Divertor Monoblock Sensitivities

This page documents sensitivity studies of different operating conditions performed on the Divertor Monoblock. The original pulsed operation of the divertor monoblock model was modified according to Hodille et al. (2021) towards a single-long tritium and heat flux pulse of an approximate equivalent total fluence. An initial sensitivity study on was performed on the divertor monoblock's boundary conditions under the steady pulse condition using the Sobol Method and other tools within the Stochastic Tools Module. Two additional sensitivity studies were performed on select transients obtained from El-Morshedy (2024) and Kessel et al. (2013).

General description of the sensitivity studies and the modified cases

Introduction

Many fusion components, especially those exposed to especially unique or extreme conditions, often lack sufficient operation data to evaluate their reliability, availability, maintainability, and inspectability (RAMI). As a proof-of-concept, we seek to construct a Probabilistic Physics of Failure (PPoF) model of a tokomak divertor system to obtain failure rate metrics and lifetime metric relevant to the RAMI of a nuclear fusion reactor.

To accomplish this, sensitivity studies on the Divertor Monoblock model described in Shimada et al. (2024) will be performed for (1) steady state conditions based on ITER data Hodille et al. (2021), (2) an inadvertent shutdown transient El-Morshedy (2024), and (3) an edge-localized mode transient Kessel et al. (2013). The sensitivity studies will be performed using the MOOSE Stochastic Module and its Sobol Method according to Saltelli (2002). The results of the sensitivity studies will be evaluated for state limits associated with heat transfer and tritium migration physics included in the Divertor Monoblock.

First, the modifications made to the Divertor Monoblock model will be described. Second, the conditions the divertor monoblock was subjected to in the steady and transient scenarios will be described. Finally, the results of the sensitivities, namely the first and second order sensitivity indices, example correlations and state frequencies will be presented.

Divertor Monoblock modifications

The pulsed operation of the divertor monoblock is relatively computationally expensive to accurately model since it requires taking small time steps to capture transients. For a sensitivity analysis study, where many simulations are required, lowering the cost of individual simulations is paramount. So as a first approximation, we replace the pulsed operation with a single steady pulse in this study.

Steady operation

The sensitivity studies must sample from a distribution of parameters to determine the net effects of those parameters on the model. Here, we vary the incident heat flux, tritium flux, coolant temperature, and coolant tritium concentration. For all these parameters, the sampling is done from uniform distributions as shown in Table 1 and Figure 1. It should be noted that the coolant temperature is unusually high in this case, as the ITER divertor is designed to have coolant run at 100°C with a 50°C temperature increase at the outlet Hirai et al. (2010); however, this is the designed 240°C temperature for the divertor to run high-pressure water in the bake-off regime for tritium unloading.

Table 1: Steady case varied boundary conditions.

Parameter	Samples	Nominal Value	Distribution	Deviation
Incident Heat Flux	100	10 MW/m $var element = document.getElementById("moose-equation-e0098106-e41b-42d4-9139-1af6745b9d2b");katex.render("^2", element, {displayMode:false,throwOnError:false});$ Pitts et al. (2009)\| Uniform	$var element = document.getElementById("moose-equation-37965e70-9e1e-48f0-aa4b-4387a1783c90");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Incident Tritium Flux	100	7.90 $var element = document.getElementById("moose-equation-d826849c-96ff-4695-ad30-3b374289b022");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-ea2bd64f-91ea-4c9c-9500-742af277a4fa");katex.render("^{-13}", element, {displayMode:false,throwOnError:false});$ (normalized)	Uniform	$var element = document.getElementById("moose-equation-11e5227c-0f34-4ace-95b4-2d43ae699c1f");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Coolant Temperature	100	552 K Hirai et al. (2010)\| Uniform	$var element = document.getElementById("moose-equation-391a8bdc-0724-42f0-922b-8c67dfe340d7");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Coolant Tritium Concentration	100	1.0 $var element = document.getElementById("moose-equation-f6a57f87-9e32-4730-9cfb-c6141ad4341d");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-ccf816bd-e1cd-4cfa-ad37-7421d5db80af");katex.render("^{-18}", element, {displayMode:false,throwOnError:false});$ (normalized)	Uniform	$var element = document.getElementById("moose-equation-c4b19240-55ab-4fe7-b3c6-93c6175a4e08");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 1%

Figure 1: Distribution of sampled steady-state parameter values.

Transient case: inadvertent shutdown

For a transient case of an inadvertent shutdown, we modify parameters as shown in Table 2 and Figure 2. This case is modified such that we assume constant operation for 5.5 hours before encountering a Peak Heat Flux (and proportionally scaled tritium flux) for the sampled Peak Duration before resuming at one-tenth the heat and tritium flux of the original steady-state.

The top boundary tritium flux and temperature flux are both adjusted to be a function of time, with functions defined in the input file as follows.

[Functions<<<{"href": "../../syntax/Functions/index.html"}>>>]
  [mobile_flux_bc_function]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../source/functions/MooseParsedFunction.html"}>>>
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 'pulse_time_function'
    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the symbol_values vector."}>>> = 'pulse_time_function'
    expression<<<{"description": "The user defined function."}>>> = "if(t<2e4, 7.90e-13, if(t<(2e4+${peak_duration}), ${peak_value}/1.0e7*7.90e-13, 7.90e-14))"
  []
[]

[Functions<<<{"href": "../../syntax/Functions/index.html"}>>>]
  [temperature_flux_bc_function]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../source/functions/MooseParsedFunction.html"}>>>
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 'pulse_time_function'
    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the symbol_values vector."}>>> = 'pulse_time_function'
    expression<<<{"description": "The user defined function."}>>> = "if(t<2e4, 1.0e7, if(t<(2e4+${peak_duration}), ${peak_value}, 1.0e6))"
  []
[]

Table 2: Inadvertent shutdown varied parameter space.

Parameter	Samples	Nominal Value	Distribution	Deviation
Peak Duration	1000	1s	Normal	$var element = document.getElementById("moose-equation-dcb6f56e-599d-4f6b-bae4-a4e1a5997d7f");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ = $var element = document.getElementById("moose-equation-787a4c12-bea0-40d4-adf1-01e22c566b02");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Peak Heat Flux	1000	20 MW/m $var element = document.getElementById("moose-equation-e221108c-be28-47b0-9c3d-7e59d5d3eec4");katex.render("^2", element, {displayMode:false,throwOnError:false});$	Normal	$var element = document.getElementById("moose-equation-1eb5040a-437e-45d1-b7dc-6b795be00c2e");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ = $var element = document.getElementById("moose-equation-64e1f02e-bc14-4e97-919a-a7b396f87241");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Coolant Temperature	1000	552 K Hirai et al. (2010)	Normal	$var element = document.getElementById("moose-equation-b359a7a7-c483-4335-8545-a104d4ec76f0");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ = $var element = document.getElementById("moose-equation-638e959a-962f-4dc7-9e29-13ebeb6f1558");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Tungsten Conductivity	1000	0.95 W/m K Shimada et al. (2024)	Uniform	$var element = document.getElementById("moose-equation-98ad98ad-ad27-401e-a78d-6721457113af");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%

Figure 2: Distribution of sampled transient shutdown characteristics.

Transient case: edge-localized mode disruption

To simulate an edge-localized mode (ELM) disruption, we simulate a set of input conditions as described in Table 3. The heat flux is much larger than the transient shutdown case, but over a shorter duration. Most of the parameters will be selected from a normal distribution, as they reflect variations around some target or estimated value. The results will be shown below in Figure 6. We use the same tungsten conductivity as in Shimada et al. (2024), but vary the conductivity by a constant factor in the range of $var element = document.getElementById("moose-equation-5d5fcdf2-ac83-4618-b302-d559f5ffe6f1");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%.

Table 3: Edge-localized mode transient case parameter space.

Parameter	Samples	Nominal Value	Distribution	Deviation
Peak Duration	1000	1.32 ms	Normal	$var element = document.getElementById("moose-equation-89318df5-5684-4533-a7f5-aa36e8600286");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ = $var element = document.getElementById("moose-equation-f584bf8e-4cbe-4339-87af-922f602f40d1");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Peak Heat Flux	1000	1147 MW/m $var element = document.getElementById("moose-equation-3b5d9e1e-5212-4aa4-9d08-d3f2e5155c51");katex.render("^2", element, {displayMode:false,throwOnError:false});$ Loarte et al. (2014)	Normal	$var element = document.getElementById("moose-equation-5af44618-6247-4e99-82be-d3a7bdf150ea");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ = $var element = document.getElementById("moose-equation-9c813c78-ac2c-4b81-8d69-ae44dbcdf418");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Coolant Temperature	1000	552 K Hirai et al. (2010)\| Normal	$var element = document.getElementById("moose-equation-40c480cf-77cd-4851-97e3-b2313727e6a8");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ = $var element = document.getElementById("moose-equation-543bc3a6-2e49-4c64-b199-9ee9585af2ad");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%
Tungsten Conductivity	1000	0.95 W/m K Shimada et al. (2024)	Uniform	$var element = document.getElementById("moose-equation-f1375bc3-38d5-4acb-9d5e-14df76b6a848");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ = $var element = document.getElementById("moose-equation-1d9a567c-d7ee-42ed-95cb-69bf10cff900");katex.render("\\pm", element, {displayMode:false,throwOnError:false});$ 5%

Results

Steady operation

For purposes of comparison, we look at the maximum observed tungsten temperature and the scaled permeation flux with regards to the heat flux in Figure 3. The maximum tungsten temperature naturally correlates with the heat flux, but also depends on thermal conductivity and coolant temperature, as expected. The link between heat flux and tritium flux is not as strong. In general, larger values of tritium flux to the coolant are obtained with higher heat flux, which is the results from increased diffusivity and temperatures more likely to be sufficient for tritium release from traps. However, small tritium flux values are still obtained when a large heat flux is applied. This is attributed to lower temperatures towards the coolant (due to cooler coolant or higher thermal conductivity). Note, however, that low tritium flux are less likely as the heat flux increases.

