Divertor Monoblock During Pulsed Operation

TMAP8 is used to model tritium transport in a divertor monoblock to elucidate the effects of pulsed operation (up to fifty 1600-second plasma discharge and cool-down cycles) on the tritium in-vessel inventory source term and ex-vessel release term (i.e., tritium retention and permeation) for safety analysis. This example reproduces the results presented in Shimada et al. (2024).

This case is also available on the Virtual Test Bed (VTB) (Charlot et al., 2025).

General description of the simulation case and corresponding input file

Introduction

In a magnetic confinement fusion system, the divertor components will be subjected to intense particle and heat fluxes from plasma, as well as 14-MeV neutrons stemming from deuterium-tritium (D-T) reactions, thus creating extremely high temperature and hydrogen concentration gradients across tens of millimeters. Furthermore, the thermomechanical properties of the divertor materials will evolve over time in response to displacement damage and gas/solid transmutations. Thus, predictive modeling of tritium transport in the divertor poses enormous challenges, as it requires simulation capabilities capable of (1) multi-material configurations, (2) high thermal cycles, (3) complex 2D and 3D geometries, (4) and microstructural evolution resulting from displacement damage and gas/solid transmutations. For example, the ITER divertor, which consists of tungsten (W) monoblocks bonded to cooling tubes made of copper-chromium-zirconium (CuCrZr) alloy, is designed to withstand high heat fluxes (∼10 MW $var element = document.getElementById("moose-equation-121dba54-ee82-4a18-bc95-e12492bf2c12");katex.render("m^{\u22122}", element, {displayMode:false,throwOnError:false});$ ) during steady-state plasma operation from intense D-T plasma flux (∼1024 $var element = document.getElementById("moose-equation-32e5d687-4d0b-4514-847a-f95ec9a1a2ec");katex.render("m^{\u22122}s^{\u22121}", element, {displayMode:false,throwOnError:false});$ ). In this example, we simulate tritium retention and permeation as well as thermal transport in an ITER-like 2D W-Cu-CuCrZr monoblock and demonstrate the first three (out of four) aforementioned required simulation capabilities for tritium transport in the divertor using TMAP8, as presented in Shimada et al. (2024). This simulation was also based on the cases published in Hodille et al. (2021).

The sections below describe the simulation details and explain how they translate into the example TMAP8 input file. Not every part of the input file is explained here. If the interested reader has more questions about this example case and how to modify this input file to adapt it to a different case, feel free to reach out to the TMAP8 development team on the TMAP8 GitHub discussion page.

Divertor monoblock geometry and mesh generation

Figure 1 shows the geometry, mesh, and temperature distribution of the 2D monoblock employed in Shimada et al. (2024) and in the present example. The 2D monoblock consisted of three materials:

W monoblock ( $var element = document.getElementById("moose-equation-8bf905b4-eb35-439d-9a17-2171454f0278");katex.render("r", element, {displayMode:false,throwOnError:false});$ (mm) > 8.5),
Cu interlayer (7.5 < $var element = document.getElementById("moose-equation-ce8d42f1-0d25-4f82-a6a4-b46d33b60c02");katex.render("r", element, {displayMode:false,throwOnError:false});$ (mm) < 8.5),
CuCrZr tube (6.0 < $var element = document.getElementById("moose-equation-62cf01b0-45b8-4f3a-b822-cd620c373458");katex.render("r", element, {displayMode:false,throwOnError:false});$ (mm) < 7.5), and
H $var element = document.getElementById("moose-equation-fd2cac27-3955-43d2-b1ff-a13097d2d3f9");katex.render("_2", element, {displayMode:false,throwOnError:false});$ O cooling ( $var element = document.getElementById("moose-equation-e559f095-dfe1-4c1a-aee4-8ef21f37ebf0");katex.render("r", element, {displayMode:false,throwOnError:false});$ (mm) < 6.0),

where $var element = document.getElementById("moose-equation-e4198d72-21ac-41b6-9705-5c355992c84f");katex.render("r=\\sqrt{x^2+y^2}", element, {displayMode:false,throwOnError:false});$ in a Cartesian coordinate system, and the center point ( $var element = document.getElementById("moose-equation-67363161-a6c5-423b-8240-770236d4f2ed");katex.render("x", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-6bad301f-289e-4d59-a42e-9832b211f2cb");katex.render("y", element, {displayMode:false,throwOnError:false});$ ) = (0, 0) is defined at the center of the CuCrZr tube.

2D monoblock: (left) geometry and mesh; (right) temperature distribution. This corresponds to Fig. 1 in Ref. [!cite](Shimada2024114438).

Figure 1: 2D monoblock: (left) geometry and mesh; (right) temperature distribution. This corresponds to Fig. 1 in Ref. Shimada et al. (2024).

MOOSE is equipped with a set of user-friendly, built-in mesh generators for creating meshes based on simple geometries (e.g., a monoblock). The built-in mesh generator ConcentricCircleMeshGenerator is used to create meshes for the 2D monoblock geometry. Only the left-hand (−14.0 < $var element = document.getElementById("moose-equation-dfae429f-b007-476b-a645-46c13896bbbe");katex.render("x", element, {displayMode:false,throwOnError:false});$ (mm) < 0 & −14.0 < $var element = document.getElementById("moose-equation-73d39db3-2947-4f5c-a1fa-5907e583afb0");katex.render("y", element, {displayMode:false,throwOnError:false});$ (mm) < 14.0) portion of the monoblock is used, such that the mesh size is halved by assuming symmetry at the $var element = document.getElementById("moose-equation-6ea2f7f6-dcb3-4e42-8326-5826a3fa6f6d");katex.render("x = 0", element, {displayMode:false,throwOnError:false});$ plane. This example uses a total of 9,418 nodes, creating 28,402 degrees of freedom in the non-linear system. The finer mesh size and time step are required to compute the trapping-limited diffusion regime, as described in TMAP8 verification case ver-1d. However, a set of relatively low detrapping energies (< 0.85 eV) is used here to reduce the computing time via the relatively coarse mesh size.

In the input file, the mesh is defined as:

[Mesh<<<{"href": "../../syntax/Mesh/index.html"}>>>]
  [ccmg]
    type = ConcentricCircleMeshGenerator<<<{"description": "This ConcentricCircleMeshGenerator source code is to generate concentric circle meshes.", "href": "../../source/meshgenerators/ConcentricCircleMeshGenerator.html"}>>>
    num_sectors<<<{"description": "num_sectors % 2 = 0, num_sectors > 0Number of azimuthal sectors in each quadrant'num_sectors' must be an even number."}>>> = ${num_sectors}
    rings<<<{"description": "Number of rings in each circle or in the enclosing square"}>>> = '${rings_H2O} ${rings_CuCrZr} ${rings_Cu} ${rings_W}'
    radii<<<{"description": "Radii of major concentric circles"}>>> = '${radius_coolant} ${radius_CuCrZr} ${radius_Cu}'
    has_outer_square<<<{"description": "It determines if meshes for a outer square are added to concentric circle meshes."}>>> = on
    pitch<<<{"description": "The enclosing square can be added to the completed concentric circle mesh.Elements are quad meshes."}>>> = '${fparse block_size}'
    portion<<<{"description": "Control of which part of mesh is created"}>>> = left_half
    preserve_volumes<<<{"description": "Volume of concentric circles can be preserved using this function."}>>> = false
    smoothing_max_it<<<{"description": "Number of Laplacian smoothing iterations"}>>> = 3
  []
  [ssbsg1]
    type = SideSetsBetweenSubdomainsGenerator<<<{"description": "MeshGenerator that creates a sideset composed of the nodes located between two or more subdomains.", "href": "../../source/meshgenerators/SideSetsBetweenSubdomainsGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = ccmg
    primary_block<<<{"description": "The primary set of blocks for which to draw a sideset between"}>>> = '4' # W
    paired_block<<<{"description": "The paired set of blocks for which to draw a sideset between"}>>> = '3' # Cu
    new_boundary<<<{"description": "The list of boundary names to create on the supplied subdomain"}>>> = '4to3'
  []
  [ssbsg2]
    type = SideSetsBetweenSubdomainsGenerator<<<{"description": "MeshGenerator that creates a sideset composed of the nodes located between two or more subdomains.", "href": "../../source/meshgenerators/SideSetsBetweenSubdomainsGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = ssbsg1
    primary_block<<<{"description": "The primary set of blocks for which to draw a sideset between"}>>> = '3' # Cu
    paired_block<<<{"description": "The paired set of blocks for which to draw a sideset between"}>>> = '4' # W
    new_boundary<<<{"description": "The list of boundary names to create on the supplied subdomain"}>>> = '3to4'
  []
  [ssbsg3]
    type = SideSetsBetweenSubdomainsGenerator<<<{"description": "MeshGenerator that creates a sideset composed of the nodes located between two or more subdomains.", "href": "../../source/meshgenerators/SideSetsBetweenSubdomainsGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = ssbsg2
    primary_block<<<{"description": "The primary set of blocks for which to draw a sideset between"}>>> = '3' # Cu
    paired_block<<<{"description": "The paired set of blocks for which to draw a sideset between"}>>> = '2' # CuCrZr
    new_boundary<<<{"description": "The list of boundary names to create on the supplied subdomain"}>>> = '3to2'
  []
  [ssbsg4]
    type = SideSetsBetweenSubdomainsGenerator<<<{"description": "MeshGenerator that creates a sideset composed of the nodes located between two or more subdomains.", "href": "../../source/meshgenerators/SideSetsBetweenSubdomainsGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = ssbsg3
    primary_block<<<{"description": "The primary set of blocks for which to draw a sideset between"}>>> = '2' # CuCrZr
    paired_block<<<{"description": "The paired set of blocks for which to draw a sideset between"}>>> = '3' # Cu
    new_boundary<<<{"description": "The list of boundary names to create on the supplied subdomain"}>>> = '2to3'
  []
  [ssbsg5]
    type = SideSetsBetweenSubdomainsGenerator<<<{"description": "MeshGenerator that creates a sideset composed of the nodes located between two or more subdomains.", "href": "../../source/meshgenerators/SideSetsBetweenSubdomainsGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = ssbsg4
    primary_block<<<{"description": "The primary set of blocks for which to draw a sideset between"}>>> = '2' # CuCrZr
    paired_block<<<{"description": "The paired set of blocks for which to draw a sideset between"}>>> = '1' # H2O
    new_boundary<<<{"description": "The list of boundary names to create on the supplied subdomain"}>>> = '2to1'
  []
  [bdg]
    type = BlockDeletionGenerator<<<{"description": "Mesh generator which removes elements from the specified subdomains", "href": "../../source/meshgenerators/BlockDeletionGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = ssbsg5
    block<<<{"description": "The list of blocks to be processed (deleted or kept)"}>>> = '1' # H2O
  []
[]

Nomenclature of variables and physical parameters

Table 1 lists the variables and physical parameters used in this example with their units.

Table 1: Nomenclature of variables and physical parameters used in this example.

warning:Typo in Shimada et al. (2024)

The equivalent table in Ambrosek and Longhurst (2008) has a typographical error: It lists the density in g m $var element = document.getElementById("moose-equation-6cc017b7-4956-4ea5-abb2-9b9ea5434d89");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$ when the correct unit, used in Table 1, is kg m $var element = document.getElementById("moose-equation-af6bc155-e2f3-4fe2-948e-09fd2c72df42");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$ . This does not, however, change the results since the value used for the density was correct and used the proper unit.

Variables and governing equations

To simulate tritium and thermal transport, we define two sets of partial differential equations (PDEs). First, the strong form of the mass conservation equation for solute (mobile) T atoms, $var element = document.getElementById("moose-equation-cf8b3edb-75e8-41a6-9635-18d42aa6d24c");katex.render("C_s", element, {displayMode:false,throwOnError:false});$ , is written as:

$(1)var element = document.getElementById("moose-equation-8cdcc6f2-84a4-42a6-afd0-36afe80af04b");katex.render(" \\frac{\\partial C_s}{\\partial t} + \\nabla \\cdot (-D \\nabla C_s) + \\alpha_t \\frac{C_t^e}{N} C_s - \\alpha_r C_t = 0.", element, {displayMode:true,throwOnError:false});$

We use three sets of mass conservation equations to calculate the behaviors of solute T atoms in three different materials (i.e., W, Cu, CuCrZr). Second, the strong form of the conservation of energy equation is written as:

$(2)var element = document.getElementById("moose-equation-0b17590d-ccaa-4dea-94f6-1b033cf822fb");katex.render(" \\rho c_p \\frac{\\partial T}{\\partial t} - \\nabla \\cdot (k_T \\nabla T) = 0.", element, {displayMode:true,throwOnError:false});$

Table 1 lists all the symbols and definitions used in Eq. (1) and Eq. (2).

The next step is to convert these two strong-form PDEs into their weak forms by multiplying with a test function, $var element = document.getElementById("moose-equation-acab70e7-33d4-42e1-9913-1f46933168c0");katex.render("\\psi", element, {displayMode:false,throwOnError:false});$ , and integrating over a domain, $var element = document.getElementById("moose-equation-14fb8b98-b8f0-47dc-b9a6-b40472dbc84f");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ with surface $var element = document.getElementById("moose-equation-9a9dca95-5e4a-473c-840f-2ff26cbdab55");katex.render("\\delta \\Omega", element, {displayMode:false,throwOnError:false});$ and outward-facing normal vector $var element = document.getElementById("moose-equation-285b032e-dc45-4b67-97cb-e0864e83b3b1");katex.render("\\hat{n}", element, {displayMode:false,throwOnError:false});$ . Using the divergence theorem, one can obtain the weak form of the mass conservation equation (Eq. (1)) as follows:

$(3)var element = document.getElementById("moose-equation-b947a845-c3d8-458b-b2df-59fed40600b9");katex.render(" \\int_{\\Omega}\\psi \\frac{\\partial C_s}{\\partial t} -\\int_{\\delta \\Omega} \\psi D \\nabla C_s \\cdot \\hat{n} +\\int_{\\Omega} \\nabla \\psi \\cdot D \\nabla C_s +\\int_{\\Omega} \\psi \\alpha_t \\frac{C_t^e}{N} C_s -\\int_{\\Omega} \\psi \\alpha_r C_t = 0.", element, {displayMode:true,throwOnError:false});$

Similarly, the weak form of the conservation of energy equation can be written by multiplying Eq. (2) with a test function, $var element = document.getElementById("moose-equation-8aa8e309-e515-4461-8b26-b3738d851a97");katex.render("\\psi", element, {displayMode:false,throwOnError:false});$ , and integrating over a domain, $var element = document.getElementById("moose-equation-d8bddfa0-a639-4132-8c48-9976cbaf4200");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ with surface $var element = document.getElementById("moose-equation-cf525fbe-c976-4b86-942a-b1e7a7e36263");katex.render("\\delta \\Omega", element, {displayMode:false,throwOnError:false});$ and outward-facing normal vector $var element = document.getElementById("moose-equation-d69ef06c-127a-43e9-9ddf-28e7806a2d1a");katex.render("\\hat{n}", element, {displayMode:false,throwOnError:false});$ . Using the divergence theorem, one can obtain the weak form of the conservation of energy equation as follows:

$(4)var element = document.getElementById("moose-equation-dc62f3c9-73ba-4265-bc55-28977fa92d99");katex.render(" \\int_{\\Omega}\\psi c_p \\frac{\\partial T}{\\partial t} -\\int_{\\delta \\Omega} \\psi k_T \\nabla T \\cdot \\hat{n} +\\int_{\\Omega} \\nabla \\psi \\cdot k_T \\nabla T = 0.", element, {displayMode:true,throwOnError:false});$

Then, to solve for the PDEs and physical phenomena, we can select appropriate kernels and boundary conditions (BCs) from MOOSE’s extensive library. The following three subsections describe each kernel, BC, and numerical method utilized in the present work.

In the input file, the variables are defined as:

[AuxVariables<<<{"href": "../../syntax/AuxVariables/index.html"}>>>]
  [flux_y]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
  []
  ############################## AuxVariables for W (block = 4)
  [Sc_C_mobile_W]
    block = 4
  []
  [Sc_C_trapped_W]
    block = 4
  []
  [C_total_W]
    block = 4
  []
  [Sc_C_total_W]
    block = 4
  []
  [S_empty_W]
    block = 4
  []
  [Sc_S_empty_W]
    block = 4
  []
  [S_trapped_W]
    block = 4
  []
  [Sc_S_trapped_W]
    block = 4
  []
  [S_total_W]
    block = 4
  []
  [Sc_S_total_W]
    block = 4
  []
  ############################## AuxVariables for Cu (block = 3)
  [Sc_C_mobile_Cu]
    block = 3
  []
  [Sc_C_trapped_Cu]
    block = 3
  []
  [C_total_Cu]
    block = 3
  []
  [Sc_C_total_Cu]
    block = 3
  []
  [S_empty_Cu]
    block = 3
  []
  [Sc_S_empty_Cu]
    block = 3
  []
  [S_trapped_Cu]
    block = 3
  []
  [Sc_S_trapped_Cu]
    block = 3
  []
  [S_total_Cu]
    block = 3
  []
  [Sc_S_total_Cu]
    block = 3
  []
  ############################## AuxVariables for CuCrZr (block = 2)
  [Sc_C_mobile_CuCrZr]
    block = 2
  []
  [Sc_C_trapped_CuCrZr]
    block = 2
  []
  [C_total_CuCrZr]
    block = 2
  []
  [Sc_C_total_CuCrZr]
    block = 2
  []
  [S_empty_CuCrZr]
    block = 2
  []
  [Sc_S_empty_CuCrZr]
    block = 2
  []
  [S_trapped_CuCrZr]
    block = 2
  []
  [Sc_S_trapped_CuCrZr]
    block = 2
  []
  [S_total_CuCrZr]
    block = 2
  []
  [Sc_S_total_CuCrZr]
    block = 2
  []
[]

Note the usage of initial_condition and block parameters in order to set initial conditions and material block applicability for each variable.

Kernels

For the mass conservation equation, we use ADTimeDerivative and MatDiffusion to represent the 1 $var element = document.getElementById("moose-equation-d07017e0-ba85-45f7-b801-88ecf264ab14");katex.render("^{\\text{st}}", element, {displayMode:false,throwOnError:false});$ and 3 $var element = document.getElementById("moose-equation-c74aae23-1f81-4628-a10d-4345f0f8a2c8");katex.render("^{\\text{rd}}", element, {displayMode:false,throwOnError:false});$ terms of Eq. (3). The TMAP8 kernels TrappingNodalKernel and ReleasingNodalKernel represent the 4 $var element = document.getElementById("moose-equation-28dafe2f-6ad3-4954-b9d3-bf8f7cb53fd5");katex.render("^{\\text{th}}", element, {displayMode:false,throwOnError:false});$ and 5 $var element = document.getElementById("moose-equation-abeadb65-ea10-489e-adc7-f8882f8edd53");katex.render("^{\\text{th}}", element, {displayMode:false,throwOnError:false});$ terms of Eq. (3), respectively, and simulate the trapping/release behavior of hydrogen isotopes in/from trap sites. For the conservation of energy equation, SpecificHeatConductionTimeDerivative and HeatConduction solve the 1 $var element = document.getElementById("moose-equation-ef1299f2-c2df-4d84-9409-560ce547ba40");katex.render("^{\\text{st}}", element, {displayMode:false,throwOnError:false});$ and 3 $var element = document.getElementById("moose-equation-1c4c7712-bc67-4599-bf8d-8aa288c05180");katex.render("^{\\text{rd}}", element, {displayMode:false,throwOnError:false});$ terms of Eq. (4). ADTimeDerivative and ADMatDiffusion are kernels from the core MOOSE framework, and SpecificHeatConductionTimeDerivative and HeatConduction are kernels from the MOOSE Heat Transfer Module. These pre-made kernels are commonly re-used to represent time derivative, diffusion, and heat conduction terms in a given material within a variety of MOOSE-based simulations.

