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InterpolationPoint_Triangle

This routine returns the interpolation points on triangle.

Interface​

INTERFACE
  MODULE FUNCTION InterpolationPoint_Triangle(order, ipType, &
    & layout, xij) RESULT(ans)
    INTEGER(I4B), INTENT(IN) :: order
    !! order
    INTEGER(I4B), INTENT(IN) :: ipType
    !! interpolation point type
    REAL(DFP), OPTIONAL, INTENT(IN) :: xij(:, :)
    !! Coord of domain in xij format
    CHARACTER(*), INTENT(IN) :: layout
    !! local node numbering layout, always VEFC
    REAL(DFP), ALLOCATABLE :: ans(:, :)
    !! xij coordinates
  END FUNCTION InterpolationPoint_Triangle
END INTERFACE
xij
  • xij contains nodal coordinates of triangle in xij format.
  • SIZE(xij,1) = nsd, and SIZE(xij,2)=3
  • If xij is absent then unit triangle is assumed
ipType

  • ipType is interpolation point type, it can take following values

  • Equidistance, uniformly/evenly distributed points

  • BlythPozChebyshev

  • BlythPozLegendre

  • GaussLegendreLobatto, which is same as IsaacLegendre

  • GaussChebyshevLobatto, which is same as IsaacChebyshev

  • GaussJacobiLobatto

  • GaussUltrasphericalLobatto

  • IsaacChebyshev

  • IsaacLegendre

  • ChenBabuska TODO

  • Hesthaven TODO

  • Feket TODO

layout
  • layout specifies the arrangement of points. The nodes are always returned in “VEFC” format (vertex, edge, face, cell). 1:3 are vertex points, then edge, and then internal nodes. The internal nodes also follow the same convention. Please read Gmsh manual on this topic.

Example of Equidistance points​

Click here to see example
program main
  use easifembase
  implicit none
  integer( i4b ) :: i1, i2, order, ipType
  real( dfp ), allocatable :: x(:,:), xij(:, :)
  CHARACTER( 20 ) :: layout
  order=5
  xij = RefTriangleCoord("UNIT")
  ipType=Equidistance
  layout="VEFC"
  x =InterpolationPoint_Triangle( order=order, &
  & ipType=ipType, layout=layout, xij=xij)
  call display( Mdencode(TRANSPOSE(x)), "xij (order="//tostring(order)//")=" )
  call blanklines(nol=2)
end program main
See results

xij (order=5)=

00
10
01
0.20
0.40
0.60
0.80
0.80.2
0.60.4
0.40.6
0.20.8
00.8
00.6
00.4
00.2
0.20.2
0.60.2
0.20.6
0.40.2
0.40.4
0.20.4

BlythPozChebyshev​

Click here to see example
program main
  use easifembase
  implicit none
  integer( i4b ) :: i1, i2, order, ipType
  real( dfp ), allocatable :: x(:,:), xij(:, :)
  CHARACTER( 20 ) :: layout
  order=5
  xij = RefTriangleCoord("UNIT")
  ipType=BlythPozChebyshev
  layout="VEFC"
  x =InterpolationPoint_Triangle( order=order, &
  & ipType=ipType, layout=layout, xij=xij)
  call display( Mdencode(TRANSPOSE(x)), "xij (order="//tostring(order)//")=" )
  call blanklines(nol=2)
end program main
See results

xij (order=5)=

00
10
01
9.54915E-02-2.02431E-14
0.34549-3.27886E-14
0.65451-3.28071E-14
0.90451-2.02431E-14
0.904519.54915E-02
0.654510.34549
0.345490.65451
9.54915E-020.90451
-2.02431E-140.90451
-3.28071E-140.65451
-3.27886E-140.34549
-2.02431E-149.54915E-02
0.146990.14699
0.706010.14699
0.146990.70601
0.416670.16667
0.416670.41667
0.166670.41667

BlythPozLegendre​

Click here to see example
program main
  use easifembase
  implicit none
  integer( i4b ) :: i1, i2, order, ipType
  real( dfp ), allocatable :: x(:,:), xij(:, :)
  CHARACTER( 20 ) :: layout
  order=5
  xij = RefTriangleCoord("UNIT")
  ipType=BlythPozLegendre
  layout="VEFC"
  x =InterpolationPoint_Triangle( order=order, &
  & ipType=ipType, layout=layout, xij=xij)
  call display( Mdencode(TRANSPOSE(x)), "xij (order="//tostring(order)//")=" )
  call blanklines(nol=2)
end program main
See results

xij (order=5)=

-1.85037E-16-1.85037E-16
1-1.85037E-16
-1.85037E-161
0.11747-1.4803E-16
0.35738-3.70074E-17
0.64262-3.70074E-17
0.88253-1.29526E-16
0.882530.11747
0.642620.35738
0.357380.64262
0.117470.88253
-1.29526E-160.88253
-3.70074E-170.64262
-3.70074E-170.35738
-1.4803E-160.11747
0.158290.15829
0.683430.15829
0.158290.68343
0.41330.17339
0.41330.4133
0.173390.4133

IsaacChebyshev​

Click here to see example
program main
  use easifembase
  implicit none
  integer( i4b ) :: i1, i2, order, ipType
  real( dfp ), allocatable :: x(:,:), xij(:, :)
  CHARACTER( 20 ) :: layout
  order=5
  xij = RefTriangleCoord("UNIT")
  ipType=IsaacLegendre
  layout="VEFC"
  x =InterpolationPoint_Triangle( order=order, &
  & ipType=ipType, layout=layout, xij=xij)
  call display( Mdencode(TRANSPOSE(x)), "xij (order="//tostring(order)//")=" )
  call blanklines(nol=2)
end program main
See results

xij (order=5)=

-8.32667E-17-8.32667E-17
1-8.32667E-17
-8.32667E-171
0.11747-4.89901E-17
0.35738-1.58335E-17
0.64262-1.58335E-17
0.88253-4.89901E-17
0.882530.11747
0.642620.35738
0.357380.64262
0.117470.88253
-4.89901E-170.88253
-1.58335E-170.64262
-1.58335E-170.35738
-4.89901E-170.11747
0.155990.15599
0.688020.15599
0.155990.68802
0.418070.16387
0.418070.41807
0.163870.41807

IssacLegendre​

Click here to see example
program main
  use easifembase
  implicit none
  integer( i4b ) :: i1, i2, order, ipType
  real( dfp ), allocatable :: x(:,:), xij(:, :)
  CHARACTER( 20 ) :: layout
  order=5
  xij = RefTriangleCoord("UNIT")
  ipType=IsaacLegendre
  layout="VEFC"
  x =InterpolationPoint_Triangle( order=order, &
  & ipType=ipType, layout=layout, xij=xij)
  call display( Mdencode(TRANSPOSE(x)), "xij (order="//tostring(order)//")=" )
  call blanklines(nol=2)
end program main
See results

xij (order=5)=

-8.32667E-17-8.32667E-17
1-8.32667E-17
-8.32667E-171
0.11747-4.89901E-17
0.35738-1.58335E-17
0.64262-1.58335E-17
0.88253-4.89901E-17
0.882530.11747
0.642620.35738
0.357380.64262
0.117470.88253
-4.89901E-170.88253
-1.58335E-170.64262
-1.58335E-170.35738
-4.89901E-170.11747
0.155990.15599
0.688020.15599
0.155990.68802
0.418070.16387
0.418070.41807
0.163870.41807
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Last updated on
(Simulated during dev for better perf)