Correlation between the incident heat flux and the maximum temperature of the tungsten, as well as the total (scaled) permeation flux of tritium into the coolant.

Figure 3: Correlation between the incident heat flux and the maximum temperature of the tungsten, as well as the total (scaled) permeation flux of tritium into the coolant.

Inadvertent shutdown

For the inadvertent shutdown case, we can correlate the input parameters with several output parameters. Perhaps most interesting is the relationship between coolant temperature and the scaled tritium flux, where tritium does not permeate effectively for coolant temperatures below about 550 K. In Figure 4, the upper plots are scatterplots, the lower are kernel density plots and the diagonal are histograms.

Figure 4: Correlation between the input parameters and several metrics of performance in an inadvertent shutdown scenario.

The results of the simulation are shown in Figure 5. As would be expected, the amount of permeation and the amount of tritium retention have an approximately inverse relationship, and the pairwise interactions across the board are shown. The time at which maximum temperatures occur are all correlated, as one might expect.

Figure 5: Distribution of sampled postprocessor values in the inadvertent shutdown case.

Edge-localized mode disruption

For the case of simulating an ELM disruption event, we show in Figure 6 that although the operating conditions vary, the trends are substantially the same.

Correlation between the incident heat flux and the maximum temperature of the tungsten, as well as the total (scaled) permeation flux of tritium.

Figure 6: Correlation between the incident heat flux and the maximum temperature of the tungsten, as well as the total (scaled) permeation flux of tritium.

Complete input files

Below are the complete input files for the various sensitivity studies. Note that none of the inputs have been optimized for computational costs.

Steady operation

Subapp input

### This case adapts the published divertor monoblock input file under steady-state
### conditions, and is used to do a sobol sensitivity analysis of the underlying model
### by running steady_state_sobol.i directly from TMAP8.
### M. Shimada, P.-C. A. Simon, C. T. Icenhour, and G. Singh, “Toward a high-fidelity
### tritium transport modeling for retention and permeation experiments,” Fusion
### Engineering and Design, Volume 203, 2024, 114438, ISSN 0920-3796,
### https://doi.org/10.1016/j.fusengdes.2024.114438.
### Nomenclatures                                                                                  # Nelson S. Comments/Annotations
###
### C_mobile_j      mobile H concentration in "j" material, where j = CuCrZr, Cu, W
### C_trapped_j     trapped H concentration in "j" material, where j = CuCrZr, Cu, W
### C_total_j       total H concentration in "j" material, where j = CuCrZr, Cu, W
###
### F_permeation    permeation flux
### F_recombination recombination flux
###
### Sc_             Scaled
### Int_            Integrated
### ScInt_          Scaled and integrated

temperature_top_val = ${units 10 MW/m^2}
C_mob_W_top_flux_val = 7.90e-13 # normalized units
temperature_tube_val = ${units 552 K}

[Controls]
  [stochastic]
    # Sends data to Stochastic
    type = SamplerReceiver
  []
[]

### This input uses the `!include` feature to incorporate other input files
!include divertor_monoblock_explicit_base.i

# Geometry and design
num_sectors := 12
rings_CuCrZr := 6
rings_Cu := 4
rings_W := 22

Functions/mobile_flux_bc_function/expression := '${fparse C_mob_W_top_flux_val}'
Functions/temperature_flux_bc_function/expression := '${fparse temperature_top_val}'
Functions/temperature_inner_function/expression := '${fparse temperature_tube_val}'
Functions/timestep_function/expression := 'if(t<100, 25, 400)'

Executioner/nl_rel_tol := 1e-2
Executioner/nl_abs_tol := 1e-1
Executioner/end_time := ${fparse plasma_cycle_time + plasma_ss_duration}
Executioner/nl_max_its := 24
Executioner/TimeStepper/growth_factor := 2.0

Postprocessors/F_recombination/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/F_permeation/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_mobile_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_trapped_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_total_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_mobile_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_trapped_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_total_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/temperature_top/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/temperature_tube/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/timestep_max_pp/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
# Need to add unused parameters from the divertor monoblock case to the dummy postprocessor
Postprocessors/unused_parameters/expression := '${num_sectors} + ${rings_H2O} + ${rings_CuCrZr} + ${rings_Cu} + ${rings_W}
                  + ${temperature_coolant_max} + ${plasma_max_heat} + ${plasma_min_heat}
                  + ${plasma_max_flux} + ${plasma_min_flux}'

Variables/temperature/initial_condition= ${temperature_initial}

[Outputs]
  # Control outputs, include exodus and csv.
  execute_on = 'none'
[]

Controller input

### This input file defines a sobol indice analysis of the monoblock input file to simulate an
### steady-state operation of the ITER divertor monoblock,
### with some simplifying assumptions that allow it
### to be run in a short amount of time.
### The physics input file is steady_state_runner.i
### This file should be called directly by TMAP8, preferably with access to multiple cpus
### Output information including summary statistics can be found in a json file

# Define parameters in one spot for easier reading
upper_heat_flux_lower = ${units 9.5 MW/m^2 -> W/m^2}
upper_heat_flux_upper = ${units 10.5 MW/m^2 -> W/m^2}
upper_tritium_flux_lower = 7.505e-13 # units
upper_tritium_flux_upper = 8.295e-13 # units
coolant_temperature_lower = ${units 524.4 K}
coolant_temperature_upper = ${units 579.6 K}
N_samples = 10

[StochasticTools<<<{"href": "../../syntax/StochasticTools/index.html"}>>>]
  # Designate as the Controller/Main Input
[]

[MultiApps<<<{"href": "../../syntax/MultiApps/index.html"}>>>]
  # Designate a subapp to control later
  [runner]
    type = SamplerFullSolveMultiApp<<<{"description": "Creates a full-solve type sub-application for each row of each Sampler matrix.", "href": "../../source/multiapps/SamplerFullSolveMultiApp.html"}>>>
    sampler<<<{"description": "The Sampler object to utilize for creating MultiApps."}>>> = sobol
    input_files<<<{"description": "The input file for each App.  If this parameter only contains one input file it will be used for all of the Apps.  When using 'positions_from_file' it is also admissable to provide one input_file per file."}>>> = 'steady_state_runner.i'
    mode<<<{"description": "The operation mode, 'normal' creates one sub-application for each row in the Sampler and 'batch-reset' and 'batch-restore' creates N sub-applications, where N is the minimum of 'num_rows' in the Sampler and floor(number of processes / min_procs_per_app). To run the rows in the Sampler, 'batch-reset' will destroy and re-create sub-apps as needed, whereas the 'batch-restore' will backup and restore sub-apps to the initial state prior to execution, without destruction."}>>> = batch-reset
    keep_full_output_history<<<{"description": "Whether or not to keep the full output history when this multiapp has multiple entries"}>>> = True
  []
[]

[Controls<<<{"href": "../../syntax/Controls/index.html"}>>>]
  # Control inputs from Main->Subapp
  [cmdline]
    type = MultiAppSamplerControl<<<{"description": "Control for modifying the command line arguments of MultiApps.", "href": "../../source/controls/MultiAppSamplerControl.html"}>>>
    multi_app<<<{"description": "The name of the MultiApp to control."}>>> = runner
    sampler<<<{"description": "The Sampler object to utilize for altering the command line options of the MultiApp."}>>> = sobol
    param_names<<<{"description": "The names of the command line parameters to set via the sampled data."}>>> = "temperature_top_val
                   C_mob_W_top_flux_val
                   temperature_tube_val"
  []
[]

[Distributions<<<{"href": "../../syntax/Distributions/index.html"}>>>]
  # Define probability distributions of parameters for sampling
  [H_top]
    type = Uniform<<<{"description": "Continuous uniform distribution.", "href": "../../source/distributions/Uniform.html"}>>>
    lower_bound<<<{"description": "Distribution lower bound"}>>> = ${upper_heat_flux_lower}
    upper_bound<<<{"description": "Distribution upper bound"}>>> = ${upper_heat_flux_upper}
  []
  [C_top]
    type = Uniform<<<{"description": "Continuous uniform distribution.", "href": "../../source/distributions/Uniform.html"}>>>
    lower_bound<<<{"description": "Distribution lower bound"}>>> = ${upper_tritium_flux_lower}
    upper_bound<<<{"description": "Distribution upper bound"}>>> = ${upper_tritium_flux_upper}
  []
  [H_bot]
    type = Uniform<<<{"description": "Continuous uniform distribution.", "href": "../../source/distributions/Uniform.html"}>>>
    lower_bound<<<{"description": "Distribution lower bound"}>>> = ${coolant_temperature_lower}
    upper_bound<<<{"description": "Distribution upper bound"}>>> = ${coolant_temperature_upper}
  []
[]

[Samplers<<<{"href": "../../syntax/Samplers/index.html"}>>>]
  # Sampling methodology using the probability distributions above
  [hypercube_1]
    type = LatinHypercube<<<{"description": "Latin Hypercube Sampler.", "href": "../../source/samplers/LatinHypercubeSampler.html"}>>>
    distributions<<<{"description": "The distribution names to be sampled, the number of distributions provided defines the number of columns per matrix."}>>> = 'H_top C_top H_bot'
    num_rows<<<{"description": "The size of the square matrix to generate."}>>> = ${N_samples}
    seed<<<{"description": "Random number generator initial seed"}>>> = 1001
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
  [hypercube_2]
    type = LatinHypercube<<<{"description": "Latin Hypercube Sampler.", "href": "../../source/samplers/LatinHypercubeSampler.html"}>>>
    distributions<<<{"description": "The distribution names to be sampled, the number of distributions provided defines the number of columns per matrix."}>>> = 'H_top C_top H_bot'
    num_rows<<<{"description": "The size of the square matrix to generate."}>>> = ${N_samples}
    seed<<<{"description": "Random number generator initial seed"}>>> = 1002
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
  [sobol]
    type = Sobol<<<{"description": "Sobol variance-based sensitivity analysis Sampler.", "href": "../../source/samplers/SobolSampler.html"}>>>
    sampler_a<<<{"description": "The 'sample' matrix."}>>> = hypercube_1
    sampler_b<<<{"description": "The 're-sample' matrix."}>>> = hypercube_2
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
[]

