NonlinearEigen

Executioner for eigenvalue problems.

Overview

Eigenvalue executioners such as this one intend on solving the eigenvalue problem described by:

var element = document.getElementById("moose-equation-8a1e9740-e0bb-4ba0-aee0-02d5d6e40df0");katex.render("Ax = \\frac{1}{k}Bx,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});

where $var element = document.getElementById("moose-equation-1e033c9d-3d0e-4b7a-ae8d-4399ff0122bc");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-33e34a15-54de-4b8a-89d6-88bcd00c9a8c");katex.render("B", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ are linear or nonlinear operators represented by kernels. To differentiate the $var element = document.getElementById("moose-equation-2a090ad0-98ec-4353-ad01-3a34a24e31a5");katex.render("B", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ kernels from the $var element = document.getElementById("moose-equation-4804fc42-806c-4c66-9877-f202e34d89d4");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ kernels, we must derive all $var element = document.getElementById("moose-equation-13eae800-a91f-4cde-a339-98c460e45fda");katex.render("B", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ kernels from EigenKernel. Currently we are only interested in the absolute minimum eigenvalue $var element = document.getElementById("moose-equation-fe768ceb-6a54-4b6a-aad0-261a89365b5d");katex.render("\\frac{1}{k}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and the corresponding eigenvector $var element = document.getElementById("moose-equation-00d2c555-3721-407a-b0c7-3d907098f9b2");katex.render("x", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ of the system. We are also not seeking the solutions of a general nonlinear eigenvalue problem, where the operators have nonlinear dependency on the eigenvalue.

The nonlinear Newton method

From the above section, we can see the eigenvalue problem can be viewed as a nonlinear problem

var element = document.getElementById("moose-equation-e2fd2ac8-9711-470b-8d07-8f621b6dc547");katex.render("\\begin{aligned} &Ax = \\frac{1}{k}Bx, \\\\ &k = |Bx|, \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});

so we can use the Newton method to solve it. However, to make the solving converge, we need to have a fairly close initial guess to the fundamental mode. This can be achieved with several free power iterations before the Newton iteration. We do not have to have $var element = document.getElementById("moose-equation-c4b0c78e-c443-488e-a4d0-ea86531351ca");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ as part of the solution vector. Instead we can apply the elimination technique and view the equation as

var element = document.getElementById("moose-equation-79c843f7-4b5b-4e9f-ab57-636a038abdba");katex.render("Ax = \\frac{Bx}{|Bx|}.", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});

Again we can use PJFNK (preconditioned Jacobian-free Newton Krylov) method to solve this nonlinear problem. The preconditioning matrix can affect the linear convergence in each Newton iteration. If there is a convergence issue, it is suggested to use $var element = document.getElementById("moose-equation-f4a4a056-0fa8-4632-83aa-e4598f1748db");katex.render("A-\\frac{B}{|Bx|}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and gradually reduce its complexity as the preconditioning matrix.

Input Parameters

bx_normTo evaluate |Bx| for the eigenvalue
C++ Type:PostprocessorName
Options:
Description:To evaluate |Bx| for the eigenvalue

