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Diffusion.f90
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!COPYRIGHT
!
!This solver is based on Nodal Integral Method for the 2-Dimesional Steady State Diffusion equation.
!Copyright (C) 2017 Rishabh Prakash Sharma.
!
! 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, see <http://www.gnu.org/licenses/>
!\***************************************************************\
Program Diffusion
!Do not use very high no. of grids.
!Very high no. of grids leads to high aspect ratio and divergence of scheme.
parameter(n=20, m=20, pt=7)
double precision Sx(m,n),Sy(m,n),Tx(m,n),Ty(m,n),SxxSyy(m,n),TxxTyy(m,n),Sx2Sy2(m,n),Tx2Ty2(m,n),SxTxSyTy(m,n),SxTx(m,n),SyTy(m,n)
double precision A2(m,n),A3(m,n),A4(m,n),A5(m,n),A6(m,n),A7(m,n),A8(m,n),A9(m,n)
double precision B2(m,n),B4(m,n),B3(m,n),B5(m,n),B6(m,n),B7(m,n),B8(m,n),B9(m,n)
double precision F11(m,n),F12(m,n),F13(m,n),F14(m,n),F15(m,n),F16(m,n),F17(m,n),F18(m,n),F29(m,n),F30(m,n)
double precision F21(m,n),F22(m,n),F23(m,n),F24(m,n),F25(m,n),F26(m,n),F28(m,n),Sm1(m,n),Sm2(m,n),Sm3(m,n),Sm4(m,n)
double precision dr,dtheta,err5(m+1,n),err6(m+1,n),f(m,n),Tm1(m,n),Tm2(m,n),Tm3(m,n),Tm4(m,n)
double precision Ts(m+1,n),Tt(m,n+1),Tc(m+1,n+1),Tst(m,n),q(m,n),rs(m,n),Thetat(m,n)
double precision tol5,tol6,temp5(m,n+1),temp6(m+1,n),temp7,temp8,temp9,temp10,temp12(m,n),temp13(m,n)
double precision temptol5,temptol6,xdiv,ydiv,tol,s(7),t(7),ws(7),wt(7)
double precision x1(m,n),x2(m,n),x3(m,n),x4(m,n),x5(m,n),x6(m,n),x7(m,n),x8(m,n),x9(m,n)
double precision y1(m,n),y2(m,n),y3(m,n),y4(m,n),y5(m,n),y6(m,n),y7(m,n),y8(m,n),y9(m,n)
double precision xs(m+1,n),ys(m+1,n),xt(m,n+1),yt(m,n+1),FSx(7,7),FSy(7,7),FTx(7,7),FTy(7,7)
double precision FSx2Sy2(7,7),FTx2Ty2(7,7),FSxTxSyTy(7,7),FSxTx(7,7),FSyTy(7,7)
double precision l1(m,n),l2(m,n),l3(m,n),l4(m,n),m1x(m,n),m1y(m,n),m2x(m,n),m2y(m,n),m3x(m,n),m3y(m,n),m4x(m,n),m4y(m,n)
real T1, T2,r1,r2,angle
integer i,j,ks,kt,var
CALL CPU_TIME(T1)
!=================================Parameters============================!
r1=1.0; !Inner radius
r2=2.0; !Outer radius
xdiv=n;
ydiv=m;
tol=1.0e-08; !tolerance
angle=2*ATAN(1.0); !Angle=Pi/2
dr=(r2-r1)/n;
dtheta=angle/m;
!-----------------------------------------------------------------------!
!============================Grid generation=============================!
