From f88580cf1ad25454f209b5cf7b3700285abb2774 Mon Sep 17 00:00:00 2001
From: "Ryan C. Cooper" <ryan.c.cooper@uconn.edu>
Date: Mon, 20 Nov 2017 13:32:40 -0500
Subject: [PATCH] added final project

---
 .../17_integrals.ipynb                        |  241 +--
 final_project/README.md                       |   97 ++
 final_project/equations/eq1.png               |  Bin 0 -> 3906 bytes
 final_project/equations/eq1.tex               |    1 +
 final_project/equations/eq2.png               |  Bin 0 -> 4190 bytes
 final_project/equations/eq2.tex               |    1 +
 final_project/equations/eq3.png               |  Bin 0 -> 6760 bytes
 final_project/equations/eq3.tex               |    4 +
 final_project/equations/eq4.png               |  Bin 0 -> 6913 bytes
 final_project/equations/eq4.tex               |    4 +
 final_project/equations/eq5.png               |  Bin 0 -> 2881 bytes
 final_project/equations/eq5.tex               |    1 +
 final_project/figures/membrane.png            |  Bin 0 -> 24282 bytes
 final_project/figures/membrane.svg            | 1533 +++++++++++++++++
 14 files changed, 1729 insertions(+), 153 deletions(-)
 create mode 100644 final_project/README.md
 create mode 100644 final_project/equations/eq1.png
 create mode 100644 final_project/equations/eq1.tex
 create mode 100644 final_project/equations/eq2.png
 create mode 100644 final_project/equations/eq2.tex
 create mode 100644 final_project/equations/eq3.png
 create mode 100644 final_project/equations/eq3.tex
 create mode 100644 final_project/equations/eq4.png
 create mode 100644 final_project/equations/eq4.tex
 create mode 100644 final_project/equations/eq5.png
 create mode 100644 final_project/equations/eq5.tex
 create mode 100644 final_project/figures/membrane.png
 create mode 100644 final_project/figures/membrane.svg

diff --git a/17_integrals_and_derivatives/17_integrals.ipynb b/17_integrals_and_derivatives/17_integrals.ipynb
index 03cd66f..2618cea 100644
--- a/17_integrals_and_derivatives/17_integrals.ipynb
+++ b/17_integrals_and_derivatives/17_integrals.ipynb
@@ -4,9 +4,10 @@
    "cell_type": "code",
    "execution_count": 1,
    "metadata": {
-    "collapsed": true,
-    "slideshow": {
-     "slide_type": "skip"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "1"
     }
    },
    "outputs": [],
@@ -18,9 +19,10 @@
    "cell_type": "code",
    "execution_count": 2,
    "metadata": {
-    "collapsed": true,
-    "slideshow": {
-     "slide_type": "skip"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "2"
     }
    },
    "outputs": [],
@@ -30,42 +32,30 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "slide"
-    }
-   },
-   "source": [
-    "# Numerical Integrals"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "subslide"
-    }
-   },
+   "metadata": {},
    "source": [
+    "# Numerical Integrals\n",
+    "\n",
     "# Integrals in practice\n",
     "\n",
     "### Example: Compare toughness of Stainless steel to Structural steel\n",
     "\n",
     "![Stress-strain plot of steel](steel_psi.jpg)\n",
     "\n",
-    "### Step 1 - G3Data to get points \n",
+    "### Step 1 - G3Data to get points\n",
     "\n",
-    "Use the plot shown to determine the toughness of stainless steel and the toughness of structural steel.\n",
-    "\n"
+    "Use the plot shown to determine the toughness of stainless steel and the\n",
+    "toughness of structural steel."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 9,
    "metadata": {
-    "collapsed": false,
-    "slideshow": {
-     "slide_type": "slide"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "9"
     }
    },
    "outputs": [
@@ -91,30 +81,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "slide"
-    }
-   },
+   "metadata": {},
    "source": [
     "# Gauss Quadrature (for functions)\n",
     "\n",
-    "Evaluating an integral, we assumed a polynomial form for each Newton-Cotes approximation.\n",
+    "Evaluating an integral, we assumed a polynomial form for each Newton-Cotes\n",
+    "approximation.\n",
     "\n",
-    "If we can evaluate the function at any point, it makes more sense to choose points more wisely rather than just using endpoints\n",
+    "If we can evaluate the function at any point, it makes more sense to choose\n",
+    "points more wisely rather than just using endpoints\n",
     "\n",
-    "![trapezoidal example](trap_example.png)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "slide"
-    }
-   },
-   "source": [
