KKSMultiACBulkF

KKS model kernel (part 1 of 2) for the Bulk Allen-Cahn. This includes all terms NOT dependent on chemical potential.

Residual

For the 3-phase KKS model, if the non-linear variable is $var element = document.getElementById("moose-equation-87472bb0-68c4-409a-85b7-dd4e90d60970");katex.render("\\eta_1", element, {displayMode:false,throwOnError:false});$ ,

$(1)var element = document.getElementById("moose-equation-21203117-445a-4d65-8e24-da4a5d9182ac");katex.render("R = \\left(\\frac{\\partial h_1}{\\partial \\eta_1} F_1 + \\frac{\\partial h_2}{\\partial \\eta_1} F_2 + \\frac{\\partial h_3}{\\partial \\eta_1} F_3 + W_1 \\frac{\\partial g_1}{\\partial \\eta_1} \\right)", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-57e6ca7b-918e-4155-89ce-59d10baa0f35");katex.render("c_i", element, {displayMode:false,throwOnError:false});$ is the phase concentration for phase $var element = document.getElementById("moose-equation-a89692cd-e40c-4c6b-afbb-bcb1d2bf3b4d");katex.render("i", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-f46c547a-68b8-4c8b-91aa-b75d3ab0c6fe");katex.render("h_i", element, {displayMode:false,throwOnError:false});$ is the interpolation function for phase $var element = document.getElementById("moose-equation-1ac4a87f-d5f5-4015-8b59-a721136910b5");katex.render("i", element, {displayMode:false,throwOnError:false});$ defined in Folch and Plapp (2005) (referred to as $var element = document.getElementById("moose-equation-30e6d4e0-a1f6-471e-884d-a2ac8102f151");katex.render("g_i", element, {displayMode:false,throwOnError:false});$ there, but we use $var element = document.getElementById("moose-equation-229b9b70-d5c5-473a-a9e7-aabce8948477");katex.render("h_i", element, {displayMode:false,throwOnError:false});$ to maintain consistency with other interpolation functions in MOOSE). Here $var element = document.getElementById("moose-equation-167817b7-cf67-4820-be97-b8eefea90392");katex.render("g_i = \\eta_i^2 (1-\\eta_i)^2", element, {displayMode:false,throwOnError:false});$ , also for consistency with notation in MOOSE. $var element = document.getElementById("moose-equation-95ae0dbd-5b93-4182-820a-6490b7474dfe");katex.render("W_1", element, {displayMode:false,throwOnError:false});$ is the free energy barrier height.

Jacobian

On-diagonal

If the non-linear variable is $var element = document.getElementById("moose-equation-ca76fcb8-6428-456f-be9d-dbf53ba32e7c");katex.render("\\eta_1", element, {displayMode:false,throwOnError:false});$ , the on-diagonal Jacobian is

$(2)var element = document.getElementById("moose-equation-565d2e8a-6917-4578-bc28-daa1443bef5d");katex.render("\\begin{aligned} J &=& \\phi_j \\frac{\\partial R}{\\partial \\eta_1} \\\\ &=& \\phi_j \\left( \\frac{\\partial ^2 h_1}{\\partial \\eta_1^2} F_1 + \\frac{\\partial ^2 h_2}{\\partial \\eta_1^2} F_2 + \\frac{\\partial ^2 h_3}{\\partial \\eta_1^2} F_3 + W_1 \\frac{\\partial ^2 g}{\\partial \\eta_1^2} \\right) \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Off-diagonal

Off-diagonal Jacobian for $var element = document.getElementById("moose-equation-73539b41-e161-4333-91a4-b33597177064");katex.render("\\eta_2", element, {displayMode:false,throwOnError:false});$ (similar for $var element = document.getElementById("moose-equation-014a2d96-b1f1-4e0b-b893-ff1a073d3df7");katex.render("\\eta_3", element, {displayMode:false,throwOnError:false});$ ):

$(3)var element = document.getElementById("moose-equation-fe291b8e-2794-42bb-aa4c-58a32a4e3ebd");katex.render("\\begin{aligned} J &=& \\phi_j \\frac{\\partial R}{\\partial \\eta_2} \\\\ &=& \\phi_j \\left( \\frac{\\partial ^2 h_1}{\\partial \\eta_1 \\partial \\eta_2} F_1 + \\frac{\\partial ^2 h_2}{\\partial \\eta_1 \\partial \\eta_2} F_2 + \\frac{\\partial ^2 h_3}{\\partial \\eta_1 \\partial \\eta_2} F_3 \\right) \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Off-diagonal Jacobian for $var element = document.getElementById("moose-equation-f27bb9e1-b601-4bb8-b17c-1b4efd41f111");katex.render("c_1", element, {displayMode:false,throwOnError:false});$ (similar for $var element = document.getElementById("moose-equation-47ddff13-7063-4cff-8de4-656ef9130dc6");katex.render("c_2, c_3", element, {displayMode:false,throwOnError:false});$ ):

