InversePowerMethod

Inverse power method for eigenvalue problems.

Overview

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

var element = document.getElementById("moose-equation-4fe479a5-1071-44b0-9b8e-5d0051524e04");katex.render("Ax = \\frac{1}{k}Bx,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});

where $var element = document.getElementById("moose-equation-565fc7ce-3ebc-41e4-a792-5434f53d28d2");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-286e5961-3576-439b-810f-2cf72e040a55");katex.render("B", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ are linear or nonlinear operators represented by kernels. To differentiate the $var element = document.getElementById("moose-equation-9d7fa9f4-4908-4239-a3ff-1e9fa1211e7e");katex.render("B", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ kernels from the $var element = document.getElementById("moose-equation-01b59884-5e56-45f6-9884-c39021af337a");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ kernels, we must derive all $var element = document.getElementById("moose-equation-e34835a4-522b-451e-a337-48908c75038c");katex.render("B", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ kernels from EigenKernel. Currently we are only interested in the absolute minimum eigenvalue $var element = document.getElementById("moose-equation-4081c350-c1ae-49b9-8a28-274f3220fade");katex.render("\\frac{1}{k}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and the corresponding eigenvector $var element = document.getElementById("moose-equation-d9defe24-7541-4875-bf32-1cf094847f51");katex.render("x", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ 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 inverse power method algorithm

Initialization
$var element = document.getElementById("moose-equation-ca3a6304-27d8-4157-9042-5ffb15d103fd");katex.render("\\begin{aligned} k^{(0)} &= k_0 \\\\ x^{(0)} &= x_0 \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$
Update x and k
$var element = document.getElementById("moose-equation-f20a4a33-a5d5-4169-9778-a77944dafc15");katex.render("\\begin{aligned} x^{(n)} &= \\frac{1}{k^{(n-1)}} A^{-1}Bx^{(n-1)} \\\\ k^{(n)} &= k^{(n-1)} \\frac{|Bx^{(n)}|}{|Bx^{(n-1)}|} \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$
Check the convergence
$var element = document.getElementById("moose-equation-5758d084-ab75-428c-b49e-da2722929f85");katex.render("\\frac{|x^{(n)}-x^{(n-1)}|}{|x^{(n)}|} < tol_x", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$
and
$var element = document.getElementById("moose-equation-5edd68a0-8ac0-466f-83bb-4ff6c099f7cd");katex.render("\\frac{|k^{(n)}-k^{(n-1)}|}{|k^{(n)}|} < tol_k", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$
When either of them is not true, return Step 2, otherwise exit.

We notice immediately that $var element = document.getElementById("moose-equation-9f5e6e6e-ce11-4c16-a22d-715b82b336f3");katex.render("\\frac{|Bx|}{k}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ remains constant during the iteration, so if we make $var element = document.getElementById("moose-equation-af443d39-5fce-44c3-92e7-1a780f103d59");katex.render("\\frac{|Bx^{(0)}|}{k^{(0)}}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ equal to 1, the algorithm can be simplified a little:

Initialization
$var element = document.getElementById("moose-equation-4d72f628-b5ad-4fab-8a4f-8a68f04e92b8");katex.render("\\begin{aligned} k^{(0)} &= k_0 \\\\ x^{(0)} &= k_0 \\frac{x_0}{|Bx_0|} \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$
Update x and k
$var element = document.getElementById("moose-equation-60207c48-3d62-47e9-a7a5-59457a437d7e");katex.render("\\begin{aligned} x^{(n)} &= \\frac{1}{k^{(n-1)}} A^{-1}Bx^{(n-1)} \\\\ k^{(n)} &= |Bx^{(n)}| \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$
Check the convergence
$var element = document.getElementById("moose-equation-1b6e3a57-03d6-4ee5-a328-e3c04d63e1cf");katex.render("\\frac{|x^{(n)}-x^{(n-1)}|}{|x^{(n)}|} < tol_x", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$
and
$var element = document.getElementById("moose-equation-c27369e3-01b4-40b1-b1a6-2509c2e5f841");katex.render("\\frac{|k^{(n)}-k^{(n-1)}|}{|k^{(n)}|} < tol_k", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$
When either of them is not true, return Step 2, otherwise exit.

