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seismic_ADEPML_2D_elastic_RK4_eighth_order.f90
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seismic_ADEPML_2D_elastic_RK4_eighth_order.f90
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!
! Copyright Universite de Pau et des Pays de l'Adour, CNRS and INRIA, France.
! Contributors: Roland Martin, roland DOT martin aT get DOT obs-mip DOT fr
! and Youshan Liu, China.
!
! This software is a computer program whose purpose is to solve
! the two-dimensional isotropic elastic wave equation
! using a finite-difference method with Auxiliary Differential
! Equation Perfectly Matched Layer (ADE-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_ADEPML_2D_elastic_RK4_eighth_order
! High order 2D explicit-semi implicit-implicit elastic finite-difference code
! in velocity and stress formulation with Auxiliary Differential
! Equation Perfectly Matched Layer (ADE-PML) absorbing conditions for
! an isotropic elastic medium. It is fourth order Runge-Kutta (RK4) in time
! and 8th order in space using Holberg spatial discretization.
! Version 1.1.3
! by Roland Martin, University of Pau, France, Jan 2010
! with a major bug fix in the Runge-Kutta implementation
! and also significant memory usage optimization by Youshan Liu, China, August 2015.
! based on seismic_CPML_2D_isotropic_second_order.f90
! by Dimitri Komatitsch and Roland Martin, University of Pau, France, 2007.
! The 8th-order staggered-grid formulation of Holberg is used:
!
! ^ y
! |
! |
!
! +-------------------+
! | |
! | |
! | |
! | |
! | v_y |
! sigma_xy +---------+ |
! | | |
! | | |
! | | |
! | | |
! | | |
! +---------+---------+ ---> x
! v_x sigma_xx
! sigma_yy
!
! The ADE-PML implementation is based in part on formulas given in Roden and Gedney (2010)
!
! If you use this code for your own research, please cite some (or all) of these articles:
!
! @ARTICLE{MaKoGeBr10,
! author = {Roland Martin and Dimitri Komatitsch and Stephen D. Gedney and Emilien Bruthiaux},
! title = {A high-order time and space formulation of the unsplit perfectly matched layer
! for the seismic wave equation using {Auxiliary Differential Equations (ADE-PML)}},
! journal = {Comput. Model. Eng. Sci.},
! year = {2010},
! volume = {56},
! pages = {17-42},
! number = {1}}
!
! @ARTICLE{MaCo10,
! author = {Roland Martin and Carlos Couder-Casta{\~n}eda},
! title = {An improved unsplit and convolutional Perfectly Matched Layer
! absorbing technique for the Navier-Stokes equations using cut-off frequency shift},
! journal = {Comput. Model. Eng. Sci.},
! pages ={47-77}
! year = {2010},
! volume = {63},
! number = {1}}
!
! @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}}
!
! @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{MaKo09,
! author = {Roland Martin and Dimitri Komatitsch},
! title = {An unsplit convolutional perfectly matched layer technique improved
! at grazing incidence for the viscoelastic wave equation},
! journal = {Geophysical Journal International},
! year = {2009},
! volume = {179},
! pages = {333-344},
! number = {1},
! doi = {10.1111/j.1365-246X.2009.04278.x}}
!
! @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}}
!
