STConvectiveMatrix example 2

!!! note "" This example shows how to USE the SUBROUTINE called ConvectiveMatrix to create a space-time convective matrix. Line2 in space and time.

Here, we want to DO the following.

M\left( {I,J,a,b} \right) = {\int_{{I_n}}^{} {\int_\Omega ^{} {{c_j}\frac{{\partial {N^I}{T_a}}}{{\partial {x_j}}} \cdot {N^J}{T_b}d\Omega dt} } }

M\left(I,J,a,b\right)=\int_{I_{n}}\int_{\Omega}N^{I}T_{a}c_{j}\frac{\partial N^{J}T_{b}}{\partial x_{j}}d\Omega dt

!!! warning "" $c$ is convective velocity, which can be a constant, or a FUNCTION of space-time, or some nonlinear FUNCTION.

In this example, convective matrix is formed for

[[ReferenceLine_]] Line2 element for space
[[ReferenceLine_]] Line2 element for time
[[QuadraturePoint_]] GaussLegendre
nodal velocity, constant in time

Modules and classes

[[ElemshapeData_]]

Usage

PROGRAM main
    USE easifemBase
    IMPLICIT NONE
    TYPE(STElemshapeData_), ALLOCATABLE :: test(:)
    TYPE(ElemshapeData_) :: time_elemsd
    TYPE(Quadraturepoint_) :: quadFortime
    TYPE(Quadraturepoint_) :: quadForspace
    TYPE(ReferenceLine_):: refelemForSpace
    TYPE(ReferenceLine_) :: refelemForTime
    INTEGER(I4B) :: ii
    INTEGER(I4B), PARAMETER :: nsd=1, nns=2, nnt=2
    INTEGER(I4B), PARAMETER :: orderForTime=2, orderForSpace=1
    REAL(DFP), PARAMETER :: tij(1, 2) = RESHAPE([-1,1], [1,2])
    REAL(DFP), PARAMETER :: xij(1, 2)=RESHAPE([-1,1], [nsd, nns])
    ! spatial nodal coordinates
    REAL(DFP), ALLOCATABLE :: xija(:, :, :), mat(:,:)
    ! spatial-temporal nodal coordinates
    REAL(DFP), parameter :: c(1,2) = reshape([1.0,1.0],[1,2])
    type(FEVariable_) :: cvar

!!! note "" First, we initiate a [[ReferenceLine_]] element for time domain. Note that nsd should be 1 when making reference element for time domain. Generate the quadrature points, and initiates an instance of [[ElemshapeData_]].

  refelemForTime= ReferenceLine(nsd=1)
  CALL Initiate(obj=quadFortime, &
& refelem=refelemForTime,&
& order=orderForTime, &
    	& quadratureType="GaussLegendre" )
  CALL Initiate( &
  	& obj=time_elemsd, &
 	& quad=quadForTime, &
& refelem=refelemForTime, &
    	& ContinuityType=typeH1,&
& InterpolType=TypeLagrangeInterpolation)
  CALL Set(obj=time_elemsd, &
& val=tiJ, N=time_elemsd%N, &
      & dNdXi=time_elemsd%dNdXi)

!!! note "" Initiate [[STElemshapeData_]].

CALL Initiate(obj=test, elemsd=time_elemsd)

!!! note "" Generating shape functions for space-elements. Here, we are selecting a triangular element

  refelemForSpace = ReferenceLine(nsd=nsd)
  CALL Initiate(obj=quadForSpace, &
& refelem=refelemForSpace, &
& order=orderForSpace, &
& quadratureType='GaussLegendre')

  DO ii = 1, SIZE(test)
    CALL Initiate( obj=test(ii), &
 	& quad=quadForSpace, &
& refelem=refelemForSpace, &
      & ContinuityType=typeH1, &
& InterpolType=TypeLagrangeInterpolation)
  END DO

!!! note "" Setting the remaining DATA in obj. Here, xija are the space-time nodal coordinates.

CALL Reallocate(xija, nsd, nns, nnt)
   DO ii = 1, nnt; xija(:, :, ii) = xij; END DO
   DO ii = 1, SIZE(test)
       CALL Set(obj=test(ii), &
           & val=xija, &
		& N=test(ii)%N, &
           & dNdXi=test(ii)%dNdXi, &
           & T=test(ii)%T)
   END DO

!!! note "" Let us now create the space-time convective matrix.

cvar = NodalVariable(c, typeFEVariableVector, typeFEVariableSpace)

mat=ConvectiveMatrix(test=test, trial=test, term1=del_none, &
    & term2=del_x, c=cvar)
CALL Display(mat, "mat:")

??? example "Results"

                mat:                  
----------------------------------------
-0.333333  0.333333  -0.166667  0.166667
-0.333333  0.333333  -0.166667  0.166667
-0.166667  0.166667  -0.333333  0.333333
-0.166667  0.166667  -0.333333  0.333333

mat=ConvectiveMatrix(test=test, trial=test, term1=del_x, &
    & term2=del_none, c=cvar)
CALL Display(mat, "mat:")

??? example "Results"

                mat:                   
------------------------------------------
-0.333333  -0.333333  -0.166667  -0.166667
0.333333   0.333333   0.166667   0.166667
-0.166667  -0.166667  -0.333333  -0.333333
0.166667   0.166667   0.333333   0.333333

!!! settings "Cleanup"

END PROGRAM main

Modules and classes​

Usage​

Modules and classes

Usage