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seismic_CPML_2D_isotropic_fourth_order.f90
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seismic_CPML_2D_isotropic_fourth_order.f90
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!
! SEISMIC_CPML Version 1.1.1, November 2009.
!
! Copyright Universite de Pau et des Pays de l'Adour, CNRS and INRIA, France.
! Contributors: Dimitri Komatitsch, komatitsch aT lma DOT cnrs-mrs DOT fr
! and Roland Martin, roland DOT martin aT get DOT obs-mip DOT fr
!
! This software is a computer program whose purpose is to solve
! the two-dimensional isotropic elastic wave equation
! using a finite-difference method with Convolutional Perfectly Matched
! Layer (C-PML) conditions.
!
! This program is free software; you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation; either version 3 of the License, or
! (at your option) any later version.
!
! This program is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License along
! with this program; if not, write to the Free Software Foundation, Inc.,
! 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
!
! The full text of the license is available in file "LICENSE".
program seismic_CPML_2D_iso_fourth
! 2D elastic finite-difference code in velocity and stress formulation
! with Convolutional-PML (C-PML) absorbing conditions for an isotropic medium
! Dimitri Komatitsch, University of Pau, France, April 2007.
! Fourth-order implementation by Dimitri Komatitsch and Roland Martin, University of Pau, France, August 2007.
! The staggered-grid formulation of Madariaga (1976) and Virieux (1986) is used:
!
! ^ y
! |
! |
!
! +-------------------+
! | |
! | |
! | |
! | |
! | v_y |
! sigma_xy +---------+ |
! | | |
! | | |
! | | |
! | | |
! | | |
! +---------+---------+ ---> x
! v_x sigma_xx
! sigma_yy
!
! but a fourth-order spatial operator is used instead of a second-order operator
! as in program seismic_CPML_2D_iso_second.f90 . You can type the following command
! to see the changes that have been made to switch from the second-order operator
! to the fourth-order operator:
!
! diff seismic_CPML_2D_isotropic_second_order.f90 seismic_CPML_2D_isotropic_fourth_order.f90
! The C-PML implementation is based in part on formulas given in Roden and Gedney (2000)
!
! If you use this code for your own research, please cite some (or all) of these articles:
!
! @ARTICLE{MaKoEz08,
! author = {Roland Martin and Dimitri Komatitsch and Abdelaaziz Ezziani},
! title = {An unsplit convolutional perfectly matched layer improved at grazing
! incidence for seismic wave equation in poroelastic media},
! journal = {Geophysics},
! year = {2008},
! volume = {73},
! pages = {T51-T61},
! number = {4},
! doi = {10.1190/1.2939484}}
!
! @ARTICLE{MaKoGe08,
! author = {Roland Martin and Dimitri Komatitsch and Stephen D. Gedney},
! title = {A variational formulation of a stabilized unsplit convolutional perfectly
! matched layer for the isotropic or anisotropic seismic wave equation},
! journal = {Computer Modeling in Engineering and Sciences},
! year = {2008},
! volume = {37},
! pages = {274-304},
! number = {3}}
!
! @ARTICLE{RoGe00,
! author = {J. A. Roden and S. D. Gedney},
! title = {Convolution {PML} ({CPML}): {A}n Efficient {FDTD} Implementation
! of the {CFS}-{PML} for Arbitrary Media},
! journal = {Microwave and Optical Technology Letters},
! year = {2000},
! volume = {27},
! number = {5},
! pages = {334-339},
! doi = {10.1002/1098-2760(20001205)27:5 < 334::AID-MOP14>3.0.CO;2-A}}
!
! @ARTICLE{KoMa07,
! author = {Dimitri Komatitsch and Roland Martin},
! title = {An unsplit convolutional {P}erfectly {M}atched {L}ayer improved
! at grazing incidence for the seismic wave equation},
! journal = {Geophysics},
! year = {2007},
! volume = {72},
! number = {5},
! pages = {SM155-SM167},
! doi = {10.1190/1.2757586}}
!
! To display the 2D results as color images, use:
!
! " display image*.gif " or " gimp image*.gif "
!
! or
!
! " montage -geometry +0+3 -rotate 90 -tile 1x21 image*Vx*.gif allfiles_Vx.gif "
! " montage -geometry +0+3 -rotate 90 -tile 1x21 image*Vy*.gif allfiles_Vy.gif "
! then " display allfiles_Vx.gif " or " gimp allfiles_Vx.gif "
! then " display allfiles_Vy.gif " or " gimp allfiles_Vy.gif "
!
! IMPORTANT : all our CPML codes work fine in single precision as well (which is significantly faster).
