probabilities are represented by each parameter's probability density function. All parameters have a independent uniform distribution, , where and are the lower and upper limit values for the property, respectively. Table 2 lists the details of each of the property's distribution.

Table 2: Uniform distribution parameters for uncertain properties

IndexProperty
12030
290110
3900011000
41.03.0
50.51.5
62.02.2
73.03.2
80.290.31
90.190.21

Quantities of Interest

There are a total of ten QoIs for the model, which involve temperature and displacement:

  1. Temperature at center of inner surface —

  2. Temperature at center of outer surface —

  3. Temperature at end of inner surface —

  4. Temperature at end of outer surface —

  5. x-displacement at center of inner surface —

  6. x-displacement at center of outer surface —

  7. x-displacement at end of inner surface —

  8. x-displacement at end of outer surface —

  9. z-displacement at end of inner surface —

  10. z-displacement at end of outer surface —

Figure 2 shows geometrically where these QoIs are located in the model.

Figure 2: Geometric description of quantities of interest

Results

In this exercise, we will use the statistics and Sobol sensitivity capabilities available in the stochastic tools module. The goal of this exercise is to understand how the uncertainty in the parameters affects the resulting QoIs. This is done through sampling the model at different perturbations of the parameters and performing statistical calculations on resulting QoI values. Two methods are used to perform this analysis. First is using the sampler system to perturb the uncertain properties and retrieve the QoIs which will undergo the analysis. The second is training a polynomial chaos surrogate and using that reduced order model to sample and perform the analysis. The idea is that many evaluations of the model are necessary to compute accurate statistical quantities and surrogate modeling speeds up this computation by requiring much fewer full model evaluations for training and is significantly faster to evaluate once trained.

Using latin hypercube sampling, the thermomechanics model was run with a total of 100,000 samples, the input file is shown by Listing 2. A order four polynomial chaos surrogate was training using a Smolyak sparse quadrature for a total of 7,344 runs of the full model. The training input is shown by Listing 3 and the evaluation input is shown by Listing 4. Table 3 shows the run-time for sampling the full order model and training and evaluating the surrogate. We see here that cumulative time for training and evaluating the surrogate is much smaller than just sampling the full order model, this is because building the surrogate required far fewer evaluations of the full model and evaluating the surrogate is much faster than evaluating the full model.

Table 3: Stochastic run-time results

SimulationSamplesCPU Time
Full-Order Sampling100,000176 hr
Polynomial Chaos — Training7,34413.7 hr
Polynomial Chaos — Evaluation100,0006.8 s

Listing 2: Latin hypercube sampling and statistics input file

[StochasticTools]
[]

[Distributions]
  [cond_inner]
    type = Uniform
    lower_bound = 20
    upper_bound = 30
  []
  [cond_outer]
    type = Uniform
    lower_bound = 90
    upper_bound = 110
  []
  [heat_source]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
  [alpha_inner]
    type = Uniform
    lower_bound = 1e-6
    upper_bound = 3e-6
  []
  [alpha_outer]
    type = Uniform
    lower_bound = 5e-7
    upper_bound = 1.5e-6
  []
  [ymod_inner]
    type = Uniform
    lower_bound = 2e5
    upper_bound = 2.2e5
  []
  [ymod_outer]
    type = Uniform
    lower_bound = 3e5
    upper_bound = 3.2e5
  []
  [prat_inner]
    type = Uniform
    lower_bound = 0.29
    upper_bound = 0.31
  []
  [prat_outer]
    type = Uniform
    lower_bound = 0.19
    upper_bound = 0.21
  []
[]

[Samplers]
  [sample]
    type = LatinHypercube
    num_rows = 100000
    distributions = 'cond_inner cond_outer heat_source alpha_inner alpha_outer ymod_inner ymod_outer prat_inner prat_outer'
    execute_on = INITIAL
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = graphite_ring_thermomechanics.i
    sampler = sample
    mode = batch-reset
  []
[]

[Transfers]
  [sub]
    type = SamplerParameterTransfer
    to_multi_app = sub
    sampler = sample
    parameters = 'Materials/cond_inner/prop_values Materials/cond_outer/prop_values
                  Postprocessors/heat_source/scale_factor
                  Materials/thermal_strain_inner/thermal_expansion_coeff Materials/thermal_strain_outer/thermal_expansion_coeff
                  Materials/elasticity_tensor_inner/youngs_modulus Materials/elasticity_tensor_outer/youngs_modulus
                  Materials/elasticity_tensor_inner/poissons_ratio Materials/elasticity_tensor_outer/poissons_ratio'
    check_multiapp_execute_on = false
  []
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = sample
    stochastic_reporter = storage
    from_reporter = 'temp_center_inner/value  temp_center_outer/value  temp_end_inner/value  temp_end_outer/value
                     dispx_center_inner/value dispx_center_outer/value dispx_end_inner/value dispx_end_outer/value
                     dispz_inner/value dispz_outer/value'
  []
[]

