README.md

# ME3255_Final_Project

# Part A
```matlab
function [w] = membrane_solution3(T,P)
% membrane_solution3: dispalacement of node for membrane with 3x3 interior
% nodes
% [w] = membrane_solution3(T,P)
% input:
% T = Tension (microNewton/micrometer)
% P = Pressure (MPa)
% output:
% w = vector of displacement of interior nodes
    
    od = ones(8,1);
    od(3:3:end) = 0;
    k = -4*diag(ones(9,1))+diag(ones(9-3,1),3)+diag(ones(9-3,1),-3)+diag(od,1)+diag(od,-1);
   
    
    y = -(10/4)^2*(P/T)*ones(9,1);
    w = k\y; 
    % Solves for displacement (micrometers)
    % Output w is a vector
    % Solution represents a 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)
    title('Membrane Displacement')
    zlabel('Displacement (micrometer)')
    % Membrane displacement is shown on chart
    
end 
```


# Part B
```matlab
% Part B Script
[w] = membrane_solution3(0.006,0.001);
```
![](https://github.uconn.edu/ltd13002/ME3255_Final_Project/blob/master/Part%20B/PartBFigure.png)


# Part C
```matlab
function [w] = membrane_solution(T,P,n)
% membrane_solution: dispalacement of node for membrane with nxn interior nodes
% [w] = membrane_solution(T,P,n)
% input:
% T = Tension (microNewton/micrometer)
% P = Pressure (MPa)
% n = number of rows and columns of interior nodes
% output:
% w = vector of displacement 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;
    % Solves for displacement (micrometers)
    % Output w is a vector
    % Solution represents a 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]);
    surf(x,y,z)
    title('Membrane Displacement')
    zlabel('Displacement (micrometer)')
    % Membrane displacement is shown on chart
end
```


# Part D
```matlab
% Part D Script
[w] = membrane_solution(0.006,0.001,10)
```
![](https://github.uconn.edu/ltd13002/ME3255_Final_Project/blob/master/Part%20D/PartDFigure.png)


# Part E
```matlab
function [pw_se,w]=SE_diff(T,P,n)
% SE_diff: calculates difference between strain energy and work done by pressure in
% membrane
% [pw_se,w]=SE_diff(T,P,n)
% input:
% T = Tension (microNewton/micrometer)
% P = Pressure (MPa)
% n = number of rows and columns of interior nodes
% output:
% pw_se = difference between strain energy and work done by pressure in
% membrane
% w = vector of displacement of interior nodes

E = 1; %TPa Units may need to be changed
v = .31; %Poissons ratio
t = .3; %nm
h = 10/(n+1); %nm
w = membrane_solution(T,P,n);
z = zeros(n+2);
z(2:end-1,2:end-1) = reshape(w,[n n]);
num = n + 1;
wbar = zeros(num);
for i = 1:num
    for j = 1:num
        wbar(i,j) = mean([z(i,j),z(i+1,j),z(i,j+1),z(i+1,j+1)]);
    end
end
pw = sum(sum(wbar.*h^2.*P));
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(0.25.*dwdx.^4+.25.*dwdy.^4+0.5.*(dwdx.*dwdy).^2));
pw_se = pw-se;
```

# Part F
```matlab
 n=[3,20:5:40];
 P=0.001; %MPa
 T = zeros(1,length(n));
 ea = zeros(1,length(n));
 for i = 1:length(n)

[T(i), ea(i)] = tension_sol(P,n(i));
 end
 ```
 ```matlab
function [T,ea] = tension_sol(P,n)
% tension_sol: outputs tension of a membrane given the pressure and number
% of nodes
% [T,ea] = tension_sol(P,n)
% input:
% P = Pressure (MPa)
% n = number of rows and columns of interior nodes
% output:
% T = Tension (microNewton/micrometer)
% ea = approximate relative error (%)
 
y =@(T) SE_diff(T,P,n);
[T,fx,ea,iter]=bisect(y,.01,1);
```
```matlab
function [root,fx,ea,iter]=bisect(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{:});
```
```matlab
function re = Rel_error (T)
 Rel_error: calculates relative error of a vector
% re = Rel_error (T)
% input:
% T = vector of numbers
% output:
% re = relative error of vector

re = zeros(1,length(T)-1);
for i = 2:length(T)
    re(i-1)= abs(T(i)-T(i-1))/T(i-1);
end
```
|number of nodes|Tension (uN/um)| rel. error|
|---|---|---|
|3 |0.0489 |n/a|
|20|0.0599|22.6%|
|25|0.0601|0.27%|
|30|0.0602|0.15%|
|35|0.0602|0.09%|
|40|0.0603|0.06%|


# Part G
```matlab
P = linspace(.001,.01,10);
n = 20;
T = zeros(1,length(P));
wmax = zeros(1,length(P));
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
clf
setDefaults
x = wmax';
y = P';
Z=x.^3;
a=Z\y;
x_fcn=linspace(min(x),max(x));
plot(x,y,'o',x_fcn,a*x_fcn.^3)
title('Pressure vs Maximum Deflection')
xlabel('Maximum Deflection (um)')
ylabel('Pressure (MPa)')
print('Part g','-dpng')
```
![](https://github.uconn.edu/ltd13002/ME3255_Final_Project/blob/master/Part%20G/Part%20g.png)