From 626d74cdd48b0ef673f3378dde8b33d5bc49d394 Mon Sep 17 00:00:00 2001
From: Robert Lanzafame <R.C.Lanzafame@tudelft.nl>
Date: Wed, 9 Oct 2024 09:50:23 +0200
Subject: [PATCH] WS 1.6 final?
---
content/Week_1_6/WS_1_6_solution.ipynb | 288 +++--
content/Week_1_6/WS_1_6_student.ipynb | 391 +++---
src/teachers/Week_1_6/WS_1_6_solution.html | 291 +++--
src/teachers/Week_1_6/WS_1_6_solution.ipynb | 1279 +++++++++++++++++++
src/teachers/Week_1_6/WS_1_6_student.html | 403 +++---
src/teachers/Week_1_6/WS_1_6_student.ipynb | 397 +++---
6 files changed, 2096 insertions(+), 953 deletions(-)
create mode 100644 src/teachers/Week_1_6/WS_1_6_solution.ipynb
diff --git a/content/Week_1_6/WS_1_6_solution.ipynb b/content/Week_1_6/WS_1_6_solution.ipynb
index 19c994e3..bf46f829 100644
--- a/content/Week_1_6/WS_1_6_solution.ipynb
+++ b/content/Week_1_6/WS_1_6_solution.ipynb
@@ -34,14 +34,12 @@
"\n",
"This assignment is aimed to develop an understanding of the **Ordinary Differential Equation (ODE)**. There will be two sections about cooling and heating scenerios, corresponding to the first-order and the second-order ODEs. Please go through the text that follows and perform all steps outlined therein.\n",
"\n",
- "## Section 1: First-order ODE\n",
- "### 1.1 Linear ODE\n",
+ "## Part 1: First-order ODE\n",
+ "\n",
"In the study of heat transfer, **Newton's law of cooling** is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. It can be expressed in the form of ODE, as below:\n",
"\n",
"$$\\frac{dT}{dt}=-k(T - T_s)$$\n",
"\n",
- "\n",
- "\n",
"where $T$ is the temperature of the object at time $t$, $T_s$ is the temperature of the surrounding and assumed to be constant, and $k$ is the constant that characterizes the ability of the object to exchange the\n",
"heat energy (unit 1/s), which depends on the specific material properties.\n",
"\n",
@@ -54,9 +52,9 @@
"id": "73781a3c",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 1:</b>\n",
+ "<b>Task 1.1:</b>\n",
" \n",
"Suppose the considered period of time is long enough (bring it to steady state), what will be the final temperature of the object? \n",
" \n",
@@ -71,8 +69,7 @@
"id": "3081d88d",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p>Your answers for Task 1:\n",
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
"<p>\n",
" \n",
"20°C \n",
@@ -96,9 +93,9 @@
"id": "18cbde20",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 2:</b>\n",
+ "<b>Task 1.2:</b>\n",
"\n",
"Write the algebraic representation of the ODE using Explicit Euler.\n",
" \n",
@@ -111,9 +108,9 @@
"id": "bec070a5",
"metadata": {},
"source": [
- " <div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ " <div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
"<p>\n",
- "Answer for Task 2:\n",
+ "Answer for Task 1.2:\n",
"<p>\n",
" \n",
" $$ T_{i+1} = T_{i}-k(T_{i} - T_s)dt$$\n",
@@ -150,14 +147,16 @@
},
{
"cell_type": "markdown",
- "id": "579a9440",
+ "id": "fb6eda82",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3:</b>\n",
+ "<b>Task 1.3:</b>\n",
+ "\n",
+ "Compute the temperature evolution in the next 60 seconds.\n",
"\n",
- "Compute the temperature evolution in the next 60 seconds. This task is defined in subtasks.\n",
+ "**Please complete the missing parts of the code in each step below, which is divided into 5 substeps (a through e).**\n",
" \n",
"</p>\n",
"</div>"
@@ -168,9 +167,9 @@
"id": "88b861bc",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.1:</b> \n",
+ "<b>Task 1.3a:</b> \n",
"\n",
"The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.\n",
" \n",
@@ -205,9 +204,9 @@
"id": "4586d4b6",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.2:</b> \n",
+ "<b>Task 1.3b:</b> \n",
"\n",
"Implement your time discretization and find the solution from $t=0$ until $t=60$ sec. \n",
" \n",
@@ -249,9 +248,9 @@
"id": "ffb4547f",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.3:</b>\n",
+ "<b>Task 1.3c:</b>\n",
"\n",
"Try different time steps to check the stability of the calculation. At which value the solution is stable?\n",
" \n",
@@ -264,7 +263,7 @@
"id": "59bcbb0a",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
"<p>\n",
"<b>Answer:</b> \n",
" \n",
@@ -279,9 +278,9 @@
"id": "8a907a09",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.4:</b>\n",
+ "<b>Task 1.3d:</b>\n",
"\n",
"Obtain the mathematical expression that proves your stability criteria.\n",
" \n",
@@ -294,7 +293,7 @@
"id": "eb00c9ab",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
"<p> Answer\n",
"<p>\n",
" \n",
@@ -335,9 +334,9 @@
"id": "1ff1e4fe",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.5:</b>\n",
+ "<b>Task 1.3e:</b>\n",
"\n",
"Now, discretize the equation using Implicit (Backward) Euler. Can you find a time step that makes the problem unstable?\n",
" \n",
@@ -382,13 +381,28 @@
"plt.grid()"
]
},
+ {
+ "cell_type": "markdown",
+ "id": "f534fb27",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Answer:</b> \n",
+ " \n",
+ "No, you cannot! This method is unconditionally stable. (a good thing to remember for the exam)\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
{
"cell_type": "markdown",
"id": "2109f010",
"metadata": {},
"source": [
- "## Section 2: Second-order ODE\n",
- "### Background\n",
+ "## Part 2: Second-order ODE\n",
+ "\n",
"The following 1D equation describes the steady state solution of the temperature along a pin that sticks out of a furnace. The rest of the pin is exposed to the ambient. \n",
"\n",
"$$\n",
@@ -415,33 +429,46 @@
},
{
"cell_type": "markdown",
- "id": "c739aaf1",
+ "id": "c74adcb4",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 5:</b>\n",
+ "<b>Task 2.1:</b>\n",
+ "\n",
+ "This task has three parts: a) discretize the analytic expression into a system of equations using central differences, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.\n",
"\n",
- "Below we already provide the code structured that solves the equation with the above Dirichlet conditions and using a central difference approximation of the second derivative. Complete the code to complete the matrix and vectors. **Write in paper (add the image) the matrix A, and the vectors T and b.**\n",
+ "<em>Parts 2.1b and 2.1c do not need to be completed in order; in fact, it may be useful to go back and forth between the two in order to understand the problem.</em> \n",
" \n"
]
},
{
"cell_type": "markdown",
- "id": "11009615",
+ "id": "0db7afc9",
"metadata": {},
- "source": []
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1a:</b>\n",
+ "\n",
+ "Discretize the analytic expression into a system of equations for a grid with 6 points using central differences.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
},
{
"cell_type": "markdown",
- "id": "4133c6e2",
+ "id": "6ae1eb87",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p> Answer for 2.1a and 2.1b\n",
"<p>\n",
- " \n",
- "Add the image.\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_{i-1} - 2T_{i} + T_{i+1}) - \\alpha T_{i} + \\alpha T_s = 0\n",
+ "$$\n",
"\n",
"**For $i = 1$:**\n",
"The finite difference approximation for the first internal node is:\n",
@@ -483,7 +510,34 @@
"(T_3 - (2 + \\alpha \\Delta x^2) T_4 ) = -\\alpha Ts \\Delta x^2 -T_5 \n",
"$$\n",
"\n",
- "We can now move the equations in the Ax=y form:\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "df2341fc",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1b:</b>\n",
+ "\n",
+ "Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c8616d1c",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ "\n",
+ "The discretized equations are presented in Ax=y form:\n",
"\n",
"$$\n",
"\\begin{bmatrix}\n",
@@ -512,6 +566,27 @@
"</div>"
]
},
+ {
+ "cell_type": "markdown",
+ "id": "0c12775e",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1c:</b>\n",
+ "\n",
+ "Implement the discretized system of equations in a code cell.\n",
+ "\n",
+ "<em>We have already done this for you! Your task is to read the cell and make sure you understand how the matrices are implemented. Reading the code should help you formulate the matrices in Task 2.1b.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "11009615",
+ "metadata": {},
+ "source": []
+ },
{
"cell_type": "code",
"execution_count": 4,
@@ -587,9 +662,9 @@
"source": [
"\n",
"\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 6:</b>\n",
+ "<b>Task 2.2:</b>\n",
"\n",
"Copy the code above below and modify it to replace the Dirichlet condition at the right BC for a Neumann condition such that there $\\frac{dT}{dx}=0$. Approximate the Neuman boundary $\\frac{dT}{dx}=0$ by using the Backward difference for first order differential equation of first order accuracy. \n",
"\n",
@@ -730,18 +805,30 @@
},
{
"cell_type": "markdown",
- "id": "0b47e9c2",
+ "id": "ac255017",
"metadata": {},
"source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.3:</b>\n",
"\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "Just as we did in Task 2.1, this task has three parts: a) discretize the analytic expression into a system of equations using <b>forward differences</b>, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.\n",
+ "</div> \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "104690ed",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 7:</b>\n",
+ "<b>Task 2.3a:</b>\n",
"\n",
- "Now, instead of using the central difference for approximating the second derivative. Use the forward difference to replace the second derivative (first order accurate). **Write the algebraic representation of the differential equation below.**\n",
+ "Discretize the analytic expression into a system of equations for a grid with 6 points using <b>forward differences</b>.\n",
"\n",
- "</p>\n",
- "</div>"
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
]
},
{
@@ -749,8 +836,7 @@
"id": "74174f3a",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
"<p>\n",
"Forward difference for second order differential equation:\n",
"\n",
@@ -767,7 +853,7 @@
"\n",
"\n",
"$$\n",
- "-2T_1 +T_2 = -(1+alpha)T_0 - \\alpha T_s \\Delta x^2\n",
+ "-2T_1 +T_2 = -(1+\\alpha)T_0 - \\alpha T_s \\Delta x^2\n",
"$$\n",
"\n",
"**For $i = 1$:**\n",
@@ -817,18 +903,17 @@
},
{
"cell_type": "markdown",
- "id": "a4e0342e",
+ "id": "7b12c86e",
"metadata": {},
"source": [
- "\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 8:</b>\n",
+ "<b>Task 2.3b:</b>\n",
"\n",
- "Now, write the resulting matrix A, the vector T and the vector b.\n",
- " \n",
- "</p>\n",
- "</div>"
+ "Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
]
},
{
@@ -862,41 +947,17 @@
},
{
"cell_type": "markdown",
- "id": "b9ca660d",
- "metadata": {},
- "source": []
- },
- {
- "cell_type": "markdown",
- "id": "8cb492f8",
- "metadata": {},
- "source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
- "<p>\n",
- " \n",
- "Add the image.\n",
- "\n",
- "</p>\n",
- "</div>"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0640b43e",
+ "id": "1b304504",
"metadata": {},
"source": [
- "\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 9:</b>\n",
+ "<b>Task 2.3c:</b>\n",
"\n",
- "Copy the code of task 5 below, modify it accordingly to your new discretization (using FD for the second derivative in space). Keep Dirichlet conditions. \n",
+ "Implement the discretized system of equations in a code cell.\n",
"\n",
- "How different are the values of the temperature with respect to task 5? Which solution is expected to be more accurate?\n",
- "\n",
- "</p>\n",
- "</div>"
+ "<b>This time we did not do it for you!</b> Copy the code from Task 2.1c and revise it to solve the system of equations using <b>Forward Differences</b>. Keep the Dirichlet conditions.\n",
+ " \n"
]
},
{
@@ -989,12 +1050,12 @@
},
{
"cell_type": "markdown",
- "id": "64d693df",
+ "id": "18d60370",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 9:</b>\n",
+ "<b>Task 2.4:</b>\n",
"\n",
"How much finer does your grid has to be in the forward difference implementation to get a similar value at x = 0.02 as in the central difference implementation? Vary your dx.\n",
"\n",
@@ -1005,22 +1066,28 @@
},
{
"cell_type": "markdown",
- "id": "e771703c",
+ "id": "7ef65db5",
"metadata": {},
"source": [
- "you can test this above make the temperature the same at x=0.02 "
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ " \n",
+ "You can test this above make the temperature the same at x=0.02\n",
+ "\n",
+ "</p>\n",
+ "</div>"
]
},
{
"cell_type": "markdown",
- "id": "031dca37",
+ "id": "b78e8e16",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
"<p>\n",
- "<b>Bonus note:</b> \n",
+ "<b>Bonus Task</b> \n",
" \n",
- "The matrix inversion using numpy is one way to solve the system, another is the gauss_jordan written below and another one is the sparse matrix based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change dx to 0.0002 of the code in task 5 and run it. Now, you should have a matrix A with a high resolution that solves the second degree ODE. Test the time each method takes.\n",
+ "The matrix inversion using numpy is one way to solve the system, another is the <code>gauss_jordan</code> method, written below, and another one is the sparse matrix-based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change <code>dx</code> to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.\n",
" \n",
"</p>\n",
"</div>"
@@ -1081,9 +1148,9 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "The time used by direct matrix inversion solution is 2.522e-03 sec\n",
- "The time used by Gauss-jordan solution is 2.020e-01 sec\n",
- "The time used by the sparse matrix solver is 3.001e-03 sec\n"
+ "The time used by direct matrix inversion solution is 1.399e-03 sec\n",
+ "The time used by Gauss-jordan solution is 1.411e-01 sec\n",
+ "The time used by the sparse matrix solver is 1.286e-03 sec\n"
]
}
],
@@ -1119,6 +1186,21 @@
"assert np.allclose(np.dot(A, u2), b), \"Oops! The calculation is wrong..\""
]
},
+ {
+ "cell_type": "markdown",
+ "id": "5d5b9189",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p> Answer\n",
+ "<p>\n",
+ " \n",
+ "The sparse matrix is fastest, but we see that the Numpy implementation is also very fast! problem still small; Gauss-Jordan slow because...???\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
{
"cell_type": "markdown",
"id": "2921b3f2",
diff --git a/content/Week_1_6/WS_1_6_student.ipynb b/content/Week_1_6/WS_1_6_student.ipynb
index 05577a25..6feca725 100644
--- a/content/Week_1_6/WS_1_6_student.ipynb
+++ b/content/Week_1_6/WS_1_6_student.ipynb
@@ -34,14 +34,12 @@
"\n",
"This assignment is aimed to develop an understanding of the **Ordinary Differential Equation (ODE)**. There will be two sections about cooling and heating scenerios, corresponding to the first-order and the second-order ODEs. Please go through the text that follows and perform all steps outlined therein.\n",
"\n",
- "## Section 1: First-order ODE\n",
- "### 1.1 Linear ODE\n",
+ "## Part 1: First-order ODE\n",
+ "\n",
"In the study of heat transfer, **Newton's law of cooling** is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. It can be expressed in the form of ODE, as below:\n",
"\n",
"$$\\frac{dT}{dt}=-k(T - T_s)$$\n",
"\n",
- "\n",
- "\n",
"where $T$ is the temperature of the object at time $t$, $T_s$ is the temperature of the surrounding and assumed to be constant, and $k$ is the constant that characterizes the ability of the object to exchange the\n",
"heat energy (unit 1/s), which depends on the specific material properties.\n",
"\n",
@@ -54,9 +52,9 @@
"id": "73781a3c",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 1:</b>\n",
+ "<b>Task 1.1:</b>\n",
" \n",
"Suppose the considered period of time is long enough (bring it to steady state), what will be the final temperature of the object? \n",
" \n",
@@ -85,9 +83,9 @@
"id": "18cbde20",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 2:</b>\n",
+ "<b>Task 1.2:</b>\n",
"\n",
"Write the algebraic representation of the ODE using Explicit Euler.\n",
" \n",
@@ -106,32 +104,26 @@
"id": "579a9440",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3:</b>\n",
+ "<b>Task 1.3:</b>\n",
"\n",
"Compute the temperature evolution in the next 60 seconds.\n",
"\n",
- "**Please complete the missing parts of the code in each step below**\n",
+ "**Please complete the missing parts of the code in each step below, which is divided into 5 substeps (a through e).**\n",
" \n",
"</p>\n",
"</div>"
]
},
- {
- "cell_type": "markdown",
- "id": "ec6e3109",
- "metadata": {},
- "source": []