[Transfers<<<{"href": "../../syntax/Transfers/index.html"}>>>]
  # Define values to extract from subapp
  [results]
    type = SamplerReporterTransfer<<<{"description": "Transfers data from Reporters on the sub-application to a StochasticReporter on the main application.", "href": "../../source/transfers/SamplerReporterTransfer.html"}>>>
    from_multi_app<<<{"description": "The name of the MultiApp to receive data from"}>>> = runner
    sampler<<<{"description": "A the Sampler object that Transfer is associated.."}>>> = sobol
    stochastic_reporter<<<{"description": "The name of the StochasticReporter object to transfer values to."}>>> = results
    #execute_on = 'MULTIAPP_FIXED_POINT_END'
    from_reporter<<<{"description": "The name(s) of the Reporter(s) on the sub-app to transfer from."}>>> = "F_permeation/value
                       Scaled_Tritium_Flux/value
                       total_retention/value
                       coolant_heat_flux/value
                       max_temperature_W/value
                       max_temperature_Cu/value
                       max_temperature_CuCrZr/value"
  []
  [matrix_results]
    type = SamplerReporterTransfer<<<{"description": "Transfers data from Reporters on the sub-application to a StochasticReporter on the main application.", "href": "../../source/transfers/SamplerReporterTransfer.html"}>>>
    from_multi_app<<<{"description": "The name of the MultiApp to receive data from"}>>> = runner
    sampler<<<{"description": "A the Sampler object that Transfer is associated.."}>>> = sobol
    stochastic_reporter<<<{"description": "The name of the StochasticReporter object to transfer values to."}>>> = matrix
    from_reporter<<<{"description": "The name(s) of the Reporter(s) on the sub-app to transfer from."}>>> = "F_permeation/value
                       Scaled_Tritium_Flux/value
                       total_retention/value
                       coolant_heat_flux/value
                       max_temperature_W/value
                       max_temperature_Cu/value
                       max_temperature_CuCrZr/value"
  []
[]

[Reporters<<<{"href": "../../syntax/Reporters/index.html"}>>>]
  [results]
    type = StochasticReporter<<<{"description": "Storage container for stochastic simulation results coming from Reporters.", "href": "../../source/reporters/StochasticReporter.html"}>>>
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL' # INITIAL TIMESTEP_END MULTIAPP_FIXED_POINT_END
  []
  [stats]
    type = StatisticsReporter<<<{"description": "Compute statistical values of a given VectorPostprocessor objects and vectors.", "href": "../../source/reporters/StatisticsReporter.html"}>>>
    reporters<<<{"description": "List of Reporter values to utilized for statistic computations."}>>> = "results/results:F_permeation:value
                   results/results:Scaled_Tritium_Flux:value
                   results/results:total_retention:value
                   results/results:coolant_heat_flux:value
                   results/results:max_temperature_W:value
                   results/results:max_temperature_Cu:value
                   results/results:max_temperature_CuCrZr:value"
    compute<<<{"description": "The statistic(s) to compute for each of the supplied vector postprocessors."}>>> = 'mean stddev'
    ci_method<<<{"description": "The method to use for computing confidence level intervals."}>>> = 'percentile'
    ci_levels<<<{"description": "A vector of confidence levels to consider, values must be in (0, 1)."}>>> = "0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
                   0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
  []
  [sobol]
    type = SobolReporter<<<{"description": "Compute SOBOL statistics values of a given VectorPostprocessor or Reporter objects and vectors.", "href": "../../source/reporters/SobolReporter.html"}>>>
    sampler<<<{"description": "SobolSampler object."}>>> = sobol
    reporters<<<{"description": "List of Reporter values to utilized for statistic computations."}>>> = "results/results:F_permeation:value
                   results/results:Scaled_Tritium_Flux:value
                   results/results:total_retention:value
                   results/results:coolant_heat_flux:value
                   results/results:max_temperature_W:value
                   results/results:max_temperature_Cu:value
                   results/results:max_temperature_CuCrZr:value"
    ci_levels<<<{"description": "A vector of confidence levels to consider, values must be in (0, 1)."}>>> = "0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
                   0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
  []
  [matrix]
    type = StochasticMatrix<<<{"description": "Tool for extracting Sampler object data and storing data from stochastic simulations.", "href": "../../source/reporters/StochasticMatrix.html"}>>>
    sampler<<<{"description": "The sample from which to extract distribution data."}>>> = sobol
    sampler_column_names<<<{"description": "Prescribed names of sampler columns, used to assign names of outputted vectors."}>>> = "BCs/temperature_top/value
                              BCs/C_mob_W_top_flux/value
                              BCs/temperature_tube/value"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
    parallel_type<<<{"description": "This parameter will determine how the stochastic data is gathered. It is common for outputting purposes that this parameter be set to ROOT, otherwise, many files will be produced showing the values on each processor. However, if there are lot of samples, gathering on root may be memory restrictive."}>>> = ROOT
  []
[]

[Outputs<<<{"href": "../../syntax/Outputs/index.html"}>>>]
  [out]
    type = JSON<<<{"description": "Output for Reporter values using JSON format.", "href": "../../source/outputs/JSONOutput.html"}>>>
    distributed<<<{"description": "Whether or not to output distributed data (data not on rank 0)"}>>> = False
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
    execute_reporters_on<<<{"description": "Control of when Reporter values are output"}>>> = 'FINAL'
  []
[]

Inadvertent shutdown

Subapp input

### This case extends the published divertor monoblock input file to simulate a
### shutdown transient, with some simplifying assumptions that allow it
### to be run in a short amount of time.
### M. Shimada, P.-C. A. Simon, C. T. Icenhour, and G. Singh, “Toward a high-fidelity
### tritium transport modeling for retention and permeation experiments,” Fusion
### Engineering and Design, Volume 203, 2024, 114438, ISSN 0920-3796,
### https://doi.org/10.1016/j.fusengdes.2024.114438.

### Nomenclatures                                                                                  # Nelson S. Comments/Annotations
###
### C_mobile_j      mobile H concentration in "j" material, where j = CuCrZr, Cu, W
### C_trapped_j     trapped H concentration in "j" material, where j = CuCrZr, Cu, W
### C_total_j       total H concentration in "j" material, where j = CuCrZr, Cu, W
###
### F_permeation    permeation flux
### F_recombination recombination flux
###
### Sc_             Scaled
### Int_            Integrated
### ScInt_          Scaled and integrated                                                          # Nelson S. Comments/Annotations

### VARIABLES ###
peak_value = ${units 20 MW/m^2 -> W/m^2}
peak_duration = ${units 1.0 s}
coolant_temperature = ${units 552 K}
W_cond_factor = 1.0

!include divertor_monoblock_explicit_base.i

Functions/timestep_function/expression := "if(t<2e4, 500, if(t<(2e4+${peak_duration}+1), 0.10, 500))"
Functions/mobile_flux_bc_function/expression := "if(t<2e4, 7.90e-13, if(t<(2e4+${peak_duration}), ${peak_value}/1.0e7*7.90e-13, "
                 "7.90e-14))"
Functions/temperature_flux_bc_function/expression := "if(t<2e4, 1.0e7, if(t<(2e4+${peak_duration}), ${peak_value}, 1.0e6))"
Functions/temperature_inner_function/expression := ${coolant_temperature}

Functions/temperature_steady_state/expression := "-1.59786e4*x^2  -1.11629611e4*x + 4.84297313e2 + 1.9491599e6*y^2 + 1.55723201e4*y "
                 "- 7.312884e5*x*y"

Executioner/petsc_options_iname := '-pc_type'
Executioner/petsc_options_value := 'lu'
Executioner/nl_rel_tol := 1e-2
Executioner/nl_abs_tol := 1e-1
Executioner/end_time := ${units 2.0002e4 s}
Executioner/TimeStepper/dt := 100
Executioner/dtmin := 1e-6
Executioner/nl_max_its := 36
Executioner/TimeStepper/optimal_iterations := 10
Executioner/TimeStepper/growth_factor := 2.0

[Executioner]
  [TimeStepper]
    time_t = '0 2e4 2.01e4'
    time_dt = '100 0.001 100'
  []
  error_on_dtmin = False
[]

Postprocessors/F_recombination/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/F_permeation/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_mobile_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_trapped_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_total_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_mobile_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_trapped_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_total_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/temperature_top/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/temperature_tube/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/timestep_max_pp/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
# Continue using the steady-state dummy postprocessor, but add unused definitions
# from the steady-state input.
Postprocessors/unused_parameters/expression := '${num_sectors} + ${rings_H2O} + ${rings_CuCrZr} + ${rings_Cu} + ${rings_W} + ${temperature_coolant_max}
                                                + ${plasma_max_heat} + ${plasma_min_heat} + ${plasma_max_flux} + ${plasma_min_flux}'

[Postprocessors]
  [time_max_T_W]
    type = TimeExtremeValue
    postprocessor = max_temperature_W
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
  [time_max_T_Cu]
    type = TimeExtremeValue
    postprocessor = max_temperature_Cu
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
  [time_max_T_CuCrZr]
    type = TimeExtremeValue
    postprocessor = max_temperature_CuCrZr
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
[]

VectorPostprocessors/line/execute_on := 'NONE'

Variables/temperature/initial_condition := '${temperature_initial}'

Materials/thermal_conductivity_W/expression := '${W_cond_factor}*(2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4)'