!-----------Co-ordinates for plotting-------------!
do i=1,m+1
do j=1,n
xs(i,j)=(r1+(dr/2.0+(j-1)*dr))*(cos((i-1)*dtheta));
ys(i,j)=(r1+(dr/2.0+(j-1)*dr))*(sin((i-1)*dtheta));
rs(i,j)=sqrt(xs(i,j)**2+ys(i,j)**2)
enddo
enddo
do i=1,m
do j=1,n+1
xt(i,j)=(r1+(j-1)*dr)*(cos(dtheta/2.0+(i-1)*dtheta));
yt(i,j)=(r1+(j-1)*dr)*(sin(dtheta/2.0+(i-1)*dtheta));
Thetat(i,j)=ATAN(yt(i,j)/xt(i,j))
enddo
enddo
!-------Discretization of Inner domain using linear elements--------------!
if (n.EQ.2) then
var=2;
else
var=n-1;
endif
do i=1,m
do j=2,var
x1(i,j)=(r1+(j)*dr)*(cos((i)*dtheta));
y1(i,j)=(r1+(j)*dr)*(sin((i)*dtheta));
x2(i,j)=(r1+(j-1)*dr)*(cos((i)*dtheta));
y2(i,j)=(r1+(j-1)*dr)*(sin((i)*dtheta));
x3(i,j)=(r1+(j-1)*dr)*(cos((i-1)*dtheta));
y3(i,j)=(r1+(j-1)*dr)*(sin((i-1)*dtheta));
x4(i,j)=(r1+(j)*dr)*(cos((i-1)*dtheta));
y4(i,j)=(r1+(j)*dr)*(sin((i-1)*dtheta));
x5(i,j)=(x1(i,j)+x2(i,j))/2;
y5(i,j)=(y1(i,j)+y2(i,j))/2;
x6(i,j)=(x3(i,j)+x2(i,j))/2;
y6(i,j)=(y3(i,j)+y2(i,j))/2;
x7(i,j)=(x3(i,j)+x4(i,j))/2;
y7(i,j)=(y3(i,j)+y4(i,j))/2;
x8(i,j)=(x1(i,j)+x4(i,j))/2;
y8(i,j)=(y1(i,j)+y4(i,j))/2;
x9(i,j)=(x1(i,j)+x2(i,j)+x3(i,j)+x4(i,j))/4;
y9(i,j)=(y1(i,j)+y2(i,j)+y3(i,j)+y4(i,j))/4;
enddo
enddo
!-------------------------------------------------------------------------!
!------------Discretization of curved boundaries using 9-noded quadratic elements---------------!
! For inner radius(r1=1)
do i=1,m
j=1;
x1(i,j)=x2(i,j+1);
y1(i,j)=y2(i,j+1);
x2(i,j)=(r1+(j-1)*dr)*(cos((i)*dtheta));
y2(i,j)=(r1+(j-1)*dr)*(sin((i)*dtheta));
x3(i,j)=(r1+(j-1)*dr)*(cos((i-1)*dtheta));
y3(i,j)=(r1+(j-1)*dr)*(sin((i-1)*dtheta));
x4(i,j)=x3(i,j+1);
y4(i,j)=y3(i,j+1);
x6(i,j)=(r1+(j-1)*dr)*(cos(dtheta/2.0+(i-1)*dtheta));
y6(i,j)=(r1+(j-1)*dr)*(sin(dtheta/2.0+(i-1)*dtheta));
x7(i,j)=(r1+((dr/2.0)+(j-1)*dr))*(cos((i-1)*dtheta));
y7(i,j)=(r1+((dr/2.0)+(j-1)*dr))*(sin((i-1)*dtheta));
x5(i,j)=(r1+((dr/2)+(j-1)*dr))*(cos((i)*dtheta));
y5(i,j)=(r1+((dr/2)+(j-1)*dr))*(sin((i)*dtheta));
x8(i,j)=(x1(i,j)+x4(i,j))/2;
y8(i,j)=(y1(i,j)+y4(i,j))/2;
