Merge branch 'master' of https://github.uconn.edu/cpl11001/07_final_p…

…roject

README.md

@@ -68,3 +68,90 @@ Solve for the vector w using the function from part C with a P value of 0.001, T
 ```
 
 ![nxn Interior Nodes, n = 10](./figures/figure2.png)
+
+### Part E
+``` matlab
+function [pw_se,w] = SE_diff(T,P,n)
+
+    % Call function from part c to obtain vector w
+    w = membrane_solution(T,P,n);
+    % Create All Nodes Matrix (Interior & Exterior)
+    m = zeros(n+2);
+    c = vec2mat(w,n); % Vector w rearranged to n x n matrix.
+    m(2:n+1,2:n+1) = c; % insert matrix c into middle of m matrix
+    % m = symmetric matrix
+
+    h = 10/(n+1); % h = delta_x = delta_y (micron)
+
+    % Compute w bar
+    w_bar = zeros(n+1,n+1);
+    for i = 1:n+1
+        for j = 1:n+1
+            w_bar(i,j) = (m(i,j)+m(i,j+1)+m(i+1,j)+m(i+1,j+1))/4;
+        end
+    end
+
+    % Left side of the equation
+    left_side = zeros(n+1,n+1);
+    for  i = 1:(n+1)^2
+        left_side(i) = P * h^2*w_bar(i);
+    end
+    L = sum(sum(left_side));
+
+    % Compute dw_dx
+    dx = zeros(n+2,n+1);
+    for i = 2:n+2
+        dx(:,i-1)= (m(:,i)-m(:,i-1))/(h);
+    end
+
+    dw_dx = zeros(n+1,n+1);
+    for i = 1:n+1 % rows
+        for j = 1:n+1 % columns
+        dw_dx(i,j) = (dx(i,j)+dx(i+1,j))/2;
+        end
+    end
+
+    % Compute dw_dy
+    dy = zeros(n+1,n+2);
+    for i = 2:n+2
+        dy(i-1,:)= (m(i-1,:)-m(i,:))/(h);
+    end
+
+    dw_dy = zeros(n+1,n+1);
+    for i = 1:n+1 % rows
+        for j = 1:n+1 % columns
+        dw_dy(i,j) = (dy(i,j)+dy(i,j+1))/2;
+        end
+    end
+
+    right_side = zeros(n+1,n+1);
+
+    for i = 1:(n+1)^2
+        right_side(i) = (0.25*(dw_dx(i))^4+0.25*(dw_dy(i))^4+0.5*(dw_dx(i)*dw_dy(i))^2);
+    end
+    R = sum(sum(right_side));
+    R = (1000*10^3*0.3*10^-3*h^2)/(2*(1-0.31^2))*R;
+
+    pw_se = R - L;
+end
+```
+
+- At 10 nodes, pw_se = 53.4902
+
+### Part F
+
+``` matlab
+[pw_se,w] = SE_diff(T,P,n);
+[root,fx,ea,iter] = bisect_final_project(@(T) SE_diff(T,0.001,25),.001,1,.1)
+```
+
+The same code was run five times, changing the value of n from 20 to 40 for intervals of 5.
+
+| number of nodes | Tension (uN/um) | rel. error|
+| --- | --- | --- |
+|   3  |  0.0489  |  n/a    |
+|  20  |  0.0599  |  18.4%  |
+|  25  |  0.0601  |  0.30%  |
+|  30  |  0.0602  |  0.17%  |
+|  35  |  0.0603  |  0.16%  |
+|  40  |  0.0603  |  0.00%  |  

bisect_final_project.m~

@@ -0,0 +1,37 @@
+function [root,fx,ea,iter]=bisect_final_project(func,xl,xu,es,maxit,varargin)
+% bisect: root location zeroes
+% [root,fx,ea,iter]=bisect(func,xl,xu,es,maxit,p1,p2,...):
+% uses bisection method to find the root of func
+% input:
+% func = name of function
+% xl, xu = lower and upper guesses
+% es = desired relative error (default = 0.0001%)
+% maxit = maximum allowable iterations (default = 50)
+% p1,p2,... = additional parameters used by func
+% output:
+% root = real root
+% fx = function value at root
+% ea = approximate relative error (%)
+% iter = number of iterations
+if nargin<3,error('at least 3 input arguments required'),end
+test = func(xl,varargin{:})*func(xu,varargin{:});
+if test>0,error('no sign change'),end
+if nargin<4||isempty(es), es=0.0001;end
+if nargin<5||isempty(maxit), maxit=50;end
+iter = 0; xr = xl; ea = 100;
+while (1)
+  xrold = xr;
+  xr = (xl + xu)/2;
+  iter = iter + 1;
+  if xr ~= 0,ea = abs((xr - xrold)/xr) * 100;end
+  test = func(xl,varargin{:})*func(xr,varargin{:});
+  if test < 0
+    xu = xr;
+  elseif test > 0
+    xl = xr;
+  else
+    ea = 0;
+  end
+  if ea <= es || iter >= maxit,break,end
+end
+root = xr; fx = func(xr, varargin{:}); 

bisect_final_project_script.m

@@ -0,0 +1,2 @@
+[pw_se,w] = SE_diff(T,P,n);
+[root,fx,ea,iter] = bisect_final_project(@(T) SE_diff(T,0.001,40),.001,1,.1)

partg.m

@@ -0,0 +1,16 @@
+
+P = linspace(0.001,0.01,10);
+
+for i = 1:length(P)
+    func = @(T) SE_diff(T,P(i),10);
+    [root(i),fx,ea,iter] = bisect_final_project(func,0.001,1,.1);
+end
+
+for i = 1:length(root)
+    w = membrane_solution(root(i),P(i),10);
+    w1(:,i) = w; % displays w values as column
+end
+
+w_max = max(w1);
+v_cub = interp1(w_max,P,'pchip');
+plot(w_max,P)