-
Notifications
You must be signed in to change notification settings - Fork 6
/
c_e_mats_sensitivity_federico.m
167 lines (129 loc) · 4.6 KB
/
c_e_mats_sensitivity_federico.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
%% Matrices for Li Diffusion in Electrolyte Phase, c_e(x,t)
% Created July 15, 2011 by Scott Moura
% Modified April 22, 2014 by Federico Bribiesca
function [A,B,C, A_corr_sens_p, A_corr_sens_n, A_corr_sens_s] = c_e_mats_sensitivity_federico(p,c_ex)
% Leading Coefficients
alpha_n = 1 / (p.L_n^2 * p.delta_x_n^2);
alpha_s = 1 / (p.L_s^2 * p.delta_x_s^2);
alpha_p = 1 / (p.L_p^2 * p.delta_x_p^2);
beta_n = (1 - p.t_plus) / (p.epsilon_e_n * p.Faraday * 2 * p.L_n * p.delta_x_n);
beta_s = (1 - p.t_plus) / (p.epsilon_e_s * p.Faraday * 2 * p.L_s * p.delta_x_s);
beta_p = (1 - p.t_plus) / (p.epsilon_e_p * p.Faraday * 2 * p.L_p * p.delta_x_p);
%% Electrolyte Diffusion
De = electrolyteDe(c_ex);
De0 = De(1);
De_n = De(2:p.Nxn);
De_ns = De(p.Nxn+1);
De_s = De(p.Nxn+2 : p.Nxn+2+p.Nxs-2);
De_sp = De(p.Nxn+2+p.Nxs-1);
De_p = De(p.Nxn+2+p.Nxs : end-1);
DeN = De(end);
%% Block Matrices
M1n = zeros(p.Nxn-1);
for idx = 1:p.Nxn-1
if(idx == 1)
M1n(idx,idx) = alpha_n * -(De0 + De_n(idx+1));
M1n(idx,idx+1) = alpha_n * De_n(idx+1);
elseif(idx == p.Nxn-1)
M1n(idx,idx-1) = alpha_n * De_n(idx-1);
M1n(idx,idx) = alpha_n * -(De_n(idx-1) + De_ns);
else
M1n(idx,idx-1) = alpha_n * De_n(idx-1);
M1n(idx,idx) = alpha_n * -(De_n(idx-1) + De_n(idx+1));
M1n(idx,idx+1) = alpha_n * De_n(idx+1);
end
end
M1s = zeros(p.Nxs-1);
for idx = 1:p.Nxs-1
if(idx == 1)
M1s(idx,idx) = alpha_s * -(De_ns + De_s(idx+1));
M1s(idx,idx+1) = alpha_s * De_s(idx+1);
elseif(idx == p.Nxs-1)
M1s(idx,idx-1) = alpha_s * De_s(idx-1);
M1s(idx,idx) = alpha_s * -(De_s(idx-1) + De_sp);
else
M1s(idx,idx-1) = alpha_s * De_s(idx-1);
M1s(idx,idx) = alpha_s * -(De_s(idx-1) + De_s(idx+1));
M1s(idx,idx+1) = alpha_s * De_s(idx+1);
end
end
M1p = zeros(p.Nxp-1);
for idx = 1:p.Nxp-1
if(idx == 1)
M1p(idx,idx) = alpha_p * -(De_sp + De_p(idx+1));
M1p(idx,idx+1) = alpha_p * De_p(idx+1);
elseif(idx == p.Nxp-1)
M1p(idx,idx-1) = alpha_p * De_p(idx-1);
M1p(idx,idx) = alpha_p * -(De_p(idx-1) + DeN);
else
M1p(idx,idx-1) = alpha_p * De_p(idx-1);
M1p(idx,idx) = alpha_p * -(De_p(idx-1) + De_p(idx+1));
M1p(idx,idx+1) = alpha_p * De_p(idx+1);
end
end
rs = [p.Nxn-1; p.Nxs-1; p.Nxp-1];
cs = rs';
M1 = sparse(blkdiagFast(rs,cs,M1n,M1s,M1p));
% M2 : c_e z
M2n = zeros(p.Nxn-1,2);
M2n(1,1) = alpha_n * De0;
M2n(end,end) = alpha_n * De_ns;
M2s = zeros(p.Nxs-1,2);
M2s(1,1) = alpha_s * De_ns;
M2s(end,end) = alpha_s * De_sp;
M2p = zeros(p.Nxp-1,2);
M2p(1,1) = alpha_p * De_sp;
M2p(end,end) = alpha_p * DeN;
M2 = [M2n, zeros(p.Nxn-1,2); ...