Required Parameters

auto_advanceFalseWhether to automatically advance sub-applications regardless of whether their solve converges.
Default:False
C++ Type:bool
Options:
Description:Whether to automatically advance sub-applications regardless of whether their solve converges.
contact_line_search_allowed_lambda_cuts2The number of times lambda is allowed to be cut in half in the contact line search. We recommend this number be roughly bounded by 0 <= allowed_lambda_cuts <= 3
Default:2
C++ Type:unsigned int
Options:
Description:The number of times lambda is allowed to be cut in half in the contact line search. We recommend this number be roughly bounded by 0 <= allowed_lambda_cuts <= 3
contact_line_search_ltolThe linear relative tolerance to be used while the contact state is changing between non-linear iterations. We recommend that this tolerance be looser than the standard linear tolerance
C++ Type:double
Options:
Description:The linear relative tolerance to be used while the contact state is changing between non-linear iterations. We recommend that this tolerance be looser than the standard linear tolerance
custom_abs_tol1e-50The absolute nonlinear residual to shoot for during Picard iterations. This check is performed based on postprocessor defined by picard_custom_pp residual.
Default:1e-50
C++ Type:double
Options:
Description:The absolute nonlinear residual to shoot for during Picard iterations. This check is performed based on postprocessor defined by picard_custom_pp residual.
custom_rel_tol1e-08The relative nonlinear residual drop to shoot for during Picard iterations. This check is performed based on postprocessor defined by picard_custom_pp residual.
Default:1e-08
C++ Type:double
Options:
Description:The relative nonlinear residual drop to shoot for during Picard iterations. This check is performed based on postprocessor defined by picard_custom_pp residual.
direct_pp_valueFalseTrue to use direct postprocessor value (scaled by value on first iteration). False (default) to use difference in postprocessor value between picard iterations.
Default:False
C++ Type:bool
Options:
Description:True to use direct postprocessor value (scaled by value on first iteration). False (default) to use difference in postprocessor value between picard iterations.
free_l_tol0.01Relative linear tolerance in free power iteration
Default:0.01
C++ Type:double
Options:
Description:Relative linear tolerance in free power iteration
free_power_iterations4The number of free power iterations
Default:4
C++ Type:unsigned int
Options:
Description:The number of free power iterations
k01Initial guess of the eigenvalue
Default:1
C++ Type:double
Options:
Description:Initial guess of the eigenvalue
line_searchdefaultSpecifies the line search type (Note: none = basic)
Default:default
C++ Type:MooseEnum
Options:basic, bt, contact, cp, default, l2, none, project, shell
Description:Specifies the line search type (Note: none = basic)
line_search_packagepetscThe solver package to use to conduct the line-search
Default:petsc
C++ Type:MooseEnum
Options:petsc, moose
Description:The solver package to use to conduct the line-search
max_xfem_update4294967295Maximum number of times to update XFEM crack topology in a step due to evolving cracks
Default:4294967295
C++ Type:unsigned int
Options:
Description:Maximum number of times to update XFEM crack topology in a step due to evolving cracks
mffd_typewpSpecifies the finite differencing type for Jacobian-free solve types. Note that the default is wp (for Walker and Pernice).
Default:wp
C++ Type:MooseEnum
Options:wp, ds
Description:Specifies the finite differencing type for Jacobian-free solve types. Note that the default is wp (for Walker and Pernice).
n_max_nonlinear_pingpong100The maximum number of times the non linear residual can ping pong before requesting halting the current evalution and requesting timestep cut
Default:100
C++ Type:unsigned int
Options:
Description:The maximum number of times the non linear residual can ping pong before requesting halting the current evalution and requesting timestep cut
nl_abs_div_tol1e+50Nonlinear Absolute Divergence Tolerance. A negative value disables this check.
Default:1e+50
C++ Type:double
Options:
Description:Nonlinear Absolute Divergence Tolerance. A negative value disables this check.
nl_div_tol1e+10Nonlinear Relative Divergence Tolerance. A negative value disables this check.
Default:1e+10
C++ Type:double
Options:
Description:Nonlinear Relative Divergence Tolerance. A negative value disables this check.
nl_forced_its0The Number of Forced Nonlinear Iterations
Default:0
C++ Type:unsigned int
Options:
Description:The Number of Forced Nonlinear Iterations
output_after_power_iterationsTrueTrue to output solution after free power iterations
Default:True
C++ Type:bool
Options:
Description:True to output solution after free power iterations
petsc_optionsSingleton PETSc options
C++ Type:MultiMooseEnum
Options:-dm_moose_print_embedding, -dm_view, -ksp_converged_reason, -ksp_gmres_modifiedgramschmidt, -ksp_monitor, -ksp_monitor_snes_lg-snes_ksp_ew, -ksp_snes_ew, -snes_converged_reason, -snes_ksp, -snes_ksp_ew, -snes_linesearch_monitor, -snes_mf, -snes_mf_operator, -snes_monitor, -snes_test_display, -snes_view
Description:Singleton PETSc options
petsc_options_inameNames of PETSc name/value pairs
C++ Type:MultiMooseEnum
Options:-ksp_atol, -ksp_gmres_restart, -ksp_max_it, -ksp_pc_side, -ksp_rtol, -ksp_type, -mat_fd_coloring_err, -mat_fd_type, -mat_mffd_type, -pc_asm_overlap, -pc_factor_levels, -pc_factor_mat_ordering_type, -pc_hypre_boomeramg_grid_sweeps_all, -pc_hypre_boomeramg_max_iter, -pc_hypre_boomeramg_strong_threshold, -pc_hypre_type, -pc_type, -snes_atol, -snes_linesearch_type, -snes_ls, -snes_max_it, -snes_rtol, -snes_divergence_tolerance, -snes_type, -sub_ksp_type, -sub_pc_type
Description:Names of PETSc name/value pairs
petsc_options_valueValues of PETSc name/value pairs (must correspond with "petsc_options_iname"
C++ Type:std::vector<std::string>
Options:
Description:Values of PETSc name/value pairs (must correspond with "petsc_options_iname"
resid_vs_jac_scaling_param0A parameter that indicates the weighting of the residual vs the Jacobian in determining variable scaling parameters. A value of 1 indicates pure residual-based scaling. A value of 0 indicates pure Jacobian-based scaling
Default:0
C++ Type:double
Options:
Description:A parameter that indicates the weighting of the residual vs the Jacobian in determining variable scaling parameters. A value of 1 indicates pure residual-based scaling. A value of 0 indicates pure Jacobian-based scaling
scaling_group_variablesName of variables that are grouped together to for determing scale factors. (Multiple groups can be provided, separated by semicolon)
C++ Type:std::vector<std::vector<std::string>>
Options:
Description:Name of variables that are grouped together to for determing scale factors. (Multiple groups can be provided, separated by semicolon)
skip_exception_checkFalseSpecifies whether or not to skip exception check
Default:False
C++ Type:bool
Options:
Description:Specifies whether or not to skip exception check
solve_typePJFNK: Preconditioned Jacobian-Free Newton Krylov JFNK: Jacobian-Free Newton Krylov NEWTON: Full Newton Solve FD: Use finite differences to compute Jacobian LINEAR: Solving a linear problem
C++ Type:MooseEnum
Options:PJFNK, JFNK, NEWTON, FD, LINEAR
Description:PJFNK: Preconditioned Jacobian-Free Newton Krylov JFNK: Jacobian-Free Newton Krylov NEWTON: Full Newton Solve FD: Use finite differences to compute Jacobian LINEAR: Solving a linear problem
splittingTop-level splitting defining a hierarchical decomposition into subsystems to help the solver.
C++ Type:std::vector<std::string>
Options:
Description:Top-level splitting defining a hierarchical decomposition into subsystems to help the solver.
update_xfem_at_timestep_beginFalseShould XFEM update the mesh at the beginning of the timestep
Default:False
C++ Type:bool
Options:
Description:Should XFEM update the mesh at the beginning of the timestep
verboseFalseSet to true to print additional information
Default:False
C++ Type:bool
Options:
Description:Set to true to print additional information