! If you want you can thus force automatic conversion to single precision at compile time
! or change all the declarations and constants in the code from double precision to single.
implicit none
! total number of grid points in each direction of the grid
integer, parameter :: NX = 101
integer, parameter :: NY = 641
! size of a grid cell
double precision, parameter :: DELTAX = 10.d0
double precision, parameter :: DELTAY = DELTAX
! flags to add PML layers to the edges of the grid
logical, parameter :: USE_PML_XMIN = .true.
logical, parameter :: USE_PML_XMAX = .true.
logical, parameter :: USE_PML_YMIN = .true.
logical, parameter :: USE_PML_YMAX = .true.
! thickness of the PML layer in grid points
integer, parameter :: NPOINTS_PML = 10
! P-velocity, S-velocity and density
double precision, parameter :: cp = 3300.d0
double precision, parameter :: cs = cp / 1.732d0
double precision, parameter :: density = 2800.d0
! total number of time steps
! the time step is twice smaller for this fourth-order simulation,
! therefore let us double the number of time steps to keep the same total duration
integer, parameter :: NSTEP = 2000 * 2
! time step in seconds
! fourth-order in space and second-order in time finite-difference schemes
! are less stable than second-order in space and second-order in time,
! therefore let us divide the time step by 2
double precision, parameter :: DELTAT = 2.d-3 / 2
! parameters for the source
double precision, parameter :: f0 = 7.d0
double precision, parameter :: t0 = 1.20d0 / f0
double precision, parameter :: factor = 1.d7
! source
integer, parameter :: ISOURCE = NX - 2*NPOINTS_PML - 1
integer, parameter :: JSOURCE = 2 * NY / 3 + 1
double precision, parameter :: xsource = (ISOURCE - 1) * DELTAX
double precision, parameter :: ysource = (JSOURCE - 1) * DELTAY
! angle of source force clockwise with respect to vertical (Y) axis
double precision, parameter :: ANGLE_FORCE = 135.d0
! receivers
integer, parameter :: NREC = 2
double precision, parameter :: xdeb = xsource - 100.d0 ! first receiver x in meters
double precision, parameter :: ydeb = 2300.d0 ! first receiver y in meters
double precision, parameter :: xfin = xsource ! last receiver x in meters
double precision, parameter :: yfin = 300.d0 ! last receiver y in meters
! display information on the screen from time to time
! the time step is twice smaller for this fourth-order simulation,
! therefore let us double the interval in time steps at which we display information
integer, parameter :: IT_DISPLAY = 100 * 2
! value of PI
double precision, parameter :: PI = 3.141592653589793238462643d0
! conversion from degrees to radians
double precision, parameter :: DEGREES_TO_RADIANS = PI / 180.d0
! zero
double precision, parameter :: ZERO = 0.d0
! large value for maximum
double precision, parameter :: HUGEVAL = 1.d+30
! velocity threshold above which we consider that the code became unstable
double precision, parameter :: STABILITY_THRESHOLD = 1.d+25
! main arrays
double precision, dimension(0:NX+1,0:NY+1) :: vx,vy,sigmaxx,sigmayy,sigmaxy,lambda,mu,rho
! to interpolate material parameters at the right location in the staggered grid cell
double precision lambda_half_x,mu_half_x,lambda_plus_two_mu_half_x,mu_half_y,rho_half_x_half_y
! for evolution of total energy in the medium
double precision epsilon_xx,epsilon_yy,epsilon_xy
double precision, dimension(NSTEP) :: total_energy_kinetic,total_energy_potential
! power to compute d0 profile
double precision, parameter :: NPOWER = 2.d0
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-11
double precision, parameter :: K_MAX_PML = 1.d0
double precision, parameter :: ALPHA_MAX_PML = 2.d0*PI*(f0/2.d0) ! from Festa and Vilotte
! arrays for the memory variables
! could declare these arrays in PML only to save a lot of memory, but proof of concept only here
double precision, dimension(0:NX+1,0:NY+1) :: &
memory_dvx_dx, &
memory_dvx_dy, &
memory_dvy_dx, &
memory_dvy_dy, &
memory_dsigmaxx_dx, &
memory_dsigmayy_dy, &
memory_dsigmaxy_dx, &
memory_dsigmaxy_dy
double precision :: &
value_dvx_dx, &
value_dvx_dy, &
value_dvy_dx, &