[AuxKernels<<<{"href": "../../syntax/AuxKernels/index.html"}>>>]
  ############################## AuxKernels for W (block = 4)
  [Scaled_mobile_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_mobile_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_mobile_W
  []
  [Scaled_trapped_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_trapped_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_W
  []
  [total_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = C_total_W
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'C_mobile_W + C_trapped_W'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'C_mobile_W C_trapped_W'
  []
  [Scaled_total_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_total_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_total_W
  []
  [empty_sites_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_empty_W
    type = EmptySitesAux<<<{"description": "Calculates the concentration of empty trapping sites.", "href": "../../source/auxkernels/EmptySitesAux.html"}>>>
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_W}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_W}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix"}>>> = ${trap_per_free_W}
    trapped_concentration_variables<<<{"description": "Variables representing trapped particle concentrations."}>>> = C_trapped_W
  []
  [scaled_empty_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_empty_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_empty_W
  []
  [trapped_sites_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_trapped_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = 1e0
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_W
  []
  [scaled_trapped_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_trapped_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_trapped_W
  []
  [total_sites_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_total_W
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'S_trapped_W + S_empty_W'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'S_trapped_W S_empty_W'
  []
  [scaled_total_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_total_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_total_W
  []
  ############################## AuxKernels for Cu (block = 3)
  [Scaled_mobile_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_mobile_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_mobile_Cu
  []
  [Scaled_trapped_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_trapped_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_Cu
  []
  [total_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = C_total_Cu
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'C_mobile_Cu + C_trapped_Cu'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'C_mobile_Cu C_trapped_Cu'
  []
  [Scaled_total_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_total_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_total_Cu
  []
  [empty_sites_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_empty_Cu
    type = EmptySitesAux<<<{"description": "Calculates the concentration of empty trapping sites.", "href": "../../source/auxkernels/EmptySitesAux.html"}>>>
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_Cu}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_Cu}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix"}>>> = ${trap_per_free_Cu}
    trapped_concentration_variables<<<{"description": "Variables representing trapped particle concentrations."}>>> = C_trapped_Cu
  []
  [scaled_empty_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_empty_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_empty_Cu
  []
  [trapped_sites_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_trapped_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = 1e0
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_Cu
  []
  [scaled_trapped_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_trapped_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_trapped_Cu
  []
  [total_sites_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_total_Cu
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'S_trapped_Cu + S_empty_Cu'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'S_trapped_Cu S_empty_Cu'
  []
  [scaled_total_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_total_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_total_Cu
  []
  ############################## AuxKernels for CuCrZr (block = 2)
  [Scaled_mobile_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_mobile_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_mobile_CuCrZr
  []
  [Scaled_trapped_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_trapped_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_CuCrZr
  []
  [total_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = C_total_CuCrZr
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'C_mobile_CuCrZr + C_trapped_CuCrZr'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'C_mobile_CuCrZr C_trapped_CuCrZr'
  []
  [Scaled_total_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_total_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_total_CuCrZr
  []
  [empty_sites_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_empty_CuCrZr
    type = EmptySitesAux<<<{"description": "Calculates the concentration of empty trapping sites.", "href": "../../source/auxkernels/EmptySitesAux.html"}>>>
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_CuCrZr}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_CuCrZr}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix"}>>> = ${trap_per_free_CuCrZr}
    trapped_concentration_variables<<<{"description": "Variables representing trapped particle concentrations."}>>> = C_trapped_CuCrZr
  []
  [scaled_empty_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_empty_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_empty_CuCrZr
  []
  [trapped_sites_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_trapped_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = 1e0
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_CuCrZr
  []
  [scaled_trapped_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_trapped_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_trapped_CuCrZr
  []
  [total_sites_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_total_CuCrZr
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'S_trapped_CuCrZr + S_empty_CuCrZr'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'S_trapped_CuCrZr S_empty_CuCrZr'
  []
  [scaled_total_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_total_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_total_CuCrZr
  []
  [flux_y_W]
    type = DiffusionFluxAux<<<{"description": "Compute components of flux vector for diffusion problems $(\\vec{J} = -D \\nabla C)$.", "href": "../../source/auxkernels/DiffusionFluxAux.html"}>>>
    diffusivity<<<{"description": "The name of the diffusivity material property that will be used in the flux computation."}>>> = diffusivity_W
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = flux_y
    diffusion_variable<<<{"description": "The name of the variable"}>>> = C_mobile_W
    component<<<{"description": "The desired component of flux."}>>> = y
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 4
  []
  [flux_y_Cu]
    type = DiffusionFluxAux<<<{"description": "Compute components of flux vector for diffusion problems $(\\vec{J} = -D \\nabla C)$.", "href": "../../source/auxkernels/DiffusionFluxAux.html"}>>>
    diffusivity<<<{"description": "The name of the diffusivity material property that will be used in the flux computation."}>>> = diffusivity_Cu
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = flux_y
    diffusion_variable<<<{"description": "The name of the variable"}>>> = C_mobile_Cu
    component<<<{"description": "The desired component of flux."}>>> = y
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 3
  []
  [flux_y_CuCrZr]
    type = DiffusionFluxAux<<<{"description": "Compute components of flux vector for diffusion problems $(\\vec{J} = -D \\nabla C)$.", "href": "../../source/auxkernels/DiffusionFluxAux.html"}>>>
    diffusivity<<<{"description": "The name of the diffusivity material property that will be used in the flux computation."}>>> = diffusivity_CuCrZr
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = flux_y
    diffusion_variable<<<{"description": "The name of the variable"}>>> = C_mobile_CuCrZr
    component<<<{"description": "The desired component of flux."}>>> = y
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
[]

NodalKernels are used for non-diffusive variables (trapped species).

[NodalKernels<<<{"href": "../../syntax/NodalKernels/index.html"}>>>]
  ############################## NodalKernels for W (block = 4)
  [time_W]
    type = TimeDerivativeNodalKernel<<<{"description": "Forms the contribution to the residual and jacobian of the time derivative term from an ODE being solved at all nodes.", "href": "../../source/nodalkernels/TimeDerivativeNodalKernel.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_W
  []
  [trapping_W]
    type = TrappingNodalKernel<<<{"description": "Implements a residual describing the trapping of a species in a material.", "href": "../../source/nodal_kernels/TrappingNodalKernel.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_W
    temperature<<<{"description": "The temperature (K)"}>>> = temperature
    alpha_t<<<{"description": "The trapping rate coefficient. This has units of 1/s (e.g. no number densities are involved)"}>>> = ${alpha_t}
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_W}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_W}
    trapping_energy<<<{"description": "The trapping energy (K)"}>>> = ${trapping_energy}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix (-)"}>>> = ${trap_per_free_W}
    mobile_concentration<<<{"description": "The variable representing the mobile concentration of solute particles (1/m^3)"}>>> = 'C_mobile_W'
    extra_vector_tags<<<{"description": "The extra tags for the vectors this Kernel should fill"}>>> = ref
  []
  [release_W]
    type = ReleasingNodalKernel<<<{"description": "Implements a residual describing the release of trapped species in a material.", "href": "../../source/nodal_kernels/ReleasingNodalKernel.html"}>>>
    alpha_r<<<{"description": "The release rate coefficient (1/s)"}>>> = ${alpha_r}
    temperature<<<{"description": "The temperature (K)"}>>> = temperature
    detrapping_energy<<<{"description": "The detrapping energy (K)"}>>> = ${detrapping_energy_W}
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_W
  []
  ############################## NodalKernels for Cu (block = 3)
  [time_Cu]
    type = TimeDerivativeNodalKernel<<<{"description": "Forms the contribution to the residual and jacobian of the time derivative term from an ODE being solved at all nodes.", "href": "../../source/nodalkernels/TimeDerivativeNodalKernel.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_Cu
  []
  [trapping_Cu]
    type = TrappingNodalKernel<<<{"description": "Implements a residual describing the trapping of a species in a material.", "href": "../../source/nodal_kernels/TrappingNodalKernel.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_Cu
    temperature<<<{"description": "The temperature (K)"}>>> = temperature
    alpha_t<<<{"description": "The trapping rate coefficient. This has units of 1/s (e.g. no number densities are involved)"}>>> = ${alpha_t}
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_Cu}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_Cu}
    trapping_energy<<<{"description": "The trapping energy (K)"}>>> = ${trapping_energy}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix (-)"}>>> = ${trap_per_free_Cu}
    mobile_concentration<<<{"description": "The variable representing the mobile concentration of solute particles (1/m^3)"}>>> = 'C_mobile_Cu'
    extra_vector_tags<<<{"description": "The extra tags for the vectors this Kernel should fill"}>>> = ref
  []
  [release_Cu]
    type = ReleasingNodalKernel<<<{"description": "Implements a residual describing the release of trapped species in a material.", "href": "../../source/nodal_kernels/ReleasingNodalKernel.html"}>>>
    alpha_r<<<{"description": "The release rate coefficient (1/s)"}>>> = ${alpha_r}
    temperature<<<{"description": "The temperature (K)"}>>> = temperature
    detrapping_energy<<<{"description": "The detrapping energy (K)"}>>> = ${detrapping_energy_Cu}
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_Cu
  []
  ############################## NodalKernels for CuCrZr (block = 2)
  [time_CuCrZr]
    type = TimeDerivativeNodalKernel<<<{"description": "Forms the contribution to the residual and jacobian of the time derivative term from an ODE being solved at all nodes.", "href": "../../source/nodalkernels/TimeDerivativeNodalKernel.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_CuCrZr
  []
  [trapping_CuCrZr]
    type = TrappingNodalKernel<<<{"description": "Implements a residual describing the trapping of a species in a material.", "href": "../../source/nodal_kernels/TrappingNodalKernel.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_CuCrZr
    temperature<<<{"description": "The temperature (K)"}>>> = temperature
    alpha_t<<<{"description": "The trapping rate coefficient. This has units of 1/s (e.g. no number densities are involved)"}>>> = ${alpha_t}
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_CuCrZr}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_CuCrZr}
    trapping_energy<<<{"description": "The trapping energy (K)"}>>> = ${trapping_energy}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix (-)"}>>> = ${trap_per_free_CuCrZr}
    mobile_concentration<<<{"description": "The variable representing the mobile concentration of solute particles (1/m^3)"}>>> = 'C_mobile_CuCrZr'
    extra_vector_tags<<<{"description": "The extra tags for the vectors this Kernel should fill"}>>> = ref
  []
  [release_CuCrZr]
    type = ReleasingNodalKernel<<<{"description": "Implements a residual describing the release of trapped species in a material.", "href": "../../source/nodal_kernels/ReleasingNodalKernel.html"}>>>
    alpha_r<<<{"description": "The release rate coefficient (1/s)"}>>> = ${alpha_r}
    temperature<<<{"description": "The temperature (K)"}>>> = temperature
    detrapping_energy<<<{"description": "The detrapping energy (K)"}>>> = ${detrapping_energy_CuCrZr}
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_trapped_CuCrZr
  []
[]

AuxVariables and AuxKernels

AuxVariables are used to track quantities that are not solved for by the PDEs, but are needed to compute materials properties, or are desirable to obtain as outputs, such as the total concentration of tritium or flux values. AuxKernels provide the expressions that define the AuxVariables.

[AuxVariables<<<{"href": "../../syntax/AuxVariables/index.html"}>>>]
  [flux_y]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
  []
  ############################## AuxVariables for W (block = 4)
  [Sc_C_mobile_W]
    block = 4
  []
  [Sc_C_trapped_W]
    block = 4
  []
  [C_total_W]
    block = 4
  []
  [Sc_C_total_W]
    block = 4
  []
  [S_empty_W]
    block = 4
  []
  [Sc_S_empty_W]
    block = 4
  []
  [S_trapped_W]
    block = 4
  []
  [Sc_S_trapped_W]
    block = 4
  []
  [S_total_W]
    block = 4
  []
  [Sc_S_total_W]
    block = 4
  []
  ############################## AuxVariables for Cu (block = 3)
  [Sc_C_mobile_Cu]
    block = 3
  []
  [Sc_C_trapped_Cu]
    block = 3
  []
  [C_total_Cu]
    block = 3
  []
  [Sc_C_total_Cu]
    block = 3
  []
  [S_empty_Cu]
    block = 3
  []
  [Sc_S_empty_Cu]
    block = 3
  []
  [S_trapped_Cu]
    block = 3
  []
  [Sc_S_trapped_Cu]
    block = 3
  []
  [S_total_Cu]
    block = 3
  []
  [Sc_S_total_Cu]
    block = 3
  []
  ############################## AuxVariables for CuCrZr (block = 2)
  [Sc_C_mobile_CuCrZr]
    block = 2
  []
  [Sc_C_trapped_CuCrZr]
    block = 2
  []
  [C_total_CuCrZr]
    block = 2
  []
  [Sc_C_total_CuCrZr]
    block = 2
  []
  [S_empty_CuCrZr]
    block = 2
  []
  [Sc_S_empty_CuCrZr]
    block = 2
  []
  [S_trapped_CuCrZr]
    block = 2
  []
  [Sc_S_trapped_CuCrZr]
    block = 2
  []
  [S_total_CuCrZr]
    block = 2
  []
  [Sc_S_total_CuCrZr]
    block = 2
  []
[]

[AuxKernels<<<{"href": "../../syntax/AuxKernels/index.html"}>>>]
  ############################## AuxKernels for W (block = 4)
  [Scaled_mobile_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_mobile_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_mobile_W
  []
  [Scaled_trapped_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_trapped_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_W
  []
  [total_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = C_total_W
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'C_mobile_W + C_trapped_W'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'C_mobile_W C_trapped_W'
  []
  [Scaled_total_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_total_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_total_W
  []
  [empty_sites_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_empty_W
    type = EmptySitesAux<<<{"description": "Calculates the concentration of empty trapping sites.", "href": "../../source/auxkernels/EmptySitesAux.html"}>>>
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_W}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_W}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix"}>>> = ${trap_per_free_W}
    trapped_concentration_variables<<<{"description": "Variables representing trapped particle concentrations."}>>> = C_trapped_W
  []
  [scaled_empty_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_empty_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_empty_W
  []
  [trapped_sites_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_trapped_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = 1e0
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_W
  []
  [scaled_trapped_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_trapped_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_trapped_W
  []
  [total_sites_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_total_W
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'S_trapped_W + S_empty_W'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'S_trapped_W S_empty_W'
  []
  [scaled_total_W]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_total_W
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_total_W
  []
  ############################## AuxKernels for Cu (block = 3)
  [Scaled_mobile_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_mobile_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_mobile_Cu
  []
  [Scaled_trapped_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_trapped_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_Cu
  []
  [total_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = C_total_Cu
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'C_mobile_Cu + C_trapped_Cu'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'C_mobile_Cu C_trapped_Cu'
  []
  [Scaled_total_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_total_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_total_Cu
  []
  [empty_sites_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_empty_Cu
    type = EmptySitesAux<<<{"description": "Calculates the concentration of empty trapping sites.", "href": "../../source/auxkernels/EmptySitesAux.html"}>>>
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_Cu}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_Cu}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix"}>>> = ${trap_per_free_Cu}
    trapped_concentration_variables<<<{"description": "Variables representing trapped particle concentrations."}>>> = C_trapped_Cu
  []
  [scaled_empty_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_empty_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_empty_Cu
  []
  [trapped_sites_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_trapped_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = 1e0
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_Cu
  []
  [scaled_trapped_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_trapped_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_trapped_Cu
  []
  [total_sites_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_total_Cu
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'S_trapped_Cu + S_empty_Cu'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'S_trapped_Cu S_empty_Cu'
  []
  [scaled_total_Cu]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_total_Cu
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_total_Cu
  []
  ############################## AuxKernels for CuCrZr (block = 2)
  [Scaled_mobile_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_mobile_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_mobile_CuCrZr
  []
  [Scaled_trapped_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_trapped_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_CuCrZr
  []
  [total_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = C_total_CuCrZr
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'C_mobile_CuCrZr + C_trapped_CuCrZr'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'C_mobile_CuCrZr C_trapped_CuCrZr'
  []
  [Scaled_total_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_C_total_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_total_CuCrZr
  []
  [empty_sites_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_empty_CuCrZr
    type = EmptySitesAux<<<{"description": "Calculates the concentration of empty trapping sites.", "href": "../../source/auxkernels/EmptySitesAux.html"}>>>
    N<<<{"description": "The atomic number density of the host material (1/m^3)"}>>> = ${N_CuCrZr}
    Ct0<<<{"description": "The fraction of host sites that can contribute to trapping as a function (-)"}>>> = ${Ct0_CuCrZr}
    trap_per_free<<<{"description": "An estimate for the ratio of the concentration magnitude of trapped species to free species. Setting a value for this can be helpful in producing a well-scaled matrix"}>>> = ${trap_per_free_CuCrZr}
    trapped_concentration_variables<<<{"description": "Variables representing trapped particle concentrations."}>>> = C_trapped_CuCrZr
  []
  [scaled_empty_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_empty_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_empty_CuCrZr
  []
  [trapped_sites_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_trapped_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = 1e0
    source_variable<<<{"description": "The variable to be normalized"}>>> = C_trapped_CuCrZr
  []
  [scaled_trapped_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_trapped_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_trapped_CuCrZr
  []
  [total_sites_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = S_total_CuCrZr
    type = ParsedAux<<<{"description": "Sets a field variable value to the evaluation of a parsed expression.", "href": "../../source/auxkernels/ParsedAux.html"}>>>
    expression<<<{"description": "Parsed function expression to compute"}>>> = 'S_trapped_CuCrZr + S_empty_CuCrZr'
    coupled_variables<<<{"description": "Vector of coupled variable names"}>>> = 'S_trapped_CuCrZr S_empty_CuCrZr'
  []
  [scaled_total_CuCrZr]
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = Sc_S_total_CuCrZr
    type = NormalizationAux<<<{"description": "Normalizes a variable based on a Postprocessor value.", "href": "../../source/auxkernels/NormalizationAux.html"}>>>
    normal_factor<<<{"description": "The normalization factor"}>>> = ${tungsten_atomic_density}
    source_variable<<<{"description": "The variable to be normalized"}>>> = S_total_CuCrZr
  []
  [flux_y_W]
    type = DiffusionFluxAux<<<{"description": "Compute components of flux vector for diffusion problems $(\\vec{J} = -D \\nabla C)$.", "href": "../../source/auxkernels/DiffusionFluxAux.html"}>>>
    diffusivity<<<{"description": "The name of the diffusivity material property that will be used in the flux computation."}>>> = diffusivity_W
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = flux_y
    diffusion_variable<<<{"description": "The name of the variable"}>>> = C_mobile_W
    component<<<{"description": "The desired component of flux."}>>> = y
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 4
  []
  [flux_y_Cu]
    type = DiffusionFluxAux<<<{"description": "Compute components of flux vector for diffusion problems $(\\vec{J} = -D \\nabla C)$.", "href": "../../source/auxkernels/DiffusionFluxAux.html"}>>>
    diffusivity<<<{"description": "The name of the diffusivity material property that will be used in the flux computation."}>>> = diffusivity_Cu
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = flux_y
    diffusion_variable<<<{"description": "The name of the variable"}>>> = C_mobile_Cu
    component<<<{"description": "The desired component of flux."}>>> = y
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 3
  []
  [flux_y_CuCrZr]
    type = DiffusionFluxAux<<<{"description": "Compute components of flux vector for diffusion problems $(\\vec{J} = -D \\nabla C)$.", "href": "../../source/auxkernels/DiffusionFluxAux.html"}>>>
    diffusivity<<<{"description": "The name of the diffusivity material property that will be used in the flux computation."}>>> = diffusivity_CuCrZr
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = flux_y
    diffusion_variable<<<{"description": "The name of the variable"}>>> = C_mobile_CuCrZr
    component<<<{"description": "The desired component of flux."}>>> = y
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
[]

Boundary conditions and history

For the mass conservation equation, we use FunctionNeumannBC and DirichletBC to solve the 2 $var element = document.getElementById("moose-equation-62632e27-a035-4071-9aeb-e1f66874db66");katex.render("^{\\text{nd}}", element, {displayMode:false,throwOnError:false});$ term of Eq. (3). FunctionNeumannBC is used to treat the time-dependent plasma exposure at the top (plasma-exposed) surface, and DirichletBC is used to set the BC of the solute T atom concentration to zero at the inner CuCrZr tube (at $var element = document.getElementById("moose-equation-90875a4f-fff0-48a7-8919-ad65ce1c934d");katex.render("r = 6", element, {displayMode:false,throwOnError:false});$ mm). For the energy conservation equation, we use FunctionNeumannBC and DirichletBC to solve the 2 $var element = document.getElementById("moose-equation-d5a11422-795b-442f-88d6-8ca1be0b8b9a");katex.render("^{\\text{nd}}", element, {displayMode:false,throwOnError:false});$ term of Eq. (4). FunctionNeumannBC is used to treat the time-dependent heat flux at the top (plasma-exposed) surface, and FunctionDirichletBC is used to treat the temperature increase in the cooling tube. FunctionNeumannBC, FunctionDirichletBC, DirichletBC, and FunctionDirichletBC are MOOSE objects commonly used to represent the BCs of the variables to be solved.

We simulate a 20,000-second plasma discharge, with each 1,600-second cycle consisting of a 100-second plasma ramp-up, a 400-second steady-state plasma discharge, a 100-second plasma ramp-down, and a 1,000-second waiting phase. Up to 50 cycles are simulated to achieve the total discharge. Figure 2 shows the integrated (solute, total and trapped) tritium concentration profiles in the monoblock. It shows that implantation fluxes and temperatures vary linearly up to (from) their steady-state values from (up to) their initial values during ramp-up (ramp-down).

Temperature profiles (orange) and integrated tritium concentration profiles (blue) during two 1,600-second-cycle plasma discharges. This corresponds to Fig. 2 in [!cite](Shimada2024114438).

Figure 2: Temperature profiles (orange) and integrated tritium concentration profiles (blue) during two 1,600-second-cycle plasma discharges. This corresponds to Fig. 2 in Shimada et al. (2024).

During the steady-state plasma discharge, we set a heat flux of 10 MW $var element = document.getElementById("moose-equation-48500679-f637-4328-8ceb-254b84b17b1d");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ m $var element = document.getElementById("moose-equation-66b82618-b810-4312-94a5-b2c3a7cccc7b");katex.render("^{-2}", element, {displayMode:false,throwOnError:false});$ at the top of the 2D monoblock (at $var element = document.getElementById("moose-equation-170fcd99-87ee-47a0-b109-499538fa3407");katex.render("y = 14.0", element, {displayMode:false,throwOnError:false});$ mm) and a cooling temperature of 552 K at the inner CuCrZr tube (at $var element = document.getElementById("moose-equation-1f59e112-b2d6-41bc-ac0a-a0cb6a4b09c2");katex.render("r = 6.0", element, {displayMode:false,throwOnError:false});$ mm). We assume a 100% T plasma with a 5.0 $var element = document.getElementById("moose-equation-fe040d0f-cb63-4492-9260-0b8c0926be6e");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-30c2d6c5-00f6-4d47-89b2-cfb53f63292e");katex.render("^{23}", element, {displayMode:false,throwOnError:false});$ m $var element = document.getElementById("moose-equation-b3be9de8-28b9-4b66-a434-f5026640f925");katex.render("^{-2}", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-7d139e96-3f8b-4f16-aa16-73f3c58db094");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-88baa1d0-2cb9-47bc-8f69-d23bef17ab28");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ plasma particle flux (which is half of the full 1.0 $var element = document.getElementById("moose-equation-302d650f-4860-4c9f-a1c4-bc66ec4c364e");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-cc0e2867-553d-4bd0-8410-df3b0c514231");katex.render("^{24}", element, {displayMode:false,throwOnError:false});$ m $var element = document.getElementById("moose-equation-44ce12ae-d1c6-4cd6-9680-1e6a44a9bc7c");katex.render("^{-2}", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-7a8db7c3-c684-48eb-9d63-af2eae675fd2");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-04902e18-376a-4437-a1ad-dd803830ce1f");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ DT plasma particle flux), and only 0.1% of the incident plasma particle flux (5.0 $var element = document.getElementById("moose-equation-f61e96ed-37b7-4625-a133-bac93e479be9");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-49f94f62-3b92-4d9e-aec2-41cd7072dfaa");katex.render("^{20}", element, {displayMode:false,throwOnError:false});$ m $var element = document.getElementById("moose-equation-a2434dac-2666-4f5a-9507-32c6f497c569");katex.render("^{-2}", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-ebc32a3a-c80f-4d9b-a5fb-15b0fcb16078");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-220f9083-217e-4808-a82e-26b600efef30");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) entered the first layer of mesh at the exposed surface ( $var element = document.getElementById("moose-equation-ddcb7752-bc55-4871-87a1-da84b788291e");katex.render("y = 14.0", element, {displayMode:false,throwOnError:false});$ mm) as in Shimada et al. (2024). The solute T atom concentration is set to zero at the inner CuCrZr tube (at $var element = document.getElementById("moose-equation-819d7e5c-c213-4c8d-b87a-3f247531dfb0");katex.render("r = 6.0", element, {displayMode:false,throwOnError:false});$ mm).

