EigenProblem

Problem object for solving an eigenvalue problem.

EigenProblem is the primary eigenvalue system in the MOOSE framework. It offers linear and nonlinear eigensolvers by leveraging SLEPc capabilities (mainly the EPS system). For linear problems, the system supports one eigenvalue and multiple eigenvalues. For nonlinear problems, we currently support solving for the smallest (in magnitude) eigenvalue. The development of nonlinear eigensolver was mainly motivated by neutron transport criticality calculations, but the implementation and algorithm are general and applicable to other applications.

A generalized (linear) eigenvalue problem can uniformly be rewritten as and are matrices, is an eigenvector, and is the eigenvalue. Generally speaking, is a nonsingular matrix, and can be singular (e.g., an eigenvalue problem with Dirichlet boundary conditions). Many solvers in SLEPc can handle this linear eigenproblem. We can choose these algorithms in the executioner block. Here is an example with Jacobi Davidson:

[Executioner]
  type = Eigenvalue
  eigen_problem_type = gen_non_hermitian
  which_eigen_pairs = SMALLEST_MAGNITUDE
  n_eigen_pairs = 1
  n_basis_vectors = 18
  solve_type = jacobi_davidson
  petsc_options = '-eps_view'
[]

Note that to use the linear solvers from SLEPc, we need to form accurate and that will be utilized to represent the underlying physics. If and are incorrect, you will likely end up with the wrong eigenvalue and eigenvector. For that reason, if you want to use linear eigenvalue solvers, you need to implement Jacobian by hand or with AD in the corresponding kernels. Details of all the linear eigensolvers can be found in SLEPc Users Manual.

In a multiphysics environment, physics are often nonlinear. For instance, neutron transport with temperature feedback. In this case, and are not constant anymore; instead, they depend on the eigenvector directly or indirectly. To describe our nonlinear solver, we write a generalized nonlinear eigenvalue problem as, where and are nonlinear functions. In the EigenProblem system, the kernels of are referred to as "noneigen," and the kernels of are referred to as "eigen." To utilize Newton method to solve this problem, we define , and rewrite the nonlinear eigen problem as Newton can be applied to solve this problem. The only issue is that the derivatives of with respective to can be dense. We use the Jacobian-free Newton method to overcome this difficulty, where the action of Jacobian on a solution vector is approximated via finite differences. To handle various use cases, variants of Newton are provided in this system. There is a current list:

  • PJFNK - Preconditioned Jacobian-free Newton Krylov: The preconditioning matrix is formed using noneigen kernels. We do have an option to allow users to incorporate eigen kernels into the preconditioning matrix. Still, in general, we won't encourage users to do that since the preconditioning matrix might be singular when close to the solution.

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
[]
  • JFNK - Jacobian-free Newton Krylov: Users are responsible for providing a preconditioner. This allows users to build a customized preconditioner, such as using a sweeper in radiation transport, that might be more efficient for their underlying problems.

[Executioner]
  type = Eigenvalue
  solve_type = JFNK
[]

[Preconditioning]
  [./PBP]
    type = PBP
    solve_order = 'u v'
    preconditioner = 'LU LU'
  [../]
[]
  • PJFNKMO- Matrix-only Preconditioned Jacobian-free Newton Krylov: Mathematically, this option is the same as PJFNK. The difference is the residual evaluation. PJFNK calls the user residual/function evaluation routine, which includes all finite element calculations. PJFNKMO forms residual vectors via matrix-vector multiplications: and . This is useful when matrices are constant. This option will be more efficient. If the problem is nonlinear, i.e. the matrices are not constant, you should not use this solve option because we do not attempt to re-evaluate and at each Newton iteration. This solve option is valid with Picard iterations when the matrices are constant at each Picard iteration with conditions only changing with Picard iterations. We do re-evaluate the two matrices at the beginning of each Picard iteration.

[Executioner]
  type = Eigenvalue
  solve_type = PJFNKMO
  constant_matrices = true
[]
  • Newton - Newton method: Both Jacobian and preconditioning matrices are . This option is added for consistency between the nonlinear system and the nonlinear eigensystem. Since this option does not account for the derivative of eigenvalue with respective to the eigenvector, this option is not efficient. Users should not use it in general.

In general, EigenProblem takes care of the following customizations of the solve:

  • normalization and scaling parameters for eigenvectors

  • setting the type of the SLEPc eigenvalue solver

  • whether to start solves with free power iterations and how many

  • selecting the eigenvector of interest for the residual

Input Parameters

  • active_eigen_index0Which eigenvector is used to compute residual and also associated to nonlinear variable

    Default:0

    C++ Type:unsigned int

    Controllable:No

    Description:Which eigenvector is used to compute residual and also associated to nonlinear variable

  • boundary_restricted_elem_integrity_checkTrueSet to false to disable checking of boundary restricted elemental object variable dependencies, e.g. are the variable dependencies defined on the selected boundaries?

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Set to false to disable checking of boundary restricted elemental object variable dependencies, e.g. are the variable dependencies defined on the selected boundaries?

  • boundary_restricted_node_integrity_checkTrueSet to false to disable checking of boundary restricted nodal object variable dependencies, e.g. are the variable dependencies defined on the selected boundaries?

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Set to false to disable checking of boundary restricted nodal object variable dependencies, e.g. are the variable dependencies defined on the selected boundaries?