value_dvy_dy, &
value_dsigmaxx_dx, &
value_dsigmayy_dy, &
value_dsigmaxy_dx, &
value_dsigmaxy_dy
! 1D arrays for the damping profiles
double precision, dimension(NX) :: d_x,K_x,alpha_x,a_x,b_x,d_x_half,K_x_half,alpha_x_half,a_x_half,b_x_half
double precision, dimension(NY) :: d_y,K_y,alpha_y,a_y,b_y,d_y_half,K_y_half,alpha_y_half,a_y_half,b_y_half
double precision :: thickness_PML_x,thickness_PML_y,xoriginleft,xoriginright,yoriginbottom,yorigintop
double precision :: Rcoef,d0_x,d0_y,xval,yval,abscissa_in_PML,abscissa_normalized
! for the source
double precision :: a,t,force_x,force_y,source_term
! for receivers
double precision xspacerec,yspacerec,distval,dist
integer, dimension(NREC) :: ix_rec,iy_rec
double precision, dimension(NREC) :: xrec,yrec
! for seismograms
double precision, dimension(NSTEP,NREC) :: sisvx,sisvy
integer :: i,j,it,irec
double precision :: Courant_number,velocnorm
!---
!--- program starts here
!---
print *
print *,'2D elastic finite-difference code in velocity and stress formulation with C-PML'
print *
! display size of the model
print *
print *,'NX = ',NX
print *,'NY = ',NY
print *
print *,'size of the model along X = ',(NX - 1) * DELTAX
print *,'size of the model along Y = ',(NY - 1) * DELTAY
print *
print *,'Total number of grid points = ',NX * NY
print *
!--- define profile of absorption in PML region
! thickness of the PML layer in meters
thickness_PML_x = NPOINTS_PML * DELTAX
thickness_PML_y = NPOINTS_PML * DELTAY
! reflection coefficient (INRIA report section 6.1) http://hal.inria.fr/docs/00/07/32/19/PDF/RR-3471.pdf
Rcoef = 0.001d0
! check that NPOWER is okay
if (NPOWER < 1) stop 'NPOWER must be greater than 1'
! compute d0 from INRIA report section 6.1 http://hal.inria.fr/docs/00/07/32/19/PDF/RR-3471.pdf
d0_x = - (NPOWER + 1) * cp * log(Rcoef) / (2.d0 * thickness_PML_x)
d0_y = - (NPOWER + 1) * cp * log(Rcoef) / (2.d0 * thickness_PML_y)
print *,'d0_x = ',d0_x
print *,'d0_y = ',d0_y
print *
d_x(:) = ZERO
d_x_half(:) = ZERO
K_x(:) = 1.d0
K_x_half(:) = 1.d0
alpha_x(:) = ZERO
alpha_x_half(:) = ZERO
a_x(:) = ZERO
a_x_half(:) = ZERO
d_y(:) = ZERO
d_y_half(:) = ZERO
K_y(:) = 1.d0
K_y_half(:) = 1.d0
alpha_y(:) = ZERO
alpha_y_half(:) = ZERO
a_y(:) = ZERO
a_y_half(:) = ZERO
! damping in the X direction
! origin of the PML layer (position of right edge minus thickness, in meters)
xoriginleft = thickness_PML_x
xoriginright = (NX-1)*DELTAX - thickness_PML_x
do i = 1,NX
! abscissa of current grid point along the damping profile
xval = DELTAX * dble(i-1)
!---------- left edge
if (USE_PML_XMIN) then
! define damping profile at the grid points
abscissa_in_PML = xoriginleft - xval
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_x
d_x(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x(i) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_x(i) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
! define damping profile at half the grid points
abscissa_in_PML = xoriginleft - (xval + DELTAX/2.d0)
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_x
d_x_half(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x_half(i) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_x_half(i) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
!---------- right edge
if (USE_PML_XMAX) then
! define damping profile at the grid points
abscissa_in_PML = xval - xoriginright
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_x
d_x(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x(i) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_x(i) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
! define damping profile at half the grid points
abscissa_in_PML = xval + DELTAX/2.d0 - xoriginright
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_x
d_x_half(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x_half(i) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_x_half(i) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
! just in case, for -5 at the end
if (alpha_x(i) < ZERO) alpha_x(i) = ZERO
if (alpha_x_half(i) < ZERO) alpha_x_half(i) = ZERO
b_x(i) = exp(- (d_x(i) / K_x(i) + alpha_x(i)) * DELTAT)
