Analysis of Steady and Unsteady State solutions of a 2D Heat conduction problem

ANALYSIS OF VARIOUS ITERATIVE SCHEMES FOR THE SOLUTION OF A 2D HEAT CONDUCTION PROBLEM:

[PART: 2/3]

In the previous part we had explained the problem statement and the MATLAB Program in detail. In this Part we are going to explain the outputs from the 2D Heat Conduction program.

The Temperature profile of the square plate is evaluated at steady state. The Steady state solution does not involve time marching. We set the number of gridpoints N_x = N_y = 51 in this case with the other conditions such as omega_(opt) = 2/(1+sin(pidx))

The steady state solutions are characterized by fixed temperature gradients that are setup in the system which do not change with time. Here in the solution we can see the distinct profiles for the given boundary conditions.

1. Gauss Jacobi Scheme:

2. Gauss Siedel Scheme:

3. Successive Over Relaxation (SOR) Scheme:

As seen above the number of iterations it takes for convergence drops drastically from the Gauss Jacobi to the SOR method converging in only 148 iterations compared to 4759 iterations for the Gauss Jacobi method and 2558 iterations for the Gauss Siedel method. The computation time is 0.494 s for the Gauss Jacobi method, 0.266 s for the Gauss Siedel method and 0.028 s only for the SOR method. It indicates that SOR method is atleast 17x faster than Jacobi method and 9.5x faster than Gauss Siedel method for this case.

B. Transient State solutions:

Transient State/ Unsteady state solutions are characterized by formation of moving temperature gradients with the progression of time due to thermal diffusion in the x and y directions. The final temperature in this case largely depends on the number of iterations for which the program is executed. If a sufficiently large number of iterations are performed then the temperature profile obtained resembles steady state where the temperature gradients within the system become constant.

1. Explicit Iteration Scheme:

The square plate is evaluated at unsteady state and fixed point iteration is used to obtain the values of temperatures from the old values. For our problem we executed the program for 20000 time-steps.

2. Implicit Iteration Scheme:

The following plots show the solution (temperature profile) of the square plate for same number of iterations (i.e. 20000) solved implicitly. The iteration counter is started once at t = 0 and it prints the total number of iterations required for convergence after 20000 time-steps.

2 (a). Gauss Jacobi Scheme:

2 (b). Gauss Siedel Scheme:

2 (c). Successive Over Relaxation (SOR) Scheme:

The SOR method gives the lowest number of iterations at 43354 followed by Gauss Siedel method 49756 iterations and Gauss Jacobi method 59427 iterations. Surprisingly the Gauss Siedel method was the fastest requiring a computation time of 2.951s followed by SOR method 3.070s and finally the Gauss Jacobi method 3.568s.

However the explicit method is the fastest with a computation time of 1.2381s compared to the fastest explicit method of Gauss Siedel i.e. 2.951s.

SUMMARY:

For the Transient/ Unsteady Heat Conduction Case:

For the Steady State Heat Conduction Case:

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