! 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
integer, parameter :: NX = 241
integer, parameter :: NY = 241
! Explicit (epsn=1,epsn=0), implicit (epsn=0,epsn1=1), semi-implicit (epsn=0.5,epsn1=0.5)
integer, parameter :: iexpl=0
integer, parameter :: iimpl=0
integer, parameter :: isemiimpl=1
double precision :: epsn,epsn1
! 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. 8th order in space imposes to
! increase the thickness of the CPML
integer, parameter :: NPOINTS_PML = 10
! P-velocity, S-velocity and density
double precision, parameter :: cp = 2000.d0
double precision, parameter :: cs = 1150.d0
double precision, parameter :: density = 2000.d0
!double precision, parameter :: cp = 3300.d0
!double precision, parameter :: cs = 1905.d0
!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 = 2501
! time step in seconds
! 8th-order in space and 4th-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 = 3.d-3
! parameters for the source
double precision, parameter :: f0 = 10.d0
double precision, parameter :: t0 = 1.0d0 / f0
double precision, parameter :: factor = 1.d4
! source
!integer, parameter :: ISOURCE = NX - 2*NPOINTS_PML - 1
integer, parameter :: ISOURCE = (NX-1)/2
integer, parameter :: JSOURCE = (NY-1)/2
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
double precision, parameter :: ANGLE_FORCE = 90.d0
! receivers
!integer, parameter :: NREC = 3
!double precision, parameter :: xdeb = xsource ! first receiver x in meters
!double precision, parameter :: ydeb = ysource - 2000.d0 ! first receiver y in meters
!double precision, parameter :: xfin = xsource ! last receiver x in meters
!double precision, parameter :: yfin = ysource - 4000.d0 ! last receiver y in meters
integer, parameter :: NREC = NX
double precision, parameter :: xdeb = 0.d0 ! first receiver x in meters
double precision, parameter :: ydeb = 50.d0 ! first receiver y in meters
double precision, parameter :: xfin = (NX-1)*DELTAX ! last receiver x in meters
double precision, parameter :: yfin = 50.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 = 200
! 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
! Holberg (1987) coefficients, taken from
! @ARTICLE{Hol87,
! author = {O. Holberg},
! title = {Computational aspects of the choice of operator and sampling interval
! for numerical differentiation in large-scale simulation of wave phenomena},
! journal = {Geophysical Prospecting},
! year = {1987},
! volume = {35},
! pages = {629-655}}
double precision, parameter :: c1 = 1.231666d0
double precision, parameter :: c2 = -1.041182d-1
double precision, parameter :: c3 = 2.063707d-2
double precision, parameter :: c4 = -3.570998d-3
! RK4 scheme coefficients, 2 per subloop, 8 in total
double precision, dimension(4) :: rk41, rk42
! main arrays
double precision, dimension(-4:NX+4,-4:NY+4) :: lambda,mu,rho,vx,vy,sigmaxx,sigmayy,sigmaxy
! variables are stored in four indices in the first dimension to implement RK4
! dv does not always indicate a derivative
double precision, dimension(3,-4:NX+4,-4:NY+4) :: dvx,dvy,dsigmaxx,dsigmayy,dsigmaxy
! 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, dimension(NSTEP) :: total_energy_kinetic,total_energy_potential
! power to compute d0 profile
double precision, parameter :: NPOWER = 2.d0
double precision, parameter :: NPOWER2 = 2.d0
! Kappa must be strong enough to absorb energy and low enough to avoid
! reflections.