b_x_half(i) = exp(- (d_x_half(i) / K_x_half(i) + alpha_x_half(i)) * DELTAT)
! this to avoid division by zero outside the PML
if (abs(d_x(i)) > 1.d-6) a_x(i) = d_x(i) * (b_x(i) - 1.d0) / (K_x(i) * (d_x(i) + K_x(i) * alpha_x(i)))
if (abs(d_x_half(i)) > 1.d-6) a_x_half(i) = d_x_half(i) * &
(b_x_half(i) - 1.d0) / (K_x_half(i) * (d_x_half(i) + K_x_half(i) * alpha_x_half(i)))
enddo
! damping in the Y direction
! origin of the PML layer (position of right edge minus thickness, in meters)
yoriginbottom = thickness_PML_y
yorigintop = NY*DELTAY - thickness_PML_y
do j = 1,NY
! abscissa of current grid point along the damping profile
yval = DELTAY * dble(j-1)
!---------- bottom edge
if (USE_PML_YMIN) then
! define damping profile at the grid points
abscissa_in_PML = yoriginbottom - yval
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_y
d_y(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y(j) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_y(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
! define damping profile at half the grid points
abscissa_in_PML = yoriginbottom - (yval + DELTAY/2.d0)
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_y
d_y_half(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y_half(j) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_y_half(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
!---------- top edge
if (USE_PML_YMAX) then
! define damping profile at the grid points
abscissa_in_PML = yval - yorigintop
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_y
d_y(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y(j) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_y(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
! define damping profile at half the grid points
abscissa_in_PML = yval + DELTAY/2.d0 - yorigintop
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_y
d_y_half(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y_half(j) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_y_half(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
b_y(j) = exp(- (d_y(j) / K_y(j) + alpha_y(j)) * DELTAT)
b_y_half(j) = exp(- (d_y_half(j) / K_y_half(j) + alpha_y_half(j)) * DELTAT)
! this to avoid division by zero outside the PML
if (abs(d_y(j)) > 1.d-6) a_y(j) = d_y(j) * (b_y(j) - 1.d0) / (K_y(j) * (d_y(j) + K_y(j) * alpha_y(j)))
if (abs(d_y_half(j)) > 1.d-6) a_y_half(j) = d_y_half(j) * &
(b_y_half(j) - 1.d0) / (K_y_half(j) * (d_y_half(j) + K_y_half(j) * alpha_y_half(j)))
enddo
! compute the Lame parameters and density
do j = 1,NY
do i = 1,NX
rho(i,j) = density
mu(i,j) = density*cs*cs
lambda(i,j) = density*(cp*cp - 2.d0*cs*cs)
enddo
enddo
! print position of the source
print *,'Position of the source:'
print *
print *,'x = ',xsource
print *,'y = ',ysource
print *
! define location of receivers
print *,'There are ',nrec,' receivers'
print *
xspacerec = (xfin-xdeb) / dble(NREC-1)
yspacerec = (yfin-ydeb) / dble(NREC-1)
do irec=1,nrec
xrec(irec) = xdeb + dble(irec-1)*xspacerec
yrec(irec) = ydeb + dble(irec-1)*yspacerec
enddo
! find closest grid point for each receiver
do irec=1,nrec
dist = HUGEVAL
do j = 1,NY
do i = 1,NX
distval = sqrt((DELTAX*dble(i-1) - xrec(irec))**2 + (DELTAY*dble(j-1) - yrec(irec))**2)
if (distval < dist) then
dist = distval
ix_rec(irec) = i
iy_rec(irec) = j
endif
enddo
enddo
print *,'receiver ',irec,' x_target,y_target = ',xrec(irec),yrec(irec)
print *,'closest grid point found at distance ',dist,' in i,j = ',ix_rec(irec),iy_rec(irec)
print *
enddo
! check the Courant stability condition for the explicit time scheme
! R. Courant et K. O. Friedrichs et H. Lewy (1928)
Courant_number = cp * DELTAT * sqrt(1.d0/DELTAX**2 + 1.d0/DELTAY**2)
print *,'Courant number is ',Courant_number
print *
if (Courant_number > 1.d0) stop 'time step is too large, simulation will be unstable'
! suppress old files (can be commented out if "call system" is missing in your compiler)
! call system('rm -f Vx_*.dat Vy_*.dat image*.pnm image*.gif')
! initialize arrays
vx(:,:) = ZERO
vy(:,:) = ZERO
sigmaxx(:,:) = ZERO
sigmayy(:,:) = ZERO
sigmaxy(:,:) = ZERO
! PML
memory_dvx_dx(:,:) = ZERO
