Comparison of steady state and transient state results
Objective: To compare the Steady state analysis vs Transient state analysis.
Description:
Basically, every system has a transient and a steady state. The steady state is the state that is established after a certain time in your system. The transient state is basically between the beginning of the event and the steady state. That means the final output of both of them is same.
let us clear the definition using a linear heat convection equation:
`{partial T}/{partial t}= alpha({partial ^{2}T}/{partial x^{2}}+{partial ^{2}T}/{partial y^{2}}+{partial ^{2}T}/{partial z^{2}})`
To define a steady state for a system, we’d observe that whatever affects the system’s behaviour are constant or unchanging in time Also, in terms of continuous-time, this means that for those properties (p) of the system, the partial derivative in regards to time remains zero.
The following equation depicts this relationship: ∂p / ∂t = 0
where T is the temperature, `alpha` is thermal diffusivity and (x,y,z,t) are space and time variables respectively.
The 2D steady state heat conduction equation with no heat generation is
`{partial ^{2}T}/{partial x^{2}}+{partial ^{2}T}/{partial y^{2}} = 0`
The 2D transient state heat conduction equation with no heat generation is
`{partial T}/{partial t}= alpha({partial ^{2}T}/{partial x^{2}}+{partial ^{2}T}/{partial y^{2}})`
Steady State Calculation Methods:
Overall, there are two methods we can use to calculate the Steady state. Firstly, we can use time-domain algorithms and secondly, we can use frequency-domain algorithms or the harmonic balance method.
Also, the time domain method is divided into two subdivisions, shooting methods (iterative methods) and time domain sensitivities (one-step methods).
Moreover, time domain sensitivities require derivatives to calculate the steady state. However, when those are not readily accessible, then we utilize the shooting methods.
Different shooting/iterative methods are
- Jacobi method
- Gauss-seidel method
- SOR method
Transient State Calculation Methods:
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations.
The generalized Crank-Nicolson method, also known as θ method, according to the generalized trapezoidal approximation, the temperatures obtained at the time level n + 1 are
`T^(n+1) = T^n +[ (1-theta){dotT }^ n + theta{dotT }^ (n+1)]triangleT`
where dt is the time step between time levels n and n+1 and the temporal parameter θ has the value between 0 and 1.
The stability of solution depends on the value of parameter θ. If θ >= ½ then the solution is stable for the arbitrary chosen time step dt. The large time step may cause the computational time low. Nevertheless, the solution can be inaccurate. Therefore, the size of the time step should be a compromise between the accuracy of temperature determination and the computational speed. By varying the parameter θ, different transient schemes are obtained.
θ = 0 – Fully explicit scheme; conditionally stable – the time step dt should be smaller than the dx.
θ = 1/2 – Semi-implicit scheme – Crank Nicolson method; unconditionally stable.
θ = 2/3 – Galerkin method; unconditionally stable.
θ = 1 – Fully implicit scheme; unconditionally stable.
The explicit method (θ = 0) is the first-order scheme. Hence, to ensure high accuracy of calculations, the time step size should be small. The allowable time step size decreases with the dx.
it is unnecessary to solve the system of linear equations because the vector `T^(n+1)` is obtained in the time marching procedure. Therefore, the calculations are straightforward and easy to program. Despite the limitations of the time step dt, the explicit method is frequently used, since it is very accurate for a small-time step, especially when the temperature change rate for a solid is large.
For the implicit method (θ = 1), the system of linear algebraic equation must be solved at every time level using direct methods (Gaussian elimination method) or iterative methods. Solution obtained is accurate.
Crank-Nicolson method (θ = 1/2) is the second-order scheme(dt^2). This method is unconditionally stable. However, with large time steps, the solution exhibits oscillations (which do not occur in reality) and becomes less accurate. Similarly, Galerkin method also gives inaccurate results.
Therefore, in CFD we use only implicit and explicit schemes.
Also, the amount of time spent in the transient state depends on the initial conditions of the system.
Comparison of Steady state methods (Jacobi Vs Gauss-seidel Vs SOR)
The Jacobi method is based on solving for every variable locally with respect to the other variables; one iteration of the method corresponds to solving for every variable once. The resulting method is easy to understand and implement, but convergence is slow.
In general, if the Jacobi method converges, the Gauss-Seidel method will converge faster than the Jacobi method, though still relatively slowly. The Gauss-Seidel method is a technique for solving the equations of the linear system of equations one at a time in sequence, and uses previously computed results as soon as they are available.
The successive overrelaxation method (SOR) is a method of solving a linear system of equations derived by extrapolating the gauss-seidel method. This extrapolation takes the form of a weighted average between the previous iterate and the computed Gauss-Seidel iterate successively for each component, where denotes a Gauss-Seidel iterate and is the extrapolation factor. The idea is to choose a value for that will accelerate the rate of convergence of the iterates to the solution. In general, Successive Overrelaxation method may converge faster than Gauss-Seidel by an order of magnitude.
The relaxation factor is not equal at all cases. The choice of relaxation factor is not necessarily easy, it should be chosen accordingly.
Computational time for Jacobi > Gauss-seidel > SOR
Number of iterations for Jacobi > Gauss-seidel > SOR
Comparison of Transient state methods (Implicit Vs Explicit)
In general, the explicit schemes are usually easier to solve. This is because in the explicit case, we have a system of linear equations (nodes) that can be solved independently without iteration. This scheme actually works exactly like the steady case. The time integration here is just the convergence loop in the steady case.
The implicit solvers require iteration until convergence in the spatial node, plus then the same time integration than the explicit solvers. They are therefore computationally more expensive.
No of time levels approached using Implicit_Jacobi > Implicit_GS > Explicit > Implicit_SOR
Computational time for Implicit_Jacobi > Implicit_GS > Implicit_SOR > Explicit
Conclusion:
1. In steady state analysis, SOR should be used because it converges in fewer iterations and it has smaller memory requirements than Jacobi.
2. In transient state analysis, explicit scheme is faster.
3. From both of them, the steady state solver is obviously faster. This is because of the mathematical simplification of the problem. There is no thermal diffusivity and no time integration. Therefore, there is no numerical diffusion and the solution will always be stable, we just need to choose a decent dx and dy.
=======================================================================
link for Matlab code of above screenshots
https://skill-lync.com/projects/solving-the-steady-and-unsteady-2d-heat-conduction-problem-62
link for stability analysis for the above statement at unstable conditions
https://skill-lync.com/projects/stability-analysis-in-the-unsteady-problem-12