[Controls]
  [stochastic]
    # Sends data to Stochastic
    type = SamplerReceiver
  []
[]

[Outputs]
  # Control outputs, include exodus and csv.
  execute_on = 'none'
[]

Controller input

### This input file defines a sobol indice analysis of the monoblock input file to simulate a
### shutdown transient, with some simplifying assumptions that allow it
### to be run in a short amount of time.
### The physics input file is elm_transient_runner.i
### This file should be called directly by TMAP8, preferably with access to multiple cpus
### Output information including summary statistics can be found in a json file

# Define parameters in one spot for easier reading
peak_flux_value = ${units 20 MW/m^2 -> W/m^2}
peak_flux_std_deviation = ${units 1 MW/m^2 -> W/m^2}
peak_flux_duration_value = ${units 1 s}
peak_flux_duration_std_deviation = ${units 0.05 s}
coolant_temperature_value = ${units 552 K}
coolant_temperature_std_deviation = ${units 27.6 K}
tungsten_kth_prefactor_lower = 0.9
tungsten_kth_prefactor_upper = 1.0
N_samples = 10

[StochasticTools<<<{"href": "../../syntax/StochasticTools/index.html"}>>>]
  # Designate as the Controller/Main Input
[]

[MultiApps<<<{"href": "../../syntax/MultiApps/index.html"}>>>]
  # Designate a subapp to control later
  [runner]
    type = SamplerFullSolveMultiApp<<<{"description": "Creates a full-solve type sub-application for each row of each Sampler matrix.", "href": "../../source/multiapps/SamplerFullSolveMultiApp.html"}>>>
    sampler<<<{"description": "The Sampler object to utilize for creating MultiApps."}>>> = sobol
    input_files<<<{"description": "The input file for each App.  If this parameter only contains one input file it will be used for all of the Apps.  When using 'positions_from_file' it is also admissable to provide one input_file per file."}>>> = 'shutdown_transient_runner.i'
    mode<<<{"description": "The operation mode, 'normal' creates one sub-application for each row in the Sampler and 'batch-reset' and 'batch-restore' creates N sub-applications, where N is the minimum of 'num_rows' in the Sampler and floor(number of processes / min_procs_per_app). To run the rows in the Sampler, 'batch-reset' will destroy and re-create sub-apps as needed, whereas the 'batch-restore' will backup and restore sub-apps to the initial state prior to execution, without destruction."}>>> = batch-reset
    keep_full_output_history<<<{"description": "Whether or not to keep the full output history when this multiapp has multiple entries"}>>> = True
    ignore_solve_not_converge<<<{"description": "True to continue main app even if a sub app's solve does not converge."}>>> = True
  []
[]

[Controls<<<{"href": "../../syntax/Controls/index.html"}>>>]
  # Control inputs from Main->Subapp
  [cmdline]
    type = MultiAppSamplerControl<<<{"description": "Control for modifying the command line arguments of MultiApps.", "href": "../../source/controls/MultiAppSamplerControl.html"}>>>
    multi_app<<<{"description": "The name of the MultiApp to control."}>>> = runner
    sampler<<<{"description": "The Sampler object to utilize for altering the command line options of the MultiApp."}>>> = sobol
    param_names<<<{"description": "The names of the command line parameters to set via the sampled data."}>>> = "peak_value
                     peak_duration
                     coolant_temperature
                     W_cond_factor"
  []
[]

[Distributions<<<{"href": "../../syntax/Distributions/index.html"}>>>]
  # Define probability distributions of parameters for sampling
  [P_val]
    type = Normal<<<{"description": "Normal distribution", "href": "../../source/distributions/Normal.html"}>>>
    mean<<<{"description": "Mean (or expectation) of the distribution."}>>> = ${peak_flux_value}
    standard_deviation<<<{"description": "Standard deviation of the distribution "}>>> = ${peak_flux_std_deviation}
  []
  [P_dur]
    type = Normal<<<{"description": "Normal distribution", "href": "../../source/distributions/Normal.html"}>>>
    mean<<<{"description": "Mean (or expectation) of the distribution."}>>> = ${peak_flux_duration_value}
    standard_deviation<<<{"description": "Standard deviation of the distribution "}>>> = ${peak_flux_duration_std_deviation}
  []
  [C_temperature]
    type = Normal<<<{"description": "Normal distribution", "href": "../../source/distributions/Normal.html"}>>>
    mean<<<{"description": "Mean (or expectation) of the distribution."}>>> = ${coolant_temperature_value}
    standard_deviation<<<{"description": "Standard deviation of the distribution "}>>> = ${coolant_temperature_std_deviation}
  []
  [W_cond]
    type = Uniform<<<{"description": "Continuous uniform distribution.", "href": "../../source/distributions/Uniform.html"}>>>
    lower_bound<<<{"description": "Distribution lower bound"}>>> = ${tungsten_kth_prefactor_lower}
    upper_bound<<<{"description": "Distribution upper bound"}>>> = ${tungsten_kth_prefactor_upper}
  []
[]

[Samplers<<<{"href": "../../syntax/Samplers/index.html"}>>>]
  # Sampling methodology using the probability distributions above
  [hypercube_1]
    type = LatinHypercube<<<{"description": "Latin Hypercube Sampler.", "href": "../../source/samplers/LatinHypercubeSampler.html"}>>>
    distributions<<<{"description": "The distribution names to be sampled, the number of distributions provided defines the number of columns per matrix."}>>> = 'P_val P_dur C_temperature W_cond'
    num_rows<<<{"description": "The size of the square matrix to generate."}>>> = ${N_samples}
    seed<<<{"description": "Random number generator initial seed"}>>> = 1001
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
  [hypercube_2]
    type = LatinHypercube<<<{"description": "Latin Hypercube Sampler.", "href": "../../source/samplers/LatinHypercubeSampler.html"}>>>
    distributions<<<{"description": "The distribution names to be sampled, the number of distributions provided defines the number of columns per matrix."}>>> = 'P_val P_dur C_temperature W_cond'
    num_rows<<<{"description": "The size of the square matrix to generate."}>>> = ${N_samples}
    seed<<<{"description": "Random number generator initial seed"}>>> = 1002
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
  [sobol]
    type = Sobol<<<{"description": "Sobol variance-based sensitivity analysis Sampler.", "href": "../../source/samplers/SobolSampler.html"}>>>
    sampler_a<<<{"description": "The 'sample' matrix."}>>> = hypercube_1
    sampler_b<<<{"description": "The 're-sample' matrix."}>>> = hypercube_2
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
[]

[Transfers<<<{"href": "../../syntax/Transfers/index.html"}>>>]
  # Define values to extract from subapp
  [results]
    type = SamplerReporterTransfer<<<{"description": "Transfers data from Reporters on the sub-application to a StochasticReporter on the main application.", "href": "../../source/transfers/SamplerReporterTransfer.html"}>>>
    from_multi_app<<<{"description": "The name of the MultiApp to receive data from"}>>> = runner
    sampler<<<{"description": "A the Sampler object that Transfer is associated.."}>>> = sobol
    stochastic_reporter<<<{"description": "The name of the StochasticReporter object to transfer values to."}>>> = results
    #execute_on = 'MULTIAPP_FIXED_POINT_END'
    from_reporter<<<{"description": "The name(s) of the Reporter(s) on the sub-app to transfer from."}>>> = "F_permeation/value
                       Scaled_Tritium_Flux/value
                       total_retention/value
                       coolant_heat_flux/value
                       max_temperature_W/value
                       max_temperature_Cu/value
                       max_temperature_CuCrZr/value
                       time_max_T_W/value
                       time_max_T_Cu/value
                       time_max_T_CuCrZr/value"
  []
  [matrix_results]
    type = SamplerReporterTransfer<<<{"description": "Transfers data from Reporters on the sub-application to a StochasticReporter on the main application.", "href": "../../source/transfers/SamplerReporterTransfer.html"}>>>
    from_multi_app<<<{"description": "The name of the MultiApp to receive data from"}>>> = runner
    sampler<<<{"description": "A the Sampler object that Transfer is associated.."}>>> = sobol
    stochastic_reporter<<<{"description": "The name of the StochasticReporter object to transfer values to."}>>> = matrix
    from_reporter<<<{"description": "The name(s) of the Reporter(s) on the sub-app to transfer from."}>>> = "F_permeation/value
                       Scaled_Tritium_Flux/value
                       total_retention/value
                       coolant_heat_flux/value
                       max_temperature_W/value
                       max_temperature_Cu/value
                       max_temperature_CuCrZr/value
                       time_max_T_W/value
                       time_max_T_Cu/value
                       time_max_T_CuCrZr/value"
  []
[]

