Kernels

The list of Achlys specific kernel implementations are given here. The following sections give the full description of the kernel selections for implementnig the governing equations shown in Eq. (1), Eq. (2), and Eq. (3). The descriptions include links to the MOOSE framework documentation where standard kernels can be employed.

$(1)var element = document.getElementById("moose-equation-fcc42ee6-b0e6-4b12-9a38-751efbd504c5");katex.render("\\frac{\\partial C_{m}}{\\partial t} = \\nabla \\cdot \\left( D \\left(T \\right) \\nabla C_{m} \\right) - \\sum \\frac{\\partial C_{t,i}}{\\partial t} + S_{ext} ", element, {displayMode:true,throwOnError:false});$

$(2)var element = document.getElementById("moose-equation-cea0992b-bf67-44fc-a7f0-99b7efca0332");katex.render("\\frac{\\partial C_{t,i}}{\\partial t} = \\nu_m \\left(T\\right) C_m \\left(n_i - C_{t,i} \\right) - \\nu_i\\left(T\\right) C_{t,i} ", element, {displayMode:true,throwOnError:false});$

$(3)var element = document.getElementById("moose-equation-59068be1-5885-4354-b2af-da0b9c96e5fb");katex.render(" \\rho_m C_p \\frac{\\partial T}{\\partial t} = \\nabla \\cdot \\left(k \\nabla T \\right)", element, {displayMode:true,throwOnError:false});$

Note that some Achlys kernels are designed for use with specific achlys materials such as the Trapping Material class which calculates the diffusion coefficient and trapping/detrapping reaction rates.

Mobile Concentration

The first of the governing equations gives the mass balance for the mobile phase of dissolved atoms. This is a balance between the source/sink terms, given by either external sources or the coupling to the trapping/detrapping process, and diffusive transport driven by a concentration gradient.

Note that the coupled time derivative here provides the coupling between the mobile and trapped phase mass balances.

$var element = document.getElementById("moose-equation-623ce119-caad-4c6e-8f16-2d3bce6e3981");katex.render("\\underbrace{\\left(\\psi, \\frac{\\partial C_{m}}{\\partial t}\\right) }_{\\text{(i) \\:ADTimeDerivative}} - \\underbrace{\\bigg(\\nabla\\psi, D\\nabla C_m \\bigg)} _{\\text{(ii) \\:ADMatDiffusion}} + \\underbrace{\\left(\\psi, \\sum \\frac{\\partial C_{t,i}}{\\partial t} \\right) }_{\\text{(iii) \\:ADCoupledTimeDerivative}} + \\underbrace{\\bigg( \\psi,-S_{ext} \\bigg)}_{\\text{(iv) \\:ADBodyForce}} = 0", element, {displayMode:true,throwOnError:false});$

Number	Description	Framework / Achlys	Link
(i)	Time derivative	Framework	ADTimeDerivative
(ii)	Diffusion	Framework	ADMatDiffusion
(iii)	Coupled time derivate	Framework	ADCoupledTimeDerivative
(iv)	Source term	Framework	ADBodyForce

Trapped Concentration, site $var element = document.getElementById("moose-equation-c3731639-b661-4d3b-a095-1aa6ddcd336d");katex.render("i", element, {displayMode:false,throwOnError:false});$

The second of the governing equations must be implemented once for each material trapping site being considered. This is achieved using the following kernels:

$var element = document.getElementById("moose-equation-39aee7df-397a-4350-809b-8b6ca2b5d0ad");katex.render("\\underbrace{\\left(\\psi, \\frac{\\partial C_{t,i}}{\\partial t}\\right) }_{\\text{(i) \\:ADTimeDerivative}} - \\underbrace{\\bigg(\\psi, \\nu_m C_m(n_i - C_{t,i}) \\bigg) + \\bigg(\\psi,\\nu_i C_{t,i} \\bigg) }_{\\text{( iii ) \\: ADTrappingEquilibriumEquation}} = 0", element, {displayMode:true,throwOnError:false});$

Number	Description	Framework / Achlys	Link
(i)	Time derivative	Framework	ADTimeDerivative
(ii)	Trapping equilibrium	Achlys	ADEquilibriumEquation

Temperature Evolution

Thermal transport calculations always use the MOOSE Heat Conduction App without any modification or enhancement.

