PenaltyDirichletBC

Enforces a Dirichlet boundary condition in a weak sense by penalizing differences between the current solution and the Dirichlet data.

Description

PenaltyDirichletBC is a IntegratedBC used for enforcing Dirichlet boundary conditions which differs from the DirichletBC class in the way in which it handles the enforcement. It is appropriate for partial differential equations (PDEs) in the form

$var element = document.getElementById("moose-equation-0c2bda09-1ff8-48f1-aab4-fccdd695d564");katex.render("\\begin{aligned} -\\nabla^2 u &= f && \\quad \\in \\Omega \\\\ u &= g && \\quad \\in \\partial \\Omega_D \\\\ \\frac{\\partial u}{\\partial n} &= h && \\quad \\in \\partial \\Omega_N \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Instead of imposing the Dirichlet condition directly on the basis by replacing the equations associated with those degrees of freedom (DOFs) by the auxiliary equation $var element = document.getElementById("moose-equation-801a7a57-7817-4795-9e2d-e99ca2a07268");katex.render("u-g=0", element, {displayMode:false,throwOnError:false});$ , the PenaltyDirichletBC is based on the variational statement: find $var element = document.getElementById("moose-equation-e8f25d26-8c22-4c76-bf9d-ab09bf2ae61e");katex.render("u \\in H^1(\\Omega)", element, {displayMode:false,throwOnError:false});$ such that $(1)var element = document.getElementById("moose-equation-23a6a573-3dc3-4bf2-a2bc-9476528a0c13");katex.render(" \\int_{\\Omega} \\left( \\nabla u \\cdot \\nabla v - fv \\right) \\,\\text{d}x -\\int_{\\partial \\Omega_N} hv \\,\\text{d}s +\\int_{\\partial \\Omega_D} \\frac{1}{\\epsilon} (u-g)v \\,\\text{d}s = 0", element, {displayMode:true,throwOnError:false});$ holds for every $var element = document.getElementById("moose-equation-c3415c73-bd09-4dbe-a98c-2f95a3810d1a");katex.render("v \\in H^1(\\Omega)", element, {displayMode:false,throwOnError:false});$ . In Eq. (1), $var element = document.getElementById("moose-equation-78111dc1-70eb-4507-a76d-88e1d9d1a097");katex.render("\\epsilon \\ll 1", element, {displayMode:false,throwOnError:false});$ is a user-selected parameter which must be taken small enough to ensure that $var element = document.getElementById("moose-equation-057f00c5-9250-4757-98cf-ffb299c59fe0");katex.render("u \\approx g", element, {displayMode:false,throwOnError:false});$ on $var element = document.getElementById("moose-equation-c3f0c4b9-144b-4811-86b0-88e3917ea2dd");katex.render("\\partial \\Omega_D", element, {displayMode:false,throwOnError:false});$ . The user-selectable class parameter penalty corresponds to $var element = document.getElementById("moose-equation-72661f13-9c58-42ae-acd8-8f4d73021278");katex.render("\\frac{1}{\\epsilon}", element, {displayMode:false,throwOnError:false});$ , and must be chosen large enough to ensure good agreement with the Dirichlet data, but not so large that the resulting Jacobian becomes ill-conditioned, resulting in failed solves and overall accuracy losses.

Benefits of the penalty-based approach include simplified Dirichlet boundary condition enforcement for non-Lagrange finite element bases, maintaining the symmetry (if any) of the original problem, and avoiding the need to zero out contributions from other rows in a special post-assembly step. Integrating by parts "in reverse" from Eq. (1), one obtains

$(2)var element = document.getElementById("moose-equation-8efa02b6-8b4c-47dc-a829-92875e8f726a");katex.render(" \\int_{\\Omega} \\left( -\\nabla^2 u - f \\right) v \\,\\text{d}x +\\int_{\\partial \\Omega_N} \\left( \\frac{\\partial u}{\\partial n} - h \\right) v \\,\\text{d}s +\\int_{\\partial \\Omega_D} \\left[ \\frac{\\partial u}{\\partial n} + \\frac{1}{\\epsilon} (u-g) \\right] v \\,\\text{d}s = 0", element, {displayMode:true,throwOnError:false});$

We therefore recover a "perturbed" version of the original problem with the flux boundary condition

