force_sph_sph_ext.m 4.66 KB
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%function [] = force_cube_cube_ext(N)

close all; clc; clear;
addpath(genpath("../../"));

% Initializing parameters for the problem
N = 30;
% Radius of spheres
Rad = 10;
% distance between the centers
% separation
s = 1e-6 * Rad;
dist = 2*Rad + s;

% Mesh for the geometry
mesh_in = mshSphere(N,Rad);
mesh_out = mesh_in;
% Translate the outer cube
N_vtcs = size(mesh_out.vtx,1);
trans = [dist * ones(N_vtcs,1), zeros(N_vtcs,1), zeros(N_vtcs,1)];
mesh_out.vtx = mesh_out.vtx + trans;
% Join to create the final mesh
mesh = union(mesh_in,mesh_out);

figure;
plot(mesh); 
title('Mesh');
%%

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

Op_in = restriction(S0_Gamma,mesh_in);
Op_out = restriction(S0_Gamma,mesh_out);

% 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
% Cutoff radius for the BC
%R = 1; 
R = dist/2;
%assert(R < Rad + s);
Vin = 10;
Vout = 10;
g = @(X) (sqrt(sum(X.^2,2)) > R)* Vout + (sqrt(sum(X.^2,2)) <= R) * Vin;

% Checking the boundary condition
figure;
plot(mesh);
hold on;
PI_g = g(S0_Gamma.dof);
plot(mesh,PI_g);
title('Boundary condition');
colorbar;

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

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

Psi_in = Op_in * Psi;
Psi_out = Op_out * Psi;

% Solving the adjoint problem to get the adjoint solution
Rho = V\(-0.5 * 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;

%%
% Classical formula for force evaluation
normals_in = mesh_in.nrm;
classical_force_in = 0.5 * sum( mesh_in.ndv .* Psi_in.^2.* normals_in, 1)

% Alternative approach to calculating the integral shown above?

charge_in = sum( mesh_in.ndv .* Psi_in, 1)

normals_out = mesh_out.nrm;

classical_force_out = 0.5 * sum( mesh_out.ndv .* Psi_out.^2.* normals_out, 1)
charge_out = sum( mesh_out.ndv .* Psi_out, 1)

% Analytical formula for s -> 0
%force_exact = force_spheres(dist,Rad,1,Vin-Vout)
coulomb_force = charge_in * charge_out / 4 / pi / dist^2

force_unit = charge_in * charge_out / 4 / pi / Rad^2;

force_ratio_unit = norm(classical_force_in)/force_unit

force_ratio = norm(classical_force_in)/coulomb_force

force_ratio_analytic = force_identical_spheres(Rad/dist)

%%
% Choice of velocity field for force computation
% Input = N X 3, output = N X 3

% Velocity fields for the far away sphere
%Nux = @(X) (sum(X.*X,2)>R*R).*[X(:,1)==X(:,1), 0 * X(:,1) , 0 * X(:,1)];
%Nuy = @(X) (sum(X.*X,2)>R*R).*[0*X(:,1), X(:,2)==X(:,2) , 0 * X(:,1)];
%Nuz = @(X) (sum(X.*X,2)>R*R).*[0* X(:,1), 0 * X(:,1) , X(:,3)==X(:,3)];

% Velocity fields for the near sphere
Nux = @(X) (sum(X.*X,2)<R*R).*[X(:,1)==X(:,1), 0 * X(:,1) , 0 * X(:,1)];
Nuy = @(X) (sum(X.*X,2)<R*R).*[0*X(:,1), X(:,2)==X(:,2) , 0 * X(:,1)];
Nuz = @(X) (sum(X.*X,2)<R*R).*[0* X(:,1), 0 * X(:,1) , X(:,3)==X(:,3)];

% Visualizing the velocity fields
figure;
plot(mesh);
hold on;
vtcs = S0_Gamma.dof;
vels = Nux(vtcs);
quiver3(vtcs(:,1),vtcs(:,2),vtcs(:,3),vels(:,1),vels(:,2),vels(:,3));
title('Perturbation field');


% Definition of the kernel for T2
kernelx = @(x,y,z) sum(z.*(Nux(x) - Nux(y)), 2)./(sqrt(sum(z.^2,2)).^3)/ (4*pi);

t2matx = panel_oriented_assembly(mesh,kernelx,S0_Gamma,S0_Gamma);

sum(sum(t2matx))

forcex = dot(Psi,t2matx * Rho)
%forcevals = [forcevals; force];

%end