-
Notifications
You must be signed in to change notification settings - Fork 0
/
HW03nolinearheatduction.m
258 lines (205 loc) · 7.84 KB
/
HW03nolinearheatduction.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
% This is the nonlinear solver for heat equation with material nonlinearity
% clean the memory and the screen
close all
clear all; clc;
% domain
omega_l = 0.0;
omega_r = 1.0;
% -------------------------------------------------------------------------
% material properties and input data
%kappa = 1.0;
% exact solution
exact = @(x) sin(x);
exact_x = @(x) cos(x);
% nonlinear kappa
fun_kappa = @(x) 1 + x*x;
fun_dkappa = @(x) 2*x;
f = @(x) -2.0*cos(x)*cos(x)*sin(x) + sin(x)*(sin(x)*sin(x) + 1.0);
h = @(x) -1.0 * fun_kappa( exact(0) );
g = @(x) sin(1);
% -------------------------------------------------------------------------
% interpolation degree
pp = 2;
nElem_vec = [12,14,16,18,20];
cn = length(nElem_vec);
hh_x = zeros(cn,1);
error_l2 = zeros(cn,1);
error_h1 = zeros(cn,1);
slop_l2 = zeros(cn,1);
slop_h1 = zeros(cn,1);
for num = 1 : cn
% number of elements
nElem = nElem_vec(num);
% quadrature rule
%nqp = pp + 1;
nqp = 6;
[qp, wq] = Gauss(nqp, -1, 1);
n_np = nElem * pp + 1; % number of nodal points
n_en = pp + 1; % number of element nodes
IEN = zeros(n_en, nElem);
for ee = 1 : nElem
for aa = 1 : n_en
IEN(aa, ee) = (ee - 1) * pp + aa;
end
end
% mesh is assumbed to have uniform size hh
hh = (omega_r - omega_l) / nElem;
x_coor = omega_l : (hh/pp) : omega_r;
% setup ID array based on the boundary condition
ID = 1 : n_np;
ID(end) = 0;
% Setup the stiffness matrix and load vector
% number of equations equals the number of nodes minus the number of
% Dirichlet nodes
n_eq = n_np - 1;
%initial guess
uh = [zeros(n_eq,1);g(omega_r)];
counter = 0;
nmax = 20;
error = 1.0;
while counter < nmax && error > 1.0e-20
% Allocate an empty stiffness matrix and load vector
K = sparse(n_eq, n_eq);
F = zeros(n_eq, 1);
% Assembly the siffness matrix and load vector
for ee = 1 : nElem
% Allocate zero element stiffness matrix and element load vector
k_ele = zeros(n_en, n_en);
f_ele = zeros(n_en, 1);
x_ele = zeros(n_en, 1);
d_ele = zeros(n_en, 1);
for aa = 1 : n_en
x_ele(aa) = x_coor( IEN(aa,ee) );
u_ele(aa) = uh( IEN(aa,ee) );
end
for qua = 1 : nqp
% geometrical mapping
x_qua = 0.0;
dx_dxi = 0.0;
u_qua = 0.0;
u_xi = 0.0;
for aa = 1 : n_en
x_qua = x_qua + x_ele(aa) * PolyBasis(pp, aa, 0, qp(qua));
dx_dxi = dx_dxi + x_ele(aa) * PolyBasis(pp, aa, 1, qp(qua));
u_qua = u_qua + u_ele(aa) * PolyBasis(pp, aa, 0, qp(qua));
u_xi = u_xi + u_ele(aa) * PolyBasis(pp, aa, 1, qp(qua));
end
dxi_dx = 1.0 / dx_dxi;
kappa = fun_kappa( u_qua );
dkappa = fun_dkappa( u_qua );
for aa = 1 : n_en
Na = PolyBasis(pp, aa, 0, qp(qua));
Na_xi = PolyBasis(pp, aa, 1, qp(qua));
f_ele(aa) = f_ele(aa) + wq(qua) * Na * f(x_qua) * dx_dxi;
f_ele(aa) = f_ele(aa) - wq(qua) * Na_xi * kappa * u_xi * dxi_dx;
for bb = 1 : n_en
Nb = PolyBasis(pp, bb, 0, qp(qua));
Nb_xi = PolyBasis(pp, bb, 1, qp(qua));
k_ele(aa,bb) = k_ele(aa,bb) + wq(qua) * Na_xi * kappa * Nb_xi * dxi_dx;
k_ele(aa,bb) = k_ele(aa,bb) + wq(qua) * Na_xi * dkappa * Nb * u_xi* dxi_dx;
end
end
end
% end of the quadrature loop
% distribute the entries to the global stiffness matrix and global load vector
for aa = 1 : n_en
LM_a = ID( IEN(aa, ee) );
if LM_a > 0
F(LM_a) = F(LM_a) + f_ele(aa);
for bb = 1 : n_en
LM_b = ID( IEN(bb, ee) );
if LM_b > 0
K(LM_a, LM_b) = K(LM_a, LM_b) + k_ele(aa, bb);
else
% x_qua = x_coor( IEN(bb,ee) ); % obtain the Dirichlet node's physical coordinates
% g_qua = g( x_qua ); % Obtain the boundary data at this point
% F( LM_a ) = F( LM_a ) - k_ele(aa, bb) * g_qua;
end
end
end
end
% Modify the load vector by the Natural BC
% Note: for multi-dimensional cases, one needs to perform line or
% surface integration for the natural BC.
