PeriodicSegmentalConstraint

PeriodicSegmentalConstraint enforces macro-micro periodic conditions between secondary and primary sides of a mortar interface using Lagrange multipliers.Must be used alongside EqualValueConstraint.

Description

This Constraint demonstrates the usage of the scalar augmentation class described in MortarScalarBase. The other terms in the weak form are handled using the EqualValueConstraint as described below.

In comparison to Dirichlet or Neumann conditions, periodic boundary conditions have been found to typically be the most accurate (fastest converging) approach for applying macro-to-micro scale transition constraints. Several methods for imposing periodic boundary conditions exist, each with pros and cons. For example, the mortar approach requires an extra Lagrange multiplier field. Alternatively, the periodic condition can be imposed by the penalty method using PenaltyPeriodicSegmentalConstraint or one of the other periodic approaches in MOOSE.

This class provides the macro-micro coupling terms to implement periodic boundary conditions using the mortar method, as proposed within Reis and Andrade Pires (2014). Alternatively, these equations impose an average value of the diffusive flux of a spatial variable over a domain using surface rather than volume integrals.

The strong form is posed over domain with opposing boundary pairs and :

(1)

where is the average diffusive gradient to be solved for, is the imposed average diffusive flux, and is the Lagrange multiplier that imposes this average constraint. The jump operator is defined for a single valued or vector valued field as and , respectively.

The corresponding weak form is (using inner-product notation):

(2)

where is the test function of the diffusive variable , is the test function for the Lagrange multiplier , and is an arbitrary test vector (spatially uniform) to impose the constraint involving the scalar variable . As is typical for mixed-field problems with Lagrange multipliers, the shape functions for and need to be chosen to satisfy the Babuska-Brezzi inf-sup condition if stabilization is not added to the system. As discussed in Reis and Andrade Pires (2014), using quadratic and piecewise linear (discontinuous at corners) provides for stable results. Note that element-discontinuous (e.g. L2_LAGRANGE or MONOMIAL basis) does not produce stable results without interpolation. An easy way to make a discontinuous field at corners is described below.

Input File Parameters

The terms in the weak form Eq. (2) are handled by several different classes. The volume integrals are handled by Diffusion or MatDiffusion. The surface terms and are computed by EqualValueConstraint. The remaining three terms are handled by this class.

Two of these objects are shown in the input file below:

[Constraints]
  [mortarlr]
    type = EqualValueConstraint
    primary_boundary = '11'
    secondary_boundary = '13'
    primary_subdomain = 'primary_right'
    secondary_subdomain = 'secondary_left'
    secondary_variable = u
    variable = lm1
    correct_edge_dropping = true
  []
  [periodiclr]
    type = PeriodicSegmentalConstraint
    primary_boundary = '11'
    secondary_boundary = '13'
    primary_subdomain = 'primary_right'
    secondary_subdomain = 'secondary_left'
    secondary_variable = u
    epsilon = epsilon
    sigma = sigma
    variable = lm1
    correct_edge_dropping = true
  []
  [mortarbt]
    type = EqualValueConstraint
    primary_boundary = '12'
    secondary_boundary = '10'
    primary_subdomain = 'primary_top'
    secondary_subdomain = 'secondary_bottom'
    secondary_variable = u
    variable = lm2
    correct_edge_dropping = true
  []
  [periodicbt]
    type = PeriodicSegmentalConstraint
    primary_boundary = '12'
    secondary_boundary = '10'
    primary_subdomain = 'primary_top'
    secondary_subdomain = 'secondary_bottom'
    secondary_variable = u
    epsilon = epsilon
    sigma = sigma
    variable = lm2
    correct_edge_dropping = true
  []
[]
(../moose/test/tests/mortar/periodic_segmental_constraint/periodic_simple2d.i)

The applied macroscale diffusive flux is applied as the sigma vector via an auxillary scalar. The computed macroscale diffusive gradient is assigned in a scalar variable epsilon. Both of these scalars should have the same number of components as the spatial dimension of . The volume integral of the gradient of the primary field will be constrained to in a weak sense.

Also, the coupled_scalar must be assigned the same scalar as epsilon.