$(4)var element = document.getElementById("moose-equation-055eb7df-93c1-4d7c-b2b6-688c8f873807");katex.render("\\begin{aligned} J &=& \\phi_j \\frac{\\partial R}{\\partial c_1} \\\\ &=& \\phi_j \\frac{\\partial h_1}{\\partial \\eta_1} \\frac{\\partial F_1}{\\partial c_1} \\end{aligned}", element, {displayMode:true,throwOnError:false});$

These statements can be generalized for non-linear variable $var element = document.getElementById("moose-equation-20a157cd-e820-4447-a606-83c1c1917614");katex.render("v", element, {displayMode:false,throwOnError:false});$ as:

$(5)var element = document.getElementById("moose-equation-55b08f27-5217-412c-b96b-8a6146dbcd8a");katex.render("\\begin{aligned} J &=& \\phi_j \\frac{\\partial R}{\\partial v} \\\\ &=& \\left( \\frac{\\partial ^2 h_1}{\\partial \\eta_1 \\partial v} F_1 + \\frac{\\partial ^2 h_2}{\\partial \\eta_1 \\partial v} F_2 + \\frac{\\partial ^2 h_3}{\\partial \\eta_1 \\partial v} F_3 + \\frac{\\partial h_1}{\\partial \\eta_1} \\frac{\\partial F_1}{\\partial v} + \\frac{\\partial h_2}{\\partial \\eta_1} \\frac{\\partial F_2}{\\partial v} + \\frac{\\partial h_3}{\\partial \\eta_1} \\frac{\\partial F_3}{\\partial v}\\right) \\end{aligned}", element, {displayMode:true,throwOnError:false});$

For the off-diagonal Jacobians we also need to multiply by $var element = document.getElementById("moose-equation-94f89341-e5a4-4c68-8c74-528ec7b33b03");katex.render("L", element, {displayMode:false,throwOnError:false});$ , the Allen-Cahn mobility.

Input Parameters

eta_iOrder parameter that derivatives are taken with respect to
C++ Type:std::vector
Options:
Description:Order parameter that derivatives are taken with respect to
wiDouble well height parameter
C++ Type:double
Options:
Description:Double well height parameter
variableThe name of the variable that this Kernel operates on
C++ Type:NonlinearVariableName
Options:
Description:The name of the variable that this Kernel operates on
gi_nameBase name for the double well function g_i(eta_i)
C++ Type:MaterialPropertyName
Options:
Description:Base name for the double well function g_i(eta_i)
hj_namesSwitching Function Materials that provide h. Place in same order as Fj_names!
C++ Type:std::vector
Options:
Description:Switching Function Materials that provide h. Place in same order as Fj_names!
Fj_namesList of free energies for each phase. Place in same order as hj_names!
C++ Type:std::vector
Options:
Description:List of free energies for each phase. Place in same order as hj_names!

Required Parameters

mob_nameLThe mobility used with the kernel
Default:L
C++ Type:MaterialPropertyName
Options:
Description:The mobility used with the kernel
argsVector of arguments of the mobility
C++ Type:std::vector
Options:
Description:Vector of arguments of the mobility
blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector
Options:
Description:The list of block ids (SubdomainID) that this object will be applied

Optional Parameters

enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
save_inThe name of auxiliary variables to save this Kernel'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
Options:
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
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
Options:
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.
control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Options:
Description:The seed for the master random number generator
diag_save_inThe name of auxiliary variables to save this Kernel'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
Options:
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Options:
Description:Determines whether this object is calculated using an implicit or explicit form

Advanced Parameters

vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime time
Description:The tag for the vectors this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector
Options:
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
Description:The tag for the matrices this Kernel should fill
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector
Options:
Description:The extra tags for the matrices this Kernel should fill

Tagging Parameters

Input Files

modules/phase_field/test/tests/KKS_system/kks_multiphase.i

Input Parameters
Input Files

Application Usage

Physics and Syntax

Application Development

Framework Development

MOOSEDocs

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Questions

Information and Tools

KKSMultiACBulkF

Residual

Jacobian

On-diagonal

Off-diagonal

Input Parameters

Required Parameters

Optional Parameters

Advanced Parameters

Tagging Parameters

Input Files

KKSMultiACBulkF

Residual

Jacobian

On-diagonal

Off-diagonal

Input Parameters

Required Parameters

Optional Parameters

Advanced Parameters

Tagging Parameters

Input Files

modules/phase_field/test/tests/KKS_system/kks_multiphase.i