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MPFC_3D_CN_SAV.m
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MPFC_3D_CN_SAV.m
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function phi = MPFC_3D_CN_SAV(pde,domain,Nx,Ny,Nz,time,option)
% Solve 2D phase field crystal equaiton
% (1) phi_tt + alpha\phi_t = \Delta \mu
% (2) \mu = (1+\Delta)^2\phi + (\phi^3 - epsilon\phi)
% (2) \mu = (1+\Delta)^2\phi + \beta*phi + \phi*(\phi^2 - epsilon - \beta)
% Qi Li
% 05/23/2019
global dt epsilon M k2 k4 beta_bar beta alpha C0 hx hy hz Lx Ly Lz sigma
if ~exist('option','var'), option = []; end
if ~isfield(option,'tol')
option.tol = 10^-14; % default tol
end
if ~isfield(option,'tolit')
option.tolit = 10^-8; % default tolit
end
if ~isfield(option,'maxit')
option.maxit = 2000; % default maxit
end
if ~isfield(option,'plotflag')
option.plotflag = 0;
end
if ~isfield(option,'saveflag')
option.saveflag = 0;
end
if ~isfield(option,'savefinal')
option.savefinal = 0;
end
if ~isfield(option,'printflag')
option.printflag = 0;
end
if ~isfield(option,'vtkflag')
option.printflag = 0;
end
if ~isfield(option,'energyflag')
option.energyflag = 0;
end
if 1 == option.energyflag
figname_mass = [pde.name,'e',num2str(pde.epsilon),'M',num2str(pde.M),'b_bar',num2str(pde.beta_bar),'b',num2str(pde.beta),'dt',num2str(time.dt),'_mass.txt'];
figname_energy = [pde.name,'e',num2str(pde.epsilon),'M',num2str(pde.M),'b_bar',num2str(pde.beta_bar),'b',num2str(pde.beta),'dt',num2str(time.dt),'_energy.txt'];
out1 = fopen(figname_mass,'w');
out2 = fopen(figname_energy,'w');
end
tol = option.tol;
tolit = option.tolit;
maxit = option.maxit;
%%
T = time.T;
t = time.t0;
dt = time.dt;
tsave = time.tsave;
dir_fig = [pde.name '/fig'];
dir_data = [pde.name '/data'];
epsilon = pde.epsilon;
M = pde.M;
alpha = pde.alpha;
beta_bar = pde.beta_bar;
beta = pde.beta;
sigma = pde.sigma;
Lx = domain.xb - domain.xa;
Ly = domain.yb - domain.ya;
Lz = domain.zb - domain.za;
hx = Lx/Nx;
hy = Ly/Ny;
hz = Ly/Nz;
x = domain.xa + hx*(0:Nx-1);
y = domain.ya + hy*(0:Ny-1);
z = domain.za + hy*(0:Nz-1);
[xx,yy,zz] = meshgrid(x,y,z);
phi0 = pde.init(xx,yy,zz);
psi0 = zeros(size(phi0));
nfigure =1;
[k_x,k_y,k_z,kx,ky,kz,kxx,kyy,kzz,k2,k4] = prepare_fft3(Lx,Ly,Lz,Nx,Ny,Nz);
nplot = round((T-t)/dt);
nsave = round(tsave/dt);
tstart = tic;
C0 = (epsilon+alpha).^2*Lx*Ly*Lz;
%% initialization phi 1
time1 = time; time1.T = dt;
[phi1, r1] = MPFC_3D_First_order_SAV(pde,domain,Nx,Ny,Nz,time1,option);
[phi1, r1] = MPFC_3D_CN_SAV_PreCor(pde,domain,Nx,Ny,Nz,time1,phi1,option);
psi1 = 2*(phi1 - phi0)/dt-psi0;
t = t+dt;
for nt = 2:nplot
t = t+dt;
phi_star = 1/2*(3*phi1-phi0);
% step 1
H = fun_H(phi_star);
if isfield(pde,'rhs') && isfield(pde,'exact')
rhs = dt*pde.rhs(xx,yy,zz,t-dt/2);
else
rhs = 0;
end
g = get_rhs(phi1,psi1,r1,H)+rhs;
psiA = inv_A(lap_diff(H));
psiB = inv_A(g);
gamma = -fftn(H.*psiA);
gamma = gamma(1,1,1)*hx*hy*hz;
% Step 2
Hphi = fftn(H.*psiB);
Hphi = Hphi(1,1,1)*hx*hy*hz/(1+dt*M*gamma/4);
% Step 3
phi = dt*M/4*Hphi.*psiA + psiB;
if 1 == option.printflag
fprintf('epsilon=%.3f,t=%.4f/%.f, dt=%.4f, Nx=%d, Ny=%d, Nz=%d \n',epsilon,t,T,dt,Nx,Ny,Nz);
end
%% update phi0
r1 = fun_r(phi,phi1,r1,H);
psi1 = fun_psi(phi,phi1,psi1);
phi0 = phi1;
phi1 = phi;
if 1 == option.energyflag
calculate_energy(out1,out2,hx,hy,hz,t,phi,psi1,r1,C0);
end
if 0 == mod(nt,nsave)
