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SiStER_thermal_solver_sparse_CFD.m
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SiStER_thermal_solver_sparse_CFD.m
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function [T, rhs, Lii, Ljj, Lvv]=SiStER_thermal_solver_sparse_CFD(x,y,Told,rho,cp,k,dt,BCtherm,H)
% [T, rhs, Lii, Ljj, Lvv]=SiStER_thermal_solver_sparse_CFD(x,y,Told,rho,cp,k,dt,BCtherm,H)
% implicit Solver for thermal diffusion
% rho cp dT/dt = div (k grad T) + H
% J.-A. Olive, November 2014
% B.Z. Klein, added sparse matrix filling, End of 2014
% Added internal heat gen - howellsm 2/2017
% Updated to fully conservative (centered finite difference)
% for variable thermal diffusivity
% by S. Howell, now accepts k on shear nodes
Nx=length(x);
Ny=length(y);
dx=diff(x);
dy=diff(y);
Li=zeros(Nx*Ny, 1);
Lj=zeros(Nx*Ny, 1);
Lv=zeros(Nx*Ny, 1);
rhs=zeros(Nx*Ny,1);
n = 1;
for i=1:Ny
for j=1:Nx
in=i+Ny*(j-1); % global index
if i==1 % top boundary
if BCtherm.top(1)==1 % Dirichlet
Li(n) = in;
Lj(n) = in;
Lv(n) = 1;
n = n+1;
rhs(in) = BCtherm.top(2);
%L(in,in)=1;
%rhs(in)=BCtherm.top(2);
elseif BCtherm.top(1)==2 % Neumann
Li(n) = in;
Lj(n) = in;
Lv(n) = -1;
n = n+1;
Li(n) = in;
Lj(n) = in+1;
Lv(n) = 1;
n = n+1;
rhs(in)=BCtherm.top(2)*dy(i);
% L(in,in)=-1;
% L(in,in+1)=1;
% rhs(in)=BCtherm.top(2)*dy(i);
end
elseif i==Ny % bottom boundary
if BCtherm.bot(1)==1 % Dirichlet
Li(n) = in;
Lj(n) = in;
Lv(n) = 1;
n = n+1;
rhs(in) = BCtherm.bot(2);
% L(in,in)=1;
% rhs(in)=BCtherm.bot(2);
elseif BCtherm.bot(1)==2 % Neumann
Li(n) = in;
Lj(n) = in;
Lv(n) = 1;
n = n+1;
Li(n) = in;
Lj(n) = in-1;
Lv(n) = -1;
n = n+1;
rhs(in) = BCtherm.bot(2)*dy(i-1);
% L(in,in)=1;
% L(in,in-1)=-1;
% rhs(in)=BCtherm.bot(2)*dy(i-1);
end
elseif j==1 % left boundary
if BCtherm.left(1)==2 % Neumann
Li(n) = in;
Lj(n) = in;
Lv(n) = -1;
n = n+1;
Li(n) = in;
Lj(n) = in+Ny;
Lv(n) = 1;
n = n+1;
rhs(in) = BCtherm.left(2)*dx(j);
% L(in,in)=-1;
% L(in,in+Ny)=1;
% rhs(in)=BCtherm.left(2)*dx(j);
elseif BCtherm.left(1)==1 % Dirichlet
Li(n) = in;
Lj(n) = in;
Lv(n) = 1;
n = n+1;
rhs(in) = BCtherm.left(2);
% L(in,in)=1;
% rhs(in)=BCtherm.left(2);
end
elseif j==Nx % right boundary
if BCtherm.right(1)==2 % Neumann
Li(n) = in;
Lj(n) = in;
Lv(n) = 1;
n = n+1;
Li(n) = in;
Lj(n) = in-Ny;
Lv(n) = -1;
n = n+1;
rhs(in) = BCtherm.right(2)*dx(j-1);
% L(in,in)=1;
% L(in,in-Ny)=-1;
% rhs(in)=BCtherm.right(2)*dx(j-1);
elseif BCtherm.right(1)==1 % Dirichlet
Li(n) = in;
Lj(n) = in;
Lv(n) = 1;
n = n+1;
rhs(in) = BCtherm.right(2);
% L(in,in)=1;
% rhs(in)=BCtherm.right(2);
end
else
% % internal nodes
ddx=dx(j-1)+dx(j);
ddy=dy(i-1)+dy(i);
% Follows Gerya lettering scheme, P139-140
kA = (2 * k(i,j-1) * k(i,j) )/( k(i,j-1) + k(i,j) );
kB = (2 * k(i,j) * k(i,j+1) )/( k(i,j) + k(i,j+1) );
kC = (2 * k(i-1,j) * k(i,j) )/( k(i-1,j) + k(i,j) );
kD = (2 * k(i,j) * k(i+1,j) )/( k(i,j) + k(i+1,j) );
Li(n) = in;
Lj(n) = in;
Lv(n) = rho(i,j)*cp(i,j)+2*dt*kB/(dx(j)*ddx) + 2*dt*kA/(dx(j-1)*ddx) + ...
2*dt*kD/(ddy*dy(i)) + 2*dt*kC/(ddy*dy(i-1));
n = n+1;
% Lv(n) = rho(i,j)*cp(i,j)+2*dt*kB/(dx(j)*ddx) + 2*dt*kA/(dx(j-1)*ddx) + ...
% 2*dt*kD/(ddy*dy(i)) + 2*dt*kC/(ddy*dy(i-1));
Li(n) = in;
Lj(n) = in+Ny;
Lv(n) = -2*dt*kB/(dx(j)*ddx);
n = n+1;
% L(in,in+Ny)=-2*dt*k(i,j)/(dx(j)*ddx);
%
Li(n) = in;
Lj(n) = in-Ny;
Lv(n) = -2*dt*kA/(dx(j-1)*ddx);
n = n+1;
%
% % L(in,in-Ny)=-2*dt*k(i,j-1)/(dx(j-1)*ddx);
%
Li(n) = in;
Lj(n) = in+1;
Lv(n) = -2*dt*kD/(dy(i)*ddy);
n = n+1;
%
% % L(in,in+1)=-2*dt*k(i,j)/(dy(i)*ddy);
%
Li(n) = in;
Lj(n) = in-1;
Lv(n) = -2*dt*kC/(dy(i-1)*ddy);
n = n+1;
% L(in,in-1)=-2*dt*k(i-1,j)/(dy(i-1)*ddy);
rhs(in)=rho(i,j)*cp(i,j)*Told(i,j) + H(i,j)*dt;
end
end
end
nn = n-1;
Lii = Li(1:nn);
Ljj = Lj(1:nn);
Lvv = Lv(1:nn);
L = sparse(Lii, Ljj, Lvv);
tvec=L\rhs;
T=reshape(tvec,Ny,Nx);