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Discretization2D.m
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Discretization2D.m
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function [Lattice,BCs,Char] = Discretization2D(NumCell,Order,Block,Dimensionless,BCs,Char,LS2D)
%% Discretization of the Dimensionless System to Lattice System
Lattice.dx = 1/NumCell; % spatial resolution
Lattice.gridx = Block(1,1):Lattice.dx:Block(1,2); % vector of position in x-direction
Lattice.gridy = Block(2,1):Lattice.dx:Block(2,2); % vector of position in y-direction
[Lattice.y,Lattice.x] = meshgrid(Lattice.gridy,Lattice.gridx); % matrix of position for (x,y)
[Lattice.Nx,Lattice.Ny] = size(Lattice.x); % size of the matrix
Lattice.Lx = Dimensionless.Lx;
Lattice.Ly = Dimensionless.Ly;
switch Order
case 2
Lattice.dt = Lattice.dx^2; % temporal resolution, second order
case 1
Lattice.dt = Lattice.dx; % temporal resolution, first order
case 0
Lattice.dt = 1e-6;
otherwise
disp('change temporal resulution by oneself here in Discreitzation')
end
Lattice.Rho = Dimensionless.Rho;
Lattice.Nu = Dimensionless.Nu*Lattice.dt/Lattice.dx^2; % lattice viscosity
Lattice.Tau = 3*Lattice.Nu + 0.5; % single relaxation time
Lattice.UMax = Lattice.dt/Lattice.dx*Dimensionless.UMax;
Lattice.TimeMax = Dimensionless.TimeMax/Lattice.dt;
Lattice.Fx = Dimensionless.Fx(Lattice.x,Lattice.y,1,Lattice.UMax);
Lattice.Fy = Dimensionless.Fy(Lattice.x,Lattice.y,1,Lattice.UMax);
Char.DimensionlessRho = Lattice.Rho;
Char.DimensionlessLength = Lattice.dx;
Char.DimensionlessTime = Lattice.dt;
Char.DimensionlessUMax = Lattice.dx/Lattice.dt;
% boundary indices for B = all boundary nodes, {L,R,U,D} = Left,Right,Up,Down boundary nodes
BCs.B = find(Lattice.x==Block(1,1)|Lattice.x==Block(1,2)|Lattice.y==Block(2,1)|Lattice.y==Block(2,2));
BCs.L = find(Lattice.x==Block(1,1));
BCs.R = find(Lattice.x==Block(1,2));
BCs.U = find(Lattice.y==Block(2,1));
BCs.D = find(Lattice.y==Block(2,2));
switch BCs.Left
case 'velocity'
BCs.UxLeftStable = BCs.UxLeft(:,Lattice.gridy,Lattice.UMax);
BCs.UyLeftStable = BCs.UyLeft(:,Lattice.gridy,Lattice.UMax);
case 'pressure'
BCs.RhoLeftStable = BCs.RhoLeft(:,Lattice.gridy,Lattice.UMax);
BCs.UyLeftStable = BCs.UyLeft(:,Lattice.gridy,Lattice.UMax);
end
switch BCs.Right
case 'velocity'
BCs.UxRightStable = BCs.UxRight(:,Lattice.gridy,Lattice.UMax);
BCs.UyRightStable = BCs.UyRight(:,Lattice.gridy,Lattice.UMax);
case 'pressure'
BCs.RhoRightStable = BCs.RhoRight(:,Lattice.gridy,Lattice.UMax);
BCs.UyRightStable = BCs.UyRight(:,Lattice.gridy,Lattice.UMax);
end
switch BCs.Up
case 'velocity'
BCs.UxUpStable = BCs.UxUp(Lattice.gridx,:,Lattice.UMax);
BCs.UyUpStable = BCs.UyUp(Lattice.gridx,:,Lattice.UMax);
case 'pressure'
BCs.RhoUpStable = BCs.RhoUp(Lattice.gridx,:,Lattice.UMax);
BCs.UxUpStable = BCs.UxUp(Lattice.gridx,:,Lattice.UMax);
end
switch BCs.Down
case 'velocity'
BCs.UxDownStable = BCs.UxDown(Lattice.gridx,:,Lattice.UMax);
BCs.UyDownStable = BCs.UyDown(Lattice.gridx,:,Lattice.UMax);
case 'pressure'
BCs.RhoDownStable = BCs.RhoDown(Lattice.gridx,:,Lattice.UMax);
BCs.UxDownStable = BCs.UxDown(Lattice.gridx,:,Lattice.UMax);
end
% switch BCs.Curve
% case 'velocity'
% BCs.UxCurveStable = BCs.UxCurve(:,Lattice.Boundary,Lattice.UMax);
% BCs.UyCurveStable = BCs.UyCurve(:,Lattice.Boundary,Lattice.UMax);
% case 'pressure'
% BCs.RhoLeftStable = BCs.RhoLeft(:,Lattice.gridy,Lattice.UMax);
% BCs.UyLeftStable = BCs.UyLeft(:,Lattice.gridy,Lattice.UMax);
% end
BCs.C = zeros(10,Lattice.Nx,Lattice.Ny);
Lattice.Region = zeros(Lattice.Nx,Lattice.Ny);
Lattice.Region(find(((Lattice.x-(Block(1,1)+Block(1,2))/2).^2+(Lattice.y-(Block(2,1)+Block(2,2))/2).^2<1) ...