Optional Parameters

accept_on_max_picard_iterationFalseTrue to treat reaching the maximum number of Picard iterations as converged.
Default:False
C++ Type:bool
Options:
Description:True to treat reaching the maximum number of Picard iterations as converged.
disable_picard_residual_norm_checkFalseDisable the Picard residual norm evaluation thus the three parameters picard_rel_tol, picard_abs_tol and picard_force_norms.
Default:False
C++ Type:bool
Options:
Description:Disable the Picard residual norm evaluation thus the three parameters picard_rel_tol, picard_abs_tol and picard_force_norms.
picard_abs_tol1e-50The absolute nonlinear residual to shoot for during Picard iterations. This check is performed based on the Master app's nonlinear residual.
Default:1e-50
C++ Type:double
Options:
Description:The absolute nonlinear residual to shoot for during Picard iterations. This check is performed based on the Master app's nonlinear residual.
picard_custom_ppPostprocessor for custom picard convergence check.
C++ Type:PostprocessorName
Options:
Description:Postprocessor for custom picard convergence check.
picard_force_normsFalseForce the evaluation of both the TIMESTEP_BEGIN and TIMESTEP_END norms regardless of the existance of active MultiApps with those execute_on flags, default: false.
Default:False
C++ Type:bool
Options:
Description:Force the evaluation of both the TIMESTEP_BEGIN and TIMESTEP_END norms regardless of the existance of active MultiApps with those execute_on flags, default: false.
picard_max_its1Specifies the maximum number of Picard iterations. Mainly used when wanting to do Picard iterations with MultiApps that are set to execute_on timestep_end or timestep_begin. Setting this parameter to 1 turns off the Picard iterations.
Default:1
C++ Type:unsigned int
Options:
Description:Specifies the maximum number of Picard iterations. Mainly used when wanting to do Picard iterations with MultiApps that are set to execute_on timestep_end or timestep_begin. Setting this parameter to 1 turns off the Picard iterations.
picard_rel_tol1e-08The relative nonlinear residual drop to shoot for during Picard iterations. This check is performed based on the Master app's nonlinear residual.
Default:1e-08
C++ Type:double
Options:
Description:The relative nonlinear residual drop to shoot for during Picard iterations. This check is performed based on the Master app's nonlinear residual.
relaxation_factor1Fraction of newly computed value to keep.Set between 0 and 2.
Default:1
C++ Type:double
Options:
Description:Fraction of newly computed value to keep.Set between 0 and 2.
relaxed_variablesList of variables to relax during Picard Iteration
C++ Type:std::vector<std::string>
Options:
Description:List of variables to relax during Picard Iteration

