GetSUPGParam

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

!!! info "opt=1"

$$\frac{1}{\tau^{2}}=\frac{1}{\tau_{1}^{2}}+\frac{1}{\tau_{2}^{2}}+\frac{1}{\tau_{3}^{2}}+\frac{1}{\tau_{4}^{2}}$$

!!! info "opt=2"

$$\frac{1}{\tau}=\frac{1}{\vert \tau_{1} \vert}+\frac{1}{\vert \tau_{2} \vert}+\frac{1}{\vert\tau_{3} \vert} + \frac{1}{\vert\tau_{4} \vert}$$

where,

$$\frac{1}{\tau_{1}}=\sum_{I}\vert\boldsymbol{u}\cdot\nabla N^{I}\vert$$ $$\frac{1}{\tau_{2}}=\frac{2}{\Delta t}$$ $$\frac{1}{\tau_{3}}=\frac{\nu}{(h/2)^{2}}$$ $$\frac{1}{\tau_{4}}=\frac{2\nu}{K_{0}}$$ $$\frac{2}{h}=\sum_{I}\vert\boldsymbol{r}\cdot\nabla N^{I}\vert$$

For scalar variable:

$$\boldsymbol{r}=\frac{\nabla\vert\phi|}{\Vert\nabla\vert\phi\vert\Vert}$$

For vector variable:

$$\boldsymbol{r}=\frac{\nabla\Vert{\bf v}\Vert}{\Vert\nabla\Vert{\bf v}\Vert\Vert}$$

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

!!! info "opt=1"

$$\frac{1}{\tau^{2}}=\frac{1}{\tau_{12}^{2}}+\frac{1}{\tau_{3}^{2}}+\frac{1}{\tau_{4}^{2}}$$

!!! info "opt=2"

$$\frac{1}{\tau}=\frac{1}{\vert \tau_{12} \vert}+\frac{1}{\vert\tau_{3} \vert} + \frac{1}{\vert\tau_{4} \vert}$$

where,

$$\frac{1}{\tau_{12}}=\sum_{a=1}^{n_{nt}}\sum_{I=1}^{n_{ns}}\left|\frac{\partial N^{I}T_{a}}{\partial t}+{\mathbf{v}}\cdot\nabla\left(N^{I}T_{a}\right)\right|$$ $$\frac{1}{\tau_{3}}=\frac{4\nu}{h_{RGN}^{2}}$$ $$\frac{1}{\tau_{4}}=\frac{2\nu}{K_{0}}$$

!!! 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 $\tau$, 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 $\nu$, 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 $\tau$, 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 $\nu$, 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