The microscale diffusion variable is specified using the primary_variable parameter. If the solution values to be matched are between different variables, the secondary_variable parameter can also be supplied. The enforcement takes place using Lagrange multipliers assigned to variable. These same parameters must be used for the micro-micro coupling terms in the EqualValueConstraint object.

The generation of the lower-dimensional mesh surfaces for and are described in the Mortar Constraint system. The projection between two separated surfaces on opposite sides of the domain are naturally handled by the system. This is true for both EqualValueConstraint and PeriodicSegmentalConstraint. In fact, the meshes can be nonconforming as long as the geometry is conforming, although the choice of discretization becomes more delicate. Note that the periodic parameter is NOT needed, but if it is applied then it should be the same for BOTH EqualValueConstraint and PeriodicSegmentalConstraint.

commentnote:Parallel offsets

Due to current restrictions on AutomaticMortarGeneration, the opposing surfaces must be directly opposite along the unit normal direction.

As mentioned above, the discretization needs to be continuous along patches of element faces (LAGRANGE, not MONOMIAL) in order to be stable, but must be discontinuous along corners of the mesh where the outward unit normal is discontinuous since it is a flux variable (see the thrid condition Eq. (1)). An easy way to do this is to make a separate LAGRANGE variable for each 'face' of the model with different , which usually corresponds with different named side-sets or boundaries used for creating lower-dimensional mesh surfaces. This approach is demonstrated in many of the test input files.

warningwarning:Solver Type NEWTON

The PJFNK solver does not perform well for discrete systems lacking terms on the diagonal of the Jacobian matrix, such as this mortar method. Thus, the NEWTON solver is recommended.

Input Parameters

  • primary_boundaryThe name of the primary boundary sideset.

    C++ Type:BoundaryName

    Controllable:No

    Description:The name of the primary boundary sideset.

  • primary_subdomainThe name of the primary subdomain.

    C++ Type:SubdomainName

    Controllable:No

    Description:The name of the primary subdomain.

  • secondary_boundaryThe name of the secondary boundary sideset.

    C++ Type:BoundaryName

    Controllable:No

    Description:The name of the secondary boundary sideset.

  • secondary_subdomainThe name of the secondary subdomain.

    C++ Type:SubdomainName

    Controllable:No

    Description:The name of the secondary subdomain.

  • sigmaControlled scalar averaging variable

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

    Controllable:No

    Description:Controlled scalar averaging variable

Required Parameters

  • aux_lmAuxiliary Lagrange multiplier variable that is utilized together with the Petrov-Galerkin approach.

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

    Controllable:No

    Description:Auxiliary Lagrange multiplier variable that is utilized together with the Petrov-Galerkin approach.

  • compute_lm_residualsTrueWhether to compute Lagrange Multiplier residuals

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Whether to compute Lagrange Multiplier residuals

  • compute_primal_residualsTrueWhether to compute residuals for the primal variable.

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Whether to compute residuals for the primal variable.

  • compute_scalar_residualsTrueWhether to compute scalar residuals

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Whether to compute scalar residuals

  • correct_edge_droppingFalseWhether to enable correct edge dropping treatment for mortar constraints. When disabled any Lagrange Multiplier degree of freedom on a secondary element without full primary contributions will be set (strongly) to 0.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether to enable correct edge dropping treatment for mortar constraints. When disabled any Lagrange Multiplier degree of freedom on a secondary element without full primary contributions will be set (strongly) to 0.

  • debug_meshFalseWhether this constraint is going to enable mortar segment mesh debug information. An exodusfile will be generated if the user sets this flag to true

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether this constraint is going to enable mortar segment mesh debug information. An exodusfile will be generated if the user sets this flag to true

  • epsilonPrimary coupled scalar variable

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

    Controllable:No

    Description:Primary coupled scalar variable

  • ghost_higher_d_neighborsFalseWhether we should ghost higher-dimensional neighbors. This is necessary when we are doing second order mortar with finite volume primal variables, because in order for the method to be second order we must use cell gradients, which couples in the neighbor cells.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether we should ghost higher-dimensional neighbors. This is necessary when we are doing second order mortar with finite volume primal variables, because in order for the method to be second order we must use cell gradients, which couples in the neighbor cells.

  • ghost_point_neighborsFalseWhether we should ghost point neighbors of secondary face elements, and consequently also their mortar interface couples.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether we should ghost point neighbors of secondary face elements, and consequently also their mortar interface couples.