We treated this plasma exposure by setting the flux BC of the solute T atom concentration at the exposed surface ( $var element = document.getElementById("moose-equation-4e2a0baa-2927-4a89-9ad3-da3c1a22a8c4");katex.render("y = 14.0", element, {displayMode:false,throwOnError:false});$ mm) as a simplification of the complex plasma implantation and recombination phenomena, which would require a very fine mesh and increase computational costs.

warning:Small typo fixed for the heat flux from Shimada et al. (2024)

Since publication, the input file has been updated to fix a small typo that has a minor, almost insignificant effect on the results. The typo set the heat flux to be equal to 300 W/m $var element = document.getElementById("moose-equation-4b181cdb-17a0-447f-88ef-e210bb3e3a71");katex.render("^2", element, {displayMode:false,throwOnError:false});$ while the pulse was off, as opposed to being equal to 0 W/m $var element = document.getElementById("moose-equation-268d8b39-f55b-49c9-a5a2-de682fbe472a");katex.render("^2", element, {displayMode:false,throwOnError:false});$ . This was likely due to a mistake in using the off-pulse temperature of the cooling channel (300 K) rather than the zero flux. The change from 300 W/m $var element = document.getElementById("moose-equation-01ab5567-65e8-4035-bb6b-40427b62f1ee");katex.render("^2", element, {displayMode:false,throwOnError:false});$ to 0 W/m $var element = document.getElementById("moose-equation-6aa353f0-2d40-46fd-9d20-2e8d56bb7ad9");katex.render("^2", element, {displayMode:false,throwOnError:false});$ was observed to be negligible, which is attributed to the fact that 300 W/m $var element = document.getElementById("moose-equation-b07c0f36-1418-43bd-b8d9-7148f98993a5");katex.render("^2", element, {displayMode:false,throwOnError:false});$ is a small value in this scenario. For context, the maximum heat flux value is 1 $var element = document.getElementById("moose-equation-b0b3698b-22f9-4835-b671-6ce29928dfba");katex.render("\\times 10^{7}", element, {displayMode:false,throwOnError:false});$ W/m $var element = document.getElementById("moose-equation-35362637-f11b-466c-8ecc-1cfb558a499a");katex.render("^2", element, {displayMode:false,throwOnError:false});$ . The current version of the input file uses 0 W/m $var element = document.getElementById("moose-equation-93726ef9-7bab-4b79-9b59-5256f15a3456");katex.render("^2", element, {displayMode:false,throwOnError:false});$ .

[BCs<<<{"href": "../../syntax/BCs/index.html"}>>>]
  [C_mob_W_top_flux]
    type = FunctionNeumannBC<<<{"description": "Imposes the integrated boundary condition $\\frac{\\partial u}{\\partial n}=h(t,\\vec{x})$, where $h$ is a (possibly) time and space-dependent MOOSE Function.", "href": "../../source/bcs/FunctionNeumannBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_mobile_W
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'top'
    function<<<{"description": "The function."}>>> = mobile_flux_bc_function
  []
  [mobile_tube]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_mobile_CuCrZr
    value<<<{"description": "Value of the BC"}>>> = ${C_mobile_CuCrZr_DirichletBC_Coolant}
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = '2to1'
  []
  [temperature_top]
    type = FunctionNeumannBC<<<{"description": "Imposes the integrated boundary condition $\\frac{\\partial u}{\\partial n}=h(t,\\vec{x})$, where $h$ is a (possibly) time and space-dependent MOOSE Function.", "href": "../../source/bcs/FunctionNeumannBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = temperature
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'top'
    function<<<{"description": "The function."}>>> = temperature_flux_bc_function
  []
  [temperature_tube]
    type = FunctionDirichletBC<<<{"description": "Imposes the essential boundary condition $u=g(t,\\vec{x})$, where $g$ is a (possibly) time and space-dependent MOOSE Function.", "href": "../../source/bcs/FunctionDirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = temperature
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = '2to1'
    function<<<{"description": "The forcing function."}>>> = temperature_inner_func
  []
[]

Functions

In this example, Functions are used to define the time-dependent tritium and heat flux at the exposed surface, as well as the coolant temperature. Functions can also be spatially-dependent. The functions are then provided to BCs, as described above.

[Functions<<<{"href": "../../syntax/Functions/index.html"}>>>]
  [t_in_cycle]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../source/functions/MooseParsedFunction.html"}>>>
    expression<<<{"description": "The user defined function."}>>> = 't % ${plasma_cycle_time}'
  []
  # pulse between 0 and 1 following the plasma operation
  [pulse_time_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."}>>> = 't_in_cycle'
    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."}>>> = 't_in_cycle'
    expression<<<{"description": "The user defined function."}>>> = 'if(t_in_cycle < ${plasma_ramp_time}, t_in_cycle/${plasma_ramp_time},
                        if(t_in_cycle < ${plasma_ss_end}, 1,
                        if(t_in_cycle < ${plasma_ramp_down_end}, 1 - (t_in_cycle-${plasma_ss_end})/${plasma_ramp_time}, 0.0)))'
  []
  [mobile_flux_bc_function]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../source/functions/MooseParsedFunction.html"}>>>
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 'pulse_time_function'
    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the symbol_values vector."}>>> = 'pulse_time_function'
    expression<<<{"description": "The user defined function."}>>> = '(${plasma_max_flux} - ${plasma_min_flux}) * pulse_time_function + ${plasma_min_flux}'
  []
  [temperature_flux_bc_function]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../source/functions/MooseParsedFunction.html"}>>>
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 'pulse_time_function'
    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the symbol_values vector."}>>> = 'pulse_time_function'
    expression<<<{"description": "The user defined function."}>>> = '(${plasma_max_heat} - ${plasma_min_heat}) * pulse_time_function + ${plasma_min_heat}'
  []
  [temperature_inner_func]
    type = ParsedFunction<<<{"description": "Function created by parsing a string", "href": "../../source/functions/MooseParsedFunction.html"}>>>
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 'pulse_time_function'
    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the symbol_values vector."}>>> = 'pulse_time_function'
    expression<<<{"description": "The user defined function."}>>> = '(${temperature_coolant_max} - ${temperature_initial}) * pulse_time_function + ${temperature_initial}'
  []
  [timestep_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."}>>> = 't_in_cycle'
    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."}>>> = 't_in_cycle'
    expression<<<{"description": "The user defined function."}>>> = 'if(t_in_cycle < ${fparse 0.1 * plasma_ramp_time}   ,  20,
                      if(t_in_cycle < ${fparse 0.9 * plasma_ramp_time}   ,  40,
                      if(t_in_cycle < ${fparse 1.1 * plasma_ramp_time}   ,  20,
                      if(t_in_cycle < ${fparse plasma_ss_end - 20}       ,  40,
                      if(t_in_cycle < ${plasma_ss_end}                   ,  20,
                      if(t_in_cycle < ${fparse plasma_ramp_down_end - 10},   4,
                      if(t_in_cycle < ${fparse plasma_ramp_down_end + 10},  20,
                      if(t_in_cycle < ${fparse plasma_cycle_time - 100}  , 200,
                      if(t_in_cycle < ${plasma_cycle_time}               ,  40,  2)))))))))'
  []
[]

Material properties

We use the tritium mass transport properties listed in Table 2 to solve the mass conservation equation (Eq. (3)) for two variables: $var element = document.getElementById("moose-equation-79ca01d0-a40b-429f-88ad-a928885e35a7");katex.render("C_s", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-6bf62a26-8179-4fc8-b242-60339748c1f1");katex.render("C_t", element, {displayMode:false,throwOnError:false});$ (i.e., solute and trapped T atom concentrations) in three materials and the thermal properties (i.e., density, temperature-dependent specific heat, and thermal conductivity) listed in Table 3 to solve the energy conservation equation (Eq. (4)) for one variable: the temperature $var element = document.getElementById("moose-equation-e2ab5840-1d9a-4c0c-b39e-77329347cbbc");katex.render("T", element, {displayMode:false,throwOnError:false});$ . The diffusivity is defined as $var element = document.getElementById("moose-equation-e2e93a98-e346-4a74-8ff3-13796aa8657b");katex.render("D=D_0 \\exp\u2061(-E_D/k_B/T)", element, {displayMode:false,throwOnError:false});$ and the solubility is defined as $var element = document.getElementById("moose-equation-a3d417e0-76c2-45cf-8fb6-dc0eb5fd5015");katex.render("K_s=K_{s,0} \\exp\u2061(-E_s/k_B/T)", element, {displayMode:false,throwOnError:false});$ .

Table 2: Tritium mass transport and trapping properties used in the W, Cu, and CuCrZr layers. $var element = document.getElementById("moose-equation-27784e36-d22a-430a-acd2-ce23429a9275");katex.render("^{**}", element, {displayMode:false,throwOnError:false});$ NOTE: Two-component solubility in W kept the maximum solubility ratio between W and Cu to 104 at low temperature. The references are provided in Shimada et al. (2024).

Material	$var element = document.getElementById("moose-equation-ec2ffaa8-552c-4e90-8b8a-596cd58c6f87");katex.render("D_0", element, {displayMode:false,throwOnError:false});$ (m $var element = document.getElementById("moose-equation-552bce08-ad55-40bd-b440-f569b345d96e");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /s)	$var element = document.getElementById("moose-equation-abf2486c-457c-4580-988f-1b572e5c3bba");katex.render("E_D", element, {displayMode:false,throwOnError:false});$ (eV)	$var element = document.getElementById("moose-equation-9667b91c-d2a2-4769-9092-40a0a67366dd");katex.render("K_{s,0}", element, {displayMode:false,throwOnError:false});$ (Pa $var element = document.getElementById("moose-equation-0526461b-cb5f-4efe-a33e-2341ec968e1e");katex.render("^{1/2}", element, {displayMode:false,throwOnError:false});$ )	$var element = document.getElementById("moose-equation-2ddac14c-1b92-4ea9-b1e0-afa82aef017d");katex.render("E_s", element, {displayMode:false,throwOnError:false});$ (eV)	Detrapping energy: $var element = document.getElementById("moose-equation-c85eb5d1-05d5-4199-bba2-6474e53495cb");katex.render("E_{dt}", element, {displayMode:false,throwOnError:false});$ (eV)	Trap density: $var element = document.getElementById("moose-equation-8753a9ba-3326-459d-9f83-7514dce7379e");katex.render("n_{trap}", element, {displayMode:false,throwOnError:false});$ (at.fr.)
W	2.4 $var element = document.getElementById("moose-equation-7eaf175e-b9fa-4011-a8ba-8492a53e7903");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-1bc574ce-33fb-4702-bd97-42b509b50acd");katex.render("^{-7}", element, {displayMode:false,throwOnError:false});$	0.39	1.87 $var element = document.getElementById("moose-equation-3bb41766-fd94-42f7-aea6-8e39d70d38b3");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-3b786018-0eaf-4b3b-9f6a-3ac2a9989fe3");katex.render("^{24}", element, {displayMode:false,throwOnError:false});$	1.04	0.85	1.0 $var element = document.getElementById("moose-equation-c93e8689-9b2d-4258-bfb8-5b155656537f");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-2d9b38e7-2459-41b4-9d09-8c908a0a532e");katex.render("^{-4}", element, {displayMode:false,throwOnError:false});$
W			3.14 $var element = document.getElementById("moose-equation-e50fd150-9354-4b1e-b654-a4ae252aee0f");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-d882701a-b69b-4b08-9466-49a6aee473e9");katex.render("^{20}", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-f806dce6-8bb6-4a99-9a1e-78bb9d447088");katex.render("^{**}", element, {displayMode:false,throwOnError:false});$	0.57 $var element = document.getElementById("moose-equation-f48ac97b-841d-44de-ba86-9c3fb65a35f2");katex.render("^{**}", element, {displayMode:false,throwOnError:false});$
Cu	6.6 $var element = document.getElementById("moose-equation-61cfa26a-40fa-498d-b12a-86ca37ae9a43");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-87635bc0-7f69-4a6f-b3cf-ca6ba620d80e");katex.render("^{-7}", element, {displayMode:false,throwOnError:false});$	0.39	3.14 $var element = document.getElementById("moose-equation-fef66b65-95ff-4fc7-a420-022227f5ed3f");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-53309bdf-14ae-4ca7-8c05-c8ecf66058c0");katex.render("^{24}", element, {displayMode:false,throwOnError:false});$	0.57	0.50	5.0 $var element = document.getElementById("moose-equation-13a5386a-b05c-40ae-a4ef-b980ee5fe623");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-2f7e9553-a073-4581-a903-99b3b8dfeced");katex.render("^{-5}", element, {displayMode:false,throwOnError:false});$
CuCrZr	3.9 $var element = document.getElementById("moose-equation-57130832-cfcd-4120-a78b-e79c67b95a18");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-c4b044b1-4034-4a69-a218-deac8dc1bc94");katex.render("^{-7}", element, {displayMode:false,throwOnError:false});$	0.42	4.28 $var element = document.getElementById("moose-equation-17a7b355-acfd-4c77-bae2-bd6a770b0591");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-d49c18f3-db3d-48ac-a168-10a58a90ccc1");katex.render("^{23}", element, {displayMode:false,throwOnError:false});$	0.39	0.83	5.0 $var element = document.getElementById("moose-equation-bc1468f2-0841-49d3-a784-b7840eaab497");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-fffb868c-4a31-4d33-aa7e-22e23b035628");katex.render("^{-5}", element, {displayMode:false,throwOnError:false});$

Table 3: Thermal properties used in the W, Cu, and CuCrZr layers. The references are provided in Shimada et al. (2024).

Material	Density: $var element = document.getElementById("moose-equation-0578c8a9-8404-4202-b867-5a3e06bbcc97");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ (g m $var element = document.getElementById("moose-equation-bbd4019f-6481-480e-902a-954b8ba67ca7");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$ )	Specific heat: $var element = document.getElementById("moose-equation-e04a8229-cf31-4e68-8f21-8f3967930983");katex.render("c_p", element, {displayMode:false,throwOnError:false});$ (J kg $var element = document.getElementById("moose-equation-a2fe1fd6-0fca-488f-a89b-51a36764d7f0");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ K $var element = document.getElementById("moose-equation-7884c0f2-c986-47b9-bb4a-271aea431715");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ )	Thermal conductivity: $var element = document.getElementById("moose-equation-17f864ba-2cb4-4453-818c-eb0a86289568");katex.render("k_T", element, {displayMode:false,throwOnError:false});$ (W m $var element = document.getElementById("moose-equation-a5df6ec6-00ea-4bf0-8c62-8c5376618d74");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ K $var element = document.getElementById("moose-equation-5c0c782c-c80b-4f98-b6ed-df02cdfe260d");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ )
W	19,300	1.16 $var element = document.getElementById("moose-equation-7e20196b-60fb-4fc7-a239-85c8859fd951");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-c1111ac3-6f31-4979-92bb-c2e7c80efc6b");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ +7.11 $var element = document.getElementById("moose-equation-c55e2d4c-2305-4502-bc5a-0e0c28eeb84d");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-4a1bb903-cc9a-4805-86ca-3199d63e0bb6");katex.render("^{-2}", element, {displayMode:false,throwOnError:false});$ T –6.58 $var element = document.getElementById("moose-equation-dd24be2e-1504-49c2-90dd-310c80212118");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-f4cc9600-e67c-438b-8ec6-97f80cbb8cf8");katex.render("^{-5}", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-8d950685-2c25-4c4a-8914-809b7566eb49");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ +3.24 $var element = document.getElementById("moose-equation-b86bf9a6-c8dd-445b-95f3-2bfad9f49b7a");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-2fc8f17b-70c7-4e55-82d4-ffc82424668b");katex.render("^{-8}", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-b046dc2a-307d-4518-b5a0-5f0024b76084");katex.render("^{3}", element, {displayMode:false,throwOnError:false});$ –5.45 $var element = document.getElementById("moose-equation-425ecddb-032e-4773-8399-51119acf74d6");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-1061b107-76ff-4273-9c2c-79c5e0cc1390");katex.render("^{-12}", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-8ac2e7f8-1ab5-43f4-b9e2-47dc1c6c26d3");katex.render("^{4}", element, {displayMode:false,throwOnError:false});$ (293 < T (K) < 2500)	2.41 $var element = document.getElementById("moose-equation-fa2c2ab0-1cdb-437a-b1bb-beb73011e890");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-bdcaf74e-8a7c-453b-beb5-483a0328d0d6");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ –2.90 $var element = document.getElementById("moose-equation-db45c1d0-c601-497b-b26d-02118fb86bdf");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-a7a37c4e-f14b-4b3a-b59c-98a41820bae0");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ T + 2.54 $var element = document.getElementById("moose-equation-5c67d71d-7757-4d6d-8e72-730054dd28e8");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-92fb0f9d-8dcd-4f74-a4d2-302ee8239d2c");katex.render("^{-4}", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-b80e3530-1521-4b86-a7a6-7e1b8e20756a");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ –1.03 $var element = document.getElementById("moose-equation-35f560df-43c4-4661-bb8d-d7aaac19a6b1");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-e3771ddd-865d-4d38-87a1-09e8c04fc1e6");katex.render("^{-7}", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-e59184c6-152d-427b-8554-cd5a3c05dc02");katex.render("^{3}", element, {displayMode:false,throwOnError:false});$ +1.52 $var element = document.getElementById("moose-equation-4e75cf32-5108-4f3d-a4f0-4711a1ab31a9");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-8c02f7f8-068b-4417-97af-a4db41577e06");katex.render("^{-11}", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-6f9e1dc7-ea9b-4eb0-ac60-1e519710e268");katex.render("^{4}", element, {displayMode:false,throwOnError:false});$ (293 < T (K) < 2500)
Cu	8,960	4.21 $var element = document.getElementById("moose-equation-1f85eb9b-ecc1-4ff4-9c35-1183c0d4054f");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-732b2b3d-3534-4598-b319-fbfed419e096");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ –6.85 $var element = document.getElementById("moose-equation-3c8e3a00-560c-4d6b-8d70-f76df7d19c32");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-8a0adb62-74ea-495b-a416-3690e1ccf29c");katex.render("^{-2}", element, {displayMode:false,throwOnError:false});$ T (293 < T (K) < 873)	3.16 $var element = document.getElementById("moose-equation-817b9dae-e0a8-410e-930d-4e84917e1a4f");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-d82feb3d-8509-4c0f-8abc-81a6cda6f6d5");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ +3.18 $var element = document.getElementById("moose-equation-f790c92d-c6da-484e-8f02-6813cb65b7ef");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-c14173eb-7d01-4f82-bdff-a23af39bc7ce");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ T –3.49 $var element = document.getElementById("moose-equation-65c9c89f-888c-4966-99b3-e52cd6cc269d");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-599ba63b-7448-49a1-b083-ad683d369f62");katex.render("^{-4}", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-9908e3f1-0ecb-41cc-9754-0fda47ce01ef");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ +1.66 $var element = document.getElementById("moose-equation-9f7b288c-25d7-440b-a50b-5e43a8ea4db0");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-defaff76-48e5-450f-acd8-d3040c85408d");katex.render("^{-7}", element, {displayMode:false,throwOnError:false});$ T $var element = document.getElementById("moose-equation-0bbf0dae-245a-44d9-94a5-f6b01d0d046f");katex.render("^{3}", element, {displayMode:false,throwOnError:false});$ (293 < T (K) < 873)
CuCrZr	8,900	390	3.87 $var element = document.getElementById("moose-equation-b50ee994-60a0-4501-853b-7e49ec29d0ab");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-4a0a12ae-f03f-43b0-9330-ffe89efeac60");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ –1.28 $var element = document.getElementById("moose-equation-ea60ab70-d041-4605-b790-e39887f127e0");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-73a650ea-404f-4d61-b892-4c15e478e06c");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ T (293 < T (K) < 927)

