use finite difference and central differences to solve plate deflection in matlab

main.m
% Plate dimensions
Lx = 10; % length in x-direction
Ly = 10; % length in y-direction

% Number of grid points
nx = 100; % number of points in x-direction
ny = 100; % number of points in y-direction

% Grid spacing
dx = Lx / (nx - 1);
dy = Ly / (ny - 1);

% Generate grid
x = linspace(0, Lx, nx);
y = linspace(0, Ly, ny);
[X, Y] = meshgrid(x, y);

% Plate properties
E = 1; % Young's modulus
h = 1; % thickness

% Apply external load (just an example)
q = 1; % load per unit area

% Discretize plate equation using central differences
K = (E * h^3 / (12 * (1 - 0.3^2))) * ((dx^2 * dy^2) / (dx^2 + dy^2));
D = zeros(nx, ny);

for i = 2:nx-1
    for j = 2:ny-1
        D(i, j) = K * (X(i+1, j) - 2*X(i, j) + X(i-1, j)) / dx^2 + K * (Y(i, j+1) - 2*Y(i, j) + Y(i, j-1)) / dy^2 + q;
    end
end

% Solve for deflection
U = D;

% Plot the deflected shape
figure;
surf(X, Y, U)
xlabel('x')
ylabel('y')
zlabel('Deflection')
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