- },
{
"cell_type": "markdown",
"id": "88b861bc",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.1:</b> \n",
+ "<b>Task 1.3a:</b> \n",
"\n",
"The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.\n",
" \n",
@@ -166,9 +158,9 @@
"id": "4586d4b6",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.2:</b> \n",
+ "<b>Task 1.3b:</b> \n",
"\n",
"Implement your time discretization and find the solution from $t=0$ until $t=60$ sec. \n",
" \n",
@@ -178,23 +170,12 @@
},
{
"cell_type": "code",
- "execution_count": 21,
+ "execution_count": null,
"id": "0bf8247e",
"metadata": {
"scrolled": false
},
- "outputs": [
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- }
- ],
+ "outputs": [],
"source": [
"for i in range(n-1):\n",
" T[i+1] = YOUR_CODE_HERE\n",
@@ -210,9 +191,9 @@
"id": "ffb4547f",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.3:</b>\n",
+ "<b>Task 1.3c:</b>\n",
"\n",
"Try different time steps to check the stability of the calculation. At which value the solution is stable?\n",
" \n",
@@ -231,9 +212,9 @@
"id": "8a907a09",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.4:</b>\n",
+ "<b>Task 1.3d:</b>\n",
"\n",
"Obtain the mathematical expression that proves your stability criteria.\n",
" \n",
@@ -252,9 +233,9 @@
"id": "1ff1e4fe",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.5:</b>\n",
+ "<b>Task 1.3e:</b>\n",
"\n",
"Now, discretize the equation using Implicit (Backward) Euler. Can you find a time step that makes the problem unstable?\n",
" \n",
@@ -264,21 +245,10 @@
},
{
"cell_type": "code",
- "execution_count": 29,
+ "execution_count": null,
"id": "18c73272",
"metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- }
- ],
+ "outputs": [],
"source": [
"dt = YOUR_CODE_HERE \n",
"t_end = 60 \n",
@@ -304,8 +274,8 @@
"id": "2109f010",
"metadata": {},
"source": [
- "## Section 2: Second-order ODE\n",
- "### Background\n",
+ "## Part 2: Second-order ODE\n",
+ "\n",
"The following 1D equation describes the steady state solution of the temperature along a pin that sticks out of a furnace. The rest of the pin is exposed to the ambient. \n",
"\n",
"$$\n",
@@ -332,63 +302,94 @@
},
{
"cell_type": "markdown",
- "id": "c739aaf1",
+ "id": "babd0424",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 5:</b>\n",
+ "<b>Task 2.1:</b>\n",
"\n",
- "Below we provide the code already implemented of the equation with the above Dirichlet conditions and using a central difference approximation of the second derivative. **Write in paper (add the image) the matrix A, and the vectors T and b.**\n",
+ "This task has three parts: a) discretize the analytic expression into a system of equations using central differences, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.\n",
+ "\n",
+ "<em>Parts 2.1b and 2.1c do not need to be completed in order; in fact, it may be useful to go back and forth between the two in order to understand the problem.</em> \n",
" \n"
]
},
{
"cell_type": "markdown",
- "id": "4133c6e2",
+ "id": "1f70b8e9",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- " \n",
- "Add the image.\n",
+ "<b>Task 2.1a:</b>\n",
"\n",
- "</p>\n",
- "</div>"
+ "Discretize the analytic expression into a system of equations for a grid with 6 points using central differences.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "81a5f165",
+ "metadata": {},
+ "source": [
+ "_Your answer here._"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c513ff65",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1b:</b>\n",
+ "\n",
+ "Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cb9ff085",
+ "metadata": {},
+ "source": [
+ "_Your answer here._"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fae61d18",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1c:</b>\n",
+ "\n",
+ "Implement the discretized system of equations in a code cell.\n",
+ "\n",
+ "<em>We have already done this for you! Your task is to read the cell and make sure you understand how the matrices are implemented. Reading the code should help you formulate the matrices in Task 2.1b.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "89fd190c",
+ "metadata": {},
+ "source": [
+ "_Add your image here._"
]
},
{
"cell_type": "code",
- "execution_count": 8,
+ "execution_count": null,
"id": "53fb4f99",
"metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "(4, 4)\n"
- ]
- },
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The estimated temperature at the nodes are: ['250.00', '168.77', '115.29', '78.86', '52.21', '30.00'] [C]\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
"import numpy as np \n",
"import matplotlib.pyplot as plt\n",
@@ -439,9 +440,9 @@
"source": [
"\n",
"\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 6:</b>\n",
+ "<b>Task 2.2:</b>\n",
"\n",
"The code below is the same as above, modify it to replace the Dirichlet condition at the right BC for a Neumann condition such that there $\\frac{dT}{dx}=0$. How does the temperature profile changes? \n",
"\n",
@@ -461,35 +462,10 @@
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": null,
"id": "cfcd00f9",
"metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "(4, 4)\n"
- ]
- },
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The estimated temperature at the nodes are: ['250.00', '174.84', '128.64', '102.18', '90.15', '90.15'] [C]\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
"Ts = 30\n",
"alpha = 500\n",
@@ -536,155 +512,86 @@
},
{
"cell_type": "markdown",
- "id": "0b47e9c2",
+ "id": "57632792",
"metadata": {},
"source": [
- "\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 7:</b>\n",
- "\n",
- "Now, instead of using the central difference for approximating the second derivative. Use the forward difference to replace the second derivative (first order accurate). Write the algebraic representation of the differential equation below.\n",
+ "<b>Task 2.3:</b>\n",
"\n",
- "</p>\n",
- "</div>"
+ "Just as we did in Task 2.1, this task has three parts: a) discretize the analytic expression into a system of equations using <b>forward differences</b>, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.\n",
+ "</div> \n"
]
},
{
"cell_type": "markdown",
- "id": "74174f3a",
+ "id": "2d4b9fb9",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- " \n",
- "Add the image.\n",
+ "<b>Task 2.3a:</b>\n",
"\n",
- "</p>\n",
- "</div>"
+ "Discretize the analytic expression into a system of equations for a grid with 6 points using <b>forward differences</b>.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
]
},
{
"cell_type": "markdown",
- "id": "a4e0342e",
+ "id": "1c5ec2f2",
"metadata": {},
"source": [
- "\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
- "<p>\n",
- "<b>Task 8:</b>\n",
- "\n",
- "Now, write the resulting matrix A, the vector T and the vector b.\n",
- " \n",
- "</p>\n",
- "</div>"
+ "_Your answer here._"
]
},
{
"cell_type": "markdown",
- "id": "b9ca660d",
+ "id": "361f1638",
"metadata": {},
- "source": []
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.3b:</b>\n",
+ "\n",
+ "Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
},
{
"cell_type": "markdown",
- "id": "8cb492f8",
+ "id": "047a38d3",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
- "<p>\n",
- " \n",
- "Add the image.\n",
- "\n",
- "</p>\n",
- "</div>"
+ "_Your answer here._"
]
},
{
"cell_type": "markdown",
- "id": "0640b43e",
+ "id": "3996b611",
"metadata": {},
"source": [
- "\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 9:</b>\n",
+ "<b>Task 2.3c:</b>\n",
"\n",
- "The original code is copied below, modify it accordingly to your new discretization. Keep Dirichlet conditions. \n",
+ "Implement the discretized system of equations in a code cell.\n",
"\n",
- "</p>\n",
- "</div>"
+ "<b>This time we did not do it for you!</b> Copy the code from Task 2.1c and revise it to solve the system of equations using <b>Forward Differences</b>. Keep the Dirichlet conditions.\n",
+ " \n"
]
},
{
"cell_type": "code",
- "execution_count": 10,
+ "execution_count": null,
"id": "1e783d3e",
"metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": 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RokWyLEu33Xaby3JjY2N1xRVXlPlK2Z49e7o8btiwoSzLUrdu3ZxtPj4+uuyyy1xqXLRokVJSUtS0aVOX5Xfp0uWCr9Q932tv5cqVkoq//1199dVq2LCh8/2vIrZBd+IE5UqgYcOG5zxB+brrrtOCBQv0j3/8Q3fccYdyc3N1+eWX6+9//7sGDhx40cuNiIhweezn53fO9jOP5Q8aNEhffPGFHnvsMV111VUKCwuTZVnq3r27cnJyzrvsovMb+vbtW2qfw4cPy7IsFRYWKjY2ttj0ktrO5nA4tHr1aj399NN65JFHdOTIEcXFxemee+7Ro48+Kl9fXx06dEg+Pj6Kiopyea5lWYqNjdWhQ4fOu5zydujQoRKvXouPj3dOP1NkZKTL46ITacvytyiLOnXqaPny5ZoyZYrS0tKUnZ2t2rVr64EHHtCDDz543uefr76i9YmJiXHp5+PjU+y55xIVFVXqa+lCxzQuLq5Y36SkJOdY9O/fX+vWrdPIkSN12WWX6YEHHtCOHTu0fPlyBQYGqnXr1s7nXejr5exlF9V2sa+Dsij68Csaj0tV9BofNWqURo0aVWKfP/74w+Xx2etdNI+i//CdzcvL9f/rQUFBLv8pkk5va2e+dx08eFDx8fHFnnv2co0xxbbHIrVr1y71uWcq6b20pBr9/PyUlZXlsvyff/651CsJzx63cynra6+k7T0+Pt65XVTENuhOhJ0q4oYbbtANN9yg3NxcrV+/XpMmTdKgQYOUnJysa665pkJryczM1KJFizRu3DiNHTvW2Z6bm6vDhw+XaR5F333y0ksvlXr1TExMjE6dOiXLspSRkVFsekltJWncuLHmzJkjY4y2bNmiGTNm6IknnlBgYKDGjh2ryMhI5efn6+DBgy6BxxijjIyMUt9oJTnftM4+AfJC3oxKEhkZqf379xdrLzqxsLy+O+bM+s+80qik+q+99lpde+21Kigo0MaNG/XSSy9p+PDhiomJ0YABAy6pjqI35N9//91lb2N+fn65hc0LHdPSrgLq2LGj/vWvf2n16tUqLCxU+/btFRoaqvj4eC1btkzLly/Xtdde6xzPi3m9nL3sovEp7XVwqV/rYIzRJ598ouDg4HK7MrRoPB9++GHddNNNJfapX7++y+Oz17toHnPnzi23vQdRUVFas2aNCgsLSw081atXl2VZ+vLLL0u8As/dV+VVr15dgYGBeuedd0qdXl6Ktq39+/cXu2p33759zmW5ext0Nw5jVTH+/v5q166d8/LQoisSyvt/8udiWZaMMcVe8G+99ZYKCgqK1VtSXW3atFG1atW0bds2tWjRosSbn5+f82qwefPmufzv7NixY/rkk08uuO4rrrhCL7zwgqpVq6Zvv/1WkpyX0L733nsu/T/++GNlZ2e7XGJ7tqIX+JYtW1zaFy5cWKzvhfyNOnbsqG3btjlrLDJz5kxZlqXU1NTzzqMsSqv/XGPr7e2tli1bOq8uO7vGi3HddddJOn3VzZnmzp2r/Pz8S56/VH5j2qlTJ/3++++aNm2aWrVqpdDQUOf858+frw0bNrgcwrqQ10tpWrVqpYCAAL3//vsu7WvXri2XQ5UTJkzQtm3b9OCDDxbb63Cx6tevr7p16+r7778v9TVeNHal6dKli3x8fPTLL7+UOo8L1a1bN508edLlqtCz9ezZU8YY/fbbbyUus3Hjxhe83AvRs2dP/fLLL4qMjCxx+WcGC39//0t63+/QoYOk4u9/GzZs0Pbt253vf+7eBt2NPTuVwNatW0t8Q69Tp46ioqL0+OOPKz09XR07dlSNGjV09OhRvfjii/L19XV+p0WdOnUUGBio999/Xw0bNlRISIji4+PLbZf0mcLCwnTdddfp2WefVfXq1ZWcnKzVq1fr7bffVrVq1Vz6pqSkSJLeeOMNhYaGKiAgQLVq1VJkZKReeuklDR48WIcPH1bfvn0VHR2tgwcP6vvvv9fBgwf12muvSZKefPJJde3aVddff71GjhypgoICTZ48WcHBwefdk7Ro0SK9+uqr6tOnj2rXri1jjObNm6ejR4/q+uuvlyRdf/316tKli8aMGaOsrCy1adNGW7Zs0bhx49SsWTPdfvvtpc4/NjZWnTp10qRJkxQeHq6kpCR98cUXmjdvXrG+RW+QkydPVrdu3eTt7a0mTZo4DxOe6aGHHtLMmTPVo0cPPfHEE0pKStKnn36qV199VUOHDi23L2zr3r27IiIidPfdd+uJJ56Qj4+PZsyYob1797r0e/3117VixQr16NFDNWvW1MmTJ53/6zzzg/1iXX755Ro4cKCef/55eXt7q0OHDvrxxx/1/PPPy+FwnPOQQ1mV15h26NDBeZn0hAkTnO2dOnXS4MGDnf8uciGvl9KEh4dr1KhReuqpp/SXv/xFt9xyi/bu3avx48df0CGEo0ePOr+SIDs72/mlgl9++aX69evnsj7l4X//93/VrVs3denSRUOGDFFCQoIOHz6s7du369tvv9VHH310zucnJyfriSee0N///nft3LlTXbt2VXh4uH7//Xd98803Cg4OvuCaBw4cqOnTp+tvf/ubduzYodTUVBUWFurrr79Ww4YNNWDAALVp00b33nuv7rzzTm3cuFHXXXedgoODtX//fq1Zs0aNGzfW0KFDL2Vozmn48OH6+OOPdd111+mhhx5SkyZNVFhYqD179mjp0qUaOXKkWrZsKen0+8qqVav0ySefKC4uTqGhocX2mJ1L/fr1de+99+qll16Sl5eXunXrpt27d+uxxx5TYmKiHnroIUnltw16jIdOjIY599VYOuNqgUWLFplu3bqZhIQE4+fnZ6Kjo0337t1dLhE15vSVKw0aNDC+vr4uV1GUdjVWSVdmSDJpaWkubSVdgZGenm5uvvlmEx4ebkJDQ03Xrl3N1q1bi11VZIwx06ZNM7Vq1TLe3t7FrlBavXq16dGjh4mIiDC+vr4mISHB9OjRo9hVHQsXLjRNmjQxfn5+pmbNmuaZZ54p9UqiM/3nP/8xAwcONHXq1DGBgYHG4XCYq6++2syYMcOlX05OjhkzZoxJSkoyvr6+Ji4uzgwdOtTlsmpjil+NZYwx+/fvN3379jURERHG4XCY2267zXkV2Jnrmpuba/7yl7+YqKgoY1mWy9VOJY3br7/+agYNGmQiIyONr6+vqV+/vnn22WedV7AYU/rVMcYUv5KmNN98841p3bq1CQ4ONgkJCWbcuHHOq46K6lu3bp258cYbTVJSkvH39zeRkZGmXbt2xa7QO3uZRdv42VeVlHQV28mTJ82IESNMdHS0CQgIMK1atTLr1q0zDofDPPTQQ+ddj5K23bNd6pgWadasmZFkvvrqK2fbb7/9ZiSZyMhI56XBRcr6eiltvIw5/RUJkyZNMomJicbPz880adLEfPLJJyVukyVJSkpyvrdYlmVCQkJM/fr1ze23324+//zzEp9z9t/zQq/GMsaY77//3vTr189ER0cbX19fExsbazp06GBef/31Mq23MaevCE1NTTVhYWHG39/fJCUlmb59+7p8RcXgwYNNcHBwseeW9D6Rk5NjHn/8cVO3bl3j5+dnIiMjTYcOHczatWtd+r3zzjumZcuWJjg42AQGBpo6deqYO+64w2zcuLHEOs9e5pmXbZ+rxnbt2pnLL7/cpe348ePm0UcfNfXr1zd+fn7G4XCYxo0bm4ceesjl0vfNmzebNm3amKCgIJeroi7ktVdQUGAmT55s6tWrZ3x9fU316tXNbbfd5vz6kiKXug16kmVMOZ+CDwDlZO3atWrTpo3ef/99DRo0yNPlAKiiCDsAKoVly5Zp3bp1at68uQIDA/X999/rmWeekcPh0JYtW8rtXBIAfz6cswOgUggLC9PSpUs1bdo0HTt2TNWrV1e3bt00adIkgg6AS8KeHQAAYGtceg4AAGyNsAMAAGyNsAMAAGyNE5QlFRYWat++fQoNDS31K+IBAEDlYozRsWPHzvt7Z4Qdnf79j8TERE+XAQAALsLevXuL/bbXmQg7kvP3Wfbu3auwsDAPVwMAAMoiKytLiYmJ5/2dNcKO/v8v7YaFhRF2AACoYs53CgonKAMAAFsj7AAAAFsj7AAAAFvjnB0AsLGCggKdOnXK02UAF8XX11fe3t6XPB/CDgDYkDFGGRkZOnr0qKdLAS5JtWrVFBsbe0nfg0fYAQAbKgo60dHRCgoK4gtTUeUYY3TixAkdOHBAkhQXF3fR8yLsAIDNFBQUOINOZGSkp8sBLlpgYKAk6cCBA4qOjr7oQ1qcoAwANlN0jk5QUJCHKwEuXdF2fCnnnhF2AMCmOHQFOyiP7ZiwAwAAbM2jYWfSpEm66qqrFBoaqujoaPXp00c7duxw6TNkyBBZluVya9WqlUuf3Nxc3X///apevbqCg4PVu3dvpaenV+SqlGhL+lENfGO9tqQf9XQpAPCnNWPGDFWrVs3TZVwQd9c8fvx4NW3a9JLnU1XG1qNhZ/Xq1UpLS9P69eu1bNky5efnq3PnzsrOznbp17VrV+3fv995W7x4scv04cOHa/78+ZozZ47WrFmj48ePq2fPniooKKjI1Slm3re/ad3OQ5r37W8erQMAqoqS/oNrWZa6du1apucnJydr2rRpLm39+/fX//3f/7mhWldV5YP/YnlybC+VR6/GWrJkicvj6dOnKzo6Wps2bdJ1113nbPf391dsbGyJ88jMzNTbb7+tWbNmqVOnTpKk9957T4mJiVq+fLm6dOnivhUoQfqREzqSfUqWJX3y/T5Jp+/7Nq8hY6TwYF/VCOekQQBVx5b0o5q0+D96uHsDNalRze3L69q1q6ZPn+7S5u/vf9HzCwwMdF7Vg/JVVca2Up2zk5mZKUmKiIhwaV+1apWio6NVr1493XPPPc5r7iVp06ZNOnXqlDp37uxsi4+PV0pKitauXVvicnJzc5WVleVyKy9tJ69Ur5fXqOdLa3Q4O0+SdDg7Tz1fWqNeL69R28kry21ZAFARKnovddF/cM+8hYeHO6ePHz9eNWvWlL+/v+Lj4/XAAw9Iktq3b69ff/1VDz30kHOPkFR8j0vRIZx33nlHNWvWVEhIiIYOHaqCggJNmTJFsbGxio6O1tNPP+1S19SpU9W4cWMFBwcrMTFRw4YN0/HjxyWd/py68847lZmZ6Vz2+PHjJUl5eXkaPXq0EhISFBwcrJYtW2rVqlUu854xY4Zq1qypoKAg3XjjjTp06NA5xygvL0/33Xef4uLiFBAQoOTkZE2aNMk5fc+ePbrhhhsUEhKisLAw9evXT7///nup82vfvr2GDx/u0tanTx8NGTLkgsZWkl577TXVqVNHfn5+ql+/vmbNmuUy3bIsvfXWW7rxxhsVFBSkunXrauHChedc30tVacKOMUYjRoxQ27ZtlZKS4mzv1q2b3n//fa1YsULPP/+8NmzYoA4dOig3N1fS6S/O8vPzc3khSFJMTIwyMjJKXNakSZPkcDict8TExHJbj2n9m8rH6/RGYIrW7b/3Pl6WpvVvWm7LAgB3ST9yQj+kZ2rrb5kue6m3/papH9IzlX7khEfqmjt3rl544QX97//+r3766SctWLBAjRs3liTNmzdPNWrU0BNPPOE87aE0v/zyiz777DMtWbJEs2fP1jvvvKMePXooPT1dq1ev1uTJk/Xoo49q/fr1zud4eXnpH//4h7Zu3ap3331XK1as0OjRoyVJrVu31rRp0xQWFuZc9qhRoyRJd955p7766ivNmTNHW7Zs0S233KKuXbvqp59+kiR9/fXXuuuuuzRs2DBt3rxZqampeuqpp845Dv/4xz+0cOFCffjhh9qxY4fee+89JScnSzr9edqnTx8dPnxYq1ev1rJly/TLL7+of//+Fz3uZR3b+fPn68EHH9TIkSO1detW/fWvf9Wdd96plStd/6M/YcIE9evXT1u2bFH37t1166236vDhwxdd3/lUmi8VvO+++7RlyxatWbPGpf3MP05KSopatGihpKQkffrpp7rppptKnZ8xptTL1R5++GGNGDHC+TgrK6vcAk+fZgm6LDpEPV9aU2zagrQ2SklwlMtyAMCdztwLXfROWrSXusjuZ3q4ZdmLFi1SSEiIS9uYMWP02GOPac+ePYqNjVWnTp3k6+urmjVr6uqrr5Z0+qiAt7e3QkNDSz31oUhhYaHeeecdhYaGqlGjRkpNTdWOHTu0ePFieXl5qX79+po8ebJWrVrlvCjmzD0ftWrV0pNPPqmhQ4fq1VdflZ+fnxwOhyzLcln2L7/8otmzZys9PV3x8fGSpFGjRmnJkiWaPn26Jk6cqBdffFFdunTR2LFjJUn16tXT2rVri53qcaY9e/aobt26atu2rSzLUlJSknPa8uXLtWXLFu3atcv52TZr1ixdfvnl2rBhg6666qrz/QmKKevYPvfccxoyZIiGDRsmSRoxYoTWr1+v5557Tqmpqc5+Q4YM0cCBAyVJEydO1EsvvaRvvvmmzOdmXahKsWfn/vvv18KFC7Vy5UrVqFHjnH3j4uKUlJTkTMSxsbHKy8vTkSNHXPodOHBAMTExJc7D399fYWFhLjd3KMpafNUFgKrGk3upU1NTtXnzZpdbWlqaJOmWW25RTk6OateurXvuuUfz589Xfn7+BS8jOTlZoaGhzscxMTFq1KiRvLy8XNrOPG1i5cqVuv7665WQkKDQ0FDdcccdOnToULGLas707bffyhijevXqKSQkxHlbvXq1fvnlF0nS9u3bdc0117g87+zHZxsyZIg2b96s+vXr64EHHtDSpUud07Zv367ExESX/8Q3atRI1apV0/bt288zMpdm+/btatOmjUtbmzZtii23SZMmzn8HBwcrNDTUZazLm0fDjjFG9913n+bNm6cVK1aoVq1a533OoUOHtHfvXudvZDRv3ly+vr5atmyZs8/+/fu1detWtW7d2m21n0tkiJ+iQvzVOMGhp29MUeMEh6JC/BUZ4ueRegDgQvVplqAFaW1KnLYgrY36NEtw27KDg4N12WWXudyKzuVMTEzUjh079MorrygwMFDDhg3Tddddd8Hfruvr6+vy2LKsEtsKCwslSb/++qu6d++ulJQUffzxx9q0aZNeeeUVSef+Zt/CwkJ5e3tr06ZNLuFt+/btevHFFyWd/iy8UFdeeaV27dqlJ598Ujk5OerXr5/69u3rnF9JRzbOdcTDy8urWB0X+43FZy+jpOWea6zdwaOHsdLS0vTPf/5T//rXvxQaGuo8x8bhcCgwMFDHjx/X+PHjdfPNNysuLk67d+/WI488ourVq+vGG2909r377rs1cuRIRUZGKiIiQqNGjVLjxo2dV2dVtDhHoNaMTZWft5csy9Kgq2sqr6BQ/j6X/jP1AFDRLEsy5v/fe1pgYKB69+6t3r17Ky0tTQ0aNNAPP/ygK6+8Un5+fm752pGNGzcqPz9fzz//vHPvz4cffujSp6RlN2vWTAUFBTpw4ICuvfbaEufdqFEjl3ODJBV7XJKwsDD1799f/fv3V9++fdW1a1cdPnxYjRo10p49e7R3717n3p1t27YpMzNTDRs2LHFeUVFRLufhFBQUaOvWrS6Hnsoytg0bNtSaNWt0xx13ONvWrl1b6nIrikfDzmuvvSbp9FneZ5o+fbqGDBkib29v/fDDD5o5c6aOHj2quLg4paam6oMPPnDZ/fjCCy/Ix8dH/fr1U05Ojjp27KgZM2Zc9A+GlYczg41lWQQdAFVO0V7quGoB6n9Voj7YsFf7j550+17q3NzcYheY+Pj4qHr16poxY4YKCgrUsmVLBQUFadasWQoMDHSes5KcnKx///vfGjBggPz9/VW9evVyqalOnTrKz8/XSy+9pF69eumrr77S66+/7tInOTlZx48f1xdffKErrrhCQUFBqlevnm699Vbdcccdev7559WsWTP98ccfWrFihRo3bqzu3bvrgQceUOvWrTVlyhT16dNHS5cuPef5OtLpz724uDg1bdpUXl5e+uijjxQbG6tq1aqpU6dOatKkiW699VZNmzZN+fn5GjZsmNq1a6cWLVqUOL8OHTpoxIgR+vTTT1WnTh298MILOnr0aLH1O9/Y/s///I/69eunK6+8Uh07dtQnn3yiefPmafny5Rc24OXNwGRmZhpJJjMz09OlAMAly8nJMdu2bTM5OTmXPK+Tp/JNYWGhMcaYwsJCc/JU/iXP81wGDx5sdPr0IJdb/fr1jTHGzJ8/37Rs2dKEhYWZ4OBg06pVK7N8+XLn89etW2eaNGli/P39TdFH3PTp043D4XD2GTdunLniiiuKLfeGG25waWvXrp158MEHnY+nTp1q4uLiTGBgoOnSpYuZOXOmkWSOHDni7PO3v/3NREZGGklm3Lhxxhhj8vLyzOOPP26Sk5ONr6+viY2NNTfeeKPZsmWL83lvv/22qVGjhgkMDDS9evUyzz33nEvNZ3vjjTdM06ZNTXBwsAkLCzMdO3Y03377rXP6r7/+anr37m2Cg4NNaGioueWWW0xGRkapY5CXl2eGDh1qIiIiTHR0tJk0aZK54YYbzODBgy9obI0x5tVXXzW1a9c2vr6+pl69embmzJku0yWZ+fPnu7Q5HA4zffr0Etf1XNtzWT+/rf8u+E8tKytLDodDmZmZbjtZGQAqysmTJ7Vr1y7VqlVLAQEBni4HuCTn2p7L+vldKa7GAgAAcBfCDgAAsDXCDgAAsDXCDgAAsDXCDgDYFNefwA7KYzsm7ACAzRR9O+2JE575sU6gPBVtx2d/6/KFqDQ/BAoAKB/e3t6qVq2a87eGgoKCSv2ZAKCyMsboxIkTOnDggKpVq3ZJXxRM2AEAGyr6ZWp3/rgiUBGqVat23l+xPx/CDgDYkGVZiouLU3R09EX/oCPgab6+vuXy00+EHQCwMW9vb4/+TiBQGXCCMgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDWPhp1JkybpqquuUmhoqKKjo9WnTx/t2LHDpY8xRuPHj1d8fLwCAwPVvn17/fjjjy59cnNzdf/996t69eoKDg5W7969lZ6eXpGrAgAAKimPhp3Vq1crLS1N69ev17Jly5Sfn6/OnTsrOzvb2WfKlCmaOnWqXn75ZW3YsEGxsbG6/vrrdezYMWef4cOHa/78+ZozZ47WrFmj48ePq2fPniooKPDEagEAgErEMsYYTxdR5ODBg4qOjtbq1at13XXXyRij+Ph4DR8+XGPGjJF0ei9OTEyMJk+erL/+9a/KzMxUVFSUZs2apf79+0uS9u3bp8TERC1evFhdunQ573KzsrLkcDiUmZmpsLAwt64jAAAoH2X9/K5U5+xkZmZKkiIiIiRJu3btUkZGhjp37uzs4+/vr3bt2mnt2rWSpE2bNunUqVMufeLj45WSkuLsc7bc3FxlZWW53AAAgD1VmrBjjNGIESPUtm1bpaSkSJIyMjIkSTExMS59Y2JinNMyMjLk5+en8PDwUvucbdKkSXI4HM5bYmJiea8OAACoJCpN2Lnvvvu0ZcsWzZ49u9g0y7JcHhtjirWd7Vx9Hn74YWVmZjpve/fuvfjCAQBApVYpws7999+vhQsXauXKlapRo4azPTY2VpKK7aE5cOCAc29PbGys8vLydOTIkVL7nM3f319hYWEuNwAAYE8eDTvGGN13332aN2+eVqxYoVq1arlMr1WrlmJjY7Vs2TJnW15enlavXq3WrVtLkpo3by5fX1+XPvv379fWrVudfQAAwJ+XjycXnpaWpn/+85/617/+pdDQUOceHIfDocDAQFmWpeHDh2vixImqW7eu6tatq4kTJyooKEiDBg1y9r377rs1cuRIRUZGKiIiQqNGjVLjxo3VqVMnT64eAACoBDwadl577TVJUvv27V3ap0+friFDhkiSRo8erZycHA0bNkxHjhxRy5YttXTpUoWGhjr7v/DCC/Lx8VG/fv2Uk5Ojjh07asaMGfL29q6oVQEAAJVUpfqeHU/he3YAAKh6quT37AAAAJQ3wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg4AALA1wg5sYUv6UQ18Y722pB/1dCkAgEqGsANbmPftb1q385Dmffubp0sBAFQyPp4uALhY6UdO6Ej2KVmW9Mn3+ySdvu/bvIaMkcKDfVUjPMjDVQIAPI2wgyqr7eSVzn9b/70/nJ2nni+tcbbvfqZHBVcFAKhsOIyFKmta/6by8Todc8x/24rufbwsTevf1BNlAQAqGfbsoMrq0yxBl0WHuOzJKbIgrY1SEhweqAoAUNmwZwe2YFmu9wAAFGHPDqq0yBA/RYX4K65agPpflagPNuzV/qMnFRni5+nSAACVhGWMMefvZm9ZWVlyOBzKzMxUWFiYp8vBBcrNL5Cft5csy5IxRnkFhfL38fZ0WQAANyvr5zd7dlDlnRlsLMsi6AAAXHDODgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDWPhp1///vf6tWrl+Lj42VZlhYsWOAyfciQIbIsy+XWqlUrlz65ubm6//77Vb16dQUHB6t3795KT0+vwLUAAACVmUfDTnZ2tq644gq9/PLLpfbp2rWr9u/f77wtXrzYZfrw4cM1f/58zZkzR2vWrNHx48fVs2dPFRQUuLt8AABQBfh4cuHdunVTt27dztnH399fsbGxJU7LzMzU22+/rVmzZqlTp06SpPfee0+JiYlavny5unTpUu41AwCAqqXSn7OzatUqRUdHq169errnnnt04MAB57RNmzbp1KlT6ty5s7MtPj5eKSkpWrt2rSfKBQAAlYxH9+ycT7du3XTLLbcoKSlJu3bt0mOPPaYOHTpo06ZN8vf3V0ZGhvz8/BQeHu7yvJiYGGVkZJQ639zcXOXm5jofZ2VluW0dAACAZ1XqsNO/f3/nv1NSUtSiRQslJSXp008/1U033VTq84wxsiyr1OmTJk3ShAkTyrVWAABQOVX6w1hniouLU1JSkn766SdJUmxsrPLy8nTkyBGXfgcOHFBMTEyp83n44YeVmZnpvO3du9etdQMAAM+pUmHn0KFD2rt3r+Li4iRJzZs3l6+vr5YtW+bss3//fm3dulWtW7cudT7+/v4KCwtzuQEAAHvy6GGs48eP6+eff3Y+3rVrlzZv3qyIiAhFRERo/PjxuvnmmxUXF6fdu3frkUceUfXq1XXjjTdKkhwOh+6++26NHDlSkZGRioiI0KhRo9S4cWPn1VkAAODPzaNhZ+PGjUpNTXU+HjFihCRp8ODBeu211/TDDz9o5syZOnr0qOLi4pSamqoPPvhAoaGhzue88MIL8vHxUb9+/ZSTk6OOHTtqxowZ8vb2rvD1AQAAlY9ljDGeLsLTsrKy5HA4lJmZySEtAACqiLJ+flepc3YAAAAuFGEHAADYGmEHAADYGmEHAADYGmEHAADYWpnDzl133aVjx465sxYAAIByV+aw8+677yonJ8edtQAAAJS7Mocdvo4HAABURRd0zs65fkkcAACgMrqgn4uoV6/eeQPP4cOHL6kgAACA8nRBYWfChAlyOBzuqgUAAKDcXVDYGTBggKKjo91VCwAAQLkr8zk7nK8DAACqIq7GAgAAtlbmw1iFhYXurAMAAMAt+LkIAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAGW2Jf2oBr6xXlvSj3q6FAAoM8IOgDKb9+1vWrfzkOZ9+5unSwGAMvPxdAEAKrf0Iyd0JPuULEv65Pt9kk7f921eQ8ZI4cG+qhEe5OEqAaB0hB0A59R28krnv63/3h/OzlPPl9Y423c/06OCqwKAsuMwFoBzmta/qXy8Tscc89+2onsfL0vT+jf1RFkAUGbs2QFwTn2aJeiy6BCXPTlFFqS1UUqCwwNVAUDZsWcHQJlZlus9AFQF7NkBcF6RIX6KCvFXXLUA9b8qUR9s2Kv9R08qMsTP06UBwHlZxhhz/m72lpWVJYfDoczMTIWFhXm6HKBSys0vkJ+3lyzLkjFGeQWF8vfx9nRZAP7Eyvr5zZ4dAGVyZrCxLIugA6DK4JwdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABga4QdAABgax4NO//+97/Vq1cvxcfHy7IsLViwwGW6MUbjx49XfHy8AgMD1b59e/34448ufXJzc3X//ferevXqCg4OVu/evZWenl6BawEAACozj4ad7OxsXXHFFXr55ZdLnD5lyhRNnTpVL7/8sjZs2KDY2Fhdf/31OnbsmLPP8OHDNX/+fM2ZM0dr1qzR8ePH1bNnTxUUFFTUagAAgErMMsYYTxchSZZlaf78+erTp4+k03t14uPjNXz4cI0ZM0bS6b04MTExmjx5sv76178qMzNTUVFRmjVrlvr37y9J2rdvnxITE7V48WJ16dKlTMvOysqSw+FQZmamwsLC3LJ+AACgfJX187vSnrOza9cuZWRkqHPnzs42f39/tWvXTmvXrpUkbdq0SadOnXLpEx8fr5SUFGefkuTm5iorK8vlBgAA7KnShp2MjAxJUkxMjEt7TEyMc1pGRob8/PwUHh5eap+STJo0SQ6Hw3lLTEws5+oBAEBlUWnDThHLslweG2OKtZ3tfH0efvhhZWZmOm979+4tl1oBAEDlU2nDTmxsrCQV20Nz4MAB596e2NhY5eXl6ciRI6X2KYm/v7/CwsJcbgAAwJ4qbdipVauWYmNjtWzZMmdbXl6eVq9erdatW0uSmjdvLl9fX5c++/fv19atW519AADAn5uPJxd+/Phx/fzzz87Hu3bt0ubNmxUREaGaNWtq+PDhmjhxourWrau6detq4sSJCgoK0qBBgyRJDodDd999t0aOHKnIyEhFRERo1KhRaty4sTp16uSp1QIAAJWIR8POxo0blZqa6nw8YsQISdLgwYM1Y8YMjR49Wjk5ORo2bJiOHDmili1baunSpQoNDXU+54UXXpCPj4/69eunnJwcdezYUTNmzJC3t3eFrw8AAKh8Ks337HgS37MDAEDVU+W/ZwcAAKA8EHYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAoJLZkn5UA99Yry3pRz1dCmALhB0AqGTmffub1u08pHnf/ubpUgBb8PF0AQAAKf3ICR3JPiXLkj75fp+k0/d9m9eQMVJ4sK9qhAd5uEqgaiLsAEAl0HbySue/rf/eH87OU8+X1jjbdz/To4KrAuyBw1gAUAlM699UPl6nY475b1vRvY+XpWn9m3qiLMAW2LMDAJVAn2YJuiw6xGVPTpEFaW2UkuDwQFWAPbBnBwAqGctyvQdwadizAwCVRGSIn6JC/BVXLUD9r0rUBxv2av/Rk4oM8fN0aUCVZhljzPm72VtWVpYcDocyMzMVFhbm6XIA/Inl5hfIz9tLlmXJGKO8gkL5+3h7uiygUirr5zd7dgCgEjkz2FiWRdABygHn7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAA/pS2pB/VwDfWa0v6UU+XAjcj7AAA/pTmffub1u08pHnf/ubpUuBmPp4uAACAipJ+5ISOZJ+SZUmffL9P0un7vs1ryBgpPNhXNcKDPFwlylul3rMzfvx4WZblcouNjXVON8Zo/Pjxio+PV2BgoNq3b68ff/zRgxUDACqztpNXqtfLa9TzpTU6nJ0nSTqcnaeeL61Rr5fXqO3klR6uEO5QqcOOJF1++eXav3+/8/bDDz84p02ZMkVTp07Vyy+/rA0bNig2NlbXX3+9jh075sGKAQCV1bT+TeXjZUmSzH/biu59vCxN69/UE2XBzSp92PHx8VFsbKzzFhUVJen0Xp1p06bp73//u2666SalpKTo3Xff1YkTJ/TPf/7Tw1UDACqjPs0StCCtTYnTFqS1UZ9mCRVcESpCpQ87P/30k+Lj41WrVi0NGDBAO3fulCTt2rVLGRkZ6ty5s7Ovv7+/2rVrp7Vr155znrm5ucrKynK5AQD+XCzL9R72VanDTsuWLTVz5kx9/vnnevPNN5WRkaHWrVvr0KFDysjIkCTFxMS4PCcmJsY5rTSTJk2Sw+Fw3hITE922DgCAyiUyxE9RIf5qnODQ0zemqHGCQ1Eh/ooM8fN0aXATyxhjzt+tcsjOzladOnU0evRotWrVSm3atNG+ffsUFxfn7HPPPfdo7969WrJkSanzyc3NVW5urvNxVlaWEhMTlZmZqbCwMLeuAwDA83LzC+Tn7SXLsmSMUV5Bofx9vD1dFi5QVlaWHA7HeT+/K/WenbMFBwercePG+umnn5xXZZ29F+fAgQPF9vaczd/fX2FhYS43AMCfh7+Pt6z/Hr+yLIugY3NVKuzk5uZq+/btiouLU61atRQbG6tly5Y5p+fl5Wn16tVq3bq1B6sEAACVSaX+UsFRo0apV69eqlmzpg4cOKCnnnpKWVlZGjx4sCzL0vDhwzVx4kTVrVtXdevW1cSJExUUFKRBgwZ5unQAAFBJVOqwk56eroEDB+qPP/5QVFSUWrVqpfXr1yspKUmSNHr0aOXk5GjYsGE6cuSIWrZsqaVLlyo0NNTDlQMAgMqiSp2g7C5lPcEJAABUHrY8QRkAAOBCEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAICtEXYAAIDbbEk/qoFvrNeW9KMeq4GwAwAA3Gbet79p3c5Dmvftbx6rwcdjSwYAALaUfuSEjmSfkmVJn3y/T9Lp+77Na8gYKTzYVzXCgyqsHsIOAAAoV20nr3T+2/rv/eHsPPV8aY2zffczPSqsHg5jAQCAcjWtf1P5eJ2OOea/bUX3Pl6WpvVvWqH1sGcHAACUqz7NEnRZdIjLnpwiC9LaKCXBUaH1sGcHAAC4jWW53nsCe3YAAEC5iwzxU1SIv+KqBaj/VYn6YMNe7T96UpEhfhVei2WMMefvZm9ZWVlyOBzKzMxUWFiYp8sBAMAWcvML5OftJcuyZIxRXkGh/H28y23+Zf38Zs8OAABwizODjWVZ5Rp0LgTn7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFsj7AAAAFvj5yIkFf08WFZWlocrAQAAZVX0uX2+n/kk7Eg6duyYJCkxMdHDlQAAgAt17NgxORyOUqfzq+eSCgsLtW/fPoWGhsqyrHKbb1ZWlhITE7V3715+Td3NGOuKwThXDMa5YjDOFcOd42yM0bFjxxQfHy8vr9LPzGHPjiQvLy/VqFHDbfMPCwvjhVRBGOuKwThXDMa5YjDOFcNd43yuPTpFOEEZAADYGmEHAADYGmHHjfz9/TVu3Dj5+/t7uhTbY6wrBuNcMRjnisE4V4zKMM6coAwAAGyNPTsAAMDWCDsAAMDWCDsAAMDWCDsAAMDWCDsX6NVXX1WtWrUUEBCg5s2b68svvzxn/9WrV6t58+YKCAhQ7dq19frrrxfr8/HHH6tRo0by9/dXo0aNNH/+fHeVX2WU9zi/+eabuvbaaxUeHq7w8HB16tRJ33zzjTtXoUpwx/ZcZM6cObIsS3369Cnnqqsed4zz0aNHlZaWpri4OAUEBKhhw4ZavHixu1ahynDHWE+bNk3169dXYGCgEhMT9dBDD+nkyZPuWoUq4ULGef/+/Ro0aJDq168vLy8vDR8+vMR+bv0sNCizOXPmGF9fX/Pmm2+abdu2mQcffNAEBwebX3/9tcT+O3fuNEFBQebBBx8027ZtM2+++abx9fU1c+fOdfZZu3at8fb2NhMnTjTbt283EydOND4+Pmb9+vUVtVqVjjvGedCgQeaVV14x3333ndm+fbu58847jcPhMOnp6RW1WpWOO8a5yO7du01CQoK59tprzQ033ODmNanc3DHOubm5pkWLFqZ79+5mzZo1Zvfu3ebLL780mzdvrqjVqpTcMdbvvfee8ff3N++//77ZtWuX+fzzz01cXJwZPnx4Ra1WpXOh47xr1y7zwAMPmHfffdc0bdrUPPjgg8X6uPuzkLBzAa6++mrzt7/9zaWtQYMGZuzYsSX2Hz16tGnQoIFL21//+lfTqlUr5+N+/fqZrl27uvTp0qWLGTBgQDlVXfW4Y5zPlp+fb0JDQ82777576QVXUe4a5/z8fNOmTRvz1ltvmcGDB//pw447xvm1114ztWvXNnl5eeVfcBXmjrFOS0szHTp0cOkzYsQI07Zt23Kquuq50HE+U7t27UoMO+7+LOQwVhnl5eVp06ZN6ty5s0t7586dtXbt2hKfs27dumL9u3Tpoo0bN+rUqVPn7FPaPO3OXeN8thMnTujUqVOKiIgon8KrGHeO8xNPPKGoqCjdfffd5V94FeOucV64cKGuueYapaWlKSYmRikpKZo4caIKCgrcsyJVgLvGum3bttq0aZPzsPfOnTu1ePFi9ejRww1rUfldzDiXhbs/C/kh0DL6448/VFBQoJiYGJf2mJgYZWRklPicjIyMEvvn5+frjz/+UFxcXKl9Spun3blrnM82duxYJSQkqFOnTuVXfBXirnH+6quv9Pbbb2vz5s3uKr1Kcdc479y5UytWrNCtt96qxYsX66efflJaWpry8/P1+OOPu219KjN3jfWAAQN08OBBtW3bVsYY5efna+jQoRo7dqzb1qUyu5hxLgt3fxYSdi6QZVkuj40xxdrO1//s9gud55+BO8a5yJQpUzR79mytWrVKAQEB5VBt1VWe43zs2DHddtttevPNN1W9evXyL7YKK+/tubCwUNHR0XrjjTfk7e2t5s2ba9++fXr22Wf/tGGnSHmP9apVq/T000/r1VdfVcuWLfXzzz/rwQcfVFxcnB577LFyrr7qcMfnljs/Cwk7ZVS9enV5e3sXS5kHDhwolkaLxMbGltjfx8dHkZGR5+xT2jztzl3jXOS5557TxIkTtXz5cjVp0qR8i69C3DHOP/74o3bv3q1evXo5pxcWFkqSfHx8tGPHDtWpU6ec16Ryc9f2HBcXJ19fX3l7ezv7NGzYUBkZGcrLy5Ofn185r0nl566xfuyxx3T77bfrL3/5iySpcePGys7O1r333qu///3v8vL6c50NcjHjXBbu/iz8c/2VLoGfn5+aN2+uZcuWubQvW7ZMrVu3LvE511xzTbH+S5cuVYsWLeTr63vOPqXN0+7cNc6S9Oyzz+rJJ5/UkiVL1KJFi/Ivvgpxxzg3aNBAP/zwgzZv3uy89e7dW6mpqdq8ebMSExPdtj6Vlbu25zZt2ujnn392hklJ+r//+z/FxcX9KYOO5L6xPnHiRLFA4+3tLXP6Ap9yXIOq4WLGuSzc/llYLqc5/0kUXW739ttvm23btpnhw4eb4OBgs3v3bmOMMWPHjjW33367s3/RZY0PPfSQ2bZtm3n77beLXdb41VdfGW9vb/PMM8+Y7du3m2eeeYZLz90wzpMnTzZ+fn5m7ty5Zv/+/c7bsWPHKnz9Kgt3jPPZuBrLPeO8Z88eExISYu677z6zY8cOs2jRIhMdHW2eeuqpCl+/ysQdYz1u3DgTGhpqZs+ebXbu3GmWLl1q6tSpY/r161fh61dZXOg4G2PMd999Z7777jvTvHlzM2jQIPPdd9+ZH3/80Tnd3Z+FhJ0L9Morr5ikpCTj5+dnrrzySrN69WrntMGDB5t27dq59F+1apVp1qyZ8fPzM8nJyea1114rNs+PPvrI1K9f3/j6+poGDRqYjz/+2N2rUemV9zgnJSUZScVu48aNq4C1qbzcsT2fibBzmjvGee3ataZly5bG39/f1K5d2zz99NMmPz/f3atS6ZX3WJ86dcqMHz/e1KlTxwQEBJjExEQzbNgwc+TIkQpYm8rrQse5pPffpKQklz7u/Cy0/lsEAACALXHODgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgDbOXjwoGJjYzVx4kRn29dffy0/Pz8tXbrUg5UB8AR+GwuALS1evFh9+vTR2rVr1aBBAzVr1kw9evTQtGnTPF0agApG2AFgW2lpaVq+fLmuuuoqff/999qwYYMCAgI8XRaACkbYAWBbOTk5SklJ0d69e7Vx40Y1adLE0yUB8ADO2QFgWzt37tS+fftUWFioX3/91dPlAPAQ9uwAsKW8vDxdffXVatq0qRo0aKCpU6fqhx9+UExMjKdLA1DBCDsAbOl//ud/NHfuXH3//fcKCQlRamqqQkNDtWjRIk+XBqCCcRgLgO2sWrVK06ZN06xZsxQWFiYvLy/NmjVLa9as0Wuvvebp8gBUMPbsAAAAW2PPDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsDXCDgAAsLX/B2g9m7pRhHEIAAAAAElFTkSuQmCC",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The estimated temperature at the nodes are: ['250.00', '168.77', '115.29', '78.86', '52.21', '30.00'] [C]\n",
- "The temperature at x=0.02 is: 168.77 [C]\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
- "Ts = 30\n",
- "alpha = 500\n",
- "dx=0.02\n",
- "\n",
- "# grid creation\n",
- "x = np.arange(0,0.1+dx,dx)\n",
- "T = np.zeros(x.shape)\n",
- "n=len(x)\n",
- "\n",
- "# boundary conditions\n",
- "T[0] = 250\n",
- "T[-1] = Ts\n",
- "\n",
- "# Building matrix A\n",
- "matrix_element = -(2+dx**2*alpha)\n",
- "A = np.zeros((len(x)-2,len(x)-2))\n",
- "np.fill_diagonal(A, matrix_element)\n",
- "A[np.arange(n-3), np.arange(1, n-2)] = 1 # Upper diagonal\n",
- "A[np.arange(1, n-2), np.arange(n-3)] = 1 # Lower diagonal\n",
- "\n",
- "# Building vector b\n",
- "b_element = -dx**2*alpha*Ts\n",
- "b = np.zeros(len(x)-2) + b_element\n",
- "b[0] = b[0] - T[0]\n",
- "b[-1] = b[-1] - T[-1]\n",
- "\n",
- "# Solving the system\n",
- "T[1:-1] = np.linalg.solve(A,b)\n",
- "\n",
- "plt.plot(x,T,'*',label='Estimated solution')\n",
- "plt.xlabel('x')\n",
- "plt.ylabel('T')\n",
- "plt.title('Estimated solution using Forward Difference method')\n",
- "plt.legend()\n",
- "plt.show()\n",
- "\n",
- "print(f'The estimated temperature at the nodes are: {[f\"{temp:.2f}\" for temp in T]} [C]')\n",
- "\n",
- "#to test your temperature at x=0.02 for task 9\n",
- "ind = np.argmin(abs(x-0.02))\n",
- "print(f'The temperature at x=0.02 is: {T[ind]:.2f} [C]')"
+ "YOUR_CODE_HERE"
]
},
{
@@ -692,9 +599,9 @@
"id": "bac6bb0f",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 9:</b>\n",
+ "<b>Task 2.4:</b>\n",
"\n",
"How much finer does your grid has to be in the forward difference implementation to get a similar value at x = 0.02 as in the central difference implementation? Vary your dx.\n",
"\n",
@@ -705,20 +612,22 @@
},
{
"cell_type": "markdown",
- "id": "db91d68b",
+ "id": "dd2ead47",
"metadata": {},
- "source": []
+ "source": [
+ "_Your answer here._"
+ ]
},
{
"cell_type": "markdown",
"id": "031dca37",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
"<p>\n",
- "<b>Bonus note:</b> \n",
+ "<b>Bonus Task</b> \n",
" \n",
- "The matrix inversion using numpy is one way to solve the system, another is the gauss_jordan written below and another one is the sparse matrix based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change dx to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.\n",
+ "The matrix inversion using numpy is one way to solve the system, another is the <code>gauss_jordan</code> method, written below, and another one is the sparse matrix-based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change <code>dx</code> to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.\n",
" \n",
"</p>\n",
"</div>"
@@ -771,20 +680,10 @@
},
{
"cell_type": "code",
- "execution_count": 106,
+ "execution_count": null,
"id": "fbd32c69",
"metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The time used by direct matrix inversion solution is 2.037e-02 sec\n",
- "The time used by Gauss-jordan solution is 2.169e+00 sec\n",
- "The time used by the sparse matrix solver is 7.204e-03 sec\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
"import time\n",
"from scipy.sparse import csc_matrix\n",
diff --git a/src/teachers/Week_1_6/WS_1_6_solution.html b/src/teachers/Week_1_6/WS_1_6_solution.html
index 9cb23b25..b468d24f 100644
--- a/src/teachers/Week_1_6/WS_1_6_solution.html
+++ b/src/teachers/Week_1_6/WS_1_6_solution.html
@@ -7531,7 +7531,7 @@ a.anchor-link {
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
<h2 id="Overview">Overview<a class="anchor-link" href="#Overview">¶</a></h2><p>This assignment is aimed to develop an understanding of the <strong>Ordinary Differential Equation (ODE)</strong>. There will be two sections about cooling and heating scenerios, corresponding to the first-order and the second-order ODEs. Please go through the text that follows and perform all steps outlined therein.</p>
-<h2 id="Section-1:-First-order-ODE">Section 1: First-order ODE<a class="anchor-link" href="#Section-1:-First-order-ODE">¶</a></h2><h3 id="1.1-Linear-ODE">1.1 Linear ODE<a class="anchor-link" href="#1.1-Linear-ODE">¶</a></h3><p>In the study of heat transfer, <strong>Newton's law of cooling</strong> is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. It can be expressed in the form of ODE, as below:</p>
+<h2 id="Part-1:-First-order-ODE">Part 1: First-order ODE<a class="anchor-link" href="#Part-1:-First-order-ODE">¶</a></h2><p>In the study of heat transfer, <strong>Newton's law of cooling</strong> is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. It can be expressed in the form of ODE, as below:</p>
$$\frac{dT}{dt}=-k(T - T_s)$$<p>where $T$ is the temperature of the object at time $t$, $T_s$ is the temperature of the surrounding and assumed to be constant, and $k$ is the constant that characterizes the ability of the object to exchange the
heat energy (unit 1/s), which depends on the specific material properties.</p>
<p>Now, Let's consider an object with the initial temperature of 50°C in a surrounding environment with constant temperature at 20°C. The constant of heat exchange between the object and the environment is 0.5 $sec^-1$.</p>
@@ -7545,9 +7545,9 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 1:</b>
+<b>Task 1.1:</b>
<p>Suppose the considered period of time is long enough (bring it to steady state), what will be the final temperature of the object?</p>
<p><strong>Write your answer in the following markdown cell.</strong></p>
</p>
@@ -7562,13 +7562,12 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
-<p>Your answers for Task 1:
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
<p>
<p>20°C</p>
<p><em>This is because the body's temperature, $T$, will tend to the ambience temperature in the long term. The decrease will be abrupt at the beginning and will slow down as time passes $T(t=large) = T_s = 20°C$.</em></p>
</p>
-</p></div>
+</div>
</div>
</div>
</div>
@@ -7590,9 +7589,9 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 2:</b>
+<b>Task 1.2:</b>
<p>Write the algebraic representation of the ODE using Explicit Euler.</p>
</p>
</div>
@@ -7606,9 +7605,9 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
<p>
-Answer for Task 2:
+Answer for Task 1.2:
<p>
<p>$$ T_{i+1} = T_{i}-k(T_{i} - T_s)dt$$</p>
<p>How to get there:</p>
@@ -7630,16 +7629,17 @@ $$<p>or from forward finite difference</p>
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=579a9440">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=fb6eda82">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3:</b>
-<p>Compute the temperature evolution in the next 60 seconds. This task is defined in subtasks.</p>
+<b>Task 1.3:</b>
+<p>Compute the temperature evolution in the next 60 seconds.</p>
+<p><strong>Please complete the missing parts of the code in each step below, which is divided into 5 substeps (a through e).</strong></p>
</p>
</div>
</div>
@@ -7652,9 +7652,10 @@ $$<p>or from forward finite difference</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3.1:</b> The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.
+<b>Task 1.3a:</b>
+<p>The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.</p>
</p>
</div>
</div>
@@ -7692,9 +7693,10 @@ $$<p>or from forward finite difference</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3.2:</b> Implement your time discretization and find the solution from $t=0$ until $t=60$ sec.
+<b>Task 1.3b:</b>
+<p>Implement your time discretization and find the solution from $t=0$ until $t=60$ sec.</p>
</p>
</div>
</div>
@@ -7739,9 +7741,9 @@ $$<p>or from forward finite difference</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3.3:</b>
+<b>Task 1.3c:</b>
<p>Try different time steps to check the stability of the calculation. At which value the solution is stable?</p>
</p>
</div>
@@ -7755,7 +7757,7 @@ $$<p>or from forward finite difference</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
<p>
<b>Answer:</b>
<p>4 seconds is the limit between an unstable and a stable solution.</p>
@@ -7771,9 +7773,9 @@ $$<p>or from forward finite difference</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3.4:</b>
+<b>Task 1.3d:</b>
<p>Obtain the mathematical expression that proves your stability criteria.</p>
</p>
</div>
@@ -7787,7 +7789,7 @@ $$<p>or from forward finite difference</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
<p> Answer
<p>
$$
@@ -7816,9 +7818,9 @@ Then $|k*dt| < 2$, thus $dt < 4$.</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3.5:</b>
+<b>Task 1.3e:</b>
<p>Now, discretize the equation using Implicit (Backward) Euler. Can you find a time step that makes the problem unstable?</p>
</p>
</div>
@@ -7868,16 +7870,32 @@ Then $|k*dt| < 2$, thus $dt < 4$.</p>
</div>
</div>
</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=f534fb27">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<p>
+<b>Answer:</b>
+<p>No, you cannot! This method is unconditionally stable. (a good thing to remember for the exam)</p>
+</p>
+</div>
+</div>
+</div>
+</div>
+</div>
<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=2109f010">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<h2 id="Section-2:-Second-order-ODE">Section 2: Second-order ODE<a class="anchor-link" href="#Section-2:-Second-order-ODE">¶</a></h2><h3 id="Background">Background<a class="anchor-link" href="#Background">¶</a></h3><p>The following 1D equation describes the steady state solution of the temperature along a pin that sticks out of a furnace. The rest of the pin is exposed to the ambient.</p>
+<h2 id="Part-2:-Second-order-ODE">Part 2: Second-order ODE<a class="anchor-link" href="#Part-2:-Second-order-ODE">¶</a></h2><p>The following 1D equation describes the steady state solution of the temperature along a pin that sticks out of a furnace. The rest of the pin is exposed to the ambient.</p>
$$
\frac{d^2T}{dx^2} -\alpha(T-T_s)=0
-$$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is $250^o$ C. The length of the pint is 0.1m. Your grid should have a spatial step of 0.02 m. Finally, $\alpha=500$.</p>
+$$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is $250^o$ C. The length of the pin is 0.1m. Your grid should have a spatial step of 0.02 m. Finally, $\alpha=500$.</p>
</div>
</div>
</div>
@@ -7901,41 +7919,48 @@ $$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=c739aaf1">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=c74adcb4">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 5:</b>
-<p>Below we already provide the code structured that solves the equation with the above Dirichlet conditions and using a central difference approximation of the second derivative. Complete the code to complete the matrix and vectors. <strong>Write in paper (add the image) the matrix A, and the vectors T and b.</strong></p>
+<b>Task 2.1:</b>
+<p>This task has three parts: a) discretize the analytic expression into a system of equations using central differences, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.</p>
+<p><em>Parts 2.1b and 2.1c do not need to be completed in order; in fact, it may be useful to go back and forth between the two in order to understand the problem.</em></p>
</p></div>
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=11009615">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0db7afc9">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<p>
+<b>Task 2.1a:</b>
+<p>Discretize the analytic expression into a system of equations for a grid with 6 points using central differences.</p>
+<p><em>Write your answer by hand using paper/tablet and include the image below.</em></p>
+</p></div>
</div>
</div>
</div>
-</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=4133c6e2">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=6ae1eb87">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
-<p> Answer
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<p> Answer for 2.1a and 2.1b
<p>
-<p>Add the image.</p>
-<p><strong>For $i = 1$:</strong>
+$$
+\frac{1}{\Delta x^2}(T_{i-1} - 2T_{i} + T_{i+1}) - \alpha T_{i} + \alpha T_s = 0
+$$<p><strong>For $i = 1$:</strong>
The finite difference approximation for the first internal node is:</p>
$$
\frac{1}{\Delta x^2}(T_0 - 2T_1 + T_2) - \alpha T_1 + \alpha T_s = 0
@@ -7957,7 +7982,36 @@ $$
$$<p>Notice that now we can plug in our other bc $T_5$ and rearrange the terms:</p>
$$
(T_3 - (2 + \alpha \Delta x^2) T_4 ) = -\alpha Ts \Delta x^2 -T_5
-$$<p>We can now move the equations in the Ax=y form:</p>
+$$</p>
+</p></div>
+</div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=df2341fc">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<p>
+<b>Task 2.1b:</b>
+<p>Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.</p>
+<p><em>Write your answer by hand using paper/tablet and include the image below.</em></p>
+</p></div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=c8616d1c">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<p>
+<p>The discretized equations are presented in Ax=y form:</p>
$$
\begin{bmatrix}
-(2 + \alpha \Delta x^2) & 1 & 0 & 0 \\
@@ -7978,10 +8032,35 @@ T_4
-\alpha Ts \Delta x^2 -T_5
\end{bmatrix}
$$</p>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0c12775e">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<p>
+<b>Task 2.1c:</b>
+<p>Implement the discretized system of equations in a code cell.</p>
+<p><em>We have already done this for you! Your task is to read the cell and make sure you understand how the matrices are implemented. Reading the code should help you formulate the matrices in Task 2.1b.</em></p>
</p></div>
</div>
</div>
</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=11009615">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+</div>
+</div>
+</div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell" id="cell-id=53fb4f99">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
@@ -8062,9 +8141,9 @@ $$</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 6:</b>
+<b>Task 2.2:</b>
<p>Copy the code above below and modify it to replace the Dirichlet condition at the right BC for a Neumann condition such that there $\frac{dT}{dx}=0$. Approximate the Neuman boundary $\frac{dT}{dx}=0$ by using the Backward difference for first order differential equation of first order accuracy.</p>
<p>How does the temperature profile changes?</p>
<p><strong>Remember to change the title of the plot to indicate that now the BC is of the Neumann type</strong></p>
@@ -8195,19 +8274,33 @@ $$
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0b47e9c2">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=ac255017">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 7:</b>
-<p>Now, instead of using the central difference for approximating the second derivative. Use the forward difference to replace the second derivative (first order accurate). <strong>Write the algebraic representation of the differential equation below.</strong></p>
-</p>
+<b>Task 2.3:</b>
+<p>Just as we did in Task 2.1, this task has three parts: a) discretize the analytic expression into a system of equations using <b>forward differences</b>, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.</p>
+</p></div>
+</div>
+</div>
+</div>
</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=104690ed">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<p>
+<b>Task 2.3a:</b>
+<p>Discretize the analytic expression into a system of equations for a grid with 6 points using <b>forward differences</b>.</p>
+<p><em>Write your answer by hand using paper/tablet and include the image below.</em></p>
+</p></div>
</div>
</div>
</div>
@@ -8217,8 +8310,7 @@ $$
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
-<p> Answer
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
<p>
Forward difference for second order differential equation:
$$
@@ -8228,7 +8320,7 @@ The finite difference approximation for the first internal node is:</p>
$$
\frac{1}{\Delta x^2}(T_0 - 2T_1 + T_2) - \alpha T_0 + \alpha T_s = 0
$$$$
--2T_1 +T_2 = -(1+alpha)T_0 - \alpha T_s \Delta x^2
+-2T_1 +T_2 = -(1+\alpha)T_0 - \alpha T_s \Delta x^2
$$<p><strong>For $i = 1$:</strong>
The finite difference approximation for the second internal node is:</p>
$$
@@ -8251,24 +8343,23 @@ $$<p>Rearranging the terms:</p>
$$
(1 - \alpha \Delta x^2)T_3 - 2T_4 = -\alpha T_s \Delta x^2 -T_5
$$</p>
-</p></div>
</div>
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=a4e0342e">
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=7b12c86e">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 8:</b>
-<p>Now, write the resulting matrix A, the vector T and the vector b.</p>
-</p>
-</div>
-</div>
+<b>Task 2.3b:</b>
+<p>Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.</p>
+<p><em>Write your answer by hand using paper/tablet and include the image below.</em></p>
+</p></div>
</div>
</div>
</div>
@@ -8303,48 +8394,20 @@ $$
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=b9ca660d">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=1b304504">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-</div>
-</div>
-</div>
-</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=8cb492f8">
-<div class="jp-Cell-inputWrapper" tabindex="0">
-<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
-</div>
-<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
-</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
-<p> Answer
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<p>Add the image.</p>
-</p>
+<b>Task 2.3c:</b>
+<p>Implement the discretized system of equations in a code cell.</p>
+<p><b>This time we did not do it for you!</b> Copy the code from Task 2.1c and revise it to solve the system of equations using <b>Forward Differences</b>. Keep the Dirichlet conditions.</p>
</p></div>
</div>
</div>
-</div>
-</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0640b43e">
-<div class="jp-Cell-inputWrapper" tabindex="0">
-<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
-</div>
-<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
-</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
-<p>
-<b>Task 9:</b>
-<p>Copy the code of task 5 below, modify it accordingly to your new discretization (using FD for the second derivative in space). Keep Dirichlet conditions.</p>
-<p>How different are the values of the temperature with respect to task 5? Which solution is expected to be more accurate?</p>
-</p>
-</div>
-</div>
-</div>
-</div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell" id="cell-id=1e783d3e">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
@@ -8439,15 +8502,15 @@ The temperature at x=0.02 is: 168.04 [C]
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=64d693df">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=18d60370">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 9:</b>
+<b>Task 2.4:</b>
<p>How much finer does your grid has to be in the forward difference implementation to get a similar value at x = 0.02 as in the central difference implementation? Vary your dx.</p>
</p>
</div>
@@ -8455,27 +8518,31 @@ The temperature at x=0.02 is: 168.04 [C]
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=e771703c">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=7ef65db5">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<p>you can test this above make the temperature the same at x=0.02</p>
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<p>
+<p>You can test this above make the temperature the same at x=0.02</p>
+</p>
+</div>
</div>
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=031dca37">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=b78e8e16">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
<p>
-<b>Bonus note:</b>
-<p>The matrix inversion using numpy is one way to solve the system, another is the gauss_jordan written below and another one is the sparse matrix based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change dx to 0.0002 of the code in task 5 and run it. Now, you should have a matrix A with a high resolution that solves the second degree ODE. Test the time each method takes.</p>
+<b>Bonus Task</b>
+<p>The matrix inversion using numpy is one way to solve the system, another is the <code>gauss_jordan</code> method, written below, and another one is the sparse matrix-based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change <code>dx</code> to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.</p>
</p>
</div>
</div>
@@ -8579,15 +8646,31 @@ The temperature at x=0.02 is: 168.04 [C]
<div class="jp-OutputArea-child">
<div class="jp-OutputPrompt jp-OutputArea-prompt"></div>
<div class="jp-RenderedText jp-OutputArea-output" data-mime-type="text/plain" tabindex="0">
-<pre>The time used by direct matrix inversion solution is 2.522e-03 sec
-The time used by Gauss-jordan solution is 2.020e-01 sec
-The time used by the sparse matrix solver is 3.001e-03 sec
+<pre>The time used by direct matrix inversion solution is 1.399e-03 sec
+The time used by Gauss-jordan solution is 1.411e-01 sec
+The time used by the sparse matrix solver is 1.286e-03 sec
</pre>
</div>
</div>
</div>
</div>
</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=5d5b9189">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<p> Answer
+<p>
+<p>The sparse matrix is fastest, but we see that the Numpy implementation is also very fast! problem still small; Gauss-Jordan slow because...???</p>
+</p>
+</p></div>
+</div>
+</div>
+</div>
+</div>
<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=2921b3f2">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
@@ -8619,7 +8702,7 @@ The time used by the sparse matrix solver is 3.001e-03 sec
</div>
</div>
</div>
-</div></main>
+</div></div></div></div></div></div></div></main>
</body>
<script type="application/vnd.jupyter.widget-state+json">
{"state": {}, "version_major": 2, "version_minor": 0}
diff --git a/src/teachers/Week_1_6/WS_1_6_solution.ipynb b/src/teachers/Week_1_6/WS_1_6_solution.ipynb
new file mode 100644
index 00000000..bf46f829
--- /dev/null
+++ b/src/teachers/Week_1_6/WS_1_6_solution.ipynb
@@ -0,0 +1,1279 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "1d48ad6c",
+ "metadata": {
+ "id": "9adbf457-797f-45b7-8f8b-0e46e0e2f5ff"
+ },
+ "source": [
+ "# WS 1.6: Understanding Ordinary Differential Equation\n",
+ "\n",
+ "<h1 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; top: 60px;right: 30px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\" style=\"width:100px\" />\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\" style=\"width:100px\" />\n",
+ "\n",
+ "</h1>\n",
+ "<h2 style=\"height: 10px\">\n",
+ "</h2>\n",
+ "\n",
+ "*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/): Week 1.6. For: 9th October, 2024.*"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "366ef404",
