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Mathematical Methods for Engineers 2 Heat diffusion group computer project Due Wednesday 9th October Late submission incurs a penalty of 10% per day, up to 40%. Work more than 4 days late will not be accepted. Graduate Qualities: This project is designed to help develop Graduate Qualities 1, 2 3, 4 and 5, namely Body of Knowledge, Lifelong Learning, Effective Problem Solving, Working Autonomously and Collab- oratively, and Ethical Action and Social Responsibility. Assessment: The assessment will take into account all of your documentation of the mathematical analysis of the problem, your Matlab file(s) (including proper parameter entry), your output, the correctness of the final solutions and the presentation of your whole report. Ground rules for internal students 1. This is a group project, and each group should have exactly two students. 2. The two students must share the work equally between them. Each group member will be awarded individually the mark allocated to the report. If the group members have not made an equal contribution according to the details provided on the Peer Assessment form, the Course Coordinator may adjust the marks accordingly. 3. We will keep a list of all groups. Please advise your prac supervisor who you will partner with. 4. Each team member must participate in all aspects of the project: mathematical calcula- tions, Matlab work, and report writing. Do not allocate one group member the Matlab work, and the other the task of report writing. Trust me, this never works out well. Do not allocate one person the first half of the project, and the other the second half. This doesn’t work either. 5. It sometimes happens that a student will come to me close to the end of the project and say “Oh, I don’t have a group yet”. Don’t be that person. It is in your interest to find a partner in good time – preferably before prac 3. 6. It sometimes happens that one partner feels as though they are carrying the group and the second partner is not contributing. If that happens, I need to be informed immediately. 7. It sometimes happens that two partners do not get along, even though both are con- tributing to the project. If you feel uncomfortable in your group, let me know as soon as possible. 8. If you wish, you may submit a peer assessment by using the form that can be found on the course webpage. Details for submission of the peer assessment can be found on the peer assessment sheet. 9. Under exceptional circumstances, we can allow a student to carry out the project indi- vidually. In this case, approval must be given by me. If a student elects to do the project individually, then that student must take full responsibility for the extra workload. 10. The University policy on plagiarism will apply between different groups. Presenting your work: • Use a group coversheet. • Your report should include: 1. A brief introduction (description of the problem, objectives of your report, etc). 2. Written worked answers to all numbered Questions where this is required. Some Tasks will also require you to write something in your report (this is clearly indi- cated). 3. Appropriately labelled figures where required. Any necessary diagrams may be neatly hand-drawn. 4. A brief conclusion. • Your Matlab files should include: 1. Code relating to each Task outlined below, together with appropriate explanatory comments. Be sure to list all group members at the top of the file(s). Submission: • One member of your group should submit (1) your report, and (2) your Matlab file(s) via learnonline. Heat diffusion The problem is simply stated: How to model the way heat spreads out inside an object. Problems of this type all go by the name of heat diffusion. The first person to propose a useful model of heat diffusion was the French mathematician J. Fourier (1768-1830), using differential equations (that is, partial differential equations, beyond the scope of our course). For this project we will implement a discrete space-time model, based on linear algebra. The project has three parts. In Part A you will implement a one-dimensional discrete model to compute the evolution of temperature on a metal rod. Your output will be temperatures along the rod for different times. You will then display the information in an animation. You will also have to answer questions about the diffusion, based on the data you produced. In Part B you will use linear algebra to gain insight into your work from Part A. For Part C you will essentially repeat the work already done in Part A, but for heat diffusion on a rectangular metal plate. The problem is two dimensional, and the plots you will produce are three-dimensional. Groundwork For the one-dimensional problem, we have a metallic rod of length L = 20 cm. At time t = 0 we