memory_dvx_dy(:,:) = ZERO
memory_dvy_dx(:,:) = ZERO
memory_dvy_dy(:,:) = ZERO
memory_dsigmaxx_dx(:,:) = ZERO
memory_dsigmayy_dy(:,:) = ZERO
memory_dsigmaxy_dx(:,:) = ZERO
memory_dsigmaxy_dy(:,:) = ZERO
! initialize seismograms
sisvx(:,:) = ZERO
sisvy(:,:) = ZERO
! initialize total energy
total_energy_kinetic(:) = ZERO
total_energy_potential(:) = ZERO
!---
!--- beginning of time loop
!---
do it = 1,NSTEP
!------------------------------------------------------------
! compute stress sigma and update memory variables for C-PML
!------------------------------------------------------------
do j = 2,NY
do i = 1,NX-1
! interpolate material parameters at the right location in the staggered grid cell
lambda_half_x = 0.5d0 * (lambda(i+1,j) + lambda(i,j))
mu_half_x = 0.5d0 * (mu(i+1,j) + mu(i,j))
lambda_plus_two_mu_half_x = lambda_half_x + 2.d0 * mu_half_x
value_dvx_dx = (27.d0*vx(i+1,j)-27.d0*vx(i,j)-vx(i+2,j)+vx(i-1,j)) / (24.d0*DELTAX)
value_dvy_dy = (27.d0*vy(i,j)-27.d0*vy(i,j-1)-vy(i,j+1)+vy(i,j-2)) / (24.d0*DELTAY)
memory_dvx_dx(i,j) = b_x_half(i) * memory_dvx_dx(i,j) + a_x_half(i) * value_dvx_dx
memory_dvy_dy(i,j) = b_y(j) * memory_dvy_dy(i,j) + a_y(j) * value_dvy_dy
value_dvx_dx = value_dvx_dx / K_x_half(i) + memory_dvx_dx(i,j)
value_dvy_dy = value_dvy_dy / K_y(j) + memory_dvy_dy(i,j)
sigmaxx(i,j) = sigmaxx(i,j) + &
(lambda_plus_two_mu_half_x * value_dvx_dx + lambda_half_x * value_dvy_dy) * DELTAT
sigmayy(i,j) = sigmayy(i,j) + &
(lambda_half_x * value_dvx_dx + lambda_plus_two_mu_half_x * value_dvy_dy) * DELTAT
enddo
enddo
do j = 1,NY-1
do i = 2,NX
! interpolate material parameters at the right location in the staggered grid cell
mu_half_y = 0.5d0 * (mu(i,j+1) + mu(i,j))
value_dvy_dx = (27.d0*vy(i,j)-27.d0*vy(i-1,j)-vy(i+1,j)+vy(i-2,j)) / (24.d0*DELTAX)
value_dvx_dy = (27.d0*vx(i,j+1)-27.d0*vx(i,j)-vx(i,j+2)+vx(i,j-1)) / (24.d0*DELTAY)
memory_dvy_dx(i,j) = b_x(i) * memory_dvy_dx(i,j) + a_x(i) * value_dvy_dx
memory_dvx_dy(i,j) = b_y_half(j) * memory_dvx_dy(i,j) + a_y_half(j) * value_dvx_dy
value_dvy_dx = value_dvy_dx / K_x(i) + memory_dvy_dx(i,j)
value_dvx_dy = value_dvx_dy / K_y(j) + memory_dvx_dy(i,j)
sigmaxy(i,j) = sigmaxy(i,j) + mu_half_y * (value_dvy_dx + value_dvx_dy) * DELTAT
enddo
enddo
!--------------------------------------------------------
! compute velocity and update memory variables for C-PML
!--------------------------------------------------------
do j = 2,NY
do i = 2,NX
value_dsigmaxx_dx = (27.d0*sigmaxx(i,j)-27.d0*sigmaxx(i-1,j)-sigmaxx(i+1,j)+sigmaxx(i-2,j)) / (24.d0*DELTAX)
value_dsigmaxy_dy = (27.d0*sigmaxy(i,j)-27.d0*sigmaxy(i,j-1)-sigmaxy(i,j+1)+sigmaxy(i,j-2)) / (24.d0*DELTAY)
memory_dsigmaxx_dx(i,j) = b_x(i) * memory_dsigmaxx_dx(i,j) + a_x(i) * value_dsigmaxx_dx
memory_dsigmaxy_dy(i,j) = b_y(j) * memory_dsigmaxy_dy(i,j) + a_y(j) * value_dsigmaxy_dy
value_dsigmaxx_dx = value_dsigmaxx_dx / K_x(i) + memory_dsigmaxx_dx(i,j)
value_dsigmaxy_dy = value_dsigmaxy_dy / K_y(j) + memory_dsigmaxy_dy(i,j)
vx(i,j) = vx(i,j) + (value_dsigmaxx_dx + value_dsigmaxy_dy) * DELTAT / rho(i,j)
enddo
enddo
do j = 1,NY-1
do i = 1,NX-1
! interpolate density at the right location in the staggered grid cell
rho_half_x_half_y = 0.25d0 * (rho(i,j) + rho(i+1,j) + rho(i+1,j+1) + rho(i,j+1))
value_dsigmaxy_dx = (27.d0*sigmaxy(i+1,j)-27.d0*sigmaxy(i,j)-sigmaxy(i+2,j)+sigmaxy(i-1,j)) / (24.d0*DELTAX)
value_dsigmayy_dy = (27.d0*sigmayy(i,j+1)-27.d0*sigmayy(i,j)-sigmayy(i,j+2)+sigmayy(i,j-1)) / (24.d0*DELTAY)
memory_dsigmaxy_dx(i,j) = b_x_half(i) * memory_dsigmaxy_dx(i,j) + a_x_half(i) * value_dsigmaxy_dx
memory_dsigmayy_dy(i,j) = b_y_half(j) * memory_dsigmayy_dy(i,j) + a_y_half(j) * value_dsigmayy_dy
value_dsigmaxy_dx = value_dsigmaxy_dx / K_x_half(i) + memory_dsigmaxy_dx(i,j)
value_dsigmayy_dy = value_dsigmayy_dy / K_y_half(j) + memory_dsigmayy_dy(i,j)
vy(i,j) = vy(i,j) + (value_dsigmaxy_dx + value_dsigmayy_dy) * DELTAT / rho_half_x_half_y
enddo
enddo
! add the source (force vector located at a given grid point)
a = pi*pi*f0*f0
t = dble(it-1)*DELTAT
! Gaussian
! source_term = factor * exp(-a*(t-t0)**2)
! first derivative of a Gaussian