x9(i,j)=(x1(i,j)+x2(i,j)+x3(i,j)+x4(i,j)+x5(i,j)+x6(i,j)+x7(i,j)+x8(i,j))/8;
y9(i,j)=(y1(i,j)+y2(i,j)+y3(i,j)+y4(i,j)+y5(i,j)+y6(i,j)+y7(i,j)+y8(i,j))/8;
enddo
!For Outer Radius(r2=2)
do i=1,m
j=n;
x1(i,j)=(r1+(j)*dr)*(cos((i)*dtheta));
y1(i,j)=(r1+(j)*dr)*(sin((i)*dtheta));
x2(i,j)=x1(i,j-1);
y2(i,j)=y1(i,j-1);
x3(i,j)=x4(i,j-1);
y3(i,j)=y4(i,j-1);
x4(i,j)=(r1+(j)*dr)*(cos((i-1)*dtheta));
y4(i,j)=(r1+(j)*dr)*(sin((i-1)*dtheta));
x6(i,j)=(x2(i,j)+x3(i,j))/2;
y6(i,j)=(y2(i,j)+y3(i,j))/2;
x7(i,j)=(r1+((dr/2.0)+(j-1)*dr))*(cos((i-1)*dtheta));
y7(i,j)=(r1+((dr/2.0)+(j-1)*dr))*(sin((i-1)*dtheta));
x5(i,j)=(r1+((dr/2)+(j-1)*dr))*(cos((i)*dtheta));
y5(i,j)=(r1+((dr/2)+(j-1)*dr))*(sin((i)*dtheta));
x8(i,j)=(r1+(j)*dr)*(cos(dtheta/2.0+(i-1)*dtheta));
y8(i,j)=(r1+(j)*dr)*(sin(dtheta/2.0+(i-1)*dtheta));
x9(i,j)=(x1(i,j)+x2(i,j)+x3(i,j)+x4(i,j)+x5(i,j)+x6(i,j)+x7(i,j)+x8(i,j))/8;
y9(i,j)=(y1(i,j)+y2(i,j)+y3(i,j)+y4(i,j)+y5(i,j)+y6(i,j)+y7(i,j)+y8(i,j))/8;
enddo
!----------------------------------------------------------------------------------------!
!=========================Transformation Coefficicents=======================!
do i=1,m
do j=1,n
A2(i,j)=(x8(i,j)-x6(i,j))/2.0;
A3(i,j)=(x5(i,j)-x7(i,j))/2.0;
A4(i,j)=(x6(i,j)+x8(i,j))/2.0-x9(i,j);
A5(i,j)=(x5(i,j)+x7(i,j))/2.0-x9(i,j);
A6(i,j)=(x1(i,j)-x2(i,j)+x3(i,j)-x4(i,j))/4.0;
A7(i,j)=(x1(i,j)+x2(i,j)-x3(i,j)-x4(i,j))/4.0-(-x7(i,j)+x5(i,j))/2.0;
A8(i,j)=(x1(i,j)-x2(i,j)-x3(i,j)+x4(i,j))/4.0-(-x6(i,j)+x8(i,j))/2.0;
A9(i,j)=(x1(i,j)+x2(i,j)+x3(i,j)+x4(i,j))/4.0-(x5(i,j)+x6(i,j)+x7(i,j)+x8(i,j))/2.0+x9(i,j);
B2(i,j)=(y8(i,j)-y6(i,j))/2.0;
B3(i,j)=(y5(i,j)-y7(i,j))/2.0;
B4(i,j)=(y6(i,j)+y8(i,j))/2.0-y9(i,j);
B5(i,j)=(y5(i,j)+y7(i,j))/2.0-y9(i,j);
B6(i,j)=(y1(i,j)-y2(i,j)+y3(i,j)-y4(i,j))/4.0;
B7(i,j)=(y1(i,j)+y2(i,j)-y3(i,j)-y4(i,j))/4.0-(-y7(i,j)+y5(i,j))/2.0;
B8(i,j)=(y1(i,j)-y2(i,j)-y3(i,j)+y4(i,j))/4.0-(-y6(i,j)+y8(i,j))/2.0;
B9(i,j)=(y1(i,j)+y2(i,j)+y3(i,j)+y4(i,j))/4.0-(y5(i,j)+y6(i,j)+y7(i,j)+y8(i,j))/2.0+y9(i,j);
enddo
enddo
!----------------------------------------------------------------------------!
