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SiStER_assemble_L_R.m
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SiStER_assemble_L_R.m
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function [L, R, Kc, Kb]=SiStER_assemble_L_R(dx,dy,etas,etan,rho,BC,PARAMS,srhs_xx,srhs_xy)
%% Fill LHS and RHS Solution Matrices
p0cell=PARAMS.p0cell;
Nx=length(dx)+1;
Ny=length(dy)+1;
gx=PARAMS.gx;
gy=PARAMS.gy;
% Vector of right part initialization
R=zeros(Nx*Ny*3,1);
Lii = zeros(10*Nx*Ny*3,1);
Ljj = zeros(10*Nx*Ny*3,1);
Lvv = zeros(10*Nx*Ny*3,1);
nn = 1;
%%%%%%%%%% SCALING THE FD MATRIX
% Computing Kc and Kb coefficients
meta=min(min(etas));
mdx=max(dx);
mdy=max(dy);
Kc=2*meta/(mdx+mdy);
Kb=4*meta/(mdx+mdy)^2;
% pressure anchor
%IP=2;
%JP=3;
JP=ceil((Nx-1)/2); %G.Ito
IP=2;
% fill in FD matrix and right-hand side
for j=1:Nx
for i=1:Ny
%for j=1:Nx
in=(j-1)*Ny+i;
inp=3*in-2;
invx=3*in-1;
invy=3*in;
%%%%%%% CONTINUITY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if (i==1) || (j==1) || (i==2 && j==2) || (i==2 && j==Nx) || ....
(i==Ny && j==2) || (i==Ny && j==Nx) || (i==IP && j==JP && BC.top(2)~=3) || ...
(BC.top(2)==3 && i==2 && j > 2 && j < Ny) %<--G.Ito
% boundary conditions
if (i==1) || (j==1) % P(i,j)=0
% L(inp,inp)=Kb;
Lii(nn) = inp;
Ljj(nn) = inp;
Lvv(nn) = Kb;
nn = nn+1;
R(inp,1)=0;
end
if (i==2 && j==2) || (i==Ny && j==2) % left corners P(i,j)-P(i,j+1)=0
% L(inp,inp)=Kb;
% L(inp,inp+3*Ny)=-Kb;
Lii(nn) = inp;
Ljj(nn) = inp;
Lvv(nn) = Kb;
nn = nn+1;
Lii(nn) = inp;
Ljj(nn) = inp+3*Ny;
Lvv(nn) = -Kb;
nn = nn+1;
R(inp,1)=0;
end
if (i==2 && j==Nx) || (i==Ny && j==Nx) % right corners P(i,j)-P(i,j-1)=0
% L(inp,inp)=Kb;
% L(inp,inp-3*Ny)=-Kb;
Lii(nn) = inp;
Ljj(nn) = inp;
Lvv(nn) = Kb;
nn = nn+1;
Lii(nn) = inp;
Ljj(nn) = inp-3*Ny;
Lvv(nn) = -Kb;
nn = nn+1;
R(inp,1)=0;
end
if (BC.top(2)==3 && i==2 && j > 2 && j < Ny) %open top effect all nodes but corners, G.Ito
% Pressure gradient between top two rows extrapolates to 0 pressure at very top
%
Lii(nn) = inp;
Ljj(nn) = inp;
Lvv(nn) = Kb.*(1+dy(i-1)/(dy(i)+dy(i-1)));
nn = nn+1;
Lii(nn) = inp;
Ljj(nn) = inp+3;
Lvv(nn) = -Kb.*dy(i-1)/(dy(i)+dy(i-1));
nn = nn+1;
R(inp,1)=0;
end
if (i==IP && j==JP && BC.top(2)~=3) % additional pressure cell i=2, j=3 P(i,j)=p0cell
% L(inp,inp)=Kb;
Lii(nn) = inp;
Ljj(nn) = inp;
Lvv(nn) = Kb;
nn = nn+1;
