LagrangeEvalAll

Evaluate Lagrange polynomials on Quadrangle.

Interface 1

܀ Interface

INTERFACE LagrangeEvalAll_Quadrangle
  MODULE FUNCTION LagrangeEvalAll_Quadrangle1( &
    & order, &
    & x, &
    & xij, &
    & refQuadrangle, &
    & coeff, &
    & firstCall, &
    & basisType, &
    & alpha, &
    & beta, &
    & lambda) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: order
    !! order of Lagrange polynomials
    REAL(DFP), INTENT(IN) :: x(2)
    !! point of evaluation
    !! x(1) is x coord
    !! x(2) is y coord
    REAL(DFP), INTENT(INOUT) :: xij(:, :)
    !! Interpolation points
    !! The number of rows in xij can be 2 or 3
    !! The number of columns in xij should be equal to total
    !! degree of freedom
    CHARACTER(*), INTENT(IN) :: refQuadrangle
    !! It should be BIUNIT
    REAL(DFP), OPTIONAL, INTENT(INOUT) :: coeff(SIZE(xij, 2), SIZE(xij, 2))
    !! coefficient of Lagrange polynomials
    LOGICAL(LGT), OPTIONAL :: firstCall
    !! If firstCall is true, then coeff will be computed and returned
    !! by this routine.
    !! If firstCall is False, then coeff should be given, which will be
    !! used.
    !! Default value of firstCall is True
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: basisType
    !! Monomials *Default
    !! Legendre
    !! Lobatto
    !! Chebyshev
    !! Jacobi
    !! Ultraspherical
    !! Heirarchical
    REAL(DFP), OPTIONAL, INTENT(IN) :: alpha
    !! Jacobi parameter
    REAL(DFP), OPTIONAL, INTENT(IN) :: beta
    !! Jacobi parameter
    REAL(DFP), OPTIONAL, INTENT(IN) :: lambda
    !! Ultraspherical parameter
    REAL(DFP) :: ans(SIZE(xij, 2))
    !! Value of n+1 Lagrange polynomials at point x
  END FUNCTION LagrangeEvalAll_Quadrangle1
END INTERFACE LagrangeEvalAll_Quadrangle

️܀ Monomial

program main
  use easifembase
  implicit none
  real(dfp), allocatable :: xij(:,:), avec(:)
  integer(i4b) :: ii, jj, cnt, n
  real(dfp), allocatable :: ans(:), coeff(:,:)
  integer(i4b) :: order, basisType


  order = 1
  basisType = Monomial
  xij = InterpolationPoint_Quadrangle( &
    & order=order, &
    & ipType=Equidistance, &
    & xij=RefQuadrangleCoord("BIUNIT"), &
    & layout="VEFC")

  call reallocate(coeff, size(xij, 2), size(xij,2))

  ans = LagrangeEvalAll_Quadrangle(&
    & order=order, &
    & x=[0.0_DFP, 0.0_DFP], &
    & xij=xij, &
    & basisType=basisType, &
    & coeff=coeff)

  call display(mdencode(ans), "ans = " // char_lf // char_lf )


  order = 2
  basisType = Monomial
  xij = InterpolationPoint_Quadrangle( &
    & order=order, &
    & ipType=Equidistance, &
    & xij=RefQuadrangleCoord("BIUNIT"), &
    & layout="VEFC")

  call reallocate(coeff, size(xij, 2), size(xij,2))

  ans = LagrangeEvalAll_Quadrangle(&
    & order=order, &
    & x=[0.0_DFP, 0.0_DFP], &
    & xij=xij, &
    & basisType=basisType, &
    & coeff=coeff)

  call display(mdencode(ans), "ans = " // char_lf // char_lf )

end program main

See results for order 1

ans =

$l_1$	$l_2$	$l_3$	$l_4$
0.25	0.25	0.25	0.25

See results for order 2

ans =

$l_1$	$l_2$	$l_3$	$l_4$					$l_9$
0	0	0	0	0	0	0	0	1

Legendre

program main
  use easifembase
  implicit none
  real(dfp), allocatable :: xij(:,:), avec(:)
  integer(i4b) :: ii, jj, cnt, n
  real(dfp), allocatable :: ans(:), coeff(:,:)
  integer(i4b) :: order, basisType
  real(dfp) :: alpha, beta

  order = 2
  basisType = Legendre
  xij = InterpolationPoint_Quadrangle( &
    & order=order, &
    & ipType=Equidistance, &
    & xij=RefQuadrangleCoord("BIUNIT"), &
    & layout="VEFC")

  call reallocate(coeff, size(xij, 2), size(xij,2))

  ans = LagrangeEvalAll_Quadrangle(&
    & order=order, &
    & x=[0.0_DFP, 0.0_DFP], &
    & xij=xij, &
    & basisType=basisType, &
    & coeff=coeff)

  call display(mdencode(ans), "ans = " // char_lf // char_lf )

end program main

See results for order 2

ans =

$l_1$	$l_2$	$l_3$	$l_4$					$l_9$
0	0	0	0	0	0	0	0	1

Chebyshev

program main
  use easifembase
  implicit none
  real(dfp), allocatable :: xij(:,:), avec(:)
  integer(i4b) :: ii, jj, cnt, n
  real(dfp), allocatable :: ans(:), coeff(:,:)
  integer(i4b) :: order, basisType
  real(dfp) :: alpha, beta

  order = 2
  basisType = Chebyshev
  xij = InterpolationPoint_Quadrangle( &
    & order=order, &
    & ipType=Equidistance, &
    & xij=RefQuadrangleCoord("BIUNIT"), &
    & layout="VEFC")

  call reallocate(coeff, size(xij, 2), size(xij,2))