Picard Parameters

auto_initializationTrueTrue to ask the solver to set initial
Default:True
C++ Type:bool
Options:
Description:True to ask the solver to set initial
control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
outputsVector of output names were you would like to restrict the output of variables(s) associated with this object
C++ Type:std::vector<OutputName>
Options:
Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object
time0System time
Default:0
C++ Type:double
Options:
Description:System time

Advanced Parameters

automatic_scalingFalseWhether to use automatic scaling for the variables.
Default:False
C++ Type:bool
Options:
Description:Whether to use automatic scaling for the variables.
compute_initial_residual_before_preset_bcsFalseUse the residual norm computed *before* preset BCs are imposed in relative convergence check
Default:False
C++ Type:bool
Options:
Description:Use the residual norm computed *before* preset BCs are imposed in relative convergence check
compute_scaling_onceTrueWhether the scaling factors should only be computed once at the beginning of the simulation through an extra Jacobian evaluation. If this is set to false, then the scaling factors will be computed during an extra Jacobian evaluation at the beginning of every time step.
Default:True
C++ Type:bool
Options:
Description:Whether the scaling factors should only be computed once at the beginning of the simulation through an extra Jacobian evaluation. If this is set to false, then the scaling factors will be computed during an extra Jacobian evaluation at the beginning of every time step.
l_abs_tol1e-50Linear Absolute Tolerance
Default:1e-50
C++ Type:double
Options:
Description:Linear Absolute Tolerance
l_max_its10000Max Linear Iterations
Default:10000
C++ Type:unsigned int
Options:
Description:Max Linear Iterations
l_tol0.01Linear Tolerance
Default:0.01
C++ Type:double
Options:
Description:Linear Tolerance
nl_abs_step_tol0Nonlinear Absolute step Tolerance
Default:0
C++ Type:double
Options:
Description:Nonlinear Absolute step Tolerance
nl_abs_tol1e-06Nonlinear Absolute Tolerance
Default:1e-06
C++ Type:double
Options:
Description:Nonlinear Absolute Tolerance
nl_max_funcs10000Max Nonlinear solver function evaluations
Default:10000
C++ Type:unsigned int
Options:
Description:Max Nonlinear solver function evaluations
nl_max_its50Max Nonlinear Iterations
Default:50
C++ Type:unsigned int
Options:
Description:Max Nonlinear Iterations
nl_rel_step_tol0Nonlinear Relative step Tolerance
Default:0
C++ Type:double
Options:
Description:Nonlinear Relative step Tolerance
nl_rel_tol1e-50Nonlinear Relative Tolerance
Default:1e-50
C++ Type:double
Options:
Description:Nonlinear Relative Tolerance
num_grids1The number of grids to use for a grid sequencing algorithm. This includes the final grid, so num_grids = 1 indicates just one solve in a time-step
Default:1
C++ Type:unsigned int
Options:
Description:The number of grids to use for a grid sequencing algorithm. This includes the final grid, so num_grids = 1 indicates just one solve in a time-step
snesmf_reuse_baseTrueSpecifies whether or not to reuse the base vector for matrix-free calculation
Default:True
C++ Type:bool
Options:
Description:Specifies whether or not to reuse the base vector for matrix-free calculation

Solver Parameters

normal_factorNormalize x to make |x| equal to this factor
C++ Type:double
Options:
Description:Normalize x to make |x| equal to this factor
normalizationTo evaluate |x| for normalization
C++ Type:PostprocessorName
Options:
Description:To evaluate |x| for normalization
output_before_normalizationTrueTrue to output a step before normalization
Default:True
C++ Type:bool
Options:
Description:True to output a step before normalization

Normalization Parameters

Restart Parameters

Input Files

(test/tests/executioners/eigen_executioners/ne_mat.i)
(test/tests/executioners/eigen_executioners/ne_coupled.i)
(test/tests/executioners/eigen_executioners/ane.i)
(test/tests/executioners/eigen_executioners/ne.i)
(test/tests/multiapps/full_solve_multiapp/master_eigen.i)
(test/tests/executioners/eigen_executioners/ne_deficient_b.i)

