Part A update

README.md

@@ -3,12 +3,38 @@
 ## Part A
 #### Problem Statement
 Create a central finite difference approximation of the gradient with 3-by-3 interior nodes of w
-for the given membrane solution in terms of P and T. `[w]=membrane_solution3(T,P);` The
-output `w` should be a vector, but the solution represents a 2D data set w(x,y).
+for the given membrane solution in terms of P and T. `[w]=membrane_solution3(T,P);`
+
 #### Approach
-Central finite difference approximation for gradient of 3x3 interior nodes of w.
 ```matlab
-[w] = membrane_solution3(T,P);
+function [w] = membrane_solution3(T,P)
+  % Central finite difference approximation of gradient
+  % with 3x3 (um^2) interior nodes in terms of P & T.
+  % Input:
+  % T = tension per unit length (uN/um)
+  % P = pressure (MPa)
+  % Output:
+  % w = displacement vector for interior nodes
+
+  od = ones(8,1);
+  od(3:3:end) = 0;
+  k = -4 * diag(ones((3^2),1)) + diag(ones((3^2)-3,1),3) + diag(ones((3^2)-3,1),-3) + diag(od,1) + diag(od,-1);
+
+  y = -(10/4)^2*(P/T)*ones(9,1);
+  w = k\y;
+
+  % Find displacement vector w
+  % which represents 2D data set w(x,y)
+  [x,y] = meshgrid(0:10/4:10,0:10/4:10);
+  z = zeros(size(x));
+  z(2:end-1,2:end-1) = reshape(w,[3 3]);
+
+  % Plot gradient of membrane displacement
+  surf(x,y,z)
+  title('Gradient of Membrane Displacement')
+  zlabel('Displacement [um] (micrometers)')
+end
+
 ```
 
