membrane_solution.m

function [w] = membrane_solution(T,P,n)
  % General Central finite difference approximation of
  % gradient with n-by-n interior nodes in terms of P & T.
  % Input:
  % T = tension per unit length (uN/um)
  % P = pressure (MPa)
  % n = # of interior node rows/columns
  % Output:
  % w = displacement vector for interior nodes

  od = ones(n^2-1,1);
  od(n:n:end) = 0;
  k = -4 * diag(ones(n^2,1)) + diag(ones((n^2)-n,1),n) + diag(ones((n^2)-n,1),-n) + diag(od,1) + diag(od,-1);

  y = -(10/(n+1))^2*(P/T)*ones(n^2,1);
  w = k\y;

  % Find displacement vector w
  % which represents 2D data set w(x,y)
  [x,y] = meshgrid(0:10/(n+1):10,0:10/(n+1):10);
  z = zeros(size(x));
  z(2:end-1,2:end-1) = reshape(w,[n n]);

  % Plot gradient of membrane displacement
  surf(x,y,z)
  title('Gradient of Membrane Displacement')
  zlabel('Displacement [um] (micrometers)')
end