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  ############################## Materials for W (block = 4)
  [diffusivity_W]
    type = ADParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = diffusivity_W
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 4
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  [solubility_W]
    type = ADParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = solubility_W
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 4
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  [converter_to_regular_W]
    type = MaterialADConverter<<<{"description": "Converts regular material properties to AD properties and vice versa", "href": "../../source/materials/MaterialADConverter.html"}>>>
    ad_props_in<<<{"description": "The names of the AD material properties to convert to regular properties"}>>> = 'diffusivity_W'
    reg_props_out<<<{"description": "The names of the output regular properties"}>>> = 'diffusivity_W_nonAD'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 4
  []
  [heat_transfer_W]
    type = GenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../../source/materials/GenericConstantMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = 'density_W'
    prop_values<<<{"description": "The values associated with the named properties"}>>> = '${density_W}'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 4
  []
  [specific_heat_W]
    type = ParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = specific_heat_W
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 4
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  [thermal_conductivity_W]
    type = ParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = thermal_conductivity_W
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 4
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  ############################## Materials for Cu (block = 3)
  [diffusivity_Cu]
    type = ADParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = diffusivity_Cu
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 3
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  [solubility_Cu]
    type = ADParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = solubility_Cu
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 3
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  [converter_to_regular_Cu]
    type = MaterialADConverter<<<{"description": "Converts regular material properties to AD properties and vice versa", "href": "../../source/materials/MaterialADConverter.html"}>>>
    ad_props_in<<<{"description": "The names of the AD material properties to convert to regular properties"}>>> = 'diffusivity_Cu'
    reg_props_out<<<{"description": "The names of the output regular properties"}>>> = 'diffusivity_Cu_nonAD'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 3
  []
  [heat_transfer_Cu]
    type = GenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../../source/materials/GenericConstantMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = 'density_Cu'
    prop_values<<<{"description": "The values associated with the named properties"}>>> = '${density_Cu}'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 3
  []
  [specific_heat_Cu]
    type = ParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = specific_heat_Cu
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 3
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  [thermal_conductivity_Cu]
    type = ParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = thermal_conductivity_Cu
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 3
    # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  ############################## Materials for CuCrZr (block = 2)
  [diffusivity_CuCrZr]
    type = ADParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = diffusivity_CuCrZr
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  [solubility_CuCrZr]
    type = ADParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = solubility_CuCrZr
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  [converter_to_regular_CuCrZr]
    type = MaterialADConverter<<<{"description": "Converts regular material properties to AD properties and vice versa", "href": "../../source/materials/MaterialADConverter.html"}>>>
    ad_props_in<<<{"description": "The names of the AD material properties to convert to regular properties"}>>> = 'diffusivity_CuCrZr'
    reg_props_out<<<{"description": "The names of the output regular properties"}>>> = 'diffusivity_CuCrZr_nonAD'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
  [heat_transfer_CuCrZr]
    type = GenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../../source/materials/GenericConstantMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = 'density_CuCrZr specific_heat_CuCrZr'
    prop_values<<<{"description": "The values associated with the named properties"}>>> = '${density_CuCrZr} ${specific_heat_CuCrZr}'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
  [thermal_conductivity_CuCrZr]
    type = ParsedMaterial<<<{"description": "Parsed expression Material.", "href": "../../source/materials/ParsedMaterial.html"}>>>
    property_name<<<{"description": "Name of the parsed material property"}>>> = thermal_conductivity_CuCrZr
    coupled_variables<<<{"description": "Vector of variables used in the parsed function"}>>> = 'temperature'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
    expression<<<{"description": "Parsed function (see FParser) expression for the parsed material"}>>> = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
  ############################## Materials for others
  [interface_jump_4to3]
    type = SolubilityRatioMaterial<<<{"description": "Calculates the jump in concentration across an interface.", "href": "../../source/materials/SolubilityRatioMaterial.html"}>>>
    solubility_primary<<<{"description": "The material property on the primary side of the interface"}>>> = solubility_W
    solubility_secondary<<<{"description": "The material property on the secondary side of the interface"}>>> = solubility_Cu
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies"}>>> = '4to3'
    concentration_primary<<<{"description": "Primary side non-linear variable for jump computation"}>>> = C_mobile_W
    concentration_secondary<<<{"description": "Secondary side non-linear variable for jump computation"}>>> = C_mobile_Cu
  []
  [interface_jump_3to2]
    type = SolubilityRatioMaterial<<<{"description": "Calculates the jump in concentration across an interface.", "href": "../../source/materials/SolubilityRatioMaterial.html"}>>>
    solubility_primary<<<{"description": "The material property on the primary side of the interface"}>>> = solubility_Cu
    solubility_secondary<<<{"description": "The material property on the secondary side of the interface"}>>> = solubility_CuCrZr
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies"}>>> = '3to2'
    concentration_primary<<<{"description": "Primary side non-linear variable for jump computation"}>>> = C_mobile_Cu
    concentration_secondary<<<{"description": "Secondary side non-linear variable for jump computation"}>>> = C_mobile_CuCrZr
  []
[]

Interface Kernels

TMAP8 assumes equilibrium between the chemical potentials of the diffusing species similar to how TMAP4 and TMAP7 treated the diffusing species across two different materials with different chemical potentials. SolubilityRatioMaterial is used to treat solute concentration differences stemming from a difference in T solubilities across the interface. The solubility ratio jump is calculated via the following: $(5)var element = document.getElementById("moose-equation-a832ea10-f7d3-475c-8f84-2ec6380748ca");katex.render(" J = \\frac{C_{s,1}}{K_{s,1}} - \\frac{C_{s,2}}{K_{s,2}}.", element, {displayMode:true,throwOnError:false});$

ADPenaltyInterfaceDiffusion is used to conserve the particle flux at this interface between two different solubilities. The extremely low tritium solubility in W at low temperature leads to an extremely low solute concentration in W, creating a large solute concentration difference between W and Cu. Two-component solubility in W is used to keep the maximum solubility ratio between W and Cu to 104 at low temperature to avoid the convergence issue associated with calculating two significantly different solute concentration in W and Cu.

[InterfaceKernels<<<{"href": "../../syntax/InterfaceKernels/index.html"}>>>]
  [tied_4to3]
    type = ADPenaltyInterfaceDiffusion<<<{"description": "A penalty-based interface condition that forcesthe continuity of variables and the flux equivalence across an interface.", "href": "../../source/interfacekernels/PenaltyInterfaceDiffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_mobile_W
    neighbor_var<<<{"description": "The variable on the other side of the interface."}>>> = C_mobile_Cu
    penalty<<<{"description": "The penalty that penalizes jump between primary and neighbor variables."}>>> = 0.05 #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
    # jump_prop_name = solubility_ratio_4to3
    jump_prop_name<<<{"description": "the name of the material property that calculates the jump."}>>> = solubility_ratio
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies"}>>> = '4to3'
  []
  [tied_3to2]
    type = ADPenaltyInterfaceDiffusion<<<{"description": "A penalty-based interface condition that forcesthe continuity of variables and the flux equivalence across an interface.", "href": "../../source/interfacekernels/PenaltyInterfaceDiffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = C_mobile_Cu
    neighbor_var<<<{"description": "The variable on the other side of the interface."}>>> = C_mobile_CuCrZr
    penalty<<<{"description": "The penalty that penalizes jump between primary and neighbor variables."}>>> = 0.05 #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
    # jump_prop_name = solubility_ratio_3to2
    jump_prop_name<<<{"description": "the name of the material property that calculates the jump."}>>> = solubility_ratio
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies"}>>> = '3to2'
  []
[]

Numerical method

We use a standard preconditioner: the “single matrix preconditioner”. The Newton method is used to model the transient tritium and thermal transport in a 2D monoblock. It is important to note that MOOSE is equipped with the built-in Message Passing Interface (MPI) protocol, as tritium and thermal transport analysis of fifty 1,600-second cycle plasma discharges in the 2D monoblock is performed in under 2 hours using a single device/computer (3.5 GHz Apple M2 Pro, 10-Core CPU/16-Core GPU) with this MPI feature.

[Executioner<<<{"href": "../../syntax/Executioner/index.html"}>>>]
  type = Transient
  scheme = bdf2
  solve_type = NEWTON
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu NONZERO'
  nl_rel_tol = 1e-6 # 1e-8 works for 1 cycle
  nl_abs_tol = 1e-7 # 1e-11 works for 1 cycle
  end_time = '${fparse 50 * plasma_cycle_time}' # 50 ITER shots
  automatic_scaling = true
  line_search = 'none'
  dtmin = 1e-4
  nl_max_its = 18
  [TimeStepper<<<{"href": "../../syntax/Executioner/TimeStepper/index.html"}>>>]
    type = IterationAdaptiveDT
    dt = 20
    optimal_iterations = 15
    iteration_window = 1
    growth_factor = 1.2
    cutback_factor = 0.8
    timestep_limiting_postprocessor = timestep_max_pp
  []
[]

Results

The simulation results from this example are shown in Figure 3 and Figure 4. For more results, information, and discussion about the results for this example case and their significance, the reader is referred to Shimada et al. (2024).

Figure 3: Tritium concentration profile in W (left), Cu (center), and CuCrZr (right) after ten 1,600-second cycles ( $var element = document.getElementById("moose-equation-bba505c7-8ec1-464b-a231-f171fd8d2771");katex.render("t = 14912", element, {displayMode:false,throwOnError:false});$ s). This corresponds to Fig. 4A in Shimada et al. (2024).

Figure 4: Tritium concentration profile in W (left), Cu (center), and CuCrZr (right) after fifty 1,600-second cycles ( $var element = document.getElementById("moose-equation-bf646753-d0be-4db2-8144-c4b9fb8b3ef9");katex.render("t = 78912", element, {displayMode:false,throwOnError:false});$ s). This corresponds to Fig. 4B in Shimada et al. (2024).

warning:The exodus file in gold is a smaller version of the output

The input file (tmap8/test/tests/divertor_monoblock/divertor_monoblock.i) returns the outputs that were used in Shimada et al. (2024). However, a slightly modified version of this input is run in (tmap8/test/tests/divertor_monoblock/tests) as part of TMAP8's Software Quality Assurance process: It simulates only one pulse cycle, has a coarser mesh, and outputs the results less regularly to limit the file size. As a result, the exodus file in the test gold directory is a smaller version of the output generated when running the full input file.

Complete input file

This case is reproduced three times:

Once with the usual TMAP8 syntax for input files, which is done in (tmap8/test/tests/divertor_monoblock/divertor_monoblock.i),
Another time using the Physics Syntax to reproduce the same exact case, but with a simpler input file, which is shown in (tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics.i), and
Last with a simpler model utilizing only two variables (mobile and trapped) for tritium concentration across the different materials instead of having different variables in each block. The input file also uses the Physics Syntax and is shown in (tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics-single-variable.i).

Note that since these input files have a lot of sections in common, we utilize the !include feature available in MOOSE/TMAP8 to template common parts of the input files. In this case:

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_common_base.i) contains all the parts of the input files common to all three cases.
(tmap8/test/tests/divertor_monoblock/divertor_monoblock_mesh_base.i) contains parts related to the geometry and mesh for all three cases.
(tmap8/test/tests/divertor_monoblock/divertor_monoblock_multi_variable_base.i) contains the parts shared by both the (tmap8/test/tests/divertor_monoblock/divertor_monoblock.i) and (tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics.i) cases.
(tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics-single-variable.i) contains the parts equivalent to the ones in (tmap8/test/tests/divertor_monoblock/divertor_monoblock_multi_variable_base.i), but for the case in (tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics-single-variable.i).

Below is the input file (tmap8/test/tests/divertor_monoblock/divertor_monoblock.i), which includes several of the base input files described above. This case can be run reliably with approximately 4 processor cores. Note that this input file has been optimized for showcasing capability rather than computational cost.

### This input file is the complete input file for the divertor monoblock case.
### This case was published in:
### 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
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_CuCrZr_DirichletBC_Coolant = 1.0e-18 # at.fraction
C_mobile_W_init = 1.0e-20 # at.fraction
C_mobile_Cu_init = 5.0e-17 # at.fraction
C_mobile_CuCrZr_init = 1.0e-15 # at.fraction

# include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_multi_variable_base.i

[Problem]
    type = ReferenceResidualProblem
    extra_tag_vectors = 'ref'
    reference_vector = 'ref'
[]

[Variables]
    [temperature]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${temperature_initial}
    []
    ######################### Variables for W (block = 4)
    [C_mobile_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_W_init}
        block = 4
    []
    [C_trapped_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 4
    []
    ######################### Variables for Cu (block = 3)
    [C_mobile_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_Cu_init}
        block = 3
    []
    [C_trapped_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 3
    []
    ######################### Variables for CuCrZr (block = 2)
    [C_mobile_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_CuCrZr_init}
        block = 2
    []
    [C_trapped_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 2
    []
[]

[Kernels]
    ############################## Kernels for W (block = 4)
    [diff_W]
        type = ADMatDiffusion
        variable = C_mobile_W
        diffusivity = diffusivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_diff_W]
        type = ADTimeDerivative
        variable = C_mobile_W
        block = 4
        extra_vector_tags = ref
    []
    [coupled_time_W]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_W
        v = C_trapped_W
        block = 4
        extra_vector_tags = ref
    []
    [heat_conduction_W]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_heat_conduction_W]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_W
        density = density_W
        block = 4
        extra_vector_tags = ref
    []
    ############################## Kernels for Cu (block = 3)
    [diff_Cu]
        type = ADMatDiffusion
        variable = C_mobile_Cu
        diffusivity = diffusivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_diff_Cu]
        type = ADTimeDerivative
        variable = C_mobile_Cu
        block = 3
        extra_vector_tags = ref
    []
    [coupled_time_Cu]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_Cu
        v = C_trapped_Cu
        block = 3
        extra_vector_tags = ref
    []
    [heat_conduction_Cu]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_heat_conduction_Cu]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_Cu
        density = density_Cu
        block = 3
        extra_vector_tags = ref
    []
    ############################## Kernels for CuCrZr (block = 2)
    [diff_CuCrZr]
        type = ADMatDiffusion
        variable = C_mobile_CuCrZr
        diffusivity = diffusivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_diff_CuCrZr]
        type = ADTimeDerivative
        variable = C_mobile_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [coupled_time_CuCrZr]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_CuCrZr
        v = C_trapped_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [heat_conduction_CuCrZr]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_heat_conduction_CuCrZr]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_CuCrZr
        density = density_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
[]

[InterfaceKernels]
    [tied_4to3]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_W
        neighbor_var = C_mobile_Cu
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_4to3
        jump_prop_name = solubility_ratio
        boundary = '4to3'
    []
    [tied_3to2]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_Cu
        neighbor_var = C_mobile_CuCrZr
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_3to2
        jump_prop_name = solubility_ratio
        boundary = '3to2'
    []
[]

[NodalKernels]
    ############################## NodalKernels for W (block = 4)
    [time_W]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_W
    []
    [trapping_W]
        type = TrappingNodalKernel
        variable = C_trapped_W
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_W}
        Ct0 = ${Ct0_W}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_W}
        mobile_concentration = 'C_mobile_W'
        extra_vector_tags = ref
    []
    [release_W]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_W}
        variable = C_trapped_W
    []
    ############################## NodalKernels for Cu (block = 3)
    [time_Cu]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_Cu
    []
    [trapping_Cu]
        type = TrappingNodalKernel
        variable = C_trapped_Cu
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_Cu}
        Ct0 = ${Ct0_Cu}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_Cu}
        mobile_concentration = 'C_mobile_Cu'
        extra_vector_tags = ref
    []
    [release_Cu]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_Cu}
        variable = C_trapped_Cu
    []
    ############################## NodalKernels for CuCrZr (block = 2)
    [time_CuCrZr]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_CuCrZr
    []
    [trapping_CuCrZr]
        type = TrappingNodalKernel
        variable = C_trapped_CuCrZr
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_CuCrZr}
        Ct0 = ${Ct0_CuCrZr}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_CuCrZr}
        mobile_concentration = 'C_mobile_CuCrZr'
        extra_vector_tags = ref
    []
    [release_CuCrZr]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_CuCrZr}
        variable = C_trapped_CuCrZr
    []
[]

[BCs]
    [C_mob_W_top_flux]
        type = FunctionNeumannBC
        variable = C_mobile_W
        boundary = 'top'
        function = mobile_flux_bc_function
    []
    [mobile_tube]
        type = DirichletBC
        variable = C_mobile_CuCrZr
        value = ${C_mobile_CuCrZr_DirichletBC_Coolant}
        boundary = '2to1'
    []
    [temperature_top]
        type = FunctionNeumannBC
        variable = temperature
        boundary = 'top'
        function = temperature_flux_bc_function
    []
    [temperature_tube]
        type = FunctionDirichletBC
        variable = temperature
        boundary = '2to1'
        function = temperature_inner_func
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_W]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_W'
        reg_props_out = 'diffusivity_W_nonAD'
        block = 4
    []
    [heat_transfer_W]
        type = GenericConstantMaterial
        prop_names = 'density_W'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ParsedMaterial
        property_name = specific_heat_W
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ParsedMaterial
        property_name = thermal_conductivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_Cu]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_Cu'
        reg_props_out = 'diffusivity_Cu_nonAD'
        block = 3
    []
    [heat_transfer_Cu]
        type = GenericConstantMaterial
        prop_names = 'density_Cu'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ParsedMaterial
        property_name = specific_heat_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ParsedMaterial
        property_name = thermal_conductivity_Cu
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_CuCrZr]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_CuCrZr'
        reg_props_out = 'diffusivity_CuCrZr_nonAD'
        block = 2
    []
    [heat_transfer_CuCrZr]
        type = GenericConstantMaterial
        prop_names = 'density_CuCrZr specific_heat_CuCrZr'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ParsedMaterial
        property_name = thermal_conductivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile_W
        concentration_secondary = C_mobile_Cu
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile_Cu
        concentration_secondary = C_mobile_CuCrZr
    []
[]

References

James Ambrosek and GR Longhurst. Verification and Validation of TMAP7. Technical Report INEEL/EXT-04-01657, Idaho National Engineering and Environmental Laboratory, December 2008.

@techreport{ambrosek2008verification,
    author = "Ambrosek, James and Longhurst, GR",
    title = "{Verification and Validation of TMAP7}",
    year = "2008",
    number = "INEEL/EXT-04-01657",
    month = "December",
    institution = "Idaho National Engineering and Environmental Laboratory"
}

L. Charlot, E. Shemon, G. Giudicelli, P. Behne, B. Spencer, Z. M. Prince, Y. Miao, Y. Cao, S. Kyu Lee, N. E. Stauff, T. Lange, O. Calvin, K. Swanson, D. Yankura, P.-C. A. Simon, M. Shimada, and T. Folk. Update on new models available on the Virtual Test Bed (VTB) in 2025. In Transactions of the American Nuclear Society. June 2025. doi:10.13182/T133-49163.

@inproceedings{Charlot2025VTB,
    author = "Charlot, L. and Shemon, E. and Giudicelli, G. and Behne, P. and Spencer, B. and Prince, Z. M. and Miao, Y. and Cao, Y. and Lee, S. Kyu and Stauff, N. E. and Lange, T. and Calvin, O. and Swanson, K. and Yankura, D. and Simon, P.-C. A. and Shimada, M. and Folk, T.",
    title = "Update on New Models Available on the {Virtual Test Bed (VTB)} in 2025",
    booktitle = "Transactions of the American Nuclear Society",
    year = "2025",
    month = "June",
    doi = "10.13182/T133-49163"
}

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.

@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"
}

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.

@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"
}

(tmap8/test/tests/divertor_monoblock/divertor_monoblock.i)

### This input file is the complete input file for the divertor monoblock case.
### This case was published in:
### 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
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_CuCrZr_DirichletBC_Coolant = 1.0e-18 # at.fraction
C_mobile_W_init = 1.0e-20 # at.fraction
C_mobile_Cu_init = 5.0e-17 # at.fraction
C_mobile_CuCrZr_init = 1.0e-15 # at.fraction

# include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_multi_variable_base.i

[Problem]
    type = ReferenceResidualProblem
    extra_tag_vectors = 'ref'
    reference_vector = 'ref'
[]

[Variables]
    [temperature]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${temperature_initial}
    []
    ######################### Variables for W (block = 4)
    [C_mobile_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_W_init}
        block = 4
    []
    [C_trapped_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 4
    []
    ######################### Variables for Cu (block = 3)
    [C_mobile_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_Cu_init}
        block = 3
    []
    [C_trapped_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 3
    []
    ######################### Variables for CuCrZr (block = 2)
    [C_mobile_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_CuCrZr_init}
        block = 2
    []
    [C_trapped_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 2
    []
[]

[Kernels]
    ############################## Kernels for W (block = 4)
    [diff_W]
        type = ADMatDiffusion
        variable = C_mobile_W
        diffusivity = diffusivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_diff_W]
        type = ADTimeDerivative
        variable = C_mobile_W
        block = 4
        extra_vector_tags = ref
    []
    [coupled_time_W]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_W
        v = C_trapped_W
        block = 4
        extra_vector_tags = ref
    []
    [heat_conduction_W]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_heat_conduction_W]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_W
        density = density_W
        block = 4
        extra_vector_tags = ref
    []
    ############################## Kernels for Cu (block = 3)
    [diff_Cu]
        type = ADMatDiffusion
        variable = C_mobile_Cu
        diffusivity = diffusivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_diff_Cu]
        type = ADTimeDerivative
        variable = C_mobile_Cu
        block = 3
        extra_vector_tags = ref
    []
    [coupled_time_Cu]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_Cu
        v = C_trapped_Cu
        block = 3
        extra_vector_tags = ref
    []
    [heat_conduction_Cu]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_heat_conduction_Cu]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_Cu
        density = density_Cu
        block = 3
        extra_vector_tags = ref
    []
    ############################## Kernels for CuCrZr (block = 2)
    [diff_CuCrZr]
        type = ADMatDiffusion
        variable = C_mobile_CuCrZr
        diffusivity = diffusivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_diff_CuCrZr]
        type = ADTimeDerivative
        variable = C_mobile_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [coupled_time_CuCrZr]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_CuCrZr
        v = C_trapped_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [heat_conduction_CuCrZr]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_heat_conduction_CuCrZr]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_CuCrZr
        density = density_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
[]

[InterfaceKernels]
    [tied_4to3]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_W
        neighbor_var = C_mobile_Cu
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_4to3
        jump_prop_name = solubility_ratio
        boundary = '4to3'
    []
    [tied_3to2]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_Cu
        neighbor_var = C_mobile_CuCrZr
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_3to2
        jump_prop_name = solubility_ratio
        boundary = '3to2'
    []
[]

[NodalKernels]
    ############################## NodalKernels for W (block = 4)
    [time_W]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_W
    []
    [trapping_W]
        type = TrappingNodalKernel
        variable = C_trapped_W
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_W}
        Ct0 = ${Ct0_W}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_W}
        mobile_concentration = 'C_mobile_W'
        extra_vector_tags = ref
    []
    [release_W]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_W}
        variable = C_trapped_W
    []
    ############################## NodalKernels for Cu (block = 3)
    [time_Cu]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_Cu
    []
    [trapping_Cu]
        type = TrappingNodalKernel
        variable = C_trapped_Cu
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_Cu}
        Ct0 = ${Ct0_Cu}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_Cu}
        mobile_concentration = 'C_mobile_Cu'
        extra_vector_tags = ref
    []
    [release_Cu]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_Cu}
        variable = C_trapped_Cu
    []
    ############################## NodalKernels for CuCrZr (block = 2)
    [time_CuCrZr]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_CuCrZr
    []
    [trapping_CuCrZr]
        type = TrappingNodalKernel
        variable = C_trapped_CuCrZr
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_CuCrZr}
        Ct0 = ${Ct0_CuCrZr}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_CuCrZr}
        mobile_concentration = 'C_mobile_CuCrZr'
        extra_vector_tags = ref
    []
    [release_CuCrZr]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_CuCrZr}
        variable = C_trapped_CuCrZr
    []
[]

