sphere_torus_exterior_laplace.m 2.99 KB
Newer Older
ppanchal's avatar
ppanchal committed
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
function [l2_err,av_err] = sphere_torus_exterior_laplace(N)
addpath(genpath("../../"));

% Initializing parameters for the problem
%N = 100;

% Radius of the sphere
Rad = 5;

% Radii for the torus
r1 = 5;
r2 = 2;

% Mesh for the geometry
mesh_out = mshSphere(N,Rad);
mesh_in = mshTorus(N,r1,r2);

% Rotate the inner torus
rng(10);
A = rand(3,3);
[Q,R] = qr(A);
mesh_in.vtx = mesh_in.vtx * Q;

% Translate the torus
tx = 15;
ty = 0;
tz = 0;
dist = norm([tx ty tz]);
N_vtcs = size(mesh_in.vtx,1);
trans = ones(N_vtcs,1) * [tx ty tz];
mesh_in.vtx = mesh_in.vtx + trans;

% Join to create the final mesh
mesh = union(mesh_in,mesh_out);

figure;
plot(mesh);
title('Mesh and normals');
hold on;
% Checking the normal direction for the mesh
ctrs = mesh.ctr;
nrms = mesh.nrm;
quiver3(ctrs(:,1),ctrs(:,2),ctrs(:,3),nrms(:,1),nrms(:,2),nrms(:,3));
scatter3(0,0,0);

%% BEM 

% Definition of FEM spaces and integration rule
S1_Gamma = fem(mesh,'P1');
S0_Gamma = fem(mesh,'P0');
order = 3;
Gamma = dom(mesh,order);

% Solving a Direct first kind BVP to get the representation of the state
% V Psi = (0.5*M+K) g_N (Interior problem)
% V Psi = (-0.5*M+K) g_N (Exterior problem)

% Getting the Single Layer matrix V using Gypsilab implementation
Gxy = @(X,Y)femGreenKernel(X,Y,'[1/r]',0); % 0 wave number
V = 1/(4*pi)*integral(Gamma,Gamma,S0_Gamma,Gxy,S0_Gamma);
V = V + 1/(4*pi)*regularize(Gamma,Gamma,S0_Gamma,'[1/r]',S0_Gamma);

% Getting the Double Layer matrix K using Gypsilab implementation
GradG = cell(3,1);
GradG{1} = @(X,Y)femGreenKernel(X,Y,'grady[1/r]1',0);
GradG{2} = @(X,Y)femGreenKernel(X,Y,'grady[1/r]2',0);
GradG{3} = @(X,Y)femGreenKernel(X,Y,'grady[1/r]3',0);
K = 1/(4*pi)*integral(Gamma,Gamma,S0_Gamma,GradG,ntimes(S0_Gamma));
K = K +1/(4*pi)*regularize(Gamma,Gamma,S0_Gamma,'grady[1/r]',ntimes(S0_Gamma));

% Defining the mass matrix M
M = integral(Gamma,S0_Gamma,S0_Gamma);

% Defining the Dirichlet boundary condition
g = @(x) 1/4/pi./vecnorm(x,2,2); % Point charge at origin

figure;
plot(mesh);
hold on;
plot(mesh,g(S0_Gamma.dof));
title('Dirichlet Boundary Condition');
colorbar;

% Constructing the RHS
g_N = integral(Gamma,S0_Gamma,g);

% Exterior problem
Psi = V\((-0.5 * M + K)* (M\g_N));

% Visualizing the Neumann trace
dofs = S0_Gamma.dof;
normals = mesh.nrm;
sPsi = -2*Psi;
figure;
plot(mesh);
hold on;
quiver3(dofs(:,1),dofs(:,2),dofs(:,3),normals(:,1).*sPsi,normals(:,2).*sPsi,normals(:,3).*sPsi,0);
title('Field on the surface');
%quiver3(dofs(:,1),dofs(:,2),dofs(:,3),normals(:,1),normals(:,2),normals(:,3));

% Visualizing the charge density on the surface of the objects
figure;
plot(mesh);
hold on;
plot(mesh,Psi);
title("Surface charge density");
colorbar;

%% Exact solution
Psi_exact = -1/4/pi./vecnorm(dofs,2,2).^3 .* (dot(dofs,nrms,2));

% Visualizing the exactness
figure;
plot(Psi./Psi_exact);

err_vec = Psi-Psi_exact;

% L2 Error
l2_err = norm(err_vec)

av_err = err_vec' * V * err_vec

close all;
end