!==============Length of each cell==========!
do i=1,m
do j=1,n
l1(i,j)=sqrt((x1(i,j)-x4(i,j))**2+(y1(i,j)-y4(i,j))**2);
l2(i,j)=sqrt((x2(i,j)-x1(i,j))**2+(y2(i,j)-y1(i,j))**2);
l3(i,j)=sqrt((x3(i,j)-x2(i,j))**2+(y3(i,j)-y2(i,j))**2);
l4(i,j)=sqrt((x4(i,j)-x3(i,j))**2+(y4(i,j)-y3(i,j))**2);
enddo
enddo
!-------------------------------------------!
!=================Normal unit vectors at each surface of a cell==================!
do i=1,m
do j=1,n
m1x(i,j)=((y1(i,j)-y4(i,j)))/l1(i,j);
m1y(i,j)=-((x1(i,j)-x4(i,j)))/l1(i,j);
m2x(i,j)=((y2(i,j)-y1(i,j)))/l2(i,j);
m2y(i,j)=-((x2(i,j)-x1(i,j)))/l2(i,j);
m3x(i,j)=((y3(i,j)-y2(i,j)))/l3(i,j);
m3y(i,j)=-((x3(i,j)-x2(i,j)))/l3(i,j);
m4x(i,j)=((y4(i,j)-y3(i,j)))/l4(i,j);
m4y(i,j)=-((x4(i,j)-x3(i,j)))/l4(i,j);
enddo
enddo
!---------------------------------------------------------------------------------!
!=============Derivatives used in transfomation of Equation from Global to local coordinate system=======!
Sx(m,n)=0.0;
Sy(m,n)=0.0;
Tx(m,n)=0.0;
Ty(m,n)=0.0;
Sx2Sy2(m,n)=0.0;
Tx2Ty2(m,n)=0.0;
SxxSyy(m,n)=0.0;
TxxTyy(m,n)=0.0;
SxTxSyTy(m,n)=0.0;
SxTx(m,n)=0.0;
SyTy(m,n)=0.0;
!-----------------Gauss-Quadrature points and corresponding weights for Numerical integration----------------!
! 7 point gauss quadrature is used
s(1)=-0.9491079123 ;
s(2)=-0.7415311856 ;
s(3)=-0.4058451514 ;
s(4)=0.0;
s(5)=0.4058451514 ;
s(6)=0.7415311856 ;
s(7)=0.9491079123 ;
t(1)=-0.9491079123;
t(2)=-0.7415311856 ;
t(3)=-0.4058451514 ;
t(4)=0.0;
t(5)=0.4058451514 ;
t(6)=0.7415311856 ;
t(7)=0.9491079123 ;
ws(1)=0.1294849662 ;
ws(2)=0.2797053915 ;
ws(3)=0.3818300505 ;
ws(4)=0.4179591837 ;
ws(5)=0.3818300505 ;
ws(6)=0.2797053915 ;
ws(7)=0.1294849662;
wt(1)=0.1294849662 ;
wt(2)=0.2797053915 ;
wt(3)=0.3818300505 ;
wt(4)=0.4179591837 ;
wt(5)=0.3818300505 ;
wt(6)=0.2797053915 ;
wt(7)=0.1294849662 ;
!-----------------------------------------------------------------------------------!
