Parts A-E

README.md

@@ -44,7 +44,7 @@ Root-finding Method used to find tension in the Membrane, given:
 
 Error
 ```
-| number of nodes | Tension (uN/um) | rel. error |
+|number of nodes |Tension (uN/um) |rel. error |
 |---|---|---|
 |3 |0.059 |n/a|
 |20|0.0618|4.5%|
@@ -54,4 +54,4 @@ Error
 ```
 
 ## Part G
-Pressure v. Maximum deflection
+Pressure v. Maximum deflection (MPa v. um)

SE_diff.m

@@ -0,0 +1,33 @@
+function [pw_se,w] = SE_diff(T,P,n)
+  E = 1; % 1 TPa ~= 10^6 MPa
+  v = 0.31;
+  t = 3*10^-4;
+  h = 10/(n+1);
+
+  w = membrane_solution(T,P,n);
+
+  z = zeros(n + 2);
+  z(2:end-1,2:end-1) = reshape(w,[n n]);
+
+  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
+
+  pw = sum(sum(wavg.*h^2.))
+  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
+
+  se = (E*t*h^2)/(2*(1-v^2)) * sum(sum((1/4).*dwdx.^4+(1/4).*dwdy.^4+(1/4).*(dwdx.*dwdy).^2));
+
+  pw_se = pw - se;

membrane_solution.m

@@ -0,0 +1,28 @@
+function [w] = membrane_solution(T,P,n)
+  %
+  % T = tension per unit length (uN/um)
+  % P = pressure (MPa)
+  % n = # of 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;
+
+  %
+  [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]);
+
+  surf(x,y,z)
+
+  title('Gradient of Membrane Displacement')
+  zlabel('Displacement [um] (micrometers)')
+end

membrane_solution3.m

@@ -0,0 +1,25 @@
+function [w] = membrane_solution3(T,P)
+  % Central finite difference approximation of gradient
+  % with 3x3 (um^2) interior nodes in terms of P & T.
+  % T = tension per unit length (uN/um)
+  % P = pressure (MPa)
+
+  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;
+
+  %
+  [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)
+
+  title('Gradient of Membrane Displacement')
+  zlabel('Displacement [um] (micrometers)')
+end