[Reporters]
  [storage]
    type = StochasticReporter
    parallel_type = ROOT
  []
  [stats]
    type = StatisticsReporter
    reporters = 'storage/data:temp_center_inner:value  storage/data:temp_center_outer:value  storage/data:temp_end_inner:value  storage/data:temp_end_outer:value
                 storage/data:dispx_center_inner:value storage/data:dispx_center_outer:value storage/data:dispx_end_inner:value storage/data:dispx_end_outer:value
                 storage/data:dispz_inner:value storage/data:dispz_outer:value'
    compute = 'mean stddev'
    ci_method = 'percentile'
    ci_levels = '0.05 0.95'
  []
[]

[Outputs]
  [out]
    type = JSON
  []
  execute_on = TIMESTEP_END
[]
(modules/combined/examples/stochastic/lhs_uniform.i)

Listing 3: Polynomial chaos training input file

[StochasticTools]
[]

[Distributions]
  [cond_inner]
    type = Uniform
    lower_bound = 20
    upper_bound = 30
  []
  [cond_outer]
    type = Uniform
    lower_bound = 90
    upper_bound = 110
  []
  [heat_source]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
  [alpha_inner]
    type = Uniform
    lower_bound = 1e-6
    upper_bound = 3e-6
  []
  [alpha_outer]
    type = Uniform
    lower_bound = 5e-7
    upper_bound = 1.5e-6
  []
  [ymod_inner]
    type = Uniform
    lower_bound = 2e5
    upper_bound = 2.2e5
  []
  [ymod_outer]
    type = Uniform
    lower_bound = 3e5
    upper_bound = 3.2e5
  []
  [prat_inner]
    type = Uniform
    lower_bound = 0.29
    upper_bound = 0.31
  []
  [prat_outer]
    type = Uniform
    lower_bound = 0.19
    upper_bound = 0.21
  []
[]

[GlobalParams]
  distributions = 'cond_inner cond_outer heat_source alpha_inner alpha_outer ymod_inner ymod_outer prat_inner prat_outer'
[]

[Samplers]
  [sample]
    type = Quadrature
    sparse_grid = smolyak
    order = 5
    execute_on = INITIAL
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = graphite_ring_thermomechanics.i
    sampler = sample
    mode = batch-reset
  []
[]

[Transfers]
  [sub]
    type = SamplerParameterTransfer
    to_multi_app = sub
    sampler = sample
    parameters = 'Materials/cond_inner/prop_values Materials/cond_outer/prop_values
                  Postprocessors/heat_source/scale_factor
                  Materials/thermal_strain_inner/thermal_expansion_coeff Materials/thermal_strain_outer/thermal_expansion_coeff
                  Materials/elasticity_tensor_inner/youngs_modulus Materials/elasticity_tensor_outer/youngs_modulus
                  Materials/elasticity_tensor_inner/poissons_ratio Materials/elasticity_tensor_outer/poissons_ratio'
    check_multiapp_execute_on = false
  []
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = sample
    stochastic_reporter = storage
    from_reporter = 'temp_center_inner/value  temp_center_outer/value  temp_end_inner/value  temp_end_outer/value
                     dispx_center_inner/value dispx_center_outer/value dispx_end_inner/value dispx_end_outer/value
                     dispz_inner/value dispz_outer/value'
  []
[]

[Reporters]
  [storage]
    type = StochasticReporter
  []
[]

[Trainers]
  [temp_center_inner]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:temp_center_inner:value
  []
  [temp_center_outer]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:temp_center_outer:value
  []
  [temp_end_inner]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:temp_end_inner:value
  []
  [temp_end_outer]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:temp_end_outer:value
  []
  [dispx_center_inner]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:dispx_center_inner:value
  []
  [dispx_center_outer]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:dispx_center_outer:value
  []
  [dispx_end_inner]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:dispx_end_inner:value
  []
  [dispx_end_outer]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:dispx_end_outer:value
  []
  [dispz_inner]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:dispz_inner:value
  []
  [dispz_outer]
    type = PolynomialChaosTrainer
    execute_on = timestep_end
    order = 4
    sampler = sample
    response = storage/data:dispz_outer:value
  []
[]

[Outputs]
  [out]
    type = SurrogateTrainerOutput
    trainers = 'temp_center_inner  temp_center_outer  temp_end_inner  temp_end_outer
                dispx_center_inner dispx_center_outer dispx_end_inner dispx_end_outer
                dispz_inner dispz_outer'
    execute_on = FINAL
  []
[]
(modules/combined/examples/stochastic/poly_chaos_train_uniform.i)

Listing 4: Polynomial chaos evaluation input file

[StochasticTools]
[]

