README.md

# ME3255_FInalProject
I noticed that I was never assigned to a group so I did this project solo
## Part A

```
function [w] = membrane_solution3(T,P)


    od = ones(8,1);
    od(3:3:end) = 0;
    a = -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 = a\y;

grid on
    [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')
    xlabel('X (\muM)')
    ylabel('Y (\muM)')
    zlabel('Displacement (\muM)')
    colormap jet
    shading interp
    disp(w)

end
```

## Part B

```
membrane_solution3(0.006,0.001)
```
![](assets/README-20ed9cd0.png)

## Part C

```
function [w] = membrane_solution3(T,P)


    od = ones(8,1);
    od(3:3:end) = 0;
    a = -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 = a\y;

grid on
    [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')
    xlabel('X (\muM)')
    ylabel('Y (\muM)')
    zlabel('Displacement (\muM)')
    colormap jet
    shading interp
    disp(w)

end
```

## Part D

```
membrane_solution(0.006,0.001,10)
```
![](assets/README-579fb915.png)

## Part E
```
function [pw_se,w]=SE_diff(T,P,n)

E = 1e6;
v = .31;
t = .0003;
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]);
nt = n + 1;
wn = zeros(nt);
for i = 1:nt
    for j = 1:nt
        wn(i,j) = mean([z(i,j),z(i+1,j),z(i,j+1),z(i+1,j+1)]);
    end
end
pw = sum(sum(wn.*h^2.*P));
dwdx = zeros(nt);
dwdy = zeros(nt);
for i = 1:nt
    for j = 1:nt
        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
```
function [root,fx,ea,iter]=bisect(func,xl,xu,es,maxit,varargin)
if nargin<3,error('at least 3 input arguments required'),end
test = func(xl,varargin{:})*func(xu,varargin{:});
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)
  xr_old = xr;
  xr = (xl + xu)/2;
  iter = iter + 1;
  if xr ~= 0,ea = abs((xr - xr_old)/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{:});
```
```
function [T,ea] = tension_sol(P,n)
y =@(T) SE_diff(T,P,n);
[T,fx,ea,iter]=bisect(y,.01,1);
```
```
n=[3,20:5:40];
P=0.001;
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
 ```

 |number of nodes|Tension (uN/um)| rel. error|
|---|---|---|
|3 |0.049 |n/a|
|20|0.0599|20.5%|
|25|0.0601|0.27%|
|30|0.0602|0.16%|
|35|0.0602|0.09%|
|40|0.0603|0.05%|

## Part G
```
No = linspace(.001,.01,10);
n = 20;

T = zeros(1,length(No));
wmax = zeros(1,length(No));

for i = 1:length(No)
    T(i) = tension_sol(No(i),n);
    w = membrane_solution(T(i),No(i),n);
    wmax(i) = max(w);
end

clf
x = wmax';
y = No';
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 (\muM)')
ylabel('Pressure (MPa)')
```
![](assets/README-bbfc19b7.png)