source_term = - factor * 2.d0*a*(t-t0)*exp(-a*(t-t0)**2)
! Ricker source time function (second derivative of a Gaussian)
! source_term = factor * (1.d0 - 2.d0*a*(t-t0)**2)*exp(-a*(t-t0)**2)
force_x = sin(ANGLE_FORCE * DEGREES_TO_RADIANS) * source_term
force_y = cos(ANGLE_FORCE * DEGREES_TO_RADIANS) * source_term
! define location of the source
i = ISOURCE
j = JSOURCE
! interpolate density at the right location in the staggered grid cell
rho_half_x_half_y = 0.25d0 * (rho(i,j) + rho(i+1,j) + rho(i+1,j+1) + rho(i,j+1))
vx(i,j) = vx(i,j) + force_x * DELTAT / rho(i,j)
vy(i,j) = vy(i,j) + force_y * DELTAT / rho_half_x_half_y
! Dirichlet conditions (rigid boundaries) on the edges or at the bottom of the PML layers
vx(1,:) = ZERO
vx(NX,:) = ZERO
vx(:,1) = ZERO
vx(:,NY) = ZERO
vy(1,:) = ZERO
vy(NX,:) = ZERO
vy(:,1) = ZERO
vy(:,NY) = ZERO
! store seismograms
do irec = 1,NREC
sisvx(it,irec) = vx(ix_rec(irec),iy_rec(irec))
sisvy(it,irec) = vy(ix_rec(irec),iy_rec(irec))
enddo
! compute total energy in the medium (without the PML layers)
! compute kinetic energy first, defined as 1/2 rho ||v||^2
! in principle we should use rho_half_x_half_y instead of rho for vy
! in order to interpolate density at the right location in the staggered grid cell
! but in a homogeneous medium we can safely ignore it
total_energy_kinetic(it) = 0.5d0 * sum( &
rho(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)*( &
vx(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)**2 + &
vy(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)**2))
! add potential energy, defined as 1/2 epsilon_ij sigma_ij
! in principle we should interpolate the medium parameters at the right location
! in the staggered grid cell but in a homogeneous medium we can safely ignore it
total_energy_potential(it) = ZERO
do j = NPOINTS_PML, NY-NPOINTS_PML+1
do i = NPOINTS_PML, NX-NPOINTS_PML+1
epsilon_xx = ((lambda(i,j) + 2.d0*mu(i,j)) * sigmaxx(i,j) - lambda(i,j) * &
sigmayy(i,j)) / (4.d0 * mu(i,j) * (lambda(i,j) + mu(i,j)))
epsilon_yy = ((lambda(i,j) + 2.d0*mu(i,j)) * sigmayy(i,j) - lambda(i,j) * &
sigmaxx(i,j)) / (4.d0 * mu(i,j) * (lambda(i,j) + mu(i,j)))
epsilon_xy = sigmaxy(i,j) / (2.d0 * mu(i,j))
total_energy_potential(it) = total_energy_potential(it) + &
0.5d0 * (epsilon_xx * sigmaxx(i,j) + epsilon_yy * sigmayy(i,j) + 2.d0 * epsilon_xy * sigmaxy(i,j))
enddo
enddo
! output information
if (mod(it,IT_DISPLAY) == 0 .or. it == 5) then
! print maximum of norm of velocity
velocnorm = maxval(sqrt(vx**2 + vy**2))
print *,'Time step # ',it,' out of ',NSTEP
print *,'Time: ',sngl((it-1)*DELTAT),' seconds'
print *,'Max norm velocity vector V (m/s) = ',velocnorm
print *,'total energy = ',total_energy_kinetic(it) + total_energy_potential(it)
print *
! check stability of the code, exit if unstable
if (velocnorm > STABILITY_THRESHOLD) stop 'code became unstable and blew up'
call create_color_image(vx,NX+2,NY+2,it,ISOURCE,JSOURCE,ix_rec,iy_rec,nrec, &
NPOINTS_PML,USE_PML_XMIN,USE_PML_XMAX,USE_PML_YMIN,USE_PML_YMAX,1)
call create_color_image(vy,NX+2,NY+2,it,ISOURCE,JSOURCE,ix_rec,iy_rec,nrec, &
NPOINTS_PML,USE_PML_XMIN,USE_PML_XMAX,USE_PML_YMIN,USE_PML_YMAX,2)
endif
enddo ! end of time loop
! save seismograms
call write_seismograms(sisvx,sisvy,NSTEP,NREC,DELTAT)
! save total energy
open(unit=20,file='energy.dat',status='unknown')
do it = 1,NSTEP
write(20,*) sngl(dble(it-1)*DELTAT),sngl(total_energy_kinetic(it)), &
sngl(total_energy_potential(it)),sngl(total_energy_kinetic(it) + total_energy_potential(it))
enddo
close(20)
! create script for Gnuplot for total energy
open(unit=20,file='plot_energy',status='unknown')
write(20,*) '# set term x11'
write(20,*) 'set term postscript landscape monochrome dashed "Helvetica" 22'
write(20,*)
write(20,*) 'set xlabel "Time (s)"'
write(20,*) 'set ylabel "Total energy"'
write(20,*)
write(20,*) 'set output "cpml_total_energy_semilog.eps"'
write(20,*) 'set logscale y'
write(20,*) 'plot "energy.dat" us 1:2 t ''Ec'' w l lc 1, "energy.dat" us 1:3 &
& t ''Ep'' w l lc 3, "energy.dat" us 1:4 t ''Total energy'' w l lc 4'
write(20,*) 'pause -1 "Hit any key..."'
