README.md

# ME 3255 Final Project
## Designing a graphene pressure sensor

[grading rubric](./rubric.md)

![Graphene membrane under
pressure](https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiG_prg5c3XAhULSyYKHQJWCPAQjRwIBw&url=https%3A%2F%2Fwww.theverge.com%2F2016%2F11%2F24%2F13740946%2Fdutch-scientists-use-color-changing-graphene-bubbles-to-create-mechanical-pixels&psig=AOvVaw1kho8L-n3x-KizdpV1bWrJ&ust=1511289329303919)

In this final project, you and your team mates will design a graphene pressure sensor.
Graphene is a single-layer of carbon atoms arranged in a hexagonal lattice. Graphene films
have a thickness of 0.3 nm, Young's modulus of 1 TPa, and Poisson's ratio of 0.31. The
film has no bending stiffness so the shape of the membrane under a given pressure, P, with
a pretension, T is given by:

![eq1](./equations/eq1.png) (1)

This equation assumes you know both the applied pressure and tension in the membrane, but
the tension will be a function of the applied pressure, assuming there is no initial
pretension. We will use an energy balance to solve the nonlinear relation between tension
and pressure. 

![eq2](./equations/eq2.png) (2)

![eq3](./equations/eq3.png) (3)

Where w(x,y) is the displacement of the membrane at a given point, (x,y), and the integral
is over the entire of area of the membrane, A. We can rewrite this exact integral as a
summation over the elements, as such.

![eq4](./equations/eq4.png) (4)

Where, ![eq5](./equations/eq5.png) is the average w of the four nodes in the element, and
the summation goes from 1 to the number of elements, nel. 

In this design problem, we will solve for
the tension and deflection of a square membrane, as seen in figure 2. 

![Figure 2. Square membrane design with sides of 10 um.](./figures/membrane.png)

*Figure 2*. Square membrane design with sides of 10 um. 2A demonstrates the node and element
numbering for 9 interior nodes and 16 elements. 2B shows element 6 and its nodal
displacments along with the aveerage displacement as the red point. 2C shows a
cross-section of the force-balance for the square membrane window and 2D shows an
isometric view of the membrane displacement solution and force-balance. 

**a\.** Create a central finite difference approximation of the gradient with 3-by-3 interior nodes of w
for the given membrane solution in terms of P and T. `[w]=membrane_solution3(T,P);` The
output `w` should be a vector, but the solution represents a 2D data set w(x,y). 

**b\.** Solve for w given a pressure, P=0.001 MPa and tension, T=0.006 uN. Plot the result with
`surf(X,Y,W)` where X, Y, and W are the x-, y-, and z-coordinates of each point on the
membrane from 0-10um. 

**c\.** Create a general central finite difference approximation of the gradient with
n-by-n interior nodes of w
for the given membrane solution in terms of P and T. `[w]=membrane_solution(T,P,n);` The
output `w` should be a vector, but the solution represents a 2D data set w(x,y). 

**d\.** Solve for w given a pressure, P=0.001 MPa and tension, T=0.006 uN with 10 interior
nodes. Plot the result with `surf(X,Y,W)` where X, Y, and W are the x-, y-, and
z-coordinates of each point on the membrane from 0-10um. Include the graph in your README.

**e\.** Create a function `SE_diff` that calculates the difference in strain energy (right hand side Eq.
4) and work done by pressure (left hand side Eq. 4) for n-by-n elements. 

`[pw_se,w]=SE_diff(T,P,n)`

Use the solution from part **c** to calculate w, then do a numerical integral over the
elements to calculate work done and strain energy.

**f\.** Use a root-finding method to calculate the tension in the membrane given a
pressure, P=0.001 MPa, and n=[20:5:40] interior nodes. 

Show that the error in tension is decreasing with a table:

```
|number of nodes|Tension (uN)| rel. error|
|---|---|---|
|20|0.08| n/a|
|25|0.006|0.5%|
|30|0.006|0.3%|
|40|0.006|0.2%|
```


**g\.** Plot the Pressure vs maximum deflection (P (y-axis) vs max(w) (x-axis)) for
P=linspace(0.001,0.01,10). Use a root-finding method to determine tension, T, at each
pressure. Use a cubic best-fit to find A, where, P(x)=A*dw^3. State how
many interior nodes were used for the graph. Plot the data and best-fit curve in your
README.

**h\.[Bonus 5 pts]** Show that the constant A is converging as the number of nodes is
increased (Similar table to **f**). 

**i\.[Bonus 10 pts]** If the square membrane sides are always equal, but have a tolerance of 0.1\%, what
should the depth of the sensor be if 2.5% of the sensors won't hit the bottom given a
maximum pressure of 0.01 MPa.