+ "metadata": {},
+ "source": [
+ "## Overview\n",
+ "\n",
+ "\n",
+ "This assignment is aimed to develop an understanding of the **Ordinary Differential Equation (ODE)**. There will be two sections about cooling and heating scenerios, corresponding to the first-order and the second-order ODEs. Please go through the text that follows and perform all steps outlined therein.\n",
+ "\n",
+ "## Part 1: First-order ODE\n",
+ "\n",
+ "In the study of heat transfer, **Newton's law of cooling** is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. It can be expressed in the form of ODE, as below:\n",
+ "\n",
+ "$$\\frac{dT}{dt}=-k(T - T_s)$$\n",
+ "\n",
+ "where $T$ is the temperature of the object at time $t$, $T_s$ is the temperature of the surrounding and assumed to be constant, and $k$ is the constant that characterizes the ability of the object to exchange the\n",
+ "heat energy (unit 1/s), which depends on the specific material properties.\n",
+ "\n",
+ "\n",
+ "Now, Let's consider an object with the initial temperature of 50°C in a surrounding environment with constant temperature at 20°C. The constant of heat exchange between the object and the environment is 0.5 $sec^-1$.\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "73781a3c",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1.1:</b>\n",
+ " \n",
+ "Suppose the considered period of time is long enough (bring it to steady state), what will be the final temperature of the object? \n",
+ " \n",
+ "**Write your answer in the following markdown cell.**\n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3081d88d",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ " \n",
+ "20°C \n",
+ " \n",
+ "*This is because the body's temperature, $T$, will tend to the ambience temperature in the long term. The decrease will be abrupt at the beginning and will slow down as time passes $T(t=large) = T_s = 20°C$.*\n",
+ "\n",
+ "</p>\n",
+ "</div>\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "06320ed9",
+ "metadata": {},
+ "source": [
+ "Next, let's evaluate the temperature of the object by checking it at a series of time points."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "18cbde20",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1.2:</b>\n",
+ "\n",
+ "Write the algebraic representation of the ODE using Explicit Euler.\n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bec070a5",
+ "metadata": {},
+ "source": [
+ " <div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ "Answer for Task 1.2:\n",
+ "<p>\n",
+ " \n",
+ " $$ T_{i+1} = T_{i}-k(T_{i} - T_s)dt$$\n",
+ "\n",
+ "\n",
+ "\n",
+ "How to get there:\n",
+ "\n",
+ "$$\n",
+ "\\int_{t}^{t+1} \\frac{dT}{dt} dt=\\int_{t}^{t+1}-k(T - T_s)dt\n",
+ "$$\n",
+ "$$\n",
+ "\\int_{t}^{t+1} T' dt=\\int_{t}^{t+1}f(t)dt\n",
+ "$$\n",
+ "\n",
+ "FTC (fundamental theorem of calculus) for left side and using left rieman to approximate right term\n",
+ "\n",
+ "$$\n",
+ "T_{t+1} - T_t = \\Delta t f(t_i)\n",
+ "$$\n",
+ "\n",
+ "$$\n",
+ "T_{t+1}= T_t + \\Delta t f(t_i)\n",
+ "$$\n",
+ "\n",
+ "$$\n",
+ "T_{t+1}= T_t - k(T_t - T_s)\n",
+ "$$\n",
+ "\n",
+ "or from forward finite difference\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fb6eda82",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1.3:</b>\n",
+ "\n",
+ "Compute the temperature evolution in the next 60 seconds.\n",
+ "\n",
+ "**Please complete the missing parts of the code in each step below, which is divided into 5 substeps (a through e).**\n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "88b861bc",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1.3a:</b> \n",
+ "\n",
+ "The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.\n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "b44aa057",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "dt =5 \n",
+ "t_end = 60 \n",
+ "Ts = 20 # [C] \n",
+ "k = 0.5 # [sec^-1]\n",
+ "\n",
+ "t = np.arange(0,t_end+dt,dt)\n",
+ "n = len(t)\n",
+ "T = np.empty(n)\n",
+ "T[0] = 50\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4586d4b6",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1.3b:</b> \n",
+ "\n",
+ "Implement your time discretization and find the solution from $t=0$ until $t=60$ sec. \n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "0bf8247e",
+ "metadata": {
+ "scrolled": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "for i in range(n-1):\n",
+ " T[i+1] = T[i]-k*(T[i]-Ts)*dt \n",
+ " \n",
+ "plt.plot(t, T, 'o-')\n",
+ "plt.xlabel('t (sec)')\n",
+ "plt.ylabel('T (deg)')\n",
+ "plt.grid()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ffb4547f",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1.3c:</b>\n",
+ "\n",
+ "Try different time steps to check the stability of the calculation. At which value the solution is stable?\n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "59bcbb0a",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Answer:</b> \n",
+ " \n",
+ "4 seconds is the limit between an unstable and a stable solution.\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8a907a09",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1.3d:</b>\n",
+ "\n",
+ "Obtain the mathematical expression that proves your stability criteria.\n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "eb00c9ab",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p> Answer\n",
+ "<p>\n",
+ " \n",
+ "$$\n",
+ "\\frac{dT}{dt}= -k(T-Ts)\n",
+ "$$\n",
+ "\n",
+ "$$\n",
+ "\\int \\frac{1}{(T-Ts)}dT=\\int -k dt\n",
+ "$$\n",
+ "\n",
+ "$$\n",
+ "ln|T-Ts| = -kt +C\n",
+ "$$\n",
+ "\n",
+ "$$\n",
+ "T-Ts = Ce^{-kt}\n",
+ "$$\n",
+ "\n",
+ "$$\n",
+ "T= Ts +Ce^{-kt}\n",
+ "$$\n",
+ "\n",
+ "\n",
+ "This is an Explicit Euler scheme, thus is conditionally stable, with the criterion: $ |1-k*dt|<1$. \n",
+ "\n",
+ "Solving for n time steps, the solution can be generalized as $T_n=T_0(1-k\\Delta t)^n + k \\Delta t T_s \\left[ (1-k \\Delta t)^{n-1} + (1-k \\Delta t)^{n-2} + ... + (1-k \\Delta t)^{0} \\right]$ \n",
+ "\n",
+ "The criterion of stability is $ |1-k*dt|<1$. Otherwise, the solution would tend to grow instead of decaying. \n",
+ "Then $|k*dt| < 2$, thus $dt < 4$. \n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1ff1e4fe",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1.3e:</b>\n",
+ "\n",
+ "Now, discretize the equation using Implicit (Backward) Euler. Can you find a time step that makes the problem unstable?\n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "18c73272",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "dt = 10 \n",
+ "t_end = 60 \n",
+ "Ts = 20 # [C] \n",
+ "k = 0.5 # [sec^-1]\n",
+ "\n",
+ "t = np.arange(0,t_end+dt,dt)\n",
+ "n = len(t)\n",
+ "T = np.empty(n)\n",
+ "T[0] = 50 \n",
+ "\n",
+ "for i in range(n-1):\n",
+ " T[i+1] = (T[i]+k*Ts*dt)/(1+k*dt) \n",
+ " \n",
+ "plt.plot(t, T, 'o-')\n",
+ "plt.xlabel('t (sec)')\n",
+ "plt.ylabel('T (deg)')\n",
+ "plt.grid()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f534fb27",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Answer:</b> \n",
+ " \n",
+ "No, you cannot! This method is unconditionally stable. (a good thing to remember for the exam)\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2109f010",
+ "metadata": {},
+ "source": [
+ "## Part 2: Second-order ODE\n",
+ "\n",
+ "The following 1D equation describes the steady state solution of the temperature along a pin that sticks out of a furnace. The rest of the pin is exposed to the ambient. \n",
+ "\n",
+ "$$\n",
+ "\\frac{d^2T}{dx^2} -\\alpha(T-T_s)=0\n",
+ "$$\n",
+ "\n",
+ "The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is $250^o$ C. The length of the pin is 0.1m. Your grid should have a spatial step of 0.02 m. Finally, $\\alpha=500$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5bf9639d",
+ "metadata": {},
+ "source": [
+ "\n",
+ "The solution includes the steps:\n",
+ "1. Use the Taylor series to obtain an approximation for the derivatives;\n",
+ "2. Discretize the equation;\n",
+ "3. Define parameters and grid;\n",
+ "4. Provide boundary conditions;\n",
+ "5. Build matrix with solution $AT=b$\n",
+ "6. Solve the matrix"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c74adcb4",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1:</b>\n",
+ "\n",
+ "This task has three parts: a) discretize the analytic expression into a system of equations using central differences, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.\n",
+ "\n",
+ "<em>Parts 2.1b and 2.1c do not need to be completed in order; in fact, it may be useful to go back and forth between the two in order to understand the problem.</em> \n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0db7afc9",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1a:</b>\n",
+ "\n",
+ "Discretize the analytic expression into a system of equations for a grid with 6 points using central differences.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6ae1eb87",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p> Answer for 2.1a and 2.1b\n",
+ "<p>\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_{i-1} - 2T_{i} + T_{i+1}) - \\alpha T_{i} + \\alpha T_s = 0\n",
+ "$$\n",
+ "\n",
+ "**For $i = 1$:**\n",
+ "The finite difference approximation for the first internal node is:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_0 - 2T_1 + T_2) - \\alpha T_1 + \\alpha T_s = 0\n",
+ "$$\n",
+ "\n",
+ "Next, we multiply by $\\Delta x^2$ and substitute the boundary condition for $T_0$. Notice that the only unknown solutions are $T_1$ and $T_2$\n",
+ "Rearranging the terms gives:\n",
+ "\n",
+ "$$\n",
+ "-(2 + \\alpha \\Delta x^2) T_1 + T_2 = -\\alpha T_s \\Delta x^2 - T_0\n",
+ "$$\n",
+ "\n",
+ "**For $i = 2$:**\n",
+ "The finite difference approximation for the second internal node is:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_1 - 2T_2 + T_3) - \\alpha T_2 + \\alpha T_s = 0\n",
+ "$$\n",
+ "\n",
+ "Rearranging the terms:\n",
+ "\n",
+ "$$\n",
+ "T_1 - (2 + \\alpha \\Delta x^2) T_2 + T_3 = -\\alpha T_s \\Delta x^2\n",
+ "$$\n",
+ "\n",
+ "**For $i = 3$:**\n",
+ "The finite difference approximation for the third internal node is:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_2 - 2T_3 + T_4) - \\alpha T_3 + \\alpha T_s = 0\n",
+ "$$\n",
+ "\n",
+ "Notice that now we can plug in our other bc $T_5$ and rearrange the terms:\n",
+ "\n",
+ "$$\n",
+ "(T_3 - (2 + \\alpha \\Delta x^2) T_4 ) = -\\alpha Ts \\Delta x^2 -T_5 \n",
+ "$$\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "df2341fc",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1b:</b>\n",
+ "\n",
+ "Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c8616d1c",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ "\n",
+ "The discretized equations are presented in Ax=y form:\n",
+ "\n",
+ "$$\n",
+ "\\begin{bmatrix}\n",
+ " -(2 + \\alpha \\Delta x^2) & 1 & 0 & 0 \\\\\n",
+ "1 & -(2 + \\alpha \\Delta x^2) & 1 & 0 \\\\\n",
+ "0 & 1 & -(2 + \\alpha \\Delta x^2) & 1 \\\\\n",
+ "0 & 0 & 1 & -(2 + \\alpha \\Delta x^2)\n",
+ "\\end{bmatrix} \\begin{bmatrix}\n",
+ "T_1 \\\\\n",
+ "T_2 \\\\\n",
+ "T_3 \\\\\n",
+ "T_4\n",
+ "\\end{bmatrix}\n",
+ "=\n",
+ "\\begin{bmatrix}\n",
+ "-\\alpha Ts \\Delta x^2 -T_0\\\\\n",
+ "-\\alpha Ts \\Delta x^2 \\\\\n",
+ "-\\alpha Ts \\Delta x^2\\\\\n",
+ "-\\alpha Ts \\Delta x^2 -T_5\n",
+ "\\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0c12775e",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1c:</b>\n",
+ "\n",
+ "Implement the discretized system of equations in a code cell.\n",
+ "\n",
+ "<em>We have already done this for you! Your task is to read the cell and make sure you understand how the matrices are implemented. Reading the code should help you formulate the matrices in Task 2.1b.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "11009615",
+ "metadata": {},
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "id": "53fb4f99",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The estimated temperature at the nodes are: ['250.00', '168.77', '115.29', '78.86', '52.21', '30.00'] [C]\n"
+ ]
+ }
+ ],
+ "source": [
+ "import numpy as np \n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "Ts = 30\n",
+ "alpha = 500\n",
+ "dx=0.02\n",
+ "\n",
+ "# grid creation\n",
+ "x = np.arange(0,0.1+dx,dx)\n",
+ "T = np.zeros(x.shape)\n",
+ "n=len(x)\n",
+ "\n",
+ "# boundary conditions\n",
+ "T[0] = 250\n",
+ "T[-1] = Ts\n",
+ "\n",
+ "# Building matrix A\n",
+ "matrix_element = -(2+dx**2*alpha)\n",
+ "A = np.zeros((len(x)-2,len(x)-2))\n",
+ "np.fill_diagonal(A, matrix_element)\n",
+ "A[np.arange(n-3), np.arange(1, n-2)] = 1 # Upper diagonal\n",
+ "A[np.arange(1, n-2), np.arange(n-3)] = 1 # Lower diagonal\n",
+ "\n",
+ "# Building vector b\n",
+ "b_element = -dx**2*alpha*Ts\n",
+ "b = np.zeros(len(x)-2) + b_element\n",
+ "b[0] = b[0] - T[0]\n",
+ "b[-1] = b[-1] - T[-1]\n",
+ "\n",
+ "# Solving the system\n",
+ "\n",
+ "T[1:-1] = np.linalg.solve(A,b)\n",
+ "\n",
+ "plt.plot(x,T,'*',label='Estimated solution')\n",
+ "plt.xlabel('x')\n",
+ "plt.ylabel('T')\n",
+ "plt.title('Estimated solution using Central Difference method')\n",
+ "plt.legend()\n",
+ "plt.show()\n",
+ "\n",
+ "print(f'The estimated temperature at the nodes are: {[f\"{temp:.2f}\" for temp in T]} [C]')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "72ed77c5",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.2:</b>\n",
+ "\n",
+ "Copy the code above below and modify it to replace the Dirichlet condition at the right BC for a Neumann condition such that there $\\frac{dT}{dx}=0$. Approximate the Neuman boundary $\\frac{dT}{dx}=0$ by using the Backward difference for first order differential equation of first order accuracy. \n",
+ "\n",
+ "How does the temperature profile changes? \n",
+ "\n",
+ "**Remember to change the title of the plot to indicate that now the BC is of the Neumann type**\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f89b7b33",
+ "metadata": {},
+ "source": [
+ "The first elements for i = 1,2,3 are the same as above in task 5\n",
+ "\n",
+ "The neuman bc is on the last node \n",
+ "\n",
+ "$$\n",
+ "\\frac{dT}{dx} = \\frac{T_i - T_{i-1}}{\\Delta x} \n",
+ "$$\n",
+ "\n",
+ "Neuman bc $$\\frac{dT_5}{dx}=0 \\approx \\frac{T_5 - T_{4}}{\\Delta x} $$\n",
+ "Therefore $T_5=T_4$\n",
+ "\n",
+ "**For $i = 4$**\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_3-2T_4+T_5) - \\alpha T_4 + \\alpha Ts =0\n",
+ "$$\n",
+ "\n",
+ "Plug in BC $T_5=T_4$\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_3-2T_4+T_4) - \\alpha T_4 + \\alpha Ts =0\n",
+ "$$\n",
+ "\n",
+ "$$\n",
+ "T_3-(1 + \\alpha \\Delta x^2)T_4 = \\alpha Ts \n",
+ "$$\n",
+ "\n",
+ "We can now move the equations in the Ax=y form:\n",
+ "\n",
+ "$$\n",
+ "\\begin{bmatrix}\n",
+ " -(2 + \\alpha \\Delta x^2) & 1 & 0 & 0 \\\\\n",
+ "1 & -(2 + \\alpha \\Delta x^2) & 1 & 0 \\\\\n",
+ "0 & 1 & -(2 + \\alpha \\Delta x^2) & 1 \\\\\n",
+ "0 & 0 & 1 & -(1 + \\alpha \\Delta x^2)\n",
+ "\\end{bmatrix} \\begin{bmatrix}\n",
+ "T_1 \\\\\n",
+ "T_2 \\\\\n",
+ "T_3 \\\\\n",
+ "T_4\n",
+ "\\end{bmatrix}\n",
+ "=\n",
+ "\\begin{bmatrix}\n",
+ "-\\alpha Ts \\Delta x^2 -T_0\\\\\n",
+ "-\\alpha Ts \\Delta x^2 \\\\\n",
+ "-\\alpha Ts \\Delta x^2\\\\\n",
+ "-\\alpha Ts \\Delta x^2 \n",
+ "\\end{bmatrix}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "cfcd00f9",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The estimated temperature at the nodes are: ['250.00', '174.84', '128.64', '102.18', '90.15', '90.15'] [C]\n"
+ ]
+ }
+ ],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ " \n",
+ "Ts = 30\n",
+ "alpha = 500\n",
+ "dx=0.02\n",
+ " \n",
+ "# grid creation\n",
+ "x = np.arange(0,0.1+dx,dx)\n",
+ "T = np.zeros(x.shape)\n",
+ "n=len(x)\n",
+ " \n",
+ "# boundary conditions\n",
+ "T[0] = 250\n",
+ " \n",
+ " \n",
+ "# Building matrix A\n",
+ "matrix_element = -(2+dx**2*alpha)\n",
+ "A = np.zeros((len(x)-2,len(x)-2))\n",
+ "np.fill_diagonal(A, matrix_element)\n",
+ "A[np.arange(n-3), np.arange(1, n-2)] = 1 # Upper diagonal\n",
+ "A[np.arange(1, n-2), np.arange(n-3)] = 1 # Lower diagonal\n",
+ " \n",
+ "A[-1,-1] = -(1+dx**2*alpha) #the lower right corner of the matrix changes\n",
+ " \n",
+ "# Building vector b\n",
+ "b_element = -dx**2*alpha*Ts\n",
+ "b = np.zeros(len(x)-2) + b_element\n",
+ "b[0] = b[0] - T[0]\n",
+ "b[-1] = b[-1] #the vector b also changes\n",
+ " \n",
+ "# Solving the system\n",
+ "\n",