[BCs]
    [C_mob_W_top_flux]
        type = FunctionNeumannBC
        variable = C_mobile_W
        boundary = 'top'
        function = mobile_flux_bc_function
    []
    [mobile_tube]
        type = DirichletBC
        variable = C_mobile_CuCrZr
        value = ${C_mobile_CuCrZr_DirichletBC_Coolant}
        boundary = '2to1'
    []
    [temperature_top]
        type = FunctionNeumannBC
        variable = temperature
        boundary = 'top'
        function = temperature_flux_bc_function
    []
    [temperature_tube]
        type = FunctionDirichletBC
        variable = temperature
        boundary = '2to1'
        function = temperature_inner_func
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_W]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_W'
        reg_props_out = 'diffusivity_W_nonAD'
        block = 4
    []
    [heat_transfer_W]
        type = GenericConstantMaterial
        prop_names = 'density_W'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ParsedMaterial
        property_name = specific_heat_W
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ParsedMaterial
        property_name = thermal_conductivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_Cu]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_Cu'
        reg_props_out = 'diffusivity_Cu_nonAD'
        block = 3
    []
    [heat_transfer_Cu]
        type = GenericConstantMaterial
        prop_names = 'density_Cu'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ParsedMaterial
        property_name = specific_heat_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ParsedMaterial
        property_name = thermal_conductivity_Cu
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_CuCrZr]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_CuCrZr'
        reg_props_out = 'diffusivity_CuCrZr_nonAD'
        block = 2
    []
    [heat_transfer_CuCrZr]
        type = GenericConstantMaterial
        prop_names = 'density_CuCrZr specific_heat_CuCrZr'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ParsedMaterial
        property_name = thermal_conductivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile_W
        concentration_secondary = C_mobile_Cu
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile_Cu
        concentration_secondary = C_mobile_CuCrZr
    []
[]

(tmap8/test/tests/divertor_monoblock/tests)

[Tests]
  issues = '#144 #359 #333'
  design = 'divertor_monoblock/index.md'
  [Shimada2024_input_check]
    type = RunApp
    input = 'divertor_monoblock.i'
    requirement = 'The system shall maintain a working input file to model heat and tritium transport in a divertor monoblock during pulsed operation.'
    check_input = True
    capabilities = 'method=opt'
  []
  [Shimada2024_run]
    type = Exodiff
    input = 'divertor_monoblock.i'
    exodiff = 'divertor_monoblock_exodus.e'
    cli_args = 'Outputs/exodus/sync_only=true
                Executioner/end_time=1600
                rings_CuCrZr=10
                rings_Cu=8
                rings_W=40
                Executioner/nl_rel_tol=1e-9
                Executioner/nl_abs_tol=1e-11'
    requirement = 'The system shall model heat and tritium transport in a divertor monoblock during pulsed operation.'
    rel_err = 8e-4 # increasing the relative error tolerance because the test is sensitive to the environment. This error is still small for physical problems.
    heavy = true
    min_parallel = 2
    max_time = 1000
  []
  [Shimada2024_run-physics]
    type = Exodiff
    input = 'divertor_monoblock_physics.i'
    exodiff = 'divertor_monoblock_physics_exodus.e'
    cli_args = 'Outputs/exodus/sync_only=true
                Executioner/end_time=1600
                rings_CuCrZr=10
                rings_Cu=8
                rings_W=40
                Executioner/nl_rel_tol=1e-9
                Executioner/nl_abs_tol=1e-11'
    requirement = 'The system shall model heat and tritium transport in a divertor monoblock during pulsed operation using a shorthand physics syntax'
    rel_err = 8e-4 # increasing the relative error tolerance because the test is sensitive to the environment. This error is still small for physical problems.
    heavy = true
    min_parallel = 2
    max_time = 1000
    custom_cmp = 'physics_comp.cmp'
  []
  [Shimada2024_run-physics_single-variable]
    type = CSVDiff
    input = 'divertor_monoblock_physics-single-variable.i'
    csvdiff = 'divertor_monoblock_physics-single-variable_out.csv'
    cli_args = 'Outputs/exodus/sync_only=true
                Executioner/end_time=100
                rings_CuCrZr=10
                rings_Cu=8
                rings_W=40
                Executioner/nl_rel_tol=1e-9
                Executioner/nl_abs_tol=1e-11'
    requirement = 'The system shall model heat and tritium transport in a divertor monoblock during pulsed operation using a shorthand physics syntax and using a single variable for tracking the tritium concentration.'
    rel_err = 8e-4 # increasing the relative error tolerance because the test is sensitive to the environment. This error is still small for physical problems.
    abs_zero = 3e-9
    heavy = true
    min_parallel = 2
    max_time = 1000
  []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock.i)

### This input file is the complete input file for the divertor monoblock case.
### This case was published in:
### 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
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_CuCrZr_DirichletBC_Coolant = 1.0e-18 # at.fraction
C_mobile_W_init = 1.0e-20 # at.fraction
C_mobile_Cu_init = 5.0e-17 # at.fraction
C_mobile_CuCrZr_init = 1.0e-15 # at.fraction

# include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_multi_variable_base.i

[Problem]
    type = ReferenceResidualProblem
    extra_tag_vectors = 'ref'
    reference_vector = 'ref'
[]

[Variables]
    [temperature]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${temperature_initial}
    []
    ######################### Variables for W (block = 4)
    [C_mobile_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_W_init}
        block = 4
    []
    [C_trapped_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 4
    []
    ######################### Variables for Cu (block = 3)
    [C_mobile_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_Cu_init}
        block = 3
    []
    [C_trapped_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 3
    []
    ######################### Variables for CuCrZr (block = 2)
    [C_mobile_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_CuCrZr_init}
        block = 2
    []
    [C_trapped_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 2
    []
[]

[Kernels]
    ############################## Kernels for W (block = 4)
    [diff_W]
        type = ADMatDiffusion
        variable = C_mobile_W
        diffusivity = diffusivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_diff_W]
        type = ADTimeDerivative
        variable = C_mobile_W
        block = 4
        extra_vector_tags = ref
    []
    [coupled_time_W]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_W
        v = C_trapped_W
        block = 4
        extra_vector_tags = ref
    []
    [heat_conduction_W]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_heat_conduction_W]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_W
        density = density_W
        block = 4
        extra_vector_tags = ref
    []
    ############################## Kernels for Cu (block = 3)
    [diff_Cu]
        type = ADMatDiffusion
        variable = C_mobile_Cu
        diffusivity = diffusivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_diff_Cu]
        type = ADTimeDerivative
        variable = C_mobile_Cu
        block = 3
        extra_vector_tags = ref
    []
    [coupled_time_Cu]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_Cu
        v = C_trapped_Cu
        block = 3
        extra_vector_tags = ref
    []
    [heat_conduction_Cu]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_heat_conduction_Cu]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_Cu
        density = density_Cu
        block = 3
        extra_vector_tags = ref
    []
    ############################## Kernels for CuCrZr (block = 2)
    [diff_CuCrZr]
        type = ADMatDiffusion
        variable = C_mobile_CuCrZr
        diffusivity = diffusivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_diff_CuCrZr]
        type = ADTimeDerivative
        variable = C_mobile_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [coupled_time_CuCrZr]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_CuCrZr
        v = C_trapped_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [heat_conduction_CuCrZr]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_heat_conduction_CuCrZr]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_CuCrZr
        density = density_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
[]

[InterfaceKernels]
    [tied_4to3]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_W
        neighbor_var = C_mobile_Cu
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_4to3
        jump_prop_name = solubility_ratio
        boundary = '4to3'
    []
    [tied_3to2]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_Cu
        neighbor_var = C_mobile_CuCrZr
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_3to2
        jump_prop_name = solubility_ratio
        boundary = '3to2'
    []
[]

[NodalKernels]
    ############################## NodalKernels for W (block = 4)
    [time_W]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_W
    []
    [trapping_W]
        type = TrappingNodalKernel
        variable = C_trapped_W
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_W}
        Ct0 = ${Ct0_W}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_W}
        mobile_concentration = 'C_mobile_W'
        extra_vector_tags = ref
    []
    [release_W]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_W}
        variable = C_trapped_W
    []
    ############################## NodalKernels for Cu (block = 3)
    [time_Cu]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_Cu
    []
    [trapping_Cu]
        type = TrappingNodalKernel
        variable = C_trapped_Cu
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_Cu}
        Ct0 = ${Ct0_Cu}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_Cu}
        mobile_concentration = 'C_mobile_Cu'
        extra_vector_tags = ref
    []
    [release_Cu]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_Cu}
        variable = C_trapped_Cu
    []
    ############################## NodalKernels for CuCrZr (block = 2)
    [time_CuCrZr]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_CuCrZr
    []
    [trapping_CuCrZr]
        type = TrappingNodalKernel
        variable = C_trapped_CuCrZr
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_CuCrZr}
        Ct0 = ${Ct0_CuCrZr}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_CuCrZr}
        mobile_concentration = 'C_mobile_CuCrZr'
        extra_vector_tags = ref
    []
    [release_CuCrZr]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_CuCrZr}
        variable = C_trapped_CuCrZr
    []
[]

[BCs]
    [C_mob_W_top_flux]
        type = FunctionNeumannBC
        variable = C_mobile_W
        boundary = 'top'
        function = mobile_flux_bc_function
    []
    [mobile_tube]
        type = DirichletBC
        variable = C_mobile_CuCrZr
        value = ${C_mobile_CuCrZr_DirichletBC_Coolant}
        boundary = '2to1'
    []
    [temperature_top]
        type = FunctionNeumannBC
        variable = temperature
        boundary = 'top'
        function = temperature_flux_bc_function
    []
    [temperature_tube]
        type = FunctionDirichletBC
        variable = temperature
        boundary = '2to1'
        function = temperature_inner_func
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_W]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_W'
        reg_props_out = 'diffusivity_W_nonAD'
        block = 4
    []
    [heat_transfer_W]
        type = GenericConstantMaterial
        prop_names = 'density_W'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ParsedMaterial
        property_name = specific_heat_W
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ParsedMaterial
        property_name = thermal_conductivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_Cu]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_Cu'
        reg_props_out = 'diffusivity_Cu_nonAD'
        block = 3
    []
    [heat_transfer_Cu]
        type = GenericConstantMaterial
        prop_names = 'density_Cu'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ParsedMaterial
        property_name = specific_heat_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ParsedMaterial
        property_name = thermal_conductivity_Cu
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_CuCrZr]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_CuCrZr'
        reg_props_out = 'diffusivity_CuCrZr_nonAD'
        block = 2
    []
    [heat_transfer_CuCrZr]
        type = GenericConstantMaterial
        prop_names = 'density_CuCrZr specific_heat_CuCrZr'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ParsedMaterial
        property_name = thermal_conductivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile_W
        concentration_secondary = C_mobile_Cu
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile_Cu
        concentration_secondary = C_mobile_CuCrZr
    []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics.i)

### This input file is the complete input file for the divertor monoblock case using the physics
### syntax with several variable.
### 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
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_CuCrZr_DirichletBC_Coolant = 1.0e-18 # at.fraction
C_mobile_W_init = 1.0e-20  # at.fraction
C_mobile_Cu_init = 5.0e-17  # at.fraction
C_mobile_CuCrZr_init = 1.0e-15  # at.fraction

# include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_multi_variable_base.i

[GlobalParams]
    species_scaling_factors = '1'
    # The heat conduction physics preconditioning does not apply well across the other physics
    preconditioning = 'defer'
[]

[Physics]
    [HeatConduction]
        [all]
            temperature_name = 'temperature'
            initial_temperature = ${temperature_initial}
            # if using AD, increase the size of the factorization space in petsc options
            # using -mat_mumps_icntl_14 300 or use superlu_dist over mumps
            use_automatic_differentiation = false

            thermal_conductivity = 'thermal_conductivity'
            specific_heat = 'specific_heat'

            heat_flux_boundaries = 'top'
            boundary_heat_fluxes = 'temperature_flux_bc_function'
            fixed_temperature_boundaries = '2to1'
            boundary_temperatures = 'temperature_inner_func'
        []
    []
    [Diffusion]
        [W]
            variable_name = 'C_mobile_W'
            block = '4'
            diffusivity_matprop = diffusivity_W
            initial_condition = ${C_mobile_W_init}

            neumann_boundaries = 'top'
            boundary_fluxes = 'mobile_flux_bc_function'
        []
        [Cu]
            variable_name = 'C_mobile_Cu'
            block = '3'
            diffusivity_matprop = diffusivity_Cu
            initial_condition = ${C_mobile_Cu_init}
        []
        [CuCrZr]
            variable_name = 'C_mobile_CuCrZr'
            block = '2'
            diffusivity_matprop = diffusivity_CuCrZr
            initial_condition = ${C_mobile_CuCrZr_init}

            dirichlet_boundaries ='2to1'
            boundary_values = '${C_mobile_CuCrZr_DirichletBC_Coolant}'
        []
    []
    [SpeciesTrapping]
        [W]
            species = 'C_trapped_W'
            mobile = 'C_mobile_W'
            block = '4'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false

            temperature = temperature
            alpha_t = ${alpha_t}
            N = ${N_W}
            Ct0 = ${Ct0_W}
            trap_per_free = ${trap_per_free_W}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_W}
        []
        [Cu]
            species = 'C_trapped_Cu'
            mobile = 'C_mobile_Cu'
            block = '3'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_Cu}
            Ct0 = ${Ct0_Cu}
            trap_per_free = ${trap_per_free_Cu}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_Cu}
        []
        [CuCrZr]
            species = 'C_trapped_CuCrZr'
            mobile = 'C_mobile_CuCrZr'
            block = '2'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_CuCrZr}
            Ct0 = ${Ct0_CuCrZr}
            trap_per_free = ${trap_per_free_CuCrZr}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_CuCrZr}
        []
    []
[]

[InterfaceKernels]
    [tied_4to3]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_W
        neighbor_var = C_mobile_Cu
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_4to3
        jump_prop_name = solubility_ratio
        boundary = '4to3'
    []
    [tied_3to2]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_Cu
        neighbor_var = C_mobile_CuCrZr
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_3to2
        jump_prop_name = solubility_ratio
        boundary = '3to2'
    []
[]


[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_W]
        type = GenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_Cu]
        type = GenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_CuCrZr]
        type = GenericConstantMaterial
        prop_names = 'density specific_heat'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile_W
        concentration_secondary = C_mobile_Cu
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile_Cu
        concentration_secondary = C_mobile_CuCrZr
    []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics-single-variable.i)

### This input file is the complete input file for the divertor monoblock case using the physics
### syntax with a single variable across materials.
### This input uses the `!include` feature to incorporate other input files

### Nomenclatures
### C_mobile        mobile H concentration
### C_trapped       trapped H concentration
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_init = 1.0e-20 # at.fraction

# Include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_single_variable_base.i

[GlobalParams]
    species_scaling_factors = '1'
    # The heat conduction physics preconditioning does not apply well across the other physics
    preconditioning = 'defer'
[]

# Define the variable outside of the Physics to prevent the Physics from defining it
# with a block restriction
[Variables]
    [C_trapped]
    []
[]

[Physics]
    [HeatConduction]
        [all]
            temperature_name = 'temperature'
            initial_temperature = ${temperature_initial}

            thermal_conductivity = 'thermal_conductivity'
            specific_heat = 'specific_heat'

            heat_flux_boundaries = 'top'
            boundary_heat_fluxes = 'temperature_flux_bc_function'
            fixed_temperature_boundaries = '2to1'
            boundary_temperatures = 'temperature_inner_func'
        []
    []
    [Diffusion]
        [all]
            variable_name = 'C_mobile'
            diffusivity_matprop = diffusivity
            initial_condition = ${C_mobile_init}

            neumann_boundaries = 'top'
            boundary_fluxes = 'mobile_flux_bc_function'
        []
    []
    [SpeciesTrapping]
        [all]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '4'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false

            temperature = temperature
            alpha_t = ${alpha_t}
            N = ${N_W}
            Ct0 = ${Ct0_W}
            trap_per_free = ${trap_per_free_W}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_W}
        []
        [Cu]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '3'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_Cu}
            Ct0 = ${Ct0_Cu}
            trap_per_free = ${trap_per_free_Cu}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_Cu}
        []
        [CuCrZr]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '2'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_CuCrZr}
            Ct0 = ${Ct0_CuCrZr}
            trap_per_free = ${trap_per_free_CuCrZr}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_CuCrZr}
        []
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_W]
        type = ADGenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ADParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_Cu]
        type = ADGenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ADParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_CuCrZr]
        type = ADGenericConstantMaterial
        prop_names = 'density specific_heat'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile
        concentration_secondary = C_mobile
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile
        concentration_secondary = C_mobile
    []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_common_base.i)

# This input file contains key pieces for the divertor monoblock case.
# It cannot be run on its own and is included in the main input files for all divertor monoblock
# cases, namely:
# - divertor_monoblock.i
# - divertor_monoblock_physics.i
# - divertor_monoblock_physics-single-variable.i

# Geometry and design
radius_coolant = ${units 6.0 mm -> m}
radius_CuCrZr = ${units 7.5 mm -> m}
radius_Cu = ${units 8.5 mm -> m}
block_size = ${units 28 mm -> m}
num_sectors = 36 # (-) defines mesh size
rings_H2O = 1 # (-)
rings_CuCrZr = 30 # (-)
rings_Cu = 20 # (-)
rings_W = 110 # (-)

# Operation conditions
plasma_ramp_time = ${units 100.0 s}
plasma_ss_duration = ${units 400.0 s}
plasma_cycle_time = ${units 1600.0 s}
plasma_ss_end = ${fparse plasma_ramp_time + plasma_ss_duration}
plasma_ramp_down_end = ${fparse plasma_ramp_time + plasma_ss_duration + plasma_ramp_time}

temperature_initial = ${units 300.0 K}
temperature_coolant_max = ${units 552.0 K}

plasma_max_heat = ${units 1.0e7 W/m^2} # Heat flux of 10 MW/m^2 at steady state
plasma_min_heat = ${units 0.0 W/m^2} # no flux while the pulse is off.

# Maximum mobile flux of 7.90e-13 at the top surface (1.0e-4 [m])
# 10e24 at/m^3/s plasma flux at steady state
# 50% of it corresponds to tritium in a DT plasma
# Assuming 0.1% of incident plasma is retained in the divertor, this results in a flux of
# 5e20 at/m^3/s for tritium
# This is then normalized by the tungsten_atomic_density and leads to a flux of ~7.90e-9 m/s
# This is then distributed in the first mesh layer, which has a thickness of 1e-4 m
plasma_max_flux = ${units 7.90e-13 1/s} # at.fraction / s
plasma_min_flux = ${units 0.0 1/s}

# Initial conditions
C_trapping_init = 1.0e-15 # at.fraction

# Materials properties - W
tungsten_atomic_density = ${units 6.338e28 atoms/m^3}
density_W = ${units 19300 kg/m^3}
N_W = ${fparse tungsten_atomic_density / tungsten_atomic_density} # usually in atoms/m^3, but here normalized by tungsten_atomic_density
Ct0_W = 1.0e-4 # at.fraction - E.A. Hodille et al 2021 Nucl. Fusion 61 1260033, trap 2 (same as trap 1)
trap_per_free_W = 1.0e0 # (-)
alpha_t = ${units 2.75e11 s} # same for all materials
trapping_energy = ${units 0 K} # same for all materials
alpha_r = ${units 8.4e12 s} # same for all materials
detrapping_energy_W = ${units 11604.6 K} # = 1.00 eV    E.A. Hodille et al 2021 Nucl. Fusion 61 126003, trap 2
# detrapping_energy_W = ${units 9863.9 K} # = 0.85 eV    E.A. Hodille et al 2021 Nucl. Fusion 61 126003, trap 1
# H diffusivity in W
diffusivity_W_D0 = ${units 2.4e-7 m^2/s}
diffusivity_W_Ea = ${units 4525.8 K}
# H solubility in W
solubility_W_1_D0 = ${units 2.95e-5 1/Pa^0.5} # (1.87e24)/(tungsten_atomic_density) 1/m^3/Pa^(1/2) / (1/m^3) = 1/Pa^(1/2)
solubility_W_1_Ea = ${units 12069.0 K}
solubility_W_2_D0 = ${units 4.95e-8 1/Pa^0.5} # (3.14e20)/(tungsten_atomic_density) 1/m^3/Pa^(1/2) / (1/m^3) = 1/Pa^(1/2)
solubility_W_2_Ea = ${units 6614.6 K}

# Materials properties - Cu
density_Cu = ${units 8960.0 kg/m^3}
N_Cu = 1.0e0 # usually in atoms/m^3, but here normalized by material's atomic density
Ct0_Cu = 5.0e-5 # at.fraction - R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038, trap 3
trap_per_free_Cu = 1.0e0 # (-)
detrapping_energy_Cu = ${units 5802.3 K} # = 0.50eV  R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038, trap 3
# H diffusivity in Cu
diffusivity_Cu_D0 = ${units 6.60e-7 m^2/s}
diffusivity_Cu_Ea = ${units 4525.8 K}
# H solubility in Cu
solubility_Cu_D0 = ${units 4.95e-5 1/Pa^0.5} # ${fparse 3.14e24 / tungsten_atomic_density} 1/m^3/Pa^(1/2) / (1/m^3) = 1/Pa^(1/2)
solubility_Cu_Ea = ${units 6614.6 K}

# Materials properties - CuCrZr
density_CuCrZr = ${units 8900.0 kg/m^3}
specific_heat_CuCrZr = ${units 390.0 J/kg/K}
N_CuCrZr = 1.0e0 # usually in atoms/m^3, but here normalized by material's atomic density
Ct0_CuCrZr = 5.0e-5 # at.fraction - R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038, trap 4
# Ct0_CuCrZr = 4.0e-2 # at.fraction - R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038, trap 5
trap_per_free_CuCrZr = 1.0e0 # (-)
detrapping_energy_CuCrZr = ${units 5802.3 K} # = 0.50eV  R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038, trap 4
# detrapping_energy_CuCrZr = ${units 9631.8 K} # = 0.83 eV  R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038, trap 5
# H diffusivity in CuCrZr
diffusivity_CuCrZr_D0 = ${units 3.90e-7 m^2/s}
diffusivity_CuCrZr_Ea = ${units 4873.9 K}
# H solubility in CuCrZr
solubility_CuCrZr_D0 = ${units 6.75e-6 1/Pa^0.5} # ${fparse 4.28e23 / tungsten_atomic_density} 1/m^3/Pa^(1/2) / (1/m^3) = 1/Pa^(1/2)
solubility_CuCrZr_Ea = ${units 4525.8 K}

# For postprocessor scaling
diffusivity_fixed = ${units 5.01e-24 g/m^2} # (3.01604928)/(6.02e23)/[gram(T)/m^2]
# diffusivity_fixed = ${units 5.508e-19 g/m^2}  # (1.0e3)*(1.0e3)/(6.02e23)/(3.01604928) [gram(T)/m^2] alternative
scaling_factor = ${units 3.491e10 g/m^2} # (1.0e3)*(1.0e3)*(${tungsten_atomic_density})/(6.02e23)/(3.01604928) [gram(T)/m^2]
scaling_factor_2 = ${units 3.44e10 g/m^2} # (1.0e3)*(1.0e3)*(${tungsten_atomic_density})/(6.02e23)/(3.01604928) [gram(T)/m^2]