! Alpha1 must be low to absorb energy and high enough to have efficiency on
! grazing incident waves.
double precision, parameter :: K_MAX_PML = 7.d0
double precision, parameter :: ALPHA_MAX_PML = 2.d0*PI*(f0/2.d0)
! arrays for the memory variables
! could declare these arrays in PML only to save a lot of memory, but proof of concept only here
!!! Youshan Liu suppressed the two comment lines below
!!!!!! not true anymore: We have as many memory variables as the number of frequency shift poles in the CPML
!!!!!! not true anymore: Indices are 1 and 2 for the 2 frequency shift poles
! ==================== revised by Youshan Liu ==================
double precision, dimension(-4:NX+4,-4:NY+4) :: 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(-4:NX+4) :: d_x,K_x,alpha_x,g_x,ksi_x
double precision, dimension(-4:NX+4) :: d_x_half,K_x_half,alpha_x_half,g_x_half,ksi_x_half
double precision, dimension(-4:NY+4) :: d_y,K_y,alpha_y,g_y,ksi_y
double precision, dimension(-4:NY+4) :: d_y_half,K_y_half,alpha_y_half,g_y_half,ksi_y_half
! coefficients that allow to reset the memory variables at each RK4 substep depend on the substepping and are then of dimension 4,
! 1D arrays for the damping profiles
double precision, dimension(4,-4:NX+4) :: a_x,b_x
double precision, dimension(4,-4:NX+4) :: a_x_half,b_x_half
double precision, dimension(4,-4:NY+4) :: a_y,b_y
double precision, dimension(4,-4:NY+4) :: 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,k,it,irec,inc
double precision :: Courant_number
!define by ysliu 8/2/2015
integer(2) head(1:120)
character(80) :: routine
real,dimension(NSTEP,NREC) :: seisvx, seisvy
real,dimension(NX,NY) :: snapvx,snapvy
!---
!--- program starts here
!---
if (iexpl == 1) then
epsn = 1.d0
epsn1 = 0.d0
endif
if (iimpl == 1) then
epsn = 0.d0
epsn1 = 1.d0
endif
if (isemiimpl == 1) then
epsn = 0.5d0
epsn1 = 0.5d0
endif
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 = 0.00001d0
! 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 *
! parameters involved in RK4 time expansion
rk41(1) = ZERO
rk41(2) = 0.5d0
rk41(3) = 0.5d0
rk41(4) = 1.d0
rk42(1) = 1.d0 / 6.d0
rk42(2) = 2.d0 / 6.d0
rk42(3) = 2.d0 / 6.d0
rk42(4) = 1.d0 / 6.d0
ksi_x(:) = ZERO
ksi_x_half(:) = ZERO
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
g_x(:) = 5.d-1
g_x_half(:) = 5.d-1
ksi_y(:) = ZERO
ksi_y_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
g_y(:) = 1.d0
g_y_half(:) = 1.d0
! 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 = -4,NX+4
! 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**NPOWER2
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**NPOWER2
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**NPOWER2
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**NPOWER2
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
! CPML damping parameters for the 4 sub time steps of RK4 algorithm
do inc=1,4
b_x(inc,i) = (1.-epsn*DELTAT*rk41(inc)*(d_x(i)/K_x(i) + alpha_x(i)))/ &
(1.+epsn1*DELTAT*rk41(inc)*(d_x(i)/K_x(i) + alpha_x(i)))
b_x_half(inc,i) = (1.-epsn*DELTAT*rk41(inc)*(d_x_half(i)/K_x_half(i) &
+ alpha_x_half(i)))/(1. +epsn1*DELTAT*rk41(inc)*(d_x_half(i)/K_x_half(i) &
+ alpha_x_half(i)))
! this to avoid division by zero outside the PML
if (abs(d_x(i)) > 1.d-6) a_x(inc,i) = - DELTAT*rk41(inc)*d_x(i) / (K_x(i)* K_x(i))/&
(1. +epsn1*DELTAT*rk41(inc)*(d_x(i)/K_x(i) + alpha_x(i)))
if (abs(d_x_half(i)) > 1.d-6) a_x_half(inc,i) =-DELTAT*rk41(inc)*d_x_half(i)/&
