Matlab homework - TMV166 VT22

A common element in the study of heat transfer is to find out what the temperature distribution in stationary conditions looks like over a thin plate where temperatures at its edges are known.

The figure below represents the cross section of a metallic beam. The heat flow is negligible in the direction perpendicular to the plate.

Let T₁, T₂, T₃ and T₄ be the temperatures of the 4 inner points in the figure below.

Picture 1.png

Assume that the temperature at a point can be approximated by the arithmetic mean of the temperature of the 4 adjacent points (i.e. the points above, below, right and left of a certain point).

To complete this homework, you have to build a system of linear equations for which the solution estimates T₁, T₂, T₃ and T₄, and to find an estimated solution for the temperatures T₁, T₂, T₃ and T₄, using an iterative method described in your book (see after).

The homework is in two parts:

PART I: write a computer program in Matlab language that implements an iterative method described in lecture 17. There are four different methods, distributed randomly by CANVAS to each student. There is an algorithm for each of the iterative methods in your book:

Algorithm 1 (p. 168) : Jacobi method

Algorithm 2 (p. 169) : Gauss-Seidel method

Algorithm 3 (p. 170) : Successive Over-Relaxation (SOR) method

Algorithm 4 (p. 171) : Conjugate Gradients method. ATTENTION! There is a typo in the book for this algorithm. You should add as a first line: $LaTeX: x\:\longleftarrow x_0$ .

Submit your Matlab program in CANVAS! NO MORE THAN 30 LINES (excluding comments)!

PART II: use the computer program you wrote in PART I to solve the system of linear equations that you determined earlier. Answer the following questions in CANVAS:

Using the variable $LaTeX: x_0=\begin{bmatrix} 10 \\ 10 \\ 10 \\ 10 \end{bmatrix}$ , , and $LaTeX: \omega=1$ (for those who are writing the code corresponding to Algorithm 3 p. 170 in the book : Successive Over-Relaxation Method), what is your estimated solution (three digits after the comma) for the temperatures T₁, T₂, T₃, and T₄?
How many iterations were necessary to reach this estimated solution?
This time, use $LaTeX: x_0=\begin{bmatrix} 100 \\ 100 \\ 100 \\ 100 \end{bmatrix}$ (same and $LaTeX: \omega=1$ ). What is your estimated solution (three digits after the comma) for the temperatures T₁, T₂, T₃, and T₄?
How many iterations were necessary to reach this estimated solution?
What do you conclude (no more than one sentence)?

Important information:

The CANVAS quiz where you will submit your work will open on Thursday 24 February at 11:40 and will close on Friday 04 March at 23:59.
This Matlab homework is graded but not mandatory. However you can get up to 2 BONUS points.
It has to be submitted in CANVAS no later than Friday 04 March 2022 at 23:59.
Grading:
- PART I : Submitted Matlab code that corresponds to the given iterative method and to the book algorithm AND returns a result = 1 BONUS point;
- PART II : Submitted correct answers to the five questions = 1 BONUS point.