Merge branch 'master' of https://github.uconn.edu/ltd13002/ME3255_Fin…

…al_Project

Part C/membrane_solution.asv

@@ -21,7 +21,9 @@ w = k\y;
 z = zeros(size(x));
 z(2:end-1,2:end-1) = reshape(w,[n n]);
 surf(x,y,z)
-title('Membrane Displacement')
-zlabel('Displacement (micrometer)')
+title('Membrane Displacement (10 x 10)')
+zlabel('Z Position (micron)')
+ylabel('Y Position (micron)')
+xlabel('X Position (micron)')
 % Membrane displacement is shown on chart
-end
+end

README.md

@@ -36,7 +36,8 @@ w = k\y;
 % Membrane displacement is shown on chart
 
 end
-   
+    
+end 
 ```
 ### Approach
 - A matrix is set up using finite element analysis of the interior nodes of a 5x5 matrix
@@ -47,6 +48,9 @@ end
 # Part B
 ```matlab
 % Part B Script
+% From problem statement:
+% T=0.006 uN/um
+% P=0.001 MPa
 [w] = membrane_solution3(0.006,0.001);
 ```
 ![](https://github.uconn.edu/ltd13002/ME3255_Final_Project/blob/master/Part%20B/PartBFigure.png)
@@ -78,10 +82,8 @@ function [w] = membrane_solution(T,P,n)
     z = zeros(size(x));
     z(2:end-1,2:end-1) = reshape(w,[n n]);
     surf(x,y,z)
-    title('Membrane Displacement (10 x 10)')
-    zlabel('Z Position (micron)')
-    ylabel('Y Position (micron)')
-    xlabel('X Position (micron)')
+    title('Membrane Displacement')
+    zlabel('Displacement (micrometer)')
     % Membrane displacement is shown on chart
 end
 ```
@@ -92,6 +94,10 @@ end
 # Part D
 ```matlab
 % Part D Script
+% From problem statement:
+% T=0.006 uN/um
+% P=0.001 MPa
+% n = 10 nodes
 [w] = membrane_solution(0.006,0.001,10)
 ```
 ![](https://github.uconn.edu/ltd13002/ME3255_Final_Project/blob/master/Part%20D/PartDFigure.png)
@@ -147,12 +153,17 @@ pw_se = pw-se;
 
 # Part F
 ```matlab
+% Part F script
+% From Problem Statement:
  n=[3,20:5:40];
  P=0.001; %MPa
+ 
+ % Sets vector length so it doesn't change every iteration in for loop
  T = zeros(1,length(n));
  ea = zeros(1,length(n));
+ 
+ % Uses tension_sol function to solve for the tension for each input
  for i = 1:length(n)
-
 [T(i), ea(i)] = tension_sol(P,n(i));
  end
  ```
@@ -238,22 +249,34 @@ end
 - The script runs through all iterations of different amounts of nodes, zeroing the SE_diff function output and saving the values for tension in the T variable as a vector
 # Part G
 ```matlab
+% Part G Script
+% From Problem Statement
 P = linspace(.001,.01,10);
 n = 20;
+
+% Sets vector length so it doesn't change every iteration in for loop
 T = zeros(1,length(P));
 wmax = zeros(1,length(P));
+
+% Uses tension_sol and membrane_solution functions to solve for the maximum displacement for all iterations
 for i = 1:length(P)
     T(i) = tension_sol(P(i),n);
     w = membrane_solution(T(i),P(i),n);
     wmax(i) = max(w);
 end
+
+% Sets up figure for plot
 clf
 setDefaults
+
+% Standard Linear Regression with equation P(x) = A*w^2
 x = wmax';
 y = P';
 Z=x.^3;
 a=Z\y;
 x_fcn=linspace(min(x),max(x));
+
+% Plots regression line with actual points
 plot(x,y,'o',x_fcn,a*x_fcn.^3)
 title('Pressure vs Maximum Deflection')
 xlabel('Maximum Deflection (um)')
@@ -267,3 +290,44 @@ print('Part g','-dpng')
 - Additionally, the max deflection for each pressure and tension is calculated at each iteration
 - From there, a general linear regression is calculated with the formula P(x) = A*w^3
 - The results are plotted
+
+# Part H
+```matlab
+% Part G Script
+% From Problem Statement
+P=0.001;
+n = 5:5:50;
+
+% Sets vector length so it doesn't change every iteration in for loop
+a = zeros(1,length(n));
+
+% Uses tension_sol and membrane_solution functions along with a regression line to solve for the constant A in the regression line
+for i = 1:length(n)
+    T = tension_sol(P,n(i));
+    w = membrane_solution(T,P,n(i));
+    wmax = max(w);
+    x = wmax';
+    y = P';
+    Z = x.^3;
+    a(i) = Z\y;
+end
+
+% Calculates the relative error between the constant A values
+Rel_error(a)
+```
+|number of nodes|vaalue of const. A| rel. error|
+|---|---|---|
+|5 |0.4423 |n/a|
+|10|0.5389|21.84%|
+|15|0.5349|0.73%|
+|20|0.5483|2.5%|
+|25|0.5454|0.52%|
+|30|0.5503|0.89%|
+|35|0.5485|0.31%|
+|40|0.5510|0.45%|
+|45|0.5499|0.2%|
+
+### Approach
+- Part H uses the same basis as Part G, but with varrying n (number of nodes) valeus instead
+- Additionally, the outputs and a few calculations were deleted to speed up the process since they were not needed here
+- In the chart, while there are some fluctuations, there is a clear decrease in the relative error of A, leading to the assumption that it is converging as the number of nodes increases