(K_x_half(i)*K_x_half(i) )/&
(1. +epsn1*DELTAT*rk41(inc)*(d_x_half(i)/K_x_half(i)&
+ alpha_x_half(i)))
enddo
enddo !do i = -4,NX+4
! damping in the Y direction
! origin of the PML layer (position of right edge minus thickness, in meters)
yoriginbottom = thickness_PML_y
yorigintop = (NY-1)*DELTAY - thickness_PML_y
do j = -4,NY+4
! 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**NPOWER2
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**NPOWER2
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**NPOWER2
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**NPOWER2
alpha_y_half(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
! just in case, for -5 at the end
if (alpha_y(j) < ZERO) alpha_y(j) = ZERO
if (alpha_y_half(j) < ZERO) alpha_y_half(j) = ZERO
! CPML damping parameters for the 4 sub time steps of RK4 algorithm
do inc=1,4
b_y(inc,j) = (1.-epsn*DELTAT*rk41(inc)*(d_y(j)/K_y(j) + alpha_y(j)))/ &
(1.+epsn1*DELTAT*rk41(inc)*(d_y(j)/K_y(j) + alpha_y(j)))
b_y_half(inc,j) = (1.-epsn*DELTAT*rk41(inc)*(d_y_half(j)/K_y_half(j) + &
alpha_y_half(j)))/(1.+epsn1*DELTAT*rk41(inc)*(d_y_half(j)/K_y_half(j) &
+ alpha_y_half(j)))
! this to avoid division by zero outside the PML
if (abs(d_y(j)) > 1.d-6) a_y(inc,j) = - DELTAT*rk41(inc)*d_y(j) &
/ (K_y(j)* K_y(j))/&
(1.+epsn1*DELTAT*rk41(inc)*(d_y(j)/K_y(j) + alpha_y(j)))
if (abs(d_y_half(j)) > 1.d-6) a_y_half(inc,j) = -DELTAT*rk41(inc)*d_y_half(j) /&
(K_y_half(j) * K_y_half(j) )/&
(1.+epsn1*DELTAT*rk41(inc)*(d_y_half(j)/K_y_half(j) + alpha_y_half(j)))
enddo
enddo !do j = -4,NY+4
! compute the Lame parameters and density
do j = -4,NY+4
do i = -4,NX+4
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 !do irec=1,nrec
! check the Courant stability condition for the explicit time scheme
! R. Courant and K. O. Friedrichs and 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
dvx(:,:,:) = ZERO
dvy(:,:,:) = ZERO
dsigmaxx(:,:,:) = ZERO
dsigmayy(:,:,:) = ZERO
dsigmaxy(:,:,:) = ZERO
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
!! v and sigma temporary variables of RK4
!======================================================
!====================revised by ysliu==================
!backup the current snapshots
dvx(2,:,:) = vx(:,:)
dvy(2,:,:) = vy(:,:)
dsigmaxx(2,:,:) = sigmaxx(:,:)
dsigmayy(2,:,:) = sigmayy(:,:)
dsigmaxy(2,:,:) = sigmaxy(:,:)
dvx(3,:,:) = vx(:,:)
dvy(3,:,:) = vy(:,:)
dsigmaxx(3,:,:) = sigmaxx(:,:)
dsigmayy(3,:,:) = sigmayy(:,:)
dsigmaxy(3,:,:) = sigmaxy(:,:)
!======================================================
! RK4 loop (loop on the four RK4 substeps)
do inc= 1,4
! ==================== revised by Youshan Liu ==================
! The new values of the different variables v and sigma are computed
dvx(1,:,:) = dvx(3,:,:) + rk41(inc) * dvx(2,:,:) * DELTAT
dvy(1,:,:) = dvy(3,:,:) + rk41(inc) * dvy(2,:,:) * DELTAT
dsigmaxx(1,:,:) = dsigmaxx(3,:,:) + rk41(inc) * dsigmaxx(2,:,:) * DELTAT
dsigmayy(1,:,:) = dsigmayy(3,:,:) + rk41(inc) * dsigmayy(2,:,:) * DELTAT
dsigmaxy(1,:,:) = dsigmaxy(3,:,:) + rk41(inc) * dsigmaxy(2,:,:) * DELTAT
!------------------
! compute velocity
!------------------
do j = 2,NY
do i = 2,NX
value_dsigmaxx_dx = ( c1 * (dsigmaxx(1,i,j) - dsigmaxx(1,i-1,j)) + c2 * (dsigmaxx(1,i+1,j) - dsigmaxx(1,i-2,j)) + &
c3 * (dsigmaxx(1,i+2,j) - dsigmaxx(1,i-3,j)) + c4 * (dsigmaxx(1,i+3,j) - dsigmaxx(1,i-4,j)) )/ DELTAX