zeros(p.Nxs-1,1), M2s, zeros(p.Nxs-1,1);...
zeros(p.Nxp-1,2), M2p];
M2 = sparse(M2);
% M3 : i_e
M3 = zeros(p.Nx-3,p.Nx+1);
for idx = 1:p.Nx-3
if(idx <= p.Nxn-1)
M3(idx,idx) = -beta_n;
M3(idx,idx+2) = beta_n;
elseif(idx <= p.Nxn+p.Nxs-2)
M3(idx,idx+1) = -beta_s;
M3(idx,idx+3) = beta_s;
else
M3(idx,idx+2) = -beta_p;
M3(idx,idx+4) = beta_p;
end
end
%% Boundary Conditions
N1 = zeros(4,p.Nx-3);
N2 = zeros(4);
N1_BC_n=zeros(4,p.Nx-3); %The normalized impact on the boundary condition is the same for De and \varepsilon (since they enter at the same point)
N1_BC_p=zeros(4,p.Nx-3);
N1_BC_s=zeros(4,p.Nx-3);
% BC1
N1(1,1) = 1/(p.L_n * p.delta_x_n);
N2(1,1) = -1/(p.L_n * p.delta_x_n);
% BC2
N1(2,p.Nxn-1) = p.epsilon_e_n/(p.L_n * p.delta_x_n);
N1(2,p.Nxn) = p.epsilon_e_s/(p.L_s * p.delta_x_s);
N2(2,2) = -p.epsilon_e_n/(p.L_n * p.delta_x_n) - p.epsilon_e_s/(p.L_s * p.delta_x_s);
N1_BC_n(2,p.Nxn-2)=-p.epsilon_e_n/(p.L_n * p.delta_x_n);
N1_BC_n(2,p.Nxn-1)=p.epsilon_e_n/(p.L_n * p.delta_x_n);
N1_BC_s(2,p.Nxn)=p.epsilon_e_n/(p.L_n * p.delta_x_n);
N1_BC_s(2,p.Nxn+1)=-p.epsilon_e_n/(p.L_n * p.delta_x_n);
% BC3
N1(3,p.Nxn+p.Nxs-2) = p.epsilon_e_s/(p.L_s * p.delta_x_s);
N1(3,p.Nxn+p.Nxs-1) = p.epsilon_e_p/(p.L_p * p.delta_x_p);
N2(3,3) = -p.epsilon_e_s/(p.L_s * p.delta_x_s) - p.epsilon_e_p/(p.L_p * p.delta_x_p);
N1_BC_p(3,p.Nxn+p.Nxs-1)=-p.epsilon_e_p/(p.L_p * p.delta_x_p);
N1_BC_p(3,p.Nxn+p.Nxs)=p.epsilon_e_p/(p.L_p * p.delta_x_p);
N1_BC_s(3,p.Nxn+p.Nxs-3)=p.epsilon_e_p/(p.L_p * p.delta_x_p);
N1_BC_s(3,p.Nxn+p.Nxs-2)=-p.epsilon_e_p/(p.L_p * p.delta_x_p);
% BC4
N1(4,end) = -1/(p.L_p * p.delta_x_p);
N2(4,4) = 1/(p.L_p * p.delta_x_p);
%% A,B Matrics
A = sparse(M1 - M2*(N2\N1));
B = sparse(M3);
C = sparse(-N2\N1);
A_corr_sens_p = sparse(-M2*(N2\N1_BC_p));
A_corr_sens_n = sparse(-M2*(N2\N1_BC_n));
A_corr_sens_s = sparse(-M2*(N2\N1_BC_s));