(test/tests/executioners/eigen_executioners/ne_mat.i)

[Mesh]
 type = GeneratedMesh
 dim = 2
 xmin = 0
 xmax = 10
 ymin = 0
 ymax = 10
 elem_type = QUAD4
 nx = 8
 ny = 8

 uniform_refine = 0
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  active = 'u'

  [./u]
    # second order is way better than first order
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  active = 'diff rhs'

  [./diff]
    type = Diffusion
    variable = u
  [../]

  [./rhs]
    type = MaterialEigenKernel
    variable = u
    mat = varmat
  [../]
[]

[Materials]
  [./var_mat]
    type = VarCouplingMaterialEigen
    block = 0
    var = u
    material_prop_name = varmat
  [../]
[]

[BCs]
  active = 'homogeneous'

  [./homogeneous]
    type = DirichletBC
    variable = u
    preset = false
    boundary = '0 1 2 3'
    value = 0
  [../]
[]

[Executioner]
  type = NonlinearEigen

  bx_norm = 'unorm'
  normalization = 'unorm'
  normal_factor = 9.990012561844

  free_power_iterations = 2
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-50
  k0 = 1.0

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'
[]

[Postprocessors]
  active = 'unorm udiff'

  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    # execute on residual is important for nonlinear eigen solver!
    execute_on = linear
  [../]

  [./udiff]
    type = ElementL2Diff
    variable = u
  [../]
[]

[Outputs]
  execute_on = 'timestep_end'
  file_base = ne_mat
  exodus = true
[]

(test/tests/executioners/eigen_executioners/ne_coupled.i)

[Mesh]
 type = GeneratedMesh
 dim = 2
 xmin = 0
 xmax = 10
 ymin = 0
 ymax = 10
 elem_type = QUAD4
 nx = 8
 ny = 8

 uniform_refine = 0
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]

  [./T]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./power]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = DiffMKernel
    variable = u
    mat_prop = diffusion
    offset = 0.0
  [../]

  [./rhs]
    type = MassEigenKernel
    variable = u
  [../]

  [./diff_T]
    type = Diffusion
    variable = T
  [../]
  [./src_T]
    type = CoupledForce
    variable = T
    v = power
  [../]
[]

[AuxKernels]
  [./power_ak]
    type = NormalizationAux
    variable = power
    source_variable = u
    normalization = unorm

# this coefficient will affect the eigenvalue.
    normal_factor = 10


    execute_on = linear
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./homogeneousT]
    type = DirichletBC
    variable = T
    boundary = '0 1 2 3'
    value = 0
  [../]
[]

[Materials]
  [./dc]
    type = VarCouplingMaterial
    var = T
    block = 0
    base = 1.0
    coef = 1.0
  [../]
[]

[Executioner]
  type = NonlinearEigen

  bx_norm = 'unorm'

  free_power_iterations = 2
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-50
  k0 = 1.0
  output_after_power_iterations = false

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'
[]

[Postprocessors]
  active = 'unorm udiff'

  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    # execute on residual is important for nonlinear eigen solver!
    execute_on = linear
  [../]

  [./udiff]
    type = ElementL2Diff
    variable = u
    outputs = console
  [../]
[]

[Outputs]
  execute_on = 'timestep_end'
  file_base = ne_coupled
  exodus = true
[]

(test/tests/executioners/eigen_executioners/ane.i)

[Mesh]
 type = GeneratedMesh
 dim = 2
 xmin = 0
 xmax = 10
 ymin = 0
 ymax = 10
 elem_type = QUAD4
 nx = 8
 ny = 8

 uniform_refine = 0
[]

# the minimum eigenvalue is (2*PI*(p-1)^(1/p)/a/p/sin(PI/p))^p;
# Its inverse is 35.349726539758187. Here a is equal to 10.