[Functions]
    [t_in_cycle]
        type = ParsedFunction
        expression = 't % ${plasma_cycle_time}'
    []
    # pulse between 0 and 1 following the plasma operation
    [pulse_time_function]
        type = ParsedFunction
        symbol_values = 't_in_cycle'
        symbol_names = 't_in_cycle'
        expression =   'if(t_in_cycle < ${plasma_ramp_time}, t_in_cycle/${plasma_ramp_time},
                        if(t_in_cycle < ${plasma_ss_end}, 1,
                        if(t_in_cycle < ${plasma_ramp_down_end}, 1 - (t_in_cycle-${plasma_ss_end})/${plasma_ramp_time}, 0.0)))'
    []
    [mobile_flux_bc_function]
        type = ParsedFunction
        symbol_values = 'pulse_time_function'
        symbol_names = 'pulse_time_function'
        expression = '(${plasma_max_flux} - ${plasma_min_flux}) * pulse_time_function + ${plasma_min_flux}'
    []
    [temperature_flux_bc_function]
        type = ParsedFunction
        symbol_values = 'pulse_time_function'
        symbol_names = 'pulse_time_function'
        expression = '(${plasma_max_heat} - ${plasma_min_heat}) * pulse_time_function + ${plasma_min_heat}'
    []
    [temperature_inner_func]
        type = ParsedFunction
        symbol_values = 'pulse_time_function'
        symbol_names = 'pulse_time_function'
        expression = '(${temperature_coolant_max} - ${temperature_initial}) * pulse_time_function + ${temperature_initial}'
    []
    [timestep_function]
        type = ParsedFunction
        symbol_values = 't_in_cycle'
        symbol_names = 't_in_cycle'
        expression = 'if(t_in_cycle < ${fparse 0.1 * plasma_ramp_time}   ,  20,
                      if(t_in_cycle < ${fparse 0.9 * plasma_ramp_time}   ,  40,
                      if(t_in_cycle < ${fparse 1.1 * plasma_ramp_time}   ,  20,
                      if(t_in_cycle < ${fparse plasma_ss_end - 20}       ,  40,
                      if(t_in_cycle < ${plasma_ss_end}                   ,  20,
                      if(t_in_cycle < ${fparse plasma_ramp_down_end - 10},   4,
                      if(t_in_cycle < ${fparse plasma_ramp_down_end + 10},  20,
                      if(t_in_cycle < ${fparse plasma_cycle_time - 100}  , 200,
                      if(t_in_cycle < ${plasma_cycle_time}               ,  40,  2)))))))))'
    []
[]

[AuxVariables]
  [flux_y]
      order = FIRST
      family = MONOMIAL
  []
  ############################## AuxVariables for W (block = 4)
  [Sc_C_mobile_W]
      block = 4
  []
  [Sc_C_trapped_W]
      block = 4
  []
  [C_total_W]
      block = 4
  []
  [Sc_C_total_W]
      block = 4
  []
  [S_empty_W]
      block = 4
  []
  [Sc_S_empty_W]
      block = 4
  []
  [S_trapped_W]
      block = 4
  []
  [Sc_S_trapped_W]
      block = 4
  []
  [S_total_W]
      block = 4
  []
  [Sc_S_total_W]
      block = 4
  []
  ############################## AuxVariables for Cu (block = 3)
  [Sc_C_mobile_Cu]
      block = 3
  []
  [Sc_C_trapped_Cu]
      block = 3
  []
  [C_total_Cu]
      block = 3
  []
  [Sc_C_total_Cu]
      block = 3
  []
  [S_empty_Cu]
      block = 3
  []
  [Sc_S_empty_Cu]
      block = 3
  []
  [S_trapped_Cu]
      block = 3
  []
  [Sc_S_trapped_Cu]
      block = 3
  []
  [S_total_Cu]
      block = 3
  []
  [Sc_S_total_Cu]
      block = 3
  []
  ############################## AuxVariables for CuCrZr (block = 2)
  [Sc_C_mobile_CuCrZr]
      block = 2
  []
  [Sc_C_trapped_CuCrZr]
      block = 2
  []
  [C_total_CuCrZr]
      block = 2
  []
  [Sc_C_total_CuCrZr]
      block = 2
  []
  [S_empty_CuCrZr]
      block = 2
  []
  [Sc_S_empty_CuCrZr]
      block = 2
  []
  [S_trapped_CuCrZr]
      block = 2
  []
  [Sc_S_trapped_CuCrZr]
      block = 2
  []
  [S_total_CuCrZr]
      block = 2
  []
  [Sc_S_total_CuCrZr]
      block = 2
  []
[]

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

[Executioner]
    type = Transient
    scheme = bdf2
    solve_type = NEWTON
    petsc_options_iname = '-pc_type -pc_factor_shift_type'
    petsc_options_value = 'lu NONZERO'
    nl_rel_tol  = 1e-6 # 1e-8 works for 1 cycle
    nl_abs_tol  = 1e-7 # 1e-11 works for 1 cycle
    end_time = ${fparse 50 * plasma_cycle_time} # 50 ITER shots
    automatic_scaling = true
    line_search = 'none'
    dtmin = 1e-4
    nl_max_its = 18
    [TimeStepper]
        type = IterationAdaptiveDT
        dt = 20
        optimal_iterations = 15
        iteration_window = 1
        growth_factor = 1.2
        cutback_factor = 0.8
        timestep_limiting_postprocessor = timestep_max_pp
    []
[]

[Outputs]
    [exodus]
        type = Exodus
        sync_only = false
        # Output at key moments in the first two cycles, and then at the end of the simulation
        sync_times = '${fparse 1.1 * plasma_ramp_time} ${fparse plasma_ss_end - 20} ${fparse plasma_ramp_down_end - 10} ${plasma_cycle_time} ${fparse plasma_cycle_time + 1.1 * plasma_ramp_time} ${fparse plasma_cycle_time + plasma_ss_end - 20} ${fparse plasma_cycle_time + plasma_ramp_down_end - 10} ${fparse 2 * plasma_cycle_time} ${fparse 50 * plasma_cycle_time}'
    []
    csv = true
    hide = 'dt
            Int_C_mobile_W Int_C_trapped_W Int_C_total_W
            Int_C_mobile_Cu Int_C_trapped_Cu Int_C_total_Cu
            Int_C_mobile_CuCrZr Int_C_trapped_CuCrZr Int_C_total_CuCrZr'
    perf_graph = true
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_mesh_base.i)

# This input file contains the geometry and mesh for the divertor monoblock case.
# It creates the geometry and mesh based on input parameters and generates the relevant interfaces between materials.
# It cannot be run on its own and is included in the main input file for this case, namely:
# - divertor_monoblock.i
# - divertor_monoblock_physics.i
# - divertor_monoblock_physics-single-variable.i

[Mesh]
  [ccmg]
      type = ConcentricCircleMeshGenerator
      num_sectors = ${num_sectors}
      rings = '${rings_H2O} ${rings_CuCrZr} ${rings_Cu} ${rings_W}'
      radii = '${radius_coolant} ${radius_CuCrZr} ${radius_Cu}'
      has_outer_square = on
      pitch = ${fparse block_size}
      portion = left_half
      preserve_volumes = false
      smoothing_max_it = 3
  []
  [ssbsg1]
      type = SideSetsBetweenSubdomainsGenerator
      input = ccmg
      primary_block = '4'     # W
      paired_block = '3'      # Cu
      new_boundary = '4to3'
  []
  [ssbsg2]
      type = SideSetsBetweenSubdomainsGenerator
      input = ssbsg1
      primary_block = '3'     # Cu
      paired_block = '4'      # W
      new_boundary = '3to4'
  []
  [ssbsg3]
      type = SideSetsBetweenSubdomainsGenerator
      input = ssbsg2
      primary_block = '3'     # Cu
      paired_block = '2'      # CuCrZr
      new_boundary = '3to2'
  []
  [ssbsg4]
      type = SideSetsBetweenSubdomainsGenerator
      input = ssbsg3
      primary_block = '2'     # CuCrZr
      paired_block = '3'      # Cu
      new_boundary = '2to3'
  []
  [ssbsg5]
      type = SideSetsBetweenSubdomainsGenerator
      input = ssbsg4
      primary_block = '2'     # CuCrZr
      paired_block = '1'      # H2O
      new_boundary = '2to1'
  []
  [bdg]
      type = BlockDeletionGenerator
      input = ssbsg5
      block = '1'             # H2O
  []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_multi_variable_base.i)

# This input file contains key pieces for the divertor monoblock case that includes multiple
# variables.
# It cannot be run on its own and is included in the main input files for two cases, namely:
# - divertor_monoblock.i
# - divertor_monoblock_physics.i

[AuxKernels]
  ############################## AuxKernels for W (block = 4)
  [Scaled_mobile_W]
      variable = Sc_C_mobile_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_mobile_W
  []
  [Scaled_trapped_W]
      variable = Sc_C_trapped_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_trapped_W
  []
  [total_W]
      variable = C_total_W
      type = ParsedAux
      expression = 'C_mobile_W + C_trapped_W'
      coupled_variables = 'C_mobile_W C_trapped_W'
  []
  [Scaled_total_W]
      variable = Sc_C_total_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_total_W
  []
  [empty_sites_W]
      variable = S_empty_W
      type = EmptySitesAux
      N = ${N_W}
      Ct0 = ${Ct0_W}
      trap_per_free = ${trap_per_free_W}
      trapped_concentration_variables = C_trapped_W
  []
  [scaled_empty_W]
      variable = Sc_S_empty_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_empty_W
  []
  [trapped_sites_W]
      variable = S_trapped_W
      type = NormalizationAux
      normal_factor = 1e0
      source_variable = C_trapped_W
  []
  [scaled_trapped_W]
      variable = Sc_S_trapped_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_trapped_W
  []
  [total_sites_W]
      variable = S_total_W
      type = ParsedAux
      expression = 'S_trapped_W + S_empty_W'
      coupled_variables = 'S_trapped_W S_empty_W'
  []
  [scaled_total_W]
      variable = Sc_S_total_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_total_W
  []
  ############################## AuxKernels for Cu (block = 3)
  [Scaled_mobile_Cu]
      variable = Sc_C_mobile_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_mobile_Cu
  []
  [Scaled_trapped_Cu]
      variable = Sc_C_trapped_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_trapped_Cu
  []
  [total_Cu]
      variable = C_total_Cu
      type = ParsedAux
      expression = 'C_mobile_Cu + C_trapped_Cu'
      coupled_variables = 'C_mobile_Cu C_trapped_Cu'
  []
  [Scaled_total_Cu]
      variable = Sc_C_total_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_total_Cu
  []
  [empty_sites_Cu]
      variable = S_empty_Cu
      type = EmptySitesAux
      N = ${N_Cu}
      Ct0 = ${Ct0_Cu}
      trap_per_free = ${trap_per_free_Cu}
      trapped_concentration_variables = C_trapped_Cu
  []
  [scaled_empty_Cu]
      variable = Sc_S_empty_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_empty_Cu
  []
  [trapped_sites_Cu]
      variable = S_trapped_Cu
      type = NormalizationAux
      normal_factor = 1e0
      source_variable = C_trapped_Cu
  []
  [scaled_trapped_Cu]
      variable = Sc_S_trapped_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_trapped_Cu
  []
  [total_sites_Cu]
      variable = S_total_Cu
      type = ParsedAux
      expression = 'S_trapped_Cu + S_empty_Cu'
      coupled_variables = 'S_trapped_Cu S_empty_Cu'
  []
  [scaled_total_Cu]
      variable = Sc_S_total_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_total_Cu
  []
  ############################## AuxKernels for CuCrZr (block = 2)
  [Scaled_mobile_CuCrZr]
      variable = Sc_C_mobile_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_mobile_CuCrZr
  []
  [Scaled_trapped_CuCrZr]
      variable = Sc_C_trapped_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_trapped_CuCrZr
  []
  [total_CuCrZr]
      variable = C_total_CuCrZr
      type = ParsedAux
      expression = 'C_mobile_CuCrZr + C_trapped_CuCrZr'
      coupled_variables = 'C_mobile_CuCrZr C_trapped_CuCrZr'
  []
  [Scaled_total_CuCrZr]
      variable = Sc_C_total_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_total_CuCrZr
  []
  [empty_sites_CuCrZr]
      variable = S_empty_CuCrZr
      type = EmptySitesAux
      N = ${N_CuCrZr}
      Ct0 = ${Ct0_CuCrZr}
      trap_per_free = ${trap_per_free_CuCrZr}
      trapped_concentration_variables = C_trapped_CuCrZr
  []
  [scaled_empty_CuCrZr]
      variable = Sc_S_empty_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_empty_CuCrZr
  []
  [trapped_sites_CuCrZr]
      variable = S_trapped_CuCrZr
      type = NormalizationAux
      normal_factor = 1e0
      source_variable = C_trapped_CuCrZr
  []
  [scaled_trapped_CuCrZr]
      variable = Sc_S_trapped_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_trapped_CuCrZr
  []
  [total_sites_CuCrZr]
      variable = S_total_CuCrZr
      type = ParsedAux
      expression = 'S_trapped_CuCrZr + S_empty_CuCrZr'
      coupled_variables = 'S_trapped_CuCrZr S_empty_CuCrZr'
  []
  [scaled_total_CuCrZr]
      variable = Sc_S_total_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_total_CuCrZr
  []
  [flux_y_W]
      type = DiffusionFluxAux
      diffusivity = diffusivity_W
      variable = flux_y
      diffusion_variable = C_mobile_W
      component = y
      block = 4
  []
  [flux_y_Cu]
      type = DiffusionFluxAux
      diffusivity = diffusivity_Cu
      variable = flux_y
      diffusion_variable = C_mobile_Cu
      component = y
      block = 3
  []
  [flux_y_CuCrZr]
      type = DiffusionFluxAux
      diffusivity = diffusivity_CuCrZr
      variable = flux_y
      diffusion_variable = C_mobile_CuCrZr
      component = y
      block = 2
  []
[]


[Postprocessors]
  ############################################################ Postprocessors for W (block = 4)
  [F_recombination]
      type = SideDiffusiveFluxAverage
      boundary = 'top'
      diffusivity = ${diffusivity_fixed}
      variable = Sc_C_total_W
  []
  [F_permeation]
      type = SideDiffusiveFluxAverage
      boundary = '2to1'
      diffusivity = ${diffusivity_fixed}
      variable = Sc_C_total_CuCrZr
  []

  [Int_C_mobile_W]
      type = ElementIntegralVariablePostprocessor
      variable = C_mobile_W
      block = 4
  []
  [ScInt_C_mobile_W]
      type = ScalePostprocessor
      value =  Int_C_mobile_W
      scaling_factor = ${scaling_factor}
  []
  [Int_C_trapped_W]
      type = ElementIntegralVariablePostprocessor
      variable = C_trapped_W
      block = 4
  []
  [ScInt_C_trapped_W]
      type = ScalePostprocessor
      value = Int_C_trapped_W
      scaling_factor = ${scaling_factor}
  []
  [Int_C_total_W]
      type = ElementIntegralVariablePostprocessor
      variable = C_total_W
      block = 4
  []
  [ScInt_C_total_W]
      type = ScalePostprocessor
      value = Int_C_total_W
      scaling_factor = ${scaling_factor}
  []
  # ############################################################ Postprocessors for Cu (block = 3)
  [Int_C_mobile_Cu]
      type = ElementIntegralVariablePostprocessor
      variable = C_mobile_Cu
      block = 3
  []
  [ScInt_C_mobile_Cu]
      type = ScalePostprocessor
      value =  Int_C_mobile_Cu
      scaling_factor = ${scaling_factor}
  []
  [Int_C_trapped_Cu]
      type = ElementIntegralVariablePostprocessor
      variable = C_trapped_Cu
      block = 3
  []
  [ScInt_C_trapped_Cu]
      type = ScalePostprocessor
      value = Int_C_trapped_Cu
      scaling_factor = ${scaling_factor_2}
  []
  [Int_C_total_Cu]
      type = ElementIntegralVariablePostprocessor
      variable = C_total_Cu
      block = 3
  []
  [ScInt_C_total_Cu]
      type = ScalePostprocessor
      value = Int_C_total_Cu
      scaling_factor = ${scaling_factor}
  []
  # ############################################################ Postprocessors for CuCrZr (block = 2)
  [Int_C_mobile_CuCrZr]
      type = ElementIntegralVariablePostprocessor
      variable = C_mobile_CuCrZr
      block = 2
  []
  [ScInt_C_mobile_CuCrZr]
      type = ScalePostprocessor
      value =  Int_C_mobile_CuCrZr
      scaling_factor = ${scaling_factor}
  []
  [Int_C_trapped_CuCrZr]
      type = ElementIntegralVariablePostprocessor
      variable = C_trapped_CuCrZr
      block = 2
  []
  [ScInt_C_trapped_CuCrZr]
      type = ScalePostprocessor
      value = Int_C_trapped_CuCrZr
      scaling_factor = ${scaling_factor_2}
  []
  [Int_C_total_CuCrZr]
      type = ElementIntegralVariablePostprocessor
      variable = C_total_CuCrZr
      block = 2
  []
  [ScInt_C_total_CuCrZr]
      type = ScalePostprocessor
      value = Int_C_total_CuCrZr
      scaling_factor = ${scaling_factor}
  []
  ############################################################ Postprocessors for others
  [dt]
      type = TimestepSize
  []
  [temperature_top]
      type = PointValue
      variable = temperature
      point = '0 ${fparse block_size / 2} 0'
  []
  [temperature_tube]
      type = PointValue
      variable = temperature
      point = '0 ${radius_coolant} 0'
  []
  # Limit timestep
  [timestep_max_pp] # s
    type = FunctionValuePostprocessor
    function = timestep_function
  []
[]

[VectorPostprocessors]
  [line]
      type = LineValueSampler
      start_point = '0 ${fparse block_size / 2} 0'
      end_point = '0 ${radius_coolant} 0'
      num_points = 100
      sort_by = 'y'
      variable = 'C_total_W C_total_Cu C_total_CuCrZr C_mobile_W C_mobile_Cu C_mobile_CuCrZr C_trapped_W C_trapped_Cu C_trapped_CuCrZr flux_y temperature'
      execute_on = timestep_end
  []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock.i)

### This input file is the complete input file for the divertor monoblock case.
### This case was published in:
### 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
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_CuCrZr_DirichletBC_Coolant = 1.0e-18 # at.fraction
C_mobile_W_init = 1.0e-20 # at.fraction
C_mobile_Cu_init = 5.0e-17 # at.fraction
C_mobile_CuCrZr_init = 1.0e-15 # at.fraction

# include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_multi_variable_base.i

[Problem]
    type = ReferenceResidualProblem
    extra_tag_vectors = 'ref'
    reference_vector = 'ref'
[]

[Variables]
    [temperature]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${temperature_initial}
    []
    ######################### Variables for W (block = 4)
    [C_mobile_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_W_init}
        block = 4
    []
    [C_trapped_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 4
    []
    ######################### Variables for Cu (block = 3)
    [C_mobile_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_Cu_init}
        block = 3
    []
    [C_trapped_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 3
    []
    ######################### Variables for CuCrZr (block = 2)
    [C_mobile_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_CuCrZr_init}
        block = 2
    []
    [C_trapped_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 2
    []
[]

[Kernels]
    ############################## Kernels for W (block = 4)
    [diff_W]
        type = ADMatDiffusion
        variable = C_mobile_W
        diffusivity = diffusivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_diff_W]
        type = ADTimeDerivative
        variable = C_mobile_W
        block = 4
        extra_vector_tags = ref
    []
    [coupled_time_W]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_W
        v = C_trapped_W
        block = 4
        extra_vector_tags = ref
    []
    [heat_conduction_W]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_heat_conduction_W]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_W
        density = density_W
        block = 4
        extra_vector_tags = ref
    []
    ############################## Kernels for Cu (block = 3)
    [diff_Cu]
        type = ADMatDiffusion
        variable = C_mobile_Cu
        diffusivity = diffusivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_diff_Cu]
        type = ADTimeDerivative
        variable = C_mobile_Cu
        block = 3
        extra_vector_tags = ref
    []
    [coupled_time_Cu]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_Cu
        v = C_trapped_Cu
        block = 3
        extra_vector_tags = ref
    []
    [heat_conduction_Cu]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_heat_conduction_Cu]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_Cu
        density = density_Cu
        block = 3
        extra_vector_tags = ref
    []
    ############################## Kernels for CuCrZr (block = 2)
    [diff_CuCrZr]
        type = ADMatDiffusion
        variable = C_mobile_CuCrZr
        diffusivity = diffusivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_diff_CuCrZr]
        type = ADTimeDerivative
        variable = C_mobile_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [coupled_time_CuCrZr]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_CuCrZr
        v = C_trapped_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [heat_conduction_CuCrZr]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_heat_conduction_CuCrZr]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_CuCrZr
        density = density_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
[]

[InterfaceKernels]
    [tied_4to3]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_W
        neighbor_var = C_mobile_Cu
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_4to3
        jump_prop_name = solubility_ratio
        boundary = '4to3'
    []
    [tied_3to2]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_Cu
        neighbor_var = C_mobile_CuCrZr
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_3to2
        jump_prop_name = solubility_ratio
        boundary = '3to2'
    []
[]

[NodalKernels]
    ############################## NodalKernels for W (block = 4)
    [time_W]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_W
    []
    [trapping_W]
        type = TrappingNodalKernel
        variable = C_trapped_W
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_W}
        Ct0 = ${Ct0_W}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_W}
        mobile_concentration = 'C_mobile_W'
        extra_vector_tags = ref
    []
    [release_W]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_W}
        variable = C_trapped_W
    []
    ############################## NodalKernels for Cu (block = 3)
    [time_Cu]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_Cu
    []
    [trapping_Cu]
        type = TrappingNodalKernel
        variable = C_trapped_Cu
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_Cu}
        Ct0 = ${Ct0_Cu}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_Cu}
        mobile_concentration = 'C_mobile_Cu'
        extra_vector_tags = ref
    []
    [release_Cu]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_Cu}
        variable = C_trapped_Cu
    []
    ############################## NodalKernels for CuCrZr (block = 2)
    [time_CuCrZr]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_CuCrZr
    []
    [trapping_CuCrZr]
        type = TrappingNodalKernel
        variable = C_trapped_CuCrZr
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_CuCrZr}
        Ct0 = ${Ct0_CuCrZr}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_CuCrZr}
        mobile_concentration = 'C_mobile_CuCrZr'
        extra_vector_tags = ref
    []
    [release_CuCrZr]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_CuCrZr}
        variable = C_trapped_CuCrZr
    []
[]