[Distributions]
  [cond_inner]
    type = Uniform
    lower_bound = 20
    upper_bound = 30
  []
  [cond_outer]
    type = Uniform
    lower_bound = 90
    upper_bound = 110
  []
  [heat_source]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
  [alpha_inner]
    type = Uniform
    lower_bound = 1e-6
    upper_bound = 3e-6
  []
  [alpha_outer]
    type = Uniform
    lower_bound = 5e-7
    upper_bound = 1.5e-6
  []
  [ymod_inner]
    type = Uniform
    lower_bound = 2e5
    upper_bound = 2.2e5
  []
  [ymod_outer]
    type = Uniform
    lower_bound = 3e5
    upper_bound = 3.2e5
  []
  [prat_inner]
    type = Uniform
    lower_bound = 0.29
    upper_bound = 0.31
  []
  [prat_outer]
    type = Uniform
    lower_bound = 0.19
    upper_bound = 0.21
  []
[]

[Samplers]
  [sample]
    type = MonteCarlo
    num_rows = 100000
    distributions = 'cond_inner cond_outer heat_source alpha_inner alpha_outer ymod_inner ymod_outer prat_inner prat_outer'
    execute_on = INITIAL
  []
[]

[Surrogates]
  [temp_center_inner]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_temp_center_inner.rd'
  []
  [temp_center_outer]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_temp_center_outer.rd'
  []
  [temp_end_inner]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_temp_end_inner.rd'
  []
  [temp_end_outer]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_temp_end_outer.rd'
  []
  [dispx_center_inner]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_dispx_center_inner.rd'
  []
  [dispx_center_outer]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_dispx_center_outer.rd'
  []
  [dispx_end_inner]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_dispx_end_inner.rd'
  []
  [dispx_end_outer]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_dispx_end_outer.rd'
  []
  [dispz_inner]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_dispz_inner.rd'
  []
  [dispz_outer]
    type = PolynomialChaos
    filename = 'poly_chaos_train_uniform_out_dispz_outer.rd'
  []
[]

[Reporters]
  [storage]
    type = EvaluateSurrogate
    sampler = sample
    model = 'temp_center_inner  temp_center_outer  temp_end_inner  temp_end_outer
             dispx_center_inner dispx_center_outer dispx_end_inner dispx_end_outer
             dispz_inner dispz_outer'
    parallel_type = ROOT
  []
  [stats]
    type = PolynomialChaosReporter
    pc_name = 'temp_center_inner  temp_center_outer  temp_end_inner  temp_end_outer
               dispx_center_inner dispx_center_outer dispx_end_inner dispx_end_outer
               dispz_inner dispz_outer'
    statistics = 'mean stddev'
    include_sobol = true
  []
[]

[Outputs]
  [out]
    type = JSON
  []
  execute_on = TIMESTEP_END
[]
(modules/combined/examples/stochastic/poly_chaos_uniform.i)

Statistics

Table 4 shows the statistical results of sampling the thermomechanics model and the polynomial chaos surrogate. and represent the mean and standard deviation of the QoI, and CI is the confidence interval. Note that the confidence interval for the polynomial chaos statistics is not relevant since these values were found analytically using integration techniques. Figure 9 to Figure 11 compares several of the probability distributions of the QoIs between sampling the full-order model and the polynomial chaos surrogate.

Table 4: Statistics Results

QoI95% CI95% CIPC – PC –
609.97(609.87,
610.06)		18.22(18.18,
18.27)		609.9718.23
586.85(586.77,
586.94)		16.64(16.60,
16.68)		586.8516.64
506.31(506.25,
506.37)		12.05(12.03,
12.08)		506.3112.05
507.92(507.85,
507.98)		12.13(12.10,
12.15)		507.9212.12
4.032E-04(4.028E-04,
4.037E-04)		8.608E-05(8.581E-05,
8.636E-05)		4.032E-048.625E-05
4.996E-04(4.990E-04,
5.001E-04)		1.078E-04(1.075E-04,
1.082E-04)		4.996E-041.081E-04
4.220E-04(4.213E-04,
4.226E-04)		1.255E-04(1.252E-04,
1.258E-04)		4.220E-041.256E-04
4.793E-04(4.786E-04,
4.800E-04)		1.352E-04(1.349E-04,
1.356E-04)		4.794E-041.354E-04
6.139E-04(6.131E-04,
6.146E-04)		1.466E-04(1.462E-04,
1.470E-04)		6.139E-041.469E-04
4.995E-04(4.989E-04,
5.000E-04)		1.067E-04(1.065E-04,
1.072E-04)		4.995E-041.071E-04

Figure 9: Histogram

Figure 10: Histogram

Figure 11: Histogram

Sobol Sensitivities

Sobol sensitivities, or Sobol indices, are a metric to compare the global sensitivity a parameter has on a QoI. This examples demonstrates several different types of the sensitivities. The first is total sensitivity, which measure the total sensitivity from a parameter, Figure 3 shows these values for each QoI and parameter. The second is a correlative sensitivity, which measures the sensitivity due to a combination of parameters, Figure 12 to Figure 14 show heat maps of these values for several QoIs.

Figure 3: Total Sobol sensitivities

Figure 12: Sobol sensitivities

Figure 13: Sobol sensitivities

Figure 14: Sobol sensitivities

Supplementary Figures

Probability Distributions for All QoIs

Sobol Sensitivity Heatmaps for All QoIs