if ee == 1
F( ID(IEN(1, ee)) ) = F( ID(IEN(1, ee)) ) + h( x_coor(IEN(1,ee)));
end
end
% Solve the stiffness matrix problem
%uh = K \ F;
increment = K \ F;
%[L,U,P] = lu(K);
%y = L\(P*F);
%uh = U \ y;
% Append the displacement vector by the Dirichlet data
%uh = [ uh; g(omega_r) ];
uh = [ uh(1:end-1) + increment; g(omega_r) ];
error = norm(F);
counter = counter + 1;
end
% -------------------------------------------------------------------------
% Now we do the postprocessing
nqp = 6;
[qp, wq] = Gauss(nqp, -1, 1);
top_l2 = 0.0; bot_l2 = 0.0; top_h1 = 0.0; bot_h1 = 0.0;
for ee = 1 : nElem
for qua = 1 : nqp
x_ele = zeros(n_en, 1);
u_ele = zeros(n_en, 1);
for aa = 1 : n_en
x_ele(aa) = x_coor(IEN(aa, ee));
u_ele(aa) = uh(IEN(aa, ee));
end
x = 0.0; dx_dxi = 0.0; duh_dxi = 0.0; uh_0 = 0.0;
for aa = 1 : n_en
x = x + x_ele(aa) * PolyBasis(pp, aa, 0, qp(qua));
dx_dxi = dx_dxi + x_ele(aa) * PolyBasis(pp, aa, 1, qp(qua));
uh_0 = uh_0 + u_ele(aa) * PolyBasis(pp, aa, 0, qp(qua));
duh_dxi = duh_dxi + u_ele(aa) * PolyBasis(pp, aa, 1, qp(qua));
end
dxi_dx = 1.0 / dx_dxi;
top_h1 = top_h1 + wq(qua) * (duh_dxi * dxi_dx - exact_x(x))^2 * dx_dxi;
bot_h1 = bot_h1 + wq(qua) * exact_x(x)^2 * dx_dxi;
top_l2 = top_l2 + wq(qua) * (uh_0 - exact(x))^2 * dx_dxi;
bot_l2 = bot_l2 + wq(qua) * exact(x)^2 * dx_dxi;
end
end
top_l2 = sqrt(top_l2);
bot_l2 = sqrt(bot_l2);
top_h1 = sqrt(top_h1);
bot_h1 = sqrt(bot_h1);
errorl2 = top_l2 / bot_l2;
errorh1 = top_h1 / bot_h1;
error_l2(num,1) = log(errorl2);
error_h1(num,1) = log(errorh1);
hh_x(num,1) = log(hh);
dx_hh = zeros( 2,1);
dy_error_l2 = zeros( 2,1);
dy_error_h1 = zeros( 2,1);
if (num >= 2)
dx_hh = [hh_x(num);hh_x(num-1)];
dy_error_l2 = [error_l2(num);error_l2(num-1)];
dy_error_h1 = [error_h1(num);error_h1(num-1)];
slop_l2(num) = (dy_error_l2(2) - dy_error_l2(1)) / (dx_hh(2)-dx_hh(1));
slop_h1(num) = (dy_error_h1(2) - dy_error_h1(1)) / (dx_hh(2)-dx_hh(1));
else
slop_l2(num) = 0;
slop_h1(num) = 0;
end
end
% -------------------------------------------------------------------------
% Postprocessing plot graph
figure
%yaxis
error_x_l2 = plot(hh_x,error_l2,'--rO','LineWidth',2);
hold on
error_x_h1 = plot(hh_x,error_h1,'--rO','LineWidth',2);
%legend(error_x_l2,error_x_h1,'L_2 error convergence analysis','H_1 error convergence analysis');
xlabel('log ||hh|| ');
%ylabel('log ||e||_{l2} ');
exportgraphics(gca,['error_x_l2_pp3' '.jpg']);
T4 = table(hh_x,error_l2,error_h1,slop_l2,slop_h1,...
'variableNames',{'hh_mesh','error_l2','error_h1','l2 convergence rate','H1 convergence rate',});
writetable(T4);
disp(T4);
% EOF