ComputeWeightedGapCartesianLMMechanicalContact

The Karush-Kuhn-Tucker conditions of mechanical contact are:

$var element = document.getElementById("moose-equation-fdd368ac-5c3f-4ce0-ad6e-96a3d99fdd2f");katex.render("\\begin{aligned} g &\\ge 0\\\\ \\lambda &\\ge 0\\\\ g\\lambda &= 0 \\end{aligned}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-66d5d83e-1f43-4b6f-9680-de07e999df92");katex.render("g", element, {displayMode:false,throwOnError:false});$ is the gap and $var element = document.getElementById("moose-equation-c584b064-928c-42bb-a7d6-11429b9e7a45");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ is the contact pressure, a Lagrange multiplier variable living on the secondary face. Per (Wohlmuth, 2011) and (Popp and Wall, 2014), the variationally consistent, discretized version of the KKT conditions are:

$var element = document.getElementById("moose-equation-560f4d0f-da26-4869-8269-4324618ed4d1");katex.render("\\begin{aligned} (\\tilde{g}_n)_j &\\ge 0\\\\ (\\lambda_n)_j &\\ge 0\\\\ (\\tilde{g}_n)_j (\\lambda_n)_j &= 0 \\end{aligned}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-503f304e-3f21-47c0-b724-923876017f50");katex.render("n", element, {displayMode:false,throwOnError:false});$ indicates the normal direction, $var element = document.getElementById("moose-equation-2993965b-b284-4280-9d34-7c4e113ea598");katex.render("j", element, {displayMode:false,throwOnError:false});$ denotes the j'th secondary contact interface node, and $var element = document.getElementById("moose-equation-a3001d89-0a51-46b5-b522-e2d2d0cc7245");katex.render("(\\tilde{g}_n)_j", element, {displayMode:false,throwOnError:false});$ is the discrete weighted gap, computed by:

$var element = document.getElementById("moose-equation-461e9ba8-a950-4530-ac46-f0978cce6030");katex.render("(\\tilde{g}_n)_j = \\int_{\\gamma_c^{(1)}} \\Phi_j g_{n,h} dA", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-da6ed6ca-fbdf-4312-a02c-c92d5bd60a95");katex.render("\\gamma_c^{(1)}", element, {displayMode:false,throwOnError:false});$ denotes the secondary contact interface, $var element = document.getElementById("moose-equation-daaf0c53-8684-4ffc-bcb3-4824a1075b6d");katex.render("\\Phi_j", element, {displayMode:false,throwOnError:false});$ is the j'th lagrange multiplier test function, and $var element = document.getElementById("moose-equation-abeaa83d-67a5-4144-8e57-18a67386550f");katex.render("g_{n,h}", element, {displayMode:false,throwOnError:false});$ is the discretized version of the gap function.

The KKT conditions are enforced using a nonlinear complementarity problem (NCP) function, in this case the most simple such function, $var element = document.getElementById("moose-equation-7f505697-c975-4f2c-8f78-63fad0eedf5b");katex.render("min(c(\\tilde{g}_n)_j, (\\lambda_n)_j)", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-d209ac8e-306c-4222-9c54-1733a39a49c0");katex.render("c", element, {displayMode:false,throwOnError:false});$ (implemented with the input parameter c) is used to balance the size of the gap and the normal contact pressure. If the contact pressure is of order 10000, and the gap is of order .01, then c should be set to 1e6 in order to bring components of the NCP function onto the same level and achieve optimal convergence in the non-linear solve.

The ComputeWeightedGapCartesianLMMechanicalContact object computes the weighted gap and applies the KKT conditions using Lagrange multipliers defined in a global Cartesian reference frame. As a consequence, the number of contact constraints at each node will be two, in two-dimensional problems, and three, in three-dimensional problems. The normal contact pressure is obtained by projecting the Lagrange multiplier vector along the normal vector computed from the mortar generation objects. The result is a normal contact constraint, which, in general, will be a function of all (two or three) Cartesian Lagrange multipliers. This methodology only constrains one degree of freedom. The other degree(s) of freedom are constrained by enforcing that tangential tractions are identically zero. Note that, if friction with Cartesian Lagrange multipliers is chosen via ComputeFrictionalForceCartesianLMMechanicalContact, those remaining nodal degrees of freedom are constraint using Coulomb constraints within a semi-smooth Newton approach. Usage of Cartesian Lagrange multipliers is recommended when condensing Lagrange multipliers via the variable condensation preconditioner (VCP) VariableCondensationPreconditioner.

The user can also employ locally oriented Lagrange multipliers ComputeWeightedGapLMMechanicalContact, which minimizes the number of contact constraints for frictionless problems.

Computes the weighted gap that will later be used to enforce the zero-penetration mechanical contact conditions

Input Parameters

disp_xThe x displacement variable
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The x displacement variable
disp_yThe y displacement variable
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The y displacement variable
lm_xMechanical contact Lagrange multiplier along the x Cartesian axis
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Mechanical contact Lagrange multiplier along the x Cartesian axis
lm_yMechanical contact Lagrange multiplier along the y Cartesian axis.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Mechanical contact Lagrange multiplier along the y Cartesian axis.
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.

Required Parameters

aux_lmAuxiliary Lagrange multiplier variable that is utilized together with the Petrov-Galerkin approach.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Auxiliary Lagrange multiplier variable that is utilized together with the Petrov-Galerkin approach.
c1e+06Parameter for balancing the size of the gap and contact pressure
Default:1e+06
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Parameter for balancing the size of the gap and contact pressure
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.
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
disp_zThe z displacement variable
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The z displacement 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_normalsFalseWhether 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:False
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
lm_zMechanical contact Lagrange multiplier along the z Cartesian axis.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Mechanical contact Lagrange multiplier along the z Cartesian axis.
matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)
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
Unit:(no unit assumed)
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.
normalize_cFalseWhether to normalize c by weighting function norm. When unnormalized the value of c effectively depends on element size since in the constraint we compare nodal Lagrange Multiplier values to integrated gap values (LM nodal value is independent of element size, where integrated values are dependent on element size).
Default:False
C++ Type:bool
Controllable:No
Description:Whether to normalize c by weighting function norm. When unnormalized the value of c effectively depends on element size since in the constraint we compare nodal Lagrange Multiplier values to integrated gap values (LM nodal value is independent of element size, where integrated values are dependent on element size).
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
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.
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
Unit:(no unit assumed)
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

Contribution To Tagged Field Data 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_meshTrueWhether 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:True
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

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

References

Alexander Popp and WA Wall. Dual mortar methods for computational contact mechanics–overview and recent developments. GAMM-Mitteilungen, 37(1):66–84, 2014.

@article{popp2014dual,
    author = "Popp, Alexander and Wall, WA",
    title = "Dual mortar methods for computational contact mechanics--overview and recent developments",
    journal = "GAMM-Mitteilungen",
    volume = "37",
    number = "1",
    pages = "66--84",
    year = "2014",
    publisher = "Wiley Online Library"
}

Barbara Wohlmuth. Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numerica, 20:569–734, 2011.

@article{wohlmuth2011variationally,
    author = "Wohlmuth, Barbara",
    title = "Variationally consistent discretization schemes and numerical algorithms for contact problems",
    journal = "Acta Numerica",
    volume = "20",
    pages = "569--734",
    year = "2011",
    publisher = "Cambridge University Press"
}

Input Parameters
References