[Variables]
  active = 'u'

  [./u]
    # second order is way better than first order
    order = FIRST
    family = LAGRANGE
  [../]
[]

[ICs]
  [./uic]
    type = RandomIC
    variable = u
  [../]
[]

[Kernels]
  active = 'diff rhs'

  [./diff]
    type = PHarmonic
    variable = u
    p = 3
  [../]

  [./rhs]
    type = PMassEigenKernel
    variable = u
    p = 3
  [../]
[]

[BCs]
  active = 'homogeneous'

  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 2'
    value = 0
  [../]
[]

[Executioner]
  type = NonlinearEigen

  bx_norm = 'unorm'

  free_power_iterations = 10
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-50
  k0 = 1.0
# important: constant initial value set by auto_initilization does not
# converge to the fundamental mode
  auto_initialization = false
  output_after_power_iterations = false

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'
  l_max_its = 100
[]

[Postprocessors]
  active = 'unorm udiff'

  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    # execute on residual is important for nonlinear eigen solver!
    execute_on = linear
  [../]

  [./udiff]
    type = ElementL2Diff
    variable = u
    outputs = console
  [../]
[]

[Outputs]
  execute_on = 'timestep_end'
  file_base = ane
  exodus = true
[]

(test/tests/executioners/eigen_executioners/ne.i)

[Mesh]
 type = GeneratedMesh
 dim = 2
 xmin = 0
 xmax = 10
 ymin = 0
 ymax = 10
 elem_type = QUAD4
 nx = 8
 ny = 8

 uniform_refine = 0
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  active = 'u'

  [./u]
    # second order is way better than first order
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  active = 'diff rhs'

  [./diff]
    type = Diffusion
    variable = u
  [../]

  [./rhs]
    type = MassEigenKernel
    variable = u
  [../]
[]

[BCs]
  active = 'homogeneous'

  [./homogeneous]
    type = DirichletBC
    variable = u
    preset = false
    boundary = '0 1 2 3'
    value = 0
  [../]
[]

[Executioner]
  type = NonlinearEigen

  bx_norm = 'unorm'
  normalization = 'unorm'
  normal_factor = 9.990012561844

  free_power_iterations = 2
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-50
  k0 = 1.0

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'
[]

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

[Postprocessors]
  active = 'unorm udiff'

  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    # execute on residual is important for nonlinear eigen solver!
    execute_on = linear
  [../]

  [./udiff]
    type = ElementL2Diff
    variable = u
  [../]
[]

[Outputs]
  execute_on = 'timestep_end'
  file_base = ne
  exodus = true
[]

(test/tests/multiapps/full_solve_multiapp/master_eigen.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables]
  [u]
  []
[]

[Kernels]
  [diff]
    type = Diffusion
    variable = u
  []
  [rhs]
    type = MassEigenKernel
    variable = u
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = u
    boundary = left
    value = 0
  []
  [right]
    type = DirichletBC
    variable = u
    boundary = right
    value = 1
  []
[]

[Executioner]
  type = NonlinearEigen

  bx_norm = 'unorm'

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Postprocessors]
  [unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    # execute on residual is important for nonlinear eigen solver!
    execute_on = linear
  []
[]

[Outputs]
  exodus = true
  perf_graph = true
[]

[MultiApps]
  [full_solve]
    type = FullSolveMultiApp
    # not setting app_type to use the same app type of master, i.e. MooseTestApp
    execute_on = initial
    positions = '0 0 0'
    input_files = sub.i
  []
[]

(test/tests/executioners/eigen_executioners/ne_deficient_b.i)

[Mesh]
 type = GeneratedMesh
 dim = 2
 xmin = 0
 xmax = 10
 ymin = 0
 ymax = 10
 elem_type = QUAD4
 nx = 8
 ny = 8

 uniform_refine = 0
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
    eigen = true
  [../]
[]

[Kernels]
  [./diff_u]
    type = Diffusion
    variable = u
  [../]
  [./diff_v]
    type = Diffusion
    variable = v
  [../]

  [./rhs]
    type = CoupledEigenKernel
    variable = u
    v = v
  [../]
  [./src_v]
    type = CoupledForce
    variable = v
    v = u
  [../]
[]

[BCs]
  [./homogeneous_u]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./homogeneous_v]
    type = DirichletBC
    variable = v
    boundary = '0 1 2 3'
    value = 0
  [../]
[]

[Executioner]
  type = NonlinearEigen

  bx_norm = 'vnorm'

  free_power_iterations = 2
  nl_abs_tol = 1e-12
  nl_rel_tol = 1e-50
  k0 = 1.0
  output_after_power_iterations = false

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'
[]

[Postprocessors]
  [./vnorm]
    type = ElementIntegralVariablePostprocessor
    variable = v
    # execute on residual is important for nonlinear eigen solver!
    execute_on = linear
  [../]

  [./udiff]
    type = ElementL2Diff
    variable = u
    outputs = console
  [../]
[]

[Outputs]
  execute_on = 'timestep_end'
  file_base = ne_deficient_b
  exodus = true
[]

Overview
The nonlinear Newton method
Input Parameters
Input Files

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