Week 5.2 - Mid term project - Steady vs unsteady state analysis
Week 5.2 - Mid term project - Steady vs unsteady state analysis
Aim : Comparison of both the steady and unsteady equation solving methods and explanation.
1. For the Steady state case:
From the previously done program, it is clear that the 2D steady state analysis of the conduction problem consumes less time for reaching to a stable solution.
Total iterations for 2D steady equation:
- Jacobi Method = 1335
- Gauss-Siedal Method = 709
- SOR Method = 78
The SOR took the least steps amongst three and the reason for the same is that the steady state equation ( or its point iterative form) does not include the 'time-derivative `(delT)/(delt)`' term and the 'diffusivity `(alpha)`' term. It processes the variation in temperatures at the grid points along x and y directions, with no additional information of diffusivity and time derivative, gives out less number of iterations.
2. For the unsteady state case:
In case of the 2D unsteady state equation, with inclusion of two previously mentioned terms, the computational steps are increased. Again the transient case is solved in two ways, i.e. by "Explicit method with no iterative methods" and by "Implicit method with three iterative methods".
In the former case, the state of a system is solved at a later time as compared to the current stage. The methodology of solving explicitly is independent of other variables and a single equation is used to evaluate temperature at grid points (the term 'T_prev_dt=T' is absent for explicit method in unsteady case).
The latter case involves finding a solution having an equation of both current state and the later one.
Total iterations for 2D unsteady equation:
- Jacobi Method = 6735
- Gauss-Siedal Method = 4664
- SOR Method = 2736
The implicit methodology has information from previously solved equation for all grid points at each time step, hence this is more efficient. The implicit method, being unconditionally stable is time consuming. As compared to Jacobi and Gauss-Siedal methods, the SOR method (comprising a relaxation factor) converges to a stable solution at a faster pace.