value_dsigmaxy_dy = ( c1 * (dsigmaxy(1,i,j) - dsigmaxy(1,i,j-1)) + c2* (dsigmaxy(1,i,j+1) - dsigmaxy(1,i,j-2)) + &
c3 * (dsigmaxy(1,i,j+2) - dsigmaxy(1,i,j-3)) + c4 * (dsigmaxy(1,i,j+3) - dsigmaxy(1,i,j-4)) )/ DELTAY
if (i <= NPOINTS_PML+2 .or. i >= NX-NPOINTS_PML-2 .or. j <= NPOINTS_PML+2 .or. j >= NY-NPOINTS_PML-2) then
! ==================== revised by Youshan Liu ==================
memory_dsigmaxx_dx(i,j) = b_x(inc,i) * memory_dsigmaxx_dx(i,j) + a_x(inc,i) * value_dsigmaxx_dx
memory_dsigmaxy_dy(i,j) = b_y(inc,j) * memory_dsigmaxy_dy(i,j) + a_y(inc,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)
endif
dvx(2,i,j) = (value_dsigmaxx_dx + value_dsigmaxy_dy) / 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 = ( c1 * (dsigmaxy(1,i+1,j) - dsigmaxy(1,i,j)) + c2 * (dsigmaxy(1,i+2,j) - dsigmaxy(1,i-1,j)) + &
c3 * (dsigmaxy(1,i+3,j) - dsigmaxy(1,i-2,j)) + c4 * (dsigmaxy(1,i+4,j) - dsigmaxy(1,i-3,j)) )/ DELTAX
value_dsigmayy_dy = ( c1 * (dsigmayy(1,i,j+1) - dsigmayy(1,i,j)) + c2 * (dsigmayy(1,i,j+2) - dsigmayy(1,i,j-1)) + &
c3 * (dsigmayy(1,i,j+3) - dsigmayy(1,i,j-2)) + c4 * (dsigmayy(1,i,j+4) - dsigmayy(1,i,j-3)) )/ DELTAY
if (i <= NPOINTS_PML+2 .or. i >= NX-NPOINTS_PML-2 .or. j <= NPOINTS_PML+2 .or. j >= NY-NPOINTS_PML-2) then
! ==================== revised by Youshan Liu ==================
memory_dsigmaxy_dx(i,j) = b_x_half(inc,i) * memory_dsigmaxy_dx(i,j) + a_x_half(inc,i) * value_dsigmaxy_dx
memory_dsigmayy_dy(i,j) = b_y_half(inc,j) * memory_dsigmayy_dy(i,j) + a_y_half(inc,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)
endif
dvy(2,i,j) = (value_dsigmaxy_dx + value_dsigmayy_dy) / 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)+ rk41(inc)) * 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))
dvx(2,i,j) = dvx(2,i,j) + force_x / rho(i,j)
dvy(2,i,j) = dvy(2,i,j) + force_y / rho_half_x_half_y
! Dirichlet conditions (rigid boundaries) on all the edges of the grid
dvx(:,-4:1,:) = ZERO
dvx(:,NX:NX+4,:) = ZERO
dvx(:,:,-4:1) = ZERO
dvx(:,:,NY:NY+4) = ZERO
dvy(:,-4:1,:) = ZERO
dvy(:,NX:NX+4,:) = ZERO
dvy(:,:,-4:1) = ZERO
dvy(:,:,NY:NY+4) = ZERO
!----------------------
! compute stress sigma
!----------------------
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 = ( c1 * (dvx(1,i+1,j) - dvx(1,i,j)) + c2 * (dvx(1,i+2,j) - dvx(1,i-1,j)) + &
c3 * (dvx(1,i+3,j) - dvx(1,i-2,j)) + c4 * (dvx(1,i+4,j) - dvx(1,i-3,j)) )/ DELTAX
value_dvy_dy = ( c1 * (dvy(1,i,j) - dvy(1,i,j-1)) + c2 * (dvy(1,i,j+1) - dvy(1,i,j-2)) + &
c3 * (dvy(1,i,j+2) - dvy(1,i,j-3)) + c4 * (dvy(1,i,j+3) - dvy(1,i,j-4)) )/ DELTAY
if (i <= NPOINTS_PML+2 .or. i >= NX-NPOINTS_PML-2 .or. j <= NPOINTS_PML+2 .or. j >= NY-NPOINTS_PML-2) then
! ==================== revised by Youshan Liu ==================
memory_dvx_dx(i,j) = b_x_half(inc,i) * memory_dvx_dx(i,j) + a_x_half(inc,i) * value_dvx_dx
memory_dvy_dy(i,j) = b_y(inc,j) * memory_dvy_dy(i,j) + a_y(inc,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)
endif
dsigmaxx(2,i,j) = (lambda_plus_two_mu_half_x * value_dvx_dx + lambda_half_x * value_dvy_dy)
dsigmayy(2,i,j) = (lambda_half_x * value_dvx_dx + lambda_plus_two_mu_half_x * value_dvy_dy)
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_dvx_dy = ( c1 * (dvx(1,i,j+1) - dvx(1,i,j)) + c2 * (dvx(1,i,j+2) - dvx(1,i,j-1)) + &
c3 * (dvx(1,i,j+3) - dvx(1,i,j-2)) + c4 * (dvx(1,i,j+4) - dvx(1,i,j-3)) )/ DELTAY
value_dvy_dx = ( c1 * (dvy(1,i,j) - dvy(1,i-1,j)) + c2 * (dvy(1,i+1,j) - dvy(1,i-2,j)) + &