$var element = document.getElementById("moose-equation-ae04803c-78f7-43dd-8893-66f8c42137a1");katex.render("\\underbrace{\\left(\\psi, \\rho C_p \\frac{\\partial T}{\\partial t}\\right) }_{\\text{ADHeatConductionTimeDerivative}} - \\underbrace{\\bigg(\\nabla\\psi, k\\nabla T \\bigg) }_{\\text{ADHeatConduction}} = 0", element, {displayMode:true,throwOnError:false});$

Number	Description	Framework / Achlys	Link
(i)	Time derivative	Heat Conduction Module	ADHeatConductionTimeDerivative
(ii)	Heat conduction (Diffusion)	Heat Conduction Module	ADHeatConduction

Example usage

Drawing together the previous sections, the following example shows how to construct the kernels input block for an (isothermal) achlys calculation with 3 material traps.

[Variables]
  [./Mobile]
  [../]
  [./Trapped_1]
  [../]
  [./Trapped_2]
  []
  [./Trapped_3]
  []
[]

[Kernels]
  [./H3_diffusion_eq1]
    type = ADMatDiffusion
    variable = Mobile
    Diffusivity = D
  [../]
  [./mobile_time_deriv]
    type = ADTimeDerivative
    variable = Mobile
  [../]
  [./trapping_equilibrium_equation1]
    type = ADTrappingEquilibriumEquation
    variable = Trapped_1
    v = Mobile
    n_traps = n1
    vi = V1
  [../]
  [./trapping_equilibrium_equation2]
    type = ADTrappingEquilibriumEquation
    variable = Trapped_2
    v = Mobile
    n_traps = n2
    vi = V2
  [../]
  [./trapping_equilibrium_equation3]
    type = ADTrappingEquilibriumEquation
    variable = Trapped_3
    v = Mobile
    n_traps = n3
    vi = V3
  [../]
  [./trapped_time_deriv_couple]
    type = ADCoupledTimeDerivative
    variable = Mobile
    v = Trapped_1
  [../]
  [./trapped_time_deriv_couple2]
    type = ADCoupledTimeDerivative
    variable = Mobile
    v = Trapped_2
  [../]
  [./trapped_time_deriv_couple3]
    type = ADCoupledTimeDerivative
    variable = Mobile
    v = Trapped_3
  [../]
  [./trapped_time_deriv]
    type = ADTimeDerivative
    variable = Trapped_1
  [../]
  [./trapped_time_deriv2]
    type = ADTimeDerivative
    variable = Trapped_2
  [../]
  [./trapped_time_deriv3]
    type = ADTimeDerivative
    variable = Trapped_3
  [../]
[]

Available Kernels

achlys App
ADTrappingEquilibriumEquationCompute the local movement of particles between the mobile phaseand the trapped phase for a specific trap type

(problems/thermal_desorption/ogorodnikova/tds_multiapp/desorp_multi.i)

[Mesh]
 [./unlabelled]
    file = /Users/sdixon/projects/blue_kite/problems/thermal_desorption/optimisation/ogorodnikova_1k.msh
    type = FileMeshGenerator
    construct_side_list_from_node_list=true
  [../]
  [./block_1]
    type= AllSideSetsByNormalsGenerator
    input = unlabelled
    #fixed_normal = false
  [../]
[]

[Problem]
  type = FEProblem # This is the "normal" type of Finite Element Problem in MOOSE
  coord_type = XYZ # cartesian
[]

[Variables]
  [./Mobile]
  [../]
  [./Trapped_1]
  [../]
  [./Trapped_2]
  []
  [./Trapped_3]
  []
[]

