ConvectiveMatrix example 3

!!! note "" This example shows how to use the subroutine called ConvectiveMatrix to create a convective matrix in space domain for Line2 element.

Here, we want to do the following.

$$M(I,J) = \int_{\Omega} N^I c_k \frac{\partial N^J}{\partial x_k} d{\Omega}$$ $$M(I,J) = \partial u_{iI} \int_{\Omega} c_k \frac{\partial N^I}{\partial x_k} N^J d{\Omega}$$

!!! warning "" $c$ is convective velocity, which can be a constant, or a function of spatial coordinates, or some nonlinear function.

In this example, convective matrix is formed for

• [[ReferenceLine_]] element
• [[QuadraturePoint_]] GaussLegendre
• velocity is given by quadrature values

Modules and classes

• [[ElemshapeData_]]
• [[QuadraturePoint_]]
• [[ReferenceLine_]]
• [[FEVariable_]]

Usage

PROGRAM main
    USE easifemBase
    IMPLICIT NONE
    TYPE(ElemshapeData_) :: test, trial
    TYPE(QuadraturePoint_) :: quad
    TYPE(ReferenceLine_) :: refelem
    type(FEVariable_) :: cvar
    REAL(DFP), PARAMETER :: c(1,1) = reshape([1.0,1.0],[1,1])
    REAL(DFP), ALLOCATABLE :: mat(:, :)
    REAL(DFP), ALLOCATABLE :: XiJ(:, :)
    integer( I4B ), PARAMETER :: order = 1

!!! note "" Let us now create the physical coordinate of the line element.

    XiJ = RESHAPE([-1, 1], [1, 2])

!!! note "" Now we create an instance of [[ReferenceLine_]].

    refelem = referenceline(nsd=1)

!!! note "" Here, we create the quadrature points.

    CALL initiate( obj=quad, refelem=refelem, order=order+order-1, &
        & quadratureType='GaussLegendre' )

!!! note "" Initiate an instance of [[ElemshapeData_]]. You can learn more about it from [[ElemshapeData_test]]

    CALL initiate(obj=test, &
        & quad=quad, &
        & refelem=refelem, &
        & ContinuityType=typeH1, &
        & InterpolType=typeLagrangeInterpolation)
    CALL Set( obj=test, val=xij, N=test%N, dNdXi=test%dNdXi)

!!! note "" Let us now create an instance of [[FEVariable_]] to wrape the nodal values of $c$. You can learn more about this at [[FEVariable_test]]

    cvar = QuadratureVariable(c, typeFEVariableVector, typeFEVariableSpace)

!!! note "" Let us now create the convective matrix.

    mat=ConvectiveMatrix(test=test, trial=test, term1=1, term2=0, c=cvar)
    CALL Display(mat, "mat:")

??? example "Results"

        mat:
--------------------
-0.500000  -0.500000
 0.500000   0.500000

!!! note "" Let us now create the convective matrix.

    mat=ConvectiveMatrix(test=test, trial=test, term1=0, term2=1, c=cvar)
    CALL Display(mat, "mat:")

??? example "Results"

        mat:
-------------------
-0.500000  0.500000
-0.500000  0.500000

!!! settings "Cleanup"

END PROGRAM main