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demo_phase_change_matlab
Commit 5564fd7f by Mehdi Slimani
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picard

parent 411e098b
Showing with 126 additions and 55 deletions
demo.m
close all; clear all; clc; close all; clear all;
%% PAREMETERS %% PAREMETERS
x_left = 0.0; x_left = 0.0;
x_right = 0.1; x_right = 0.1;
%A = 0.01^2; cross_section = 0.01^2;
global rho c_p k diffusivity lamma L T_m T_l T_s T_0; run_type = "picard"; %picard, nr
rho = 4510; % cross_section = 1;
c_p = 520; global rho c_p k diffusivity lamma l T_m T_l T_s T_0 gpoints_ref gpoints_weights stl sts;
k = 16; rho = 4510*cross_section;
c_p = 52;
k = 16*cross_section;
diffusivity = k / rho / c_p; diffusivity = k / rho / c_p;
lamma = 0.388150542167233; l = 325000;
L = 325;
T_m = 1670; T_m = 1670;
T_l = 2000; T_l = 2000;
T_s = 1500; T_s = 1500;
T_0 = T_s; T_0 = T_s;
stl = c_p*(T_l - T_m) / l;
sts = c_p*(T_m - T_s) / l;
lamma = fzero(@transcendental_fun, 0.388150542167233);
time = 0.0; time = 0.0;
S = 4; S = 20;
max_nr_iters = 10;
dt = 1; dt = 1;
num_els = 1000; num_time_steps = 1;
num_els = 400;
num_nodes = num_els + 1; num_nodes = num_els + 1;
num_time_steps = 2;
wait_time_plots = 0.5;%seconds wait_time_plots = 0.5;%seconds
gpoints_ref = [-0.93246 -0.66120 -0.23861 0.23861 0.66120 0.993246];
gpoints_weights = [0.17132 0.36076 0.6791 0.6791 0.36076 0.17132];
get_gpoints = @(xl,xr) (xl + xr)/2.0 + (xr-xl)/2.0*gpoints_ref;
%% PAREMETERS %% PAREMETERS
Xs = linspace(x_left,x_right,num_nodes); Xs = linspace(x_left,x_right,num_nodes);
x_av = (Xs(2:end)+Xs(1:end-1))/2.0;
node2dof = [-1, 1:(num_nodes-2), -1]; node2dof = [-1, 1:(num_nodes-2), -1];
% node2dof = 1:(num_nodes); % node2dof = 1:(num_nodes);
mask = find(node2dof>0); mask = find(node2dof>0);
...@@ -40,7 +49,7 @@ u_np1 = f0(Xs'); ...@@ -40,7 +49,7 @@ u_np1 = f0(Xs');
ylims = [T_s-100, T_l+100]; ylims = [T_s-100, T_l+100];
fig = figure; fig = figure('Position', [10 10 1810 1210]);
plot(Xs, u_np1); plot(Xs, u_np1);
ylim(ylims); ylim(ylims);
pause(wait_time_plots); pause(wait_time_plots);
...@@ -54,55 +63,100 @@ for time_iter=1:num_time_steps ...@@ -54,55 +63,100 @@ for time_iter=1:num_time_steps
dirichlet_vals = nan([num_nodes,1]); dirichlet_vals = nan([num_nodes,1]);
dirichlet_vals(1) = T_l; dirichlet_vals(1) = T_l;
dirichlet_vals(end) = T_s; dirichlet_vals(end) = T_s;
% ASSEMBLY for nr_iter=1:max_nr_iters
A = sparse(num_dofs,num_dofs); fprintf("About to solve for iter %i of NR, tstep %i \n\n", nr_iter, time_iter);
M = sparse(num_dofs,num_dofs); u_i = u_np1;% newton-raphson
b = zeros([num_dofs,1]); % ASSEMBLY
for ielem=1:num_els A = sparse(num_dofs,num_dofs);
inodes_global = [ielem, ielem+1]; M = sparse(num_dofs,num_dofs);
xr = Xs(ielem+1); L = sparse(num_dofs,num_dofs);
xl = Xs(ielem); Lp = sparse(num_dofs,num_dofs);
h = xr - xl; b = zeros([num_dofs,1]);
M_loc = h / 6.0 * [2.0, 1.0; 1.0, 2.0]; app_c_p = zeros([num_els,1]);
M_loc = M_loc * rho * c_p / dt; for ielem=1:num_els
A_loc = 1 / h * [1.0, -1.0; -1.0, 1.0]; inodes_global = [ielem, ielem+1];
A_loc = A_loc * k; xr = Xs(ielem+1);
for inode=1:2 xl = Xs(ielem);
inodeg = inodes_global(inode); h = xr - xl;
idof = node2dof(inodeg); u_av = mean(u_i(inodes_global));
if (idof<0) app_c_p(ielem ) = rho * l * flp(u_av) + rho * c_p;
continue; gpoints = get_gpoints(xl,xr);
M_loc = zeros(2);
L_loc = zeros(2);
Lp_loc = zeros(2);
A_loc = zeros(2);
b_loc = zeros([2,1]);
phis = get_basis_funcs_p1_d1(xl,xr);
grad_phis = get_grad_basis_funcs_p1_d1(xl,xr);
for igpoint=1:length(gpoints)
weight = gpoints_weights(igpoint) * h;
x = gpoints(igpoint);
u_gp = sum([phis{1}(x), phis{2}(x)]'.*u_i(inodes_global));
for inode=1:2
for jnode=1:2
A_loc(inode,jnode) = A_loc(inode,jnode) + weight * k * grad_phis{inode}(x)*grad_phis{jnode}(x);