  • error_on_jacobian_nonzero_reallocationFalseThis causes PETSc to error if it had to reallocate memory in the Jacobian matrix due to not having enough nonzeros

    Default:False

    C++ Type:bool

    Controllable:No

    Description:This causes PETSc to error if it had to reallocate memory in the Jacobian matrix due to not having enough nonzeros

  • extra_tag_matricesExtra matrices to add to the system that can be filled by objects which compute residuals and Jacobians (Kernels, BCs, etc.) by setting tags on them.

    C++ Type:std::vector<TagName>

    Controllable:No

    Description:Extra matrices to add to the system that can be filled by objects which compute residuals and Jacobians (Kernels, BCs, etc.) by setting tags on them.

  • extra_tag_solutionsExtra solution vectors to add to the system that can be used by objects for coupling variable values stored in them.

    C++ Type:std::vector<TagName>

    Controllable:No

    Description:Extra solution vectors to add to the system that can be used by objects for coupling variable values stored in them.

  • extra_tag_vectorsExtra vectors to add to the system that can be filled by objects which compute residuals and Jacobians (Kernels, BCs, etc.) by setting tags on them.

    C++ Type:std::vector<TagName>

    Controllable:No

    Description:Extra vectors to add to the system that can be filled by objects which compute residuals and Jacobians (Kernels, BCs, etc.) by setting tags on them.

  • force_restartFalseEXPERIMENTAL: If true, a sub_app may use a restart file instead of using of using the master backup file

    Default:False

    C++ Type:bool

    Controllable:No

    Description:EXPERIMENTAL: If true, a sub_app may use a restart file instead of using of using the master backup file

  • fv_bcs_integrity_checkTrueSet to false to disable checking of overlapping Dirichlet and Flux BCs and/or multiple DirichletBCs per sideset

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Set to false to disable checking of overlapping Dirichlet and Flux BCs and/or multiple DirichletBCs per sideset

  • ignore_zeros_in_jacobianFalseDo not explicitly store zero values in the Jacobian matrix if true

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Do not explicitly store zero values in the Jacobian matrix if true

  • kernel_coverage_checkTrueSet to false to disable kernel->subdomain coverage check

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Set to false to disable kernel->subdomain coverage check

  • material_coverage_checkTrueSet to false to disable material->subdomain coverage check

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Set to false to disable material->subdomain coverage check

  • material_dependency_checkTrueSet to false to disable material dependency check

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Set to false to disable material dependency check

  • near_null_space_dimension0The dimension of the near nullspace

    Default:0

    C++ Type:unsigned int

    Controllable:No

    Description:The dimension of the near nullspace

  • negative_sign_eigen_kernelTrueWhether or not to use a negative sign for eigenvalue kernels. Using a negative sign makes eigenvalue kernels consistent with a nonlinear solver

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Whether or not to use a negative sign for eigenvalue kernels. Using a negative sign makes eigenvalue kernels consistent with a nonlinear solver

  • null_space_dimension0The dimension of the nullspace

    Default:0

    C++ Type:unsigned int

    Controllable:No

    Description:The dimension of the nullspace

  • parallel_barrier_messagingFalseDisplays messaging from parallel barrier notifications when executing or transferring to/from Multiapps (default: false)

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Displays messaging from parallel barrier notifications when executing or transferring to/from Multiapps (default: false)

  • previous_nl_solution_requiredFalseTrue to indicate that this calculation requires a solution vector for storing the prvious nonlinear iteration.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:True to indicate that this calculation requires a solution vector for storing the prvious nonlinear iteration.

  • restart_file_baseFile base name used for restart (e.g. / or /LATEST to grab the latest file available)

    C++ Type:FileNameNoExtension

    Controllable:No

    Description:File base name used for restart (e.g. / or /LATEST to grab the latest file available)

  • skip_additional_restart_dataFalseTrue to skip additional data in equation system for restart. It is useful for starting a transient calculation with a steady-state solution

    Default:False

    C++ Type:bool

    Controllable:No

    Description:True to skip additional data in equation system for restart. It is useful for starting a transient calculation with a steady-state solution

  • skip_nl_system_checkFalseTrue to skip the NonlinearSystem check for work to do (e.g. Make sure that there are variables to solve for).

    Default:False

    C++ Type:bool

    Controllable:No

    Description:True to skip the NonlinearSystem check for work to do (e.g. Make sure that there are variables to solve for).

  • solveTrueWhether or not to actually solve the Nonlinear system. This is handy in the case that all you want to do is execute AuxKernels, Transfers, etc. without actually solving anything

    Default:True

    C++ Type:bool

    Controllable:Yes

    Description:Whether or not to actually solve the Nonlinear system. This is handy in the case that all you want to do is execute AuxKernels, Transfers, etc. without actually solving anything

  • transpose_null_space_dimension0The dimension of the transpose nullspace

    Default:0

    C++ Type:unsigned int

    Controllable:No

    Description:The dimension of the transpose nullspace

  • use_nonlinearTrueDetermines whether to use a Nonlinear vs a Eigenvalue system (Automatically determined based on executioner)

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Determines whether to use a Nonlinear vs a Eigenvalue system (Automatically determined based on executioner)

  • verbose_multiappsFalseSet to True to enable verbose screen printing related to MultiApps

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Set to True to enable verbose screen printing related to MultiApps

Optional Parameters

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector<std::string>

    Controllable:No

    Description:Adds user-defined labels for accessing object parameters via control logic.

  • default_ghostingFalseWhether or not to use libMesh's default amount of algebraic and geometric ghosting

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether or not to use libMesh's default amount of algebraic and geometric ghosting

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Set the enabled status of the MooseObject.

Advanced Parameters