+ "T[1:-1] = np.linalg.solve(A,b)\n",
+ "T[-1] = T[-2] \n",
+ " \n",
+ "plt.plot(x,T,'*',label='Estimated solution')\n",
+ "plt.xlabel('x')\n",
+ "plt.ylabel('T')\n",
+ "plt.title('Estimated solution using Central Difference method with Neumann BC')\n",
+ "plt.legend()\n",
+ "plt.show()\n",
+ " \n",
+ "print(f'The estimated temperature at the nodes are: {[f\"{temp:.2f}\" for temp in T]} [C]')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ac255017",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.3:</b>\n",
+ "\n",
+ "Just as we did in Task 2.1, this task has three parts: a) discretize the analytic expression into a system of equations using <b>forward differences</b>, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.\n",
+ "</div> \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "104690ed",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.3a:</b>\n",
+ "\n",
+ "Discretize the analytic expression into a system of equations for a grid with 6 points using <b>forward differences</b>.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "74174f3a",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ "Forward difference for second order differential equation:\n",
+ "\n",
+ "$$\n",
+ "\\frac{T_i-2T_{i+1}+T_{i+2}}{\\Delta x^2}-\\alpha(T_i-T_s) = 0\n",
+ "$$\n",
+ "\n",
+ "**For $i = 0$:**\n",
+ "The finite difference approximation for the first internal node is:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_0 - 2T_1 + T_2) - \\alpha T_0 + \\alpha T_s = 0\n",
+ "$$\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "-2T_1 +T_2 = -(1+\\alpha)T_0 - \\alpha T_s \\Delta x^2\n",
+ "$$\n",
+ "\n",
+ "**For $i = 1$:**\n",
+ "The finite difference approximation for the second internal node is:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_1 - 2T_2 + T_3) - \\alpha T_1 + \\alpha T_s = 0\n",
+ "$$\n",
+ "\n",
+ "Rearranging the terms:\n",
+ "\n",
+ "$$\n",
+ "(1 - \\alpha \\Delta x^2)T_1 - 2T_2 + T_3 = -\\alpha T_s \\Delta x^2\n",
+ "$$\n",
+ "\n",
+ "**For $i = 2$:**\n",
+ "The finite difference approximation for the second internal node is:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_2 - 2T_3 + T_4) - \\alpha T_2 + \\alpha T_s = 0\n",
+ "$$\n",
+ "\n",
+ "Rearranging the terms:\n",
+ "\n",
+ "$$\n",
+ "(1 - \\alpha \\Delta x^2)T_2 - T_3 + T_4 = -\\alpha T_s \\Delta x^2\n",
+ "$$\n",
+ "\n",
+ "**For $i = 3$:**\n",
+ "The finite difference approximation for the second internal node is:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{\\Delta x^2}(T_3 - 2T_4 + T_5) - \\alpha T_3 + \\alpha T_s = 0\n",
+ "$$\n",
+ "\n",
+ "Rearranging the terms:\n",
+ "\n",
+ "$$\n",
+ "(1 - \\alpha \\Delta x^2)T_3 - 2T_4 = -\\alpha T_s \\Delta x^2 -T_5\n",
+ "$$\n",
+ "\n",
+ "\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7b12c86e",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.3b:</b>\n",
+ "\n",
+ "Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8d0bbab6",
+ "metadata": {},
+ "source": [
+ "We can now move the equations in the Ax=y form:\n",
+ "\n",
+ "$$\n",
+ "\\begin{bmatrix}\n",
+ " -2 & 1 & 0 & 0 \\\\\n",
+ "(1- \\alpha \\Delta x^2) & -2 & 1 & 0 \\\\\n",
+ "0 & (1- \\alpha \\Delta x^2) & -2 & 1 \\\\\n",
+ "0 & 0 & (1- \\alpha \\Delta x^2) & -2\n",
+ "\\end{bmatrix} \\begin{bmatrix}\n",
+ "T_1 \\\\\n",
+ "T_2 \\\\\n",
+ "T_3 \\\\\n",
+ "T_4\n",
+ "\\end{bmatrix}\n",
+ "=\n",
+ "\\begin{bmatrix}\n",
+ "-\\alpha Ts \\Delta x^2 -T_0\\\\\n",
+ "-\\alpha Ts \\Delta x^2 \\\\\n",
+ "-\\alpha Ts \\Delta x^2\\\\\n",
+ "-\\alpha Ts \\Delta x^2 -T_5\n",
+ "\\end{bmatrix}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1b304504",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.3c:</b>\n",
+ "\n",
+ "Implement the discretized system of equations in a code cell.\n",
+ "\n",
+ "<b>This time we did not do it for you!</b> Copy the code from Task 2.1c and revise it to solve the system of equations using <b>Forward Differences</b>. Keep the Dirichlet conditions.\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "id": "1e783d3e",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[[-2. 1. 0. ... 0. 0. 0. ]\n",
+ " [ 0.99992 -2. 1. ... 0. 0. 0. ]\n",
+ " [ 0. 0.99992 -2. ... 0. 0. 0. ]\n",
+ " ...\n",
+ " [ 0. 0. 0. ... -2. 1. 0. ]\n",
+ " [ 0. 0. 0. ... 0.99992 -2. 1. ]\n",
+ " [ 0. 0. 0. ... 0. 0.99992 -2. ]]\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The x values =============================: ['0.0000', '0.0004', '0.0008', '0.0012', '0.0016', '0.0020', '0.0024', '0.0028', '0.0032', '0.0036', '0.0040', '0.0044', '0.0048', '0.0052', '0.0056', '0.0060', '0.0064', '0.0068', '0.0072', '0.0076', '0.0080', '0.0084', '0.0088', '0.0092', '0.0096', '0.0100', '0.0104', '0.0108', '0.0112', '0.0116', '0.0120', '0.0124', '0.0128', '0.0132', '0.0136', '0.0140', '0.0144', '0.0148', '0.0152', '0.0156', '0.0160', '0.0164', '0.0168', '0.0172', '0.0176', '0.0180', '0.0184', '0.0188', '0.0192', '0.0196', '0.0200', '0.0204', '0.0208', '0.0212', '0.0216', '0.0220', '0.0224', '0.0228', '0.0232', '0.0236', '0.0240', '0.0244', '0.0248', '0.0252', '0.0256', '0.0260', '0.0264', '0.0268', '0.0272', '0.0276', '0.0280', '0.0284', '0.0288', '0.0292', '0.0296', '0.0300', '0.0304', '0.0308', '0.0312', '0.0316', '0.0320', '0.0324', '0.0328', '0.0332', '0.0336', '0.0340', '0.0344', '0.0348', '0.0352', '0.0356', '0.0360', '0.0364', '0.0368', '0.0372', '0.0376', '0.0380', '0.0384', '0.0388', '0.0392', '0.0396', '0.0400', '0.0404', '0.0408', '0.0412', '0.0416', '0.0420', '0.0424', '0.0428', '0.0432', '0.0436', '0.0440', '0.0444', '0.0448', '0.0452', '0.0456', '0.0460', '0.0464', '0.0468', '0.0472', '0.0476', '0.0480', '0.0484', '0.0488', '0.0492', '0.0496', '0.0500', '0.0504', '0.0508', '0.0512', '0.0516', '0.0520', '0.0524', '0.0528', '0.0532', '0.0536', '0.0540', '0.0544', '0.0548', '0.0552', '0.0556', '0.0560', '0.0564', '0.0568', '0.0572', '0.0576', '0.0580', '0.0584', '0.0588', '0.0592', '0.0596', '0.0600', '0.0604', '0.0608', '0.0612', '0.0616', '0.0620', '0.0624', '0.0628', '0.0632', '0.0636', '0.0640', '0.0644', '0.0648', '0.0652', '0.0656', '0.0660', '0.0664', '0.0668', '0.0672', '0.0676', '0.0680', '0.0684', '0.0688', '0.0692', '0.0696', '0.0700', '0.0704', '0.0708', '0.0712', '0.0716', '0.0720', '0.0724', '0.0728', '0.0732', '0.0736', '0.0740', '0.0744', '0.0748', '0.0752', '0.0756', '0.0760', '0.0764', '0.0768', '0.0772', '0.0776', '0.0780', '0.0784', '0.0788', '0.0792', '0.0796', '0.0800', '0.0804', '0.0808', '0.0812', '0.0816', '0.0820', '0.0824', '0.0828', '0.0832', '0.0836', '0.0840', '0.0844', '0.0848', '0.0852', '0.0856', '0.0860', '0.0864', '0.0868', '0.0872', '0.0876', '0.0880', '0.0884', '0.0888', '0.0892', '0.0896', '0.0900', '0.0904', '0.0908', '0.0912', '0.0916', '0.0920', '0.0924', '0.0928', '0.0932', '0.0936', '0.0940', '0.0944', '0.0948', '0.0952', '0.0956', '0.0960', '0.0964', '0.0968', '0.0972', '0.0976', '0.0980', '0.0984', '0.0988', '0.0992', '0.0996', '0.1000'] [m]\n",
+ "The estimated temperature at the nodes are: ['250.00', '247.99', '245.99', '244.01', '242.05', '240.11', '238.18', '236.27', '234.38', '232.50', '230.64', '228.80', '226.97', '225.16', '223.36', '221.58', '219.81', '218.06', '216.33', '214.61', '212.90', '211.21', '209.54', '207.88', '206.23', '204.60', '202.98', '201.37', '199.78', '198.21', '196.64', '195.09', '193.56', '192.03', '190.52', '189.03', '187.54', '186.07', '184.61', '183.16', '181.73', '180.31', '178.90', '177.50', '176.11', '174.74', '173.38', '172.03', '170.69', '169.36', '168.04', '166.74', '165.44', '164.16', '162.88', '161.62', '160.37', '159.13', '157.90', '156.68', '155.47', '154.27', '153.08', '151.90', '150.73', '149.57', '148.42', '147.28', '146.15', '145.02', '143.91', '142.81', '141.71', '140.63', '139.55', '138.48', '137.42', '136.37', '135.33', '134.30', '133.27', '132.26', '131.25', '130.25', '129.26', '128.27', '127.30', '126.33', '125.37', '124.42', '123.47', '122.54', '121.61', '120.68', '119.77', '118.86', '117.96', '117.07', '116.18', '115.30', '114.43', '113.56', '112.70', '111.85', '111.00', '110.16', '109.33', '108.50', '107.68', '106.87', '106.06', '105.26', '104.47', '103.68', '102.89', '102.12', '101.34', '100.58', '99.82', '99.06', '98.32', '97.57', '96.83', '96.10', '95.37', '94.65', '93.94', '93.22', '92.52', '91.82', '91.12', '90.43', '89.74', '89.06', '88.38', '87.71', '87.04', '86.38', '85.72', '85.07', '84.42', '83.78', '83.13', '82.50', '81.87', '81.24', '80.61', '80.00', '79.38', '78.77', '78.16', '77.56', '76.96', '76.36', '75.77', '75.18', '74.60', '74.01', '73.44', '72.86', '72.29', '71.72', '71.16', '70.60', '70.04', '69.49', '68.94', '68.39', '67.85', '67.30', '66.77', '66.23', '65.70', '65.17', '64.64', '64.12', '63.60', '63.08', '62.56', '62.05', '61.54', '61.03', '60.53', '60.02', '59.52', '59.02', '58.53', '58.03', '57.54', '57.05', '56.57', '56.08', '55.60', '55.12', '54.64', '54.17', '53.69', '53.22', '52.75', '52.28', '51.81', '51.35', '50.89', '50.42', '49.96', '49.51', '49.05', '48.59', '48.14', '47.69', '47.24', '46.79', '46.34', '45.90', '45.45', '45.01', '44.56', '44.12', '43.68', '43.24', '42.80', '42.37', '41.93', '41.50', '41.06', '40.63', '40.20', '39.77', '39.34', '38.91', '38.48', '38.05', '37.62', '37.19', '36.77', '36.34', '35.92', '35.49', '35.07', '34.64', '34.22', '33.80', '33.37', '32.95', '32.53', '32.11', '31.69', '31.26', '30.84', '30.42', '30.00'] [C]\n",
+ "The temperature at x=0.02 is: 168.04 [C]\n"
+ ]
+ }
+ ],
+ "source": [
+ "import numpy as np \n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "Ts = 30\n",
+ "alpha = 500\n",
+ "dx=0.0004 # The grid size is reduced to (0.02^2 = 0.0004)\n",
+ "\n",
+ "# grid creation\n",
+ "x = np.arange(0,0.1+dx,dx)\n",
+ "T = np.zeros(x.shape)\n",
+ "n=len(x)\n",
+ "\n",
+ "# boundary conditions\n",
+ "T[0] = 250\n",
+ "T[-1] = Ts\n",
+ "\n",
+ "# Building matrix A\n",
+ "matrix_element = 1-alpha*dx**2 # The matrix element changes \n",
+ "A = np.zeros((len(x)-2,len(x)-2))\n",
+ "np.fill_diagonal(A, -2)\n",
+ "A[np.arange(n-3), np.arange(1, n-2)] = 1 # Upper diagonal\n",
+ "A[np.arange(1, n-2), np.arange(n-3)] = matrix_element # Lower diagonal\n",
+ "print(A)\n",
+ "\n",
+ "# Building vector b\n",
+ "b_element = -dx**2*alpha*Ts\n",
+ "b = np.zeros(len(x)-2) + b_element\n",
+ "b[0] = b[0] - matrix_element * T[0]\n",
+ "b[-1] = b[-1] - T[-1]\n",
+ "\n",
+ "# Solving the system\n",
+ "\n",
+ "T[1:-1] = np.linalg.solve(A,b)\n",
+ "\n",
+ "plt.plot(x,T,'*',label='Estimated solution')\n",
+ "plt.xlabel('x')\n",
+ "plt.ylabel('T')\n",
+ "plt.title('Estimated solution using Forward Difference method')\n",
+ "plt.legend()\n",
+ "plt.show()\n",
+ "\n",
+ "print(f'The x values =============================: {[f\"{x_loc:.4f}\" for x_loc in x]} [m]')\n",
+ "print(f'The estimated temperature at the nodes are: {[f\"{temp:.2f}\" for temp in T]} [C]')\n",
+ "\n",
+ "ind = np.argmin(abs(x-0.02))\n",
+ "print(f'The temperature at x=0.02 is: {T[ind]:.2f} [C]')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "18d60370",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.4:</b>\n",
+ "\n",
+ "How much finer does your grid has to be in the forward difference implementation to get a similar value at x = 0.02 as in the central difference implementation? Vary your dx.\n",
+ "\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7ef65db5",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ " \n",
+ "You can test this above make the temperature the same at x=0.02\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b78e8e16",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Bonus Task</b> \n",
+ " \n",
+ "The matrix inversion using numpy is one way to solve the system, another is the <code>gauss_jordan</code> method, written below, and another one is the sparse matrix-based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change <code>dx</code> to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.\n",
+ " \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "5f7cdec7",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def gauss_jordan(A, b):\n",
+ " \"\"\"\n",
+ " Solves the system of linear equations Ax = b using Gauss-Jordan elimination.\n",
+ " \n",
+ " Parameters:\n",
+ " A (numpy.ndarray): Coefficient matrix (n x n).\n",
+ " b (numpy.ndarray): Right-hand side vector (n).\n",
+ " \n",
+ " Returns:\n",
+ " numpy.ndarray: Solution vector (x) if the system has a unique solution.\n",
+ " \"\"\"\n",
+ " # Form the augmented matrix [A | b]\n",
+ " A = np.array(A, dtype=float)\n",
+ " b = np.array(b, dtype=float)\n",
+ " aug_matrix = np.hstack([A, b.reshape(-1, 1)])\n",
+ " \n",
+ " n = len(b) # Number of rows (or variables)\n",
+ " \n",
+ " for i in range(n):\n",
+ " # Partial pivoting to handle zero diagonal elements (optional, but more robust)\n",
+ " max_row = np.argmax(np.abs(aug_matrix[i:, i])) + i\n",
+ " if aug_matrix[max_row, i] == 0:\n",
+ " raise ValueError(\"The matrix is singular and cannot be solved.\")\n",
+ " if max_row != i:\n",
+ " aug_matrix[[i, max_row]] = aug_matrix[[max_row, i]]\n",
+ " \n",
+ " # Make the diagonal element 1\n",
+ " aug_matrix[i] = aug_matrix[i] / aug_matrix[i, i]\n",
+ " \n",
+ " # Make all other elements in the current column 0\n",
+ " for j in range(n):\n",
+ " if j != i:\n",
+ " aug_matrix[j] -= aug_matrix[j, i] * aug_matrix[i]\n",
+ " \n",
+ " # Extract the solution (last column of the augmented matrix)\n",
+ " return aug_matrix[:, -1]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "id": "fbd32c69",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The time used by direct matrix inversion solution is 1.399e-03 sec\n",
+ "The time used by Gauss-jordan solution is 1.411e-01 sec\n",
+ "The time used by the sparse matrix solver is 1.286e-03 sec\n"
+ ]
+ }
+ ],
+ "source": [
+ "import time\n",
+ "from scipy.sparse import csc_matrix\n",
+ "from scipy.sparse.linalg import spsolve\n",
+ "\n",
+ "# Inverted matrix solution\n",
+ "start_time = time.time()\n",
+ "A_inv = np.linalg.inv(A)\n",
+ "T[1:-1] = A_inv @ b\n",
+ "time_used_0 = time.time() - start_time\n",
+ "print(f\"The time used by direct matrix inversion solution is {time_used_0: 0.3e} sec\")\n",
+ "assert np.allclose(np.dot(A, T[1:-1]), b), \"Oops! The calculation is wrong..\"\n",
+ "\n",
+ "\n",
+ "# Gauss-jordan solution\n",
+ "start_time = time.time()\n",
+ "u1 = gauss_jordan(A, b)\n",
+ "time_used_1 = time.time() - start_time\n",
+ "print(f\"The time used by Gauss-jordan solution is {time_used_1: 0.3e} sec\")\n",
+ "#Check if the solution is correct:\n",
+ "assert np.allclose(np.dot(A, u1), b), \"Oops! The calculation is wrong..\"\n",
+ "\n",
+ "# Solution by a sparse matrix solver \n",
+ "start_time = time.time()\n",
+ "A_sparse = csc_matrix(A)# Convert A to a compressed sparse column (CSC) matrix\n",
+ "u2 = spsolve(A_sparse, b)\n",
+ "time_used_2 = time.time() - start_time\n",
+ "print(f\"The time used by the sparse matrix solver is {time_used_2: 0.3e} sec\")\n",
+ "#Check if the solution is correct:\n",
+ "assert np.allclose(np.dot(A, u2), b), \"Oops! The calculation is wrong..\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5d5b9189",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<p> Answer\n",
+ "<p>\n",
+ " \n",
+ "The sparse matrix is fastest, but we see that the Numpy implementation is also very fast! problem still small; Gauss-Jordan slow because...???\n",
+ "\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2921b3f2",
+ "metadata": {},
+ "source": [
+ "**End of notebook.**\n",
+ "<h2 style=\"height: 60px\">\n",
+ "</h2>\n",
+ "<h3 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; bottom: 60px; right: 50px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">\n",
+ " <img alt=\"Creative Commons License\" style=\"border-width:; width:88px; height:auto; padding-top:10px\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" />\n",
+ " </a>\n",
+ " <a rel=\"TU Delft\" href=\"https://www.tudelft.nl/en/ceg\">\n",
+ " <img alt=\"TU Delft\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\"/>\n",
+ " </a>\n",
+ " <a rel=\"MUDE\" href=\"http://mude.citg.tudelft.nl/\">\n",
+ " <img alt=\"MUDE\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\"/>\n",
+ " </a>\n",
+ " \n",
+ "</h3>\n",
+ "<span style=\"font-size: 75%\">\n",
+ "© Copyright 2023 <a rel=\"MUDE Team\" href=\"https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595\">MUDE Teaching Team</a> TU Delft. This work is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>."
