Radially Symmetric Heatmap in a Sphere

I'm doing a project on the neutron diffusion equation in a spherical coordinate system. I'm looking to make an animation of what is essentially just a heatmap showing heat (representing flux) radiating outward from the center of the sphere, and I would like to be able to change paramaters like the absorption cross-section and the fission cross-section to show differing intensities of diffusion. I'm pretty new to MatLab and I don't really know how I would go about doing this. Is this feasible to do in MatLab?

Answer 1

[Sanju]

Hi Tom,

I understand that you want to visualize heat diffusion in the spherical coordinate system,

Here's an outline on how to implement the same,

• Define the neutron diffusion equation in spherical coordinates, considering parameters like absorption and fission cross-sections.
• Generate a grid representing the spherical domain and initialize it with initial conditions and parameters.
• Iterate over time steps, solving the diffusion equation numerically for each step.
• Visualize the results as a heatmap using MATLAB's plotting functions, such as "surf".
• Implement sliders or input parameters to dynamically adjust absorption and fission cross-sections, allowing for the visualization of different diffusion intensities.

Here's an example implementation you may refer to,

% Define parameters
radius = 1; % Radius of the sphere
num_points = 100; % Number of points in each dimension
time_steps = 100; % Number of time steps

% Create spherical grid
theta = linspace(0, pi, num_points);
phi = linspace(0, 2*pi, num_points);
[Theta, Phi] = meshgrid(theta, phi);

% Initialize heat distribution
heat = zeros(num_points, num_points, time_steps);
heat(:,:,1) = sin(Theta) .* cos(Phi); % Example initial condition

% Define diffusion parameters
absorption_cross_section = 0.1; % Example absorption cross-section
fission_cross_section = 0.05; % Example fission cross-section

% Perform simulation
for t = 2:time_steps
    % Apply diffusion equation
    heat(:,:,t) = heat(:,:,t-1) + absorption_cross_section * heat(:,:,t-1) - fission_cross_section * heat(:,:,t-1);
end

% Visualize heat distribution
for t = 1:time_steps
    % Plot heatmap
    surf(Theta, Phi, heat(:,:,t));
    shading interp;
    colormap('hot');
    colorbar;
    title(['Heatmap at Time Step ', num2str(t)]);
    xlabel('\theta');
    ylabel('\phi');
    zlabel('Heat');
    pause(0.1); % Pause to visualize each time step
end

This code initializes a spherical grid, sets initial conditions for heat distribution, iterates over time steps to simulate heat diffusion, and visualizes the results as a heatmap. You can adjust parameters like absorption and fission cross-sections to observe different diffusion behaviors.

Note: You can modify the equations and code according to your specifications.

You can also refer to the following odcumentation links for more information,

https://www.mathworks.com/help/phased/ug/spherical-coordinates.html

https://www.mathworks.com/help/matlab/ref/surf.html

Hope this helps!