do i=1,m
do j=1,n
do ks=1,pt
do kt=1,pt
FSx(ks,kt)=(B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt))))-&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))));
FSy(ks,kt)=(-A3(i,j) - s(ks)*(A6(i,j) + A7(i,j)*s(ks)) - 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))) -&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))));
FTx(ks,kt)=(-B2(i,j) - 2*B4(i,j)*s(ks) - t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt))))/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))) -&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))));
FTy(ks,kt)= (A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt))))/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))) -&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))));
FSxTx(ks,kt)=((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(-B2(i,j) - 2*B4(i,j)*s(ks) - t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))))/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))) -&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))))**2;
FSyTy(ks,kt)=((-A3(i,j) - s(ks)*(A6(i,j) + A7(i,j)*s(ks)) - 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))))/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))) -&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))))**2;
enddo
enddo
do ks=1,pt
do kt=1,pt
temp7=ws(ks)*wt(kt)*FSx(ks,kt);
Sx(i,j)=Sx(i,j)+temp7;
temp8=ws(ks)*wt(kt)*FSy(ks,kt);
Sy(i,j)=Sy(i,j)+temp8;
temp9=ws(ks)*wt(kt)*FTx(ks,kt);
Tx(i,j)=Tx(i,j)+temp9;
temp10=ws(ks)*wt(kt)*FTy(ks,kt);
Ty(i,j)=Ty(i,j)+temp10;
temp12(i,j)=ws(ks)*wt(kt)*FSxTx(ks,kt);
SxTx(i,j)=SxTx(i,j)+temp12(i,j);
temp13(i,j)=ws(ks)*wt(kt)*FSyTy(ks,kt);
SyTy(i,j)=SyTy(i,j)+temp13(i,j);
enddo
enddo
enddo
enddo
Sx=Sx/4.0;
Sy=Sy/4.0;
Tx=Tx/4.0;
Ty=Ty/4.0;
SxxSyy=SxxSyy/4.0;
TxxTyy=TxxTyy/4.0
do i=1,m
do j=1,n
do ks=1,pt
do kt=1,pt
FSx2Sy2(ks,kt)=((A3(i,j)+s(ks)*(A6(i,j)+A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))**2+ (B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))**2)/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))) -&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))))**2;
FTx2Ty2(ks,kt)=((A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt))))**2+ (B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt))))**2)/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))) -&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))))**2;
FSxTxSyTy(ks,kt)=(-((A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt))))) -&
(B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))))/&
((B3(i,j) + s(ks)*(B6(i,j) + B7(i,j)*s(ks)) + 2*(B5(i,j) + s(ks)*(B8(i,j) + B9(i,j)*s(ks)))*t(kt))*(A2(i,j) + 2*A4(i,j)*s(ks) + t(kt)*(A6(i,j) + A8(i,j)*t(kt) + 2*s(ks)*(A7(i,j) + A9(i,j)*t(kt)))) -&
(A3(i,j) + s(ks)*(A6(i,j) + A7(i,j)*s(ks)) + 2*(A5(i,j) + s(ks)*(A8(i,j) + A9(i,j)*s(ks)))*t(kt))*(B2(i,j) + 2*B4(i,j)*s(ks) + t(kt)*(B6(i,j) + B8(i,j)*t(kt) + 2*s(ks)*(B7(i,j) + B9(i,j)*t(kt)))))**2;
enddo
enddo
do ks=1,pt
do kt=1,pt
temp7=ws(ks)*wt(kt)*FSx2Sy2(ks,kt);
Sx2Sy2(i,j)=Sx2Sy2(i,j)+temp7;
temp8=ws(ks)*wt(kt)*FTx2Ty2(ks,kt);
Tx2Ty2(i,j)=Tx2Ty2(i,j)+temp8;
temp11=ws(ks)*wt(kt)*FSxTxSyTy(ks,kt);
SxTxSyTy(i,j)=SxTxSyTy(i,j)+temp11;
enddo
enddo
enddo
enddo
Sx2Sy2=Sx2Sy2/4.0;
Tx2Ty2=Tx2Ty2/4.0;
SxTxSyTy=SxTxSyTy/4.0;
!------------------------------------------------------------------------------------------------------!
