📄 blend.f90
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! will only be called for "sides", that is, for values (R,S) where
! at least one of R and S is either 0.0E+00 or 1.0. BOUND_RS has the
! form:
!
! subroutine bound_rs ( r, s, i, xi )
!
implicit none
!
integer n
!
integer i
real r
real s
real x(n)
real x00
real x01
real x10
real x11
real xr0
real xr1
real x0s
real x1s
!
external bound_rs
!
do i = 1, n
!
! Get the I-th coordinate component at the four corners.
!
call bound_rs ( 0.0E+00, 0.0E+00, i, x00 )
call bound_rs ( 0.0E+00, 1.0E+00, i, x01 )
call bound_rs ( 1.0E+00, 0.0E+00, i, x10 )
call bound_rs ( 1.0E+00, 1.0E+00, i, x11 )
!
! Get the I-th coordinate component at the sides.
!
call bound_rs ( r, 0.0E+00, i, xr0 )
call bound_rs ( r, 1.0E+00, i, xr1 )
call bound_rs ( 0.0E+00, s, i, x0s )
call bound_rs ( 1.0E+00, s, i, x1s )
!
! Interpolate the I-th coordinate component of the interior point.
!
call blend_112 ( r, s, x00, x01, x10, x11, xr0, xr1, x0s, x1s, x(i) )
end do
return
end
subroutine blend_rst_0dn ( r, s, t, x, n, bound_rst )
!
!*******************************************************************************
!
!! BLEND_RST_0DN extends vector data at corners into a cube.
!
!
! Diagram:
!
! 010-----r10-----110 011-----r11-----111
! | . | | . |
! | . | | . |
! 0s0.....rs0.....1s0 0s1.....rs1.....1s1 S
! | . | | . | |
! | . | | . | |
! 000-----r00-----100 001-----r01-----101 +----R
! BOTTOM TOP
!
! 011-----0s1-----001 111-----1s1-----101
! | . | | . |
! | . | | . |
! 01t.....0st.....00t 11t.....1st.....10t T
! | . | | . | |
! | . | | . | |
! 010-----0s0-----000 110-----1s0-----100 S----+
! LEFT RIGHT
!
! 001-----r01-----101 011-----r11-----111
! | . | | . |
! | . | | . |
! 00t.....r0t.....100 01t.....r1t.....11t T
! | . | | . | |
! | . | | . | |
! 000-----r00-----100 010-----r10-----110 +----R
! FRONT BACK
!
! Note:
!
! BLEND_RST_0DN is equivalent to a trilinear finite element method.
! Data along the edges, faces, and interior of the cube is
! interpolated from the data at the corners.
!
! Reference:
!
! William Gordon,
! Blending-Function Methods of Bivariate and Multivariate Interpolation
! and Approximation,
! SIAM Journal on Numerical Analysis,
! Volume 8, Number 1, March 1971, pages 158-177.
!
! William Gordon and Charles Hall,
! Transfinite Element Methods: Blending-Function Interpolation over
! Arbitrary Curved Element Domains,
! Numerische Mathematik,
! Volume 21, Number 1, 1973, pages 109-129.
!
! William Gordon and Charles Hall,
! Construction of Curvilinear Coordinate Systems and Application to
! Mesh Generation,
! International Journal of Numerical Methods in Engineering,
! Volume 7, 1973, pages 461-477.
!
! Joe Thompson, Bharat Soni, Nigel Weatherill,
! Handbook of Grid Generation,
! CRC Press, 1999.
!
! Modified:
!
! 14 December 1998
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real R, S, T, the (R,S,T) coordinates of the point to be
! evaluated.
!
! Output, real X(N), the interpolated value at the point (R,S,T).
!
! Input, integer N, the dimension of the vector space.
!
! External, BOUND_RST, is a subroutine which is given (R,S,T)
! coordinates and an component value I, and returns XI, the value
! of the I-th component of the N-vector at that point. BOUND_RST
! will only be called for "corners", that is, for values (R,S,T)
! where R, S and T are either 0.0E+00 or 1.0. BOUND_RST has the form:
!
! subroutine bound_rst ( r, s, t, i, xi )
!
implicit none
!
integer n
!
integer i
real r
real s
real t
real x(n)
real x000
real x001
real x010
real x011
real x100
real x101
real x110
real x111
real xr00
real xr01
real xr10
real xr11
real x0s0
real x0s1
real x1s0
real x1s1
real x00t
real x01t
real x10t
real x11t
real x0st
real x1st
real xr0t
real xr1t
real xrs0
real xrs1
!
external bound_rst
!
do i = 1, n
!
! Get the I-th coordinate component at the corners.