 ## Part B
@@ -67,50 +93,50 @@ elements to calculate work done and strain energy.
 #### Approach
 ```matlab
 function [pw_se,w] = SE_diff(T,P,n)
-% function that calculates the difference between
-% strain energy and work done by pressure on the membrane.
-% Input:
-% T = tension per unit length (uN/um)
-% P = pressure (MPa)
-% n = # of interior nodes
-% Output:
-% pw_se = Absolute value of difference between strain energy and work done by pressure
-% w = displacement vector for interior nodes
-
-E = 1e6;        % 1 TPa ~= 10^6 MPa
-t = 3*10^-4;    % thickness [um]
-h = 10/(n+1);   % height [um]
-v = 0.31;       % Poisson's Ratio
-
-% Displacement vector w found using Part C
-w = membrane_solution(T,P,n);
-z = zeros(n + 2);
-z(2:end-1,2:end-1) = reshape(w,[n n]);
-
-% Calculate average displacement, wavg, for each element by taking the displacement at each
-% corner and then average the found values.
-num = n + 1;
-wavg = zeros(num);
-for i = 1:num
-  for j = 1:num
-    wavg(i,j) = mean([z(i,j),z(i+1,j),z(i,j+1),z(i+1,j+1)]);
-  end
-end
-
-% final work done by pressure
-pw = sum(sum(wavg.*h^2.*P))
-
-% to find= change in displacement, find the change in displacement on
-% the x-axis, dwdx, and the change in displacement on the y-axis, dwdy, and
-% average the found values.
-dwdx = zeros(num);
-dwdy = zeros(num);
-for i = 1:num
+  % function that calculates the difference between
+  % strain energy and work done by pressure on the membrane.
+  % Input:
+  % T = tension per unit length (uN/um)
+  % P = pressure (MPa)
+  % n = # of interior node rows/columns
+  % Output:
+  % pw_se = Absolute value of difference between strain energy and work done by pressure
+  % w = displacement vector for interior nodes
+
+  E = 1e6;        % 1 TPa ~= 10^6 MPa
+  t = 3*10^-4;    % thickness [um]
+  h = 10/(n+1);   % height [um]
+  v = 0.31;       % Poisson's Ratio
+
+  % Displacement vector w found using Part C
+  w = membrane_solution(T,P,n);
+  z = zeros(n + 2);
+  z(2:end-1,2:end-1) = reshape(w,[n n]);
+
+  % Calculate average displacement, wavg, for each element by taking the displacement at each
+  % corner and then average the found values.
+  num = n + 1;
+  wavg = zeros(num);
+  for i = 1:num
     for j = 1:num
-        dwdx(i,j) = mean([z(i+1,j)-z(i,j),z(i+1,j+1)-z(i,j+1)]);
-        dwdy(i,j) = mean([z(i,j+1)-z(i,j),z(i+1,j+1)-z(i+1,j)]);
+      wavg(i,j) = mean([z(i,j),z(i+1,j),z(i,j+1),z(i+1,j+1)]);
     end
-end
+  end
+
+  % final work done by pressure
+  pw = sum(sum(wavg.*h^2.*P))
+
+  % to find= change in displacement, find the change in displacement on
+  % the x-axis, dwdx, and the change in displacement on the y-axis, dwdy, and
+  % average the found values.
+  dwdx = zeros(num);
+  dwdy = zeros(num);
+  for i = 1:num
+      for j = 1:num
+          dwdx(i,j) = mean([z(i+1,j)-z(i,j),z(i+1,j+1)-z(i,j+1)]);
+          dwdy(i,j) = mean([z(i,j+1)-z(i,j),z(i+1,j+1)-z(i+1,j)]);
+      end
+  end
 
 % Using dwdx and dwdy, calculate the strain energy, se.
 se = (E*t*h^2)/(2*(1-v^2)) * sum(sum((1/4).*dwdx.^4+(1/4).*dwdy.^4+(1/4).*(dwdx.*dwdy).^2));
@@ -121,8 +147,7 @@ pw_se = pw - se;
 
 ## Part F
 #### Problem Statement
-Use a root-finding method to calculate the tension in the membrane given a
-pressure, P=0.001 MPa, and n=[20:5:40] interior nodes.
+Use a root-finding method to calculate the tension in the membrane given a pressure, P=0.001 MPa, and n=[20:5:40] interior nodes.
 
 Show that the error in tension is decreasing with a table:
 #### Approach

SE_diff.m

@@ -4,7 +4,7 @@
 % Input:
 % T = tension per unit length (uN/um)
 % P = pressure (MPa)
-% n = # of interior nodes
+% n = # of interior node rows/columns
 % Output:
 % pw_se = Absolute value of difference between strain energy and work done by pressure
 % w = displacement vector for interior nodes

membrane_solution3.m

@@ -1,25 +1,27 @@
 function [w] = membrane_solution3(T,P)
   % Central finite difference approximation of gradient
   % with 3x3 (um^2) interior nodes in terms of P & T.
+  % Input:
   % T = tension per unit length (uN/um)
   % P = pressure (MPa)
+  % Output:
+  % w = displacement vector for interior nodes
 
   od = ones(8,1);
   od(3:3:end) = 0;
   k = -4 * diag(ones((3^2),1)) + diag(ones((3^2)-3,1),3) + diag(ones((3^2)-3,1),-3) + diag(od,1) + diag(od,-1);
 
-  %
   y = -(10/4)^2*(P/T)*ones(9,1);
-  %
   w = k\y;
 
-  %
+  % Find displacement vector w
+  % which represents 2D data set w(x,y)
   [x,y] = meshgrid(0:10/4:10,0:10/4:10);
   z = zeros(size(x));
-  %
   z(2:end-1,2:end-1) = reshape(w,[3 3]);
-  surf(x,y,z)
 
+  % Plot gradient of membrane displacement
+  surf(x,y,z)
   title('Gradient of Membrane Displacement')
   zlabel('Displacement [um] (micrometers)')
 end