R(inp,1)=Kb*p0cell/Kc;
end
else
% internal nodes
% coeffs for vx
% L(inp,invx-3)=Kc/dx(j-1);
% L(inp,invx-3*Ny-3)=-Kc/dx(j-1);
Lii(nn) = inp;
Ljj(nn) = invx-3;
Lvv(nn) = Kc/dx(j-1);
nn = nn+1;
Lii(nn) = inp;
Ljj(nn) = invx-3*Ny-3;
Lvv(nn) = -Kc/dx(j-1);
nn = nn+1;
% coeffs for vy
% L(inp,invy-3*Ny)=Kc/dy(i-1);
% L(inp,invy-3*Ny-3)=-Kc/dy(i-1);
Lii(nn) = inp;
Ljj(nn) = invy-3*Ny;
Lvv(nn) = Kc/dy(i-1);
nn = nn+1;
Lii(nn) = inp;
Ljj(nn) = invy-3*Ny-3;
Lvv(nn) = -Kc/dy(i-1);
nn = nn+1;
% RHS
R(inp,1)=0;%-drhodt(i,j);
end
%%%%%%% X-STOKES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if ((j==1) && (i<=Ny-1)) || ((j==Nx) && (i<=Ny-1)) || (i==1 && j<=Nx-1 && j>=2) || (i==Ny-1 && j<=Nx-1 && j>=2) || (i==Ny)
if ((j==1) && (i<=Ny-1)) %|| ((j==Nx) && (i<=Ny-1)) % X-STOKES: left boundary
% L(invx,invx)=Kb;
Lii(nn) = invx;
Ljj(nn) = invx;
Lvv(nn) = Kb;
nn = nn+1;
R(invx,1)=Kb*BC.left(3);
end
if ((j==Nx) && (i<=Ny-1)) % X-STOKES: right boundary
% L(invx,invx)=Kb;
Lii(nn) = invx;
Ljj(nn) = invx;
Lvv(nn) = Kb;
nn = nn+1;
R(invx,1)=Kb*BC.right(3);
end
if i==1 && j<=Nx-1 && j>=2 % X-STOKES: upper boundary not including sides
if BC.top(1)==1 % free slip
% L(invx,invx+3)=Kc/(0.5*(dy(i)+dy(i+1)));
% L(invx,invx)=-Kc/(0.5*(dy(i)+dy(i+1)));
Lii(nn) = invx;
Ljj(nn) = invx+3;
Lvv(nn) = Kc/(0.5*(dy(i)+dy(i+1)));
nn = nn+1;
Lii(nn) = invx;
Ljj(nn) = invx;
Lvv(nn) = -Kc/(0.5*(dy(i)+dy(i+1)));
nn = nn+1;
R(invx,1)=0;
elseif BC.top(1)==0 % no slip
% L(invx,invx+3)=Kc/(dy(i)+dy(i+1));
% L(invx,invx)=-Kc*(1/dy(i)+1/(dy(i)+dy(i+1)));
% Multiply whole eq. by 2 so coefficients are closer to
% the others
Lii(nn) = invx;
Ljj(nn) = invx+3;
Lvv(nn) = -2*Kc/(dy(i)+dy(i+1));
nn = nn+1;
Lii(nn) = invx;
Ljj(nn) = invx;
Lvv(nn) = 2*Kc*(1/dy(i)+1/(dy(i)+dy(i+1)));
nn = nn+1;
R(invx,1)=2*Kc*(1/dy(i))*BC.top_profile(j); %G.Ito x-velocity at top boundary;
end
end
if i==Ny-1 && j<=Nx-1 && j>=2 % X-STOKES: lower boundary not including sides
if BC.bot(1)==0 % no slip
% L(invx,invx)=Kc*(1/dy(i)+1/(dy(i-1)+dy(i)));
% L(invx,invx-3)=-Kc/(dy(i-1)+dy(i));
Lii(nn) = invx;
Ljj(nn) = invx;
Lvv(nn) = 2*Kc*(1/dy(i)+1/(dy(i-1)+dy(i)));
nn = nn+1;
Lii(nn) = invx;
Ljj(nn) = invx-3;
Lvv(nn) = -2*Kc/(dy(i-1)+dy(i));
nn = nn+1;
R(invx,1)=2*Kc*(1/dy(i))*BC.bot_profile(j);
elseif BC.bot(1)==1 % free slip
% L(invx,invx)=Kc/(0.5*(dy(i-1)+dy(i)));