!================Direction Cosine for each surface of a cell=================!
do i=1,m
do j=1,n
Sm1(i,j)=(m1x(i,j)*Sx(i,j)+m1y(i,j)*Sy(i,j)); ! For edge 1
Sm2(i,j)=(m2x(i,j)*Sx(i,j)+m2y(i,j)*Sy(i,j)); ! For edge 2
Sm3(i,j)=-(m3x(i,j)*Sx(i,j)+m3y(i,j)*Sy(i,j)); ! For edge3
Sm4(i,j)=-(m4x(i,j)*Sx(i,j)+m4y(i,j)*Sy(i,j)); ! For edge4
Tm1(i,j)=(m1x(i,j)*Tx(i,j)+m1y(i,j)*Ty(i,j)); ! For edge 1
Tm2(i,j)=(m2x(i,j)*Tx(i,j)+m2y(i,j)*Ty(i,j)); ! For edge 2
Tm3(i,j)=-(m3x(i,j)*Tx(i,j)+m3y(i,j)*Ty(i,j)); ! For edge3
Tm4(i,j)=-(m4x(i,j)*Tx(i,j)+m4y(i,j)*Ty(i,j)); ! For edge4
enddo
enddo
!-----------------------------------------------------------------------------!
!=================Coefficients of final discrete equation=================!
do i=1,m-1
do j=1,n
F12(i,j)=(1*(Tm2(i,j))/2+1*(Tm4(i+1,j))/2+(3*(Tm2(i,j))/2)*(Sx2Sy2(i,j))/(((Sx2Sy2(i,j))+(Tx2Ty2(i,j))))+(3*(Tm4(i+1,j))/2)*(Sx2Sy2(i+1,j))/(((Sx2Sy2(i+1,j))+Tx2Ty2(i+1,j))));
enddo
enddo
do i=1,m
do j=1,n
F11(i,j)=(1*(Tm4(i,j))/2-(3*Tm4(i,j)/2)*(Sx2Sy2(i,j))/((Sx2Sy2(i,j)+Tx2Ty2(i,j))));
F13(i,j)=(1*(Tm2(i,j))/2-(3*Tm2(i,j)/2)*(Sx2Sy2(i,j))/((Sx2Sy2(i,j)+Tx2Ty2(i,j))));
F14(i,j)=(3*(Tm2(i,j))/2)*(Sx2Sy2(i,j))/(((Sx2Sy2(i,j))+(Tx2Ty2(i,j))));
F15(i,j)=(3*(Tm4(i,j))/2)*(Sx2Sy2(i,j))/(((Sx2Sy2(i,j))+Tx2Ty2(i,j)));
enddo
enddo
do i=1,m
do j=1,n
F16(i,j)=1*(Tm2(i,j))/(((Sx2Sy2(i,j))+(Tx2Ty2(i,j))));
F26(i,j)=1*(Tm4(i,j))/(((Sx2Sy2(i,j))+(Tx2Ty2(i,j))));
F17(i,j)=2*(SxTxSyTy(i,j));
F18(i,j)=(1/2+(3/2)*(Sx2Sy2(i,j))/((Sx2Sy2(i,j))+(Tx2Ty2(i,j))));
enddo
enddo
do i=1,m
do j=1,n-1
F22(i,j)=(1*(Sm1(i,j))/2+1*(Sm3(i,j+1))/2+(3*(Sm1(i,j))/2)*(Tx2Ty2(i,j))/(((Sx2Sy2(i,j))+(Tx2Ty2(i,j))))+(3*(Sm3(i,j+1))/2)*(Tx2Ty2(i,j+1))/(((Sx2Sy2(i,j+1))+Tx2Ty2(i,j+1))));
enddo
enddo
do i=1,m
do j=1,n
F21(i,j)=(1*(Sm3(i,j))/2-(3*Sm3(i,j))/2*(Tx2Ty2(i,j))/(((Sx2Sy2(i,j))+Tx2Ty2(i,j))));
F23(i,j)=(1*(Sm1(i,j))/2-(3*Sm1(i,j))/2*(Tx2Ty2(i,j))/(((Sx2Sy2(i,j))+Tx2Ty2(i,j))));
F24(i,j)=(3*(Sm1(i,j))/2)*(Tx2Ty2(i,j))/(((Sx2Sy2(i,j))+Tx2Ty2(i,j)));
F25(i,j)=(3*(Sm3(i,j))/2)*(Tx2Ty2(i,j))/(((Sx2Sy2(i,j))+Tx2Ty2(i,j)));
F28(i,j)=(1/2+(3/2)*(Tx2Ty2(i,j))/((Sx2Sy2(i,j))+(Tx2Ty2(i,j))));
F29(i,j)=(1/2-(3/2)*(Tx2Ty2(i,j))/((Sx2Sy2(i,j))+(Tx2Ty2(i,j))));
F30(i,j)=((3/2)*(Tx2Ty2(i,j)))/(((Sx2Sy2(i,j))+Tx2Ty2(i,j)));
enddo
enddo
!---------------------------------------------------------------------------------!