  • interpolate_normalsTrueWhether to interpolate the nodal normals (e.g. classic idea of evaluating field at quadrature points). If this is set to false, then non-interpolated nodal normals will be used, and then the _normals member should be indexed with _i instead of _qp

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Whether to interpolate the nodal normals (e.g. classic idea of evaluating field at quadrature points). If this is set to false, then non-interpolated nodal normals will be used, and then the _normals member should be indexed with _i instead of _qp

  • minimum_projection_angle40Parameter to control which angle (in degrees) is admissible for the creation of mortar segments. If set to a value close to zero, very oblique projections are allowed, which can result in mortar segments solving physics not meaningfully, and overprojection of primary nodes onto the mortar segment mesh in extreme cases. This parameter is mostly intended for mortar mesh debugging purposes in two dimensions.

    Default:40

    C++ Type:double

    Controllable:No

    Description:Parameter to control which angle (in degrees) is admissible for the creation of mortar segments. If set to a value close to zero, very oblique projections are allowed, which can result in mortar segments solving physics not meaningfully, and overprojection of primary nodes onto the mortar segment mesh in extreme cases. This parameter is mostly intended for mortar mesh debugging purposes in two dimensions.

  • periodicFalseWhether this constraint is going to be used to enforce a periodic condition. This has the effect of changing the normals vector for projection from outward to inward facing

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether this constraint is going to be used to enforce a periodic condition. This has the effect of changing the normals vector for projection from outward to inward facing

  • primary_variablePrimal variable on primary surface. If this parameter is not provided then the primary variable will be initialized to the secondary variable

    C++ Type:VariableName

    Controllable:No

    Description:Primal variable on primary surface. If this parameter is not provided then the primary variable will be initialized to the secondary variable

  • 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

    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.

  • quadratureDEFAULTQuadrature rule to use on mortar segments. For 2D mortar DEFAULT is recommended. For 3D mortar, QUAD meshes are integrated using triangle mortar segments. While DEFAULT quadrature order is typically sufficiently accurate, exact integration of QUAD mortar faces requires SECOND order quadrature for FIRST variables and FOURTH order quadrature for SECOND order variables.

    Default:DEFAULT

    C++ Type:MooseEnum

    Options:DEFAULT, FIRST, SECOND, THIRD, FOURTH

    Controllable:No

    Description:Quadrature rule to use on mortar segments. For 2D mortar DEFAULT is recommended. For 3D mortar, QUAD meshes are integrated using triangle mortar segments. While DEFAULT quadrature order is typically sufficiently accurate, exact integration of QUAD mortar faces requires SECOND order quadrature for FIRST variables and FOURTH order quadrature for SECOND order variables.

  • secondary_variablePrimal variable on secondary surface.

    C++ Type:VariableName

    Controllable:No

    Description:Primal variable on secondary surface.

  • use_petrov_galerkinFalseWhether to use the Petrov-Galerkin approach for the mortar-based constraints. If set to true, we use the standard basis as the test function and dual basis as the shape function for the interpolation of the Lagrange multiplier variable.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether to use the Petrov-Galerkin approach for the mortar-based constraints. If set to true, we use the standard basis as the test function and dual basis as the shape function for the interpolation of the Lagrange multiplier variable.

  • variableThe name of the lagrange multiplier variable that this constraint is applied to. This parameter may not be supplied in the case of using penalty methods for example

    C++ Type:NonlinearVariableName

    Controllable:No

    Description:The name of the lagrange multiplier variable that this constraint is applied to. This parameter may not be supplied in the case of using penalty methods for example

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>

    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>

    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>

    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

    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

    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>

    Controllable:No

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

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    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

    Controllable:No

    Description:Determines whether this object is calculated using an implicit or explicit form

  • seed0The seed for the master random number generator

    Default:0

    C++ Type:unsigned int

    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

    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

References

  1. F. J. P. Reis and F. M. Andrade Pires. A mortar based approach for the enforcement of periodic boundary conditions on arbitrarily generated meshes. Computer Methods in Applied Mechanics and Engineering, 274:168–191, June 2014. URL: http://www.sciencedirect.com/science/article/pii/S0045782514000528, doi:10.1016/j.cma.2014.01.029.[BibTeX]