% L(invx,invx-3)=-Kc/(0.5*(dy(i-1)+dy(i)));
Lii(nn) = invx;
Ljj(nn) = invx;
Lvv(nn) = Kc/(0.5*(dy(i-1)+dy(i)));
nn = nn+1;
Lii(nn) = invx;
Ljj(nn) = invx-3;
Lvv(nn) = -Kc/(0.5*(dy(i-1)+dy(i)));
nn = nn+1;
R(invx,1)=0;
end
end
if i==Ny % lower boundary, ghost vx=0
% L(invx,invx)=Kb;
Lii(nn) = invx;
Ljj(nn) = invx;
Lvv(nn) = Kb;
nn = nn+1;
R(invx,1)=0;
end
else
% X-STOKES: internal nodes
% X-STOKES: coeffs for vx
% L(invx,invx-3*Ny)=4*etan(i+1,j)/(dx(j-1)*(dx(j-1)+dx(j))); %
Lii(nn) = invx;
Ljj(nn) = invx-3*Ny;
Lvv(nn) = 4*etan(i+1,j)/(dx(j-1)*(dx(j-1)+dx(j)));
nn = nn+1;
% L(invx,invx)=(-4/(dx(j-1)+dx(j)))*(etan(i+1,j+1)/dx(j)+etan(i+1,j)/dx(j-1))-(2/dy(i))*(etas(i+1,j)/(dy(i)+dy(i+1))+etas(i,j)/(dy(i-1)+dy(i))); %
Lii(nn) = invx;
Ljj(nn) = invx;
Lvv(nn) = (-4/(dx(j-1)+dx(j)))*(etan(i+1,j+1)/dx(j)+etan(i+1,j)/dx(j-1))-(2/dy(i))*(etas(i+1,j)/(dy(i)+dy(i+1))+etas(i,j)/(dy(i-1)+dy(i)));
nn = nn+1;
% L(invx,invx+3*Ny)=4*etan(i+1,j+1)/(dx(j)*(dx(j-1)+dx(j))); %
Lii(nn) = invx;
Ljj(nn) = invx+3*Ny;
Lvv(nn) = 4*etan(i+1,j+1)/(dx(j)*(dx(j-1)+dx(j)));
nn = nn+1;
% L(invx,invx-3)=2*etas(i,j)/(dy(i)*(dy(i-1)+dy(i))); %
Lii(nn) = invx;
Ljj(nn) = invx-3;
Lvv(nn) = 2*etas(i,j)/(dy(i)*(dy(i-1)+dy(i)));
nn = nn+1;
% L(invx,invx+3)=2*etas(i+1,j)/(dy(i)*(dy(i)+dy(i+1))); %
Lii(nn) = invx;
Ljj(nn) = invx+3;
Lvv(nn) = 2*etas(i+1,j)/(dy(i)*(dy(i)+dy(i+1)));
nn = nn+1;
% X-STOKES: coeffs for vy
% L(invx,invy+3)=2*etas(i+1,j)/(dy(i)*(dx(j-1)+dx(j))); %
Lii(nn) = invx;
Ljj(nn) = invy+3;
Lvv(nn) = 2*etas(i+1,j)/(dy(i)*(dx(j-1)+dx(j)));
nn = nn+1;
% L(invx,invy+3-3*Ny)=-2*etas(i+1,j)/(dy(i)*(dx(j-1)+dx(j))); %
Lii(nn) = invx;
Ljj(nn) = invy+3-3*Ny;
Lvv(nn) = -2*etas(i+1,j)/(dy(i)*(dx(j-1)+dx(j)));
nn = nn+1;
% L(invx,invy)=-2*etas(i,j)/(dy(i)*(dx(j-1)+dx(j))); %
Lii(nn) = invx;
Ljj(nn) = invy;
Lvv(nn) = -2*etas(i,j)/(dy(i)*(dx(j-1)+dx(j)));
nn = nn+1;
% L(invx,invy-3*Ny)=2*etas(i,j)/(dy(i)*(dx(j-1)+dx(j))); %
Lii(nn) = invx;
Ljj(nn) = invy-3*Ny;
Lvv(nn) = 2*etas(i,j)/(dy(i)*(dx(j-1)+dx(j)));
nn = nn+1;
% X-STOKES: coeffs for Pp
% L(invx,inp+3*Ny+3)=-2*Kc/(dx(j-1)+dx(j)); %
Lii(nn) = invx;
Ljj(nn) = inp+3*Ny+3;
Lvv(nn) = -2*Kc/(dx(j-1)+dx(j));
nn = nn+1;
% L(invx,inp+3)=2*Kc/(dx(j-1)+dx(j)); %
Lii(nn) = invx;
Ljj(nn) = inp+3;
Lvv(nn) = 2*Kc/(dx(j-1)+dx(j));
nn = nn+1;
% RHS
R(invx,1)=-gx*(rho(i,j)+rho(i+1,j))/2 - ...