  ans = LagrangeEvalAll_Quadrangle(&
    & order=order, &
    & x=[0.0_DFP, 0.0_DFP], &
    & xij=xij, &
    & basisType=basisType, &
    & coeff=coeff)

  call display(mdencode(ans), "ans = " // char_lf // char_lf )

end program main

See results for order 2

ans =

$l_1$	$l_2$	$l_3$	$l_4$					$l_9$
0	0	0	0	0	0	0	0	1

↢

Interface 2

܀ Interface

INTERFACE LagrangeEvalAll_Quadrangle
  MODULE FUNCTION LagrangeEvalAll_Quadrangle2( &
    & order, &
    & x, &
    & xij, &
    & refQuadrangle, &
    & coeff, &
    & firstCall, &
    & basisType, &
    & alpha, &
    & beta, &
    & lambda &
    & ) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: order
    !! Order of Lagrange polynomials
    REAL(DFP), INTENT(IN) :: x(:, :)
    !! Point of evaluation
    !! x(1, :) is x coord
    !! x(2, :) is y coord
    REAL(DFP), INTENT(INOUT) :: xij(:, :)
    !! Interpolation points
    CHARACTER(*), INTENT(IN) :: refQuadrangle
    !! Reference quadrangle, it should be BIUNIT
    REAL(DFP), OPTIONAL, INTENT(INOUT) :: coeff(SIZE(xij, 2), SIZE(xij, 2))
    !! Coefficient of Lagrange polynomials
    LOGICAL(LGT), OPTIONAL :: firstCall
    !! If firstCall is true, then coeff will be made
    !! If firstCall is False, then coeff will be used
    !! Default value of firstCall is True
    INTEGER(I4B), OPTIONAL, INTENT(IN) :: basisType
    !! Monomials *Default
    !! Jacobi=Dubiner
    !! Heirarchical
    REAL(DFP), OPTIONAL, INTENT(IN) :: alpha
    !! Jacobi parameter
    REAL(DFP), OPTIONAL, INTENT(IN) :: beta
    !! Jacobi parameter
    REAL(DFP), OPTIONAL, INTENT(IN) :: lambda
    !! Ultraspherical parameter
    REAL(DFP) :: ans(SIZE(x, 2), SIZE(xij, 2))
    !! Value of n+1 Lagrange polynomials at point x
  END FUNCTION LagrangeEvalAll_Quadrangle2
END INTERFACE LagrangeEvalAll_Quadrangle

️܀ Monomials

program main
  use easifembase
  implicit none
  real(dfp), allocatable :: xij(:,:), avec(:)
  integer(i4b) :: ii, jj, cnt, n
  real(dfp), allocatable :: ans(:, :), coeff(:,:)
  integer(i4b) :: order, basisType
  real(dfp) :: alpha, beta

  order = 2
  basisType = Monomial
  xij = InterpolationPoint_Quadrangle( &
    & order=order, &
    & ipType=Equidistance, &
    & xij=RefQuadrangleCoord("BIUNIT"), &
    & layout="VEFC")

  call reallocate(coeff, size(xij, 2), size(xij,2))

  ans = LagrangeEvalAll_Quadrangle(&
    & order=order, &
    & x=xij, &
    & xij=xij, &
    & basisType=basisType, &
    & coeff=coeff)

  call display(mdencode(ans), "ans = " // char_lf // char_lf )

end program main

See results for order 2

ans =

1	0	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0
0	0	0	0	1	0	0	0	0
0	0	0	0	0	1	0	0	0
0	0	0	0	0	0	1	0	0
0	0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	0	1

Legendre

program main
  use easifembase
  implicit none
  real(dfp), allocatable :: xij(:,:), avec(:)
  integer(i4b) :: ii, jj, cnt, n
  real(dfp), allocatable :: ans(:, :), coeff(:,:)
  integer(i4b) :: order, basisType
  real(dfp) :: alpha, beta

  order = 2
  basisType = Legendre
  xij = InterpolationPoint_Quadrangle( &
    & order=order, &
    & ipType=Equidistance, &
    & xij=RefQuadrangleCoord("BIUNIT"), &
    & layout="VEFC")

  call reallocate(coeff, size(xij, 2), size(xij,2))

  ans = LagrangeEvalAll_Quadrangle(&
    & order=order, &
    & x=xij, &
    & xij=xij, &
    & basisType=basisType, &
    & coeff=coeff)

  call display(mdencode(ans), "ans = " // char_lf // char_lf )

end program main

See results for order 2

ans =

1	0	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0
0	0	0	0	1	0	0	0	0
0	0	0	0	0	1	0	0	0
0	0	0	0	0	0	1	0	0
0	0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	0	1

Heirarchical

program main
  use easifembase
  implicit none
  real(dfp), allocatable :: xij(:,:), avec(:)
  integer(i4b) :: ii, jj, cnt, n
  real(dfp), allocatable :: ans(:, :), coeff(:,:)
  integer(i4b) :: order, basisType
  real(dfp) :: alpha, beta

  order = 2
  basisType = Heirarchical
  xij = InterpolationPoint_Quadrangle( &
    & order=order, &
    & ipType=Equidistance, &
    & xij=RefQuadrangleCoord("BIUNIT"), &
    & layout="VEFC")

  call reallocate(coeff, size(xij, 2), size(xij,2))

  ans = LagrangeEvalAll_Quadrangle(&
    & order=order, &
    & x=xij, &
    & xij=xij, &
    & basisType=basisType, &
    & coeff=coeff)

  call display(mdencode(ans), "ans = " // char_lf // char_lf )

end program main

See results for order 2

ans =

1	0	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0
0	0	0	0	1	0	0	0	0
0	0	0	0	0	1	0	0	0
0	0	0	0	0	0	1	0	0
0	0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	0	1

↢