!
call bound_rst ( 0.0E+00, 0.0E+00, 0.0E+00, i, x000 )
call bound_rst ( 0.0E+00, 0.0E+00, 1.0E+00, i, x001 )
call bound_rst ( 0.0E+00, 1.0E+00, 0.0E+00, i, x010 )
call bound_rst ( 0.0E+00, 1.0E+00, 1.0E+00, i, x011 )
call bound_rst ( 1.0E+00, 0.0E+00, 0.0E+00, i, x100 )
call bound_rst ( 1.0E+00, 0.0E+00, 1.0E+00, i, x101 )
call bound_rst ( 1.0E+00, 1.0E+00, 0.0E+00, i, x110 )
call bound_rst ( 1.0E+00, 1.0E+00, 1.0E+00, i, x111 )
!
! Interpolate the I-th coordinate component at the edges.
!
call blend_101 ( r, x000, x100, xr00 )
call blend_101 ( r, x001, x101, xr01 )
call blend_101 ( r, x010, x110, xr10 )
call blend_101 ( r, x011, x111, xr11 )
call blend_101 ( s, x000, x010, x0s0 )
call blend_101 ( s, x001, x011, x0s1 )
call blend_101 ( s, x100, x110, x1s0 )
call blend_101 ( s, x101, x111, x1s1 )
call blend_101 ( t, x000, x001, x00t )
call blend_101 ( t, x010, x011, x01t )
call blend_101 ( t, x100, x101, x10t )
call blend_101 ( t, x110, x111, x11t )
!
! Interpolate the I-th component on the faces.
!
call blend_112 ( s, t, x000, x001, x010, x011, x0s0, x0s1, x00t, x01t, x0st )
call blend_112 ( s, t, x100, x101, x110, x111, x1s0, x1s1, x10t, x11t, x1st )
call blend_112 ( r, t, x000, x001, x100, x101, xr00, xr01, x00t, x10t, xr0t )
call blend_112 ( r, t, x010, x011, x110, x111, xr10, xr11, x01t, x11t, xr1t )
call blend_112 ( r, s, x000, x010, x100, x110, xr00, xr10, x0s0, x1s0, xrs0 )
call blend_112 ( r, s, x001, x011, x101, x111, xr01, xr11, x0s1, x1s1, xrs1 )
!
! Interpolate the I-th coordinate component of the interior point.
!
call blend_123 ( r, s, t, x000, x001, x010, x011, x100, x101, x110, x111, &
xr00, xr01, xr10, xr11, x0s0, x0s1, x1s0, x1s1, x00t, x01t, x10t, x11t, &
x0st, x1st, xr0t, xr1t, xrs0, xrs1, x(i) )
end do
return
end
subroutine blend_rst_1dn ( r, s, t, x, n, bound_rst )
!
!*******************************************************************************
!
!! BLEND_RST_1DN extends vector data on edges into a cube.
!
!
! Diagram:
!
! 010-----r10-----110 011-----r11-----111
! | . | | . |
! | . | | . |
! 0s0.....rs0.....1s0 0s1.....rs1.....1s1 S
! | . | | . | |
! | . | | . | |
! 000-----r00-----100 001-----r01-----101 +----R
! BOTTOM TOP
!
! 011-----0s1-----001 111-----1s1-----101
! | . | | . |
! | . | | . |
! 01t.....0st.....00t 11t.....1st.....10t T
! | . | | . | |
! | . | | . | |
! 010-----0s0-----000 110-----1s0-----100 S----+
! LEFT RIGHT
!
! 001-----r01-----101 011-----r11-----111
! | . | | . |
! | . | | . |
! 00t.....r0t.....100 01t.....r1t.....11t T
! | . | | . | |
! | . | | . | |
! 000-----r00-----100 010-----r10-----110 +----R
! FRONT BACK
!
! Note:
!
! BLEND_RST_1D is NOT equivalent to a trilinear finite element method,
! since the data is sampled everywhere along the corners and edges,
! rather than at a finite number of nodes.
!
! Reference:
!
! William Gordon,
! Blending-Function Methods of Bivariate and Multivariate Interpolation
! and Approximation,
! SIAM Journal on Numerical Analysis,
! Volume 8, Number 1, March 1971, pages 158-177.
!
! William Gordon and Charles Hall,
! Transfinite Element Methods: Blending-Function Interpolation over
! Arbitrary Curved Element Domains,
! Numerische Mathematik,
! Volume 21, Number 1, 1973, pages 109-129.
!
! William Gordon and Charles Hall,
! Construction of Curvilinear Coordinate Systems and Application to
! Mesh Generation,
! International Journal of Numerical Methods in Engineering,
! Volume 7, 1973, pages 461-477.
!
! Joe Thompson, Bharat Soni, Nigel Weatherill,
! Handbook of Grid Generation,
! CRC Press, 1999.
!
! Modified:
!
! 15 December 1998
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real R, S, T, the (R,S,T) coordinates of the point to be
! evaluated.
!
! Output, real X(N), the interpolated value at the point (R,S,T).
!
! Input, integer N, the dimension of the vector space.
!
