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STConvectiveMatrix example 26

!!! note "" This example shows how to USE the SUBROUTINE called STConvectiveMatrix to create a space-time convective matrix. Triangle3 in space and Line2 in time.

Here, we want to DO the following.

M(I,J,a,b)=InΩcNITaxNJTbtdΩdtM(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial x}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt M(I,J,a,b)=InΩcNITayNJTbtdΩdtM(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial y}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt M(I,J,a,b)=InΩcNITazNJTbtdΩdtM(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial z}\frac{\partial N^{J}T_{b}}{\partial t}d\Omega dt M(I,J,a,b)=InΩcNITatNJTbxdΩdtM(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial t}\frac{\partial N^{J}T_{b}}{\partial x}d\Omega dt M(I,J,a,b)=InΩcNITatNJTbydΩdtM(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial t}\frac{\partial N^{J}T_{b}}{\partial y}d\Omega dt M(I,J,a,b)=InΩcNITatNJTbzdΩdtM(I,J,a,b)=\int_{I_{n}}\int_{\Omega}c\frac{\partial N^{I}T_{a}}{\partial t}\frac{\partial N^{J}T_{b}}{\partial z}d\Omega dt

!!! warning "" cc is scalar [[FEVariable_]], which can be a constant, or a FUNCTION of space-time, or some nonlinear FUNCTION.

In this example, convective matrix is formed for

  • [[ReferenceTriangle_]] Triangle3 element for space
  • [[ReferenceLine_]] Line2 element for time
  • [[QuadraturePoint_]] GaussLegendre
  • constant value of cc

Modules and classes

  • [[ElemshapeData_]]
  • [[STElemshapeData_]]
  • [[QuadraturePoint_]]
  • [[ReferenceLine_]]
  • [[ReferenceTriangle_]]

Usage

PROGRAM main
USE easifemBase
IMPLICIT NONE
TYPE(STElemshapeData_), ALLOCATABLE :: test(:)
TYPE(ElemshapeData_) :: time_elemsd
TYPE(Quadraturepoint_) :: quadFortime
TYPE(Quadraturepoint_) :: quadForspace
TYPE(ReferenceTriangle_):: refelemForSpace
TYPE(ReferenceLine_) :: refelemForTime
INTEGER(I4B) :: ii
INTEGER(I4B), PARAMETER :: nsd=2, nns=3, nnt=2
INTEGER(I4B), PARAMETER :: orderForTime=2, orderForSpace=1
REAL(DFP), PARAMETER :: tij(1, 2) = RESHAPE([-1,1], [1,2])
REAL(DFP), PARAMETER :: xij(2, 3)=RESHAPE([0,0,1,0,0,1], [nsd, nns])
! spatial nodal coordinates
REAL(DFP), ALLOCATABLE :: xija(:, :, :), mat(:,:)
! spatial-temporal nodal coordinates
REAL(DFP), parameter :: c(2)=[1.0, 1.0]
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 = ReferenceTriangle(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, typeFEVariableConstant)
    mat=ConvectiveMatrix(test=test, trial=test, &
& term1=del_x_all, term2=del_x, &
& c=cvar, projectOn='trial' )
CALL Display(mat, "mat:")

??? example "Results"

                                mat:                                   
--------------------------------------------------------------------------
0.666667 -0.333333 -0.333333 0.333333 -0.166667 -0.166667
-0.666667 0.333333 0.333333 -0.333333 0.166667 0.166667
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.666667 -0.333333 -0.333333 0.333333 -0.166667 -0.166667
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
-0.666667 0.333333 0.333333 -0.333333 0.166667 0.166667
0.333333 -0.166667 -0.166667 0.666667 -0.333333 -0.333333
-0.333333 0.166667 0.166667 -0.666667 0.333333 0.333333
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.333333 -0.166667 -0.166667 0.666667 -0.333333 -0.333333
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
-0.333333 0.166667 0.166667 -0.666667 0.333333 0.333333
    mat=convectivematrix(test=test, trial=test, &
& term1=del_x, term2=del_x_all, &
& c=cvar, projecton='test' )
CALL Display(mat, "mat:")

??? example "Results"

                                    mat:                                    
----------------------------------------------------------------------------
0.666667 -0.666667 0.000000 0.666667 0.000000 -0.666667 0.333333 -0.333333 0.000000 0.333333 0.000000 -0.333333
-0.333333 0.333333 0.000000 -0.333333 0.000000 0.333333 -0.166667 0.166667 0.000000 -0.166667 0.000000 0.166667
-0.333333 0.333333 0.000000 -0.333333 0.000000 0.333333 -0.166667 0.166667 0.000000 -0.166667 0.000000 0.166667
0.333333 -0.333333 0.000000 0.333333 0.000000 -0.333333 0.666667 -0.666667 0.000000 0.666667 0.000000 -0.666667
-0.166667 0.166667 0.000000 -0.166667 0.000000 0.166667 -0.333333 0.333333 0.000000 -0.333333 0.000000 0.333333
-0.166667 0.166667 0.000000 -0.166667 0.000000 0.166667 -0.333333 0.333333 0.000000 -0.333333 0.000000 0.333333

!!! settings "Cleanup"

END PROGRAM main