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GetSUPGParam

When obj is an instance of ElemshapeData_, the stabilization parameter is given by:

!!! info "opt=1"

1τ2=1τ12+1τ22+1τ32+1τ42\frac{1}{\tau^{2}}=\frac{1}{\tau_{1}^{2}}+\frac{1}{\tau_{2}^{2}}+\frac{1}{\tau_{3}^{2}}+\frac{1}{\tau_{4}^{2}}

!!! info "opt=2"

1τ=1∣τ1∣+1∣τ2∣+1∣τ3∣+1∣τ4∣\frac{1}{\tau}=\frac{1}{\vert \tau_{1} \vert}+\frac{1}{\vert \tau_{2} \vert}+\frac{1}{\vert\tau_{3} \vert} + \frac{1}{\vert\tau_{4} \vert}

where,

1τ1=∑I∣u⋅∇NI∣\frac{1}{\tau_{1}}=\sum_{I}\vert\boldsymbol{u}\cdot\nabla N^{I}\vert 1τ2=2Δt\frac{1}{\tau_{2}}=\frac{2}{\Delta t} 1τ3=ν(h/2)2\frac{1}{\tau_{3}}=\frac{\nu}{(h/2)^{2}} 1τ4=2νK0\frac{1}{\tau_{4}}=\frac{2\nu}{K_{0}} 2h=∑I∣r⋅∇NI∣\frac{2}{h}=\sum_{I}\vert\boldsymbol{r}\cdot\nabla N^{I}\vert

For scalar variable:

r=∇∣ϕ∣∥∇∣ϕ∣∥\boldsymbol{r}=\frac{\nabla\vert\phi|}{\Vert\nabla\vert\phi\vert\Vert}

For vector variable:

r=∇∥v∥∥∇∥v∥∥\boldsymbol{r}=\frac{\nabla\Vert{\bf v}\Vert}{\Vert\nabla\Vert{\bf v}\Vert\Vert}

When obj is an instance of [[STElemshapeData_]], the stabilization parameter is given by:

!!! info "opt=1"

1τ2=1τ122+1τ32+1τ42\frac{1}{\tau^{2}}=\frac{1}{\tau_{12}^{2}}+\frac{1}{\tau_{3}^{2}}+\frac{1}{\tau_{4}^{2}}

!!! info "opt=2"

1τ=1∣τ12∣+1∣τ3∣+1∣τ4∣\frac{1}{\tau}=\frac{1}{\vert \tau_{12} \vert}+\frac{1}{\vert\tau_{3} \vert} + \frac{1}{\vert\tau_{4} \vert}

where,

1τ12=∑a=1nnt∑I=1nns∣∂NITa∂t+v⋅∇(NITa)∣\frac{1}{\tau_{12}}=\sum_{a=1}^{n_{nt}}\sum_{I=1}^{n_{ns}}\left|\frac{\partial N^{I}T_{a}}{\partial t}+{\mathbf{v}}\cdot\nabla\left(N^{I}T_{a}\right)\right| 1τ3=4νhRGN2\frac{1}{\tau_{3}}=\frac{4\nu}{h_{RGN}^{2}} 1τ4=2νK0\frac{1}{\tau_{4}}=\frac{2\nu}{K_{0}}

!!! example "" You can learn about this method from expample [[_ElemshapeData_test_14]]

!!! info "Interface-1"

MODULE PURE SUBROUTINE GetSUPGParam(obj, tau, c, val, nu, k, &
  & phi, dt, opt)
  CLASS(ElemshapeData_), INTENT(IN) :: obj
  !! It can be an instance of ElemshapeData_ or STElemshapeData_
  TYPE(FEVariable_), INTENT(INOUT) :: tau
  !! Stabilization parameter
  !! Quadrature FEVariable
  !! varType=Space
  TYPE(FEVariable_), INTENT(IN) :: c
  !! Convective velocity => Vector variable
  TYPE(FEVariable_), INTENT(IN) :: val
  !! solution, it can be scalar or vector variable
  TYPE(FEVariable_), INTENT(IN) :: nu
  !! diffusivity
  !! In case of NSE it should be mu/rho
  TYPE(FEVariable_), OPTIONAL, INTENT(IN) :: k
  !! permeability
  TYPE(FEVariable_), OPTIONAL, INTENT(IN) :: phi
  !! porosity
  REAL(DFP), OPTIONAL, INTENT(IN) :: dt
  !! time step size
  INTEGER(I4B), OPTIONAL, INTENT(IN) :: opt
  !! futuristic options
END SUBROUTINE GetSUPGParam