[BCs]
    [C_mob_W_top_flux]
        type = FunctionNeumannBC
        variable = C_mobile_W
        boundary = 'top'
        function = mobile_flux_bc_function
    []
    [mobile_tube]
        type = DirichletBC
        variable = C_mobile_CuCrZr
        value = ${C_mobile_CuCrZr_DirichletBC_Coolant}
        boundary = '2to1'
    []
    [temperature_top]
        type = FunctionNeumannBC
        variable = temperature
        boundary = 'top'
        function = temperature_flux_bc_function
    []
    [temperature_tube]
        type = FunctionDirichletBC
        variable = temperature
        boundary = '2to1'
        function = temperature_inner_func
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_W]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_W'
        reg_props_out = 'diffusivity_W_nonAD'
        block = 4
    []
    [heat_transfer_W]
        type = GenericConstantMaterial
        prop_names = 'density_W'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ParsedMaterial
        property_name = specific_heat_W
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ParsedMaterial
        property_name = thermal_conductivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_Cu]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_Cu'
        reg_props_out = 'diffusivity_Cu_nonAD'
        block = 3
    []
    [heat_transfer_Cu]
        type = GenericConstantMaterial
        prop_names = 'density_Cu'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ParsedMaterial
        property_name = specific_heat_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ParsedMaterial
        property_name = thermal_conductivity_Cu
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_CuCrZr]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_CuCrZr'
        reg_props_out = 'diffusivity_CuCrZr_nonAD'
        block = 2
    []
    [heat_transfer_CuCrZr]
        type = GenericConstantMaterial
        prop_names = 'density_CuCrZr specific_heat_CuCrZr'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ParsedMaterial
        property_name = thermal_conductivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile_W
        concentration_secondary = C_mobile_Cu
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile_Cu
        concentration_secondary = C_mobile_CuCrZr
    []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics.i)

### This input file is the complete input file for the divertor monoblock case using the physics
### syntax with several variable.
### 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
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_CuCrZr_DirichletBC_Coolant = 1.0e-18 # at.fraction
C_mobile_W_init = 1.0e-20  # at.fraction
C_mobile_Cu_init = 5.0e-17  # at.fraction
C_mobile_CuCrZr_init = 1.0e-15  # at.fraction

# include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_multi_variable_base.i

[GlobalParams]
    species_scaling_factors = '1'
    # The heat conduction physics preconditioning does not apply well across the other physics
    preconditioning = 'defer'
[]

[Physics]
    [HeatConduction]
        [all]
            temperature_name = 'temperature'
            initial_temperature = ${temperature_initial}
            # if using AD, increase the size of the factorization space in petsc options
            # using -mat_mumps_icntl_14 300 or use superlu_dist over mumps
            use_automatic_differentiation = false

            thermal_conductivity = 'thermal_conductivity'
            specific_heat = 'specific_heat'

            heat_flux_boundaries = 'top'
            boundary_heat_fluxes = 'temperature_flux_bc_function'
            fixed_temperature_boundaries = '2to1'
            boundary_temperatures = 'temperature_inner_func'
        []
    []
    [Diffusion]
        [W]
            variable_name = 'C_mobile_W'
            block = '4'
            diffusivity_matprop = diffusivity_W
            initial_condition = ${C_mobile_W_init}

            neumann_boundaries = 'top'
            boundary_fluxes = 'mobile_flux_bc_function'
        []
        [Cu]
            variable_name = 'C_mobile_Cu'
            block = '3'
            diffusivity_matprop = diffusivity_Cu
            initial_condition = ${C_mobile_Cu_init}
        []
        [CuCrZr]
            variable_name = 'C_mobile_CuCrZr'
            block = '2'
            diffusivity_matprop = diffusivity_CuCrZr
            initial_condition = ${C_mobile_CuCrZr_init}

            dirichlet_boundaries ='2to1'
            boundary_values = '${C_mobile_CuCrZr_DirichletBC_Coolant}'
        []
    []
    [SpeciesTrapping]
        [W]
            species = 'C_trapped_W'
            mobile = 'C_mobile_W'
            block = '4'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false

            temperature = temperature
            alpha_t = ${alpha_t}
            N = ${N_W}
            Ct0 = ${Ct0_W}
            trap_per_free = ${trap_per_free_W}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_W}
        []
        [Cu]
            species = 'C_trapped_Cu'
            mobile = 'C_mobile_Cu'
            block = '3'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_Cu}
            Ct0 = ${Ct0_Cu}
            trap_per_free = ${trap_per_free_Cu}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_Cu}
        []
        [CuCrZr]
            species = 'C_trapped_CuCrZr'
            mobile = 'C_mobile_CuCrZr'
            block = '2'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_CuCrZr}
            Ct0 = ${Ct0_CuCrZr}
            trap_per_free = ${trap_per_free_CuCrZr}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_CuCrZr}
        []
    []
[]

[InterfaceKernels]
    [tied_4to3]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_W
        neighbor_var = C_mobile_Cu
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_4to3
        jump_prop_name = solubility_ratio
        boundary = '4to3'
    []
    [tied_3to2]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_Cu
        neighbor_var = C_mobile_CuCrZr
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_3to2
        jump_prop_name = solubility_ratio
        boundary = '3to2'
    []
[]


[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_W]
        type = GenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_Cu]
        type = GenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_CuCrZr]
        type = GenericConstantMaterial
        prop_names = 'density specific_heat'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile_W
        concentration_secondary = C_mobile_Cu
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile_Cu
        concentration_secondary = C_mobile_CuCrZr
    []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics-single-variable.i)

### This input file is the complete input file for the divertor monoblock case using the physics
### syntax with a single variable across materials.
### This input uses the `!include` feature to incorporate other input files

### Nomenclatures
### C_mobile        mobile H concentration
### C_trapped       trapped H concentration
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_init = 1.0e-20 # at.fraction

# Include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_single_variable_base.i

[GlobalParams]
    species_scaling_factors = '1'
    # The heat conduction physics preconditioning does not apply well across the other physics
    preconditioning = 'defer'
[]

# Define the variable outside of the Physics to prevent the Physics from defining it
# with a block restriction
[Variables]
    [C_trapped]
    []
[]

[Physics]
    [HeatConduction]
        [all]
            temperature_name = 'temperature'
            initial_temperature = ${temperature_initial}

            thermal_conductivity = 'thermal_conductivity'
            specific_heat = 'specific_heat'

            heat_flux_boundaries = 'top'
            boundary_heat_fluxes = 'temperature_flux_bc_function'
            fixed_temperature_boundaries = '2to1'
            boundary_temperatures = 'temperature_inner_func'
        []
    []
    [Diffusion]
        [all]
            variable_name = 'C_mobile'
            diffusivity_matprop = diffusivity
            initial_condition = ${C_mobile_init}

            neumann_boundaries = 'top'
            boundary_fluxes = 'mobile_flux_bc_function'
        []
    []
    [SpeciesTrapping]
        [all]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '4'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false

            temperature = temperature
            alpha_t = ${alpha_t}
            N = ${N_W}
            Ct0 = ${Ct0_W}
            trap_per_free = ${trap_per_free_W}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_W}
        []
        [Cu]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '3'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_Cu}
            Ct0 = ${Ct0_Cu}
            trap_per_free = ${trap_per_free_Cu}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_Cu}
        []
        [CuCrZr]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '2'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_CuCrZr}
            Ct0 = ${Ct0_CuCrZr}
            trap_per_free = ${trap_per_free_CuCrZr}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_CuCrZr}
        []
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_W]
        type = ADGenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ADParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_Cu]
        type = ADGenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ADParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_CuCrZr]
        type = ADGenericConstantMaterial
        prop_names = 'density specific_heat'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile
        concentration_secondary = C_mobile
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile
        concentration_secondary = C_mobile
    []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_multi_variable_base.i)

# This input file contains key pieces for the divertor monoblock case that includes multiple
# variables.
# It cannot be run on its own and is included in the main input files for two cases, namely:
# - divertor_monoblock.i
# - divertor_monoblock_physics.i

[AuxKernels]
  ############################## AuxKernels for W (block = 4)
  [Scaled_mobile_W]
      variable = Sc_C_mobile_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_mobile_W
  []
  [Scaled_trapped_W]
      variable = Sc_C_trapped_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_trapped_W
  []
  [total_W]
      variable = C_total_W
      type = ParsedAux
      expression = 'C_mobile_W + C_trapped_W'
      coupled_variables = 'C_mobile_W C_trapped_W'
  []
  [Scaled_total_W]
      variable = Sc_C_total_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_total_W
  []
  [empty_sites_W]
      variable = S_empty_W
      type = EmptySitesAux
      N = ${N_W}
      Ct0 = ${Ct0_W}
      trap_per_free = ${trap_per_free_W}
      trapped_concentration_variables = C_trapped_W
  []
  [scaled_empty_W]
      variable = Sc_S_empty_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_empty_W
  []
  [trapped_sites_W]
      variable = S_trapped_W
      type = NormalizationAux
      normal_factor = 1e0
      source_variable = C_trapped_W
  []
  [scaled_trapped_W]
      variable = Sc_S_trapped_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_trapped_W
  []
  [total_sites_W]
      variable = S_total_W
      type = ParsedAux
      expression = 'S_trapped_W + S_empty_W'
      coupled_variables = 'S_trapped_W S_empty_W'
  []
  [scaled_total_W]
      variable = Sc_S_total_W
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_total_W
  []
  ############################## AuxKernels for Cu (block = 3)
  [Scaled_mobile_Cu]
      variable = Sc_C_mobile_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_mobile_Cu
  []
  [Scaled_trapped_Cu]
      variable = Sc_C_trapped_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_trapped_Cu
  []
  [total_Cu]
      variable = C_total_Cu
      type = ParsedAux
      expression = 'C_mobile_Cu + C_trapped_Cu'
      coupled_variables = 'C_mobile_Cu C_trapped_Cu'
  []
  [Scaled_total_Cu]
      variable = Sc_C_total_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_total_Cu
  []
  [empty_sites_Cu]
      variable = S_empty_Cu
      type = EmptySitesAux
      N = ${N_Cu}
      Ct0 = ${Ct0_Cu}
      trap_per_free = ${trap_per_free_Cu}
      trapped_concentration_variables = C_trapped_Cu
  []
  [scaled_empty_Cu]
      variable = Sc_S_empty_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_empty_Cu
  []
  [trapped_sites_Cu]
      variable = S_trapped_Cu
      type = NormalizationAux
      normal_factor = 1e0
      source_variable = C_trapped_Cu
  []
  [scaled_trapped_Cu]
      variable = Sc_S_trapped_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_trapped_Cu
  []
  [total_sites_Cu]
      variable = S_total_Cu
      type = ParsedAux
      expression = 'S_trapped_Cu + S_empty_Cu'
      coupled_variables = 'S_trapped_Cu S_empty_Cu'
  []
  [scaled_total_Cu]
      variable = Sc_S_total_Cu
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_total_Cu
  []
  ############################## AuxKernels for CuCrZr (block = 2)
  [Scaled_mobile_CuCrZr]
      variable = Sc_C_mobile_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_mobile_CuCrZr
  []
  [Scaled_trapped_CuCrZr]
      variable = Sc_C_trapped_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_trapped_CuCrZr
  []
  [total_CuCrZr]
      variable = C_total_CuCrZr
      type = ParsedAux
      expression = 'C_mobile_CuCrZr + C_trapped_CuCrZr'
      coupled_variables = 'C_mobile_CuCrZr C_trapped_CuCrZr'
  []
  [Scaled_total_CuCrZr]
      variable = Sc_C_total_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = C_total_CuCrZr
  []
  [empty_sites_CuCrZr]
      variable = S_empty_CuCrZr
      type = EmptySitesAux
      N = ${N_CuCrZr}
      Ct0 = ${Ct0_CuCrZr}
      trap_per_free = ${trap_per_free_CuCrZr}
      trapped_concentration_variables = C_trapped_CuCrZr
  []
  [scaled_empty_CuCrZr]
      variable = Sc_S_empty_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_empty_CuCrZr
  []
  [trapped_sites_CuCrZr]
      variable = S_trapped_CuCrZr
      type = NormalizationAux
      normal_factor = 1e0
      source_variable = C_trapped_CuCrZr
  []
  [scaled_trapped_CuCrZr]
      variable = Sc_S_trapped_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_trapped_CuCrZr
  []
  [total_sites_CuCrZr]
      variable = S_total_CuCrZr
      type = ParsedAux
      expression = 'S_trapped_CuCrZr + S_empty_CuCrZr'
      coupled_variables = 'S_trapped_CuCrZr S_empty_CuCrZr'
  []
  [scaled_total_CuCrZr]
      variable = Sc_S_total_CuCrZr
      type = NormalizationAux
      normal_factor = ${tungsten_atomic_density}
      source_variable = S_total_CuCrZr
  []
  [flux_y_W]
      type = DiffusionFluxAux
      diffusivity = diffusivity_W
      variable = flux_y
      diffusion_variable = C_mobile_W
      component = y
      block = 4
  []
  [flux_y_Cu]
      type = DiffusionFluxAux
      diffusivity = diffusivity_Cu
      variable = flux_y
      diffusion_variable = C_mobile_Cu
      component = y
      block = 3
  []
  [flux_y_CuCrZr]
      type = DiffusionFluxAux
      diffusivity = diffusivity_CuCrZr
      variable = flux_y
      diffusion_variable = C_mobile_CuCrZr
      component = y
      block = 2
  []
[]


[Postprocessors]
  ############################################################ Postprocessors for W (block = 4)
  [F_recombination]
      type = SideDiffusiveFluxAverage
      boundary = 'top'
      diffusivity = ${diffusivity_fixed}
      variable = Sc_C_total_W
  []
  [F_permeation]
      type = SideDiffusiveFluxAverage
      boundary = '2to1'
      diffusivity = ${diffusivity_fixed}
      variable = Sc_C_total_CuCrZr
  []

  [Int_C_mobile_W]
      type = ElementIntegralVariablePostprocessor
      variable = C_mobile_W
      block = 4
  []
  [ScInt_C_mobile_W]
      type = ScalePostprocessor
      value =  Int_C_mobile_W
      scaling_factor = ${scaling_factor}
  []
  [Int_C_trapped_W]
      type = ElementIntegralVariablePostprocessor
      variable = C_trapped_W
      block = 4
  []
  [ScInt_C_trapped_W]
      type = ScalePostprocessor
      value = Int_C_trapped_W
      scaling_factor = ${scaling_factor}
  []
  [Int_C_total_W]
      type = ElementIntegralVariablePostprocessor
      variable = C_total_W
      block = 4
  []
  [ScInt_C_total_W]
      type = ScalePostprocessor
      value = Int_C_total_W
      scaling_factor = ${scaling_factor}
  []
  # ############################################################ Postprocessors for Cu (block = 3)
  [Int_C_mobile_Cu]
      type = ElementIntegralVariablePostprocessor
      variable = C_mobile_Cu
      block = 3
  []
  [ScInt_C_mobile_Cu]
      type = ScalePostprocessor
      value =  Int_C_mobile_Cu
      scaling_factor = ${scaling_factor}
  []
  [Int_C_trapped_Cu]
      type = ElementIntegralVariablePostprocessor
      variable = C_trapped_Cu
      block = 3
  []
  [ScInt_C_trapped_Cu]
      type = ScalePostprocessor
      value = Int_C_trapped_Cu
      scaling_factor = ${scaling_factor_2}
  []
  [Int_C_total_Cu]
      type = ElementIntegralVariablePostprocessor
      variable = C_total_Cu
      block = 3
  []
  [ScInt_C_total_Cu]
      type = ScalePostprocessor
      value = Int_C_total_Cu
      scaling_factor = ${scaling_factor}
  []
  # ############################################################ Postprocessors for CuCrZr (block = 2)
  [Int_C_mobile_CuCrZr]
      type = ElementIntegralVariablePostprocessor
      variable = C_mobile_CuCrZr
      block = 2
  []
  [ScInt_C_mobile_CuCrZr]
      type = ScalePostprocessor
      value =  Int_C_mobile_CuCrZr
      scaling_factor = ${scaling_factor}
  []
  [Int_C_trapped_CuCrZr]
      type = ElementIntegralVariablePostprocessor
      variable = C_trapped_CuCrZr
      block = 2
  []
  [ScInt_C_trapped_CuCrZr]
      type = ScalePostprocessor
      value = Int_C_trapped_CuCrZr
      scaling_factor = ${scaling_factor_2}
  []
  [Int_C_total_CuCrZr]
      type = ElementIntegralVariablePostprocessor
      variable = C_total_CuCrZr
      block = 2
  []
  [ScInt_C_total_CuCrZr]
      type = ScalePostprocessor
      value = Int_C_total_CuCrZr
      scaling_factor = ${scaling_factor}
  []
  ############################################################ Postprocessors for others
  [dt]
      type = TimestepSize
  []
  [temperature_top]
      type = PointValue
      variable = temperature
      point = '0 ${fparse block_size / 2} 0'
  []
  [temperature_tube]
      type = PointValue
      variable = temperature
      point = '0 ${radius_coolant} 0'
  []
  # Limit timestep
  [timestep_max_pp] # s
    type = FunctionValuePostprocessor
    function = timestep_function
  []
[]

[VectorPostprocessors]
  [line]
      type = LineValueSampler
      start_point = '0 ${fparse block_size / 2} 0'
      end_point = '0 ${radius_coolant} 0'
      num_points = 100
      sort_by = 'y'
      variable = 'C_total_W C_total_Cu C_total_CuCrZr C_mobile_W C_mobile_Cu C_mobile_CuCrZr C_trapped_W C_trapped_Cu C_trapped_CuCrZr flux_y temperature'
      execute_on = timestep_end
  []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock_physics-single-variable.i)

### This input file is the complete input file for the divertor monoblock case using the physics
### syntax with a single variable across materials.
### This input uses the `!include` feature to incorporate other input files

### Nomenclatures
### C_mobile        mobile H concentration
### C_trapped       trapped H concentration
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_init = 1.0e-20 # at.fraction

# Include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_single_variable_base.i

[GlobalParams]
    species_scaling_factors = '1'
    # The heat conduction physics preconditioning does not apply well across the other physics
    preconditioning = 'defer'
[]

# Define the variable outside of the Physics to prevent the Physics from defining it
# with a block restriction
[Variables]
    [C_trapped]
    []
[]

[Physics]
    [HeatConduction]
        [all]
            temperature_name = 'temperature'
            initial_temperature = ${temperature_initial}

            thermal_conductivity = 'thermal_conductivity'
            specific_heat = 'specific_heat'

            heat_flux_boundaries = 'top'
            boundary_heat_fluxes = 'temperature_flux_bc_function'
            fixed_temperature_boundaries = '2to1'
            boundary_temperatures = 'temperature_inner_func'
        []
    []
    [Diffusion]
        [all]
            variable_name = 'C_mobile'
            diffusivity_matprop = diffusivity
            initial_condition = ${C_mobile_init}

            neumann_boundaries = 'top'
            boundary_fluxes = 'mobile_flux_bc_function'
        []
    []
    [SpeciesTrapping]
        [all]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '4'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false

            temperature = temperature
            alpha_t = ${alpha_t}
            N = ${N_W}
            Ct0 = ${Ct0_W}
            trap_per_free = ${trap_per_free_W}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_W}
        []
        [Cu]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '3'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_Cu}
            Ct0 = ${Ct0_Cu}
            trap_per_free = ${trap_per_free_Cu}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_Cu}
        []
        [CuCrZr]
            species = 'C_trapped'
            mobile = 'C_mobile'
            block = '2'
            species_initial_concentrations = ${C_trapping_init}
            separate_variables_per_component = false
            temperature = temperature

            alpha_t = ${alpha_t}
            N = ${N_CuCrZr}
            Ct0 = ${Ct0_CuCrZr}
            trap_per_free = ${trap_per_free_CuCrZr}
            trapping_energy = ${trapping_energy}

            alpha_r = ${alpha_r}
            detrapping_energy = ${detrapping_energy_CuCrZr}
        []
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_W]
        type = ADGenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ADParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_Cu]
        type = ADGenericConstantMaterial
        prop_names = 'density'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ADParsedMaterial
        property_name = specific_heat
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [heat_transfer_CuCrZr]
        type = ADGenericConstantMaterial
        prop_names = 'density specific_heat'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ADParsedMaterial
        property_name = thermal_conductivity
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile
        concentration_secondary = C_mobile
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile
        concentration_secondary = C_mobile
    []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock.i)

### This input file is the complete input file for the divertor monoblock case.
### This case was published in:
### 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
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_CuCrZr_DirichletBC_Coolant = 1.0e-18 # at.fraction
C_mobile_W_init = 1.0e-20 # at.fraction
C_mobile_Cu_init = 5.0e-17 # at.fraction
C_mobile_CuCrZr_init = 1.0e-15 # at.fraction

# include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_multi_variable_base.i

[Problem]
    type = ReferenceResidualProblem
    extra_tag_vectors = 'ref'
    reference_vector = 'ref'
[]

[Variables]
    [temperature]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${temperature_initial}
    []
    ######################### Variables for W (block = 4)
    [C_mobile_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_W_init}
        block = 4
    []
    [C_trapped_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 4
    []
    ######################### Variables for Cu (block = 3)
    [C_mobile_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_Cu_init}
        block = 3
    []
    [C_trapped_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 3
    []
    ######################### Variables for CuCrZr (block = 2)
    [C_mobile_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_CuCrZr_init}
        block = 2
    []
    [C_trapped_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 2
    []
[]