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "mude-base",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.12.4"
+ },
+ "latex_envs": {
+ "LaTeX_envs_menu_present": true,
+ "autoclose": false,
+ "autocomplete": true,
+ "bibliofile": "biblio.bib",
+ "cite_by": "apalike",
+ "current_citInitial": 1,
+ "eqLabelWithNumbers": true,
+ "eqNumInitial": 1,
+ "hotkeys": {
+ "equation": "Ctrl-E",
+ "itemize": "Ctrl-I"
+ },
+ "labels_anchors": false,
+ "latex_user_defs": false,
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+ "user_envs_cfg": false
+ },
+ "widgets": {
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+ "state": {},
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+ "version_minor": 0
+ }
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/src/teachers/Week_1_6/WS_1_6_student.html b/src/teachers/Week_1_6/WS_1_6_student.html
index 77e6c519..adfe3233 100644
--- a/src/teachers/Week_1_6/WS_1_6_student.html
+++ b/src/teachers/Week_1_6/WS_1_6_student.html
@@ -7531,7 +7531,7 @@ a.anchor-link {
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
<h2 id="Overview">Overview<a class="anchor-link" href="#Overview">¶</a></h2><p>This assignment is aimed to develop an understanding of the <strong>Ordinary Differential Equation (ODE)</strong>. There will be two sections about cooling and heating scenerios, corresponding to the first-order and the second-order ODEs. Please go through the text that follows and perform all steps outlined therein.</p>
-<h2 id="Section-1:-First-order-ODE">Section 1: First-order ODE<a class="anchor-link" href="#Section-1:-First-order-ODE">¶</a></h2><h3 id="1.1-Linear-ODE">1.1 Linear ODE<a class="anchor-link" href="#1.1-Linear-ODE">¶</a></h3><p>In the study of heat transfer, <strong>Newton's law of cooling</strong> is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. It can be expressed in the form of ODE, as below:</p>
+<h2 id="Part-1:-First-order-ODE">Part 1: First-order ODE<a class="anchor-link" href="#Part-1:-First-order-ODE">¶</a></h2><p>In the study of heat transfer, <strong>Newton's law of cooling</strong> is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. It can be expressed in the form of ODE, as below:</p>
$$\frac{dT}{dt}=-k(T - T_s)$$<p>where $T$ is the temperature of the object at time $t$, $T_s$ is the temperature of the surrounding and assumed to be constant, and $k$ is the constant that characterizes the ability of the object to exchange the
heat energy (unit 1/s), which depends on the specific material properties.</p>
<p>Now, Let's consider an object with the initial temperature of 50°C in a surrounding environment with constant temperature at 20°C. The constant of heat exchange between the object and the environment is 0.5 $sec^-1$.</p>
@@ -7545,9 +7545,9 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 1:</b>
+<b>Task 1.1:</b>
<p>Suppose the considered period of time is long enough (bring it to steady state), what will be the final temperature of the object?</p>
<p><strong>Write your answer in the following markdown cell.</strong></p>
</p>
@@ -7583,9 +7583,9 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 2:</b>
+<b>Task 1.2:</b>
<p>Write the algebraic representation of the ODE using Explicit Euler.</p>
</p>
</div>
@@ -7609,36 +7609,27 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
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+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3:</b>
+<b>Task 1.3:</b>
<p>Compute the temperature evolution in the next 60 seconds.</p>
-<p><strong>Please complete the missing parts of the code in each step below</strong></p>
+<p><strong>Please complete the missing parts of the code in each step below, which is divided into 5 substeps (a through e).</strong></p>
</p>
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+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3.1:</b> The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.
+<b>Task 1.3a:</b>
+<p>The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.</p>
</p>
</div>
</div>
@@ -7676,20 +7667,21 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
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+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3.2:</b> Implement your time discretization and find the solution from $t=0$ until $t=60$ sec.
+<b>Task 1.3b:</b>
+<p>Implement your time discretization and find the solution from $t=0$ until $t=60$ sec.</p>
</p>
</div>
</div>
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-<div class="jp-InputPrompt jp-InputArea-prompt">In [21]:</div>
+<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
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<div class="highlight hl-ipython3"><pre><span></span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
@@ -7704,18 +7696,6 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
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@@ -7723,9 +7703,9 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
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+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 3.3:</b>
+<b>Task 1.3c:</b>
<p>Try different time steps to check the stability of the calculation. At which value the solution is stable?</p>
</p>
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@@ -7749,9 +7729,9 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
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<p>
-<b>Task 3.4:</b>
+<b>Task 1.3d:</b>
<p>Obtain the mathematical expression that proves your stability criteria.</p>
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@@ -7775,21 +7755,21 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
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<p>
-<b>Task 3.5:</b>
+<b>Task 1.3e:</b>
<p>Now, discretize the equation using Implicit (Backward) Euler. Can you find a time step that makes the problem unstable?</p>
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-<div class="jp-InputPrompt jp-InputArea-prompt">In [29]:</div>
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<div class="highlight hl-ipython3"><pre><span></span><span class="n">dt</span> <span class="o">=</span> <span class="n">YOUR_CODE_HERE</span>
@@ -7814,18 +7794,6 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
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"/>
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@@ -7833,10 +7801,10 @@ heat energy (unit 1/s), which depends on the specific material properties.</p>
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-<h2 id="Section-2:-Second-order-ODE">Section 2: Second-order ODE<a class="anchor-link" href="#Section-2:-Second-order-ODE">¶</a></h2><h3 id="Background">Background<a class="anchor-link" href="#Background">¶</a></h3><p>The following 1D equation describes the steady state solution of the temperature along a pin that sticks out of a furnace. The rest of the pin is exposed to the ambient.</p>
+<h2 id="Part-2:-Second-order-ODE">Part 2: Second-order ODE<a class="anchor-link" href="#Part-2:-Second-order-ODE">¶</a></h2><p>The following 1D equation describes the steady state solution of the temperature along a pin that sticks out of a furnace. The rest of the pin is exposed to the ambient.</p>
$$
\frac{d^2T}{dx^2} -\alpha(T-T_s)=0
-$$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is $250^o$ C. The length of the pint is 0.1m. Your grid should have a spatial step of 0.02 m. Finally, $\alpha=500$.</p>
+$$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is $250^o$ C. The length of the pin is 0.1m. Your grid should have a spatial step of 0.02 m. Finally, $\alpha=500$.</p>
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@@ -7860,41 +7828,104 @@ $$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is
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-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 5:</b>
-<p>Below we provide the code already implemented of the equation with the above Dirichlet conditions and using a central difference approximation of the second derivative. <strong>Write in paper (add the image) the matrix A, and the vectors T and b.</strong></p>
+<b>Task 2.1:</b>
+<p>This task has three parts: a) discretize the analytic expression into a system of equations using central differences, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.</p>
+<p><em>Parts 2.1b and 2.1c do not need to be completed in order; in fact, it may be useful to go back and forth between the two in order to understand the problem.</em></p>
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-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
-<p> Answer
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<p>Add the image.</p>
-</p>
+<b>Task 2.1a:</b>
+<p>Discretize the analytic expression into a system of equations for a grid with 6 points using central differences.</p>
+<p><em>Write your answer by hand using paper/tablet and include the image below.</em></p>
+</p></div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=81a5f165">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<p><em>Your answer here.</em></p>
+</div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=c513ff65">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<p>
+<b>Task 2.1b:</b>
+<p>Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.</p>
+<p><em>Write your answer by hand using paper/tablet and include the image below.</em></p>
+</p></div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=cb9ff085">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<p><em>Your answer here.</em></p>
+</div>
+</div>
+</div>
+</div>
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=fae61d18">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<p>
+<b>Task 2.1c:</b>
+<p>Implement the discretized system of equations in a code cell.</p>
+<p><em>We have already done this for you! Your task is to read the cell and make sure you understand how the matrices are implemented. Reading the code should help you formulate the matrices in Task 2.1b.</em></p>
</p></div>
</div>
</div>
</div>
-</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell" id="cell-id=53fb4f99">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=89fd190c">
+<div class="jp-Cell-inputWrapper" tabindex="0">
+<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
+</div>
+<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
+</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<p><em>Add your image here.</em></p>
+</div>
+</div>
+</div>
+</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=53fb4f99">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea">
-<div class="jp-InputPrompt jp-InputArea-prompt">In [8]:</div>
+<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
<div class="cm-editor cm-s-jupyter">
<div class="highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
@@ -7942,32 +7973,6 @@ $$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is
</div>
</div>
</div>
-<div class="jp-Cell-outputWrapper">
-<div class="jp-Collapser jp-OutputCollapser jp-Cell-outputCollapser">
-</div>
-<div class="jp-OutputArea jp-Cell-outputArea">
-<div class="jp-OutputArea-child">
-<div class="jp-OutputPrompt jp-OutputArea-prompt"></div>
-<div class="jp-RenderedText jp-OutputArea-output" data-mime-type="text/plain" tabindex="0">
-<pre>(4, 4)
-</pre>
-</div>
-</div>
-<div class="jp-OutputArea-child">
-<div class="jp-OutputPrompt jp-OutputArea-prompt"></div>
-<div class="jp-RenderedImage jp-OutputArea-output" tabindex="0">
-<img alt="No description has been provided for this image" class="" 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-<pre>The estimated temperature at the nodes are: ['250.00', '168.77', '115.29', '78.86', '52.21', '30.00'] [C]
-</pre>
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@@ -7975,9 +7980,9 @@ $$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is
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<p>
-<b>Task 6:</b>
+<b>Task 2.2:</b>
<p>The code below is the same as above, modify it to replace the Dirichlet condition at the right BC for a Neumann condition such that there $\frac{dT}{dx}=0$. How does the temperature profile changes?</p>
<p>Approximate the Neuman boundary $\frac{dT}{dx}=0$ by using the Backward difference for first order differential equation of first order accuracy.</p>
</p>
@@ -7995,12 +8000,12 @@ $$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is
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+<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
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<div class="highlight hl-ipython3"><pre><span></span><span class="n">Ts</span> <span class="o">=</span> <span class="mi">30</span>
@@ -8049,196 +8054,102 @@ $$<p>The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is
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-<pre>(4, 4)
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"/>
-</div>
-</div>
-<div class="jp-OutputArea-child">
-<div class="jp-OutputPrompt jp-OutputArea-prompt"></div>
-<div class="jp-RenderedText jp-OutputArea-output" data-mime-type="text/plain" tabindex="0">
-<pre>The estimated temperature at the nodes are: ['250.00', '174.84', '128.64', '102.18', '90.15', '90.15'] [C]
-</pre>
-</div>
-</div>
-</div>
-</div>
-</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0b47e9c2">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=57632792">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 7:</b>
-<p>Now, instead of using the central difference for approximating the second derivative. Use the forward difference to replace the second derivative (first order accurate). Write the algebraic representation of the differential equation below.</p>
-</p>
-</div>
+<b>Task 2.3:</b>
+<p>Just as we did in Task 2.1, this task has three parts: a) discretize the analytic expression into a system of equations using <b>forward differences</b>, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.</p>
+</p></div>
</div>
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=74174f3a">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=2d4b9fb9">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
-<p> Answer
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<p>Add the image.</p>
-</p>
+<b>Task 2.3a:</b>
+<p>Discretize the analytic expression into a system of equations for a grid with 6 points using <b>forward differences</b>.</p>
+<p><em>Write your answer by hand using paper/tablet and include the image below.</em></p>
</p></div>
</div>
</div>
</div>
-</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=a4e0342e">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=1c5ec2f2">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
-<p>
-<b>Task 8:</b>
-<p>Now, write the resulting matrix A, the vector T and the vector b.</p>
-</p>
+<p><em>Your answer here.</em></p>
</div>
</div>
</div>
</div>
-</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=b9ca660d">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=361f1638">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<p>
+<b>Task 2.3b:</b>
+<p>Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.</p>
+<p><em>Write your answer by hand using paper/tablet and include the image below.</em></p>
+</p></div>
</div>
</div>
</div>
-</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=8cb492f8">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=047a38d3">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
-<p> Answer
-<p>
-<p>Add the image.</p>
-</p>
-</p></div>
+<p><em>Your answer here.</em></p>
</div>
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0640b43e">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=3996b611">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 9:</b>
-<p>The original code is copied below, modify it accordingly to your new discretization. Keep Dirichlet conditions.</p>
-</p>
-</div>
-</div>
+<b>Task 2.3c:</b>
+<p>Implement the discretized system of equations in a code cell.</p>
+<p><b>This time we did not do it for you!</b> Copy the code from Task 2.1c and revise it to solve the system of equations using <b>Forward Differences</b>. Keep the Dirichlet conditions.</p>
+</p></div>
</div>
</div>
-</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell" id="cell-id=1e783d3e">
+</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=1e783d3e">
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-<div class="jp-InputPrompt jp-InputArea-prompt">In [10]:</div>
+<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
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-<div class="highlight hl-ipython3"><pre><span></span><span class="n">Ts</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="n">alpha</span> <span class="o">=</span> <span class="mi">500</span>
-<span class="n">dx</span><span class="o">=</span><span class="mf">0.02</span>
-
-<span class="c1"># grid creation</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.1</span><span class="o">+</span><span class="n">dx</span><span class="p">,</span><span class="n">dx</span><span class="p">)</span>
-<span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
-<span class="n">n</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-<span class="c1"># boundary conditions</span>
-<span class="n">T</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">250</span>
-<span class="n">T</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ts</span>
-
-<span class="c1"># Building matrix A</span>
-<span class="n">matrix_element</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="n">dx</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">alpha</span><span class="p">)</span>
-<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">2</span><span class="p">))</span>
-<span class="n">np</span><span class="o">.</span><span class="n">fill_diagonal</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">matrix_element</span><span class="p">)</span>
-<span class="n">A</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">3</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">2</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># Upper diagonal</span>
-<span class="n">A</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">3</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># Lower diagonal</span>
-
-<span class="c1"># Building vector b</span>
-<span class="n">b_element</span> <span class="o">=</span> <span class="o">-</span><span class="n">dx</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">alpha</span><span class="o">*</span><span class="n">Ts</span>
-<span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">b_element</span>
-<span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">T</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">b</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">T</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-<span class="c1"># Solving the system</span>
-<span class="n">T</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">b</span><span class="p">)</span>
-
-<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">T</span><span class="p">,</span><span class="s1">'*'</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">'Estimated solution'</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'T'</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Estimated solution using Forward Difference method'</span><span class="p">)</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-
-<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'The estimated temperature at the nodes are: </span><span class="si">{</span><span class="p">[</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">temp</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">"</span><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="ow">in</span><span class="w"> </span><span class="n">T</span><span class="p">]</span><span class="si">}</span><span class="s1"> [C]'</span><span class="p">)</span>
-
-<span class="c1">#to test your temperature at x=0.02 for task 9</span>
-<span class="n">ind</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mf">0.02</span><span class="p">))</span>
-<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'The temperature at x=0.02 is: </span><span class="si">{</span><span class="n">T</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1"> [C]'</span><span class="p">)</span>
+<div class="highlight hl-ipython3"><pre><span></span><span class="n">YOUR_CODE_HERE</span>
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"/>
-</div>
-</div>
-<div class="jp-OutputArea-child">
-<div class="jp-OutputPrompt jp-OutputArea-prompt"></div>
-<div class="jp-RenderedText jp-OutputArea-output" data-mime-type="text/plain" tabindex="0">
-<pre>The estimated temperature at the nodes are: ['250.00', '168.77', '115.29', '78.86', '52.21', '30.00'] [C]
-The temperature at x=0.02 is: 168.77 [C]
-</pre>
-</div>
-</div>
-</div>
-</div>
</div>
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@@ -8246,9 +8157,9 @@ The temperature at x=0.02 is: 168.77 [C]
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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
-<div style="background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px">
<p>
-<b>Task 9:</b>
+<b>Task 2.4:</b>
<p>How much finer does your grid has to be in the forward difference implementation to get a similar value at x = 0.02 as in the central difference implementation? Vary your dx.</p>
</p>
</div>
@@ -8256,12 +8167,13 @@ The temperature at x=0.02 is: 168.77 [C]
</div>
</div>
</div>
-<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=db91d68b">
+<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=dd2ead47">
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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
+<p><em>Your answer here.</em></p>
</div>
</div>
</div>
@@ -8272,10 +8184,10 @@ The temperature at x=0.02 is: 168.77 [C]
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
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-<div style="background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
+<div style="background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px">
<p>
-<b>Bonus note:</b>
-<p>The matrix inversion using numpy is one way to solve the system, another is the gauss_jordan written below and another one is the sparse matrix based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change dx to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.</p>
+<b>Bonus Task</b>
+<p>The matrix inversion using numpy is one way to solve the system, another is the <code>gauss_jordan</code> method, written below, and another one is the sparse matrix-based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change <code>dx</code> to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.</p>
</p>
</div>
</div>
@@ -8330,12 +8242,12 @@ The temperature at x=0.02 is: 168.77 [C]
</div>
</div>
</div>
-</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell" id="cell-id=fbd32c69">
+</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=fbd32c69">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea">
-<div class="jp-InputPrompt jp-InputArea-prompt">In [106]:</div>
+<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
<div class="cm-editor cm-s-jupyter">
<div class="highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">time</span>
@@ -8372,21 +8284,6 @@ The temperature at x=0.02 is: 168.77 [C]
</div>
</div>
</div>
-<div class="jp-Cell-outputWrapper">
-<div class="jp-Collapser jp-OutputCollapser jp-Cell-outputCollapser">
-</div>
-<div class="jp-OutputArea jp-Cell-outputArea">
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-<div class="jp-OutputPrompt jp-OutputArea-prompt"></div>
-<div class="jp-RenderedText jp-OutputArea-output" data-mime-type="text/plain" tabindex="0">
-<pre>The time used by direct matrix inversion solution is 2.037e-02 sec
-The time used by Gauss-jordan solution is 2.169e+00 sec
-The time used by the sparse matrix solver is 7.204e-03 sec
-</pre>
-</div>
-</div>
-</div>
-</div>
</div>
<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=2921b3f2">
<div class="jp-Cell-inputWrapper" tabindex="0">
@@ -8403,14 +8300,14 @@ The time used by the sparse matrix solver is 7.204e-03 sec
article { position: relative }
</style>
<a href="http://creativecommons.org/licenses/by-nc-sa/4.0/" rel="license">
-<img alt="Creative Commons License" src="https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png" style="border-width:; width:88px; height:auto; padding-top:10px">
-</img></a>
+<img alt="Creative Commons License" src="https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png" style="border-width:; width:88px; height:auto; padding-top:10px"/>
+</a>
<a href="https://www.tudelft.nl/en/ceg" rel="TU Delft">
-<img alt="TU Delft" src="https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png" style="border-width:0; width:100px; height:auto; padding-bottom:0px">
-</img></a>
+<img alt="TU Delft" src="https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png" style="border-width:0; width:100px; height:auto; padding-bottom:0px"/>
+</a>
<a href="http://mude.citg.tudelft.nl/" rel="MUDE">
-<img alt="MUDE" src="https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png" style="border-width:0; width:100px; height:auto; padding-bottom:0px">
-</img></a>
+<img alt="MUDE" src="https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png" style="border-width:0; width:100px; height:auto; padding-bottom:0px"/>
+</a>
</h3>
<span style="font-size: 75%">
© Copyright 2023 <a href="https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595" rel="MUDE Team">MUDE Teaching Team</a> TU Delft. This work is licensed under a <a href="http://creativecommons.org/licenses/by-nc-sa/4.0/" rel="license">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>.