!!! note "" obj can be an instance of [[ElemshapeData_]] or [[STElemshapeData_]]

In interface-1 the meaning of variables is given below:

  • obj instance of ElemshapeData_
  • tau, stabilization parameter τ\tau, it is an instance of [[FEVariable_]].
    • Rank=Scalar
    • definedOn=Quadrature
    • varType=Space (changes in space)
  • c, is convective velocity, it is an instance of [[FEVariable_]] with following properties
    • Rank=Vector
    • definedOn=Quadrature or Nodes
    • varType=Constant or Space (if obj is an instance of [[ElemshapeData_]])
    • varType=Constant, Space, SpaceTime (if obj is an instance of [[STElemshapeData_]]
  • val, is solution, an instance of [[FEVariable_]], with following properties::
    • Rank=Scalar (for convection-diffusion problems)
    • Rank=Vector (for Navier-Stokes equations)
    • varType=Constant or Space (if obj is an instance of [[ElemshapeData_]])
    • varType=Constant, Space, SpaceTime (if obj is an instance of [[STElemshapeData_]]
  • nu, denotes diffusivity or kinematic viscosity ν\nu, an instance of [[FEVariable_]]
    • Rank=Scalar
    • definedOn=Quadrature or Nodes
    • varType=Constant or Space (if obj is an instance of [[ElemshapeData_]])
    • varType=Constant, Space, SpaceTime (if obj is an instance of [[STElemshapeData_]]
  • k, denotes permeability of porous medium, an instance of [[FEVariable_]]
    • Rank=Scalar
    • definedOn=Quadrature or Nodes
    • varType=Constant or Space (if obj is an instance of [[ElemshapeData_]])
    • varType=Constant, Space, SpaceTime (if obj is an instance of [[STElemshapeData_]]
  • phi, denotes porosity of porous medium, an instance of [[FEVariable_]]
    • Rank=Scalar
    • definedOn=Quadrature or Nodes
    • varType=Constant or Space (if obj is an instance of [[ElemshapeData_]])
    • varType=Constant, Space, SpaceTime (if obj is an instance of [[STElemshapeData_]]
  • dt, time step

!!! info "Interface-2"

MODULE PURE SUBROUTINE GetSUPGParam(obj, tau, c, val, nu, k, &
  & phi, dt, opt)
  CLASS(STElemshapeData_), INTENT(IN) :: obj(:)
  !! space-time shape data
  TYPE(FEVariable_), INTENT(INOUT) :: tau
  !! Stabilization parameter
  !! Quadrature type
  !! SpaceTime
  TYPE(FEVariable_), INTENT(IN) :: c
  !! convective velocity
  !! vector FEVariable
  TYPE(FEVariable_), INTENT(IN) :: val
  !! solution
  !! scalar or vector FEVariable
  TYPE(FEVariable_), INTENT(IN) :: nu
  !! kinematic diffusivity
  TYPE(FEVariable_), OPTIONAL, INTENT(IN) :: k
  !! permeability
  !! Scalar FEVariable
  TYPE(FEVariable_), OPTIONAL, INTENT(IN) :: phi
  !! porosity
  !! Scalar FEVariable
  REAL(DFP), OPTIONAL, INTENT(IN) :: dt
  !! time step size
  INTEGER(I4B), OPTIONAL, INTENT(IN) :: opt
  !! futuristic options
END SUBROUTINE GetSUPGParam

In interface-2 the meaning of variables is given below:

  • obj a vector of [[STElemshapeData_]]
  • tau, stabilization parameter τ\tau, it is an instance of [[FEVariable_]].
    • Rank=Scalar
    • definedOn=Quadrature
    • varType=SpaceTime (changes in space and time)
  • c, is convective velocity, it is an instance of [[FEVariable_]] with following properties
    • Rank=Vector
    • definedOn=Quadrature or Nodes
    • varType=Constant, Space, SpaceTime
  • val, is solution, an instance of [[FEVariable_]], with following properties::
    • Rank=Scalar (for convection-diffusion problems)
    • Rank=Vector (for Navier-Stokes equations)
    • varType=Constant, Space, SpaceTime
  • nu, denotes diffusivity or kinematic viscosity ν\nu, an instance of [[FEVariable_]]
    • Rank=Scalar
    • definedOn=Quadrature or Nodes
    • varType=Constant, Space, SpaceTime
  • k, denotes permeability of porous medium, an instance of [[FEVariable_]]
    • Rank=Scalar
    • definedOn=Quadrature or Nodes
    • varType=Constant, Space, SpaceTime
  • phi, denotes porosity of porous medium, an instance of [[FEVariable_]]
    • Rank=Scalar
    • definedOn=Quadrature or Nodes
    • varType=Constant, Space, SpaceTime
  • dt, time step

!!! info "Interface-3" This interface is similar to the interface-1, but in this interface material properties nu, k, phi are constant real values.

MODULE PURE SUBROUTINE GetSUPGParam(obj, tau, c, val, nu, k, &
  & phi, dt, opt)
  CLASS(ElemshapeData_), INTENT(IN) :: obj
  !! It can be an instance of ElemshapeData_ or STElemshapeData_
  TYPE(FEVariable_), INTENT(INOUT) :: tau
  !! Stabilization parameter
  !! Quadrature FEVariable
  !! varType=Space
  TYPE(FEVariable_), INTENT(IN) :: c
  !! Convective velocity => Vector variable
  TYPE(FEVariable_), INTENT(IN) :: val
  !! solution, it can be scalar or vector variable
  REAL( DFP ), INTENT(IN) :: nu
  !! In case of NSE it should be mu/rho
  !! diffusivity
  REAL( DFP ), OPTIONAL, INTENT(IN) :: k
  !! permeability
  REAL( DFP ), OPTIONAL, INTENT(IN) :: phi
  !! porosity
  REAL(DFP), OPTIONAL, INTENT(IN) :: dt
  !! time step size
  INTEGER(I4B), OPTIONAL, INTENT(IN) :: opt
  !! futuristic options
END SUBROUTINE GetSUPGParam

!!! info "Interface-4" This interface is similar to the Interface-2, but in this interface material properties nu, k, phi are constant real values.

MODULE PURE SUBROUTINE GetSUPGParam(obj, tau, c, val, nu, k, &
  & phi, dt, opt)
  CLASS(STElemshapeData_), INTENT(IN) :: obj(:)
  !! space-time shape data
  TYPE(FEVariable_), INTENT(INOUT) :: tau
  !! Stabilization parameter
  !! Quadrature type
  !! SpaceTime
  TYPE(FEVariable_), INTENT(IN) :: c
  !! convective velocity
  !! vector FEVariable
  TYPE(FEVariable_), INTENT(IN) :: val
  !! solution
  !! scalar or vector FEVariable
  REAL( DFP ), INTENT(IN) :: nu
  !! kinematic diffusivity
  REAL( DFP ), OPTIONAL, INTENT(IN) :: k
  !! permeability
  !! Scalar FEVariable
  REAL( DFP ), OPTIONAL, INTENT(IN) :: phi
  !! porosity
  !! Scalar FEVariable
  REAL(DFP), OPTIONAL, INTENT(IN) :: dt
  !! time step size
  INTEGER(I4B), OPTIONAL, INTENT(IN) :: opt
  !! futuristic options
END SUBROUTINE GetSUPGParam
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