2.*(srhs_xx(i+1,j+1)-srhs_xx(i+1,j))/(dx(j)+dx(j-1)) - ...
(srhs_xy(i+1,j)-srhs_xy(i,j))/dy(i);
end
%%%%%%% Y-STOKES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if (j==1 && i<=Ny-1 && i>=2) || (j==Nx-1 && i<=Ny-1 && i>=2) || (j==Nx) || (i==1 && j<=Nx-1) || (i==Ny && j<=Nx-1)
if j==1 && i>=2 && i<=Ny-1 % left boundary without edges
if BC.left(1)==1 % free slip
% L(invy,invy)=-2*Kc/(dx(j)+dx(j+1));
% L(invy,invy+3*Ny)=2*Kc/(dx(j)+dx(j+1));
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = -2*Kc/(dx(j)+dx(j+1));
nn = nn+1;
Lii(nn) = invy;
Ljj(nn) = invy+3*Ny;
Lvv(nn) = 2*Kc/(dx(j)+dx(j+1));
nn = nn+1;
R(invy,1)=0;
elseif BC.left(1)==0 % no slip
% L(invy,invy)=-2*Kc*(1/(dx(j)+dx(j+1))+1/dx(j));
% L(invy,invy+3*Ny)=2*Kc/(dx(j+1)+dx(j));
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = -2*Kc*(1/(dx(j)+dx(j+1))+1/dx(j));
nn = nn+1;
Lii(nn) = invy;
Ljj(nn) = invy+3*Ny;
Lvv(nn) = 2*Kc/(dx(j+1)+dx(j));
nn = nn+1;
R(invy,1)=0;
end
end
if j==Nx-1 && i<=Ny-1 && i>=2 % right boundary without edges
if BC.right(1)==1 % free slip
% L(invy,invy)=Kc/(0.5*(dx(j)+dx(j-1)));
% L(invy,invy-3*Ny)=-Kc/(0.5*(dx(j)+dx(j-1)));
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = Kc/(0.5*(dx(j)+dx(j-1)));
nn = nn+1;
Lii(nn) = invy;
Ljj(nn) = invy-3*Ny;
Lvv(nn) = -Kc/(0.5*(dx(j)+dx(j-1)));
nn = nn+1;
R(invy,1)=0;
elseif BC.right(1)==0 % no slip
% L(invy,invy)=Kc*(1/(dx(j)+dx(j-1))+1/dx(j));
% L(invy,invy-3*Ny)=-Kc/(dx(j)+dx(j-1));
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = Kc*(1/(dx(j)+dx(j-1))+1/dx(j));
nn = nn+1;
Lii(nn) = invy;
Ljj(nn) = invy-3*Ny;
Lvv(nn) = -Kc/(dx(j)+dx(j-1));
nn = nn+1;
R(invy,1)=0;
end
end
if j==Nx % right boundary, ghost vy=0
% L(invy,invy)=Kb;
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = Kb;
nn = nn+1;
R(invy,1)=0;
end
if i==1 && j<=Nx-1 % upper boundary
if BC.top(2)==0
% L(invy,invy)=Kb;
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = Kb;
nn = nn+1;
R(invy,1)=Kb*BC.top(3);
elseif BC.top(2)==3
% open top dvy/dy=0, G.Ito
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = Kb;
nn = nn+1;
Lii(nn) = invy;
Ljj(nn) = invy+3;
Lvv(nn) = -Kb;
nn = nn+1;
R(invy,1)=0;
end
end
if i==Ny && j<=Nx-1 % lower boundary without edges
if BC.bot(2)==0
% L(invy,invy)=Kb;
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = Kb;
nn = nn+1;
R(invy,1)=Kb*BC.bot(3);
elseif BC.bot(2) == 2 %% external boundary
dL=300e3;
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = Kc*(1/dL+1/dy(i-1));