c3 * (dvy(1,i+2,j) - dvy(1,i-3,j)) + c4 * (dvy(1,i+3,j) - dvy(1,i-4,j)) )/ DELTAX
if (i <= NPOINTS_PML+2 .or. i >= NX-NPOINTS_PML-2 .or. j <= NPOINTS_PML+2 .or. j >= NY-NPOINTS_PML-2) then
! ==================== revised by Youshan Liu ==================
memory_dvy_dx(i,j) = b_x(inc,i) * memory_dvy_dx(i,j) + a_x(inc,i) * value_dvy_dx
memory_dvx_dy(i,j) = b_y_half(inc,j) * memory_dvx_dy(i,j) + a_y_half(inc,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_half(j) + memory_dvx_dy(i,j)
endif
dsigmaxy(2,i,j) = mu_half_y * (value_dvy_dx + value_dvx_dy)
enddo
enddo
! ==================== revised by Youshan Liu ==================
! the new values of the different variables v and sigma are computed
vx(:,:) = vx(:,:) + rk42(inc) * dvx(2,:,:) * DELTAT
vy(:,:) = vy(:,:) + rk42(inc) * dvy(2,:,:) * DELTAT
sigmaxx(:,:) = sigmaxx(:,:) + rk42(inc) * dsigmaxx(2,:,:) * DELTAT
sigmayy(:,:) = sigmayy(:,:) + rk42(inc) * dsigmayy(2,:,:) * DELTAT
sigmaxy(:,:) = sigmaxy(:,:) + rk42(inc) * dsigmaxy(2,:,:) * DELTAT
!! Dirichlet conditions (rigid boundaries) on all the edges of the grid
vx(-4:1,:) = ZERO
vx(:,-4:1) = ZERO
vy(-4:1,:) = ZERO
vy(:,-4:1) = ZERO
vx(NX:NX+4,:) = ZERO
vx(:,NY:NY+4) = ZERO
vy(NX:NX+4,:) = ZERO
vy(:,NY:NY+4) = ZERO
enddo
! end of RK4 loop
! store seismograms
do irec = 1,NREC
sisvx(it,irec) = (vx(ix_rec(irec),iy_rec(irec))+ &
vx(ix_rec(irec)+1,iy_rec(irec))+ &
vx(ix_rec(irec),iy_rec(irec)+1)+ &
vx(ix_rec(irec)+1,iy_rec(irec)+1))/4.d0
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+1:NX-NPOINTS_PML,NPOINTS_PML+1:NY-NPOINTS_PML)*( &
! vx(NPOINTS_PML+1:NX-NPOINTS_PML,NPOINTS_PML+1:NY-NPOINTS_PML)**2 + &
! vy(NPOINTS_PML+1:NX-NPOINTS_PML,NPOINTS_PML+1:NY-NPOINTS_PML)**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+1, NY-NPOINTS_PML
! do i = NPOINTS_PML+1, NX-NPOINTS_PML
! 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
if (mod(it,IT_DISPLAY) == 0) then
write(*,*) it, ' of ', nstep
head=0
head(58) = NY
head(59) = DELTAY * 1E3
snapvx = vx(1:NX,1:NY)
snapvy = vy(1:NX,1:NY)
write(routine,'(a12,i5.5,a9)') './snapshots/',it,'snapVx.su'
open(21,file=routine,access='stream')
do j = 1,NX,1
write(21) head,(real(snapvx(k,j)),k=1,NY)
enddo
close(21)
write(routine,'(a12,i5.5,a9)') './snapshots/',it,'snapVy.su'
open(21,file=routine,access='stream')
do j = 1,NX,1
write(21) head,(real(snapvy(k,j)),k=1,NY)
enddo
close(21)
endif
!! 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
! 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(1:NX,1:NY),NX,NY,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(1:NX,1:NY),NX,NY,it,ISOURCE,JSOURCE,ix_rec,iy_rec,nrec, &
! NPOINTS_PML,USE_PML_XMIN,USE_PML_XMAX,USE_PML_YMIN,USE_PML_YMAX,2)
! open(unit=20,file='energy.dat',status='unknown')
! do it2 = 1,NSTEP
! write(20,*) sngl(dble(it2-1)*DELTAT),sngl(total_energy_kinetic(it2)), &
! sngl(total_energy_potential(it2)),sngl(total_energy_kinetic(it2) + total_energy_potential(it2))
! enddo
! close(20)
! call write_seismograms(sisvx,sisvy,NSTEP,NREC,DELTAT)
!
! endif
enddo ! end of time loop
! save seismograms
!save seismogram in SU format
write(*,*) NREC,nstep
seisvx = sisvx
seisvy = sisvy
head=0
head(58)=nstep
head(59)=deltat*1e6
open(21,file='./seismograms/seisVx.su',access='stream')
do j=1,NREC,1
write(21) head,(real(seisvx(k,j)),k=1,nstep)
enddo
close(21)
open(21,file='./seismograms/seisVy.su',access='stream')
do j=1,NREC,1