BulkEnergyDerivativeSixthCoupledT

Calculates the residual for the local free energy which is an sixth order expansion in the polarization coupled to the thermal field through the first Landau coefficient.

Overview

Computes the residual and jacobian contributions for each component of the polarization vector . Note that if the structure as additional order parameters as is the case for perovskite with its oxygen octahedral antiphase tilts, then this object can also be used for those components. The bulk free energy density, , up to sixth order is defined as,

Repeated indices are summed on for . Here, we include temperature as an explicit MOOSE Variable. The relaxation time-evolution of the polarization is given by,

so we need to compute the variational derivatives of . Moving the RHS over, multiplying by a test function and integrating over the computational volume , we are left with the residual vector,

The variational derivatives of only involve derivatives with respect to each component so we can write explicitly the residual contribution as,

and so on. To compute the on-diagonal jacobian contributions, we look for,

with a shape function of the finite element method. The off-diagonal jacobian contributions provide the coupling to the other components of ,

Since the Kernel also has coupling to temperature as a Variable, we also need the off-diagonal jacobian contribution,

If the material has a cubic parent phase (in the case of conventional perovskite ferroelectrics such as , and ), then most of these terms vanish and we are left with (in Voight notation) nonzero. Typically for , the phenomenological theory only has temperature dependence on the first Landau coefficient. However, for , this is not the case. We should note here in this overview that the sixth order expression with temperature dependence on the first material coefficient for cubic phase materials has been 'hard-coded', but following our example other lower symmetry parent phases can be coded into FERRET.

Example Input File Syntax

Input Parameters

  • TcTransition temperature of unstrained ferroelectric

    C++ Type:double

    Controllable:No

    Description:Transition temperature of unstrained ferroelectric

  • alpha0The coefficients of the Landau expansion

    C++ Type:double

    Controllable:No

    Description:The coefficients of the Landau expansion

  • alpha11The coefficients of the Landau expansion

    C++ Type:double

    Controllable:No

    Description:The coefficients of the Landau expansion

  • alpha111The coefficients of the Landau expansion

    C++ Type:double

    Controllable:No

    Description:The coefficients of the Landau expansion

  • alpha112The coefficients of the Landau expansion

    C++ Type:double

    Controllable:No

    Description:The coefficients of the Landau expansion

  • alpha12The coefficients of the Landau expansion

    C++ Type:double

    Controllable:No

    Description:The coefficients of the Landau expansion

  • alpha123The coefficients of the Landau expansion

    C++ Type:double

    Controllable:No

    Description:The coefficients of the Landau expansion

  • componentAn integer corresponding to the direction in order parameter space this kernel acts in (e.g. for unrotated functionals 0 for q_x, 1 for q_y, 2 for q_z).

    C++ Type:unsigned int

    Controllable:No

    Description:An integer corresponding to the direction in order parameter space this kernel acts in (e.g. for unrotated functionals 0 for q_x, 1 for q_y, 2 for q_z).

  • polar_xThe x component of the polarization

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

    Controllable:No

    Description:The x component of the polarization

  • polar_yThe y component of the polarization

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

    Controllable:No

    Description:The y component of the polarization

  • temperatureThe temperature at the grid point

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

    Controllable:No

    Description:The temperature at the grid point

  • variableThe name of the variable that this residual object operates on

    C++ Type:NonlinearVariableName

    Controllable:No

    Description:The name of the variable that this residual object operates on

Required Parameters

  • 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

  • displacementsThe displacements

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

    Controllable:No

    Description:The displacements

  • len_scale1the len_scale of the unit

    Default:1

    C++ Type:double

    Controllable:No

    Description:the len_scale of the unit

  • polar_zThe z component of the polarization

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

    Controllable:No

    Description:The z component of the polarization

  • 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.

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.

  • 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<AuxVariableName>

    Controllable:No

    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.)

  • 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

  • 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<AuxVariableName>

    Controllable:No

    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.)

  • 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

Input Files