[Reporters<<<{"href": "../../syntax/Reporters/index.html"}>>>]
  [results]
    type = StochasticReporter<<<{"description": "Storage container for stochastic simulation results coming from Reporters.", "href": "../../source/reporters/StochasticReporter.html"}>>>
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL' # INITIAL TIMESTEP_END MULTIAPP_FIXED_POINT_END
  []
  [stats]
    type = StatisticsReporter<<<{"description": "Compute statistical values of a given VectorPostprocessor objects and vectors.", "href": "../../source/reporters/StatisticsReporter.html"}>>>
    reporters<<<{"description": "List of Reporter values to utilized for statistic computations."}>>> = "results/results:F_permeation:value
                   results/results:Scaled_Tritium_Flux:value
                   results/results:total_retention:value
                   results/results:coolant_heat_flux:value
                   results/results:max_temperature_W:value
                   results/results:max_temperature_Cu:value
                   results/results:max_temperature_CuCrZr:value
                   results/results:time_max_T_W:value
                   results/results:time_max_T_Cu:value
                   results/results:time_max_T_CuCrZr:value"
    compute<<<{"description": "The statistic(s) to compute for each of the supplied vector postprocessors."}>>> = 'mean stddev'
    ci_method<<<{"description": "The method to use for computing confidence level intervals."}>>> = 'percentile'
    ci_levels<<<{"description": "A vector of confidence levels to consider, values must be in (0, 1)."}>>> = "0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
                   0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
  []
  [sobol]
    type = SobolReporter<<<{"description": "Compute SOBOL statistics values of a given VectorPostprocessor or Reporter objects and vectors.", "href": "../../source/reporters/SobolReporter.html"}>>>
    sampler<<<{"description": "SobolSampler object."}>>> = sobol
    reporters<<<{"description": "List of Reporter values to utilized for statistic computations."}>>> = "results/results:F_permeation:value
                   results/results:Scaled_Tritium_Flux:value
                   results/results:total_retention:value
                   results/results:coolant_heat_flux:value
                   results/results:max_temperature_W:value
                   results/results:max_temperature_Cu:value
                   results/results:max_temperature_CuCrZr:value
                   results/results:time_max_T_W:value
                   results/results:time_max_T_Cu:value
                   results/results:time_max_T_CuCrZr:value"
    ci_levels<<<{"description": "A vector of confidence levels to consider, values must be in (0, 1)."}>>> = "0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
                   0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
  []
  [matrix]
    type = StochasticMatrix<<<{"description": "Tool for extracting Sampler object data and storing data from stochastic simulations.", "href": "../../source/reporters/StochasticMatrix.html"}>>>
    sampler<<<{"description": "The sample from which to extract distribution data."}>>> = sobol
    sampler_column_names<<<{"description": "Prescribed names of sampler columns, used to assign names of outputted vectors."}>>> = "peak_value
                              peak_duration
                              coolant_temperature
                              W_cond_factor"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
    parallel_type<<<{"description": "This parameter will determine how the stochastic data is gathered. It is common for outputting purposes that this parameter be set to ROOT, otherwise, many files will be produced showing the values on each processor. However, if there are lot of samples, gathering on root may be memory restrictive."}>>> = ROOT
  []
[]

[Outputs<<<{"href": "../../syntax/Outputs/index.html"}>>>]
  [out]
    type = JSON<<<{"description": "Output for Reporter values using JSON format.", "href": "../../source/outputs/JSONOutput.html"}>>>
    distributed<<<{"description": "Whether or not to output distributed data (data not on rank 0)"}>>> = True
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
    execute_reporters_on<<<{"description": "Control of when Reporter values are output"}>>> = 'FINAL'
  []
[]

Edge-localized mode disruption

Subapp input

### This case extends the published divertor monoblock input file to simulate an
### Edge-Localized Mode transient, with some simplifying assumptions that allow it
### to be run in a short amount of time.
### M. Shimada, P.-C. A. Simon, C. T. Icenhour, and G. Singh, “Toward a high-fidelity
### tritium transport modeling for retention and permeation experiments,” Fusion
### Engineering and Design, Volume 203, 2024, 114438, ISSN 0920-3796,
### https://doi.org/10.1016/j.fusengdes.2024.114438.

### This input uses the `!include` feature to incorporate other input files
### Nomenclatures
### C_mobile_j      mobile H concentration in "j" material, where j = CuCrZr, Cu, W
### C_trapped_j     trapped H concentration in "j" material, where j = CuCrZr, Cu, W
### C_total_j       total H concentration in "j" material, where j = CuCrZr, Cu, W
###
### F_permeation    permeation flux
### F_recombination recombination flux
###
### Sc_             Scaled
### Int_            Integrated
### ScInt_          Scaled and integrated

!include divertor_monoblock_explicit_base.i
base_power = 10e6 # ${units 10 MW/m^2 -> W/m^2}
coolant_temperature = ${units 552 K}
elm_value = ${units 1147 MW/m^2 -> W/m^2}
elm_duration = ${units 1.32 ms -> s}
W_cond_factor = 1.0

Functions/mobile_flux_bc_function/expression := "if(t<2e2, (${base_power}*7.9e-13/1e7),
if(t<(2e2+${elm_duration}*1/3),
(${base_power}*7.9e-13/1e7)+(t-2e2)/(${elm_duration}*1/3)*((${elm_value}*7.9e-13/1e7)-(${base_power}*7.9e-13/1e7)),
if(t<(2e2+(${elm_duration}*1/3)+(${elm_duration}*2/3)),
(${elm_value}*7.9e-13/1e7)-(t-2e2-(${elm_duration}*1/3))/(${elm_duration}*2/3)*((${elm_value}*7.9e-13/1e7)-(${base_power}*7.9e-13/1e7)), (${base_power}*7.9e-13/1e7))))"
Functions/temperature_inner_function/expression := "${coolant_temperature}"
Executioner/petsc_options_iname := '-pc_type'
Executioner/petsc_options_value := 'lu'
Executioner/nl_rel_tol := 1e-5
Executioner/nl_abs_tol := 1e-6
Executioner/end_time := 2.01e2
Executioner/dtmin := 1e-6
Executioner/nl_max_its := 36
Executioner/TimeStepper/dt := 125
Executioner/TimeStepper/growth_factor := 2.0

[Executioner]
  [TimeStepper]
    time_t = '0 2e2 2.01e2'
    time_dt = '100 0.0001 100'
  []
  error_on_dtmin = False
[]

Postprocessors/F_recombination/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/F_permeation/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_mobile_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_trapped_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_total_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_mobile_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_trapped_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_total_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/temperature_top/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/temperature_tube/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/timestep_max_pp/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
# There is a lot of overlap with the divertor monoblock case, but the terms defined there for the temperature profile and tritium flux are unused here.
# Add them to the dummy postprocessor so the parser does not complain.
Postprocessors/unused_parameters/expression := '${plasma_max_flux} + ${plasma_min_flux} + ${temperature_initial} + ${temperature_coolant_max} + ${num_sectors}
 + ${rings_H2O} + ${rings_CuCrZr} + ${rings_Cu} + ${rings_W}'

[Postprocessors]
  [time_max_T_W]
    type = TimeExtremeValue
    postprocessor = max_temperature_W
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
  [time_max_T_Cu]
    type = TimeExtremeValue
    postprocessor = max_temperature_Cu
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
  [time_max_T_CuCrZr]
    type = TimeExtremeValue
    postprocessor = max_temperature_CuCrZr
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
[]

VectorPostprocessors/line/execute_on := 'NONE'
Materials/thermal_conductivity_W/expression := '${W_cond_factor}*(2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4)'

[Outputs]
  execute_on = 'none'
[]

[ICs]
  [temperature_ic]
    type = FunctionIC
    function = temperature_steady_state
    variable = temperature
  []
[]

Controller input

### This input file defines a sobol indice analysis of the monoblock input file to simulate an
### Edge-Localized Mode transient, with some simplifying assumptions that allow it
### to be run in a short amount of time.
### The physics input file is elm_transient_runner.i
### This file should be called directly by TMAP8, preferably with access to multiple cpus
### Output information including summary statistics can be found in a json file

# Define parameters in one spot for easier reading
peak_flux_value = ${units 1147 MW/m^2 -> W/m^2}
peak_flux_std_deviation = ${units 57.35 MW/m^2 -> W/m^2}
peak_flux_duration_value = ${units 1.32 ms -> s}
peak_flux_duration_std_deviation = ${units 0.066 ms -> s}
coolant_temperature_value = ${units 552 K}
coolant_temperature_std_deviation = ${units 27.6 K}
tungsten_kth_prefactor_lower = 0.9
tungsten_kth_prefactor_upper = 1.0

[StochasticTools<<<{"href": "../../syntax/StochasticTools/index.html"}>>>]
  # Designate as the Controller/Main Input
[]

[MultiApps<<<{"href": "../../syntax/MultiApps/index.html"}>>>]
  # Designate a subapp to control later
  [runner]
    type = SamplerFullSolveMultiApp<<<{"description": "Creates a full-solve type sub-application for each row of each Sampler matrix.", "href": "../../source/multiapps/SamplerFullSolveMultiApp.html"}>>>
    sampler<<<{"description": "The Sampler object to utilize for creating MultiApps."}>>> = sobol
    input_files<<<{"description": "The input file for each App.  If this parameter only contains one input file it will be used for all of the Apps.  When using 'positions_from_file' it is also admissable to provide one input_file per file."}>>> = 'elm_transient_runner.i'
    mode<<<{"description": "The operation mode, 'normal' creates one sub-application for each row in the Sampler and 'batch-reset' and 'batch-restore' creates N sub-applications, where N is the minimum of 'num_rows' in the Sampler and floor(number of processes / min_procs_per_app). To run the rows in the Sampler, 'batch-reset' will destroy and re-create sub-apps as needed, whereas the 'batch-restore' will backup and restore sub-apps to the initial state prior to execution, without destruction."}>>> = batch-reset
    keep_full_output_history<<<{"description": "Whether or not to keep the full output history when this multiapp has multiple entries"}>>> = True
  []
[]

[Controls<<<{"href": "../../syntax/Controls/index.html"}>>>]
  # Control inputs from Main->Subapp
  [cmdline]
    type = MultiAppSamplerControl<<<{"description": "Control for modifying the command line arguments of MultiApps.", "href": "../../source/controls/MultiAppSamplerControl.html"}>>>
    multi_app<<<{"description": "The name of the MultiApp to control."}>>> = runner
    sampler<<<{"description": "The Sampler object to utilize for altering the command line options of the MultiApp."}>>> = sobol
    param_names<<<{"description": "The names of the command line parameters to set via the sampled data."}>>> = "elm_value
                     elm_duration
                     coolant_temperature
                     W_cond_factor"
  []
[]

