README.md

# ME3255_Final_Project

# Part A
```matlab
function [w] = membrane_solution3(T,P)
    % T = Tension (microNewton/micrometer)
    % P = Pressure (MPa)
    
    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)
    % T = Tension (microNewton/micrometer)
    % 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;
    % 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)
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;
```