MassLumpedTimeDerivative

Description

Lumping of the time derivative can be useful for a couple of reasons. Lumping helps ensure conservation of mass at a node. In a standard node-based Galerkin approximation, fluxes from spatial terms can be thought of as "entering" nodes. If there is no flux to a node, then the mass at that node should stay fixed. However, if the standard Galerkin method is applied to a time derivative term, $var element = document.getElementById("moose-equation-ce545f7f-1fb9-4cb1-a731-0f58a5dc5e6f");katex.render("(\\psi_i, \\frac{\\partial u_h}{\\partial t})", element, {displayMode:false,throwOnError:false});$ the corresponding coefficient matrix is tri-diagonal and the mass at a node is affected by fluxes to neighboring nodes. This can lead to violation of local mass conservation and generation of spurious oscillations with unphysical under- and over-shoot phenomena. Lumping fixes this problem by isolating a nodal solution from neighboring nodal solutions in the time derivative term. Mathematically, lumping looks like this. We start with our governing equation $var element = document.getElementById("moose-equation-80a781ea-37e6-4eba-9b0d-fcaffa3c1011");katex.render("u_t = Au", element, {displayMode:false,throwOnError:false});$ where A is a differential operator. We write our finite element solution as

$var element = document.getElementById("moose-equation-c199a497-e3a1-4070-9d88-7447c56dd072");katex.render(" u(t, x) = \\sum u_j(t) \\phi_j(x)", element, {displayMode:true,throwOnError:false});$

Substituting into our governing equation, we have:

$var element = document.getElementById("moose-equation-521180e2-88e8-4b48-acea-daa224e53ef2");katex.render(" \\sum u_j'\\phi_j = \\sum u_jA\\phi_j", element, {displayMode:true,throwOnError:false});$

Now we apply our test functions $var element = document.getElementById("moose-equation-e63eeaa1-572f-4f3b-bda0-e0a5de16cd11");katex.render("\\psi_i", element, {displayMode:false,throwOnError:false});$ and integrate over the volume:

$var element = document.getElementById("moose-equation-a74d3785-0e60-4f8b-ab53-1e0fd8526e06");katex.render(" \\sum u_j' (\\psi_i,\\phi_j) = \\sum u_j (\\psi_i, A\\phi_j)", element, {displayMode:true,throwOnError:false});$

After applying all of our test functions, we have the matrix system

$var element = document.getElementById("moose-equation-cfa01627-783b-4030-992f-5e1ab4778de5");katex.render(" \\widetilde{M}\\vec{u'} = \\widetilde{K}\\vec{u}", element, {displayMode:true,throwOnError:false});$

where here $var element = document.getElementById("moose-equation-f926586a-53b6-4e69-b190-1f302cbfaac2");katex.render("\\vec{u}", element, {displayMode:false,throwOnError:false});$ denotes the vector of coefficients $var element = document.getElementById("moose-equation-d7390edc-2f84-4adc-954c-a915d9ca7e23");katex.render("u_j", element, {displayMode:false,throwOnError:false});$ . $var element = document.getElementById("moose-equation-79a75efd-03e8-4f7c-94ce-d03cd6d99bc2");katex.render("\\widetilde{M}", element, {displayMode:false,throwOnError:false});$ is the mass matrix and $var element = document.getElementById("moose-equation-e5ac5dd5-1d47-4c54-b8f9-7bd4323d2b25");katex.render("\\widetilde{K}", element, {displayMode:false,throwOnError:false});$ is the stiffness matrix. Note that neither of these matrices are ever explicitly formed in MOOSE but they are still directly relevant. As mentioned previously, a standard Galerkin procedure results in a tri-diagonal $var element = document.getElementById("moose-equation-169f5782-731b-40c9-bc82-f7676851f2e0");katex.render("\\widetilde{M}", element, {displayMode:false,throwOnError:false});$ . As it turns out, the sum of the matrix row elements in $var element = document.getElementById("moose-equation-af28371a-8cb6-4cba-933f-5c46226b3aff");katex.render("\\widetilde{M}", element, {displayMode:false,throwOnError:false});$ is one. Lumping then consists of summing the matrix row elements (result 1) and placing the sums on the diagonals. The result is the identity matrix and $var element = document.getElementById("moose-equation-d3099f85-cd4b-46be-b65c-b336b2a719e2");katex.render("\\widetilde{M}\\vec{u'}", element, {displayMode:false,throwOnError:false});$ becomes simply $var element = document.getElementById("moose-equation-37a8b83a-1cf0-486b-806d-e3dde5886728");katex.render("\\vec{u'}", element, {displayMode:false,throwOnError:false});$ :

$var element = document.getElementById("moose-equation-affb03e8-1e34-44a9-a008-9079ab6f481d");katex.render(" \\vec{u'} = \\widetilde{K}\\vec{u}", element, {displayMode:true,throwOnError:false});$

As seen by the above equation, besides helping with local conservation of mass, mass lumping makes explicit time stepping feasible because it removes the need for solving a linear system.

Example Syntax

The MassLumpedTimeDerivative syntax is simple, only taking type and variable as shown in the kernel block below. This particular test file, from which the kernel block is taken, demonstrates a case where the concentration of $var element = document.getElementById("moose-equation-3a18097f-6bd1-4cf7-a2bf-16ca6f374610");katex.render("u", element, {displayMode:false,throwOnError:false});$ would become negative in a non-lumped scheme for sufficiently small time steps.

[Kernels]
  [./time_deriv]
    type = MassLumpedTimeDerivative
    variable = u
  [../]
  [./diff]
    type = FuncCoefDiffusion
    variable = u
    coef = diff_f
  [../]
[]

Input Parameters

variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
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
Unit:(no unit assumed)
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.

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>
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystem timeThe tag for the matrices this Kernel should fill
Default:system time
C++ Type:MultiMooseEnum
Unit:(no unit assumed)
Options:nontime, system, time
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagstimeThe tag for the vectors this Kernel should fill
Default:time
C++ Type:MultiMooseEnum
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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>
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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
Unit:(no unit assumed)
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

(moose/test/tests/kernels/mass_lumping/mass_lumping.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = -1
  xmax = 1
  nx = 2
[]

[Variables]
  [./u]
  [../]
[]

[ICs]
  [./u_init]
    type = FunctionIC
    variable = u
    function = init_f
  [../]
[]

[Kernels]
  [./time_deriv]
    type = MassLumpedTimeDerivative
    variable = u
  [../]
  [./diff]
    type = FuncCoefDiffusion
    variable = u
    coef = diff_f
  [../]
[]

[Functions]
  [./init_f]
    type = ParsedFunction
    expression = max(x,0) #(x>0)
  [../]
  [./diff_f]
    type = ParsedFunction
    expression = max(x,0)
  [../]
[]


[Executioner]
  type = Transient
  end_time = 1
  solve_type = 'NEWTON'
[]

[Outputs]
  exodus = true
[]

Description
Example Syntax
Input Parameters