Also in this simplified algorithm, the solution is automatically normalized making $var element = document.getElementById("moose-equation-7344175a-484a-4100-90b9-bb3f921d1865");katex.render("|Bx|=k", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ . We can do postprocessing to normalize the solution so that $var element = document.getElementById("moose-equation-e3ede7f6-3cd7-420a-b391-a37079da13e1");katex.render("|x|=c", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , where $var element = document.getElementById("moose-equation-a23fc5d0-a69a-41de-95bd-e3180b58e19d");katex.render("|.|", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ can be any norm and $var element = document.getElementById("moose-equation-32c3de78-dd34-49ee-913b-ab7f5ed483e0");katex.render("c", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is a scalar constant.

If the minimum eigenvalue and the second smallest eigenvalue are close, i.e. the dominance ratio is about equal to one, the inverse power iteration converges very slowly. In such a case, we can apply accelerations, such as Chebyshev acceleration, based on the on-the-fly estimation of the dominance ratio.

The inverse power method is appealing because we can apply matrix-free schemes on evaluating $var element = document.getElementById("moose-equation-dac96430-ae7d-4881-b446-9a9494976ad4");katex.render("Ax - \\frac{1}{k}Bx", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ . We can use PJFNK for inverting $var element = document.getElementById("moose-equation-f06b610b-bc5f-48a1-8938-9a1afca6845d");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and we do not have to exactly assemble matrix $var element = document.getElementById("moose-equation-38fab5b7-4084-4f4a-a48d-adebf8237015");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ for the preconditioning purpose.

Input Parameters

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

Required Parameters

Chebyshev_acceleration_onTrueIf Chebyshev acceleration is turned on
Default:True
C++ Type:bool
Options:
Description:If Chebyshev acceleration is turned on
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 fixed point iterations. This check is performed based on postprocessor defined by the custom_pp residual.
Default:1e-50
C++ Type:double
Options:
Description:The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on postprocessor defined by the custom_pp residual.
custom_rel_tol1e-08The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the postprocessor defined by custom_pp residual.
Default:1e-08
C++ Type:double
Options:
Description:The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the postprocessor defined by custom_pp residual.
eig_check_tol1e-06Eigenvalue convergence tolerance
Default:1e-06
C++ Type:double
Options:
Description:Eigenvalue convergence tolerance
fixed_point_algorithmpicardThe fixed point algorithm to converge the sequence of problems.
Default:picard
C++ Type:MooseEnum
Options:picard, secant, steffensen
Description:The fixed point algorithm to converge the sequence of problems.
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_power_iterations300The maximum number of power iterations
Default:300
C++ Type:unsigned int
Options:
Description:The maximum number of power iterations
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).
min_power_iterations1Minimum number of power iterations
Default:1
C++ Type:unsigned int
Options:
Description:Minimum number of power iterations
n_max_nonlinear_pingpong100The maximum number of times the non linear residual can ping pong before requesting halting the current evaluation 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 evaluation 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
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 for determining 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 for determining 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
sol_check_tol1.79769e+308Convergence tolerance on |x-x_previous| when provided
Default:1.79769e+308
C++ Type:double
Options:
Description:Convergence tolerance on |x-x_previous| when provided
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.
verboseFalseSet to true to print additional information
Default:False
C++ Type:bool
Options:
Description:Set to true to print additional information
xdiffTo evaluate |x-x_previous| for power iterations
C++ Type:PostprocessorName
Options:
Description:To evaluate |x-x_previous| for power iterations