@@ -8419,7 +8316,7 @@ The time used by the sparse matrix solver is 7.204e-03 sec
</div>
</div>
</div>
-</div></main>
+</div></div></div></div></div></div></div></main>
</body>
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diff --git a/src/teachers/Week_1_6/WS_1_6_student.ipynb b/src/teachers/Week_1_6/WS_1_6_student.ipynb
index d0f829a5..6feca725 100644
--- a/src/teachers/Week_1_6/WS_1_6_student.ipynb
+++ b/src/teachers/Week_1_6/WS_1_6_student.ipynb
@@ -34,14 +34,12 @@
"\n",
"This assignment is aimed to develop an understanding of the **Ordinary Differential Equation (ODE)**. There will be two sections about cooling and heating scenerios, corresponding to the first-order and the second-order ODEs. Please go through the text that follows and perform all steps outlined therein.\n",
"\n",
- "## Section 1: First-order ODE\n",
- "### 1.1 Linear ODE\n",
+ "## Part 1: First-order ODE\n",
+ "\n",
"In the study of heat transfer, **Newton's law of cooling** is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. It can be expressed in the form of ODE, as below:\n",
"\n",
"$$\\frac{dT}{dt}=-k(T - T_s)$$\n",
"\n",
- "\n",
- "\n",
"where $T$ is the temperature of the object at time $t$, $T_s$ is the temperature of the surrounding and assumed to be constant, and $k$ is the constant that characterizes the ability of the object to exchange the\n",
"heat energy (unit 1/s), which depends on the specific material properties.\n",
"\n",
@@ -54,9 +52,9 @@
"id": "73781a3c",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 1:</b>\n",
+ "<b>Task 1.1:</b>\n",
" \n",
"Suppose the considered period of time is long enough (bring it to steady state), what will be the final temperature of the object? \n",
" \n",
@@ -85,9 +83,9 @@
"id": "18cbde20",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 2:</b>\n",
+ "<b>Task 1.2:</b>\n",
"\n",
"Write the algebraic representation of the ODE using Explicit Euler.\n",
" \n",
@@ -106,32 +104,28 @@
"id": "579a9440",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3:</b>\n",
+ "<b>Task 1.3:</b>\n",
"\n",
"Compute the temperature evolution in the next 60 seconds.\n",
"\n",
- "**Please complete the missing parts of the code in each step below**\n",
+ "**Please complete the missing parts of the code in each step below, which is divided into 5 substeps (a through e).**\n",
" \n",
"</p>\n",
"</div>"
]
},
- {
- "cell_type": "markdown",
- "id": "ec6e3109",
- "metadata": {},
- "source": []
- },
{
"cell_type": "markdown",
"id": "88b861bc",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.1:</b> The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.\n",
+ "<b>Task 1.3a:</b> \n",
+ "\n",
+ "The time step of 5 seconds is constant. Discretize the time points, the solution vector $T$ and define the initial condition.\n",
" \n",
"</p>\n",
"</div>"
@@ -164,9 +158,11 @@
"id": "4586d4b6",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.2:</b> Implement your time discretization and find the solution from $t=0$ until $t=60$ sec. \n",
+ "<b>Task 1.3b:</b> \n",
+ "\n",
+ "Implement your time discretization and find the solution from $t=0$ until $t=60$ sec. \n",
" \n",
"</p>\n",
"</div>"
@@ -174,23 +170,12 @@
},
{
"cell_type": "code",
- "execution_count": 21,
+ "execution_count": null,
"id": "0bf8247e",
"metadata": {
"scrolled": false
},
- "outputs": [
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- }
- ],
+ "outputs": [],
"source": [
"for i in range(n-1):\n",
" T[i+1] = YOUR_CODE_HERE\n",
@@ -206,9 +191,9 @@
"id": "ffb4547f",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.3:</b>\n",
+ "<b>Task 1.3c:</b>\n",
"\n",
"Try different time steps to check the stability of the calculation. At which value the solution is stable?\n",
" \n",
@@ -227,9 +212,9 @@
"id": "8a907a09",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.4:</b>\n",
+ "<b>Task 1.3d:</b>\n",
"\n",
"Obtain the mathematical expression that proves your stability criteria.\n",
" \n",
@@ -248,9 +233,9 @@
"id": "1ff1e4fe",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 3.5:</b>\n",
+ "<b>Task 1.3e:</b>\n",
"\n",
"Now, discretize the equation using Implicit (Backward) Euler. Can you find a time step that makes the problem unstable?\n",
" \n",
@@ -260,21 +245,10 @@
},
{
"cell_type": "code",
- "execution_count": 29,
+ "execution_count": null,
"id": "18c73272",
"metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- }
- ],
+ "outputs": [],
"source": [
"dt = YOUR_CODE_HERE \n",
"t_end = 60 \n",
@@ -300,15 +274,15 @@
"id": "2109f010",
"metadata": {},
"source": [
- "## Section 2: Second-order ODE\n",
- "### Background\n",
+ "## Part 2: Second-order ODE\n",
+ "\n",
"The following 1D equation describes the steady state solution of the temperature along a pin that sticks out of a furnace. The rest of the pin is exposed to the ambient. \n",
"\n",
"$$\n",
"\\frac{d^2T}{dx^2} -\\alpha(T-T_s)=0\n",
"$$\n",
"\n",
- "The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is $250^o$ C. The length of the pint is 0.1m. Your grid should have a spatial step of 0.02 m. Finally, $\\alpha=500$."
+ "The ambient temperature is $T_s= 30^o$ C and the temperature at the wall is $250^o$ C. The length of the pin is 0.1m. Your grid should have a spatial step of 0.02 m. Finally, $\\alpha=500$."
]
},
{
@@ -328,63 +302,94 @@
},
{
"cell_type": "markdown",
- "id": "c739aaf1",
+ "id": "babd0424",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 5:</b>\n",
+ "<b>Task 2.1:</b>\n",
"\n",
- "Below we provide the code already implemented of the equation with the above Dirichlet conditions and using a central difference approximation of the second derivative. **Write in paper (add the image) the matrix A, and the vectors T and b.**\n",
+ "This task has three parts: a) discretize the analytic expression into a system of equations using central differences, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.\n",
+ "\n",
+ "<em>Parts 2.1b and 2.1c do not need to be completed in order; in fact, it may be useful to go back and forth between the two in order to understand the problem.</em> \n",
" \n"
]
},
{
"cell_type": "markdown",
- "id": "4133c6e2",
+ "id": "1f70b8e9",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- " \n",
- "Add the image.\n",
+ "<b>Task 2.1a:</b>\n",
"\n",
- "</p>\n",
- "</div>"
+ "Discretize the analytic expression into a system of equations for a grid with 6 points using central differences.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "81a5f165",
+ "metadata": {},
+ "source": [
+ "_Your answer here._"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c513ff65",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1b:</b>\n",
+ "\n",
+ "Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cb9ff085",
+ "metadata": {},
+ "source": [
+ "_Your answer here._"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fae61d18",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.1c:</b>\n",
+ "\n",
+ "Implement the discretized system of equations in a code cell.\n",
+ "\n",
+ "<em>We have already done this for you! Your task is to read the cell and make sure you understand how the matrices are implemented. Reading the code should help you formulate the matrices in Task 2.1b.</em>\n",
+ " \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "89fd190c",
+ "metadata": {},
+ "source": [
+ "_Add your image here._"
]
},
{
"cell_type": "code",
- "execution_count": 8,
+ "execution_count": null,
"id": "53fb4f99",
"metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "(4, 4)\n"
- ]
- },
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The estimated temperature at the nodes are: ['250.00', '168.77', '115.29', '78.86', '52.21', '30.00'] [C]\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
"import numpy as np \n",
"import matplotlib.pyplot as plt\n",
@@ -435,9 +440,9 @@
"source": [
"\n",
"\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 6:</b>\n",
+ "<b>Task 2.2:</b>\n",
"\n",
"The code below is the same as above, modify it to replace the Dirichlet condition at the right BC for a Neumann condition such that there $\\frac{dT}{dx}=0$. How does the temperature profile changes? \n",
"\n",
@@ -457,35 +462,10 @@
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": null,
"id": "cfcd00f9",
"metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "(4, 4)\n"
- ]
- },
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The estimated temperature at the nodes are: ['250.00', '174.84', '128.64', '102.18', '90.15', '90.15'] [C]\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
"Ts = 30\n",
"alpha = 500\n",
@@ -532,155 +512,86 @@
},
{
"cell_type": "markdown",
- "id": "0b47e9c2",
+ "id": "57632792",
"metadata": {},
"source": [
- "\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 7:</b>\n",
+ "<b>Task 2.3:</b>\n",
"\n",
- "Now, instead of using the central difference for approximating the second derivative. Use the forward difference to replace the second derivative (first order accurate). Write the algebraic representation of the differential equation below.\n",
- "\n",
- "</p>\n",
- "</div>"
+ "Just as we did in Task 2.1, this task has three parts: a) discretize the analytic expression into a system of equations using <b>forward differences</b>, b) write the system of equations by hand (e.g., the matrices), and c) implement the discretized system of equations in a code cell.\n",
+ "</div> \n"
]
},
{
"cell_type": "markdown",
- "id": "74174f3a",
+ "id": "2d4b9fb9",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- " \n",
- "Add the image.\n",
+ "<b>Task 2.3a:</b>\n",
"\n",
- "</p>\n",
- "</div>"
+ "Discretize the analytic expression into a system of equations for a grid with 6 points using <b>forward differences</b>.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
]
},
{
"cell_type": "markdown",
- "id": "a4e0342e",
+ "id": "1c5ec2f2",
"metadata": {},
"source": [
- "\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
- "<p>\n",
- "<b>Task 8:</b>\n",
- "\n",
- "Now, write the resulting matrix A, the vector T and the vector b.\n",
- " \n",
- "</p>\n",
- "</div>"
+ "_Your answer here._"
]
},
{
"cell_type": "markdown",
- "id": "b9ca660d",
+ "id": "361f1638",
"metadata": {},
- "source": []
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2.3b:</b>\n",
+ "\n",
+ "Write the system of equations by hand for a grid with 6 points. In other words, construct the matrix A and vectors T and b.\n",
+ "\n",
+ "<em>Write your answer by hand using paper/tablet and include the image below.</em>\n",
+ " \n"
+ ]
},
{
"cell_type": "markdown",
- "id": "8cb492f8",
+ "id": "047a38d3",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
- "<p> Answer\n",
- "<p>\n",
- " \n",
- "Add the image.\n",
- "\n",
- "</p>\n",
- "</div>"
+ "_Your answer here._"
]
},
{
"cell_type": "markdown",
- "id": "0640b43e",
+ "id": "3996b611",
"metadata": {},
"source": [
- "\n",
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%;vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 9:</b>\n",
+ "<b>Task 2.3c:</b>\n",
"\n",
- "The original code is copied below, modify it accordingly to your new discretization. Keep Dirichlet conditions. \n",
+ "Implement the discretized system of equations in a code cell.\n",
"\n",
- "</p>\n",
- "</div>"
+ "<b>This time we did not do it for you!</b> Copy the code from Task 2.1c and revise it to solve the system of equations using <b>Forward Differences</b>. Keep the Dirichlet conditions.\n",
+ " \n"
]
},
{
"cell_type": "code",
- "execution_count": 10,
+ "execution_count": null,
"id": "1e783d3e",
"metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": 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",
- "text/plain": [
- "<Figure size 640x480 with 1 Axes>"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The estimated temperature at the nodes are: ['250.00', '168.77', '115.29', '78.86', '52.21', '30.00'] [C]\n",
- "The temperature at x=0.02 is: 168.77 [C]\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
- "Ts = 30\n",
- "alpha = 500\n",
- "dx=0.02\n",
- "\n",
- "# grid creation\n",
- "x = np.arange(0,0.1+dx,dx)\n",
- "T = np.zeros(x.shape)\n",
- "n=len(x)\n",
- "\n",
- "# boundary conditions\n",
- "T[0] = 250\n",
- "T[-1] = Ts\n",
- "\n",
- "# Building matrix A\n",
- "matrix_element = -(2+dx**2*alpha)\n",
- "A = np.zeros((len(x)-2,len(x)-2))\n",
- "np.fill_diagonal(A, matrix_element)\n",
- "A[np.arange(n-3), np.arange(1, n-2)] = 1 # Upper diagonal\n",
- "A[np.arange(1, n-2), np.arange(n-3)] = 1 # Lower diagonal\n",
- "\n",
- "# Building vector b\n",
- "b_element = -dx**2*alpha*Ts\n",
- "b = np.zeros(len(x)-2) + b_element\n",
- "b[0] = b[0] - T[0]\n",
- "b[-1] = b[-1] - T[-1]\n",
- "\n",
- "# Solving the system\n",
- "T[1:-1] = np.linalg.solve(A,b)\n",
- "\n",
- "plt.plot(x,T,'*',label='Estimated solution')\n",
- "plt.xlabel('x')\n",
- "plt.ylabel('T')\n",
- "plt.title('Estimated solution using Forward Difference method')\n",
- "plt.legend()\n",
- "plt.show()\n",
- "\n",
- "print(f'The estimated temperature at the nodes are: {[f\"{temp:.2f}\" for temp in T]} [C]')\n",
- "\n",
- "#to test your temperature at x=0.02 for task 9\n",
- "ind = np.argmin(abs(x-0.02))\n",
- "print(f'The temperature at x=0.02 is: {T[ind]:.2f} [C]')"
+ "YOUR_CODE_HERE"
]
},
{
@@ -688,9 +599,9 @@
"id": "bac6bb0f",
"metadata": {},
"source": [
- "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
"<p>\n",
- "<b>Task 9:</b>\n",
+ "<b>Task 2.4:</b>\n",
"\n",
"How much finer does your grid has to be in the forward difference implementation to get a similar value at x = 0.02 as in the central difference implementation? Vary your dx.\n",
"\n",
@@ -701,20 +612,22 @@
},
{
"cell_type": "markdown",
- "id": "db91d68b",
+ "id": "dd2ead47",
"metadata": {},
- "source": []
+ "source": [
+ "_Your answer here._"
+ ]
},
{
"cell_type": "markdown",
"id": "031dca37",
"metadata": {},
"source": [
- "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
+ "<div style=\"background-color:#AABAB2; color: black; width:95%; vertical-align: middle; padding:15px; margin: 10px; width:95%; border-radius: 10px\">\n",
"<p>\n",
- "<b>Bonus note:</b> \n",
+ "<b>Bonus Task</b> \n",
" \n",
- "The matrix inversion using numpy is one way to solve the system, another is the gauss_jordan written below and another one is the sparse matrix based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change dx to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.\n",
+ "The matrix inversion using numpy is one way to solve the system, another is the <code>gauss_jordan</code> method, written below, and another one is the sparse matrix-based method in the cell afterwards. Here, we will just have a brief comparison to see how these solvers perform when the matrix is large. Change <code>dx</code> to 0.0002 of the original code that solves the second degree ODE and test the time it takes by each method.\n",
" \n",
"</p>\n",
"</div>"
@@ -767,20 +680,10 @@
},
{
"cell_type": "code",
- "execution_count": 106,
+ "execution_count": null,
"id": "fbd32c69",
"metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "The time used by direct matrix inversion solution is 2.037e-02 sec\n",
- "The time used by Gauss-jordan solution is 2.169e+00 sec\n",
- "The time used by the sparse matrix solver is 7.204e-03 sec\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
"import time\n",
"from scipy.sparse import csc_matrix\n",
--
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