write(20,*)
close(20)
open(unit=20,file='plot_comparison',status='unknown')
write(20,*) '# set term x11'
write(20,*) 'set term postscript landscape monochrome dashed "Helvetica" 22'
write(20,*)
write(20,*) 'set xlabel "Time (s)"'
write(20,*) 'set ylabel "Total energy"'
write(20,*)
write(20,*) 'set output "compare_total_energy_semilog.eps"'
write(20,*) 'set logscale y'
write(20,*) 'plot "energy.dat" us 1:4 t ''Total energy CPML'' w l lc 1, &
& "../collino/energy.dat" us 1:4 t ''Total energy Collino'' w l lc 2'
write(20,*) 'pause -1 "Hit any key..."'
write(20,*)
close(20)
! create script for Gnuplot
open(unit=20,file='plotgnu',status='unknown')
write(20,*) 'set term x11'
write(20,*) '# set term postscript landscape monochrome dashed "Helvetica" 22'
write(20,*)
write(20,*) 'set xlabel "Time (s)"'
write(20,*) 'set ylabel "Amplitude (m / s)"'
write(20,*)
write(20,*) 'set output "v_sigma_Vx_receiver_001.eps"'
write(20,*) 'plot "Vx_file_001.dat" t ''Vx C-PML'' w l lc 1'
write(20,*) 'pause -1 "Hit any key..."'
write(20,*)
write(20,*) 'set output "v_sigma_Vy_receiver_001.eps"'
write(20,*) 'plot "Vy_file_001.dat" t ''Vy C-PML'' w l lc 1'
write(20,*) 'pause -1 "Hit any key..."'
write(20,*)
write(20,*) 'set output "v_sigma_Vx_receiver_002.eps"'
write(20,*) 'plot "Vx_file_002.dat" t ''Vx C-PML'' w l lc 1'
write(20,*) 'pause -1 "Hit any key..."'
write(20,*)
write(20,*) 'set output "v_sigma_Vy_receiver_002.eps"'
write(20,*) 'plot "Vy_file_002.dat" t ''Vy C-PML'' w l lc 1'
write(20,*) 'pause -1 "Hit any key..."'
write(20,*)
close(20)
print *
print *,'End of the simulation'
print *
end program seismic_CPML_2D_iso_fourth
!----
!---- save the seismograms in ASCII text format
!----
subroutine write_seismograms(sisvx,sisvy,nt,nrec,DELTAT)
implicit none
integer nt,nrec
double precision DELTAT
double precision sisvx(nt,nrec)
double precision sisvy(nt,nrec)
integer irec,it
character(len=100) file_name
! X component
do irec=1,nrec
write(file_name,"('Vx_file_',i3.3,'.dat')") irec
open(unit=11,file=file_name,status='unknown')
do it=1,nt
write(11,*) sngl(dble(it-1)*DELTAT),' ',sngl(sisvx(it,irec))
enddo
close(11)
enddo
! Y component
do irec=1,nrec
write(file_name,"('Vy_file_',i3.3,'.dat')") irec
open(unit=11,file=file_name,status='unknown')
do it=1,nt
write(11,*) sngl(dble(it-1)*DELTAT),' ',sngl(sisvy(it,irec))
enddo
close(11)
enddo
end subroutine write_seismograms
!----
!---- routine to create a color image of a given vector component
!---- the image is created in PNM format and then converted to GIF
!----
subroutine create_color_image(image_data_2D,NX,NY,it,ISOURCE,JSOURCE,ix_rec,iy_rec,nrec, &
NPOINTS_PML,USE_PML_XMIN,USE_PML_XMAX,USE_PML_YMIN,USE_PML_YMAX,field_number)
implicit none
! non linear display to enhance small amplitudes for graphics
double precision, parameter :: POWER_DISPLAY = 0.30d0
! amplitude threshold above which we draw the color point
double precision, parameter :: cutvect = 0.01d0
! use black or white background for points that are below the threshold
logical, parameter :: WHITE_BACKGROUND = .true.