nn = nn+1;
Lii(nn) = invy;
Ljj(nn) = invy-3;
Lvv(nn) = Kc*(-1/dy(i-1));
nn = nn+1;
R(invy, 1) = 0;
end
end
else
% internal nodes
% coeffs for vy
% L(invy,invy-3*Ny)=2*etas(i,j)/(dx(j)*(dx(j-1)+dx(j))); %
Lii(nn) = invy;
Ljj(nn) = invy-3*Ny;
Lvv(nn) = 2*etas(i,j)/(dx(j)*(dx(j-1)+dx(j)));
nn = nn+1;
% L(invy,invy)=(-4/(dy(i-1)+dy(i)))*(etan(i+1,j+1)/dy(i)+etan(i,j+1)/dy(i-1))-(2/dx(j))*(etas(i,j+1)/(dx(j)+dx(j+1))+etas(i,j)/(dx(j-1)+dx(j))); %
Lii(nn) = invy;
Ljj(nn) = invy;
Lvv(nn) = (-4/(dy(i-1)+dy(i)))*(etan(i+1,j+1)/dy(i)+etan(i,j+1)/dy(i-1))-(2/dx(j))*(etas(i,j+1)/(dx(j)+dx(j+1))+etas(i,j)/(dx(j-1)+dx(j)));
nn = nn+1;
% L(invy,invy+3*Ny)=2*etas(i,j+1)/(dx(j)*(dx(j)+dx(j+1))); %
Lii(nn) = invy;
Ljj(nn) = invy+3*Ny;
Lvv(nn) = 2*etas(i,j+1)/(dx(j)*(dx(j)+dx(j+1)));
nn = nn+1;
% L(invy,invy-3)=4*etan(i,j+1)/(dy(i-1)*(dy(i-1)+dy(i))); %
Lii(nn) = invy;
Ljj(nn) = invy-3;
Lvv(nn) = 4*etan(i,j+1)/(dy(i-1)*(dy(i-1)+dy(i)));
nn = nn+1;
% L(invy,invy+3)=4*etan(i+1,j+1)/(dy(i)*(dy(i-1)+dy(i))); %
Lii(nn) = invy;
Ljj(nn) = invy+3;
Lvv(nn) = 4*etan(i+1,j+1)/(dy(i)*(dy(i-1)+dy(i)));
nn = nn+1;
% coeffs for vx
% L(invy,invx+3*Ny)=2*etas(i,j+1)/(dx(j)*(dy(i-1)+dy(i))); %
Lii(nn) = invy;
Ljj(nn) = invx+3*Ny;
Lvv(nn) = 2*etas(i,j+1)/(dx(j)*(dy(i-1)+dy(i)));
nn = nn+1;
% L(invy,invx+3*Ny-3)=-2*etas(i,j+1)/(dx(j)*(dy(i-1)+dy(i))); %
Lii(nn) = invy;
Ljj(nn) = invx+3*Ny-3;
Lvv(nn) = -2*etas(i,j+1)/(dx(j)*(dy(i-1)+dy(i)));
nn = nn+1;
% L(invy,invx)=-2*etas(i,j)/(dx(j)*(dy(i-1)+dy(i))); %
Lii(nn) = invy;
Ljj(nn) = invx;
Lvv(nn) = -2*etas(i,j)/(dx(j)*(dy(i-1)+dy(i)));
nn = nn+1;
% L(invy,invx-3)=2*etas(i,j)/(dx(j)*(dy(i-1)+dy(i))); %
Lii(nn) = invy;
Ljj(nn) = invx-3;
Lvv(nn) = 2*etas(i,j)/(dx(j)*(dy(i-1)+dy(i)));
nn=nn+1;
% coeffs for Pp
% L(invy,inp+3*Ny+3)=-2*Kc/(dy(i-1)+dy(i)); %
Lii(nn) = invy;
Ljj(nn) = inp+3*Ny+3;
Lvv(nn) = -2*Kc/(dy(i-1)+dy(i));
nn=nn+1;
% L(invy,inp+3*Ny)=2*Kc/(dy(i-1)+dy(i)); %
Lii(nn) = invy;
Ljj(nn) = inp+3*Ny;
Lvv(nn) = 2*Kc/(dy(i-1)+dy(i));
nn=nn+1;
% RHS (note: sxx = -syy because deviatoric)
R(invy,1)=-gy*(rho(i,j)+rho(i,j+1))/2 + ...
2.*(srhs_xx(i+1,j+1)-srhs_xx(i,j+1))/(dy(i)+dy(i-1)) - ...
(srhs_xy(i,j+1)-srhs_xy(i,j))/dx(j); %
end
end % for j
end % for i
nn = nn-1;
%%% Build Sparse Matrix
Li = Lii(1:nn);
Lj = Ljj(1:nn);
Lv = Lvv(1:nn);
L = sparse(Li,Lj,Lv);
% SOLVE LINEAR SYSTEM now down in SiStER_flow_solve G.Ito
% Matlab direct solver
% tic
% S=L\R;
% toc
%
end