know the temperatures at any point along the rod, and we want to know how the temperatures change as time goes by. To model the problem, we place the rod along the x-axis, from x = −10 to x = 10. We will use x to represent the position along the rod, so −10 ≤ x ≤ 10. We will use t to represent time, and we will start at time 0, so t ≥ 0. We will use u to indicate temperature, so u(x, t) = temperature on the rod at position x and time t. We will make simplifying assumptions. The first simplifying assumption is that the rod is insulated along its length. This means that there’s no trade in temperature between the inside of the rod and the environment around it, that goes through the side surface of the rod. The second assumption is that the rod has no thickness. The problem becomes one-dimensional with this assumption. The last assumption is that the temperature at the endpoints of the rod (at x = −10 and x = 10) are kept fixed and equal to zero at all times. This guarantees that the only way the rod can trade temperature (energy) with the outside is through the endpoints. The temperature at time t = 0 will be given: u(x, 0) = 1 10 (100− x2)(x− 3) cos(x) sin(3x). Part A We will use a discrete model for heat diffusion. This means that the space variable x will be discretised, and the time variable t will be discretised. We will take ∆x = 0.1 to be the gap between discrete values of x. This means that the only allowed values of x are x(1) = −10, x(2) = −10+ 0.1, x(3) = −10+ 0.2, · · · x(201) = −10+ (200) · (0.1) = 10. Note that there are only 201 allowed values for x. We will also discretise the variable t, and the allowed values of t will be t = 0, 1, 2, 3, . . . . The vector of temperatures will be indexed by two numbers m and k, with 1 ≤ m ≤ 201 (for the space variable), and k = 0, 1, 2, etc, for the time variable. So u(m, k) = temperature at position x(m), at discrete time k. The initial temperature is given as u(m, 0) = 1 10 (100− x(m)2)(x(m)− 3) cos(x(m)) sin(3x(m)). The endpoint temperatures are kept constant and equal to zero: u(0, k) = 0, u(201, k) = 0, for all k ≥ 0. Now comes an important part. We need to describe the mechanism by which temperature changes from place to place, and time to time. We already know what happens when m = 1 and m = 201, and when k = 0. We only need to describe the temperature changes for 2 ≤ m ≤ 200, and k ≥ 1. That is, for points that are interior to the rod, at future times. Consider a point x inside the rod. Since the sides of the rod are insulated, x can only trade temperature with the points immediately next to it (in our model). That is, x can only trade temperatures with x−∆x and with x+∆x. Also, because temperature has a tendency to spread evenly, x will trade temperature equally to its neighbours. That is, the amount of temperature that goes from x to x − ∆x, is equal to the amount that goes from x to x + ∆x. Our basic assumption is the following: From time t to time t+∆t, x keeps 70% of its own temperature. Of the remaining 30%, x sends 15% to x−∆x, and 15% to x+∆x. This means that x itself will receive 15% of the temperature in x − ∆x, and 15% of the temperature in x+∆x. We have the following formula: u(x, t+∆t) = 70 100 u(x, t) + 15 100 u(x−∆x, t) + 15 100 u(x+∆x, t). This is the formula that updates the temperature from what happens at time t, to what happens at time t +∆t. We want to generate the temperatures along the rod from time k = 0 to time k = 1000. 1. Plot the initial temperature u(x, 0) = f(x) using MATLAB. 2. Use MATLAB to generate the matrix u(m, k) of temperatures, for 1 ≤ m ≤ 201 and 1 ≤ k ≤ 1001. You will need to use a double for loop to produce this matrix. Do not display the full results. Instead, once the matrix is generated, produce a printout of the first 9 columns and 5 rows of the matrix. 3. The initial temperature profile u(x, 0) = f(x) has a maximum temperature. It is surmised that, as time changes, the location of the maximum remains the same. Does your data support or contradict that statement? 4. Find the earliest time for which the absolute value of the temperature along the rod is less than 10 degrees. 5. Find the earliest time for which the difference between the largest and smallest temper- atures along the rod is less than 5 degrees. 6. Produce a sequence of plots (an animation) to illustrate the changing temperatures with time. For the purposes of the animation, the time gap between frames should be set to 0.07 seconds. MATLAB has built-in commands to animate plots, but you can improvise an animation as well. I put an example at the end of this document. 7. What is the behaviour of the temperature as time grows? Use your animation to justify your answer. Part B We want to see how linear algebra is behind the work you’ve done in Part A. For that purpose, we will use ∆x