A metal plate (illustrated in the figure) is part of a piece of machinery. The circular cut-out touches an active part of the machinery, and is heated to Celsius. The right hand side touches another piece of metal (although made of a different material). The remaining edges are insulated. The metal plate is 1 inch wide and 1.6 inches high. The hole in the middle (through which wires are passed) is 0.1 inches wide and 0.8 inches high; it is centered in the plate. The circular cut-out corresponds to a quarter of a circle with a radius of 0.05 inches.

The goal of this project is to examine how heat is transmitted through the plate and (at the higher grade levels) to position the hole so as to impede the transmission of heat from the lower-left to the upper-right corner.

The heat transmission is governed by the heat equation:

where u(x,y,t) is the temperature. The initial temperature of the plate is assumed to be zero. On the right-hand side, heat can be transmitted, and the appropriate boundary condition on this edge is

On the insulated edges, .

C Grade:  Solve this problem for an “appropriate” range of time values. Develop an animation of the solution. Use the “hot” color palette. How does the solution behave?

B Grade:  Modify the position of the hole, by moving it vertically and horizontally over a grid of positions. Use four positions in each direction, for 16 locations in all. The corners of the hole must not get less than 0.1 inches from the outer edges of the plate. (If you are in “draw” mode, then you can double-click on the rectangle and specify its coordinates.) Solve the differential equation for each position,  and determine the temperature at the upper-right corner. (Number the nodes, and then “extract” the solution, in order to do this.) Use interp2 to interpolate the results, and create a surface plot of your answers.

A Grade:  Based on your B-grade results, determine which location of the hole keeps the upper-right corner of the plate as cool as possible. For this location, consider rotations of the hole (with the above restrictions on its location) and determine the angle of rotation which keeps the upper-right corner of the plate as cool as possible. (Use a relatively small number of trial values, and interpolate your results to determine an answer.) Develop an animation of the solution for this position of the hole.

Deadlines: All course work must be turned in by 9:30am on Monday, December 21.

Note: All work in this course is due by 9:30am on Monday, December 21. I will be out of town during exam week, and I will not be able to answer questions during this period. To submit animations, use the ``save'' command in the File menu of the PDE toolbox to save a Matlab script to a file. Send me the Matlab script, which I can then use to re-create the movie.