[Kernels]
    ############################## Kernels for W (block = 4)
    [diff_W]
        type = ADMatDiffusion
        variable = C_mobile_W
        diffusivity = diffusivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_diff_W]
        type = ADTimeDerivative
        variable = C_mobile_W
        block = 4
        extra_vector_tags = ref
    []
    [coupled_time_W]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_W
        v = C_trapped_W
        block = 4
        extra_vector_tags = ref
    []
    [heat_conduction_W]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_heat_conduction_W]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_W
        density = density_W
        block = 4
        extra_vector_tags = ref
    []
    ############################## Kernels for Cu (block = 3)
    [diff_Cu]
        type = ADMatDiffusion
        variable = C_mobile_Cu
        diffusivity = diffusivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_diff_Cu]
        type = ADTimeDerivative
        variable = C_mobile_Cu
        block = 3
        extra_vector_tags = ref
    []
    [coupled_time_Cu]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_Cu
        v = C_trapped_Cu
        block = 3
        extra_vector_tags = ref
    []
    [heat_conduction_Cu]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_heat_conduction_Cu]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_Cu
        density = density_Cu
        block = 3
        extra_vector_tags = ref
    []
    ############################## Kernels for CuCrZr (block = 2)
    [diff_CuCrZr]
        type = ADMatDiffusion
        variable = C_mobile_CuCrZr
        diffusivity = diffusivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_diff_CuCrZr]
        type = ADTimeDerivative
        variable = C_mobile_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [coupled_time_CuCrZr]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_CuCrZr
        v = C_trapped_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [heat_conduction_CuCrZr]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_heat_conduction_CuCrZr]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_CuCrZr
        density = density_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
[]

[InterfaceKernels]
    [tied_4to3]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_W
        neighbor_var = C_mobile_Cu
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_4to3
        jump_prop_name = solubility_ratio
        boundary = '4to3'
    []
    [tied_3to2]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_Cu
        neighbor_var = C_mobile_CuCrZr
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_3to2
        jump_prop_name = solubility_ratio
        boundary = '3to2'
    []
[]

[NodalKernels]
    ############################## NodalKernels for W (block = 4)
    [time_W]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_W
    []
    [trapping_W]
        type = TrappingNodalKernel
        variable = C_trapped_W
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_W}
        Ct0 = ${Ct0_W}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_W}
        mobile_concentration = 'C_mobile_W'
        extra_vector_tags = ref
    []
    [release_W]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_W}
        variable = C_trapped_W
    []
    ############################## NodalKernels for Cu (block = 3)
    [time_Cu]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_Cu
    []
    [trapping_Cu]
        type = TrappingNodalKernel
        variable = C_trapped_Cu
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_Cu}
        Ct0 = ${Ct0_Cu}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_Cu}
        mobile_concentration = 'C_mobile_Cu'
        extra_vector_tags = ref
    []
    [release_Cu]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_Cu}
        variable = C_trapped_Cu
    []
    ############################## NodalKernels for CuCrZr (block = 2)
    [time_CuCrZr]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_CuCrZr
    []
    [trapping_CuCrZr]
        type = TrappingNodalKernel
        variable = C_trapped_CuCrZr
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_CuCrZr}
        Ct0 = ${Ct0_CuCrZr}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_CuCrZr}
        mobile_concentration = 'C_mobile_CuCrZr'
        extra_vector_tags = ref
    []
    [release_CuCrZr]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_CuCrZr}
        variable = C_trapped_CuCrZr
    []
[]

[BCs]
    [C_mob_W_top_flux]
        type = FunctionNeumannBC
        variable = C_mobile_W
        boundary = 'top'
        function = mobile_flux_bc_function
    []
    [mobile_tube]
        type = DirichletBC
        variable = C_mobile_CuCrZr
        value = ${C_mobile_CuCrZr_DirichletBC_Coolant}
        boundary = '2to1'
    []
    [temperature_top]
        type = FunctionNeumannBC
        variable = temperature
        boundary = 'top'
        function = temperature_flux_bc_function
    []
    [temperature_tube]
        type = FunctionDirichletBC
        variable = temperature
        boundary = '2to1'
        function = temperature_inner_func
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_W]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_W'
        reg_props_out = 'diffusivity_W_nonAD'
        block = 4
    []
    [heat_transfer_W]
        type = GenericConstantMaterial
        prop_names = 'density_W'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ParsedMaterial
        property_name = specific_heat_W
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ParsedMaterial
        property_name = thermal_conductivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_Cu]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_Cu'
        reg_props_out = 'diffusivity_Cu_nonAD'
        block = 3
    []
    [heat_transfer_Cu]
        type = GenericConstantMaterial
        prop_names = 'density_Cu'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ParsedMaterial
        property_name = specific_heat_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ParsedMaterial
        property_name = thermal_conductivity_Cu
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_CuCrZr]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_CuCrZr'
        reg_props_out = 'diffusivity_CuCrZr_nonAD'
        block = 2
    []
    [heat_transfer_CuCrZr]
        type = GenericConstantMaterial
        prop_names = 'density_CuCrZr specific_heat_CuCrZr'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ParsedMaterial
        property_name = thermal_conductivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile_W
        concentration_secondary = C_mobile_Cu
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile_Cu
        concentration_secondary = C_mobile_CuCrZr
    []
[]

(tmap8/test/tests/divertor_monoblock/divertor_monoblock.i)

### This input file is the complete input file for the divertor monoblock case.
### This case was published in:
### 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
###
### S_empty_j       empty site concentration in "j" material, where j = CuCrZr, Cu, W
### S_trapped_j     trapped site concentration in "j" material, where j = CuCrZr, Cu, W
### S_total_j       total site 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

C_mobile_CuCrZr_DirichletBC_Coolant = 1.0e-18 # at.fraction
C_mobile_W_init = 1.0e-20 # at.fraction
C_mobile_Cu_init = 5.0e-17 # at.fraction
C_mobile_CuCrZr_init = 1.0e-15 # at.fraction

# include sections of the input file shared with other inputs
!include divertor_monoblock_common_base.i
!include divertor_monoblock_mesh_base.i
!include divertor_monoblock_multi_variable_base.i

[Problem]
    type = ReferenceResidualProblem
    extra_tag_vectors = 'ref'
    reference_vector = 'ref'
[]

[Variables]
    [temperature]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${temperature_initial}
    []
    ######################### Variables for W (block = 4)
    [C_mobile_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_W_init}
        block = 4
    []
    [C_trapped_W]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 4
    []
    ######################### Variables for Cu (block = 3)
    [C_mobile_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_Cu_init}
        block = 3
    []
    [C_trapped_Cu]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 3
    []
    ######################### Variables for CuCrZr (block = 2)
    [C_mobile_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_mobile_CuCrZr_init}
        block = 2
    []
    [C_trapped_CuCrZr]
        order = FIRST
        family = LAGRANGE
        initial_condition = ${C_trapping_init}
        block = 2
    []
[]

[Kernels]
    ############################## Kernels for W (block = 4)
    [diff_W]
        type = ADMatDiffusion
        variable = C_mobile_W
        diffusivity = diffusivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_diff_W]
        type = ADTimeDerivative
        variable = C_mobile_W
        block = 4
        extra_vector_tags = ref
    []
    [coupled_time_W]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_W
        v = C_trapped_W
        block = 4
        extra_vector_tags = ref
    []
    [heat_conduction_W]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_W
        block = 4
        extra_vector_tags = ref
    []
    [time_heat_conduction_W]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_W
        density = density_W
        block = 4
        extra_vector_tags = ref
    []
    ############################## Kernels for Cu (block = 3)
    [diff_Cu]
        type = ADMatDiffusion
        variable = C_mobile_Cu
        diffusivity = diffusivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_diff_Cu]
        type = ADTimeDerivative
        variable = C_mobile_Cu
        block = 3
        extra_vector_tags = ref
    []
    [coupled_time_Cu]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_Cu
        v = C_trapped_Cu
        block = 3
        extra_vector_tags = ref
    []
    [heat_conduction_Cu]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_Cu
        block = 3
        extra_vector_tags = ref
    []
    [time_heat_conduction_Cu]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_Cu
        density = density_Cu
        block = 3
        extra_vector_tags = ref
    []
    ############################## Kernels for CuCrZr (block = 2)
    [diff_CuCrZr]
        type = ADMatDiffusion
        variable = C_mobile_CuCrZr
        diffusivity = diffusivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_diff_CuCrZr]
        type = ADTimeDerivative
        variable = C_mobile_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [coupled_time_CuCrZr]
        type = ScaledCoupledTimeDerivative
        variable = C_mobile_CuCrZr
        v = C_trapped_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [heat_conduction_CuCrZr]
        type = HeatConduction
        variable = temperature
        diffusion_coefficient = thermal_conductivity_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
    [time_heat_conduction_CuCrZr]
        type = SpecificHeatConductionTimeDerivative
        variable = temperature
        specific_heat = specific_heat_CuCrZr
        density = density_CuCrZr
        block = 2
        extra_vector_tags = ref
    []
[]

[InterfaceKernels]
    [tied_4to3]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_W
        neighbor_var = C_mobile_Cu
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_4to3
        jump_prop_name = solubility_ratio
        boundary = '4to3'
    []
    [tied_3to2]
        type = ADPenaltyInterfaceDiffusion
        variable = C_mobile_Cu
        neighbor_var = C_mobile_CuCrZr
        penalty = 0.05            #  it will not converge with > 0.1, but it creates negative C_mobile _Cu with << 0.1
        # jump_prop_name = solubility_ratio_3to2
        jump_prop_name = solubility_ratio
        boundary = '3to2'
    []
[]

[NodalKernels]
    ############################## NodalKernels for W (block = 4)
    [time_W]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_W
    []
    [trapping_W]
        type = TrappingNodalKernel
        variable = C_trapped_W
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_W}
        Ct0 = ${Ct0_W}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_W}
        mobile_concentration = 'C_mobile_W'
        extra_vector_tags = ref
    []
    [release_W]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_W}
        variable = C_trapped_W
    []
    ############################## NodalKernels for Cu (block = 3)
    [time_Cu]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_Cu
    []
    [trapping_Cu]
        type = TrappingNodalKernel
        variable = C_trapped_Cu
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_Cu}
        Ct0 = ${Ct0_Cu}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_Cu}
        mobile_concentration = 'C_mobile_Cu'
        extra_vector_tags = ref
    []
    [release_Cu]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_Cu}
        variable = C_trapped_Cu
    []
    ############################## NodalKernels for CuCrZr (block = 2)
    [time_CuCrZr]
        type = TimeDerivativeNodalKernel
        variable = C_trapped_CuCrZr
    []
    [trapping_CuCrZr]
        type = TrappingNodalKernel
        variable = C_trapped_CuCrZr
        temperature = temperature
        alpha_t = ${alpha_t}
        N = ${N_CuCrZr}
        Ct0 = ${Ct0_CuCrZr}
        trapping_energy = ${trapping_energy}
        trap_per_free = ${trap_per_free_CuCrZr}
        mobile_concentration = 'C_mobile_CuCrZr'
        extra_vector_tags = ref
    []
    [release_CuCrZr]
        type = ReleasingNodalKernel
        alpha_r = ${alpha_r}
        temperature = temperature
        detrapping_energy = ${detrapping_energy_CuCrZr}
        variable = C_trapped_CuCrZr
    []
[]

[BCs]
    [C_mob_W_top_flux]
        type = FunctionNeumannBC
        variable = C_mobile_W
        boundary = 'top'
        function = mobile_flux_bc_function
    []
    [mobile_tube]
        type = DirichletBC
        variable = C_mobile_CuCrZr
        value = ${C_mobile_CuCrZr_DirichletBC_Coolant}
        boundary = '2to1'
    []
    [temperature_top]
        type = FunctionNeumannBC
        variable = temperature
        boundary = 'top'
        function = temperature_flux_bc_function
    []
    [temperature_tube]
        type = FunctionDirichletBC
        variable = temperature
        boundary = '2to1'
        function = temperature_inner_func
    []
[]

[Materials]
    ############################## Materials for W (block = 4)
    [diffusivity_W]
        type = ADParsedMaterial
        property_name = diffusivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${diffusivity_W_D0}*exp(-${diffusivity_W_Ea}/temperature)'
        outputs = all
    []
    [solubility_W]
        type = ADParsedMaterial
        property_name = solubility_W
        coupled_variables = 'temperature'
        block = 4
        expression = '${solubility_W_1_D0}*exp(-${solubility_W_1_Ea}/temperature) + ${solubility_W_2_D0}*exp(-${solubility_W_2_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_W]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_W'
        reg_props_out = 'diffusivity_W_nonAD'
        block = 4
    []
    [heat_transfer_W]
        type = GenericConstantMaterial
        prop_names = 'density_W'
        prop_values = '${density_W}'
        block = 4
    []
    [specific_heat_W]
        type = ParsedMaterial
        property_name = specific_heat_W
        coupled_variables = 'temperature'
        block = 4
        expression = '1.16e2 + 7.11e-2 * temperature - 6.58e-5 * temperature^2 + 3.24e-8 * temperature^3 -5.45e-12 * temperature^4' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_W]
        type = ParsedMaterial
        property_name = thermal_conductivity_W
        coupled_variables = 'temperature'
        block = 4
        expression = '2.41e2 - 2.90e-1 * temperature + 2.54e-4 * temperature^2 - 1.03e-7 * temperature^3 + 1.52e-11 * temperature^4' # [W/m-K]
        outputs = all
    []
    ############################## Materials for Cu (block = 3)
    [diffusivity_Cu]
        type = ADParsedMaterial
        property_name = diffusivity_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${diffusivity_Cu_D0}*exp(-${diffusivity_Cu_Ea}/temperature)'
        outputs = all
    []
    [solubility_Cu]
        type = ADParsedMaterial
        property_name = solubility_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '${solubility_Cu_D0}*exp(-${solubility_Cu_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_Cu]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_Cu'
        reg_props_out = 'diffusivity_Cu_nonAD'
        block = 3
    []
    [heat_transfer_Cu]
        type = GenericConstantMaterial
        prop_names = 'density_Cu'
        prop_values = '${density_Cu}'
        block = 3
    []
    [specific_heat_Cu]
        type = ParsedMaterial
        property_name = specific_heat_Cu
        coupled_variables = 'temperature'
        block = 3
        expression = '3.16e2 + 3.18e-1 * temperature - 3.49e-4 * temperature^2 + 1.66e-7 * temperature^3' # [J/kg-K]
        outputs = all
    []
    [thermal_conductivity_Cu]
        type = ParsedMaterial
        property_name = thermal_conductivity_Cu
        coupled_variables = 'temperature'
        block = 3
        # expression = '-3.9e-8 * temperature^3 + 3.8e-5 * temperature^2 - 7.9e-2 * temperature + 4.0e2'    # ~ 401.0  [ W/m-K] from R. Delaporte-Mathurin et al 2021 Nucl. Fusion 61 036038,
        expression = '4.21e2 - 6.85e-2 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for CuCrZr (block = 2)
    [diffusivity_CuCrZr]
        type = ADParsedMaterial
        property_name = diffusivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${diffusivity_CuCrZr_D0}*exp(-${diffusivity_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [solubility_CuCrZr]
        type = ADParsedMaterial
        property_name = solubility_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '${solubility_CuCrZr_D0}*exp(-${solubility_CuCrZr_Ea}/temperature)'
        outputs = all
    []
    [converter_to_regular_CuCrZr]
        type = MaterialADConverter
        ad_props_in = 'diffusivity_CuCrZr'
        reg_props_out = 'diffusivity_CuCrZr_nonAD'
        block = 2
    []
    [heat_transfer_CuCrZr]
        type = GenericConstantMaterial
        prop_names = 'density_CuCrZr specific_heat_CuCrZr'
        prop_values = '${density_CuCrZr} ${specific_heat_CuCrZr}'
        block = 2
    []
    [thermal_conductivity_CuCrZr]
        type = ParsedMaterial
        property_name = thermal_conductivity_CuCrZr
        coupled_variables = 'temperature'
        block = 2
        expression = '3.87e2 - 1.28e-1 * temperature' # [W/m-K]
        outputs = all
    []
    ############################## Materials for others
    [interface_jump_4to3]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_W
        solubility_secondary = solubility_Cu
        boundary = '4to3'
        concentration_primary = C_mobile_W
        concentration_secondary = C_mobile_Cu
    []
    [interface_jump_3to2]
        type = SolubilityRatioMaterial
        solubility_primary = solubility_Cu
        solubility_secondary = solubility_CuCrZr
        boundary = '3to2'
        concentration_primary = C_mobile_Cu
        concentration_secondary = C_mobile_CuCrZr
    []
[]

General description of the simulation case and corresponding input file
Results
Complete input file
References

Symbol	Variable or Physical Property	Unit
$var element = document.getElementById("moose-equation-043e2fd8-ae5f-4d84-949e-8178d2e09a42");katex.render("C_s", element, {displayMode:false,throwOnError:false});$	Concentration of solute (mobile) species	m $var element = document.getElementById("moose-equation-09575df7-e98a-4723-8a4b-e28b57fa575d");katex.render("^{\u22123}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-e8512bda-1cbb-4cf7-84ae-c14053aa01e5");katex.render("C_t", element, {displayMode:false,throwOnError:false});$	Concentration of trapped species	m $var element = document.getElementById("moose-equation-3bea3897-5d7c-4053-81e3-7192c62958c1");katex.render("^{\u22123}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-1e590057-bf74-4a65-b42b-3cc970e3388c");katex.render("T", element, {displayMode:false,throwOnError:false});$	Temperature	K
$var element = document.getElementById("moose-equation-196cdbe4-fb4e-4b2c-9805-e319ae3d1953");katex.render("D", element, {displayMode:false,throwOnError:false});$	Diffusivity of solute (mobile) species	m $var element = document.getElementById("moose-equation-6ab370b6-7731-4bab-be4c-f9dc19227efc");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$ s $var element = document.getElementById("moose-equation-1a7e5ae0-0ca0-4a5f-88c3-7ec59530a59b");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-dc4df25e-706a-4013-9e36-9ce75add8aca");katex.render("K_s", element, {displayMode:false,throwOnError:false});$	Solubility of solute (mobile) species	m $var element = document.getElementById("moose-equation-468a7a3e-f61d-425c-91fa-33a6e18b4107");katex.render("^{\u22123}", element, {displayMode:false,throwOnError:false});$ Pa $var element = document.getElementById("moose-equation-306480f2-807c-4b7d-89c8-5342f66591e7");katex.render("^{-1/2}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-2dcdda10-ced5-4bfc-9d69-2319a50e9399");katex.render("C_{total}", element, {displayMode:false,throwOnError:false});$	Total concentration of species $var element = document.getElementById("moose-equation-efb15f79-0f51-41dc-b6e1-3de1b8049412");katex.render("C_{total}=C_s + C_t", element, {displayMode:false,throwOnError:false});$	m $var element = document.getElementById("moose-equation-b8760900-a9c6-47ba-844c-32668e3d82a7");katex.render("^{\u22123}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-9e67a138-a5ca-4015-bafd-e90b12a5cfcb");katex.render("C_t^e", element, {displayMode:false,throwOnError:false});$	Concentration of empty trapping sites	m $var element = document.getElementById("moose-equation-23ce934d-4d59-4418-8810-b8c684770ed6");katex.render("^{\u22123}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-2f1e8364-bbd8-496a-82c0-238dcc554e63");katex.render("C_t^0", element, {displayMode:false,throwOnError:false});$	Concentration of trapping sites $var element = document.getElementById("moose-equation-1cc1386d-ed07-48e7-ad55-6110af9d7d9c");katex.render("C_t^e=C_t^0-C_t", element, {displayMode:false,throwOnError:false});$	m $var element = document.getElementById("moose-equation-771115cc-388f-464d-b57d-1fd301767637");katex.render("^{\u22123}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-9a974569-24d2-45f3-8c2b-9df7628ef0ac");katex.render("\\alpha_t", element, {displayMode:false,throwOnError:false});$	Trapping rate coefficient, $var element = document.getElementById("moose-equation-5081fc62-8cfd-4fd8-9827-3905520bb6db");katex.render("2.75 \\times 10^{11}", element, {displayMode:false,throwOnError:false});$	s $var element = document.getElementById("moose-equation-a7512039-dd13-45c7-b735-af867925e889");katex.render("^{\u22121}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-6fc6a746-3876-48f9-9413-895d7975e8a5");katex.render("\\alpha_r", element, {displayMode:false,throwOnError:false});$	Release rate coefficient $var element = document.getElementById("moose-equation-881936b3-dc01-4a53-89cf-7ea3a6d2fd59");katex.render("\\alpha_r=\\alpha_{r0} \\exp\u2061((-E_{dt})/(k_b T))", element, {displayMode:false,throwOnError:false});$	s $var element = document.getElementById("moose-equation-3fdea4f8-860e-4eae-8c95-ede218949a60");katex.render("^{\u22121}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-427e294b-70f8-4d36-b9b9-f050d7899110");katex.render("\\alpha_{r0}", element, {displayMode:false,throwOnError:false});$	Pre-exponential factor, $var element = document.getElementById("moose-equation-103c8c7d-eb6d-476e-b8db-f5217985908a");katex.render("8.40 \\times 10^{12}", element, {displayMode:false,throwOnError:false});$	s $var element = document.getElementById("moose-equation-3686a504-4bd8-4634-888c-431dcea773fa");katex.render("^{\u22121}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-159ca42a-7403-4180-9025-5e2321516bf7");katex.render("E_{dt}", element, {displayMode:false,throwOnError:false});$	Detrapping energy	eV
$var element = document.getElementById("moose-equation-8493f0d6-4943-4017-86ef-5938fa067c40");katex.render("N", element, {displayMode:false,throwOnError:false});$	Atomic number density	m $var element = document.getElementById("moose-equation-e1c5864e-57dc-4e60-82f8-3fbb6236265d");katex.render("^{\u22123}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-8d77a78f-fd30-4da1-8d69-8bad8cb5a9d3");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$	Density	kg m $var element = document.getElementById("moose-equation-966ce6d6-403f-45f3-9a3e-7bf182097bee");katex.render("^{-3}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-d3f14cca-dcd4-49ce-af9c-3c00047b7526");katex.render("c_p", element, {displayMode:false,throwOnError:false});$	Specific heat	J kg $var element = document.getElementById("moose-equation-0af6ff1d-4af6-4ee0-a902-a76f278add42");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ K $var element = document.getElementById("moose-equation-e5136d8d-e13d-45c4-92e9-318ea8888642");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-24a7b714-084f-4c99-8a4b-0b8ba721b5ee");katex.render("k_T", element, {displayMode:false,throwOnError:false});$	Thermal conductivity	W m $var element = document.getElementById("moose-equation-a4298302-53ff-48e2-9bf1-ac6301182d47");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ K $var element = document.getElementById("moose-equation-f3ffcc12-08cf-412d-b553-bf6a4c9261f7");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$