InterfaceNormalCurvatures

Computes the two normal curvatures kappa_1 and kappa_2 of a diffuse interface from an order parameter eta.

Overview

InterfaceNormalCurvatures is a Material object that computes two normal curvatures of a diffuse interface defined by an order parameter .

The two normal curvatures characterize how the interface bends in two orthogonal directions tangent to the surface defined by the level set of .

PropertySymbolDirection
kappa1In-plane tangent ( is chosen to lie in the -plane)
kappa2Out-of-plane tangent

As a consistency check, their mean equals the mean curvature: .

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Theory

Interface Normal

The unit normal to the diffuse interface is defined from the order parameter gradient:

(1)

This is well-defined in the interfacial region where . To prevent unreliable calculations when , curvatures are set to zero when is less than the user-settable input parameter gradient_threshold.

Local Tangent Frame

A right-handed orthonormal frame is constructed at every quadrature point.

The first tangent vector is chosen to lie in the -plane by taking the cross product with :

(2)

When the cross product degenerates; in this case falls back to so that the frame remains well-defined everywhere.

The second tangent vector (the binormal) is then:

(3)

which is automatically unit length and orthogonal to both and .

Shape Operator

The curvature of the interface is encoded in the shape operator (Weingarten map), defined as the negative surface gradient of the unit normal:

Differentiating Eq. (1) with the quotient rule gives the full Jacobian of :

(4)

where is the Hessian of .

In compact tensor notation:

Normal Curvature

The normal curvature in a direction tangent to the interface is:

Substituting Eq. (4) and using (since is tangent), the cross-term vanishes identically and the expression simplifies to:

(5)

This is the key formula implemented at each quadrature point. The two scalar outputs are:

Mean Curvature (Diagnostic)

The mean curvature is also declared as a material property for verification. It equals the trace of the shape operator:

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Material Properties Declared

NameTypeDescription
kappa1RealNormal curvature along the tangent (lies in -plane)
kappa2RealNormal curvature along the out-of-plane tangent
kappa_meanRealMean curvature

All property names can be prefixed by setting base_name.

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Requirements

This object requires the second spatial derivatives of to be available at quadrature points. This is only possible when:

  • uses second-order (or higher) Lagrange shape functions (family LAGRANGE, order SECOND), or

  • uses a -continuous basis (e.g. Hermite).

First-order elements will yield zero Hessian values and should not be used with this material.

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Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [curvatures]
    type = InterfaceNormalCurvatures<<<{"description": "Computes the two normal curvatures kappa_1 and kappa_2 of a diffuse interface from an order parameter eta.", "href": "InterfaceNormalCurvatures.html"}>>>
    eta<<<{"description": "Order parameter that defines the interface"}>>> = c
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = 'exodus'
  []
[]
(modules/phase_field/test/tests/misc/interface_normal_curvatures.i)

A minimal [Materials] block:


[Materials]
  [curvature]
    type = InterfaceNormalCurvatures
    eta = eta
    regularization = 1e-8
    base_name = ''
  []
[]

The curvature properties can be visualised by coupling them to MaterialRealAux kernels:


[AuxVariables]
  [kappa1]  [../]
  [kappa2]  [../]
[]

[AuxKernels]
  [kappa1_aux]
    type = MaterialRealAux
    variable = kappa1
    property = kappa1
    execute_on = TIMESTEP_END
  []
  [kappa2_aux]
    type = MaterialRealAux
    variable = kappa2
    property = kappa2
    execute_on = TIMESTEP_END
  []
[]

—

Parameters

Input Parameters

  • etaOrder parameter that defines the interface

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

    Unit:(no unit assumed)

    Controllable:No

    Description:Order parameter that defines the interface

Required Parameters

  • base_nameOptional prefix for all material property names

    C++ Type:std::string

    Controllable:No

    Description:Optional prefix for all material property names

  • blockThe list of blocks (ids or names) that this object will be applied

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

    Controllable:No

    Description:The list of blocks (ids or names) that this object will be applied

  • boundaryThe list of boundaries (ids or names) from the mesh where this object applies

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

    Controllable:No

    Description:The list of boundaries (ids or names) from the mesh where this object applies

  • computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.

    Default:True

    C++ Type:bool

    Controllable:No

    Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.

  • constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

    Default:NONE

    C++ Type:MooseEnum

    Options:NONE, ELEMENT, SUBDOMAIN

    Controllable:No

    Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

  • declare_suffixAn optional suffix parameter that can be appended to any declared 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 declared properties. The suffix will be prepended with a '_' character.

  • gradient_threshold1e-06If |grad(eta)| is less than this threshold at a point, curvatures are set to zero there.

    Default:1e-06

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:If |grad(eta)| is less than this threshold at a point, curvatures are set to zero there.

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.

  • 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

  • search_methodnearest_node_connected_sidesChoice of search algorithm. All options begin by finding the nearest node in the primary boundary to a query point in the secondary boundary. In the default nearest_node_connected_sides algorithm, primary boundary elements are searched iff that nearest node is one of their nodes. This is fast to determine via a pregenerated node-to-elem map and is robust on conforming meshes. In the optional all_proximate_sides algorithm, primary boundary elements are searched iff they touch that nearest node, even if they are not topologically connected to it. This is more CPU-intensive but is necessary for robustness on any boundary surfaces which has disconnections (such as Flex IGA meshes) or non-conformity (such as hanging nodes in adaptively h-refined meshes).

    Default:nearest_node_connected_sides

    C++ Type:MooseEnum

    Options:nearest_node_connected_sides, all_proximate_sides

    Controllable:No

    Description:Choice of search algorithm. All options begin by finding the nearest node in the primary boundary to a query point in the secondary boundary. In the default nearest_node_connected_sides algorithm, primary boundary elements are searched iff that nearest node is one of their nodes. This is fast to determine via a pregenerated node-to-elem map and is robust on conforming meshes. In the optional all_proximate_sides algorithm, primary boundary elements are searched iff they touch that nearest node, even if they are not topologically connected to it. This is more CPU-intensive but is necessary for robustness on any boundary surfaces which has disconnections (such as Flex IGA meshes) or non-conformity (such as hanging nodes in adaptively h-refined meshes).

  • 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

  • output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

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

    Controllable:No

    Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

  • outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object

    Default:none

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

    Controllable:No

    Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

  • 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

    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.

Material Property Retrieval Parameters

Input Files

Input Files

Children Objects

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References

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