-    "Set up two unknown constants, $c_{0}$ and $x_{0}$ and determine a *wise* place to evaluate f(x) such that \n",
+    "![trapezoidal example](trap_example.png)\n",
+    "\n",
+    "Set up two unknown constants, $c_{0}$ and $x_{0}$ and determine a *wise* place\n",
+    "to evaluate f(x) such that\n",
     "\n",
     "$I=c_{0}f(x_{0})$\n",
     "\n",
@@ -133,9 +113,10 @@
    "cell_type": "code",
    "execution_count": 10,
    "metadata": {
-    "collapsed": false,
-    "slideshow": {
-     "slide_type": "subslide"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "10"
     }
    },
    "outputs": [
@@ -321,9 +302,10 @@
    "cell_type": "code",
    "execution_count": 11,
    "metadata": {
-    "collapsed": false,
-    "slideshow": {
-     "slide_type": "slide"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "11"
     }
    },
    "outputs": [
@@ -348,13 +330,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "slide"
-    }
-   },
+   "metadata": {},
    "source": [
-    "This process is called **Gauss Quadrature**. Usually, the bounds are fixed at -1 and 1 instead of a and b\n",
+    "This process is called **Gauss Quadrature**. Usually, the bounds are fixed at -1\n",
+    "and 1 instead of a and b\n",
     "\n",
     "$I=c_{0}f(x_{0})$\n",
     "\n",
@@ -366,17 +345,8 @@
     "\n",
     "so $c_{0}=2$ and $x_{0}=0$\n",
     "\n",
-    "$I=\\int_{-1}^{1}f(x)dx \\approx 2f\\left(0\\right)$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "subslide"
-    }
-   },
-   "source": [
+    "$I=\\int_{-1}^{1}f(x)dx \\approx 2f\\left(0\\right)$\n",
+    "\n",
     "Now, integrals can be performed with a change of variable\n",
     "\n",
     "a=2\n",
@@ -391,18 +361,9 @@
     "\n",
     "at $x_{d}=-1$, x=a\n",
     "\n",
-    "at $x_{d}=1$, x=b"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "subslide"
-    }
-   },
-   "source": [
-    "so \n",
+    "at $x_{d}=1$, x=b\n",
+    "\n",
+    "so\n",
     "\n",
     "$x=\\frac{(b+a) +(b-a)x_{d}}{2}$\n",
     "\n",
@@ -414,27 +375,20 @@
     "\n",
     "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{5 +x_{d}}{2}\n",
     "+1\\right)\n",
-    "\\frac{3-2}{2}dx_{d}$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "fragment"
-    }
-   },
-   "source": [
-    "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{7}{4}+\\frac{1}{4}x_{d}\\right)dx_{d}=2\\frac{7}{4}=3.5$"
+    "\\frac{3-2}{2}dx_{d}$\n",
+    "\n",
+    "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{7}{4}+\\frac{1}{4}x_{d}\\right)dx_{d}=\n",
+    "2\\frac{7}{4}=3.5$"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 12,
    "metadata": {
-    "collapsed": true,
-    "slideshow": {
-     "slide_type": "slide"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "12"
     }
    },
    "outputs": [],
@@ -454,9 +408,10 @@
    "cell_type": "code",
    "execution_count": 14,
    "metadata": {
-    "collapsed": false,
-    "slideshow": {
-     "slide_type": "fragment"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "14"
     }
    },
    "outputs": [
@@ -474,53 +429,55 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "slide"
-    }
-   },
+   "metadata": {},
    "source": [
     "## General Gauss weights and points\n",
     "\n",
-    "![Gauss quadrature table](gauss_weights.png)"
+    "![Gauss quadrature table](gauss_weights.png)\n",
+    "\n",
+    "### If you need to evaluate an integral, to increase accuracy, increase number\n",
+    "of Gauss points\n",
+    "\n",
+    "### Adaptive Quadrature\n",
+    "\n",
+    "Matlab/Octave built-in functions use two types of adaptive quadrature to\n",
+    "increase accuracy of integrals of functions.\n",
+    "\n",
+    "1. `quad`: Simpson quadrature good for nonsmooth functions\n",
+    "\n",
+    "2. `quadl`: Lobatto quadrature good for smooth functions"
    ]
   },
   {
-   "cell_type": "markdown",
+   "cell_type": "code",
+   "execution_count": null,
    "metadata": {
-    "slideshow": {
-     "slide_type": "slide"
+    "attributes": {
+     "classes": [
+      "matlab"
+     ],
+     "id": ""
     }
    },
+   "outputs": [],
    "source": [
-    "### If you need to evaluate an integral, to increase accuracy, increase number of Gauss points\n",
-    "\n",
-    "### Adaptive Quadrature\n",
-    "\n",
-    "Matlab/Octave built-in functions use two types of adaptive quadrature to increase accuracy of integrals of functions. \n",
-    "\n",
-    "1. `quad`: Simpson quadrature good for nonsmooth functions\n",
-    "\n",
-    "2. `quadl`: Lobatto quadrature good for smooth functions\n",
-    "\n",
-    "```matlab\n",