|((Lattice.x<=(Block(1,1)+Block(1,2))/2)&(Lattice.y>=(Block(2,1)+Block(2,2))/2)))) = 1;
Lattice.Obstacle = find(Lattice.Region==0);
for i = 1:Lattice.Nx
for j = 1:Lattice.Ny
if Lattice.Region(i,j)==1
for k = 1:9
m = LS2D.cx(k);
n = LS2D.cy(k);
if i+m<=Lattice.Nx&&i+m>=1&&j+n<=Lattice.Ny&&j+n>=1
if Lattice.Region(i+m,j+n)==0
delta = 1/2;
for s = 1:10
xx = Lattice.x(i,j) + delta*LS2D.cx(k)*Lattice.dx;
yy = Lattice.y(i,j) + delta*LS2D.cy(k)*Lattice.dx;
pn = (xx-(Block(1,1)+Block(1,2))/2).^2+(yy-(Block(2,1)+Block(2,2))/2).^2>1;
delta = delta +1/2^(s+1) - pn*1/2^s;
end
BCs.C(k,i,j) = delta;
BCs.C(10,i,j) = BCs.C(10,i,j)+Lattice.Region(i+m,j+n);
end
BCs.C(10,i,j) = BCs.C(10,i,j)+Lattice.Region(i+m,j+n);
else
% BCs.C(k,i,j) = 1;
BCs.C(10,i,j)=10;
end
end
if BCs.C(10,i,j)<9
BCs.C(10,i,j)=1;
elseif BCs.C(10,i,j)>10
BCs.C(10,i,j)=2;
end
end
end
end
Lattice.CurveBoundary = find(BCs.C(10,:,:)==1);
Lattice.Boundary = find(reshape(BCs.C(10,:,:),Lattice.Nx,Lattice.Ny)~=9&Lattice.Region==1);
Lattice.InnerRegion = Lattice.Region;
Lattice.InnerRegion(Lattice.Boundary)=0;
% for i = 1:Lattice.Nx
% for j = 1:Lattice.Ny
% if Lattice.Region(i,j)==1
% for k = 1:9
% m = LS2D.cx(k);
% n = LS2D.cy(k);
% if i+m<=Lattice.Nx&&i+m>=1&&j+n<=Lattice.Ny&&j+n>=1
% if Lattice.Region(i+m,j+n)==0
% BCs.C(k,i,j)=1;
% end
% BCs.C(10,i,j) = BCs.C(10,i,j)+Lattice.Region(i+m,j+n);
% else
% BCs.C(k,i,j)=1;
% end
% end
% if BCs.C(10,i,j)<9
% BCs.C(10,i,j)=1;
% else
% BCs.C(10,i,j)=0;
% end
% end
% end
% end
% Lattice.Boundary = find(BCs.C(10,:,:)==1);
% Lattice.CurveBoundary = find(reshape(BCs.C(10,:,:),Lattice.Nx,Lattice.Ny)==1&Lattice.x~=Block(1,1)&Lattice.y~=Block(2,2));
% Lattice.InnerRegion = Lattice.Region;
% Lattice.InnerRegion(Lattice.Boundary)=0;