$var element = document.getElementById("moose-equation-c122401a-100b-42e7-8ce7-232abfd8c1f6");katex.render(" \\frac{\\partial u}{\\partial n} = -\\frac{1}{\\epsilon} (u-g) \\, \\in \\partial \\Omega_D", element, {displayMode:true,throwOnError:false});$

replacing the original Dirichlet boundary condition. It has been shown Juntunen and Stenberg (2009) that in order for the solution to this perturbed problem to converge to the solution of the original problem in the limit as $var element = document.getElementById("moose-equation-13fbaffc-f3e8-47a5-9707-8f0701ea0c38");katex.render("\\epsilon \\rightarrow 0", element, {displayMode:false,throwOnError:false});$ , the penalty parameter must depend on the mesh size, and that as we refine the mesh, the problem becomes increasingly ill-conditioned. A related method for imposing Dirichlet boundary conditions, known as Nitsche's method Juntunen and Stenberg (2009), does not suffer from the same ill-conditioning issues, and is slated for inclusion in MOOSE some time in the future.

References

M. Juntunen and R. Stenberg. Nitsche's method for general boundary conditions. Mathematics of Computation, 78(267):1353–1374, July 2009. URL: http://dx.doi.org/10.1090/S0025-5718-08-02183-2.

@article{juntunen2009nitsche,
    author = "Juntunen, M. and Stenberg, R.",
    title = "{Nitsche's method for general boundary conditions}",
    journal = "Mathematics of Computation",
    month = "July",
    year = "2009",
    volume = "78",
    number = "267",
    pages = "1353--1374",
    url = "http://dx.doi.org/10.1090/S0025-5718-08-02183-2"
}

Example Input Syntax

[BCs]
  active = 'bc_all'
  [./bc_all]
    type = PenaltyDirichletBC
    variable = u
    value = 0
    boundary = 'top left right bottom'
    penalty = 1e5
  [../]
[]

Input Parameters

boundaryThe list of boundary IDs from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Unit:(no unit assumed)
Controllable:No
Description:The list of boundary IDs from the mesh where this object applies
penaltyPenalty scalar
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Penalty scalar
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
value0Boundary value of the variable
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:Yes
Description:Boundary value of the variable

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Unit:(no unit assumed)
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Unit:(no unit assumed)
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Unit:(no unit assumed)
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Unit:(no unit assumed)
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Unit:(no unit assumed)
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Unit:(no unit assumed)
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
diag_save_inThe name of auxiliary variables to save this BC's diagonal jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this BC's diagonal jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Unit:(no unit assumed)
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
save_inThe name of auxiliary variables to save this BC's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this BC's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Unit:(no unit assumed)
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

(moose/test/tests/bcs/penalty_dirichlet_bc/penalty_dirichlet_bc_test.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  nx = 10
  ny = 10
  elem_type = QUAD9
[]

[Functions]
  [./forcing_fn]
    type = ParsedFunction
    expression = -2*(x*x+y*y-2)+(1-x*x)*(1-y*y)
  [../]

  [./solution]
    type = ParsedGradFunction
    value = (1-x*x)*(1-y*y)
    grad_x = 2*(x*y*y-x)
    grad_y = 2*(x*x*y-y)
  [../]
[]

[Variables]
  [./u]
    order = SECOND
    family = HIERARCHIC
  [../]
[]

[Kernels]
  active = 'diff forcing reaction'
  [./diff]
    type = Diffusion
    variable = u
  [../]

  [./reaction]
    type = Reaction
    variable = u
  [../]

  [./forcing]
    type = BodyForce
    variable = u
    function = forcing_fn
  [../]
[]

[BCs]
  active = 'bc_all'
  [./bc_all]
    type = PenaltyDirichletBC
    variable = u
    value = 0
    boundary = 'top left right bottom'
    penalty = 1e5
  [../]
[]

[Postprocessors]
  [./dofs]
    type = NumDOFs
  [../]

  [./h]
    type = AverageElementSize
  [../]

  [./L2error]
    type = ElementL2Error
    variable = u
    function = solution
  [../]
  [./H1error]
    type = ElementH1Error
    variable = u
    function = solution
  [../]
  [./H1Semierror]
    type = ElementH1SemiError
    variable = u
    function = solution
  [../]
[]

[Executioner]
  type = Steady
  solve_type = 'PJFNK'
  nl_rel_tol = 1e-14
[]

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

Description
References
Example Input Syntax
Input Parameters