     "q = quad(fun, a, b, tol, trace, p1, p2, …)\n",
     "fun : function to be integrates\n",
     "a, b: integration bounds\n",
     "tol: desired absolute tolerance (default: 10-6)\n",
     "trace: flag to display details or not\n",
     "p1, p2, …: extra parameters for fun\n",
-    "quadl has the same arguments\n",
-    "```"
+    "quadl has the same arguments"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 15,
    "metadata": {
-    "collapsed": false,
-    "slideshow": {
-     "slide_type": "subslide"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "15"
     }
    },
    "outputs": [
@@ -545,9 +502,10 @@
    "cell_type": "code",
    "execution_count": 16,
    "metadata": {
-    "collapsed": false,
-    "slideshow": {
-     "slide_type": "subslide"
+    "attributes": {
+     "classes": [],
+     "id": "",
+     "n": "16"
     }
    },
    "outputs": [
@@ -573,36 +531,13 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {
-    "slideshow": {
-     "slide_type": "slide"
-    }
-   },
+   "metadata": {},
    "source": [
     "# Thanks"
    ]
   }
  ],
- "metadata": {
-  "celltoolbar": "Slideshow",
-  "kernelspec": {
-   "display_name": "Octave",
-   "language": "octave",
-   "name": "octave"
-  },
-  "language_info": {
-   "file_extension": ".m",
-   "help_links": [
-    {
-     "text": "MetaKernel Magics",
-     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
-    }
-   ],
-   "mimetype": "text/x-octave",
-   "name": "octave",
-   "version": "0.19.14"
-  }
- },
+ "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 2
 }
diff --git a/final_project/README.md b/final_project/README.md
new file mode 100644
index 0000000..e237797
--- /dev/null
+++ b/final_project/README.md
@@ -0,0 +1,97 @@
+# ME 3255 Final Project
+## Designing a graphene pressure sensor
+
+[grading rubric](./rubric.md)
+
+![Graphene membrane under
+pressure](http://appliedmechanics.asmedigitalcollection.asme.org/data/journals/jamcav/926983/jam_80_4_040905_f001.png)
+
+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. 
diff --git a/final_project/equations/eq1.png b/final_project/equations/eq1.png
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diff --git a/final_project/equations/eq1.tex b/final_project/equations/eq1.tex
new file mode 100644
index 0000000..13c2295
--- /dev/null
+++ b/final_project/equations/eq1.tex
@@ -0,0 +1 @@
+\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}=-\frac{P}{T}
diff --git a/final_project/equations/eq2.png b/final_project/equations/eq2.png
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diff --git a/final_project/equations/eq2.tex b/final_project/equations/eq2.tex
new file mode 100644
index 0000000..a8d1d57
--- /dev/null
+++ b/final_project/equations/eq2.tex
@@ -0,0 +1 @@
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diff --git a/final_project/equations/eq3.png b/final_project/equations/eq3.png
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diff --git a/final_project/equations/eq3.tex b/final_project/equations/eq3.tex
new file mode 100644
index 0000000..4a33e25
--- /dev/null
+++ b/final_project/equations/eq3.tex
@@ -0,0 +1,4 @@
+\int_{A}wP dA=\frac{Et}{2(1-v^2)}\int_{A}\frac{1}{4}\left(\frac{\partial w}{\partial
+x}\right)^{4}+\frac{1}{4}\left(\frac{\partial w}{\partial
+y}\right)^{4}+\frac{1}{2}\left(\frac{\partial w \partial w}{\partial
+x \partial y}\right)^{2}dA
diff --git a/final_project/equations/eq4.png b/final_project/equations/eq4.png
new file mode 100644
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diff --git a/final_project/equations/eq4.tex b/final_project/equations/eq4.tex
new file mode 100644
index 0000000..b806301
--- /dev/null
+++ b/final_project/equations/eq4.tex
@@ -0,0 +1,4 @@
+\sum_{el=1}^{nel}Ph^2\bar{w}=\frac{Eth^{2}}{2(1-v^2)}\sum_{el=1}^{nel}\left[\frac{1}{4}\left(\frac{\partial w}{\partial
+x}\right)^{4}+\frac{1}{4}\left(\frac{\partial w}{\partial
+y}\right)^{4}+\frac{1}{2}\left(\frac{\partial w \partial w}{\partial
+x \partial y}\right)^{2}\right]
diff --git a/final_project/equations/eq5.png b/final_project/equations/eq5.png
new file mode 100644
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diff --git a/final_project/equations/eq5.tex b/final_project/equations/eq5.tex
new file mode 100644
index 0000000..0387eb3
--- /dev/null
+++ b/final_project/equations/eq5.tex
@@ -0,0 +1 @@
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diff --git a/final_project/figures/membrane.png b/final_project/figures/membrane.png
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