[Functions]
  [./dts]
    type = PiecewiseLinear
    x = '0 1e-4 5e-4 1e-3'
    y = '1e-4 1e-4 5e-4 5e-4'
  [../]
  [./Gaussian_implant]
    type = ParsedFunction
    value = 'scale * exp( -0.5 * ((x - mean) / sd)^2)'
    vars = 'scale mean sd'
    vals = '0.93e8 4.5e-9 4.5e-9'
  [../]
  [./Exponweib_implant]
    type = ParsedFunction
    value = 'peak * (a * c / scale) * (( 1 - exp(-1 * (((x*1e9)-loc)/scale)^c))^(a-1) ) * exp(-1 * (((x*1e9)-loc)/scale)^c)*(((x*1e9)-loc)/scale)^(c-1)'
    vars = 'peak a c loc scale'
    vals = '5.82e8 0.90 1.96 -0.013 4.909'
  [../]
[]

[MultiApps]
  [sub_app]
    type = FullSolveMultiApp
    #app_type = BlueKiteApp
    input_files = 'resting_multi.i'
   # positions = '0   0.029   0
   #              0.015 0.0.029 0
   #              0.015 0.014 0
   #              0.0 0.014 0'
  execute_on = INITIAL
  clone_master_mesh = true
  []
[]

[Transfers]
#   [to_sub] 
#    type = MultiAppCopyTransfer
#    multi_app = sub_app
#    source_variable = 'surface_temp'
#    variable = 'surface_temp'
#    direction = to_multiapp
#  []
  [mobile_transfer]
    type =  MultiAppCopyTransfer #MultiAppScalarToAuxScalarTransfer
    multi_app = sub_app
    source_variable = 'Mobile'
    variable = 'Mobile'
    direction = from_multiapp
  []
  [trap_1_transfer]
    type =  MultiAppCopyTransfer #MultiAppScalarToAuxScalarTransfer
    multi_app = sub_app
    source_variable = 'Trapped_1'
    variable = 'Trapped_1'
    direction = from_multiapp
  []
  [trap_2_transfer]
    type =  MultiAppCopyTransfer #MultiAppScalarToAuxScalarTransfer
    multi_app = sub_app
    source_variable = 'Trapped_2'
    variable = 'Trapped_2'
    direction = from_multiapp
  []
  [trap_3_transfer]
    type =  MultiAppCopyTransfer #MultiAppScalarToAuxScalarTransfer
    multi_app = sub_app
    source_variable = 'Trapped_3'
    variable = 'Trapped_3'
    direction = from_multiapp
  []
[]

[Kernels]
  [./H3_diffusion_eq1]
    type = ADMatDiffusion
    variable = Mobile
    Diffusivity = D
  [../]
  [./mobile_time_deriv]
    type = ADTimeDerivative
    variable = Mobile
  [../]
  [./trapping_equilibrium_equation1]
    type = ADTrappingEquilibriumEquation
    variable = Trapped_1
    v = Mobile
    n_traps = n1
    vi = V1
  [../]
  [./trapping_equilibrium_equation2]
    type = ADTrappingEquilibriumEquation
    variable = Trapped_2
    v = Mobile
    n_traps = n2
    vi = V2
  [../]
  [./trapping_equilibrium_equation3]
    type = ADTrappingEquilibriumEquation
    variable = Trapped_3
    v = Mobile
    n_traps = n3
    vi = V3
  [../]
  [./trapped_time_deriv_couple]
    type = ADCoupledTimeDerivative
    variable = Mobile
    v = Trapped_1
  [../]
  [./trapped_time_deriv_couple2]
    type = ADCoupledTimeDerivative
    variable = Mobile
    v = Trapped_2
  [../]
  [./trapped_time_deriv_couple3]
    type = ADCoupledTimeDerivative
    variable = Mobile
    v = Trapped_3
  [../]
  [./trapped_time_deriv]
    type = ADTimeDerivative
    variable = Trapped_1
  [../]     
  [./trapped_time_deriv2]
    type = ADTimeDerivative
    variable = Trapped_2
  [../]
  [./trapped_time_deriv3]
    type = ADTimeDerivative
    variable = Trapped_3
  [../]
[]

[Postprocessors]
  [/pfc_flux]
    type = ADSideFluxIntegral
    variable = Mobile
    boundary = 1
    diffusivity = D
  [../]
  [/back_flux]
    type = ADSideFluxIntegral
    variable = Mobile
    boundary = 2
    diffusivity = D
  [../]
  [./total_mobile]
    type = VariableIntegral
    variable = Mobile
  [../]
  [./total_trap_1]
    type = VariableIntegral
    variable = Trapped_1
  [../]
    [./total_trap_2]
    type = VariableIntegral
    variable = Trapped_2
  [../]
  [./total_trap_3]
    type = VariableIntegral
    variable = Trapped_3
  [../]
[]