[Distributions<<<{"href": "../../syntax/Distributions/index.html"}>>>]
  # Define probability distributions of parameters for sampling
  [P_val]
    type = Normal<<<{"description": "Normal distribution", "href": "../../source/distributions/Normal.html"}>>>
    mean<<<{"description": "Mean (or expectation) of the distribution."}>>> = ${peak_flux_value}
    standard_deviation<<<{"description": "Standard deviation of the distribution "}>>> = ${peak_flux_std_deviation}
  []
  [P_dur]
    type = Normal<<<{"description": "Normal distribution", "href": "../../source/distributions/Normal.html"}>>>
    mean<<<{"description": "Mean (or expectation) of the distribution."}>>> = ${peak_flux_duration_value}
    standard_deviation<<<{"description": "Standard deviation of the distribution "}>>> = ${peak_flux_duration_std_deviation}
  []
  [C_tem]
    type = Normal<<<{"description": "Normal distribution", "href": "../../source/distributions/Normal.html"}>>>
    mean<<<{"description": "Mean (or expectation) of the distribution."}>>> = ${coolant_temperature_value}
    standard_deviation<<<{"description": "Standard deviation of the distribution "}>>> = ${coolant_temperature_std_deviation}
  []
  [W_cond]
    type = Uniform<<<{"description": "Continuous uniform distribution.", "href": "../../source/distributions/Uniform.html"}>>>
    lower_bound<<<{"description": "Distribution lower bound"}>>> = ${tungsten_kth_prefactor_lower}
    upper_bound<<<{"description": "Distribution upper bound"}>>> = ${tungsten_kth_prefactor_upper}
  []
[]

[Samplers<<<{"href": "../../syntax/Samplers/index.html"}>>>]
  # Sampling methodology using the probability distributions above
  [hypercube_1]
    type = LatinHypercube<<<{"description": "Latin Hypercube Sampler.", "href": "../../source/samplers/LatinHypercubeSampler.html"}>>>
    distributions<<<{"description": "The distribution names to be sampled, the number of distributions provided defines the number of columns per matrix."}>>> = 'P_val P_dur C_tem W_cond'
    num_rows<<<{"description": "The size of the square matrix to generate."}>>> = 10
    seed<<<{"description": "Random number generator initial seed"}>>> = 1001
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
  [hypercube_2]
    type = LatinHypercube<<<{"description": "Latin Hypercube Sampler.", "href": "../../source/samplers/LatinHypercubeSampler.html"}>>>
    distributions<<<{"description": "The distribution names to be sampled, the number of distributions provided defines the number of columns per matrix."}>>> = 'P_val P_dur C_tem W_cond'
    num_rows<<<{"description": "The size of the square matrix to generate."}>>> = 10
    seed<<<{"description": "Random number generator initial seed"}>>> = 1002
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
  [sobol]
    type = Sobol<<<{"description": "Sobol variance-based sensitivity analysis Sampler.", "href": "../../source/samplers/SobolSampler.html"}>>>
    sampler_a<<<{"description": "The 'sample' matrix."}>>> = hypercube_1
    sampler_b<<<{"description": "The 're-sample' matrix."}>>> = hypercube_2
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'PRE_MULTIAPP_SETUP'
  []
[]

[Transfers<<<{"href": "../../syntax/Transfers/index.html"}>>>]
  # Define values to extract from subapp
  [results]
    type = SamplerReporterTransfer<<<{"description": "Transfers data from Reporters on the sub-application to a StochasticReporter on the main application.", "href": "../../source/transfers/SamplerReporterTransfer.html"}>>>
    from_multi_app<<<{"description": "The name of the MultiApp to receive data from"}>>> = runner
    sampler<<<{"description": "A the Sampler object that Transfer is associated.."}>>> = sobol
    stochastic_reporter<<<{"description": "The name of the StochasticReporter object to transfer values to."}>>> = results
    #execute_on = 'MULTIAPP_FIXED_POINT_END'
    from_reporter<<<{"description": "The name(s) of the Reporter(s) on the sub-app to transfer from."}>>> = "F_permeation/value
                       Scaled_Tritium_Flux/value
                       total_retention/value
                       coolant_heat_flux/value
                       max_temperature_W/value
                       max_temperature_Cu/value
                       max_temperature_CuCrZr/value
                       time_max_T_W/value
                       time_max_T_Cu/value
                       time_max_T_CuCrZr/value"
  []
  [matrix_results]
    type = SamplerReporterTransfer<<<{"description": "Transfers data from Reporters on the sub-application to a StochasticReporter on the main application.", "href": "../../source/transfers/SamplerReporterTransfer.html"}>>>
    from_multi_app<<<{"description": "The name of the MultiApp to receive data from"}>>> = runner
    sampler<<<{"description": "A the Sampler object that Transfer is associated.."}>>> = sobol
    stochastic_reporter<<<{"description": "The name of the StochasticReporter object to transfer values to."}>>> = matrix
    from_reporter<<<{"description": "The name(s) of the Reporter(s) on the sub-app to transfer from."}>>> = "F_permeation/value
                       Scaled_Tritium_Flux/value
                       total_retention/value
                       coolant_heat_flux/value
                       max_temperature_W/value
                       max_temperature_Cu/value
                       max_temperature_CuCrZr/value
                       time_max_T_W/value
                       time_max_T_Cu/value
                       time_max_T_CuCrZr/value"
  []
[]

[Reporters<<<{"href": "../../syntax/Reporters/index.html"}>>>]
  [results]
    type = StochasticReporter<<<{"description": "Storage container for stochastic simulation results coming from Reporters.", "href": "../../source/reporters/StochasticReporter.html"}>>>
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL' # INITIAL TIMESTEP_END MULTIAPP_FIXED_POINT_END
  []
  [stats]
    type = StatisticsReporter<<<{"description": "Compute statistical values of a given VectorPostprocessor objects and vectors.", "href": "../../source/reporters/StatisticsReporter.html"}>>>
    reporters<<<{"description": "List of Reporter values to utilized for statistic computations."}>>> = "results/results:F_permeation:value
                   results/results:Scaled_Tritium_Flux:value
                   results/results:total_retention:value
                   results/results:coolant_heat_flux:value
                   results/results:max_temperature_W:value
                   results/results:max_temperature_Cu:value
                   results/results:max_temperature_CuCrZr:value
                   results/results:time_max_T_W:value
                   results/results:time_max_T_Cu:value
                   results/results:time_max_T_CuCrZr:value"
    compute<<<{"description": "The statistic(s) to compute for each of the supplied vector postprocessors."}>>> = 'mean stddev'
    ci_method<<<{"description": "The method to use for computing confidence level intervals."}>>> = 'percentile'
    ci_levels<<<{"description": "A vector of confidence levels to consider, values must be in (0, 1)."}>>> = "0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
                   0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
  []
  [sobol]
    type = SobolReporter<<<{"description": "Compute SOBOL statistics values of a given VectorPostprocessor or Reporter objects and vectors.", "href": "../../source/reporters/SobolReporter.html"}>>>
    sampler<<<{"description": "SobolSampler object."}>>> = sobol
    reporters<<<{"description": "List of Reporter values to utilized for statistic computations."}>>> = "results/results:F_permeation:value
                   results/results:Scaled_Tritium_Flux:value
                   results/results:total_retention:value
                   results/results:coolant_heat_flux:value
                   results/results:max_temperature_W:value
                   results/results:max_temperature_Cu:value
                   results/results:max_temperature_CuCrZr:value
                   results/results:time_max_T_W:value
                   results/results:time_max_T_Cu:value
                   results/results:time_max_T_CuCrZr:value"
    ci_levels<<<{"description": "A vector of confidence levels to consider, values must be in (0, 1)."}>>> = "0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
                   0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
  []
  [matrix]
    type = StochasticMatrix<<<{"description": "Tool for extracting Sampler object data and storing data from stochastic simulations.", "href": "../../source/reporters/StochasticMatrix.html"}>>>
    sampler<<<{"description": "The sample from which to extract distribution data."}>>> = sobol
    sampler_column_names<<<{"description": "Prescribed names of sampler columns, used to assign names of outputted vectors."}>>> = "peak_value
                              peak_duration
                              coolant_temperature
                              W_cond_factor"
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
    parallel_type<<<{"description": "This parameter will determine how the stochastic data is gathered. It is common for outputting purposes that this parameter be set to ROOT, otherwise, many files will be produced showing the values on each processor. However, if there are lot of samples, gathering on root may be memory restrictive."}>>> = ROOT
  []
[]

[Outputs<<<{"href": "../../syntax/Outputs/index.html"}>>>]
  [out]
    type = JSON<<<{"description": "Output for Reporter values using JSON format.", "href": "../../source/outputs/JSONOutput.html"}>>>
    distributed<<<{"description": "Whether or not to output distributed data (data not on rank 0)"}>>> = False
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'FINAL'
    execute_reporters_on<<<{"description": "Control of when Reporter values are output"}>>> = 'FINAL'
  []
[]

References

Salah El-Din El-Morshedy. Thermal-hydraulic simulation of ITER tungsten divertor monoblock for loss of flow transient. Nuclear Materials and Energy, 38:101616, 2024. URL: https://www.sciencedirect.com/science/article/pii/S2352179124000383, doi:https://doi.org/10.1016/j.nme.2024.101616.

BibTeX

@article{ELMORSHEDY2024101616,
    author = "El-Morshedy, Salah El-Din",
    title = "{Thermal-hydraulic simulation of ITER tungsten divertor monoblock for loss of flow transient}",
    journal = "Nuclear Materials and Energy",
    volume = "38",
    pages = "101616",
    year = "2024",
    issn = "2352-1791",
    doi = "https://doi.org/10.1016/j.nme.2024.101616",
    url = "https://www.sciencedirect.com/science/article/pii/S2352179124000383"
}

RIS

TY  - JOUR
AU  - El-Morshedy, Salah El-Din
TI  - Thermal-hydraulic simulation of ITER tungsten divertor monoblock for loss of flow transient
JO  - Nuclear Materials and Energy
PY  - 2024
VL  - 38
DO  - https://doi.org/10.1016/j.nme.2024.101616
UR  - https://www.sciencedirect.com/science/article/pii/S2352179124000383
SP  - 101616
ER  -

Plain Text

Salah El-Din El-Morshedy. Thermal-hydraulic simulation of ITER tungsten divertor monoblock for loss of flow transient. Nuclear Materials and Energy, 38:101616, 2024. URL: https://www.sciencedirect.com/science/article/pii/S2352179124000383, doi:https://doi.org/10.1016/j.nme.2024.101616.

