Week 5.1 - Mid term project - Solving the steady and unsteady 2D heat conduction problem

In this assignment, we will understand the 2D heat conduction equations in both steady and transient state.

• Steady-state

First, we understand the steady-state condition,

solving by three methods,

1. Jacobi

2. Gauss seidel

3. successive over-relaxation

we will use central differencing for the derivation of these equations,

conclusion- In all the figures for the steady-state condition the SOR method gives us better results in comparison to the other two methods in terms of iterations carries out and simulation time.

• Transient state

In transient condition/simulation we work on implicit and explicit schemes

Explicit methods are the state of the system at a future time from a state of system at the current time, while implicit methods calculate the system status using both future and current time at the future time.

Mathematically if X(t) is the current system and X(t+Δt) is the state at the later time Δt then, the explicit state Y(t+Δt)=F(Y(t)) while the implicit method solves as G(Y(t), Y(t+Δt))=0

conclusion- The results are the same as the simulation in steady-state, and seen that SOR has better results in terms of

the number of simulations and simulation time.

I have uploaded the required formulas in a pdf.