!=================Initialization of Main loop==================!
tol5=1
tol6=1
do while ((tol5.GT.tol).OR.(tol6.GT.tol))
!----------------calculation do cross derivatives-----!
Tc(1,1)=(Tt(1,1)+Ts(1,1))/2.0;
Tc(1,n+1)=(Ts(1,n)+Tt(1,n+1))/2.0;
Tc(m+1,1)=(Tt(m,1)+Ts(m+1,1))/2.0;
Tc(m+1,n+1)=(Tt(m,n+1)+Ts(m+1,n))/2.0;
do j=2,n
Tc(1,j)=(Ts(1,j-1)+Ts(1,j))/2.0;
Tc(m+1,j)=(Ts(m+1,j-1)+Ts(m+1,j))/2.0;
enddo
do i=2,m
Tc(i,1)=(Tt(i-1,1)+Tt(i,1))/2.0;
Tc(i,n+1)=(Tt(i-1,n+1)+Tt(i,n+1))/2.0;
enddo
do i=2,m
do j=2,n
Tc(i,j)=(Ts(i,j-1)+Ts(i,j)+Tt(i-1,j)+Tt(i,j))/4.0;
enddo
enddo
do i=1,m
do j=1,n
Tst(i,j)=((Tc(i+1,j+1)-Tc(i,j+1)-Tc(i+1,j)+Tc(i,j))/4.0);
f(i,j)=q(i,j)+(F17(i,j)*Tst(i,j));
enddo
enddo
!-----------------------------------------------------!
!============================Boundary Conditions=========================!
!Neumann boundary condition at Inner radius of the domain
do i=1,m
Tt(i,1)=(F29(i,1)*Tt(i,2)+F30(i,1)*(Ts(i,1)+Ts(i+1,1))+0.5*(Tc(i+1,1)-Tc(i,1))*(Tm3(i,1)/Sm3(i,1)))/F28(i,1);
enddo
!Sinosuidal function is applied at outer radius
!Frequency of the applied sinosuidal boundary condition can be changed
!by changing the "feq" parameter given below
feq=2
do i=1,m
Tt(i,n+1)=COS(feq*(4*ATAN(1.0))*ATAN(yt(i,n+1)/xt(i,n+1))/angle);
enddo
!Dirichlet B.C with "zero" value at Theta=0deg
do j=1,n
Ts(1,j)=0;
enddo
!Dirichlet B.C with "zero" value at Theta=90deg
do j=1,n
Ts(m+1,j)=0;
enddo
!-------------------------------------------------------------------------!
!=============Loop for temperature(both s-averaged(Ts) and t-averaged(Tt))================!