! External, BOUND_RST, is a subroutine which is given (R,S,T)
! coordinates and an component value I, and returns XI, the value
! of the I-th component of the N-vector at that point. BOUND_RST
! will only be called for "edges", that is, for values (R,S,T)
! where at least two of R, S and T are either 0.0E+00 or 1.0.
! BOUND_RST has the form:
!
! subroutine bound_rst ( r, s, t, i, xi )
!
implicit none
!
integer n
!
integer i
real r
real s
real t
real x(n)
real x000
real x001
real x010
real x011
real x100
real x101
real x110
real x111
real xr00
real xr01
real xr10
real xr11
real x0s0
real x0s1
real x1s0
real x1s1
real x00t
real x01t
real x10t
real x11t
real x0st
real x1st
real xr0t
real xr1t
real xrs0
real xrs1
!
external bound_rst
!
do i = 1, n
!
! Get the I-th coordinate component at the corners.
!
call bound_rst ( 0.0E+00, 0.0E+00, 0.0E+00, i, x000 )
call bound_rst ( 0.0E+00, 0.0E+00, 1.0E+00, i, x001 )
call bound_rst ( 0.0E+00, 1.0E+00, 0.0E+00, i, x010 )
call bound_rst ( 0.0E+00, 1.0E+00, 1.0E+00, i, x011 )
call bound_rst ( 1.0E+00, 0.0E+00, 0.0E+00, i, x100 )
call bound_rst ( 1.0E+00, 0.0E+00, 1.0E+00, i, x101 )
call bound_rst ( 1.0E+00, 1.0E+00, 0.0E+00, i, x110 )
call bound_rst ( 1.0E+00, 1.0E+00, 1.0E+00, i, x111 )
!
! Get the I-th coordinate component at the edges.
!
call bound_rst ( r, 0.0E+00, 0.0E+00, i, xr00 )
call bound_rst ( r, 0.0E+00, 1.0E+00, i, xr01 )
call bound_rst ( r, 1.0E+00, 0.0E+00, i, xr10 )
call bound_rst ( r, 1.0E+00, 1.0E+00, i, xr11 )
call bound_rst ( 0.0E+00, s, 0.0E+00, i, x0s0 )
call bound_rst ( 0.0E+00, s, 1.0E+00, i, x0s1 )
call bound_rst ( 1.0E+00, s, 0.0E+00, i, x1s0 )
call bound_rst ( 1.0E+00, s, 1.0E+00, i, x1s1 )
call bound_rst ( 0.0E+00, 0.0E+00, t, i, x00t )
call bound_rst ( 0.0E+00, 1.0E+00, t, i, x01t )
call bound_rst ( 1.0E+00, 0.0E+00, t, i, x10t )
call bound_rst ( 1.0E+00, 1.0E+00, t, i, x11t )
!
! Interpolate the I-th component on the faces.
!
call blend_112 ( s, t, x000, x001, x010, x011, x0s0, x0s1, x00t, x01t, x0st )
call blend_112 ( s, t, x100, x101, x110, x111, x1s0, x1s1, x10t, x11t, x1st )
call blend_112 ( r, t, x000, x001, x100, x101, xr00, xr01, x00t, x10t, xr0t )
call blend_112 ( r, t, x010, x011, x110, x111, xr10, xr11, x01t, x11t, xr1t )
call blend_112 ( r, s, x000, x010, x100, x110, xr00, xr10, x0s0, x1s0, xrs0 )
call blend_112 ( r, s, x001, x011, x101, x111, xr01, xr11, x0s1, x1s1, xrs1 )
!
! Interpolate the I-th coordinate component of the interior point.
!
call blend_123 ( r, s, t, x000, x001, x010, x011, x100, x101, x110, x111, &
xr00, xr01, xr10, xr11, x0s0, x0s1, x1s0, x1s1, x00t, x01t, x10t, x11t, &
x0st, x1st, xr0t, xr1t, xrs0, xrs1, x(i) )
end do
return
end
subroutine blend_rst_2dn ( r, s, t, x, n, bound_rst )
!
!*******************************************************************************
!
!! BLEND_RST_2DN extends vector data on faces into a cube.
!
!
! Diagram:
!
! 010-----r10-----110 011-----r11-----111
! | . | | . |
! | . | | . |
! 0s0.....rs0.....1s0 0s1.....rs1.....1s1 S
! | . | | . | |
! | . | | . | |
! 000-----r00-----100 001-----r01-----101 +----R
! BOTTOM TOP
!
! 011-----0s1-----001 111-----1s1-----101
! | . | | . |
! | . | | . |
! 01t.....0st.....00t 11t.....1st.....10t T
! | . | | . | |
! | . | | . | |
! 010-----0s0-----000 110-----1s0-----100 S----+
! LEFT RIGHT
!
! 001-----r01-----101 011-----r11-----111
! | . | | . |
! | . | | . |
! 00t.....r0t.....100 01t.....r1t.....11t T
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