! size of cross and square in pixels drawn to represent the source and the receivers
integer, parameter :: width_cross = 5, thickness_cross = 1, size_square = 3
integer NX,NY,it,field_number,ISOURCE,JSOURCE,NPOINTS_PML,nrec
logical USE_PML_XMIN,USE_PML_XMAX,USE_PML_YMIN,USE_PML_YMAX
double precision, dimension(NX,NY) :: image_data_2D
integer, dimension(nrec) :: ix_rec,iy_rec
integer :: ix,iy,irec
character(len=100) :: file_name,system_command
integer :: R, G, B
double precision :: normalized_value,max_amplitude
! open image file and create system command to convert image to more convenient format
! use the "convert" command from ImageMagick http://www.imagemagick.org
if (field_number == 1) then
write(file_name,"('image',i6.6,'_Vx.pnm')") it
write(system_command,"('convert image',i6.6,'_Vx.pnm image',i6.6,'_Vx.gif ; rm image',i6.6,'_Vx.pnm')") it,it,it
else if (field_number == 2) then
write(file_name,"('image',i6.6,'_Vy.pnm')") it
write(system_command,"('convert image',i6.6,'_Vy.pnm image',i6.6,'_Vy.gif ; rm image',i6.6,'_Vy.pnm')") it,it,it
endif
open(unit=27, file=file_name, status='unknown')
write(27,"('P3')") ! write image in PNM P3 format
write(27,*) NX,NY ! write image size
write(27,*) '255' ! maximum value of each pixel color
! compute maximum amplitude
max_amplitude = maxval(abs(image_data_2D))
! image starts in upper-left corner in PNM format
do iy=NY,1,-1
do ix=1,NX
! define data as vector component normalized to [-1:1] and rounded to nearest integer
! keeping in mind that amplitude can be negative
normalized_value = image_data_2D(ix,iy) / max_amplitude
! suppress values that are outside [-1:+1] to avoid small edge effects
if (normalized_value < -1.d0) normalized_value = -1.d0
if (normalized_value > 1.d0) normalized_value = 1.d0
! draw an orange cross to represent the source
if ((ix >= ISOURCE - width_cross .and. ix <= ISOURCE + width_cross .and. &
iy >= JSOURCE - thickness_cross .and. iy <= JSOURCE + thickness_cross) .or. &
(ix >= ISOURCE - thickness_cross .and. ix <= ISOURCE + thickness_cross .and. &
iy >= JSOURCE - width_cross .and. iy <= JSOURCE + width_cross)) then
R = 255
G = 157
B = 0
! display two-pixel-thick black frame around the image
else if (ix <= 2 .or. ix >= NX-1 .or. iy <= 2 .or. iy >= NY-1) then
R = 0
G = 0
B = 0
! display edges of the PML layers
else if ((USE_PML_XMIN .and. ix == NPOINTS_PML) .or. &
(USE_PML_XMAX .and. ix == NX - NPOINTS_PML) .or. &
(USE_PML_YMIN .and. iy == NPOINTS_PML) .or. &
(USE_PML_YMAX .and. iy == NY - NPOINTS_PML)) then
R = 255
G = 150
B = 0
! suppress all the values that are below the threshold
else if (abs(image_data_2D(ix,iy)) <= max_amplitude * cutvect) then
! use a black or white background for points that are below the threshold
if (WHITE_BACKGROUND) then
R = 255
G = 255
B = 255
else
R = 0
G = 0
B = 0
endif
! represent regular image points using red if value is positive, blue if negative
else if (normalized_value >= 0.d0) then
R = nint(255.d0*normalized_value**POWER_DISPLAY)
G = 0
B = 0
else
R = 0
G = 0
B = nint(255.d0*abs(normalized_value)**POWER_DISPLAY)
endif
! draw a green square to represent the receivers
do irec = 1,nrec
if ((ix >= ix_rec(irec) - size_square .and. ix <= ix_rec(irec) + size_square .and. &
iy >= iy_rec(irec) - size_square .and. iy <= iy_rec(irec) + size_square) .or. &
(ix >= ix_rec(irec) - size_square .and. ix <= ix_rec(irec) + size_square .and. &
iy >= iy_rec(irec) - size_square .and. iy <= iy_rec(irec) + size_square)) then
! use dark green color
R = 30
G = 180
B = 60
endif
enddo
! write color pixel
write(27,"(i3,' ',i3,' ',i3)") R,G,B
enddo
enddo
! close file
close(27)
! call the system to convert image to Gif (can be commented out if "call system" is missing in your compiler)
! call system(system_command)
end subroutine create_color_image