if 1 == option.printflag
timeElapsed = toc(tstart);
fprintf('epsilon=%.3f,t=%.4f/%.f, dt=%.4f, Nx=%d, Ny=%d, Nz=%d timeElapsed=%f\n',epsilon,t,T,dt,Nx,Ny,Nz,timeElapsed);
end
if 1 == option.saveflag
ss = [dir_data '/phi_t=' num2str(t) '.txt'];
fid = fopen(ss, 'wt');
fprintf(fid, '%f\n', phi(:));
fclose(fid);
end
nfigure = nfigure +1;
if 1 == option.plotflag
if 1 == option.vtkflag
write_vtk_grid_values(dir_data,x,y,nt,phi0);
end
if 1 == option.saveflag
showsolution_3D_2(nfigure,xx,yy,zz,phi,t,dir_fig);
else
showsolution_3D_2(nfigure,xx,yy,zz,phi,t);
end
end
end
end
if 1 == option.savefinal
name=['phi_e',num2str(epsilon),'Nx=',num2str(Nx),'Ny=',num2str(Ny),'dt=',num2str(dt)];
filename=[name '.mat'];
save(filename,'epsilon','x','y','z','Nx','Ny','Nz','dt','T','phi','domain');
% showsolution_2D(nfigure,xx,yy,phi,t,'./','.fig');
end
if 1 == option.energyflag
fclose(out1);
fclose(out2);
end
end
function r = fun_psi(phi,phi0,psi1)
global dt
r = 2*(phi-phi0)/dt-psi1;
end
function r = fun_r_init(phi)
global C0 hx hy hz
E1 = fftn(F(phi));
r = sqrt(E1(1,1,1)*hx*hy*hz + C0);
end
function r = fun_r(phi,phi0,r0,b)
global hx hy hz
bphi0 = fftn(b.*phi0);
bphi0 = bphi0(1,1,1)*hx*hy*hz;
bphi1 = fftn(b.*phi);
bphi1 = bphi1(1,1,1)*hx*hy*hz;
g1 = r0 - 1/2*bphi0;
r = 1/2*bphi1+g1;
end
function r = fun_H(phi)
global C0 hx hy hz
E1 = fftn(F(phi));
r = F_derivative(phi)./sqrt(E1(1,1,1)*hx*hy*hz+C0);
end
function r = get_rhs(phi1,psi1,r1,b)
global beta_bar beta dt M hx hy hz alpha sigma Lx Ly Lz
bphiphi = fftn(b.*phi1);
bphiphi = bphiphi(1,1,1)*hx*hy*hz;
h2 = -2/dt*phi1-psi1;
gg = dt/(2*beta_bar+dt*beta)*fftn(-(beta_bar+dt*beta/2)*h2 + (beta_bar-dt*beta/2)*psi1);
gg = (gg + fftn(phi1))/2;
r = -(beta_bar+beta/2*dt)*h2 +(beta_bar-dt*beta/2)*psi1 ...
+dt/2*lap_diff(phi1+2*lap_diff(phi1)+lap_diff(lap_diff(phi1))) ...
+alpha/2*dt*lap_diff(phi1) ...
+dt/2*(2*r1-1/2*bphiphi).*lap_diff(b)...
-sigma*dt*M/2*phi1 + sigma*dt*M*gg(1,1,1)*hx*hy*hz/(Lx*Ly*Lz);
end
function lap=lap_diff(phi)
global k2
lap=real(ifftn((k2.*fftn(phi))));
end
function r = inv_A(phi)
global dt M k2 alpha beta_bar beta sigma
r = real(ifftn(fftn(phi)./((beta_bar+dt*beta/2)*2/dt-dt*M/2*k2.*(1+k2).^2-dt*M/2*alpha.*k2+sigma*dt*M/2)));
end
function [] = calculate_energy(out1,out2,hx,hy,hz,t,phi,psi,r,C0)
global epsilon alpha M k2 beta_bar Lx Ly Lz sigma
energy1 = hx*hy*hz*(sum(sum(sum( 1/4*phi.^4 - epsilon/2*phi.^2+ 1/2*phi.^2 + lap_diff(phi).*phi + 1/2*lap_diff(phi).^2 + alpha/2*phi.^2))));
Fpsi = fftn(psi)./k2; Fpsi(1,1,1) = 0; psi_inv_lap = real(ifftn(Fpsi));
ene = -0.5/M*beta_bar*hx*hy*hz*(sum(sum(sum( psi_inv_lap.*psi))));
phi_hat = fftn(phi);
psi_OK = phi - phi_hat(1,1,1)*hx*hy*hz/(Lx*Ly*Lz);
Fpsi = fftn(-psi_OK)./k2; Fpsi(1,1,1) = 0; psi_OK = real(ifftn(Fpsi));
psi_hat = fftn(-lap_diff(psi_OK).*psi_OK);
ene2 = 0.5*sigma*psi_hat(1,1,1)*hx*hy*hz;
energy2 = energy1 + ene;
energy3 = hx*hy*hz*sum(sum(sum(1/2*phi.^2 + lap_diff(phi).*phi + 1/2*lap_diff(phi).^2 + alpha/2*phi.^2))) + r.^2 - C0 + ene;
mass = hx*hy*hz*sum(sum(sum( phi )));
fprintf(out1,'%14.6e %f \n',t,mass);
fprintf(out2,'%14.6e %f %f %f \n',t,energy1+ene2,energy2+ene2,energy3+ene2);
end
function r = F_derivative(phi0)
global alpha epsilon
r = phi0.*(phi0.^2 - epsilon - alpha);
end
function r = F(phi)
global alpha epsilon
r = 1/4*phi.^4 - epsilon/2*phi.^2 - alpha/2*phi.^2;
end