M_loc(inode,jnode) = M_loc(inode,jnode) + weight * rho * c_p * phis{inode}(x)*phis{jnode}(x);
L_loc(inode,jnode) = L_loc(inode,jnode) + weight * rho * l * flp(u_gp) * phis{inode}(x)*phis{jnode}(x);
Lp_loc(inode,jnode) = Lp_loc(inode,jnode) + weight * rho * l * flpp(u_gp) * u_gp * phis{inode}(x)*phis{jnode}(x);
end
end
end end
for jnode=1:2 M_loc = M_loc / dt;
jnodeg = inodes_global(jnode); L_loc = L_loc / dt;
jdof = node2dof(jnodeg); Lp_loc = Lp_loc / dt;
if (jdof>0) for inode=1:2
M(idof,jdof) = M(idof,jdof) + M_loc(inode,jnode); inodeg = inodes_global(inode);
A(idof,jdof) = A(idof,jdof) + A_loc(inode,jnode); idof = node2dof(inodeg);
else if (idof<0)
dval = dirichlet_vals(jnodeg); continue;
b(idof) = b(idof) - A_loc(inode,jnode)*dval; end
b(idof) = b(idof) - M_loc(inode,jnode)*u_n(jnodeg); for jnode=1:2
jnodeg = inodes_global(jnode);
jdof = node2dof(jnodeg);
if (jdof>0)
M(idof,jdof) = M(idof,jdof) + M_loc(inode,jnode);
L(idof,jdof) = L(idof,jdof) + L_loc(inode,jnode);
Lp(idof,jdof) = Lp(idof,jdof) + Lp_loc(inode,jnode);
A(idof,jdof) = A(idof,jdof) + A_loc(inode,jnode);
else
dval = dirichlet_vals(jnodeg);
b(idof) = b(idof) - M_loc(inode,jnode)*u_n(jnodeg);
b(idof) = b(idof) - L_loc(inode,jnode)*u_n(jnodeg);
b(idof) = b(idof) - A_loc(inode,jnode)*dval;
end
end end
end end
end end
% SOLVE
b = b + (M+L) * u_n(mask);
stiffness_matrix = (M + L) + A;
r = b - stiffness_matrix * u_i(mask);
r_norm = sqrt(trapz(Xs(mask),r.^2));
if run_type=="picard"
u_np1(mask) = mldivide(stiffness_matrix, b);%Picard
elseif run_type=="nr"
stiffness_matrix = stiffness_matrix;
else
disp("invalid run type");
exit();
end
u_np1(forced_nodes) = dirichlet_vals(forced_nodes);
err = sqrt(sum((u_np1-u_exact).^2));
fprintf("Residual norm is %g, L2 error is %g\n\n", r_norm, err);
clf(fig);
plot(Xs, u_np1,'DisplayName','uh');
hold on
plot(Xs, u_exact,'DisplayName','exact');
ylim(ylims);
%yyaxis right
%plot(x_av, app_c_p,'DisplayName','apparent c_p');
legend;
pause(wait_time_plots)
end end
% SOLVE
b = b + M * u_n(mask);
stiffness_matrix = M + A;
u_np1(mask) = mldivide(stiffness_matrix, b);
u_np1(forced_nodes) = dirichlet_vals(forced_nodes);
clf(fig);
plot(Xs, u_np1,'DisplayName','uh');
hold on
plot(Xs, u_exact,'DisplayName','exact');
ylim(ylims);
legend;
pause(wait_time_plots)
end end
function [sol] = exact_sol(x,t) function [sol] = exact_sol(x,t)
global rho c_p k diffusivity lamma L T_m T_l T_s T_0; global rho c_p k diffusivity lamma l T_m T_l T_s T_0;
interface_pos = locate_interface(t); interface_pos = locate_interface(t);
sol = zeros(flip(size(x))); sol = zeros(flip(size(x)));
sol(x<=interface_pos) = T_l - (T_l - T_m)*erf(x(x<=interface_pos)/(2*sqrt(diffusivity*t)))/erf(lamma); sol(x<=interface_pos) = T_l - (T_l - T_m)*erf(x(x<=interface_pos)/(2*sqrt(diffusivity*t)))/erf(lamma);
...@@ -110,6 +164,23 @@ function [sol] = exact_sol(x,t) ...@@ -110,6 +164,23 @@ function [sol] = exact_sol(x,t)
end end
function [interface_pos] = locate_interface(t) function [interface_pos] = locate_interface(t)
global rho c_p k diffusivity lamma L T_m T_l T_s T_0; global rho c_p k diffusivity lamma l T_m T_l T_s T_0;
interface_pos = 2*lamma*sqrt(diffusivity*t); interface_pos = 2*lamma*sqrt(diffusivity*t);
end end
function [phis] = get_basis_funcs_p1_d1(xl,xr)
phi_l = @(x) -(x-xr) / (xr-xl);
phi_r = @(x) +(x-xl) / (xr-xl);
phis = {phi_l, phi_r};
end
function [grad_phis] = get_grad_basis_funcs_p1_d1(xl,xr)
grad_phi_l = @(x) -1 / (xr-xl);
grad_phi_r = @(x) +1 / (xr-xl);
grad_phis = {grad_phi_l, grad_phi_r};
end
function [y] = transcendental_fun(x)
global diffusivity sts stl;
y = sqrt(pi)*x - ...
(stl / exp(x^2) / erf(x) ) + ...
(sts / exp(x^2) / erfc(x) );
end
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