Optional Parameters

accept_on_max_fixed_point_iterationFalseTrue to treat reaching the maximum number of fixed point iterations as converged.
Default:False
C++ Type:bool
Options:
Description:True to treat reaching the maximum number of fixed point iterations as converged.
auto_advanceFalseWhether to automatically advance sub-applications regardless of whether their solve converges, for transient executioners only.
Default:False
C++ Type:bool
Options:
Description:Whether to automatically advance sub-applications regardless of whether their solve converges, for transient executioners only.
custom_ppPostprocessor for custom fixed point convergence check.
C++ Type:PostprocessorName
Options:
Description:Postprocessor for custom fixed point convergence check.
direct_pp_valueFalseTrue to use direct postprocessor value (scaled by value on first iteration). False (default) to use difference in postprocessor value between fixed point 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 fixed point iterations.
disable_fixed_point_residual_norm_checkFalseDisable the residual norm evaluation thus the three parameters fixed_point_rel_tol, fixed_point_abs_tol and fixed_point_force_norms.
Default:False
C++ Type:bool
Options:
Description:Disable the residual norm evaluation thus the three parameters fixed_point_rel_tol, fixed_point_abs_tol and fixed_point_force_norms.
fixed_point_abs_tol1e-50The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.
Default:1e-50
C++ Type:double
Options:
Description:The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.
fixed_point_force_normsFalseForce the evaluation of both the TIMESTEP_BEGIN and TIMESTEP_END norms regardless of the existence 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 existence of active MultiApps with those execute_on flags, default: false.
fixed_point_max_its1Specifies the maximum number of fixed point iterations.
Default:1
C++ Type:unsigned int
Options:
Description:Specifies the maximum number of fixed point iterations.
fixed_point_min_its1Specifies the minimum number of fixed point iterations.
Default:1
C++ Type:unsigned int
Options:
Description:Specifies the minimum number of fixed point iterations.
fixed_point_rel_tol1e-08The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.
Default:1e-08
C++ Type:double
Options:
Description:The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the main 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.
transformed_postprocessorsList of main app postprocessors to transform during fixed point iterations
C++ Type:std::vector<PostprocessorName>
Options:
Description:List of main app postprocessors to transform during fixed point iterations
transformed_variablesList of main app variables to transform during fixed point iterations
C++ Type:std::vector<std::string>
Options:
Description:List of main app variables to transform during fixed point iterations

Fixed Point Iterations 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-50Nonlinear Absolute Tolerance
Default:1e-50
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-08Nonlinear Relative Tolerance
Default:1e-08
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
off_diagonals_in_auto_scalingFalseWhether to consider off-diagonals when determining automatic scaling factors.
Default:False
C++ Type:bool
Options:
Description:Whether to consider off-diagonals when determining automatic scaling factors.
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

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
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

Xfem Fixed Point Iterations 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/ipm.i)

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

[Mesh]
 type = GeneratedMesh
 dim = 2
 xmin = 0
 xmax = 100
 ymin = 0
 ymax = 100
 elem_type = QUAD4
 nx = 8
 ny = 8

 uniform_refine = 0

 displacements = 'x_disp y_disp'
[]

#The minimum eigenvalue for this problem is 2*(pi/a)^2 + 2 with a = 100.
#Its inverse will be 0.49950700634518.

[Variables]
  active = 'u'

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

[AuxVariables]
  [./x_disp]
  [../]
  [./y_disp]
  [../]
[]

[AuxKernels]
  [./x_disp]
    type = FunctionAux
    variable = x_disp
    function = x_disp_func
  [../]
  [./y_disp]
    type = FunctionAux
    variable = y_disp
    function = y_disp_func
  [../]
[]

[Functions]
  [./x_disp_func]
    type = ParsedFunction
    value = 0
  [../]
  [./y_disp_func]
    type = ParsedFunction
    value = 0
  [../]
[]

[Kernels]
  active = 'diff rea rhs'

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

  [./rea]
    type = CoefReaction
    variable = u
    coefficient = 2.0
    use_displaced_mesh = true
  [../]

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

[BCs]
  active = 'homogeneous'

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

[Executioner]
  type = InversePowerMethod

  min_power_iterations = 11
  max_power_iterations = 400
  Chebyshev_acceleration_on = true
  eig_check_tol = 1e-12
  k0 = 0.5

  bx_norm = 'unorm'
  xdiff = 'udiff'
  normalization = 'unorm'

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

[Postprocessors]
  active = 'unorm udiff'

  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    execute_on = linear
    use_displaced_mesh = true
  [../]

  [./udiff]
    type = ElementL2Diff
    variable = u
    execute_on = 'linear timestep_end'
    use_displaced_mesh = true
  [../]
[]

[Outputs]
  file_base = ipm
  exodus = true
  hide = 'x_disp y_disp'
[]

Overview
The inverse power method algorithm
Input Parameters
Input Files

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