do i=1,m-1
do j=1,n-1
Ts(i+1,j)=(F11(i1+1,j)*Ts(i+2,j)+F13(i,j)*Ts(i,j)+F14(i,j)*(Tt(i,j)+Tt(i,j+1))+F15(i+1,j)*(Tt(i+1,j)+Tt(i+1,j+1))-(f(i,j)*F16(i,j)+f(i+1,j)*F16(i+1,j))+((Tc(i+1,j+1)-Tc(i+1,j))/2)*(Sm4(i+1,j)-Sm2(i,j)))/F12(i,j);
Ts(i+1,n)=(F11(i+1,n)*Ts(i+2,n)+F13(i,n)*Ts(i,n)+F14(i,n)*(Tt(i,n)+Tt(i,n+1))+F15(i+1,n)*(Tt(i+1,n)+Tt(i+1,n+1))-(f(i,n)*F16(i,n)+f(i+1,n)*F16(i+1,n))+((Tc(i+1,n+1)-Tc(i+1,n))/2)*(Sm4(i+1,n)-Sm2(i,n)))/F12(i,n);
Tt(i,j+1)=(F21(i,j+1)*Tt(i,j+2)+F23(i,j)*Tt(i,j)+F24(i,j)*(Ts(i,j)+Ts(i+1,j))+F25(i,j+1)*(Ts(i,j+1)+Ts(i+1,j+1))-(f(i,j)*F26(i,j)+f(i,j+1)*F26(i,j+1))+((Tc(i+1,j+1)-Tc(i,j+1))/2)*(Tm3(i,j+1)-Tm1(i,j)))/F22(i,j);
Tt(m,j+1)=(F21(m,j+1)*Tt(m,j+2)+F23(m,j)*Tt(m,j)+F24(m,j)*(Ts(m,j)+Ts(m+1,j))+F25(m,j+1)*(Ts(m,j+1)+Ts(m+1,j+1))-(f(m,j)*F26(m,j)+f(m,j+1)*F26(m,j+1))+((Tc(m+1,j+1)-Tc(m,j+1))/2)*(Tm3(m,j+1)-Tm1(m,j)))/F22(m,j);
enddo
enddo
!-----------------------------------------------------------------------------------------!
!=======================Residual==========================!
do i=1,m
do j=1,n+1
err5(i,j)=(Tt(i,j)-temp5(i,j));
enddo
enddo
do i=1,m+1
do j=1,n
err6(i,j)=(Ts(i,j)-temp6(i,j));
enddo
enddo
do i=1,m
do j=1,n+1
temp5(i,j)=Tt(i,j);
enddo
enddo
do i=1,m+1
do j=1,n
temp6(i,j)=Ts(i,j);
enddo
enddo
temptol5=0.0
temptol6=0.0
do i=1,m
do j=1,n+1
temptol5=temptol5+err5(i,j)**2;
enddo
enddo
tol5=sqrt(temptol5)/sqrt(xdiv*ydiv);
do i=1,m+1
do j=1,n
temptol6=temptol6+err6(i,j)**2;
enddo
enddo
tol6=sqrt(temptol6)/sqrt(xdiv*ydiv);
print *,'Error in Tt is ', tol5
print *,'Error in Ts is ', tol6
!-------------------------------------------------------------!
enddo !end of while loop
CALL CPU_TIME(T2)
!----------------------------End of Main loop--------------------------------------!
!================Output Files===============!
!S-averged surface
open(1,file='Xs')
do i=1,m+1
write(1,*) xs(i,:)
enddo
close(1)
open(2,file='Ys')
do i=1,m+1
write(2,*) ys(i,:)
enddo
close(2)
open(3,file='Ts')
do i=1,m+1
write(3,*) Ts(i,:)
enddo
close(3)
!T-averaged surface
open(4,file='Xt')
do i=1,m
write(4,*) xt(i,:)
enddo
close(4)
open(5,file='Yt')
do i=1,m
write(5,*) yt(i,:)
enddo
close(5)
open(6,file='Tt')
do i=1,m
write(6,*) Tt(i,:)
enddo
close(6)
!Temperature profile at Theta=Pi/4
open(7,file='T_P1')
do j=1,n
write(7,*) rs((m/2)+1,j),Ts((m/2)+1,j)
enddo
close(7)
!Temperature profile at r=1.5
open(8,file='T_P2')
do i=1,m
write(8,*) Thetat(i,(n/2)+1),Tt(i,(n/2)+1)
enddo
close(8)
!-------------------------------------------!
end program