Takeshi Hirai, V. Barabash, F. Escourbiac, A. Fedosov, L. Ferrand, S. Gicquel, T. Jokinen, V. Komarov, A. Martin, and M. Merola. Design and integration of ITER divertor components. Advances in Science and Technology, 73:1–10, 2010. doi:10.4028/www.scientific.net/AST.73.1.

BibTeX

@article{hirai2010iter,
    author = "Hirai, Takeshi and Barabash, V. and Escourbiac, F. and Fedosov, A. and Ferrand, L. and Gicquel, S. and Jokinen, T. and Komarov, V. and Martin, A. and Merola, M.",
    title = "Design and Integration of {ITER} Divertor Components",
    journal = "Advances in Science and Technology",
    volume = "73",
    pages = "1–10",
    DOI = "10.4028/www.scientific.net/AST.73.1",
    year = "2010",
    type = "Journal Article"
}

RIS

TY  - JOUR
AU  - Hirai, Takeshi
AU  - Barabash, V.
AU  - Escourbiac, F.
AU  - Fedosov, A.
AU  - Ferrand, L.
AU  - Gicquel, S.
AU  - Jokinen, T.
AU  - Komarov, V.
AU  - Martin, A.
AU  - Merola, M.
TI  - Design and Integration of ITER Divertor Components
JO  - Advances in Science and Technology
PY  - 2010
VL  - 73
DO  - 10.4028/www.scientific.net/AST.73.1
SP  - 1–10
ER  -

Plain Text

Takeshi Hirai, V. Barabash, F. Escourbiac, A. Fedosov, L. Ferrand, S. Gicquel, T. Jokinen, V. Komarov, A. Martin, and M. Merola. Design and integration of ITER divertor components. Advances in Science and Technology, 73:1–10, 2010. doi:10.4028/www.scientific.net/AST.73.1.

E. A. Hodille, R. Delaporte-Mathurin, J. Denis, M. Pecovnik, E. Bernard, Y. Ferro, R. Sakamoto, Y. Charles, J. Mougenot, A. De Backer, C. S. Becquart, S. Markelj, and C. Grisolia. Modelling of hydrogen isotopes trapping, diffusion and permeation in divertor monoblocks under ITER-like conditions. Nuclear Fusion, 61:126003, 10 2021. URL: https://iopscience.iop.org/article/10.1088/1741-4326/ac2abc, doi:10.1088/1741-4326/AC2ABC.

BibTeX

@article{Hodille2021126003,
    author = "Hodille, E. A. and Delaporte-Mathurin, R. and Denis, J. and Pecovnik, M. and Bernard, E. and Ferro, Y. and Sakamoto, R. and Charles, Y. and Mougenot, J. and Backer, A. De and Becquart, C. S. and Markelj, S. and Grisolia, C.",
    doi = "10.1088/1741-4326/AC2ABC",
    issn = "0029-5515",
    issue = "12",
    journal = "Nuclear Fusion",
    month = "10",
    pages = "126003",
    publisher = "IOP Publishing",
    title = "Modelling of hydrogen isotopes trapping, diffusion and permeation in divertor monoblocks under {ITER}-like conditions",
    volume = "61",
    url = "https://iopscience.iop.org/article/10.1088/1741-4326/ac2abc",
    year = "2021"
}

RIS

TY  - JOUR
AU  - Hodille, E. A.
AU  - Delaporte-Mathurin, R.
AU  - Denis, J.
AU  - Pecovnik, M.
AU  - Bernard, E.
AU  - Ferro, Y.
AU  - Sakamoto, R.
AU  - Charles, Y.
AU  - Mougenot, J.
AU  - Backer, A. De
AU  - Becquart, C. S.
AU  - Markelj, S.
AU  - Grisolia, C.
TI  - Modelling of hydrogen isotopes trapping, diffusion and permeation in divertor monoblocks under ITER-like conditions
JO  - Nuclear Fusion
PY  - 2021
VL  - 61
PB  - IOP Publishing
DO  - 10.1088/1741-4326/AC2ABC
UR  - https://iopscience.iop.org/article/10.1088/1741-4326/ac2abc
SP  - 126003
ER  -

Plain Text

E. A. Hodille, R. Delaporte-Mathurin, J. Denis, M. Pecovnik, E. Bernard, Y. Ferro, R. Sakamoto, Y. Charles, J. Mougenot, A. De Backer, C. S. Becquart, S. Markelj, and C. Grisolia. Modelling of hydrogen isotopes trapping, diffusion and permeation in divertor monoblocks under ITER-like conditions. Nuclear Fusion, 61:126003, 10 2021. URL: https://iopscience.iop.org/article/10.1088/1741-4326/ac2abc, doi:10.1088/1741-4326/AC2ABC.

C. E. Kessel, M. S. Tillack, and J. P. Blanchard. The Evaluation of the Heat Loading from Steady, Transient and Off-Normal Conditions in ARIES Power Plants. Fusion Science and Technology, 64(3):440–448, 2013. URL: https://doi.org/10.13182/FST12-538, arXiv:https://doi.org/10.13182/FST12-538, doi:10.13182/FST12-538.

BibTeX

@article{Kessel01092013,
    author = "Kessel, C. E. and Tillack, M. S. and Blanchard, J. P.",
    title = "{The Evaluation of the Heat Loading from Steady, Transient and Off-Normal Conditions in ARIES Power Plants}",
    journal = "Fusion Science and Technology",
    volume = "64",
    number = "3",
    pages = "440--448",
    year = "2013",
    publisher = "American Nuclear Society",
    doi = "10.13182/FST12-538",
    URL = "https://doi.org/10.13182/FST12-538",
    eprint = "https://doi.org/10.13182/FST12-538"
}

RIS

TY  - JOUR
AU  - Kessel, C. E.
AU  - Tillack, M. S.
AU  - Blanchard, J. P.
TI  - The Evaluation of the Heat Loading from Steady, Transient and Off-Normal Conditions in ARIES Power Plants
JO  - Fusion Science and Technology
PY  - 2013
VL  - 64
IS  - 3
PB  - American Nuclear Society
DO  - 10.13182/FST12-538
UR  - https://doi.org/10.13182/FST12-538
SP  - 440
EP  - 448
ER  -

Plain Text

C. E. Kessel, M. S. Tillack, and J. P. Blanchard. The Evaluation of the Heat Loading from Steady, Transient and Off-Normal Conditions in ARIES Power Plants. Fusion Science and Technology, 64(3):440–448, 2013. URL: https://doi.org/10.13182/FST12-538, arXiv:https://doi.org/10.13182/FST12-538, doi:10.13182/FST12-538.

A. Loarte, G. Huijsmans, S. Futatani, L. R. Baylor, T. E. Evans, D. M. Orlov, O. Schmitz, M. Becoulet, P. Cahyna, Y. Gribov, A. Kavin, A. Sashala Naik, D. J. Campbell, T. Casper, E. Daly, H. Frerichs, A. Kischner, R. Laengner, S. Lisgo, R. A. Pitts, G. Saibene, and A. Wingen. Progress on the application of ELM control schemes to ITER scenarios from the non-active phase to DT operation. Nuclear Fusion, 2014. doi:10.1088/0029-5515/54/3/033007.

BibTeX

@article{loarte2017elms,
    author = "Loarte, A. and Huijsmans, G. and Futatani, S. and Baylor, L. R. and Evans, T. E. and Orlov, D. M. and Schmitz, O. and Becoulet, M. and Cahyna, P. and Gribov, Y. and Kavin, A. and Sashala Naik, A. and Campbell, D. J. and Casper, T. and Daly, E. and Frerichs, H. and Kischner, A. and Laengner, R. and Lisgo, S. and Pitts, R. A. and Saibene, G. and Wingen, A.",
    title = "Progress on the application of {ELM} control schemes to {ITER} scenarios from the non-active phase to {DT} operation",
    journal = "Nuclear Fusion",
    volume = "54",
    number = "3",
    ISSN = "0029-5515 1741-4326",
    DOI = "10.1088/0029-5515/54/3/033007",
    year = "2014",
    type = "Journal Article"
}

RIS

TY  - JOUR
AU  - Loarte, A.
AU  - Huijsmans, G.
AU  - Futatani, S.
AU  - Baylor, L. R.
AU  - Evans, T. E.
AU  - Orlov, D. M.
AU  - Schmitz, O.
AU  - Becoulet, M.
AU  - Cahyna, P.
AU  - Gribov, Y.
AU  - Kavin, A.
AU  - Sashala Naik, A.
AU  - Campbell, D. J.
AU  - Casper, T.
AU  - Daly, E.
AU  - Frerichs, H.
AU  - Kischner, A.
AU  - Laengner, R.
AU  - Lisgo, S.
AU  - Pitts, R. A.
AU  - Saibene, G.
AU  - Wingen, A.
TI  - Progress on the application of ELM control schemes to ITER scenarios from the non-active phase to DT operation
JO  - Nuclear Fusion
PY  - 2014
VL  - 54
IS  - 3
DO  - 10.1088/0029-5515/54/3/033007
ER  -

Plain Text

A. Loarte, G. Huijsmans, S. Futatani, L. R. Baylor, T. E. Evans, D. M. Orlov, O. Schmitz, M. Becoulet, P. Cahyna, Y. Gribov, A. Kavin, A. Sashala Naik, D. J. Campbell, T. Casper, E. Daly, H. Frerichs, A. Kischner, R. Laengner, S. Lisgo, R. A. Pitts, G. Saibene, and A. Wingen. Progress on the application of ELM control schemes to ITER scenarios from the non-active phase to DT operation. Nuclear Fusion, 2014. doi:10.1088/0029-5515/54/3/033007.