[BCs]
  [./desorption]
    type =  ADDirichletBC
    variable = Mobile
    boundary = 1
    value = 0
  [../]
  [./back_desorption]
    type =  ADDirichletBC
    variable = Mobile
    boundary = 2
    value = 0
  [../]
[]

[Materials]
   [./implant]
    type = ExtrinsicStaticTrappingMaterialRampingT
# Energies
    E_diff = 0.39
    E1 = 0.87
    E2 = 1.0
    E3 = 1.50
    k_boltz = 8.617333E-5
# pre-exponential rate constants
    v0 = 1.0E13
    D0 = 4.1E-7
    lambda = 1.1E-10
# unused
    rho = 1 # unecessary scaling factor, do not use
# site densities
    n_sol = 6
    n1 = 1E-3 #1.0E25
    n2 = 4e-4
    n3a_max = 1e-1
    n3b_max = 1e-2
# trap creation rates
    eta_a = 6e-4
    eta_b = 2e-4
    trap_evolution_time = 400
# flux distribution parameters
    flux = 4e-10
    function = Gaussian_implant
    xp = 1e-6
#Temperature
    initial_T = 300
    beta = 8
    block = 'Tungsten' # 'Stopping_zone' # Plastic_region
# thermal properties
    conductivity = 150 # W/K
    Cp = 137 # J/(kg K)
    density = 19300 # kg/m3
  [../] 
[]
[Executioner]
  type = Transient
  solve_type = NEWTON
  scheme = bdf2
  automatic_scaling=True
 compute_scaling_once=False
  scaling_group_variables = 'Trapped_3; Trapped_2; Trapped_1; Mobile'
  resid_vs_jac_scaling_param = 0.8
 # l_tol = 1e-4
  l_max_its = 100
  nl_max_funcs = 7
  #nl_rel_tol = 1e-7
  nl_abs_tol = 1e-29

  # Set PETSc parameters to optimize solver efficiency
  petsc_options_iname = '-ksp_type -pc_type -pc_factor_shift_type'
  petsc_options_value = 'bcgs lu  NONZERO'
#  dt = 0.25 #0.001 #5e-2
  end_time = 62.5
  timestep_tolerance = 0.01
    [TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 5
    cutback_factor = 0.5
    growth_factor = 1.1
    dt = 0.1
  []
[]


[Outputs]
  #execute_on = 'timestep_end'
  exodus = false # Output Exodus format
  csv = true
[]
[Debug]
  show_material_props = true
  show_var_residual_norms = true
[]

Mobile Concentration
Trapped Concentration , site var element = document.getElementById("moose-equation-c3731639-b661-4d3b-a095-1aa6ddcd336d");katex.render("i", element, {displayMode:false,throwOnError:false});
Temperature Evolution
Example usage
Available Kernels

Module

Examples

Source Code

Comparison to Analytical Solution

Thermal Desorption Spectrum

Kernels

BCs

Materials

Postprocessors

Kernels

Mobile Concentration

Trapped Concentration, site $var element = document.getElementById("moose-equation-c3731639-b661-4d3b-a095-1aa6ddcd336d");katex.render("i", element, {displayMode:false,throwOnError:false});$

Temperature Evolution

Example usage

Available Kernels

Kernels

Mobile Concentration

Trapped Concentration, site var element = document.getElementById("moose-equation-c3731639-b661-4d3b-a095-1aa6ddcd336d");katex.render("i", element, {displayMode:false,throwOnError:false});

Temperature Evolution

Example usage

Available Kernels

(problems/thermal_desorption/ogorodnikova/tds_multiapp/desorp_multi.i)

Trapped Concentration, site $var element = document.getElementById("moose-equation-c3731639-b661-4d3b-a095-1aa6ddcd336d");katex.render("i", element, {displayMode:false,throwOnError:false});$