R. A. Pitts, A. Kukushkin, A. Loarte, A. Martin, M. Merola, C. E. Kessel, V. Komarov, and M. Shimada. Status and physics basis of the ITER divertor. Physica Scripta, 2009. doi:10.1088/0031-8949/2009/t138/014001.

BibTeX

@article{pitts2009iter,
    author = "Pitts, R. A. and Kukushkin, A. and Loarte, A. and Martin, A. and Merola, M. and Kessel, C. E. and Komarov, V. and Shimada, M.",
    title = "Status and physics basis of the {ITER} divertor",
    journal = "Physica Scripta",
    volume = "T138",
    ISSN = "0031-8949 1402-4896",
    DOI = "10.1088/0031-8949/2009/t138/014001",
    year = "2009",
    type = "Journal Article"
}

RIS

TY  - JOUR
AU  - Pitts, R. A.
AU  - Kukushkin, A.
AU  - Loarte, A.
AU  - Martin, A.
AU  - Merola, M.
AU  - Kessel, C. E.
AU  - Komarov, V.
AU  - Shimada, M.
TI  - Status and physics basis of the ITER divertor
JO  - Physica Scripta
PY  - 2009
VL  - T138
DO  - 10.1088/0031-8949/2009/t138/014001
ER  -

Plain Text

R. A. Pitts, A. Kukushkin, A. Loarte, A. Martin, M. Merola, C. E. Kessel, V. Komarov, and M. Shimada. Status and physics basis of the ITER divertor. Physica Scripta, 2009. doi:10.1088/0031-8949/2009/t138/014001.

Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications, 145(2):280–297, 2002. URL: https://www.sciencedirect.com/science/article/pii/S0010465502002801, doi:https://doi.org/10.1016/S0010-4655(02)00280-1.

BibTeX

@article{SALTELLI2002280,
    author = "Saltelli, Andrea",
    title = "Making best use of model evaluations to compute sensitivity indices",
    journal = "Computer Physics Communications",
    volume = "145",
    number = "2",
    pages = "280-297",
    year = "2002",
    issn = "0010-4655",
    doi = "https://doi.org/10.1016/S0010-4655(02)00280-1",
    url = "https://www.sciencedirect.com/science/article/pii/S0010465502002801",
    keywords = "Sensitivity analysis, Sensitivity measures, Sensitivity indices, Importance measures"
}

RIS

TY  - JOUR
AU  - Saltelli, Andrea
TI  - Making best use of model evaluations to compute sensitivity indices
JO  - Computer Physics Communications
PY  - 2002
VL  - 145
IS  - 2
DO  - https://doi.org/10.1016/S0010-4655(02)00280-1
UR  - https://www.sciencedirect.com/science/article/pii/S0010465502002801
SP  - 280
EP  - 297
ER  -

Plain Text

Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications, 145(2):280-297, 2002. URL: https://www.sciencedirect.com/science/article/pii/S0010465502002801, doi:https://doi.org/10.1016/S0010-4655(02)00280-1.

Masashi Shimada, Pierre-Clément A. Simon, Casey T. Icenhour, and Gyanender Singh. Toward a high-fidelity tritium transport modeling for retention and permeation experiments. Fusion Engineering and Design, 203:114438, 6 2024. URL: https://linkinghub.elsevier.com/retrieve/pii/S0920379624002916, doi:10.1016/J.FUSENGDES.2024.114438.

BibTeX

@article{Shimada2024114438,
    author = "Shimada, Masashi and Simon, Pierre-Clément A. and Icenhour, Casey T. and Singh, Gyanender",
    doi = "10.1016/J.FUSENGDES.2024.114438",
    issn = "0920-3796",
    journal = "Fusion Engineering and Design",
    month = "6",
    pages = "114438",
    publisher = "North-Holland",
    title = "Toward a high-fidelity tritium transport modeling for retention and permeation experiments",
    volume = "203",
    url = "https://linkinghub.elsevier.com/retrieve/pii/S0920379624002916",
    year = "2024"
}

RIS

TY  - JOUR
AU  - Shimada, Masashi
AU  - Simon, Pierre-Clément A.
AU  - Icenhour, Casey T.
AU  - Singh, Gyanender
TI  - Toward a high-fidelity tritium transport modeling for retention and permeation experiments
JO  - Fusion Engineering and Design
PY  - 2024
VL  - 203
PB  - North-Holland
DO  - 10.1016/J.FUSENGDES.2024.114438
UR  - https://linkinghub.elsevier.com/retrieve/pii/S0920379624002916
SP  - 114438
ER  -

Plain Text

Masashi Shimada, Pierre-Clément A. Simon, Casey T. Icenhour, and Gyanender Singh. Toward a high-fidelity tritium transport modeling for retention and permeation experiments. Fusion Engineering and Design, 203:114438, 6 2024. URL: https://linkinghub.elsevier.com/retrieve/pii/S0920379624002916, doi:10.1016/J.FUSENGDES.2024.114438.

(test/tests/divertor_monoblock/shutdown_transient_runner.i)

### This case extends the published divertor monoblock input file to simulate a
### shutdown transient, with some simplifying assumptions that allow it
### to be run in a short amount of time.
### M. Shimada, P.-C. A. Simon, C. T. Icenhour, and G. Singh, “Toward a high-fidelity
### tritium transport modeling for retention and permeation experiments,” Fusion
### Engineering and Design, Volume 203, 2024, 114438, ISSN 0920-3796,
### https://doi.org/10.1016/j.fusengdes.2024.114438.

### Nomenclatures                                                                                  # Nelson S. Comments/Annotations
###
### C_mobile_j      mobile H concentration in "j" material, where j = CuCrZr, Cu, W
### C_trapped_j     trapped H concentration in "j" material, where j = CuCrZr, Cu, W
### C_total_j       total H concentration in "j" material, where j = CuCrZr, Cu, W
###
### F_permeation    permeation flux
### F_recombination recombination flux
###
### Sc_             Scaled
### Int_            Integrated
### ScInt_          Scaled and integrated                                                          # Nelson S. Comments/Annotations

### VARIABLES ###
peak_value = ${units 20 MW/m^2 -> W/m^2}
peak_duration = ${units 1.0 s}
coolant_temperature = ${units 552 K}
W_cond_factor = 1.0

!include divertor_monoblock_explicit_base.i

Functions/timestep_function/expression := "if(t<2e4, 500, if(t<(2e4+${peak_duration}+1), 0.10, 500))"
Functions/mobile_flux_bc_function/expression := "if(t<2e4, 7.90e-13, if(t<(2e4+${peak_duration}), ${peak_value}/1.0e7*7.90e-13, "
                 "7.90e-14))"
Functions/temperature_flux_bc_function/expression := "if(t<2e4, 1.0e7, if(t<(2e4+${peak_duration}), ${peak_value}, 1.0e6))"
Functions/temperature_inner_function/expression := ${coolant_temperature}

Functions/temperature_steady_state/expression := "-1.59786e4*x^2  -1.11629611e4*x + 4.84297313e2 + 1.9491599e6*y^2 + 1.55723201e4*y "
                 "- 7.312884e5*x*y"

Executioner/petsc_options_iname := '-pc_type'
Executioner/petsc_options_value := 'lu'
Executioner/nl_rel_tol := 1e-2
Executioner/nl_abs_tol := 1e-1
Executioner/end_time := ${units 2.0002e4 s}
Executioner/TimeStepper/dt := 100
Executioner/dtmin := 1e-6
Executioner/nl_max_its := 36
Executioner/TimeStepper/optimal_iterations := 10
Executioner/TimeStepper/growth_factor := 2.0

[Executioner]
  [TimeStepper]
    time_t = '0 2e4 2.01e4'
    time_dt = '100 0.001 100'
  []
  error_on_dtmin = False
[]

Postprocessors/F_recombination/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/F_permeation/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_W/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_mobile_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_trapped_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_total_Cu/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_mobile_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_mobile_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_trapped_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_trapped_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/Int_C_total_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/ScInt_C_total_CuCrZr/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/temperature_top/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/temperature_tube/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
Postprocessors/timestep_max_pp/execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
# Continue using the steady-state dummy postprocessor, but add unused definitions
# from the steady-state input.
Postprocessors/unused_parameters/expression := '${num_sectors} + ${rings_H2O} + ${rings_CuCrZr} + ${rings_Cu} + ${rings_W} + ${temperature_coolant_max}
                                                + ${plasma_max_heat} + ${plasma_min_heat} + ${plasma_max_flux} + ${plasma_min_flux}'

[Postprocessors]
  [time_max_T_W]
    type = TimeExtremeValue
    postprocessor = max_temperature_W
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
  [time_max_T_Cu]
    type = TimeExtremeValue
    postprocessor = max_temperature_Cu
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
  [time_max_T_CuCrZr]
    type = TimeExtremeValue
    postprocessor = max_temperature_CuCrZr
    output_type = extreme_value
    value_type = max
    execute_on = 'MULTIAPP_FIXED_POINT_END FINAL'
  []
[]

VectorPostprocessors/line/execute_on := 'NONE'

Variables/temperature/initial_condition := '${temperature_initial}'

Materials/thermal_conductivity_W/expression := '${W_cond_factor}*(2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4)'

[Controls]
  [stochastic]
    # Sends data to Stochastic
    type = SamplerReceiver
  []
[]

[Outputs]
  # Control outputs, include exodus and csv.
  execute_on = 'none'
[]

General description of the sensitivity studies and the modified cases
Results
Complete input files
References